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This document contains all material for #1, as provided in
Semester #4 by #3 at the University of Queensland. The
handbook description of the course is ``#5''.
#6
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\title{A Refinement of Shor's Algorithm}
\author{David McAnally\\Distributed Systems Technology Centre\thanks{The work reported in this paper has been funded in part by the Co-operative Research Centre for Enterprise Distributed Systems Technology (DSTC) through the Australian Federal Government's CRC Programme (Department of Industry, Science \& Resources)}\\ QUT, Brisbane, Qld 4001, Australia\\ Email: {\tt [email protected]}}
\begin{document}
\maketitle
\begin{quote} A refinement of Shor's Algorithm for determining order is introduced, which determines a divisor of the order after any one run of a quantum computer with almost absolute certainty. The information garnered from each run is accumulated to determine the order, and for any $k$ greater than $1$, there is a guaranteed minimum positive probability that the order will be determined after at most $k$ runs. The probability of determination of the order after at most $k$ runs exponentially approaches a value negligibly less than one, so that the accumulated information determines the order with almost absolute certainty. The probability of determining the order after at most two runs is more than 60\%, and the probability of determining the order after at most four runs is more than 90\%. \end{quote}
\section{Introduction}
In quantum computing, there are a few algorithms which can be performed more efficiently than their most efficient known classical counterparts. One such example is Grover's algorithm which improves the efficiency of searching an unsorted list to the order of the theoretical limit of efficiency, at a cost of $O(\sqrt{N})$, where $N$ is the length of the list (see for example, \cite{grover1,grover2}). Another example is supplied by Shor's algorithms for determining order and for determining discrete logarithms, both of which can be performed in polynomial time with the aid of both a quantum computer and a classical computer. A consequence of the fact that Shor's algorithm determines order in polynomial time is that composite numbers can be factorized in polynomial time. Since Shor's algorithms aid in factorizing composite numbers and in solving the discrete logarithm problem, both in polynomial time, then their implementation on a quantum computer would challenge the security of many of today's cryptographic algorithms ({\it e.g.} RSA, ElGamal, DSA, ECC).
Shor's original algorithm had the property that the number of runs on the quantum computer needed to determine the order of $x$ modulo $n$ was $O(\log \log n)$. In Knill's modification \cite{knill}, the probability of success was improved, but on any single run of the quantum computer, the probability that the value output by the computers would be a divisor of the order may still be significantly less than $1$. Knill did, however, introduce the concept of accumulating information from various runs of the quantum computer.
It is the purpose of this paper to refine the algorithm to the point that after any one run on the quantum computer, the probability that the value output by the computers is a divisor of the order is negligibly less than $1$. When this refinement is combined with the accumulation of information, as discussed above, the number of required runs on the quantum computer is reduced to $O(1)$ (assuming ideal working of the quantum computer, including extra demands on the Quantum Fourier Transform). The refinement to the algorithm is introduced in \secref{refine}, and it is demonstrated in \secref{final} that the probability of finding the required order with not more than $k$ runs on the quantum computer is greater than $1/\zeta(k) - O(n^{-\epsilon})$ in the asymptotic limit as $n \to \infty$, where $\zeta$ is the Riemann zeta function, $\epsilon$ is a positive number, and the statement $f > g - O(h)$ means that there exists a function $F$ such that $f > g - F$ in the asymptotic limit, and $F/h$ is bounded in the same limit.
The refinement is effected by increasing the number of qubits in the first register by a factor of about $1.5$, thus increasing the requirements of space and time on the quantum computer by a constant factor, and increasing the accuracy required in performing the Quantum Fourier Transform on the first register.
In \secref{mod}, the modular metric, which measures distances between elements of $\@ifnextchar[\@BZoptarg{\Bbb Z}/q\@ifnextchar[\@BZoptarg{\Bbb Z}$ is introduced for all $q$. The purpose for introducing the modular metric is in order to obtaining a proper and invariant concept of proximity.
In \secref{shor}, Shor's original algorithm is discussed.
In \secref{refine}, a refinement of Shor's algorithm is introduced in which each run of the quantum computer determines a divisor of the required order with almost absolute certainty, and the number of required runs on the quantum computer is $O(1)$.
In \secref{firstreg}, an analysis of the probabilities of the measured value of the first register falling in some specific subsets of $\{ 0,1,\dots,q-1 \}$ is given.
In \secref{continued}, some facts about continued fractions (which are used in the classical part of the algorithm to determine information about the order) are given, with a new result determining sufficient conditions to guarantee that the classical part of the algorithm will yield a divisor of the required order.
In \secref{algo_analysis}, the results of \secref{firstreg} and \secref{continued} are united to demonstrate that the refinement guarantees, with probability negligibly less than $1$ that each run of the quantum yields a divisor of the required order, and the Section also specifies sufficient information to determine approximate probabilities for each divisor.
In \secref{prob}, an idealized version of the probability distribution is investigated in order to determine the probability that the order will be known after at most $k$ runs of the quantum computer.
In \secref{final}, the properties of the idealized probability distribution are modified to the more concrete distribution associated with the refinement of Shor's Algorithm.
\section{Modular Metrics} \seclabel{mod}
For $q \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, $q > 1$, let $I_q = \{ 0,1,\dots,q-1 \}$.
\begin{theorem}} \def\et{\end{theorem} \thmlabel{modmet} For $q \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, $q > 1$, define $\rho_q : I_q \times I_q \to \@ifnextchar[\@BZoptarg{\Bbb Z}$ by \begin{equation}} \def\ese{\end{equation} \eqlabel{mod}
\rho_q(x,y) = \min(|x-y|,q-|x-y|), \ese then $\rho_q$ is a metric on $I_q$, and $\rho_q(x,y) \leq \frac{q}{2}$ for all $x, y \in I_q$. \et
This is proven in \appref{modmet}.
The modular metric $\rho_q$ is equivalent to a metric $s_q$ on $\@ifnextchar[\@BZoptarg{\Bbb Z}/q \@ifnextchar[\@BZoptarg{\Bbb Z}$ determined by the smallest distance between representatives of the respective cosets:
\begin{eqnarray*}} \def\ee{\end{eqnarray*} s_q(\bar x,\bar y) = \min\{ |x-y| : x \in \bar x, y \in \bar y \}, \ee for $\bar x, \bar y \in \@ifnextchar[\@BZoptarg{\Bbb Z}/q \@ifnextchar[\@BZoptarg{\Bbb Z}$.
The modular metric gives a distance function on $\{ 0,1,\dots,q-1 \}$ which is invariant under cyclic symmetries, and can be thought of an arc length on a circle around which the elements have been evenly spaced.
\section{Shor's Algorithm} \seclabel{shor}
The purpose of the quantum part of Shor's Algorithm is to determine the order $r$ of $x$ modulo $n$, where $0 < x < n$, and $x$ and $n$ are relatively prime, in other words, $r$ is the smallest positive integer such that $x^r \equiv 1 \bmod n$ (note that $0 < r < n$). In Shor's paper, this was achieved in the following manner. \begin{enumerate}} \def\een{\end{enumerate} \item The state vector of the system is set to an initial state of \begin{eqnarray*}} \def\ee{\end{eqnarray*}
| \psi_0 \rangle = \frac{1}{q^{\frac{1}{2}}} \sum_{a=0}^{q-1}
| a \rangle \otimes | 0 \rangle, \ee where $q$ is an appropriate power of $2$ (the first register is composed of $l$ qubits, where $q = 2^l$). In Shor's paper, $q$ is taken to be that unique power of $2$ such that $n^2 \leq q < 2 n^2$. The state vector
$| \psi_0 \rangle$ arises from the state $| \phi_0 \rangle
= | 0 \rangle \otimes | 0 \rangle$ by taking a quantum Fourier transform on the first register, or alternatively by applying a gate of $H^{\otimes l}$ to the first register (so $H$ is applied individually to each qubit), where $H$ is the Hadamard gate. \item The next step is to perform a modular exponentiation, so that
$| \psi_0 \rangle$ is mapped to \begin{eqnarray*}} \def\ee{\end{eqnarray*}
| \psi_1 \rangle &=& \frac{1}{q^{\frac{1}{2}}} \sum_{a=0}^{q-1}
| a \rangle \otimes \left| x^a \bmod n \right\rangle\\ &=& \frac{1}{q^{\frac{1}{2}}} \sum_{k=0}^{r-1} \sum_{b=0}^{\left\lfloor\frac{q-1-k}{r}\right\rfloor}
\left| br+k \right\rangle \otimes \left| x^{br+k} \bmod n \right\rangle\\ &=& \frac{1}{q^{\frac{1}{2}}} \sum_{k=0}^{r-1} \sum_{b=0}^{\left\lfloor\frac{q-1-k}{r}\right\rfloor}
\left| br+k \right\rangle \otimes \left| x^k \bmod n \right\rangle, \ee where for real $y$, $\lfloor y \rfloor$ is the greatest integer less than or equal to $y$. The final equality follows from the fact that $r$ is the order of $x$ modulo $n$. \item The next step is to take the quantum Fourier transform on the first register, so that the state becomes \begin{eqnarray*}} \def\ee{\end{eqnarray*}
| \psi_2 \rangle &=& \frac{1}{q} \sum_{c=0}^{q-1} \sum_{a=0}^{q-1}
\exp\left(\frac{2\pi iac}{q}\right) \left| c \right\rangle \otimes
\left| x^a \bmod n \right\rangle\\ &=& \frac{1}{q} \sum_{c=0}^{q-1} \sum_{k=0}^{r-1} \sum_{b=0}^{\left\lfloor\frac{q-1-k}{r}\right\rfloor}
\exp\left(\frac{2\pi i(br+k)c}{q}\right) \left| c \right\rangle \otimes
\left| x^k \bmod n \right\rangle. \ee \item The final step is to measure the value $c$ of the first register. The value of $c$ is then input into a classical computer (which already has values for $q$ and $n$), and a value for the fraction $d'/r'$ satisfying the following conditions is found: \begin{itemize}} \def\ei{\end{itemize} \item $d'/r'$ is in lowest terms ($d'$ and $r'$ have no common factors); \item $0 \leq d'/r' \leq 1$; \item $0 < r' < n$; \item $d'/r'$ is the nearest fraction to $c/q$ which satisfies the other three conditions. \ei This is done with the use of continued fractions. \een Shor noted that the probability that $c$ ($c \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, $0 \leq c < q$) is some given value which varies from an integral multiple of $\frac{q}{r}$ by at most $\frac{1}{2}$ (this is equivalent to equation (5.11) of Shor's paper \cite{shor}), and that the value of the second register is $x^k \bmod n$ for some given $k$, is greater than $\frac{1}{3r^2}$. This observation can be formally expressed as follows: let $X$ be the random variable denoting the result of the measurement of the first register, and let $Y$ be the random variable denoting the result of a measurement of the second register, then for any given $d = 0,1,\dots,r-1$ and $k = 0,1,\dots,r-1$, \begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left(\rho_q\left(X,\frac{dq}{r}\right) \leq \frac{1}{2} \mbox{ and } Y = x^k \bmod n\right) > \frac{1}{3r^2}. \ee It follows that the probability that $c$ is the value as given above is greater than $\frac{1}{3r}$, and so the probability that there exists an integer $d$ such that $0 < d < r$, $d$ is relatively prime to $r$ ($d$ and $r$ have no common factor), and $c$ differs from $\frac{dq}{r}$ by at most $\frac{1}{2}$, is greater than $\phi(r)/(3r)$, where $\phi$ is Euler's totient function, defined by \begin{eqnarray*}} \def\ee{\end{eqnarray*} \phi(r) = \#\{ d \in \@ifnextchar[\@BZoptarg{\Bbb Z} : 0 < d < r, \ \mbox{$d$ and $r$ are relatively prime} \}. \ee A formula for $\phi$ is given by \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\phi(r) = r \prod_{p \mbox{\small{ prime, }} p | r} \left(1 - \frac{1}{p} \right). \ee The requirement that $d$ and $r$ be relatively prime comes from the fact that the only information about $d/r$ that can be be derived from $c$ is its expression in lowest terms (no common factor for numerator and denominator), so that in order for the denominator to be the order of $x$, $d$ and $r$ can have no common factors. Shor used the theorem that $\phi(r)/r > \delta_1/\log \log r$ for some $\delta_1$ to yield the result that the probability above is greater than $\delta/\log \log r$, for some $\delta$, so that the number of trials required on the quantum computer is $O(\log \log n)$.
\section{Refinement of Shor's Algorithm} \seclabel{refine}
The refinement of Shor's Algorithm to be introduced in this paper incorporates a modification of the value of the parameter $q$, and an accumulation of information in a similar manner to that suggested by Knill \cite{knill}.
Take a positive real number $\epsilon$, and let $w = n^\epsilon$. Under the refinement, the algorithm for determining $r$ is as follows. All steps except step 2 are performed on a classical computer. \begin{enumerate}} \def\een{\end{enumerate} \item Set $s := 1$ and $q \geq2 w n^3$ ({\it e.g.} set $q$ to be that unique power of 2 such that $2 w n^3 \leq q < 4 w n^3$); \item Perform the quantum algorithm on the quantum computer with $q$ as specified in Step 1, and measure the value $c$ of the first register; \item Determine the continued fraction expansion for $\frac{c}{q}$; \item Determine all denominators of convergents of the continued fraction expansion up to the first denominator greater than or equal to $n$; \item Let $r'$ be the last denominator less than $n$, and set $s := \mathop{\rm lcm}\nolimits(s,r')$; \item Calculate $x^s \bmod n$; \item If $x^s \not\equiv 1 \bmod n$, then go to Step 2; \item Output $s$. \een Note that the algorithm accumulates the information garnered from each measurement of $c$. Note also that only the denominators of the convergents are calculated. There is no need to calculate their numerators.
For the size of $n$ that would be typically used in RSA encryption, the algorithm above determines the order $r$ with probability negligibly less than one (the probability of determining a nontrivial multiple of $r$ instead of the correct value is $O(n^{-1})$). The probability that the correct value of $r$ will be found after at most 2 runs of the quantum computer is at least 60\%, the probability after at most 4 runs is at least 90\%, the probability after at most 6 runs is at least 98\%, and the probability after at most 8 runs is at least 99.5\%.
The rest of the paper is devoted to analysing the above algorithm in order to demonstrate the properties claimed for it.
\section{Probabilities of specified values for the first register} \seclabel{firstreg}
The essential feature of the profile of probabilities of the measured value $c$ of the first register is that when $q \gg r$, then the probability concentrates in the vicinities of $\frac{dq}{r}$, where $d$ is an integer with a probability of about $\frac{1}{r}$ in each vicinity. Further, if $\frac{dq}{r}$ is an integer, then the full probability of $\frac{1}{r}$ effectively concentrates itself at $c = \frac{dq}{r}$, and if $\frac{dq}{r}$ is not an integer, then the probability of $c$ in the vicinity of $\frac{dq}{r}$ is essentially inversely proportional to $(c - \frac{dq}{r})^2$. It follows that for $q \gg r$, the only dependence that the probability profile in the vicinity of $\frac{dq}{r}$ has on $q$ is on the fractional part of $\frac{dq}{r}$ ({\it i.e.} the full set of profiles is determined completely by the fractional part of $\frac{q}{r}$). This means qualitatively that as $q$ increases, the concentrated areas of probability recede from each other, but the individual profiles do not ``spread". These observations are made more rigourous in this Section.
All results presented in section without proof will be proven in \appref{firstreg}.
Since Shor's Algorithm relies on measuring the value in the first register, and then entering the result of the measurement into the classical computer, then it is useful to have information about the probability distribution for the values taken by the first register in order to determine the probabilities of various outputs of the classical computer.
The parameter $q$ will now be taken to be an arbitrary positive integer, and a measurement of the first register will be taken when the computer is in the state \begin{eqnarray*}} \def\ee{\end{eqnarray*}
| \psi_2 \rangle = \frac{1}{q} \sum_{c=0}^{q-1} \sum_{k=0}^{r-1} \sum_{b=0}^{\left\lfloor\frac{q-1-k}{r}\right\rfloor}
\exp\left(\frac{2\pi i(br+k)c}{q}\right) \left| c \right\rangle \otimes
\left| x^k \bmod n \right\rangle.
\ee Note that $| \psi_2 \rangle$ is the final form of the state vector before measurement in the quantum algorithm in Shor's algorithm. The parameter $q$ is generally taken to be a power of $2$ as a result of the requirement of the usage of qubits in the quantum algorithms for addition, multiplication and modular exponentiation. Modification to qudits (with a higher number of levels) of the algorithms for addition, multiplication and modular exponentiation will allow for a wider range of values for $q$. Also, $q$ is typically taken to be larger than $n$, although the results below are true for all possible values of $q$.
Let $X$ be the random variable describing the result of the measurement of the first register in the final step of the algorithm on the quantum computer, then $X$ must take the value of an integer between $0$ and $q-1$, inclusive, and for $0 \leq c \leq q-1$, the probability that $X = c$ is given by $P(X = c) = \langle \chi_c | \chi_c \rangle$, where \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\left| \chi_c \right\rangle = \frac{1}{q} \sum_{k=0}^{r-1} \sum_{b=0}^{\left\lfloor\frac{q-1-k}{r}\right\rfloor}
\exp\left(\frac{2\pi i(br+k)c}{q}\right) \left| x^k \bmod n \right\rangle, \ee and so, since $x^k \not\equiv x^{k'} \bmod n$ for $k$ and $k'$ such that $0 \leq k < r$, $0 \leq k' < r$ and $k \neq k'$, then
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) &=& \frac{1}{q^2} \sum_{k=0}^{r-1} \left| \sum_{b=0}^{\left\lfloor\frac{q-1-k}{r}\right\rfloor}
\exp\left(\frac{2\pi i(br+k)c}{q}\right) \right|^2\\
&=& \frac{1}{q^2} \sum_{k=0}^{r-1} \left| \sum_{b=0}^{\left\lfloor\frac{q-1-k}{r}\right\rfloor}
\exp\left(\frac{2\pi ibcr}{q}\right) \right|^2\\
&=& \frac{1}{q^2} \sum_{k=0}^{r-1} \left| \sum_{b=0}^{\left\lfloor\frac{q+k}{r}\right\rfloor-1}
\exp\left(\frac{2\pi ibcr}{q}\right) \right|^2, \ee where the last equality is obtained by substituting $r-1-k$ for $k$.
If $\frac{cr}{q} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then \begin{eqnarray*}} \def\ee{\end{eqnarray*} \exp\left(\frac{2\pi icr}{q}\right) = 1, \ee so that
\begin{eqnarray}} \def\ene{\end{eqnarray} P(X = c) &=& \frac{1}{q^2} \sum_{k=0}^{r-1} \left|
\sum_{b=0}^{\left\lfloor\frac{q+k}{r}\right\rfloor-1} 1 \right|^2 \eqlabel{cr_int}\\ &=& \frac{1}{q^2} \sum_{k=0}^{r-1} \left\lfloor\frac{q+k}{r}\right\rfloor^2. \nonumber \ene
On the other hand, if $\frac{cr}{q} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$, then
\begin{eqnarray}} \def\ene{\end{eqnarray} P(X = c) &=& \frac{1}{q^2} \sum_{k=0}^{r-1} \left| \sum_{b=0}^{\left\lfloor\frac{q+k}{r}\right\rfloor-1}
\exp\left(\frac{2\pi ibcr}{q}\right) \right|^2 \nonumber\\
&=& \frac{1}{q^2} \sum_{k=0}^{r-1} \left| \frac{\exp\left(\frac{2\pi icr}{q}\left\lfloor\frac{q+k}{r}\right\rfloor
\right)-1}{\exp\left(\frac{2\pi icr}{q}\right)-1} \right|^2 \eqlabel{cr_non_int}\\ &=& \frac{1}{q^2} \sum_{k=0}^{r-1} \frac{\sin^2\left(\frac{\pi cr}{q}\left\lfloor\frac{q+k}{r}\right\rfloor \right)}{\sin^2 \frac{\pi cr}{q}}. \nonumber \ene The second equality above follows from the evaluation of the geometric progression \begin{eqnarray*}} \def\ee{\end{eqnarray*} \sum_{b=0}^{\left\lfloor\frac{q+k}{r}\right\rfloor-1} \exp\left(\frac{2\pi ibcr}{q}\right) = \frac{\exp\left(\frac{2\pi icr}{q}\left\lfloor\frac{q+k}{r}\right\rfloor \right)-1}{\exp\left(\frac{2\pi icr}{q}\right)-1}. \ee
Much, if not all, of this is already known (e.g. page 17 of \cite{volovich}).
If $\frac{q}{r} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) = \left\{ \begin{array}} \def\ea{\end{array}{ll} \frac{1}{r}, & \frac{cr}{q} \in \@ifnextchar[\@BZoptarg{\Bbb Z},\\ 0, & \mbox{otherwise}, \ea \right. \ee so that $c$ is guaranteed to be a multiple of $\frac{q}{r}$. Since $c = \frac{dq}{r}$ for some integer $0 \leq d < r$, then $c/q$ is guaranteed to be equal to $d/r$ for some $0 \leq d < r$, and all values of $d$ occur with equal probability $1/r$.
\subsection{The Case That $q/r$ is not an Integer}
The case where $\frac{q}{r} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$ is more difficult.
In the case that $1 \leq q \leq r$, then \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) = \frac{1}{q}, \ee for all $c$, so that all possible values of the first register occur with equal probability, and so no useful information can be obtained, as the behaviour is independent of $r \geq q$.
Since no useful information can be obtained if $q \leq r$, then from now it will be assumed that $q > r$.
If $\frac{cr}{q} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then \begin{equation}} \def\ese{\end{equation} \eqlabel{non_int_int} \frac{1}{r} - \frac{2}{q} + \frac{r}{q^2} < P(X = c) < \frac{1}{r} + \frac{2}{q} + \frac{r}{q^2}. \ese Note that $P(X = c) = \frac{1}{r} + O(\frac{1}{q})$. If $q$ is much larger than $r$, then it follows that $P(X = c)$ is very close to $\frac{1}{r}$.
Suppose $\frac{cr}{q} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$, then \begin{equation}} \def\ese{\end{equation} \eqlabel{upper_bound} P(X = c) \leq \frac{r}{q^2 \sin^2\frac{\pi cr}{q}}. \ese This gives an upper bound for $P(X = c)$, and demonstrates that as the distance between $c$ and the nearest integral multiple of $\frac{q}{r}$ increases, the maximum possible probability that $X = c$ decreases. Specifically, the measured value of the first register is more likely to be in the neighbourhood of some multiple of $\frac{q}{r}$ than it is not to be in any such neighbourhood.
Suppose that $c = \frac{dq}{r} + \Delta$, where $d \in \@ifnextchar[\@BZoptarg{\Bbb Z}$ and $0 <
| \Delta | \leq \frac{q}{2r}$, so that \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) &\leq& \frac{r}{q^2 \sin^2\left(\pi d+\frac{\pi \Delta r} {q}\right)}\\ &=& \frac{r}{q^2 \sin^2\frac{\pi \Delta r}{q}}, \ee by straightforward substitution for $c$ in \eqref{upper_bound}.
If $\frac{dq}{r} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, so that $\Delta \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then \begin{equation}} \def\ese{\end{equation} \eqlabel{non_int_int_delta} P(X = c) < \frac{r}{q^2}. \ese Since $P(X = c) = O(\frac{r}{q^2})$, then for $q$ much larger than $r$, $P(X = c)$ is approximately equal to zero.
If $\frac{dq}{r} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$, so that $\Delta \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$, then \begin{equation}} \def\ese{\end{equation} \eqlabel{delta_upper} P(X = c) < \frac{1}{\left(1 - \frac{\pi^2 \Delta^2 r^2}{6 q^2}\right)^2} \left(\frac{\sin^2\frac{\pi dq}{r}}{\pi^2 \Delta^2 r}
+ \frac{2 \left|\sin\frac{\pi dq}{r}\right|}{\pi |\Delta| q} + \frac{r}{q^2}\right).
\ese Further, if $|\Delta| \leq \frac{q}{\pi r} |\sin\frac{\pi dq}{r}|$, then \begin{equation}} \def\ese{\end{equation} \eqlabel{delta_lower} P(X = c) > \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 \Delta^2 r}
- \frac{2 \left|\sin\frac{\pi dq}{r}\right|}{\pi |\Delta| q} + \frac{r}{q^2}. \ese It follows that \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) = \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 \Delta^2 r} + O\left(\frac{1}{q}\right), \ee so that if $q$ is much larger than $r$, then \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) \sim \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 \Delta^2 r}, \ee and so $P(X = c)$ is inversely proportional to $(c - \frac{dq}{r})^2$ in the asymptotic limit. Note that the asymptotic profile is dependent only on the fractional part of $\frac{dq}{r}$, and not on the size of $q$.
\subsection{Probabilities for Certain Subsets}
Since the fraction which interests us, as far as determining the order is concerned, is not $\frac{c}{q}$ (where $c$ is the measured value of the first register), but $\frac{d}{r}$, then the probability that $\frac{c}{q}$ falls in the proximity of $\frac{d}{r}$ is important, and the probability that $\frac{c}{q}$ falls within a certain distance of $\frac{d}{r}$ (or, equivalently, that $c$ falls within a certain distance of $\frac{dq}{r}$), will be determined for a certain range of distances.
If $\frac{dq}{r} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then \begin{equation}} \def\ese{\end{equation} \eqlabel{int_full} \frac{1}{r} - \frac{2}{q} + \frac{r}{q^2} < P\left(\rho_q\left(X,\frac{dq}{r}\right) \leq \frac{q}{2r} \right) < \frac{1}{r} + \frac{3}{q} + \frac{r}{q^2}, \ese where $\rho_q$ is the modular metric \eqref{mod}. Note that $P(\rho_q(X,\frac{dq}{r}) \leq \frac{q}{2r}) = \frac{1}{r} + O(\frac{1}{q})$. If $q$ is much larger than $r$, then it follows that $P(\rho_q(X,\frac{dq}{r}) \leq \frac{q}{2r})$ is very close to $\frac{1}{r}$.
The value of the parameter $q$ will now be restricted so that $q > 2r$.
From now, for $c \in \{ 0,1,\dots,q-1 \}$, $d_c$ and $\Delta_c$ will be uniquely determined by the following conditions: \begin{enumerate}} \def\een{\end{enumerate} \item $d_c \in \{ 0,1,\dots,r \}$; \item $c = \frac{d_c q}{r} + \Delta_c$; \item $- \frac{q}{2r} < \Delta_c \leq \frac{q}{2r}$. \een
For $0 < u \leq \frac{q}{2r} - 1$, \begin{equation}} \def\ese{\end{equation} \eqlabel{far}
P(|\Delta_X| \geq u + 1) < \frac{2}{\pi^2 u}. \ese This determines a hard upper bound independent of $q$ for the probability that the distance between the measured value of the first register and the nearest multiple of $\frac{q}{r}$ exceeds any given value greater than $1$ but no greater than $\frac{q}{2r}$, and demonstrates that the measured value of the first register will tend to be close to a multiple of $\frac{q}{r}$. Specifically, for any large fixed distance, the probability that the difference between the measured value of the first register and the nearest multiple of $\frac{q}{r}$ exceeds this distance is small, independent of the size of $q$.
Let $u$ now be fixed subject to $0 < u \leq \frac{q}{2r} - 1$.
If $\frac{dq}{r} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then \begin{equation}} \def\ese{\end{equation} \eqlabel{int} \frac{1}{r} - \frac{2}{q} + \frac{r}{q^2} < P\left(\rho_q\left(X,\frac{dq}{r}\right) < u + 1\right) < \frac{1}{r} + \frac{2}{q} + \frac{(2u+3)r}{q^2}. \ese
If $\frac{dq}{r} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$, then it follows from the bounds already determined on $P(X = c)$ for $c$ such that \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\left|c - \frac{dq}{r}\right| < u + 1, \ee that if
\begin{eqnarray*}} \def\ee{\end{eqnarray*} u \leq \frac{q}{\pi r} \left| \sin\frac{\pi dq}{r} \right| - 1, \ee then \begin{eqnarray}} \def\ene{\end{eqnarray} && \frac{1}{r} - \frac{2}{\pi^2 r u} - \frac{2}{\pi q} \left(\frac{r^2}{r-1} + \ln\frac{r^2(u+1)^2}{r-1}\right) + \frac{r (2u+1)}{q^2} \nonumber\\
&<& P\left(\left| X - \frac{dq}{r} \right| < u+1\right) \eqlabel{non_int}\\ &<& \frac{1}{\left(1 - \frac{\pi^2 (u+1)^2 r^2}{6 q^2}\right)^2} \left(\frac{1}{r} + \frac{2}{\pi q} \left(\frac{r^2}{r-1} + \ln\frac{r^2(u+1)^2}{r-1}\right) + \frac{r (2u+3)}{q^2}\right). \nonumber
\ene Note that if $u$ is large, and if $q$ is much larger than $r u$, then $P(|X-\frac{dq}{r}| < u+1)$ is very close to $\frac{1}{r}$.
The probability that the measured value of the first register will be in the proximity of any specified multiple of $\frac{q}{r}$ has been determined to be very close to $\frac{1}{r}$ for any given multiple, and so the probability that $\frac{c}{q}$ (where $c$ is the measured value of the first register) is close to $\frac{d}{r}$ is approximately $\frac{1}{r}$ for any given value of $d$.
In summary, for $q$ very large, the nett probability of $1$ is equally divided amongst the vicinities of $\frac{dq}{r}$ for $d \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, with the probability effectively concentrated within vicinities of fixed maximum width, so that as $q$ increases, the vicinities recede from each other while maintaining their maximum widths.
\section{Continued Fractions} \seclabel{continued}
The determination of an appropriate rational number $\frac{d'}{r'}$ from the measured value of $c$ is done on a classical computer with the use of continued fractions (see \cite{shor}, for example). In the context of the refinement of Shor's Algorithm, we are interested in the width of the vicinity of $\frac{dq}{r}$ which will, with certainty, identify $\frac{d}{r}$ as the correct approximation to $\frac{c}{q}$ where $c$ is the measured value of the first register. The width is linearly dependent on $q$.
The definition of a continued fraction is given here, along with some useful properties.
The definition of continued fractions and most of the consequences, as drawn below, can be found in \cite{hardy} and \cite{knuth}.
For integers $a_0,a_1,a_2,\dots,a_N$, where $a_0 \geq 0$ and $a_i > 0$ for $i = 1,2,\dots,N$, define the continued fraction $[a_0,a_1,a_2,\dots, a_N]$ by \begin{eqnarray*}} \def\ee{\end{eqnarray*} [a_0,a_1,a_2,\dots,a_N] = a_0 + \frac{1}{a_1+\frac{1}{a_2+\frac{1}{\dots +\frac{1}{a_N}}}}, \ee so that $[a_0,a_1,a_2,\dots,a_N]$ is a rational number. Alternatively, a finite continued fraction can be defined by induction on the number of terms as follows. For a non-negative integer $a_0$, define $[a_0] = a_0$, and for integers $a_0,a_1,a_2,\dots,a_N$, as above, define \begin{eqnarray*}} \def\ee{\end{eqnarray*} [a_0,a_1,a_2,\dots,a_N] = a_0 + \frac{1}{[a_1,a_2,\dots,a_N]}. \ee For any $0 \leq k \leq N$, $\xi_k = [a_0,a_1,\dots,a_k]$ is called a convergent of the continued fraction expansion.
If $\xi = \frac{p}{q}$ is rational, then define $a_i$ and $\zeta_i$ by induction on $i$ by \begin{eqnarray*}} \def\ee{\end{eqnarray*} \zeta_i &=& \left\{ \begin{array}} \def\ea{\end{array}{ll} \xi, & i = 0,\\ \frac{1}{\zeta_{i-1}-a_{i-1}}, & \mbox{otherwise, if $\zeta_{i-1} \ne a_{i-1}$}, \ea \right.\\ a_i &=& \lfloor \zeta_i \rfloor, \ee terminating when $\zeta_i = a_i$ ({\it i.e.} when $\zeta_i$ is an integer). This gives a continued fraction expansion $\xi = [a_0,a_1,a_2,\dots,a_N]$, where $a_N > 1$. Alternatively, $\xi = [a_0,a_1,a_2,\dots,a_N-1,1]$, yielding two distinct continued fraction expansions for $\xi$. It is known that for any rational number $\xi$, these two continued fraction expansions are the only possible expansions (for irrational numbers, there is exactly one continued fraction expansion, which is infinite).
Define integers $p_k$ and $q_k$ for $k \geq -1$ by induction on $k$ as follows. Let \begin{eqnarray}} \def\ene{\end{eqnarray} p_k &=& \left\{ \begin{array}} \def\ea{\end{array}{ll} 1, & k = -1,\\a_0, & k = 0,\\ p_{k-2} + a_k p_{k-1}, & k > 0, \ea \right. \eqlabel{numer}\\ q_k &=& \left\{ \begin{array}} \def\ea{\end{array}{ll} 0, & k = -1,\\1, & k = 0,\\ q_{k-2} + a_k q_{k-1}, & k > 0, \ea\right. \eqlabel{denom} \ene then by standard results from the theory of continued fractions, \begin{itemize}} \def\ei{\end{itemize} \item $\xi_k = p_k/q_k$ for $k = 0,1,2,\dots$; \item $p_{k-1} q_k - p_k q_{k-1} = (-1)^k$ for $k = 0,1,2,\dots$; \item $\xi_{k+1} - \xi_k = \frac{(-1)^k}{q_k q_{k+1}}$; \item $\gcd(p_k,q_k) = 1$. \ei The first two statements are very easily proved by induction, and the third and fourth statements are trivial consequences of the first two.
It is also well-known that if $\xi$ is a positive real number, $p$ and $q$
are positive integers, and $|\xi - \frac{p}{q}| \leq \frac{1}{2q^2}$, then $\frac{p}{q}$ is a convergent of the continued fraction expansion for $\xi$ (see for example, \cite{hardy,knuth}).
It is proven in \cite{hardy} that \begin{theorem}} \def\et{\end{theorem} \thmlabel{hardy} For $k > 1$, let $\frac{p_k}{q_k}$ be the corresponding convergent of the continued fraction expansion for $\xi$, so that $p_k$ and $q_k$ are defined by \eqref{numer} and \eqref{denom}, then for $0 < q \leq q_k$ and $p \in \@ifnextchar[\@BZoptarg{\Bbb Z}$ such that $\frac{p}{q} \ne \frac{p_k}{q_k}$, \begin{eqnarray*}} \def\ee{\end{eqnarray*}
| p - q \xi | \geq | p_k - q_k \xi |, \ee and \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\left| \xi - \frac{p}{q} \right| > \left| \xi - \frac{p_k}{q_k} \right|. \ee \et
We now come to the principal result that will be of use in analysing the refinement of Shor's Algorithm, since it gives a sufficient condition on $c$ (the result of measuring the first register) that will guarantee that the nearest fraction to $\frac{c}{q}$ with denominator less than $n$ is $\frac{d}{r}$ for some integer $d$, and that $\frac{d}{r}$ is a convergent of the continued fraction expansion for $\frac{c}{q}$. \begin{theorem}} \def\et{\end{theorem} \thmlabel{square} Suppose $r,n \in \@ifnextchar[\@BZoptarg{\Bbb Z}$ and $0 < r < n$. For $v > 1$, let $q$ be an integer greater than or equal to
$2 v n^2$. Suppose $d \in \{ 0,1,\dots,r \}$ and $|c-\frac{dq}{r}| \leq v$. Let $d'/r'$ be the fraction satisfying the following conditions: \begin{itemize}} \def\ei{\end{itemize} \item $d'/r'$ is in lowest terms ($d'$ and $r'$ have no common factors); \item $0 \leq d'/r' \leq 1$; \item $0 < r' < n$; \item $d'/r'$ is the nearest fraction to $c/q$ which satisfies the other three conditions. \ei Then $d'/r' = d/r$, and $d/r$ is a convergent of the continued fraction expansion for $c/q$. Define $p_k$ and $q_k$ for $k = 0,1,\dots$, by \eqref{numer} and \eqref{denom}, respectively. Let $N = \max \{ k : q_k < n \}$, then $\xi_N = p_N/q_N = d/r$, so that $d/r$ is the last convergent of the expansion which has denominator less than $n$. \et
\begin{proof}} \def\ep{\end{proof} Since $|c-\frac{dq}{r}| \leq v$, then \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\left|\frac{c}{q} - \frac{d}{r}\right| \leq \frac{v}{q} \leq \frac{1}{2n^2} < \frac{1}{2r^2}, \ee so that $d/r$ is a convergent of the continued fraction expansion for $c/q$.
Suppose $f,s \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, $0 < s < n$, and $0 \leq f \leq s$. If $f/s \ne d/r$, then \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\left|\frac{f}{s} - \frac{d}{r}\right| = \left|\frac{fr-ds}{rs}\right| \geq \frac{1}{rs} > \frac{1}{n^2}, \ee so that \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\left|\frac{c}{q} - \frac{f}{s}\right| \geq
\left|\frac{d}{r} - \frac{f}{s}\right|
- \left|\frac{c}{q} - \frac{d}{r}\right| > \frac{1}{n^2} - \frac{1}{2n^2} = \frac{1}{2n^2}
\geq \left|\frac{c}{q} - \frac{d}{r}\right|, \ee so that $d/r$ is the nearest fraction to $c/q$ which satisfies the requisite three conditions.
Since $d/r = d'/r'$ is a convergent of the continued fraction expansion for $c/q$, then there exists $N$ such that $p_N = d'$ and $q_N = r' < n$. If $q_{N+1} < n$, then, as a consequence of \thmref{hardy}, \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\left|\frac{c}{q} - \frac{p_{N+1}}{q_{N+1}}\right|
< \left|\frac{c}{q} - \frac{p_N}{q_N}\right|
= \left|\frac{c}{q} - \frac{d'}{r'}\right|, \ee contradicting the fact that $d/r$ is the nearest fraction to $c/q$ with denominator less than $n$. It follows that $q_{N+1} \geq n$, and so $d/r$ is the last convergent of the expansion which has denominator less than $n$. \ep
\section{Some Analysis of the Refinement of Shor's algorithm} \seclabel{algo_analysis}
Recall the refinement of the Shor's Algorithm as given earlier. All steps except step 2 are performed on a classical computer. \begin{enumerate}} \def\een{\end{enumerate} \item Set $s := 1$ and $q \geq 2 w n^3$ ({\it e.g.} set $q$ to be that unique power of 2 such that $2 w n^3 \leq q < 4 w n^3$); \item Perform the quantum algorithm on the quantum computer with $q$ as specified in Step 1, and measure the value $c$ of the first register; \item Determine the continued fraction expansion for $\frac{c}{q}$; \item Determine all denominators of convergents of the continued fraction expansion up to the first denominator greater than or equal to $n$; \item Let $r'$ be the last denominator less than $n$, and set $s := \mathop{\rm lcm}\nolimits(s,r')$; \item Calculate $x^s \bmod n$; \item If $x^s \not\equiv 1 \bmod n$, then go to Step 2; \item Output $s$. \een
The significant results of the last two sections can be summarised as follows: \begin{itemize}} \def\ei{\end{itemize} \item For $q$ very large, the nett probability of $1$ is essentially equally divided amongst vicinities of $\frac{dq}{r}$ of fixed finite maximum width for $d \in \@ifnextchar[\@BZoptarg{\Bbb Z}$; \item The width of the vicinity of $\frac{dq}{r}$ which will, with certainty, identify $\frac{d}{r}$ as the correct approximation to $\frac{c}{q}$, is linearly dependent on $q$. \ei This means that if a large enough value for $q$ is taken, then the vicinity which will, with certainty, identify $\frac{d}{r}$ as the correct approximation to $\frac{c}{q}$, will encompass the entire vicinity of $\frac{dq}{r}$ in which the probability is effectively concentrated. This is the {\it raison d'\^etre} for choosing $q$ with the value as given in the refinement.
In the refinement of Shor's Algorithm, then
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P(|\Delta_X| \geq w n) < \frac{2}{\pi^2 (w n - 1)}, \ee as a consequence of \eqref{far}, so that
\begin{equation}} \def\ese{\end{equation} P(|\Delta_X| < w n) > 1 - \frac{2}{\pi^2 (w n - 1)}, \eqlabel{less}
\ese and so if $n$ is large, then $P(|\Delta_X| \geq w n)$ is very small, and
$P(|\Delta_X| < w n)$ is very close to 1. If $\frac{dq}{r} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then by \eqref{int}, \begin{equation}} \def\ese{\end{equation} \eqlabel{ref1} \frac{1}{r} - \frac{1}{w n^3} < P\left(\rho_q\left(X,\frac{dq}{r}\right) < w n\right) < \frac{1}{r} + \frac{1}{w n^3} + \frac{1}{2 w n^4}. \ese This result is proven in \appref{algo_analysis}.
If $\frac{dq}{r} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$, then, by \eqref{non_int}, \begin{eqnarray}} \def\ene{\end{eqnarray} && \frac{1}{r} - \frac{2}{\pi^2 (wn - 1)} - \frac{1}{\pi w n^3} \left(n + 1 + \ln\frac{w^2 n^4}{n-1}\right) \nonumber\\
&<& P\left(\left| X - \frac{dq}{r} \right| < w n\right) \eqlabel{ref2}\\ &<& \frac{1}{\left(1 - \frac{\pi^2}{24 n^2}\right)^2} \left(\frac{1}{r} + \frac{1}{\pi w n^3} \left(n + 1 + \ln\frac{w^2 n^4}{n-1}\right) + \frac{1}{2 w n^4}\right). \nonumber \ene This result is also proven in \appref{algo_analysis}.
It follows that if $n$ is large, as in the case for any practical RSA encryption algorithm, then $P(\rho_q(\Delta_X,\frac{dq}{r}) < w n)$ is close to $1/r$ for all $d$.
For the refinement, the probability that $|\Delta_c| < wn$, where $c$ is the measured value of the first register, is greater than $1 - \frac{2}{\pi^2 (wn - 1)}$ by \eqref{less}. By \thmref{square},
if $|\Delta_c| < wn$, then the last convergent of the continued fraction expansion for $c/q$ with denominator less than $n$ is necessarily of the form $d/r$ for some $d \in \@ifnextchar[\@BZoptarg{\Bbb Z}$ such that $0\leq d \leq r$, so that, in the refinement, $r'$ necessarily divides $r$, as this is the convergent which is determined by the refinement (or rather, its denominator is determined by the refinement). It follows that after each run on the quantum computer, the probability that $r'$ divides $r$ is greater than $1 - \frac{2}{\pi^2 (wn - 1)}$. Since the runs on the quantum computer are, in effect, independent random samples with replacement, then the probability that $s$ still divides $r$ after $k$ runs on the quantum computer is greater than $(1 - \frac{2}{\pi^2 (wn - 1)})^k$. Specifically, for the size of $n$ that would typically be used in RSA encryption, the probability that $s$ will not divide $r$ after $k$ runs on the quantum computer is negligibly small (of the same order of magnitude as $\frac{k}{wn}$). Since the value of $s$ is almost guaranteed to be a divisor of $r$ after $k$ runs of the quantum computer, and $x^s \equiv 1 \bmod n$ iff $s$ is a multiple of $r$, then it is almost guaranteed that when the refinement terminates, $s$ will be equal to $r$ ($s$ is certainly a multiple of $r$ on termination, and it is almost certain to be a divisor of $r$).
This can be expressed formally as follows. Let $A_k$ denote the random variable describing the result of the measurement of the first register after the $k$-th run of the quantum computer, let $B_k$ denote the random variable describing the corresponding value of $r'$ calculated by the classical computer, and let $C_k$ be the random variable defined by \begin{eqnarray*}} \def\ee{\end{eqnarray*} C_k = \mathop{\rm lcm}\nolimits(B_1,\dots,B_k), \ee so that $C_k$ describes the value of $s$ after $k$ runs of the quantum computer, then, by \eqref{less},
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P(|\Delta_{A_k}| < w n) > 1 - \frac{2}{\pi^2 (w n - 1)}, \ee for all $k$, so that by \thmref{square},
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P(B_k | r) > 1 - \frac{2}{\pi^2 (w n - 1)}, \ee for all $k$, and so
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P(C_k | r) > \left(1 - \frac{2}{\pi^2 (w n - 1)}\right)^k, \ee for all $k$, by the independence of the random variables $A_k$ (from which the independence of the random variables $B_k$ follows).
\section{Some Results in Probability} \seclabel{prob}
It was noted in \secref{algo_analysis} that the probability that $\frac{d}{r}$ is the fraction with denominator less than $n$ which is closest to $\frac{c}{q}$ (where $c$ is the measured value of the first register) is close to $\frac{1}{r}$ (and in fact, it approaches $\frac{1}{r}$ in the limit as $q \to \infty$). The properties of the probability distributions of certain random variables (which are analogues of important random variables related to the refinement) associated with the idealized distribution follow. The purpose here is to get some idea of the probability that the refinement of Shor's Algorithm will terminate after at most $k$ runs of the quantum computer and output the required order.
Let a natural number $s$ have prime factorization \begin{eqnarray*}} \def\ee{\end{eqnarray*} s = \prod_{j \in J} p_j^{a_j}, \ee where $J$ is some index set, $p_j$ are distinct primes, and $a_j \geq 1$ for all $j \in J$. Let $Z_i, \ i = 1,2,3,\dots$, denote independent uniformly distributed random variables from the sample space $\{ 0,1,\dots,s-1 \}$, so that for all $i$, and for all $d$ in the sample space, $P(Z_i = d) = \frac{1}{s}$. Let $R_i$ be the random variable defined by \begin{eqnarray*}} \def\ee{\end{eqnarray*} R_i = \frac{s}{\gcd(Z_i,s)}, \ee so that $R_i$ are independent random variables, and $R_i$ is the denominator of $\frac{Z_i}{s}$, when expressed in lowest terms. For $k = 1,2,3,\dots$, define the random variable $S_k$ by \begin{eqnarray*}} \def\ee{\end{eqnarray*} S_k = \mathop{\rm lcm}\nolimits(R_1,R_2,\dots,R_k). \ee Note that $s$ is a parameter for the probability distributions of $Z_i$, $R_i$, and $S_k$. \begin{theorem}} \def\et{\end{theorem} \thmlabel{zeta} For all values of the parameter $s$, \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(S_k = s) > \frac{1}{\zeta(k)}, \ee for all $k \geq 2$, where $\zeta$ is the Riemann zeta function defined by \begin{eqnarray*}} \def\ee{\end{eqnarray*} \zeta(z) = \sum_{n=1}^\infty \frac{1}{n^z} = \prod_{p \mbox{\small{ prime}}} \frac{1}{1 - \frac{1}{p^z}}, \ee for $\Re(z) > 1$. \et Specifically, \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(S_2 = s) > \frac{6}{\pi^2} > \frac{3}{5}, \quad P(S_4 = s) > \frac{90}{\pi^4} > \frac{9}{10}, \ee so that the probability that $S_2$ is equal to $s$ is greater than 60\%, and the probability that $S_4$ is equal to $s$ is greater than 90\%. Similarly, \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(S_6 = s) > \frac{945}{\pi^6} > \frac{49}{50}, \quad P(S_8 = s) > \frac{9450}{\pi^8} > \frac{199}{200}, \ee so that the probability that $S_6$ is equal to $s$ is greater than 98\%, and the probability that $S_8$ is equal to $s$ is greater than 99.5\%.
The proof of \thmref{zeta} is given in \appref{zeta}
\section{More Analysis of the Refinement of Shor's Algorithm} \seclabel{final}
As before, let $A_k$ denote the random variable describing the result of the measurement of the first register after the $k$-th run of the quantum computer, let $B_k$ describe the corresponding value of $r'$ as determined by the refinement, and let the random variable $C_k$ be defined by \begin{eqnarray*}} \def\ee{\end{eqnarray*} C_k = \mathop{\rm lcm}\nolimits(B_1,\dots,B_k). \ee Further, let the random variable $D_k$ be defined by \begin{eqnarray*}} \def\ee{\end{eqnarray*} D_k = \left\lfloor \frac{rA_k}{q} + \frac{1}{2} \right\rfloor, \ee so that $D_k$ describes the nearest integer to $\frac{rc}{q}$, where $c$ is the measured value of the first register after the $k$-th run of the quantum computer. Note that $A_k, \ k = 1,2,3,\dots$, are independent random variables, and that for $c = 0,\dots,q-1$, \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(A_k = c) = \left\{ \begin{array}} \def\ea{\end{array}{ll} \frac{1}{q^2} \sum_{m=0}^{r-1} \left\lfloor\frac{q+m}{r}\right\rfloor^2, & \frac{cr}{q} \in \@ifnextchar[\@BZoptarg{\Bbb Z},\\ \\ \frac{1}{q^2} \sum_{m=0}^{r-1} \frac{\sin^2\left(\frac{\pi cr}{q}\left\lfloor\frac{q+m}{r}\right\rfloor \right)}{\sin^2 \frac{\pi cr}{q}}, & \frac{cr}{q} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}, \ea \right. \ee and as noted previously, for the refinement,
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P(|\Delta_{A_k}| < w n) > 1 - \frac{2}{\pi^2 (w n - 1)}, \ee so that
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P(B_k | r) > 1 - \frac{2}{\pi^2 (w n - 1)}, \ee by \thmref{square}, and so
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P(C_k | r) > \left(1 - \frac{2}{\pi^2 (w n - 1)}\right)^k. \ee
It follows that for any given $d$, by \eqref{ref1} and \eqref{ref2}, \begin{eqnarray*}} \def\ee{\end{eqnarray*} \frac{1}{r} - \frac{1}{w n^3} < P\left(\rho_q\left(A_k,\frac{dq}{r}\right) < w n\right) < \frac{1}{r} + \frac{1}{w n^3} + \frac{1}{2 w n^4}, \ee if $\frac{dq}{r} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, and \begin{eqnarray*}} \def\ee{\end{eqnarray*} && \frac{1}{r} - \frac{2}{\pi^2 (wn - 1)} - \frac{1}{\pi w n^3} \left(n + 1 + \ln\frac{w^2 n^4}{n-1}\right)\\
&<& P\left(\left| A_k - \frac{dq}{r} \right| < w n\right)\\ &<& \frac{1}{\left(1 - \frac{\pi^2}{24 n^2}\right)^2} \left(\frac{1}{r} + \frac{1}{\pi w n^3} \left(n + 1 + \ln\frac{w^2 n^4}{n-1}\right) + \frac{1}{2 w n^4}\right), \ee if $\frac{dq}{r} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$.
This concludes the summary of what is already known.
Note that $D_k = d$ if \begin{eqnarray*}} \def\ee{\end{eqnarray*} \rho_q\left(A_k,\frac{dq}{r}\right) < wn, \ee for $d = 1,\dots,r-1$, and $D_k = 0$ or $D_k = r$ if \begin{eqnarray*}} \def\ee{\end{eqnarray*} \rho_q(A_k,0) < wn. \ee
Given $\rho_q(A_k,\frac{dq}{r}) < wn$, for some $d = 0,1,\dots,r$, then $D_k = d$, and \begin{eqnarray*}} \def\ee{\end{eqnarray*} B_k = \frac{r}{\gcd(D_k,r)}. \ee It follows that for all $d$, \begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left(\rho_q\left(A_k,\frac{dq}{r}\right) < wn \mbox{ and } D_k = d\right) = \frac{1}{r} + O\left(\frac{1}{wn}\right), \ee and \begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left( \rho_q\left(A_k,\frac{dq}{r}\right) \geq wn \mbox{ for all } d \right) = O\left(\frac{1}{wn}\right). \ee This means that the probability distribution for $D_k$ becomes uniform in the asymptotic limit, and the results of the last Section become exact in the asymptotic limit. Here, $D_k$ plays the same role as $Z_i$, $B_k$ plays the same role as $R_i$, and $C_k$ plays the same role as $S_k$. This means that the asymptotic limit of $P(C_k = r)$ as $n$ becomes large should be greater than $\frac{1}{\zeta(k)}$ for $k \geq 2$.
By a similar argument to that used in the proof of \thmref{zeta} (in \appref{zeta}), for $k \geq 2$ and $k$ small, \begin{eqnarray}} \def\ene{\end{eqnarray} P(C_k = r) &=& \prod_{j \in J} \left(1 - \frac{1}{p_j^k}\right) + O\left(\frac{1}{w}\right) \nonumber\\ &=& \prod_{j \in J} \left(1 - \frac{1}{p_j^k}\right) + O(n^{-\epsilon}), \eqlabel{ck=r} \ene since $w = n^\epsilon$. Finally, since \begin{eqnarray*}} \def\ee{\end{eqnarray*} \prod_{j \in J} \left(1 - \frac{1}{p_j^k}\right) > \prod_{p \mbox{\small{ prime}}} \left(1 - \frac{1}{p^k}\right) = \frac{1}{\zeta(k)}, \ee then \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(C_k = r) > \frac{1}{\zeta(k)} - O(n^{-\epsilon}), \ee where the statement $f > g - O(h)$ means that there exists a function $F$ such that $f > g - F$ in the asymptotic limit, and $F/h$ is bounded in the same limit.
The derivation of \eqref{ck=r} is given in \appref{prob}.
This means that for the size of $n$ that would typically be used in RSA encryption, the probability that the correct value for $r$ will be found after at most $2$ runs of the quantum computer is at least 60\%, and the probability that the correct value for $r$ will be found after at most $4$ runs of the quantum computer is at least 90\%, {\it etc}.
\section{Conclusion}
There are various advantages and disadvantages to the refinement of Shor's algorithm as detailed in this paper. The advantages include the facts that each run of the quantum computer is almost certain to evaluate $r'$ as a divisor of $r$, and that the probability that the actual value of $r$ will be found after at most $k$ runs of the quantum computer is greater than $1/\zeta(k)$, so that the probability is greater than 60\% that no more than 2 runs will be necessary, and greater than 90\% that no more than 4 runs will be necessary. On the other hand, the quantum computer requires more space and time to run the refinement (the space and time requirements are each multiplied by approximately a constant), and the Quantum Fourier Transform requires more delicate rotations of angles (of the order of $\frac{\pi}{n^3}$, rather than the order of $\frac{\pi}{n^2}$, which is all that Shor's original algorithm would require). Also, the number of runs needed by Shor's original algorithm is $O(\log \log n)$, and $\log \log n$ is a very slowly growing function. For a value of $n = 10^{400}$, if the logarithms are to base 2, $\log \log n$ is between 10 and 11. These are questions which will have to be investigated in greater detail if the case of which algorithm is preferable is to be decided.
\section{Proof of \thmref{modmet}} \applabel{modmet}
\begin{proof}} \def\ep{\end{proof} There are three conditions to be checked in order to show that $\rho_q$ is a metric. \begin{enumerate}} \def\een{\end{enumerate}
\item Note that $|x-y| \geq 0$ for all $x,y \in I_q$. Since $0 \leq x \leq q-1$ and $0 \leq y \leq q-1$, then $|x-y| \leq q-1$, and so $q-|x-y| \geq 1 > 0$, so that for all $x, y \in I_q$, \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\rho_q(x,y) = \min(|x-y|,q-|x-y|) \geq 0. \ee
For all $x \in I_q$, $|x-x| = 0$, so $q-|x-x| = q$, and so $\rho_q(x,x) = 0$. Conversely, suppose $x,y \in I_q$ and that $\rho_q(x,y) = 0$. Since $\rho_q(x,y) = \min(|x-y|,q-|x-y|)$, then either $|x-y| = 0$ or
$q-|x-y| = 0$. If $|x-y| = 0$, then $x = y$. If $q-|x-y| = 0$, then
$|x-y| = q$, contradicting $|x-y| \leq q-1$.
It follows that $\rho_q(x,y) \geq 0$ for all $x,y \in I_q$, and that $\rho_q(x,y) = 0$ iff $x = y$.
\item Since $|x-y| = |y-x|$, then $\rho_q(x,y) = \rho_q(y,x)$, so that $\rho_q$ is symmetric.
\item If $\rho_q(x,y) = |x-y|$ and $\rho_q(y,z) = |y-z|$, then \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\rho_q(x,z) \leq |x-z| \leq |x-y| + |y-z| = \rho_q(x,y) + \rho_q(y,z). \ee
If $\rho_q(x,y) = |x-y|$ and $\rho_q(y,z) = q-|y-z|$, then, since \begin{eqnarray*}} \def\ee{\end{eqnarray*}
|y-z| \leq |x-y| + |x-z|, \ee it follows that \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\rho_q(x,z) \leq q-|x-z| \leq q+|x-y|-|y-z| = \rho_q(x,y) + \rho_q(y,z).
\ee Similarly, if $\rho_q(x,y) = q-|x-y|$ and $\rho_q(y,z) = |y-z|$, then $\rho_q(x,z) \leq \rho_q(x,y) + \rho_q(y,z)$.
If $\rho_q(x,y) = q-|x-y|$ and $\rho_q(y,z) = q-|y-z|$, then $q-|x-y| \leq |x-y|$
and $q-|y-z| \leq |y-z|$, so that $2 |x-y| \geq q$ and $2 |y-z| \geq q$, and so $|x-y| \geq \frac{q}{2}$ and $|y-z| \geq \frac{q}{2}$. There are two cases. \begin{itemize}} \def\ei{\end{itemize}
\item Case 1 ($0 \leq y < \frac{q}{2}$): Since $|x-y| \geq \frac{q}{2}$, then $y + \frac{q}{2} \leq x \leq q-1$. Similarly, $y + \frac{q}{2} \leq z
\leq q-1$. Since $|x-y| = x-y$ and $|y-z| = z-y$, then $\rho_q(x,y) = q+y-x \geq q-x > z-x$ and $\rho_q(y,z) = q+y-z \geq q-z > x-z$. It follows that \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\rho_q(x,z) = |x-z| < \rho_q(x,y) + \rho_q(y,z). \ee
\item Case 2 ($\frac{q}{2} \leq y < q-1$): Since $|x-y| \geq \frac{q}{2}$, then $0 \leq x \leq y - \frac{q}{2} < \frac{q}{2}$. Similarly, $0 \leq z \leq y - \frac{q}{2} < \frac{q}{2}$. Since $|x-y| = y-x$ and $|y-z| = y-z$, then $\rho_q(x,y) = q+x-y > x \geq x-z$ and $\rho_q(y,z) = q+z-y > z \geq z-x$. It follows that \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\rho_q(x,z) = |x-z| < \rho_q(x,y) + \rho_q(y,z). \ee \ei
It follows that the Triangle Inequality holds. \een It follows from these three facts that $\rho_q$ is a metric on $I_q$, to be called the modular metric.
Recall that $|x-y| \leq q-1$ for $x,y \in I_q$. If $|x-y| \leq \frac{q}{2}$, then $\rho_q(x,y) \leq |x-y| \leq \frac{q}{2}$. On the other hand, if $\frac{q}{2} \leq |x-y| \leq q-1$, then
$\rho_q(x,y) \leq q-|x-y| \leq \frac{q}{2}$. In either case, $\rho_q(x,y) \leq \frac{q}{2}$. \ep
\section{Proof of results presented in \secref{firstreg}} \applabel{firstreg}
The first result to prove is the result that if $\frac{q}{r} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) = \left\{ \begin{array}} \def\ea{\end{array}{ll} \frac{1}{r}, & \frac{cr}{q} \in \@ifnextchar[\@BZoptarg{\Bbb Z},\\ 0, & \mbox{otherwise}. \ea \right. \ee
In this case, then \begin{equation}} \def\ese{\end{equation} \eqlabel{qr_int} \left\lfloor\frac{q+k}{r}\right\rfloor = \frac{q}{r}, \ese for $k = 0,\dots,r-1$. If $\frac{cr}{q} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then substitution of \eqref{qr_int} into \eqref{cr_int} immediately yields \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) = \frac{1}{r}. \ee On the other hand, in the case that $\frac{cr}{q} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$, then $\sin(\frac{\pi cr}{q} \frac{r}{q}) = \sin(\pi c) = 0$ (as $c \in \@ifnextchar[\@BZoptarg{\Bbb Z}$), so that substitution of \eqref{qr_int} into \eqref{cr_non_int} immediately yields \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) = 0. \ee
The next result is that if $1 \leq q \leq r$, then \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) = \frac{1}{q}, \ee for all $c$ (thus yielding no useful information).
In this case, \begin{eqnarray*}} \def\ee{\end{eqnarray*} \left\lfloor\frac{q+k}{r}\right\rfloor = \left\{ \begin{array}} \def\ea{\end{array}{ll} 0, & 0 \leq k < r-q,\\1, & r-q \leq k \leq r-1, \ea \right. \ee so that
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) &=& \frac{1}{q^2} \sum_{k=0}^{r-1} \left| \sum_{b=0}^{\left\lfloor\frac{q+k}{r}\right\rfloor-1}
\exp\left(\frac{2\pi ibcr}{q}\right) \right|^2\\
&=& \frac{1}{q^2} \sum_{k=r-q}^{r-1} \left| \sum_{b=0}^{0}
\exp\left(\frac{2\pi ibcr}{q}\right) \right|^2\\
&=& \frac{1}{q^2} \sum_{k=r-q}^{r-1} | 1 |^2\\ &=& \frac{1}{q^2} q\\ &=& \frac{1}{q}. \ee Alternatively, \eqref{cr_int} and \eqref{cr_non_int} can be used, and they yield the same result.
In the case that $q > r$, then for $k = 0,\dots,r-1$, $q \leq q+k < q+r$, so that \begin{eqnarray*}} \def\ee{\end{eqnarray*} \frac{q}{r} - 1 < \left\lfloor\frac{q+k}{r}\right\rfloor < \frac{q}{r} + 1. \ee
\eqref{non_int_int} is now proven as follows. If $\frac{cr}{q} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then it follows from \eqref{cr_int} that \begin{eqnarray*}} \def\ee{\end{eqnarray*} \frac{1}{q^2} \sum_{k=0}^{r-1} \left(\frac{q}{r} - 1\right)^2 < P(X = c) = \frac{1}{q^2} \sum_{k=0}^{r-1} \left\lfloor\frac{q+k}{r}\right\rfloor^2 < \frac{1}{q^2} \sum_{k=0}^{r-1} \left(\frac{q}{r} + 1\right)^2, \ee and so, expanding the squares, \begin{eqnarray*}} \def\ee{\end{eqnarray*} \frac{1}{r} - \frac{2}{q} + \frac{r}{q^2} < P(X = c) < \frac{1}{r} + \frac{2}{q} + \frac{r}{q^2}. \ee
On the other hand, if $\frac{cr}{q} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$, then it follows from \eqref{cr_non_int} that \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) &=& \frac{1}{q^2} \sum_{k=0}^{r-1} \frac{\sin^2\left(\frac{\pi cr}{q}\left\lfloor\frac{q+k}{r}\right\rfloor \right)}{\sin^2\frac{\pi cr}{q}}\\ &\leq& \frac{1}{q^2} \sum_{k=0}^{r-1} \frac{1}{\sin^2\frac{\pi cr}{q}}\\ &=& \frac{r}{q^2 \sin^2\frac{\pi cr}{q}}, \ee thus demonstrating \eqref{upper_bound}.
Suppose that $c = \frac{dq}{r} + \Delta$, where $d \in \@ifnextchar[\@BZoptarg{\Bbb Z}$ and $0 <
| \Delta | < \frac{q}{r}$, then (substituting for $c$ in \eqref{cr_non_int}), \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) &=& \frac{1}{q^2} \sum_{k=0}^{r-1} \frac{\sin^2\left(\pi d \left\lfloor\frac{q+k}{r}\right\rfloor + \frac{\pi \Delta r}{q} \frac{q}{r} + \frac{\pi \Delta r}{q}\left( \left\lfloor\frac{q+k}{r}\right\rfloor - \frac{q}{r} \right)\right)}{\sin^2\left(\pi d + \frac{\pi \Delta r}{q}\right)}\\ &=& \frac{1}{q^2} \sum_{k=0}^{r-1} \frac{\sin^2\left(\pi \Delta + \frac{\pi \Delta r}{q}\left( \left\lfloor\frac{q+k}{r}\right\rfloor - \frac{q}{r} \right)\right)}{\sin^2 \frac{\pi \Delta r}{q}}, \ee exploiting the periodicity of $\sin$.
If $\frac{dq}{r} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then $\Delta \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, so that \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) &=& \frac{1}{q^2} \sum_{k=0}^{r-1} \frac{\sin^2\left(\pi \Delta + \frac{\pi \Delta r}{q}\left( \left\lfloor\frac{q+k}{r}\right\rfloor - \frac{q}{r} \right)\right)}{\sin^2 \frac{\pi \Delta r}{q}}\\ &=& \frac{1}{q^2} \sum_{k=0}^{r-1} \frac{\sin^2\left( \frac{\pi \Delta r}{q}\left( \left\lfloor\frac{q+k}{r}\right\rfloor - \frac{q}{r} \right)\right)}{\sin^2 \frac{\pi \Delta r}{q}}. \ee In particular, since \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\left| \left\lfloor\frac{q+k}{r}\right\rfloor - \frac{q}{r} \right| < 1,
\ee for all $k = 0,\dots,r-1$, then if $0 < | \Delta | \leq \frac{q}{2r}$, then \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\left| \frac{\pi \Delta r}{q}\left( \left\lfloor\frac{q+k}{r}\right\rfloor
- \frac{q}{r} \right) \right| < \left| \frac{\pi \Delta r}{q} \right| \leq \frac{\pi}{2}, \ee and so \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) = \frac{1}{q^2} \sum_{k=0}^{r-1} \frac{\sin^2\left( \frac{\pi \Delta r}{q}\left( \left\lfloor\frac{q+k}{r}\right\rfloor - \frac{q}{r} \right)\right)}{\sin^2\frac{\pi \Delta r}{q}} < \frac{1}{q^2} \sum_{k=0}^{r-1} 1 = \frac{r}{q^2}, \ee as $\sin^2 y$ is a monotonic increasing function on $y \in [0,\frac{\pi}{2}]$, thus yielding \eqref{non_int_int_delta}. This means that the probability that $X$ will differ from $\frac{dq}{r}$, for some integer $d$ such that $\frac{dq}{r}$ is also an integer, by at most $\frac{q}{2r}$ and by more than $0$, is negligible if $q$ is much larger than $r$.
If $\frac{dq}{r} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$, then $\Delta \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$. Since \begin{eqnarray*}} \def\ee{\end{eqnarray*} \frac{q}{r} - 1 < \left\lfloor\frac{q+k}{r}\right\rfloor < \frac{q}{r} + 1, \ee for $k = 0,\dots,r-1$, so that \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\left| \left\lfloor\frac{q+k}{r}\right\rfloor - \frac{q}{r} \right| < 1,
\ee and since $|\cos y| \leq 1$ for $y \in \@ifnextchar[\@BRoptarg{\Bbb R}$, it follows that \begin{eqnarray*}} \def\ee{\end{eqnarray*}
|\sin(\pi \Delta)| - \frac{\pi |\Delta| r}{q} <
\left| \sin\left(\pi \Delta + \frac{\pi \Delta r}{q}\left(
\left\lfloor\frac{q+k}{r}\right\rfloor - \frac{q}{r} \right)\right) \right|
< |\sin(\pi \Delta)| + \frac{\pi |\Delta| r}{q}, \ee for $k = 0,\dots,r-1$, thus giving bounds on the square root of the numerator of the summand in \eqref{cr_non_int}. Since $\frac{dq}{r} + \Delta$ is an integer, then \begin{eqnarray*}} \def\ee{\end{eqnarray*}
|\sin(\pi \Delta)| = \left| \sin\frac{\pi dq}{r} \right|, \ee so that \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\left| \sin\frac{\pi dq}{r} \right| - \frac{\pi |\Delta| r}{q} <
\left| \sin\left(\pi \Delta + \frac{\pi \Delta r}{q}\left(
\left\lfloor\frac{q+k}{r}\right\rfloor - \frac{q}{r} \right)\right) \right|
< \left| \sin\frac{\pi dq}{r} \right| + \frac{\pi |\Delta| r}{q}, \ee for $k = 0,\dots,r-1$. Upon taking the square (so we now have the numerator of the summand), it follows that \begin{eqnarray*}} \def\ee{\end{eqnarray*} && \sin^2\left(\pi \Delta + \frac{\pi \Delta r}{q}\left( \left\lfloor\frac{q+k}{r}\right\rfloor - \frac{q}{r} \right)\right)\\
&<& \left(\left|\sin\frac{\pi dq}{r}\right|
+ \frac{\pi |\Delta| r}{q}\right)^2\\ &=& \sin^2\frac{\pi dq}{r}
+ \frac{2\pi |\Delta| r}{q} \left|\sin\frac{\pi dq}{r}\right| + \frac{\pi^2 \Delta^2 r^2}{q^2},
\ee and that if $|\Delta| \leq \frac{q}{\pi r} |\sin\frac{\pi dq}{r}|$, then \begin{eqnarray*}} \def\ee{\end{eqnarray*} && \sin^2\left(\pi \Delta + \frac{\pi \Delta r}{q}\left( \left\lfloor\frac{q+k}{r}\right\rfloor - \frac{q}{r} \right)\right)\\
&>& \left(\left|\sin\frac{\pi dq}{r}\right|
- \frac{\pi |\Delta| r}{q}\right)^2\\ &=& \sin^2\frac{\pi dq}{r}
- \frac{2\pi |\Delta| r}{q} \left|\sin\frac{\pi dq}{r}\right| + \frac{\pi^2 \Delta^2 r^2}{q^2}, \ee and so, substituting into \eqref{cr_non_int}, \begin{eqnarray}} \def\ene{\end{eqnarray} P(X = c) &<& \frac{r}{q^2 \sin^2\frac{\pi \Delta r}{q}}
\left(\left|\sin\frac{\pi dq}{r}\right|
+ \frac{\pi |\Delta| r}{q}\right)^2 \nonumber\\ &<& \frac{1}{\pi^2 \Delta^2 r \left(1 - \frac{\pi^2 \Delta^2 r^2}{6 q^2}\right)^2}
\left(\left|\sin\frac{\pi dq}{r}\right| + \frac{\pi |\Delta| r}{q}\right)^2 \eqlabel{non_int_upper}\\ &=& \frac{1}{\pi^2 \Delta^2 r \left(1 - \frac{\pi^2 \Delta^2 r^2}{6 q^2}\right)^2} \left(\sin^2\frac{\pi dq}{r}
+ \frac{2 \pi |\Delta| r \left|\sin\frac{\pi dq}{r}\right|}{q} + \frac{\pi^2 \Delta^2 r^2}{q^2}\right) \nonumber\\ &=& \frac{1}{\left(1 - \frac{\pi^2 \Delta^2 r^2}{6 q^2}\right)^2} \left(\frac{\sin^2\frac{\pi dq}{r}}{\pi^2 \Delta^2 r}
+ \frac{2 \left|\sin\frac{\pi dq}{r}\right|}{\pi |\Delta| q} + \frac{r}{q^2}\right), \nonumber
\ene thus yielding \eqref{delta_upper}, if $|\Delta| \leq \frac{q}{2r}$, since \begin{eqnarray*}} \def\ee{\end{eqnarray*} 0 < y \left(1 - \frac{1}{6} y^2\right) < \sin y < y,
\ee for $y \in (0,\frac{\pi}{2}]$. Similarly, if $|\Delta| \leq \frac{q}{\pi r} |\sin\frac{\pi dq}{r}|$, then \begin{eqnarray}} \def\ene{\end{eqnarray} P(X = c) &>& \frac{r}{q^2 \sin^2\frac{\pi \Delta r}{q}}
\left(\left|\sin\frac{\pi dq}{r}\right|
- \frac{\pi |\Delta| r}{q}\right)^2 \nonumber\\ &>& \frac{1}{\pi^2 \Delta^2 r}
\left(\left|\sin\frac{\pi dq}{r}\right| - \frac{\pi |\Delta| r}{q}\right)^2 \eqlabel{non_int_lower}\\ &=& \frac{1}{\pi^2 \Delta^2 r} \left(\sin^2\frac{\pi dq}{r}
- \frac{2 \pi |\Delta| r \left|\sin\frac{\pi dq}{r}\right|}{q} + \frac{\pi^2 \Delta^2 r^2}{q^2}\right) \nonumber\\ &=& \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 \Delta^2 r}
- \frac{2 \left|\sin\frac{\pi dq}{r}\right|}{\pi |\Delta| q} + \frac{r}{q^2}, \nonumber \ene thus yielding \eqref{delta_lower}.
If $\frac{dq}{r} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then by \eqref{non_int_int} and \eqref{non_int_int_delta}, \begin{eqnarray*}} \def\ee{\end{eqnarray*} && \frac{1}{r} - \frac{2}{q} + \frac{r}{q^2}\\ &<& P\left(X = \frac{dq}{r}\right)\\ &\leq& P\left(\rho_q\left(X,\frac{dq}{r}\right) \leq \frac{q}{2r} \right)\\ &=& P\left( X = \frac{dq}{r} \right) + P\left(0 < \rho_q\left(X,\frac{dq}{r}\right) \leq \frac{q}{2r} \right)\\ &<& \frac{1}{r} + \frac{2}{q} + \frac{r}{q^2} + 2 \left\lfloor \frac{q}{2r} \right\rfloor \frac{r}{q^2}, \ee where $\rho_q$ is the modular metric \eqref{mod}, since $P(X = c) < \frac{r}{q^2}$ for all $c$ such that $0 < \rho_q(c,\frac{dq}{r}) \leq \frac{q}{2r}$ and \begin{eqnarray*}} \def\ee{\end{eqnarray*} \#\left\{ c : 0 < \rho_q(c,\frac{dq}{r}) \leq \frac{q}{2r} \right\} = 2 \left\lfloor \frac{q}{2r} \right\rfloor. \ee It follows that \begin{eqnarray*}} \def\ee{\end{eqnarray*} && \frac{1}{r} - \frac{2}{q} + \frac{r}{q^2}\\ &<& P\left(\rho_q\left(X,\frac{dq}{r}\right) \leq \frac{q}{2r} \right)\\ &<& \frac{1}{r} + \frac{2}{q} + \frac{r}{q^2} + 2 \frac{q}{2r} \frac{r}{q^2}\\ &=& \frac{1}{r} + \frac{2}{q} + \frac{r}{q^2} + \frac{1}{q}\\ &=& \frac{1}{r} + \frac{3}{q} + \frac{r}{q^2}, \ee this yielding \eqref{int_full}.
The value of the parameter $q$ will now be restricted so that $q > 2r$.
We are interested in the probability that the measured value $c$ of the first register will fall inside a specified distance from an integral multiple of $\frac{q}{r}$, so we are also interested in the probability that it will fall outside the specified distance. This is the motivation behind the following calculations.
Since \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(X = c) \leq \frac{r}{q^2 \sin^2\frac{\pi cr}{q}}, \ee by \eqref{upper_bound}, if $c \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, $0 \leq c \leq q$, and $\frac{cr}{q} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$, and since $\sin^2 y$ is a monotonic increasing function on $y \in [0,\frac{\pi}{2}]$, then the following hold by straightforward substitution. \begin{itemize}} \def\ei{\end{itemize} \item If $d \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, $d \in \{ 0,1,\dots,r-1 \}$, $1 < C \leq \frac{q}{2r}$, and $\frac{dq}{r} + C \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then \begin{eqnarray}} \def\ene{\end{eqnarray} P\left(X = \frac{dq}{r} + C\right) &\leq& \frac{r}{q^2 \sin^2\left(\pi d + \frac{\pi Cr}{q}\right)} \nonumber\\ &=& \frac{r}{q^2 \sin^2\frac{\pi Cr}{q}} \eqlabel{integral1}\\ &<& \int_{C-1}^C \frac{r}{q^2 \sin^2\frac{\pi \xi r}{q}} \, d\xi, \nonumber \ene the equality following from the fact that $d \in \@ifnextchar[\@BZoptarg{\Bbb Z}$; \item If $d \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, $d \in \{ 1,\dots,r \}$, $1 < C \leq \frac{q}{2r}$, and $\frac{dq}{r} - C \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then \begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left(X = \frac{dq}{r} - C\right) &\leq& \frac{r}{q^2 \sin^2\left(\pi d - \frac{\pi Cr}{q}\right)}\\ &=& \frac{r}{q^2 \sin^2\frac{\pi Cr}{q}}\\ &<& \int_{C-1}^C \frac{r}{q^2 \sin^2\frac{\pi \xi r}{q}} \, d\xi, \ee the equality following from the fact that $d \in \@ifnextchar[\@BZoptarg{\Bbb Z}$. \ei
If $d \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, $d \in \{ 0,1,\dots,r-1 \}$, $1 < A \leq A' \leq \frac{q}{2r}$, and $\frac{dq}{r} + A, \frac{dq}{r} + A' \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then by \eqref{integral1}, \begin{eqnarray}} \def\ene{\end{eqnarray} && P\left(\frac{dq}{r} + A \leq X \leq \frac{dq}{r} + A'\right) \nonumber\\ &\leq& \sum_{c = \frac{dq}{r} + A}^{\frac{dq}{r} + A'} \frac{r}{q^2 \sin^2\frac{\pi cr}{q}} \nonumber\\ &<& \int_{A - 1}^{A'} \frac{r}{q^2} \csc^2\frac{\pi \xi r}{q} \, d\xi \eqlabel{integral2}\\ &=& - \frac{1}{\pi q} \left[ \cot\frac{\pi \xi r}{q} \right]_{A - 1}^{A'} \nonumber\\ &=& \frac{1}{\pi q} \left( \cot\frac{\pi (A - 1) r}{q} - \cot\frac{\pi A' r}{q} \right). \nonumber \ene This gives an upper bound on the probability that $X$ will fall between $\frac{dq}{r} + A$ and $\frac{dq}{r} + A'$.
Similarly, if $d \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, $d \in \{ 1,\dots,r \}$, $1 < B \leq B' \leq \frac{q}{2r}$, and $\frac{dq}{r} - B, \frac{dq}{r} - B' \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then \begin{equation}} \def\ese{\end{equation} \eqlabel{integral3} P\left(\frac{dq}{r} - B' \leq X \leq \frac{dq}{r} - B\right) < \frac{1}{\pi q} \left( \cot\frac{\pi (B - 1) r}{q} - \cot\frac{\pi B' r}{q} \right). \ese
If $d \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, $d \in \{ 0,1,\dots,r-1 \}$, $1 < A \leq \frac{q}{2r}$, $1 < B \leq \frac{q}{2r}$, and $\frac{dq}{r} + A, \frac{(d+1)q}{r} - B \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, the let \begin{eqnarray*}} \def\ee{\end{eqnarray*} C = \left\lfloor \frac{(2d+1)q}{2r} \right\rfloor - \frac{dq}{r}, \ee so that $C \leq \frac{q}{2r} < C + 1$, and then, by \eqref{integral2} and \eqref{integral3}, \begin{eqnarray*}} \def\ee{\end{eqnarray*} && P\left(\frac{dq}{r} + A \leq X \leq \frac{(d+1)q}{r} - B\right)\\ &=& P\left(\frac{dq}{r} + A \leq X \leq \frac{dq}{r} + C\right) + P\left(\frac{(d+1)q}{r} - \left(\frac{q}{r} - (C + 1)\right) \leq X \leq \frac{(d+1)q}{r} - B\right)\\ &<& \frac{1}{\pi q} \left( \cot\frac{\pi (A - 1) r}{q} - \cot\frac{\pi C r}{q} \right) + \frac{1}{\pi q} \left( \cot\frac{\pi (B - 1) r}{q} - \cot\left(\pi - \frac{\pi (C + 1) r}{q}\right) \right)\\ &=& \frac{1}{\pi q} \left( \cot\frac{\pi (A - 1) r}{q} + \cot\frac{\pi (B - 1) r}{q} - \cot\frac{\pi C r}{q} + \cot\frac{\pi (C + 1) r}{q} \right)\\ &<& \frac{1}{\pi q} \left( \cot\frac{\pi (A - 1) r}{q} + \cot\frac{\pi (B - 1) r}{q} \right), \ee since $\frac{\pi C r}{q} \leq \frac{\pi}{2} < \frac{\pi (C + 1) r}{q}$, so that $\cot\frac{\pi C r}{q}$ is positive, and $\cot\frac{\pi (C + 1) r}{q}$ is negative. Since $\cot y < \frac{1}{y}$ for $0 < y \leq \frac{\pi}{2}$, then it follows that \begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left(\frac{dq}{r} + A \leq X \leq \frac{(d+1)q}{r} - B\right) < \frac{1}{\pi^2 r} \left( \frac{1}{A - 1} + \frac{1}{B - 1} \right). \ee Suppose $0 < u \leq \frac{q}{2r} - 1$, then it follows that \begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left(\frac{dq}{r} + u + 1 \leq X \leq \frac{(d+1)q}{r} - u - 1 \right) < \frac{1}{\pi^2 r} \left( \frac{1}{u} + \frac{1}{u} \right) = \frac{2}{\pi^2 r u}, \ee thus demonstrating \eqref{far}.
Adopting the same definitions of $d_c$ and $\Delta_c$ that were used in \secref{firstreg}, then it follows that
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P(|\Delta_X| \geq u + 1) = \sum_{d=0}^{r-1} P\left(\frac{dq}{r} + u + 1 \leq X \leq \frac{(d+1)q}{r} - u - 1 \right) < \sum_{d=0}^{r-1} \frac{2}{\pi^2 r u} = \frac{2}{\pi^2 u}. \ee Therefore it follows that the probability that the measured value of the first register falls outside a specified distance from a multiple of $\frac{q}{r}$ is bounded above by a quantity which inversely proportional to one less than the distance, with no dependence of the upper bound on the size of $q$, thus making the possibility unlikely if the specified distance is large.
For each value of $d$, we are interested in the values of \begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left(\rho_q\left(X,\frac{dq}{r}\right) < u + 1\right). \ee
Upper and lower bounds can be easily determined for \begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left(\rho_q\left(X,\frac{dq}{r}\right) < u + 1\right), \ee if $\frac{dq}{r} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, specifically, by \eqref{non_int_int} and \eqref{non_int_int_delta}, \begin{eqnarray*}} \def\ee{\end{eqnarray*} && \frac{1}{r} - \frac{2}{q} + \frac{r}{q^2}\\ &<& P\left(X = \frac{dq}{r}\right)\\ &\leq& P\left(\rho_q\left(X,\frac{dq}{r}\right) < u + 1\right)\\ &=& P\left( X = \frac{dq}{r} \right) + P\left(0 < \rho_q\left( X , \frac{dq}{r} \right) < u + 1 \right)\\ &<& \frac{1}{r} + \frac{2}{q} + \frac{r}{q^2} + 2 \lfloor u + 1 \rfloor \frac{r}{q^2}\\ &\leq& \frac{1}{r} + \frac{2}{q} + \frac{r}{q^2} + \frac{2(u+1)r}{q^2}\\ &=& \frac{1}{r} + \frac{2}{q} + \frac{(2u+3)r}{q^2}, \ee thus leading to \eqref{int}.
If $\frac{dq}{r} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$, then
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left(\left| X - \frac{dq}{r}\right| < u + 1\right)
= \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop \left| c - \frac{dq}{r}\right| < u + 1} P\left( X = c \right), \ee so that, by \eqref{non_int_upper}, \begin{eqnarray*}} \def\ee{\end{eqnarray*}
&& P\left(\left| X - \frac{dq}{r}\right| < u + 1\right)\\
&<& \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop \left| c - \frac{dq}{r}\right| < u + 1} \frac{1}{\left(1 - \frac{\pi^2 (u+1)^2 r^2}{6 q^2}\right)^2} \left( \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 \left(c - \frac{dq}{r}\right)^2 r}
+ \frac{2 \left|\sin\frac{\pi dq}{r}\right|}{\pi
\left|c - \frac{dq}{r}\right| q} + \frac{r}{q^2} \right),
\ee since $|\Delta_c| < u + 1$ for all $c$ in the sum, and by \eqref{non_int_lower}, \begin{eqnarray*}} \def\ee{\end{eqnarray*}
&& P\left(\left| X - \frac{dq}{r}\right| < u + 1\right)\\
&>& \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z}, \ \left| c - \frac{dq}{r} \right| < u + 1
\atop \left| c - \frac{dq}{r} \right| < \frac{q}{\pi r}
\left| \sin\frac{\pi dq}{r} \right|} \left( \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 (c - \frac{dq}{r})^2 r}
- \frac{2 \left|\sin\frac{\pi dq}{r}\right|}{\pi \left| c - \frac{dq}{r}
\right| q} + \frac{r}{q^2} \right). \ee
It follows that \begin{eqnarray*}} \def\ee{\end{eqnarray*}
&& \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z}, \ \left| c - \frac{dq}{r} \right| < u + 1
\atop \left| c - \frac{dq}{r} \right| < \frac{q}{\pi r}
\left| \sin\frac{\pi dq}{r} \right|} \left( \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 (c - \frac{dq}{r})^2 r}
- \frac{2 \left|\sin\frac{\pi dq}{r}\right|}{\pi \left| c - \frac{dq}{r}
\right| q} + \frac{r}{q^2} \right)\\
&<& P\left(\left| X - \frac{dq}{r}\right| < u + 1\right)\\ &<& \frac{1}{\left(1 - \frac{\pi^2 (u+1)^2 r^2}{6 q^2}\right)^2}
\sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop \left| c - \frac{dq}{r} \right| < u + 1} \left( \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 (c - \frac{dq}{r})^2 r}
+ \frac{2 \left|\sin\frac{\pi dq}{r}\right|}{\pi \left| c - \frac{dq}{r}
\right| q} + \frac{r}{q^2} \right). \ee Specifically, if \begin{equation}} \def\ese{\end{equation} \eqlabel{smaller_u}
u \leq \frac{q}{\pi r} \left| \sin\frac{\pi dq}{r} \right| - 1, \ese then \begin{eqnarray}} \def\ene{\end{eqnarray}
&& \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop \left| c - \frac{dq}{r} \right| < u + 1} \left( \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 (c - \frac{dq}{r})^2 r}
- \frac{2 \left|\sin\frac{\pi dq}{r}\right|}{\pi \left| c - \frac{dq}{r}
\right| q} + \frac{r}{q^2} \right)\nonumber\\
&<& P\left(\left| X - \frac{dq}{r}\right| < u + 1\right)\nonumber\\ &<& \frac{1}{\left(1 - \frac{\pi^2 (u+1)^2 r^2}{6 q^2}\right)^2}
\sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop \left| c - \frac{dq}{r} \right| < u + 1} \left( \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 (c - \frac{dq}{r})^2 r}
+ \frac{2 \left|\sin\frac{\pi dq}{r}\right|}{\pi \left| c - \frac{dq}{r}
\right| q} + \frac{r}{q^2} \right). \eqlabel{near} \ene
The terms inside the sums on the upper and lower bounds in \eqref{near} will now be investigated, one at a time.
By the Mittag-Leffler expansion into partial fractions for $\csc^2(\pi z)$ from complex analysis (which can be found in many books on complex analysis, such as \cite{remm,moore}, or by differentiating the Mittag-Leffler expansion for $\cot(\pi z)$, which can also be found in books on complex analysis, such as \cite{beren,fuchs}), \begin{eqnarray*}} \def\ee{\end{eqnarray*} \sum_{n=-\infty}^\infty \frac{1}{\pi^2 (z-n)^2} = \frac{1}{\sin^2(\pi z)}, \ee it follows that \begin{eqnarray*}} \def\ee{\end{eqnarray*} \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z}} \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 (c - \frac{dq}{r})^2 r} = \frac{1}{r}, \ee so that \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop \left| c - \frac{dq}{r} \right| < u + 1} \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 (c - \frac{dq}{r})^2 r} = \frac{1}{r}
- \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop \left| c - \frac{dq}{r} \right| \geq u + 1} \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 (c - \frac{dq}{r})^2 r}. \ee For $z \in \@ifnextchar[\@BRoptarg{\Bbb R} \backslash \@ifnextchar[\@BZoptarg{\Bbb Z}$, \begin{eqnarray*}} \def\ee{\end{eqnarray*} \sum_{n \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop n-z \geq u+1} \frac{1}{(n-z)^2} &=& \sum_{n \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop n \geq z+u+1} \frac{1}{(n-z)^2}\\ &=& \sum_{n = \lceil z+u+1 \rceil}^\infty \frac{1}{(n-z)^2}\\ &<& \int_{\lceil z+u \rceil}^\infty \frac{1}{(\xi-z)^2} \, d\xi\\ &=& - \left[\frac{1}{\xi-z}\right]_{\lceil z+u \rceil}^\infty\\ &=& \frac{1}{\lceil z+u \rceil - z}\\ &\leq& \frac{1}{u}, \ee where for real $y$, $\lceil y \rceil$ is the least integer greater than or equal to $y$. Similarly, for $z \in \@ifnextchar[\@BRoptarg{\Bbb R} \backslash \@ifnextchar[\@BZoptarg{\Bbb Z}$, \begin{eqnarray*}} \def\ee{\end{eqnarray*} \sum_{n \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop n-z \geq u+1} \frac{1}{(n-z)^2} &>& \frac{1}{\lceil z+u+1 \rceil - z}\\ &>& \frac{1}{u+2}. \ee It follows that \begin{eqnarray*}} \def\ee{\end{eqnarray*} \frac{1}{u+2} < \sum_{n \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop n-z \geq u+1} \frac{1}{(n-z)^2} < \frac{1}{u}. \ee Similarly, \begin{eqnarray*}} \def\ee{\end{eqnarray*} \frac{1}{u+2} < \sum_{n \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop n-z \leq -(u+1)} \frac{1}{(n-z)^2} < \frac{1}{u}, \ee so that \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\frac{2}{u+2} < \sum_{n \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop |n-z| \geq u+1} \frac{1}{(n-z)^2} < \frac{2}{u}, \ee and so \begin{eqnarray*}} \def\ee{\end{eqnarray*}
\sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop \left| c - \frac{dq}{r} \right| < u + 1} \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 (c - \frac{dq}{r})^2 r} = \frac{1}{r}
- \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop \left| c - \frac{dq}{r} \right| \geq u + 1} \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 (c - \frac{dq}{r})^2 r} \ee satisfies \begin{eqnarray*}} \def\ee{\end{eqnarray*} \frac{1}{r} - \frac{2 \sin^2\frac{\pi dq}{r}}{\pi^2 r u}
< \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop \left| c - \frac{dq}{r} \right| < u + 1} \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 (c - \frac{dq}{r})^2 r} < \frac{1}{r} - \frac{2 \sin^2\frac{\pi dq}{r}}{\pi^2 r (u+2)}, \ee since $\frac{dq}{r} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$, and so \begin{equation}} \def\ese{\end{equation} \eqlabel{dist_squared} \frac{1}{r} - \frac{2}{\pi^2 r u}
< \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop \left| c - \frac{dq}{r} \right| < u + 1} \frac{\sin^2\frac{\pi dq}{r}}{\pi^2 (c - \frac{dq}{r})^2 r} < \frac{1}{r}. \ese This accounts for the first term in the sums in the upper and lower bounds in \eqref{near}.
Since $\lceil 2u+1 \rceil \leq \#\{ c \in \@ifnextchar[\@BZoptarg{\Bbb Z} : |c-\frac{dq}{r}| < u+1 \} \leq \lceil 2u+2 \rceil$, then \begin{equation}} \def\ese{\end{equation} \eqlabel{constant} \frac{r (2u+1)}{q^2} \leq \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop
\left| c - \frac{dq}{r} \right| < u + 1} \frac{r}{q^2} < \frac{r (2u+3)}{q^2}. \ese This accounts for the third term in the sums in the upper and lower bounds in \eqref{near}.
All that remains is the second term in the sums. Since \begin{eqnarray*}} \def\ee{\end{eqnarray*} && \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop 0 < c - \frac{dq}{r} < u+1} \frac{1}{c - \frac{dq}{r}}\\ &=& \frac{1}{\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}} + \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop 1 < c - \frac{dq}{r} < u+1} \frac{1}{c - \frac{dq}{r}}\\ &=& \frac{1}{\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}} + \sum_{c = \left\lceil \frac{dq}{r} \right\rceil + 1}^{\left\lfloor u + \frac{dq}{r} \right\rfloor + 1} \frac{1}{c - \frac{dq}{r}}\\ &<& \frac{1}{\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}} + \int_{\left\lceil \frac{dq}{r} \right\rceil}^{\left\lfloor u + \frac{dq}{r} \right\rfloor + 1} \frac{1}{\xi - \frac{dq}{r}} \, d\xi\\ &=& \frac{1}{\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}} + \left[ \ln(\xi - \frac{dq}{r}) \right]_{\left\lceil \frac{dq}{r} \right\rceil}^{\left\lfloor u + \frac{dq}{r} \right\rfloor + 1}\\ &=& \frac{1}{\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}} + \ln\left(\frac{\left\lfloor u + \frac{dq}{r} \right\rfloor + 1 - \frac{dq}{r}}{\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}}\right)\\ &\leq& \frac{1}{\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}} + \ln\left(\frac{u + 1}{\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}}\right), \ee and similarly, \begin{eqnarray*}} \def\ee{\end{eqnarray*} && \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop 0 > c - \frac{dq}{r} > -(u+1)} \frac{1}{\frac{dq}{r} - c}\\ &<& \frac{1}{\frac{dq}{r} - \left\lfloor \frac{dq}{r} \right\rfloor} + \ln\left(\frac{\frac{dq}{r} - \left\lceil \frac{dq}{r} - u \right\rceil + 1} {\frac{dq}{r} - \left\lfloor \frac{dq}{r} \right\rfloor}\right)\\ &<& \frac{1}{\frac{dq}{r} - \left\lfloor \frac{dq}{r} \right\rfloor} + \ln\left(\frac{u + 1}{\frac{dq}{r} - \left\lfloor \frac{dq}{r} \right\rfloor}\right), \ee then \begin{eqnarray*}} \def\ee{\end{eqnarray*}
&& \sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop |c - \frac{dq}{r}| < u+1}
\frac{1}{|c - \frac{dq}{r}|}\\ &<& \frac{1}{\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}} + \frac{1}{\frac{dq}{r} - \left\lfloor \frac{dq}{r} \right\rfloor} + \ln\left(\frac{\left\lfloor u + \frac{dq}{r} \right\rfloor + 1 - \frac{dq}{r}}{\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}}\right) + \ln\left(\frac{\frac{dq}{r} - \left\lceil \frac{dq}{r} - u \right\rceil + 1}{\frac{dq}{r} - \left\lfloor \frac{dq}{r} \right\rfloor}\right)\\ &\leq& \frac{1}{\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}} + \frac{1}{\frac{dq}{r} - \left\lfloor \frac{dq}{r} \right\rfloor} + \ln\left(\frac{u + 1}{\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}}\right) + \ln\left(\frac{u + 1}{\frac{dq}{r} - \left\lfloor \frac{dq}{r}\right\rfloor}\right). \ee Since $\frac{1}{r} \leq \frac{dq}{r} - \lfloor \frac{dq}{r} \rfloor \leq \frac{r-1}{r}$ (and equivalently, $\frac{1}{r} \leq \lceil \frac{dq}{r} \rceil - \frac{dq}{r} \leq \frac{r-1}{r}$, noting that for $y \in \@ifnextchar[\@BRoptarg{\Bbb R} \backslash \@ifnextchar[\@BZoptarg{\Bbb Z}$, $\lceil y \rceil - \lfloor y \rfloor = 1$), then \begin{eqnarray*}} \def\ee{\end{eqnarray*} \frac{1}{\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}} + \frac{1}{\frac{dq}{r} - \left\lfloor \frac{dq}{r} \right\rfloor} \leq r + \frac{r}{r-1} = \frac{r^2}{r-1}. \ee Similarly, \begin{eqnarray*}} \def\ee{\end{eqnarray*} \left(\frac{dq}{r} - \left\lfloor \frac{dq}{r} \right\rfloor\right) \left(\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}\right) \geq \frac{1}{r} \frac{r-1}{r} = \frac{r-1}{r^2}, \ee so that \begin{eqnarray*}} \def\ee{\end{eqnarray*} \ln\left[\left(\frac{dq}{r} - \left\lfloor \frac{dq}{r} \right\rfloor\right) \left(\left\lceil \frac{dq}{r} \right\rceil - \frac{dq}{r}\right)\right] \geq \ln\frac{r-1}{r^2}, \ee and so \begin{equation}} \def\ese{\end{equation} \eqlabel{dist}
\sum_{c \in \@ifnextchar[\@BZoptarg{\Bbb Z} \atop |c - \frac{dq}{r}| < u+1}
\frac{1}{|c - \frac{dq}{r}|} < \frac{r^2}{r-1} + \ln\frac{r^2(u+1)^2}{r-1}. \ese
Gathering all the information about individual terms, it follows that if
\begin{eqnarray*}} \def\ee{\end{eqnarray*} u \leq \frac{q}{\pi r} \left| \sin\frac{\pi dq}{r} \right| - 1, \ee then \begin{eqnarray*}} \def\ee{\end{eqnarray*} && \frac{1}{r} - \frac{2}{\pi^2 r u} - \frac{2}{\pi q} \left(\frac{r^2}{r-1} + \ln\frac{r^2(u+1)^2}{r-1}\right) + \frac{r (2u+1)}{q^2}\\
&<& P\left(\left| X - \frac{dq}{r} \right| < u+1\right)\\ &<& \frac{1}{\left(1 - \frac{\pi^2 (u+1)^2 r^2}{6 q^2}\right)^2} \left(\frac{1}{r} + \frac{2}{\pi q} \left(\frac{r^2}{r-1} + \ln\frac{r^2(u+1)^2}{r-1}\right) + \frac{r (2u+3)}{q^2}\right), \ee as a consequence of \eqref{near}, \eqref{dist_squared}, \eqref{constant} and \eqref{dist}, thus yielding \eqref{non_int}.
\section{Proof of Inequalities in the Analysis of the Refinement} \applabel{algo_analysis}
If $\frac{dq}{r} \in \@ifnextchar[\@BZoptarg{\Bbb Z}$, then by \eqref{int} and the fact that $q \geq 2 w n^3$, \begin{eqnarray}} \def\ene{\end{eqnarray} \frac{1}{r} - \frac{1}{w n^3} &<& \frac{1}{r} - \frac{1}{w n^3} + \frac{r}{4 w^2 n^6} \nonumber\\ &\leq& \frac{1}{r} - \frac{2}{q} + \frac{r}{q^2} \nonumber\\ &<& P\left(\rho_q\left(X,\frac{dq}{r}\right) < w n\right) \eqlabel{_ref1}\\ &<& \frac{1}{r} + \frac{2}{q} + \frac{(2wn+1)r}{q^2} \nonumber\\ &\leq& \frac{1}{r} + \frac{1}{w n^3} + \frac{(2wn+1)r}{4 w^2 n^6} \nonumber\\ &<& \frac{1}{r} + \frac{1}{w n^3} + \frac{1}{2 w n^4}, \nonumber \ene the final statement following from the fact that $r < n$, so that $r \leq n-1$, and so \begin{eqnarray*}} \def\ee{\end{eqnarray*} (2wn+1)r \leq (2wn+1)(n-1) = 2wn^2 - 2wn + n - 1 = 2wn^2 - (2w - 1)n - 1. \ee This demonstrates \eqref{ref1}.
If $\frac{dq}{r} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$, then, by \eqref{non_int} and the fact that $q \geq 2 w n^3$, \begin{eqnarray}} \def\ene{\end{eqnarray} && \frac{1}{r} - \frac{2}{\pi^2 (wn - 1)} - \frac{1}{\pi w n^3} \left(n + 1 + \ln\frac{w^2 n^4}{n-1}\right) \nonumber\\ &<& \frac{1}{r} - \frac{2}{\pi^2 r (wn - 1)} - \frac{1}{\pi w n^3} \left(\frac{r^2}{r-1} + \ln\frac{r^2 w^2 n^2}{r-1}\right) \nonumber\\ &<& \frac{1}{r} - \frac{2}{\pi^2 r (wn - 1)} - \frac{2}{\pi q} \left(\frac{r^2}{r-1} + \ln\frac{r^2 w^2 n^2}{r-1}\right) + \frac{r (2wn - 1)}{q^2} \nonumber\\
&<& P\left(\left| X - \frac{dq}{r} \right| < w n\right) \eqlabel{_ref2}\\ &<& \frac{1}{\left(1 - \frac{\pi^2 w^2 n^2 r^2}{6 q^2}\right)^2} \left(\frac{1}{r} + \frac{2}{\pi q} \left(\frac{r^2}{r-1} + \ln\frac{r^2 w^2 n^2}{r-1}\right) + \frac{r (2wn + 1)}{q^2}\right) \nonumber\\ &\leq& \frac{1}{\left(1 - \frac{\pi^2 w^2 n^2 r^2}{24 w^2 n^6}\right)^2} \left(\frac{1}{r} + \frac{1}{\pi w n^3} \left(\frac{r^2}{r-1} + \ln\frac{r^2 w^2 n^2}{r-1}\right) + \frac{r (2wn + 1)}{4 w^2 n^6}\right) \nonumber\\ &=& \frac{1}{\left(1 - \frac{\pi^2 r^2}{24 n^4}\right)^2} \left(\frac{1}{r} + \frac{1}{\pi w n^3} \left(\frac{r^2}{r-1} + \ln\frac{r^2 w^2 n^2}{r-1}\right) + \frac{r (2wn + 1)}{4 w^2 n^6}\right) \nonumber\\ &<& \frac{1}{\left(1 - \frac{\pi^2}{24 n^2}\right)^2} \left(\frac{1}{r} + \frac{1}{\pi w n^3} \left(n + 1 + \ln\frac{w^2 n^4}{n-1}\right) + \frac{1}{2 w n^4}\right). \nonumber \ene The first inequality above follows from \begin{itemize}} \def\ei{\end{itemize} \item the fact that \begin{eqnarray*}} \def\ee{\end{eqnarray*} (n+1)(r-1) - r^2 = (n+1)(r-1) - (r+1)(r-1) - 1 = (n-r)(r-1) - 1 \geq 0, \ee since $n-r \geq 1$ and $r > 1$ (which is required by the fact that $\frac{dq}{r} \notin \@ifnextchar[\@BZoptarg{\Bbb Z}$), \item the fact that \begin{eqnarray*}} \def\ee{\end{eqnarray*} n^2(r-1) - r^2(n-1) = (n-r)(nr - n - r) = (n-r)[(n-1)(r-1)-1] \geq 1 (2-1) = 1, \ee since $r \geq 2$ and $n \geq r+1 \geq 3$, and so \begin{eqnarray*}} \def\ee{\end{eqnarray*} \frac{n^2}{n-1} > \frac{r^2}{r-1}. \ee \ei This demonstrates \eqref{ref2}.
\section{Proof of \thmref{zeta}} \applabel{zeta}
\begin{lemma}} \def\el{\end{lemma} For each $j \in J$, define the random variable $B_{ij}$ with sample space $\{ 0,1,2,\dots,a_j \}$ by setting \begin{eqnarray*}} \def\ee{\end{eqnarray*} R_i = \prod_{j \in J} p_j^{B_{ij}}. \ee Specifically, $B_{ij}$ is the power to which $p_j$ is raised in the prime factorization of $R_i$. Then: \begin{itemize}} \def\ei{\end{itemize} \item The probability distribution for $B_{ij}$ is given by \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(B_{ij} = b) = \left\{ \begin{array}} \def\ea{\end{array}{ll} \frac{p_j^b - p_j^{b-1}}{p_j^{a_j}}, & b > 0,\\ \\ \frac{1}{p_j^{a_j}}, & b = 0. \ea \right. \ee \item $B_{ij}$ for $i = 1,2,3,\dots$, and $j \in J$, are independent random variables. \ei \el \begin{proof}} \def\ep{\end{proof} Since $B_{ij} = b$ iff $b$ is the power to which $p_j$ is raised in the prime factorization of $R_i$, then
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P(B_{ij} = b) = P(p_j^b | R_i \wedge p_j^{b+1} \nmid R_i), \ee and so since $R_i = s/\gcd(Z_i,s)$, it follows that
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P(B_{ij} = b) &=& P(p_j^{a_j-b} | \gcd(Z_i,s) \wedge p_j^{a_j-b+1} \nmid \gcd(Z_i,s))\\
&=& \left\{ \begin{array}} \def\ea{\end{array}{l} P(p_j^{a_j-b} | Z_i \wedge p_j^{a_j-b+1} \nmid Z_i), \ b > 0,\\ \\
P(p_j^{a_j} | Z_i), \ b = 0. \ea \right. \ee The number of elements of $\{ 0,1,2,\dots,s-1 \}$ which are divisible by $p_j^{a_j-b}$ is $s/p_j^{a_j-b}$ for all $b = 0,1,2,\dots,a_j$, so that, since $Z_i$ is uniformly distributed,
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P(p_j^{a_j-b} | Z_i) = \frac{1}{p_j^{a_j-b}} = \frac{p_j^b}{p_j^{a_j}}, \ee and so \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(B_{ij} = b) = \left\{ \begin{array}} \def\ea{\end{array}{l} \frac{p_j^b-p_j^{b-1}}{p_j^{a_j}}, \ b > 0,\\ \\ \frac{1}{p_j^{a_j}}, \ b = 0. \ea \right. \ee This proves the required formula for $P(B_{ij} = b)$.
For the independence of $B_{ij}$, one can invoke the Chinese Remainder Theorem (as Knill did in \cite{knill}, for example). Alternatively, one can also take the following approach. For $j_1,\dots,j_l \in J$, and for $b_m = 0,1,\dots,a_{j_m}$ for $m = 1,\dots,l$, then \begin{eqnarray*}} \def\ee{\end{eqnarray*} && P(B_{ij_1} = b_1, B_{ij_2} = b_2, \dots, B_{ij_l} = b_l)\\
&=& P(p_{j_m}^{b_m} | R_i \wedge p_{j_m}^{b_m+1} \nmid R_i, \mbox{ for all } m = 1,\dots,l )\\
&=& P(p_{j_m}^{a_{j_m}-b_m} | \gcd(Z_i,s) \wedge p_{j_m}^{a_{j_m}-b_m+1} \nmid \gcd(Z_i,s), \mbox{ for all } m = 1,\dots,l). \ee For any divisor $t$ of $s$, then the number of elements of $\{ 0,1,\dots,s-1 \}$ which are divisible by $t$ is $\frac{s}{t}$, so that, for all $i$,
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P(t | Z_i) = \frac{1}{t}. \ee It follows that for $b_m = 0,1,\dots,a_{j_m}$, $m = 1,\dots,l$, then \begin{equation}} \def\ese{\end{equation} \eqlabel{indep}
P\left(\prod_{l=1}^m p_{j_m}^{a_{j_m}-b_m} | Z_i\right) = \prod_{m=1}^l \frac{1}{p_{j_m}^{a_{j_m}-b_m}}
= \prod_{m=1}^l P(p_{j_m}^{a_{j_m}-b_m} | Z_i). \ese For $m = 1,\dots,l, $ define $f_m : \{ -1,0,1,\dots,a_{j_m} \} \to \@ifnextchar[\@BRoptarg{\Bbb R}$ by \begin{eqnarray*}} \def\ee{\end{eqnarray*} f_m(b) = \left\{ \begin{array}} \def\ea{\end{array}{ll} \frac{1}{p_j^{a_{j_m}-b}}, & b \geq 0,\\ 0, & b = -1, \ea \right. \ee then
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left(p_{j_m}^{a_{j_m}-b} | \gcd(Z_i,s)\right) = f_m(b), \ee for $b = -1,0,1,\dots,a_{j_m}$, and so, with the aid of \eqref{indep},
\begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left(\prod_{m=1}^l p_{j_m}^{a_{j_m}-b_m} | \gcd(Z_i,s)\right) = \prod_{m=1}^l f_m(b_m), \ee for $b_m = -1,0,1,\dots,a_{j_m}$, $m = 1,\dots,l$. It follows that \begin{eqnarray*}} \def\ee{\end{eqnarray*} && P(B_{ij_1} = b_1, B_{ij_2} = b_2, \dots, B_{ij_l} = b_l)\\
&=& P(p_{j_m}^{a_{j_m}-b_m} | \gcd(Z_i,s) \wedge p_{j_m}^{a_{j_m}-b_m+1} \nmid \gcd(Z_i,s), \mbox{ for all } m = 1,\dots,l)\\ &=& \prod_{m=1}^l (f_m(b_m) - f_m(b_m-1))\\ &=& \prod_{m=1}^l P(B_{ij_m} = b_m), \ee for $b_m = 0,1,\dots,a_{j_m}$, $m = 1,\dots,l$. For example, one can use a proof by induction on $q$ to demonstrate that for $b_m = 0,1,\dots,a_{j_m}$, $m = 1,\dots,q$, and for $b_m = -1,0,1,\dots,a_{j_m}$, $m = q+1,\dots,l$, \begin{eqnarray*}} \def\ee{\end{eqnarray*} && P\left(F_m(b_m,Z_i) \mbox{ for } m = 1,\dots,q, \mbox{ and } G_m(b_m,Z_i) \mbox{ for } m = q+1,\dots,l\right)\\ &=& \prod_{m=1}^q P(F_m(b_m,Z_i)) \prod_{m=q+1}^l P(G_m(b_m,Z_i))\\ &=& \prod_{m=1}^q (f_m(b_m) - f_m(b_m-1)) \prod_{m=q+1}^l f_m(b_m), \ee where $G_m(b,Z_i)$ denotes the proposition denoting that $p_{j_m}^{a_{j_m}-b}$ divides $\gcd(Z_i,s)$, and $F_m(b,Z_i)$ denotes the proposition $G_m(b,Z_i) \wedge \neg G_m(b-1,Z_i)$, so that $F_m(b,Z_i)$ is equivalent to the proposition that $p_{j_m}^{a_{j_m}-b}$ divides $\gcd(Z_i,s)$ and $p_{j_m}^{a_{j_m}-b+1}$ does not divide $\gcd(Z_i,s)$. It follows that $B_{ij_1}, B_{ij_2}, \dots, B_{ij_l}$ are independent random variables. Since the set $\{ j_1,j_2,\dots,j_l \}$ was arbitrary, and since $Z_i$ are independent random variables, it follows that $B_{ij}$ for $i = 1,2,\dots$, and for $j \in J$, are independent random variables. \ep The proof of \thmref{zeta} can now be given. \begin{proof}} \def\ep{\end{proof} Let $B_{ij}$ be random variables as in the proof of the Lemma, and define random variables $C_{kj}$ by \begin{eqnarray*}} \def\ee{\end{eqnarray*} C_{kj} = \max(B_{1j},\dots,B_{kj}), \ee then \begin{eqnarray*}} \def\ee{\end{eqnarray*} S_k = \prod_{j \in J} p_j^{C_{kj}}. \ee Since the sample space for $B_{ij}$ is $\{0,1,\dots,a_j\}$ for all $i, j$, then the sample space for $C_{kj}$ is also $\{0,1,\dots,a_j\}$ for all $k, j$. From the result in the Lemma that \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(B_{ij} = a_j) = \frac{p_j^{a_j}-p_j^{a_j-1}}{p_j^{a_j}} = 1 - \frac{1}{p_j}, \ee for all $i, j$, then \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(B_{ij} < a_j) = \frac{1}{p_j}, \ee for all $i, j$, and so \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(C_{kj} < a_j) = P(B_{ij} < a_j \mbox{ for } i = 1,\dots,k) = \prod_{i=1}^k P(B_{ij} < a_j) = \frac{1}{p_j^k}, \ee as a consequence of the independence of $B_{1j}, B_{2j}, \dots, B_{kj}$ (which follows from the independence of $Z_i$ for $i = 1,\dots,k$). It follows that \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(C_{kj} = a_j) = 1 - \frac{1}{p_j^k}, \ee and so \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(S_k = s) = P(C_{kj} = a_j \mbox{ for all } j \in J) = \prod_{j \in J} P(C_{kj} = a_j) = \prod_{j \in J} \left(1 - \frac{1}{p_j^k}\right), \ee as a consequence of the independence of $C_{kj}$ for $j \in J$ (which follows from the independence of $B_{ij}$ for $i = 1,2,\dots,k$ and $j \in J$). Therefore \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(S_k = s) = \prod_{j \in J} \left(1 - \frac{1}{p_j^k}\right) > \prod_{p \mbox{\small{ prime}}} \left(1 - \frac{1}{p^k}\right) = \frac{1}{\zeta(k)}, \ee for $k \geq 2$. \ep
\section{Proof of \eqref{ck=r}} \applabel{prob}
A similar method to the proof of \thmref{zeta} can be used. Let $r$ have prime factorization \begin{eqnarray*}} \def\ee{\end{eqnarray*} r = \prod_{j \in J} p_j^{a_j}, \ee where $J$ is some index set, $p_j$ are distinct primes, and $a_j \geq 1$ for all $j \in J$. For each prime $p$, define the random variable $E_{kp}$, with sample space $\{ 0,1,2,\dots \}$, by setting \begin{eqnarray*}} \def\ee{\end{eqnarray*} B_k = \prod_{p \mbox{\small{ prime}}} p^{E_{kp}}. \ee Specifically, $E_{kp}$ is the power to which $p$ is raised in the prime factorization of $B_k$. By similar arguments to the ideal case, treated in \secref{prob} and \appref{zeta}, then: \begin{itemize}} \def\ei{\end{itemize} \item For a finite set $J_0$ of primes, and for non-negative integers $b_p$ for $p \in J_0$, \begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left(E_{kp} = b_p \mbox{ for all } p \in J_0\right) = O\left(\frac{1}{wn}\right), \ee if $b_p > 0$ for some $p$ not dividing $r$, or $b_{p_j} > a_j$ for some $j \in J$ such that $p_j \in J_0$; \item For a finite set $J_0$ of primes, and for non-negative integers $b_p$ for $p \in J_0$, \begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left(E_{kp} = b_p \mbox{ for all } p \in J_0\right) = \prod_{j \in J \atop p_j \in J_0, \ b_{p_j} > 0} \frac{p_j^{b_{p_j}} \left(1 - \frac{1}{p_j}\right)}{p_j^{a_j}} \prod_{j \in J \atop p_j \in J_0, \ b_{p_j} = 0} \frac{1}{p_j^{a_j}} \left(1+O\left(\frac{r}{wn}\right) \right), \ee if $b_p = 0$ for all $p$ not dividing $r$, and $b_{p_j} \leq a_j$ for all $j \in J$ such that $p_j \in J_0$. \ei It follows that for any subset $J_0 \subseteq J$, \begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left(\mbox{$E_{kp_j} < a_j$ for all $j \in J_0$, and $E_{kp} = 0$ for all $p \nmid r$} \right) = \prod_{j \in J_0} \frac{1}{p_j} \left(1+O\left(\frac{r}{wn}\right)\right). \ee For all $k$ and primes $p$, define the random variable $F_{kp}$ by \begin{eqnarray*}} \def\ee{\end{eqnarray*} F_{kp} = \max(E_{1p},\dots,E_{kp}), \ee so that \begin{eqnarray*}} \def\ee{\end{eqnarray*} C_k = \prod_{p \mbox{\small{ prime}}} p^{F_{kp}}. \ee It follows that for a finite set $J_0$ of primes, and for non-negative integers $b_p$ for $p \in J_0$, \begin{eqnarray*}} \def\ee{\end{eqnarray*} P\left(F_{kp} = b_p \mbox{ for all } p \in J_0\right) = O\left(\frac{k}{wn}\right), \ee if $b_p > 0$ for some $p$ not dividing $r$, or $b_{p_j} > a_j$ for some $j \in J$ such that $p_j \in J_0$.
Since $A_k$ are independent random variables, then $B_k$ are independent random variables, so that for any subset $J_0 \subseteq J$, \begin{eqnarray*}} \def\ee{\end{eqnarray*} && P\left(\mbox{$F_{kp_j} < a_j$ for all $j \in J_0$, and $F_{kp} = 0$ for all $p \nmid r$}\right)\\ &=& P\left(\mbox{For all $i = 1,\dots,k$, $E_{ip_j} < a_j$ for all $j \in J_0$, and $E_{ip} = 0$ for all $p \nmid r$}\right)\\ &=& \prod_{i=1}^k P\left(\mbox{$E_{ip_j} < a_j$ for all $j \in J_0$, and $E_{ip} = 0$ for all $p \nmid r$}\right)\\ &=& \prod_{i=1}^k \prod_{j \in J_0} \frac{1}{p_j} \left(1+O\left(\frac{r}{wn}\right)\right)\\ &=& \prod_{j \in J_0} \frac{1}{p_j^k} \left(1+O\left(\frac{kr}{wn}\right)\right). \ee Therefore, similarly to the idealized case, \begin{eqnarray*}} \def\ee{\end{eqnarray*} && P(C_k = r)\\ &=& P\left(\mbox{$F_{kp_j} = a_j$ for all $j \in J$, and $F_{kp} = 0$ for all $p \nmid r$}\right)\\ &=& \sum_{J_0 \subseteq J} (-1)^{\#(J_0)} P\left(\mbox{$F_{kp_j} < a_j$ for all $j \in J_0$, and $F_{kp} = 0$ for all $p \nmid r$}\right)\\ &=& \sum_{J_0 \subseteq J} (-1)^{\#(J_0)} \prod_{j \in J_0} \frac{1}{p_j^k} \left(1+O\left(\frac{kr}{wn}\right)\right)\\ &=& \prod_{j \in J} \left(1 - \frac{1}{p_j^k}\right) + \prod_{j \in J} \left(1 + \frac{1}{p_j^k}\right) O\left(\frac{kr}{wn}\right). \ee For $k$ bounded and greater than or equal to $2$, since \begin{eqnarray*}} \def\ee{\end{eqnarray*} \prod_{j \in J} \left(1 + \frac{1}{p_j^k}\right) < \sum_{n=1}^\infty \frac{1}{n^k} = \zeta(k), \ee and since $r < n$, then for $k \geq 2$ and $k$ small, \begin{eqnarray*}} \def\ee{\end{eqnarray*} P(C_k = r) &=& \prod_{j \in J} \left(1 - \frac{1}{p_j^k}\right) + O\left(\frac{1}{w}\right)\\ &=& \prod_{j \in J} \left(1 - \frac{1}{p_j^k}\right) + O(n^{-\epsilon}), \ee since $w = n^\epsilon$, thus demonstrating \eqref{ck=r}.
\end{document} | arXiv |
\begin{document}
\title{On the Containment Problem for Unambiguous Single-Register Automata with Guessing}
\author{Antoine Mottet\\ Charles University Prague \and Karin Quaas \\ Universit\"at Leipzig }
\date{} \maketitle \begin{abstract} Register automata extend classical finite automata with a finite set of registers that can store data from an infinite data domain for later equality comparisons with data from an input data word. While the registers in the original model of register automata, introduced in 1994 by Kaminski and Francez, can only store data occurring in the data word processed so far, we study here the more expressive class of register automata \emph{with guessing}, where registers can nondeterministically take any value from the infinite data domain, even if this data does not occur in the input data word. It is well known that the containment problem, i.e., the problem of deciding for two given register automata with guessing $\mathcal{A}$ and $\mathcal{B}$, whether the language $L(\mathcal{A})$ accepted by $\mathcal{A}$ is contained in the language $L(\mathcal{B})$ accepted by $\mathcal{B}$, is undecidable, even if $\mathcal{B}$ only uses a single register. We prove that the problem is decidable if $\mathcal{B}$ is unambiguous and uses a single register. \end{abstract}
\section{Introduction} Register automata~\cite{DBLP:journals/tcs/KaminskiF94, KaminskiZeitlin} are a widely studied computational model that extend classical finite automata with finitely many \emph{registers} that can take values from an infinite set and perform equality comparisons with data from the input word. Register automata accept \emph{data languages}, that is sets of \emph{data words} over $\Sigma\times\mathbb{D}$, where $\Sigma$ is a finite alphabet, and $\mathbb{D}$ is an infinite set called the \emph{data domain}.
As an example, consider the register automaton in Figure \ref{fig:gura} using a single register $r$ ($\dot r$ refers to the future value of $r$). This automaton processes finite data words over $\Sigma\times\mathbb{D}$. We assume that $\Sigma=\{\sigma\}$ is a singleton, so that we omit the letter $\sigma$ from all transitions and input words, and $\mathbb{D}=\mN$. Let us study the behaviour of the automaton: starting in the initial location $\ell_0$ and processing the first input letter $d$, the automaton can only move to $\ell'$ if it satisfies the register constraint $\dotr \neq$. This constraint requires the register $r$, when reaching $\ell_1$, to store a data value $d'\in\mN$ such that $d'\neq d$. The automaton can nondeterministically \emph{guess} such a datum $d'$. Being in $\ell_1$ with the register holding the value $d'$, by the constraint $=r$, it can only move to the accepting location $\ell_2$ if it reads the input letter $d'$; for every other input letter, satisfying the constraint $\neq r$, the automaton stays in $\ell_1$, keeping the register value (indicated by the constraint $\dotr=r$). For instance, for the input data word $w_\mathit{ad}=1 \, 2 \, 2 \, 3$, there are infinitely many distinct runs (one for each guessed datum different from $1$), but only one accepting run, namely $$(\ell_0,\bot)\xrightarrow{1}(\ell_1,3)\xrightarrow{2}(\ell_1,3)\xrightarrow{2}(\ell_1,3)\xrightarrow{3}(\ell_2,3).$$ We write
$L_{\mathit{ad}} = \{d_1\dots d_k\mid \forall k\geq 2 \, \, \forall 1\leq i<k. \, \, d_i\neq d_k\}$ to denote the set of data words that is accepted by the automaton in Figure \ref{fig:gura} ($\mathit{ad}$ standing for \emph{all different}).
\begin{figure}
\caption{
A $\textup{GURA}$ with a single register $r$ and over a singleton alphabet (we omit the labels at the edges). }
\label{fig:gura}
\end{figure} We remark that the nondeterministic \emph{guessing} of data values to store them into registers for future comparisons is not allowed in the original model of register automata, introduced by Kaminski and Francez~\cite{DBLP:journals/tcs/KaminskiF94}, and studied \eg \ in~\cite{DBLP:journals/tocl/DemriL09,DBLP:conf/stacs/MottetQ19,DBLP:journals/tocl/QuaasS19}. In fact, the model that we study here is strictly more expressive (with respect to acceptance of data languages) than the classical model. In order to distinguish the model with guessing from the classical model without guessing, we explicitly refer to the former by \emph{register automata with guessing}, \textup{GRA} \ for short.
The \textup{GRA} \ in Figure \ref{fig:gura} is not \emph{deterministic}: being in $\ell_0$ and processing the first input datum $d$, it can nondeterministically \emph{guess} any datum $d'$ such that $d'\neq d$ for storage in $r$. However, one can easily see that for every input data word there is \emph{at most one accepting run}, uniquely determined by the single guessed datum $d'$. We call automata that have, for every input word, at most one accepting run, \emph{unambiguous}. One of the main open problems concerning unambiguous register automata with guessing (GURA, for short) is whether the class of data languages accepted by GURA is closed under complementation\footnote{In Theorem 12 in~\cite{DBLP:conf/dcfs/Colcombet15}, it is claimed that the class of data languages accepted by GURA is effectively closed under complement; however, to the best of our knowledge, this claim remains unproved.}. (In contrast, it is known that data languages accepted by unambiguous register automata (without guessing) are \emph{not} closed under complementation; for instance, the complement of $L_{\mathit{ad}}$ can be accepted by an unambiguous register automata with a single register, but $L_{\mathit{ad}}$ cannot even be accepted by any nondeterministic register automaton (without guessing)~\cite{KaminskiZeitlin}.)
In this paper, we study the \emph{containment problem}: given two $\textup{GRA}$ $\mathcal{A}$ and $\mathcal{B}$, does $L(\mathcal{A})\subseteq L(\mathcal{B})$ hold? Here, $L(\mathcal{A})$ and $L(\mathcal{B})$, respectively, denote the set of data words accepted by $\mathcal{A}$ and $\mathcal{B}$, respectively. This problem, playing a central role in formal verification, has been studied a lot for register automata, see \eg~\cite{DBLP:journals/tocl/NevenSV04,DBLP:journals/tocl/DemriL09,DBLP:conf/stacs/MottetQ19}. For $\textup{GRA}$, it is well known that the problem is undecidable~\cite{DBLP:journals/tocl/DemriL09}. More detailed, the special case of deciding whether a single given $\textup{GRA}$ $\mathcal{B}$ over $\Sigma$ and $\mathbb{D}$ accepts the set $(\Sigma\times\mathbb{D})^*$ of all data words, is undecidable, even if $\mathcal{B}$ only uses a single register\footnote{A proof for undecidability can be done using a reduction from the undecidable reachability problem for Minsky machines, following the lines of the proof of Theorem 5.2 in~\cite{DBLP:journals/tocl/DemriL09}. The nondeterministic guessing can be used to express that there exists some decrement for which there is no matching preceding increment.}.
In this short note, we prove that the containment problem $L(\mathcal{A})\subseteq L(\mathcal{B})$ is decidable in $\EXPSPACE$ if $\mathcal{B}$ is unambiguous and uses a single register (and no restriction on $\mathcal{A}$).
\section{Main Definitions} In this section, we define \emph{register automata with guessing} as introduced by Kaminski and Zeitlin~\cite{KaminskiZeitlin}.
We start with some preliminary notions. We use $\Sigma$ to denote a finite alphabet, and $\mathbb{D}$ to denote an infinite data domain. A \emph{data word} is a finite sequence $(\sigma_1,d_1)\dots(\sigma_k,d_k) \in (\Sigma\times\mathbb{D})^*$. We use $\varepsilon$ to denote the empty data word. A \emph{data language} is a set of data words. We use $\textup{data}(w)$ to denote the set $\{d_1,\dots,d_k\}$ of all data occurring in $w$.
Let $\mathbb{D}_\bot$ denote the set $\mathbb{D}\cup\{\bot\}$, where $\bot\not\in\mathbb{D}$. We let $\bot \neq d$ for all $d\in\mathbb{D}$. We use boldface lower-case letters like $\vect{a}, \vect{b}, \dots $ to denote tuples in $\mathbb{D}_\bot^n$, where $n\in\mN$. Given a tuple $\vect{a}\in\mathbb{D}_\bot^n$, we write $a_i$ for its $i$-th component, and $\textup{data}(\vect{a})$ denotes the set $\{a_1,\dots,a_n\}\subseteq\mathbb{D}_\bot$ of all data occurring in $\vect{a}$.
Let $R=\{r_1,\dots,r_n\}$ be a finite set of \emph{registers}. A \emph{register valuation} is a mapping $\vect{u}:R\to\mathbb{D}_\bot$; we may write $u_i$ as shorthand for $\vect{u}(r_i)$. Let $\mathbb{D}_\bot^R$ denote the set of all register valuations. A \emph{register constraint over $R$} is defined by the grammar \begin{align*} \phi ::= \texttt{true} \,\mid\, \, =r_i \,\mid\, \dotr_i=r_j \,\mid\, \dotr_i= \, \,\mid\, \neg\phi \,\mid\,
\phi \wedge \phi \end{align*} where $r_i,r_j\inR$. Intuitively, $r_i$ refers to the current value of the register $r_i$, while $\dotr_i$ refers to the future value of the register $r_i$. We use $\Phi(R)$ to denote the set of all register operations over $R$. The satisfaction relation $\models$ on $\mathbb{D}_\bot^R\times\mathbb{D}\times\mathbb{D}_\bot^R$ is defined by structural induction as follows. We only give the atomic cases; the cases for the Boolean formulas are as usual. We have $(\vect{u},d,\vect{v}) \models \phi$ if \begin{itemize} \item $\phi$ is of the form $\texttt{true}$, \item $\phi$ is of the form $=r_i$ and $u_i = d$, \item $\phi$ is of the form $\dotr_i=r_j$ and $v_i = u_j$, \item $\phi$ is of the form $\dotr_i=$ and $v_i = d$. \end{itemize} For instance, $(1,2,1)\models (\neqr)\wedge(\dotr=r)$, while $(1,2,3)$ does not. Note that only register constraints of the form $\dotr_i=r_j$ and $\dotr=$ uniquely determine the new value of $r_i$. In absence of such a register constraint, the new value of $r_i$ can be equal to (almost\footnote{The register constraint $\dotr\neq$ requires that the new value of $r$ is different from the current input datum, so that $r$ may take any of the infinitely data in $\mathbb{D}$ except for the input datum. Likewise, the register constraint $\dotr_i\neqr_j$ requires that $r_i$ takes any of the infinitely data in $\mathbb{D}$ except for the current value of $r_j$.}) any of the infinitely many data values in $\mathbb{D}$. Register automata that allow for such nondeterministic \emph{guessings} of future register values are called \emph{register automaton with guessing}. Formally, a register automaton with guessing (GRA) over $\Sigma$ is a tuple $\mathcal{A} = (R,\mathcal{L}, \ell_{\textup{in}}, \mathcal{L}_{\textup{acc}}, E)$, where \begin{itemize} \item $R$ is a finite set of registers, \item $\mathcal{L}$ is a finite set of locations, \item $\ell_{\textup{in}}\in\mathcal{L}$ is the initial location, \item $\mathcal{L}_{\textup{acc}}\subseteq\mathcal{L}$ is the set of accepting locations, \item $E\subseteq\mathcal{L}\times\Sigma\times\Phi(R)\times\mathcal{L}$ is a finite set of edges. \end{itemize} A \emph{state} of $\mathcal{A}$ is a pair $(\ell,\vect{u})\in \mathcal{L}\times\mathbb{D}_\bot^R$, where $\ell$ is the current location and $\vect{u}$ is the current register valuation. Given two states $(\ell,\vect{u})$ and $(\ell',\vect{u'})$ and some input letter $(\sigma,d)\in\Sigma\times\mathbb{D}$, we postulate a transition $(\ell,\vect{u})\xrightarrow{\sigma,d}_\mathcal{A}(\ell',\vect{u'})$ if there exists some edge $(\ell,\sigma,\phi,\ell')\inE$ such that $(\vect{u},d,\vect{u'})\models\phi$. A \emph{run} of $\mathcal{A}$ on the data word $(\sigma_1,d_1)\dots(\sigma_k,d_k)$ is a sequence $(\ell_0,\vect{u^0}) \xrightarrow{\sigma_1,d_1}_\mathcal{A} (\ell_1,\vect{u^1}) \xrightarrow{\sigma_2,d_2}_\mathcal{A} \dots \xrightarrow{\sigma_k,d_k}_\mathcal{A} (\ell_n,\vect{u^k})$ of such transitions. We say that a run like above \emph{starts in $(\ell,\vect{u})$} if $(\ell_0,\vect{u^0})=(\ell,\vect{u})$. A run is \emph{initialized} if starts in $(\ell_\textup{in},\{\bot\}^k)$, and a run is \emph{accepting} if $\ell_k\in\mathcal{L}_\textup{acc}$. The data language \emph{accepted} by $\mathcal{A}$, denoted by $L(\mathcal{A})$, is the set of data words for which there exists an initialized, accepting run of $\mathcal{A}$. A $\textup{GRA}$ is unambiguous ($\textup{GURA}$) if for every input data word $w$ there is at most one initialized accepting run. The \emph{containment problem} is the following decision problem: given two $\textup{GRA}$ $\mathcal{A}$ and $\mathcal{B}$, does $L(\mathcal{A})\subseteq L(\mathcal{B})$ hold?
\section{Some Facts about Register Automata} \subsection{Unambiguous Register Automata with Guessing} Fix a GURA $\mathcal{B}=(R,\mathcal{L},\ell_{\textup{in}},\mathcal{L}_\textup{acc},E)$ with a single register $r$. Let $C\subseteq (\mathcal{L}\times\mathbb{D}_\bot)$ be a set of states of $\mathcal{B}$, and let $(\sigma,d)\in(\Sigma\times\mathbb{D})$. We use $\textup{Succ}_\mathcal{B}(C,(\sigma,d))$ to denote the \emph{successor of $C$ on the input $(\sigma, d)$}, formally defined by \begin{align*} \textup{Succ}_\mathcal{B}(C,(\sigma,d)) := \{(\ell,u) \in (\mathcal{L}\times\mathbb{D}_\bot) \mid \exists (\ell',u')\in C. (\ell',u')\xrightarrow{\sigma,d}_\mathcal{B}(\ell,u)\}. \end{align*} In order to extend this definition to data words, we define inductively $\textup{Succ}_\mathcal{B}(C,\varepsilon):=C$ and $\textup{Succ}_\mathcal{B}(C, w \cdot (\sigma,d)) := \textup{Succ}_\mathcal{B}(\textup{Succ}_\mathcal{B}(C,w),(\sigma,d))$. We say that a set $C\subseteq (\mathcal{L}\times\mathbb{D})$ of states is \emph{reachable in $\mathcal{B}$} if there exists some data word $w$ such that $C = \textup{Succ}_\mathcal{B}(C_\textup{in}, w)$, where $C_\textup{in}=\{(\ell_\textup{in},\bot)\}$.
A \emph{configuration of $\mathcal{B}$} is a finite union of finite or cofinite subsets of $\mathcal{L}\times\mathbb{D}_\bot$. Hence the set $C_\textup{in}:=\{(\ell_\textup{in},\bot)\}$ is a configuration, henceforth called the \emph{initial configuration}. Note that for all configurations $C$ and data words $w$, the successor $\textup{Succ}_\mathcal{B}(C,w)$ is a configuration, too. This implies that every reachable set $C\subseteq (\mathcal{L}\times\mathbb{D})$ of states is a configuration. Given a configuration $C$, we use $\textup{data}(C)$ to denote the set $\{d\in\mathbb{D}_\bot \mid \exists \ell\in\mathcal{L}. (\ell,d)\inC\}$ of data occurring in $C$.
The \emph{support} of a configuration $C$ is the set $\textup{supp}(C)$ of data $d$ such that at least one of the following holds: \begin{itemize} \item $(\ell, d)\inC$ for some $\ell$ such that $(\{\ell\}\times \mathbb{D})\capC$ is finite, \item $(\ell,d)\not\inC$ for some $\ell$ such that $(\{\ell\}\times \mathbb{D})\capC$ is cofinite. \end{itemize}
We say that a configuration $C$ is \emph{coverable} if there exists some configuration $C'\supseteq C$ such that $C'$ is reachable in $\mathcal{B}$. We say that a configuration $C$ is \emph{accepting} if there exists $(\ell,u)\inC$ such that $\ell\in\mathcal{L}_\textup{acc}$; otherwise we say that $C$ is \emph{non-accepting}. The following proposition follows immediately from the definition of \textup{GURA}. \begin{proposition} \label{prop:gura_implied_badness} If $C,C'$ are two configurations of $\mathcal{B}$ such that $C\capC'=\emptyset$ and $C\cupC'$ is coverable, then for every data word $w$ the following holds: if $\textup{Succ}_\mathcal{B}(C,w)$ is accepting, then $\textup{Succ}_\mathcal{B}(C',w)$ is non-accepting. \end{proposition} A \emph{partial isomorphism of $\mathbb{D}_\bot$} is an injective mapping $\pi:D\to\mathbb{D}_\bot$ with domain $\textup{dom}(\pi):=D\subseteq\mathbb{D}$ such that if $\bot\in D$ then $\pi(\bot)=\bot$. Let $\pi$ be a partial isomorphism of $\mathbb{D}_\bot$ and let $C$ be a configuration such that $\textup{data}(C)\subseteq\textup{dom}(\pi)$. We define the configuration $\pi(C) := \{(\ell,\pi(d))\mid (\ell,d)\inC\}$; likewise, if $\{d_1,\dots,d_k\}\subseteq\textup{dom}(\pi)$, we define the data word $\pi(w)=(\sigma_1,\pi(d_1))\dots(\pi_k,\pi(d_k))$. We say that $C, w$ and $C',w'$ \emph{are equivalent with respect to $\pi$}, written $C,w \sim_\pi C',w'$, if $\pi(C)=C' \text{ and } \pi(w)=w'$. If $w=w'=\varepsilon$, we may write $C\sim_\piC'$. We write $C,w\simC',w'$ if $C,w\sim_\piC',w'$ for some partial isomorphism $\pi$ of $\mathbb{D}_\bot$. \begin{proposition} \label{prop:ra_equivalence} If $C,w\simC',w'$, then $\textup{Succ}(C,w) \sim \textup{Succ}(C',w')$. \end{proposition} As an immediate consequence of Proposition \ref{prop:ra_equivalence}, we obtain that $\sim$ preserves the configuration properties of being \emph{accepting} respectively \emph{non-accepting}. \begin{corollary} \label{corollary:gnra_bad} If $C,w\simC',w'$ and $\textup{Succ}_\mathcal{B}(C,w)$ is non-accepting (accepting, respectively), then $\textup{Succ}_\mathcal{B}(C',w')$ is non-accepting (accepting, respectively). \end{corollary} Combining the last corollary with Proposition \ref{prop:gura_implied_badness}, we obtain \begin{corollary} \label{corollary:gura_bad} If $C,C'$ are two configurations such that $C\capC'=\emptyset$ and $C\cupC'$ is coverable in $\mathcal{B}$, then for every data word $w$ such that $C,w\simC',w$, the configurations $\textup{Succ}_\mathcal{B}(C,w)$ and $\textup{Succ}_\mathcal{B}(C',w)$ are non-accepting. \end{corollary}
\subsection{The Synchronized State Space} For the rest of this paper, let $\mathcal{A}=(R^\mathcal{A},\mathcal{L}^\mathcal{A},\ell^\mathcal{A}_{\textup{in}},\mathcal{L}^\mathcal{A}_{\textup{acc}},E^\mathcal{A})$ be a \textup{GNRA} \ over $\Sigma$ with $R^A=\{r_1,\dots,r_m\}$, and let $\mathcal{B}=(R^\mathcal{B},\mathcal{L}^\mathcal{B},\ell^\mathcal{B}_{\textup{in}},\mathcal{L}^\mathcal{B}_{\textup{acc}},E^\mathcal{B})$ be a \textup{GURA} \ over $\Sigma$ with a single register $r$. A \emph{synchronized configuration of $\mathcal{A}$ and $\mathcal{B}$} is a pair $((\ell,\vect{d}),C)$, where $(\ell,\vect{d})\in (\mathcal{L}^\mathcal{A}\times\mathbb{D}^{R^\mathcal{A}}_\bot)$ is a single state of $\mathcal{A}$, and $C\subseteq (\mathcal{L}^\mathcal{B}\times\mathbb{D}_\bot)$ is a configuration of $\mathcal{B}$. We define $S_{\textup{in}}:=((\ell_{\textup{in}}^\mathcal{A},\{\bot\}^m), C_\textup{in})$ to be the \emph{initial synchronized configuration of $\mathcal{A}$ and $\mathcal{B}$}. We define the \emph{synchronized state space of $\mathcal{A}$ and $\mathcal{B}$} to be the (infinite) state-transition system $(\mathbb{S},\Rightarrow)$, where $\mathbb{S}$ is the set of all synchronized configurations of $\mathcal{A}$ and $\mathcal{B}$, and $\Rightarrow$ is defined as follows. If $S=((\ell,\vect{d}),C)$ and $S'=((\ell',\vect{d'}),C')$, then $S\sToS'$ if there exists a letter $(\sigma,d)\in(\Sigma\times\mathbb{D})$ such that $(\ell,\vect{d})\xrightarrow{\sigma,d}_\mathcal{A}(\ell',\vect{d'})$, and $\textup{Succ}_\mathcal{B}(C,(\sigma,d))=C'$. We say that a synchronized configuration \emph{$S$ reaches a synchronized configuration $S'$ in $(\mathbb{S},\Rightarrow)$} if there exists a path in $(\mathbb{S},\Rightarrow)$ from $S$ to $S'$. We say that a synchronized configuration $S$ is \emph{reachable in $(\mathbb{S},\Rightarrow)$} if $S_\textup{in}$ reaches $S$. We say that a synchronized configuration $S=((\ell,\vect{d}),C)$ is \emph{coverable in $(\mathbb{S},\Rightarrow)$} if there exists some configuration $C'\supseteqC$ such that $((\ell,\vect{d}),C')$ is reachable in $(\mathbb{S},\Rightarrow)$.
We aim to reduce the containment problem $L(\mathcal{A})\subseteq L(\mathcal{B})$ to a reachability problem in $(\mathbb{S},\Rightarrow)$. For this, call a synchronized configuration $((\ell,\vect{d}),C)$ \emph{bad} if $\ell\in\mathcal{L}_\textup{acc}^\mathcal{A}$ is an accepting location and $C$ is non-accepting, i.e., $\ell'\not\in\mathcal{L}_\textup{acc}^\mathcal{B}$ for all $(\ell',u)\inC$. The following proposition is easy to prove, cf.~\cite{DBLP:conf/lics/OuaknineW04}.
\begin{proposition} \label{prop:reductionToReach} $L(\mathcal{A})\subseteq L(\mathcal{B})$ does not hold if, and only if, some bad synchronized configuration is reachable in $(\mathbb{S},\Rightarrow)$. \end{proposition}
We extend the equivalence relation $\sim$ defined above to synchronized configurations in a natural manner, i.e, given a partial isomorphism $\pi$ of $\mathbb{D}_\bot$ such that $\textup{data}(\vect{d})\cup\textup{data}(C)\subseteq\textup{dom}(\pi)$, we define $((\ell,\vect{d}),C) \sim_\pi ((\ell,\vect{d}'),C')$ if $\pi(C)=C'$ and $\pi(\vect{d})=\vect{d}'$. We shortly write $S\simS'$ if there exists a partial isomorphism $\pi$ of $\mathbb{D}_\bot$ such that $S\sim_\piS'$. Clearly, an analogon of Proposition~\ref{prop:ra_equivalence} holds for this extended relation. In particular, we have the following: \begin{proposition}\label{prop:equivalence-relation-synch-compatible} Let $S,S'$ be two synchronized configurations of $(\mathbb{S},\Rightarrow)$ such that $S\simS'$. If $S$ reaches a bad synchronized configuration, so does $S'$. \end{proposition} Note that $(\mathbb{S},\Rightarrow)$ is infinite so that \emph{a priori} it is not clear how to exploit Proposition \ref{prop:reductionToReach} to solving the containment problem. First of all, $(\mathbb{S},\Rightarrow)$ is not finitely branching: for every synchronized configuration $S=((\ell,\vect{d}),C)$ in $\mathbb{S}$, every input datum $d\in\mathbb{D}$ and every \emph{guessed} new value of the registers may give rise to its own individual synchronized configuration $S_d$ such that $S\sToS_d$. However, it is well known that, using standard techniques, one can define an \emph{abstract} finite-branching state-transition system that is bisimilar to $(\mathbb{S},\Rightarrow)$ with respect to $\sim$, \cf~\cite{DBLP:conf/stacs/MottetQ19}. Second, and potentially more harmful, the data needed to define the configuration $C$ in a synchronized configuration $((\ell,\vect{d}),C)$ can grow unboundedly. As an example, consider the $\textup{GURA}$ in Figure \ref{fig:gura}. For every $k\geq 1$, the configuration $\{(\ell_1,d)\mid d\in\mN\backslash\{d_1,\dots,d_k\}\}\cup\{(\ell_2,d_k)\}$ with pairwise distinct data values $d_1,\dots,d_k$ is reachable by inputting the data word $d_1 \, \dots d_k$. In the next section, we prove that one can solve the reachability problem from Proposition \ref{prop:reductionToReach} by focussing on a subset of configurations of $\mathcal{B}$ that can be defined by a bounded number of data. The approach follows the ideas presented in~\cite{DBLP:conf/stacs/MottetQ19} for unambiguous register automata (without guessing); however, the main technical proposition in~\cite{DBLP:conf/stacs/MottetQ19} does not apply to $\textup{GRA}$ and is substituted by Proposition \ref{prop:collapse_gura} below.
\section{Decidability of the Containment Problem}
\subsection{Bounding the Size of the Supports}
Recall the equivalence relation $\sim$ on $k$-tuples, where for $\vect a,\vect b\in\mathbb{D}_\bot^k$ we have $\vect a\sim \vect b$ if there exists a partial isomorphism $\pi$ of $\mathbb{D}_\bot$ such that $\pi(\vect a)=\vect b$. Note that this equivalence relation has finitely many equivalence classes for all $k\in\mN$. As an example, for $k=3$ the equivalence classes of triples are the classes of $(d_0,d_0,d_0),(d_0,d_0,d_1),(d_0,d_1,d_0),(d_1,d_0,d_0),(d_0,d_1,d_2)$, where $d_0,d_1,d_2$ are pairwise distinct data values.
Let $S=((\ell,\vect{d}),C)$ be a synchronized configuration, and let $a,b\in\textup{supp}(C)$ be two data values in the support of $C$. We say that \emph{$a$ and $b$ are indistinguishable in $S$}, written $\indiscernible{a}{b}{S}$, if $a,b\not\in\textup{data}(\vect{d})$ and $\{\ell \in\mathcal{L} \mid (\ell,a)\in C\} = \{\ell\in\mathcal{L}\mid (\ell,b)\in C\}$.
Let $S$ be a synchronized configuration $((\ell,\vect d),C)$ and let $a,b\in\textup{supp}(C)\setminus\textup{data}(\vect d)$.
Given a configuration $C$, we define for every datum $d\in\mathbb{D}$ the sets \begin{align*} C_d^+ := & \, \{(\ell,d)\in \mathcal{L}\times\{d\} \mid (\ell,d)\inC \text{ and } \textup{data}(C\cap (\{\ell\}\times\mathbb{D})) \text{ is finite}\}, \text{ and} \\ C_d^- := & \, \{(\ell,d)\in \mathcal{L}\times\{d\} \mid (\ell,d)\not\inC \text{ and } \textup{data}(C\cap (\{\ell\}\times\mathbb{D})) \text{ is infinite}\} \end{align*} For later reference, we state the following simple fact. \begin{fact} \label{fact:minusandconfignointersect}
$C\cap C^-_d=\emptyset$, for all configurations $C$ and data $d\in\mathbb{D}$. \end{fact}
\begin{example} Let $C = \{(\ell_1,0),(\ell_1,1)\} \cup \{(\ell_2,d) \mid d\in\mN\backslash\{1,2\}\}\cup\{(\ell_3,d)\mid d\in\mN\backslash\{0,1\}\}$. Then \begin{center} \begin{tabular}{lll} $C^+_0 = \{(\ell_1,0)\}$ & $C^+_1 = \{(\ell_1,1)\}$ & $C^+_2=\emptyset$ \\ $C^-_0=\{(\ell_3,0)\}$ & $C^-_1 = \{(\ell_2,1),(\ell_3,1)\}$ & $C^-_2=\{(\ell_2,2)\}$ \end{tabular} \end{center} \end{example}
We say that a configuration $C$ is \emph{essentially coverable} if for every two $(\ell,u),(\ell',u')\inC$, the set $\{(\ell,u),(\ell',u')\}$ is coverable.
\begin{proposition} \label{prop:esscov} Let $C$ be an essentially coverable configuration, and let $b\in\textup{supp}(C)$. Then $((C\backslashC^+_b)\cupC^-_b)$ is essentially coverable, too. \end{proposition} \begin{proof} Let $(\ell,c),(\ell',c')\in ((C\backslashC^+_b)\cupC^-_b)$. If $(\ell,c),(\ell',c')\inC\backslashC^+_b$, then $\{(\ell,c),(\ell',c')\}$ is coverable by essential coverability of $C$. Suppose $(\ell,c),(\ell',c')\inC^-_b$. By definition of $C^-_b$, $c=c'=b$.
Pick some value $e\in\mathbb{D}\backslash\{b\}$ such that $(\ell,e),(\ell',e)\inC$. Note that such a value $e$ must exist, as by definition of $C^-_b$, the sets $\textup{data}((\{\ell\}\times\mathbb{D})\cap C)$ and $\textup{data}((\{\ell'\}\times\mathbb{D})\cap C)$ are cofinite, and hence their intersection is non-empty. By essential coverability of $C$, $\{(\ell,e),(\ell',e)\}$ is coverable. There must thus exist some data word $w$ such that $\{(\ell,e),(\ell',e)\}\subseteq \textup{Succ}((\ell_\textup{in},\bot),w)$. Let $\pi$ be any partial isomorphism satisfying $\pi(e)=b$ and whose domain contains $\textup{data}(w)$. Clearly, $\{(\ell,b),(\ell',b)\} \subseteq \textup{Succ}((\ell_\textup{in},\bot),\pi(w))$, and hence $\{(\ell,b\},(\ell',b)\}$ is coverable. Finally, suppose $(\ell,c)\inC\setminus C_b^+$ and $(\ell',c')\inC^-_b$. The proof that $\{(\ell,c),(\ell',c')\}$ is coverable is very similar to the proof for the preceding case and left as an exercise. \end{proof}
\begin{proposition}\label{prop:collapse_gura} Let $S=((\ell^\mathcal{A},\vect{d}),C)$ be a synchronized configuration of $\mathcal{A}$ and $\mathcal{B}$ such that $C$ is essentially coverable, and let $a\neq b$ be such that $a,b\in\textup{supp}(C)$ and $\indiscernible{a}{b}{S}$.
$S$ reaches a bad configuration in $(\mathbb{S},\Rightarrow)$ if, and only if, $S':=((\ell^\mathcal{A},\vect{d}),(C\setminus C_b^+)\cup C_b^-)$ reaches a bad configuration in $(\mathbb{S},\Rightarrow)$. \end{proposition} \begin{proof}
$(\Leftarrow)$
Suppose there exists some data word $w$ such that
there exists an accepting run of $\mathcal{A}$ on $w$ that starts in $(\ell^\mathcal{A},\vect{d})$, and $\textup{Succ}_\mathcal{B}(C\backslash C_b^+\cup C_b^-,w)$ is non-accepting.
We assume in the following that $\textup{Succ}_\mathcal{B}(C^+_b,w)$ is accepting; otherwise we are done.
Let $(\ell^+,b)\in C_b^+$ be the unique state such that $\textup{Succ}_\mathcal{B}((\ell^+,b),w)$ is accepting.
In the following, we prove that we can without loss of generality assume that $w$ does not contain any $a$'s.
Pick some $a'\in\mathbb{D}$ such that $a'\not\in\textup{data}(w)\cup\textup{supp}(C)\cup\textup{data}(\vect{d})$.
Let $\pi$ be the isomorphism defined by $\pi(a)=a'$, $\pi(a')=a$, and $\pi(d)=d$ for all $d\in\mathbb{D}_\bot\backslash\{a,a'\}$.
Then
$(\ell^\mathcal{A},\vect{d}),w \sim_\pi(\ell^\mathcal{A},\vect{d}),\pi(w)$ (as $a\not\in\textup{data}(\vect{d})$ by $\indiscernible{a}{b}{S}$), and
$(\ell^+,b),w \sim_\pi (\ell^+,b),\pi(w)$.
By Corollary \ref{corollary:gnra_bad}, there exists an accepting run of $\mathcal{A}$ on $\pi(w)$ that starts in $(\ell^\mathcal{A},\vect{d})$, and $\textup{Succ}_\mathcal{B}((\ell^+,b),\pi(w))$ is accepting.
We prove that $\textup{Succ}_\mathcal{B}((\ell,c),\pi(w))$ is non-accepting, for every $(\ell,c)\inC\setminus\{(\ell^+,b)\}\cupC^-_b$:
first, let $(\ell,c)\in C\setminus\{(\ell^+,b)\}$.
By essential coverability of $C$, $\{(\ell^+,b),(\ell,c)\}$ is coverable.
By Proposition \ref{prop:gura_implied_badness},
$\textup{Succ}_\mathcal{B}((\ell,c),\pi(w))$ must be non-accepting.
Second, let $(\ell,c)\inC^-_b$. But then $c=b$, and hence $(\ell,c),w \sim_\pi (\ell,c),\pi(w)$. By assumption, $\textup{Succ}_\mathcal{B}((\ell,c),w)$ is non-accepting, so that by Corollary \ref{corollary:gnra_bad}, $\textup{Succ}_\mathcal{B}((\ell,c),\pi(w))$ is non-accepting, too.
Note that $\pi(w)$ indeed does not contain any $a$'s.
We can hence continue the proof assuming that
$w$ does not contain any $a$'s.
Next, we prove that if we replace all $b$'s occurring in $w$ by some fresh datum not occurring in $\textup{supp}(C)\cup\textup{data}(w)\cup\textup{data}(\vect{d})$, we
obtain a data word that guides $S$ to a bad synchronized configuration.
Formally, pick some datum $b'\not\in\textup{data}(w)\cup\textup{supp}(C)\cup\textup{data}(\vect{d})$, and let $\pi$ be the isomorphism defined by
$\pi(b)=b'$, $\pi(b')=b$, and $\pi(d)=d$ for all $d\in\mathbb{D}_\bot\backslash\{b,b'\}$.
Note that $\pi(w)$ does not contain any $a$'s or $b$'s.
Clearly,
$(\ell^\mathcal{A},\vect{d}),w \sim_\pi(\ell^\mathcal{A},\vect{d}),\pi(w)$. By Corollary \ref{corollary:gnra_bad}, there still exists an accepting run of $\mathcal{A}$ on $\pi(w)$ that starts in $(\ell^\mathcal{A},\vect{d})$.
We prove that $\textup{Succ}_\mathcal{B}(C,\pi(w))$ is non-accepting.
Let $(\ell,c)\in C$.
We distinguish three cases.
\begin{enumerate}
\item Let $c\not\in\{b,b'\}$.
Then $(\ell,c),w\sim_\pi (\ell,c),\pi(w)$.
Since $\textup{Succ}_\mathcal{B}((\ell,c),w)$ is non-accepting by assumption, so that by Corollary \ref{corollary:gnra_bad} also $\textup{Succ}_\mathcal{B}((\ell,c),\pi(w))$ is non-accepting.
\item Let $c=b$. By $\indiscernible{a}{b}{C}$, the state $(\ell,a)$ is in $C$ and $(\ell, a),\pi(w)\sim (\ell,c),\pi(w)$
since $a$ and $c$ do not appear in $w$. By essential coverability of $C$, $\{(\ell,a),(\ell,c)\}\subseteqC$ is coverable.
By Corollary \ref{corollary:gura_bad} we obtain that $\textup{Succ}_\mathcal{B}((\ell,c),\pi(w))$ is non-accepting.
\item Let $c=b'$. Note that $(\ell,b),w \sim_\pi (\ell,b'),\pi(w)$.
Recall that $b'\not\in\textup{supp}(C)$.
This implies that $\textup{data}(C\cap(\{\ell\}\times\mathbb{D}_\bot))$ is cofinite.
We distinguish two cases.
\begin{itemize}
\item $b\in\textup{data}(C\cap(\{\ell\}\times\mathbb{D}_\bot))$, i.e., $(\ell,b)\inC$. But note that $(\ell,b)\not\inC^+_b$ by cofiniteness of $\textup{data}(C\cap(\{\ell\}\times\mathbb{D}_\bot))$.
Hence $(\ell,b)\in C\backslash\{(\ell^+,b)\}$.
\item $b\not\in\textup{data}(C\cap(\{\ell\}\times\mathbb{D}_\bot))$, i.e., $(\ell,b)\inC^-_b$.
\end{itemize}
In both cases, we have proved above that $\textup{Succ}((\ell,b),w)$ is non-accepting. By $(\ell,b),w \sim_\pi (\ell,b'),\pi(w)$ and Corollary \ref{corollary:gnra_bad}, $\textup{Succ}_\mathcal{B}((\ell,b'),\pi(w))$ is non-accepting, too.
\end{enumerate}
Altogether we have proved that $\textup{Succ}_\mathcal{B}(C,\pi(w))$ is non-accepting, while there exists some accepting run of $\mathcal{A}$ on $\pi(w)$ starting in $(\ell^\mathcal{A},\vect{d})$.
This concludes the proof for the $(\Leftarrow)$-direction.
$(\Rightarrow)$
Suppose there exists some data word $w$ such that there exists some accepting run of $\mathcal{A}$ on $w$ starting in $(\ell^\mathcal{A},\vect{d})$, and $\textup{Succ}_\mathcal{B}(C,w)$ is non-accepting.
We assume in the following that $\textup{Succ}_\mathcal{B}(C\setminus C_b^+\cup C_b^-,w)$ is accepting; otherwise we are done.
Let $(\ell^-,b)$ be a state in $C_b^-$ such that $\textup{Succ}_\mathcal{B}((\ell^-,b),w)$ is accepting.
Pick some datum $a'\in\mathbb{D}_\bot$ such that $a'\not\in\textup{data}(w) \cup \textup{supp}(C)\cup\textup{data}(\vect{d})$.
Let $\pi$ be the isomorphism defined by $\pi(b)=a$, $\pi(a)=a'$, $\pi(a')=b$, and $\pi(d)=d$ for all $d\in\mathbb{D}\backslash\{a,b,a'\}$.
Clearly, $(\ell^\mathcal{A},\vect{d}),w \sim_\pi(\ell^\mathcal{A},\vect{d}),\pi(w)$, so that by Corollary \ref{corollary:gnra_bad}, there exists some accepting run of $\mathcal{A}$ on $\pi(w)$ starting in $(\ell^\mathcal{A},\vect{d})$.
We prove that $\textup{Succ}_\mathcal{B}(C\backslashC^+_b\cupC^-_b,\pi(w))$ is non-accepting.
Let $(\ell,c)\in C\backslashC^+_b\cupC^-_b$. We distinguish the following cases:
\begin{enumerate}
\item Let $c=a$, i.e., $(\ell,a)\inC$.
By $\indiscernible{a}{b}{S}$, we also have $(\ell,b)\inC$.
Note that $(\ell,b),w\sim_\pi(\ell,a),\pi(w)$.
Note that $(\ell,b)\neq (\ell^-,b)$ by Fact \ref{fact:minusandconfignointersect}.
By assumption, $\textup{Succ}_\mathcal{B}((\ell,b),w)$ is non-accepting.
By Corollary \ref{corollary:gura_bad}, $\textup{Succ}_\mathcal{B}((\ell,a),\pi(w))$ is non-accepting, too.
\item Let $c\neq a$.
Note that also $(\ell^-,b),w \sim_\pi (\ell^-,a),\pi(w)$.
Recall that $\textup{Succ}_\mathcal{B}((\ell^-,b),w)$ is accepting.
By Corollary \ref{corollary:gnra_bad}, $\textup{Succ}_\mathcal{B}((\ell^-,a),\pi(w))$ is accepting.
We prove below that $\{(\ell^-,a),(\ell,c)\}$ is coverable. Proposition \ref{prop:gura_implied_badness} then implies that
$\textup{Succ}_\mathcal{B}((\ell,c),\pi(w))$ is non-accepting.
Recall that $\textup{data}((\{\ell^-\}\times\mathbb{D})\cap C)$ is cofinite.
Pick some datum $d\in\mathbb{D}\backslash\{c\}$ such that $(\ell^-,d)\inC$. We distinguish two cases.
\begin{itemize}
\item Assume
$(\ell,c)\inC\backslashC^+_b$.
Since $C$ is essentially coverable,
the set $\{(\ell^-,d), (\ell,c)\}$ is coverable.
Hence there must exist some data word $u$ such that $\{(\ell^-,d),(\ell,c)\}\subseteq \textup{Succ}_\mathcal{B}((\ell_\textup{in},\bot),u)$.
Let $\pi'$ be a partial isomorphism satisfying $\pi'(d)=a$, $\pi'(a)=d$, and $\pi'(e)=e$ for all $e\in\textup{data}(u)\cup\{c\}$.
Then $\{(\ell^-,a),(\ell,c)\}\subseteq \textup{Succ}_\mathcal{B}((\ell_\textup{in},\bot),\pi'(u))$, hence $\{(\ell^-,a),(\ell,c)\}$ is coverable.
\item Second suppose $(\ell,c)\inC^-_b$, i.e., $c=b$.
This implies that $\textup{data}(C \cap (\{\ell\}\times\mathbb{D}))$ is cofinite.
Pick some datum $e\in\mathbb{D}\backslash\{d\}$ such that $(\ell,e)\inC$.
Since $C$ is essentially coverable,
the set $\{(\ell^-,d),(\ell,e)\}$ is coverable.
Hence there must exist some data word $u$ such that $\{(\ell^-,d),(\ell,e)\}\subseteq\textup{Succ}_\mathcal{B}((\ell_\textup{in},\bot),u)$.
Let $\pi'$ be a partial isomorphism satisfying $\pi'(d)=a$, $\pi'(a)=d$, $\pi'(b)=e$, $\pi'(e)=b$, and $\pi'(f)=f$ for all $f\in\textup{data}(u)$. Then $\{(\ell,b),(\ell^-,a)\}\subseteq \textup{Succ}_\mathcal{B}((\ell_\textup{in},\bot),\pi'(u))$, hence $\{(\ell,c),(\ell^-,a)\}$ is coverable.
\end{itemize}
\end{enumerate}
Altogether we have proved that $\textup{Succ}_\mathcal{B}((C\backslashC^+_b)\cupC^-_b,\pi(w))$ is non-accepting, while there is an accepting run of $\mathcal{A}$ on $\pi(w)$ starting in $(\ell^\mathcal{A},\vect{d})$.
This finishes the proof for the $(\Rightarrow)$-direction, and thus the proof of the Proposition. \end{proof}
\subsection{The Algorithm} When a synchronized configuration $S'$ is obtained from some essentially coverable synchronized configuration $S=((\ell,\vect{d}),C)$ by applying Proposition~\ref{prop:collapse_gura} to two distinct data values $a,b\in\textup{supp}(C)$, we say that $S$ \emph{collapses to} $S'$. We say that $S$ is \emph{maximally collapsed} if one cannot find two distinct data values $a,b\in\textup{supp}(C)$ that satisfy the assumptions of Proposition~\ref{prop:collapse_gura}. Note that, by Proposition \ref{prop:esscov}, the synchronized configuration $S'$ in Proposition~\ref{prop:collapse_gura} is again essentially coverable. By iterating Proposition~\ref{prop:collapse_gura}, one obtains that an essentially coverable synchronized configuration reaches a bad synchronized configuration if, and only if, it collapses in finitely many steps to a maximally collapsed synchronized configuration that also reaches a bad synchronized configuration.
The number of maximally collapsed configurations is asymptotically bounded by $2^{k\log(k)2^{|\mathcal{L}|}}$. Indeed, a maximally collapsed configuration $((\ell,\vect d),C)$ can be recovered up to $\sim$ by: \begin{itemize}
\item The location $\ell$ and the equivalence class of $\vect d$,
\item A list $L_\bot,L_1,\dots,L_k$ of subsets of $\mathcal{L}$,
\item A set $\{L_{k+1},\dots,L_p\}$ of subsets of $\mathcal{L}$,
\item For each location $\ell\in\mathcal{L}$, a bit $b_\ell\in\{0,1\}$. \end{itemize} From this, one can constitute a configuration $S=((\ell,\vect d'), C)$ where: \begin{itemize}
\item $\vect d'$ is an arbitrary tuple in the equivalence class of $\vect d$, using only data from $\{\bot,1,\dots,k\}$,
\item For every $i\in\{\bot,1,\dots,k\}$ and $\ell'\in L_i$, $C$ contains $(\ell',d'_i)$,
\item For every $i\in\{k+1,\dots,p\}$ and $\ell'\in L_i$, $C$ contains $(\ell',i)$,
\item For each $d\in\mathbb{D}\setminus\{1,\dots,p\}$, $(\ell',d)$ is in $C$ iff $b_{\ell'}=1$. That is, the bit $b_{\ell'}$ is set to 1 to indicate that $(\{\ell'\}\times\mathbb{D}) \cap C$ is cofinite. \end{itemize}
Thus, one can bound the number of maximally covered configurations by $|\mathcal{L}|\times k^k \times (k+1)2^{|\mathcal{L}|} \times 2^{2^{|\mathcal{L}|}}\times 2^{|\mathcal{L}|}$
which is asymptotically $2^{k\log(k)2^{|\mathcal{L}|}}$. Consider the graph whose vertices are the maximally collapsed synchronized configurations and which contains an edge $S\leadstoS'$ iff there exists an $S''$ such that $S\sToS''$ and $S''$ collapses to $S'$. This graph has doubly-exponential size in $\mathcal{A}$ and $\mathcal{B}$, and the relation $\leadsto$ can be decided in polynomial space~\cite{DBLP:conf/stacs/MottetQ19}. Thus, one obtains that the reachability problem in this graph can be decided in exponential space, so that the containment problem for 1-register \textup{GURA}\ is in \EXPSPACE.
\begin{theorem} The containment problem $L(\mathcal{A})\subseteq L(\mathcal{B})$ is in $\textup{\EXPSPACE}$, if $\mathcal{A}$ is a $\textup{GRA}$ and $\mathcal{B}$ is an unambiguous $\textup{GRA}$ with a single register. \end{theorem}
\end{document} | arXiv |
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Bibliography: Undecidability in Philosophy of Mathematics
What is Absolute Undecidability?†.Justin Clarke-Doane - 2013 - Noûs 47 (3):467-481.details
It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability "absolute undecidability". In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) (...) if a mathematical hypothesis is absolutely undecidable, then it is indeterminate. I shall argue that on no understanding of absolute undecidability could one hope to establish all of (a)–(c). However, I will identify one understanding of absolute undecidability on which one might hope to establish both (a) and (c) to the exclusion of (b). This suggests that a new style of mathematical antirealism deserves attention—one that does not depend on familiar epistemological or ontological concerns. The key idea behind this view is that typical mathematical hypotheses are indeterminate because they are relevantly similar to CH. (shrink)
Axioms of Set Theory, Misc in Philosophy of Mathematics
Independence Results in Set Theory in Philosophy of Mathematics
Indeterminacy in Mathematics in Philosophy of Mathematics
Mathematical Proof in Philosophy of Mathematics
New Axioms in Set Theory in Philosophy of Mathematics
Objectivity Of Mathematics in Philosophy of Mathematics
The Continuum Hypothesis in Philosophy of Mathematics
Alethic Undecidability Doesn'T Solve the Liar.Mark Jago - 2016 - Analysis 76 (3):278-283.details
Stephen Barker presents a novel approach to solving semantic paradoxes, including the Liar and its variants and Curry's paradox. His approach is based around the concept of alethic undecidability. His approach, if successful, renders futile all attempts to assign semantic properties to the paradoxical sentences, whilst leaving classical logic fully intact. And, according to Barker, even the T-scheme remains valid, for validity is not undermined by undecidable instances. Barker's approach is innovative and worthy of further consideration, particularly by those (...) of us who aim to find a solution without logical revisionism. As it stands, however, the approach is unsuccessful, as I shall demonstrate below. (shrink)
Liar Paradox in Logic and Philosophy of Logic
Undecidability and the Problem of Outcomes in Quantum Measurements.Rodolfo Gambini, Luis Pedro García Pintos & Jorge Pullin - 2010 - Foundations of Physics 40 (1):93-115.details
We argue that it is fundamentally impossible to recover information about quantum superpositions when a quantum system has interacted with a sufficiently large number of degrees of freedom of the environment. This is due to the fact that gravity imposes fundamental limitations on how accurate measurements can be. This leads to the notion of undecidability: there is no way to tell, due to fundamental limitations, if a quantum system evolved unitarily or suffered wavefunction collapse. This in turn provides a (...) solution to the problem of outcomes in quantum measurement by providing a sharp criterion for defining when an event has taken place. We analyze in detail in examples two situations in which in principle one could recover information about quantum coherence: (a) "revivals" of coherence in the interaction of a system with the measurement apparatus and the environment and (b) the measurement of global observables of the system plus apparatus plus environment. We show in the examples that the fundamental limitations due to gravity and quantum mechanics in measurement prevent both revivals from occurring and the measurement of global observables. It can therefore be argued that the emerging picture provides a complete resolution to the measurement problem in quantum mechanics. (shrink)
Interpretation of Quantum Mechanics in Philosophy of Physical Science
Quantum Indeterminacy in Philosophy of Physical Science
Undecidability in the Imitation Game.Y. Sato & T. Ikegami - 2004 - Minds and Machines 14 (2):133-43.details
This paper considers undecidability in the imitation game, the so-called Turing Test. In the Turing Test, a human, a machine, and an interrogator are the players of the game. In our model of the Turing Test, the machine and the interrogator are formalized as Turing machines, allowing us to derive several impossibility results concerning the capabilities of the interrogator. The key issue is that the validity of the Turing test is not attributed to the capability of human or machine, (...) but rather to the capability of the interrogator. In particular, it is shown that no Turing machine can be a perfect interrogator. We also discuss meta-imitation game and imitation game with analog interfaces where both the imitator and the interrogator are mimicked by continuous dynamical systems. (shrink)
The Turing Test in Philosophy of Cognitive Science
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The Undecidability of the Spatialized Prisoner's Dilemma.Patrick Grim - 1997 - Theory and Decision 42 (1):53-80.details
In the spatialized Prisoner's Dilemma, players compete against their immediate neighbors and adopt a neighbor's strategy should it prove locally superior. Fields of strategies evolve in the manner of cellular automata (Nowak and May, 1993; Mar and St. Denis, 1993a,b; Grim 1995, 1996). Often a question arises as to what the eventual outcome of an initial spatial configuration of strategies will be: Will a single strategy prove triumphant in the sense of progressively conquering more and more territory without opposition, or (...) will an equilibrium of some small number of strategies emerge? Here it is shown, for finite configurations of Prisoner's Dilemma strategies embedded in a given infinite background, that such questions are formally undecidable: there is no algorithm or effective procedure which, given a specification of a finite configuration, will in all cases tell us whether that configuration will or will not result in progressive conquest by a single strategy when embedded in the given field. The proof introduces undecidability into decision theory in three steps: by (1) outlining a class of abstract machines with familiar undecidability results, by (2) modelling these machines within a particular family of cellular automata, carrying over undecidability results for these, and finally by (3) showing that spatial configurations of Prisoner's Dilemma strategies will take the form of such cellular automata. (shrink)
Prisoner's Dilemma in Philosophy of Action
Undecidability of the Real-Algebraic Structure of Scott's Model.Miklós Erdélyi-Szabó - 1998 - Mathematical Logic Quarterly 44 (3):344-348.details
We show that true first-order arithmetic of the positive integers is interpretable over the real-algebraic structure of Scott's topological model for intuitionistic analysis. From this the undecidability of the structure follows.
Areas of Mathematics in Philosophy of Mathematics
'Undecidability' or 'Anticipatory Resoluteness' Caputo in Conversation with Heidegger.Sylvie Avakian - 2015 - International Journal for Philosophy of Religion 77 (2):123-139.details
In this article I will consider John D. Caputo's 'radical hermeneutics', with 'undecidability' as its major theme, in conversation with Martin Heidegger's notion of 'anticipatory resoluteness'. Through an examination of the positions of Caputo and Heidegger I argue that Heidegger's notion of 'anticipatory resoluteness' reaches far beyond the claims of 'radical hermeneutics', and that it assumes a reconstructive process which carries within its scope the overtones of deconstruction, the experience of repetition and authenticity and also the implications of Gelassenheit. (...) Further, I am arguing that Caputo's 'radical hermeneutics' is problematic and even erroneous when it comes to criticize Heidegger's thought portraying it as being founded on 'the myth of the early Greeks'. (shrink)
Hans-Georg Gadamer in Continental Philosophy
Hermeneutics, Misc in Continental Philosophy
Paul Ricoeur in Continental Philosophy
Undecidability of the Real-Algebraic Structure of Models of Intuitionistic Elementary Analysis.Miklós Erdélyi-Szabó - 2000 - Journal of Symbolic Logic 65 (3):1014-1030.details
We show that true first-order arithmetic is interpretable over the real-algebraic structure of models of intuitionistic analysis built upon a certain class of complete Heyting algebras. From this the undecidability of the structures follows. We also show that Scott's model is equivalent to true second-order arithmetic. In the appendix we argue that undecidability on the language of ordered rings follows from intuitionistically plausible properties of the real numbers.
Intuitionistic Logic in Logic and Philosophy of Logic
Noncomputability, Unpredictability, Undecidability, and Unsolvability in Economic and Finance Theories.Ying-Fang Kao, V. Ragupathy, K. Vela Velupillai & Stefano Zambelli - 2013 - Complexity 18 (1):51-55.details
A Note on the Undecidability of the Reachability Problem for o‐Minimal Dynamical Systems.Thomas Brihaye - 2006 - Mathematical Logic Quarterly 52 (2):165-170.details
In this paper we prove that the reachability problem is BSS-undecidable for o-minimal dynamical systems.
Dynamical Systems in Philosophy of Cognitive Science
Undecidability and the Problem of Outcomes in Quantum Measurements.Rodolfo Gambini, Luis Pedro Garcia Pintos & Jorge Pullin - forthcoming - Foundations of Physics:93-115.details
We argue that it is fundamentally impossible to recover information about quantum superpositions when a quantum system has interacted with a sufficiently large number of degrees of freedom of the environment. This is due to the fact that gravity imposes fundamental limitations on how accurate measurements can be. This leads to the notion of undecidability: there is no way to tell, due to fundamental limitations, if a quantum system evolved unitarily or suffered wavefunction collapse. This in turn provides a (...) solution to the problem of outcomes in quantum measurement by providing a sharp criterion for defining when an event has taken place. We analyze in detail in examples two situations in which in principle one could recover information about quantum coherence: a) "revivals" of coherence in the interaction of a system with the measurement apparatus and the environment and b) the measurement of global observables of the system plus apparatus plus environment. We show in the examples that the fundamental limitations due to gravity and quantum mechanics in measurement prevent both revivals from occurring and the measurement of global observables. It can therefore be argued that the emerging picture provides a complete resolution to the measurement problem in quantum mechanics. (shrink)
Interpretations of Quantum Mechanics, Misc in Philosophy of Physical Science
On the Question of Absolute Undecidability.Peter Koellner - 2010 - In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Philosophia Mathematica. Association for Symbolic Logic. pp. 153-188.details
The paper begins with an examination of Gödel's views on absolute undecidability and related topics in set theory. These views are sharpened and assessed in light of recent developments. It is argued that a convincing case can be made for axioms that settle many of the questions undecided by the standard axioms and that in a precise sense the program for large cardinals is a complete success "below" CH. It is also argued that there are reasonable scenarios for settling (...) CH and that there is not currently a convincing case to the effect that a given statement is absolutely undecidable. (shrink)
Large Cardinals in Philosophy of Mathematics
The Axiom of Determinacy in Philosophy of Mathematics
$37.00 used $55.75 new $124.00 direct from Amazon (collection) Amazon page
Undecidability Without Arithmetization.Andrzej Grzegorczyk - 2005 - Studia Logica 79 (2):163-230.details
In the present paper the well-known Gödels – Churchs argument concerning the undecidability of logic (of the first order functional calculus) is exhibited in a way which seems to be philosophically interestingfi The natural numbers are not used. (Neither Chinese Theorem nor other specifically mathematical tricks are applied.) Only elementary logic and very simple set-theoretical constructions are put into the proof. Instead of the arithmetization I use the theory of concatenation (formalized by Alfred Tarski). This theory proves to be (...) an appropriate tool. The decidability is defined directly as the property of graphical discernibility of formulas. (shrink)
Logics in Logic and Philosophy of Logic
Polish Philosophy in European Philosophy
Algorithmic Information Theory and Undecidability.Panu Raatikainen - 2000 - Synthese 123 (2):217-225.details
Chaitin's incompleteness result related to random reals and the halting probability has been advertised as the ultimate and the strongest possible version of the incompleteness and undecidability theorems. It is argued that such claims are exaggerations.
Information Theory in Philosophy of Computing and Information
Logic and Philosophy of Logic, Misc in Logic and Philosophy of Logic
Undecidability Results on Two-Variable Logics.Erich Grädel, Martin Otto & Eric Rosen - 1999 - Archive for Mathematical Logic 38 (4-5):313-354.details
It is a classical result of Mortimer that $L^2$ , first-order logic with two variables, is decidable for satisfiability. We show that going beyond $L^2$ by adding any one of the following leads to an undecidable logic:– very weak forms of recursion, viz.¶(i) transitive closure operations¶(ii) (restricted) monadic fixed-point operations¶– weak access to cardinalities, through the Härtig (or equicardinality) quantifier¶– a choice construct known as Hilbert's $\epsilon$ -operator.In fact all these extensions of $L^2$ prove to be undecidable both for satisfiability, (...) and for satisfiability in finite structures. Moreover most of them are hard for $\Sigma^1_1$ , the first level of the analytical hierachy, and thus have a much higher degree of undecidability than first-order logic. (shrink)
The Undecidability of K-Provability.Samuel R. Buss - 1991 - Annals of Pure and Applied Logic 53 (1):75-102.details
Buss, S.R., The undecidability of k-provability, Annals of Pure and Applied Logic 53 75-102. The k-provability problem is, given a first-order formula ø and an integer k, to determine if ø has a proof consisting of k or fewer lines . This paper shows that the k-provability problem for the sequent calculus is undecidable. Indeed, for every r.e. set X there is a formula ø and an integer k such that for all n,ø has a proof of k sequents (...) if and only if n ε X. (shrink)
Proof Theory in Logic and Philosophy of Logic
Undecidability and 1-Types in the Recursively Enumerable Degrees.Klaus Ambos-Spies & Richard A. Shore - 1993 - Annals of Pure and Applied Logic 63 (1):3-37.details
Ambos-Spies, K. and R.A. Shore, Undecidability and 1-types in the recursively enumerable degrees, Annals of Pure and Applied Logic 63 3–37. We show that the theory of the partial ordering of recursively enumerable Turing degrees is undecidable and has uncountably many 1-types. In contrast to the original proof of the former which used a very complicated O''' argument our proof proceeds by a much simpler infinite injury argument. Moreover, it combines with the permitting technique to get similar results for (...) any ideal of the r.e. degrees. (shrink)
Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic
Model Theory in Logic and Philosophy of Logic
Habituality and Undecidability: A Comparison of Merleau-Ponty and Derrida on the Decision.Jack Reynolds - 2002 - International Journal of Philosophical Studies 10 (4):449 – 466.details
This essay examines the relationship that obtains between Merleau-Ponty and Derrida through exploring an interesting point of dissension in their respective accounts of decision-making. Merleau-Ponty's early philosophy emphasizes the body-subject's tendency to seek an equilibrium with the world (by acquiring skills and establishing what he refers to as 'intentional arcs'), and towards deciding in an embodied and habitual manner that minimizes any confrontation with what might be termed a decision-making aporia. On the other hand, in his later writings, Derrida frequently (...) points towards a constitutive 'undecidability' involved in decision-making. He insists that a decision, if it is genuinely to be a decision, must involve a leap beyond all prior preparations, and this ensures that an aporia surrounds any attempt to decide. One must always decide without any equilibrium or stability, and yet these are precisely the things that Merleau-Ponty claims that our body moves us towards. Most of this essay will explore the significance of this disparity, and it will be argued that many of Merleau-Ponty's insights challenge the Derridean conception of the undecidability involved in decision-making. This becomes most obvious when comparing the decision-making processes of those expert in a particular field to those who are merely competent (for example chess), and this essay will attempt to establish that the aporia that Derrida discerns can actually be seen to constrict. (shrink)
Derrida: Ethics in Continental Philosophy
Maurice Merleau-Ponty in Continental Philosophy
The Undecidability of the First-Order Theory of Diagonalizable Algebras.Franco Montagna - 1980 - Studia Logica 39 (4):355 - 359.details
The undecidability of the first-order theory of diagonalizable algebras is shown here.
A Communicative Constitutive Perspective on Corporate Social Responsibility: Ventriloquism, Undecidability, and Surprisability.François Cooren - 2020 - Business and Society 59 (1):175-197.details
Adopting a communication as constitutive of organization perspective on ethics and corporate social responsibility invites us to create the conditions of a dialogue, discussion, or debate between various stakeholders, who can then try to confront their respective positions on a given issue, and possibly come to a decision regarding how a situation should be evaluated and/or responded to. As shown in this article, getting human stakeholders to voice their concerns about a specific situation is a way not only to rationally (...) confront multiple viewpoints on what should be done about it but also to allow elements of this situation to reveal themselves in a discussion. This is why any ethical decision has to experience a form of undecidability and surprisability. When beliefs and opinions can be called into question, it means that participants are ready to become attentive to what a situation might surprisingly call for. Communication is therefore a matter of ventriloquism, as human participants should also be considered the means by which elements of a situation manage to express themselves in a discussion, dictating specific courses of action over others. This concretely means that multiple voices can be heard in such a scene: not only the voices of the human stakeholders who speak to each other but also the voices of facts, principles, future generations, ecosystems, populations that are made to say things through these various turns of talk. (shrink)
Elementary Theories and Hereditary Undecidability for Semilattices of Numberings.Nikolay Bazhenov, Manat Mustafa & Mars Yamaleev - 2019 - Archive for Mathematical Logic 58 (3-4):485-500.details
A major theme in the study of degree structures of all types has been the question of the decidability or undecidability of their first order theories. This is a natural and fundamental question that is an important goal in the analysis of these structures. In this paper, we study decidability for theories of upper semilattices that arise from the theory of numberings. We use the following approach: given a level of complexity, say \, we consider the upper semilattice \ (...) of all \-computable numberings of all \-computable families of subsets of \. We prove that the theory of the semilattice of all computable numberings is computably isomorphic to first order arithmetic. We show that the theory of the semilattice of all numberings is computably isomorphic to second order arithmetic. We also obtain a lower bound for the 1-degree of the theory of the semilattice of all \-computable numberings, where \ is a computable successor ordinal. Furthermore, it is shown that for any of the theories T mentioned above, the \-fragment of T is hereditarily undecidable. Similar results are obtained for the structure of all computably enumerable equivalence relations on \, equipped with composition. (shrink)
Unity and Undecidability: The Subject of Kant's First Critique.Stuart Dalton - 1998 - Philosophy in the Contemporary World 5 (4):25-32.details
This essay argues that, in the first Critique, the need for unity leads Kant to re-inscribe the subject in a situation of multiplicity and undecidability. The result, however, is not a relativization that negates the meaning of the subject's existence, but rather a contextualization that makes meaning possible. This reading clarifies some of the connections between Kant and contemporary postmodernism, especially the work of Jacques Derrida.
Derrida: Value Theory in Continental Philosophy
Jacques Derrida in Continental Philosophy
Undecidability of the Equational Theory of Some Classes of Residuated Boolean Algebras with Operators.I. Nemeti, I. Sain & A. Simon - 1995 - Logic Journal of the IGPL 3 (1):93-105.details
We show the undecidability of the equational theories of some classes of BAOs with a non-associative, residuated binary extra-Boolean operator. These results solve problems in Jipsen [9], Pratt [21] and Roorda [22], [23]. This paper complements Andréka-Kurucz-Németi-Sain-Simon [3] where the emphasis is on BAOs with an associative binary operator.
The Undecidability of Quantified Announcements.T. Ågotnes, H. van Ditmarsch & T. French - 2016 - Studia Logica 104 (4):597-640.details
This paper demonstrates the undecidability of a number of logics with quantification over public announcements: arbitrary public announcement logic, group announcement logic, and coalition announcement logic. In APAL we consider the informative consequences of any announcement, in GAL we consider the informative consequences of a group of agents all of which are simultaneously making known announcements. So this is more restrictive than APAL. Finally, CAL is as GAL except that we now quantify over anything the agents not in that (...) group may announce simultaneously as well. The logic CAL therefore has some features of game logic and of ATL. We show that when there are multiple agents in the language, the satisfiability problem is undecidable for APAL, GAL, and CAL. In the single agent case, the satisfiability problem is decidable for all three logics. (shrink)
Doxastic and Epistemic Logic in Logic and Philosophy of Logic
Modal and Intensional Logic in Logic and Philosophy of Logic
The Undecidability of Quantified Announcements.T. French, H. Ditmarsch & T. Ågotnes - 2016 - Studia Logica 104 (4):597-640.details
The Undecidability of the Lattice of R.E. Closed Subsets of an Effective Topological Space.Sheryl Silibovsky Brady & Jeffrey B. Remmel - 1987 - Annals of Pure and Applied Logic 35 (2):193-203.details
The first-order theory of the lattice of recursively enumerable closed subsets of an effective topological space is proved undecidable using the undecidability of the first-order theory of the lattice of recursively enumerable sets. In particular, the first-order theory of the lattice of recursively enumerable closed subsets of Euclidean n -space, for all n , is undecidable. A more direct proof of the undecidability of the lattice of recursively enumerable closed subsets of Euclidean n -space, n ⩾ 2, is (...) provided using the method of reduction and the recursive inseparability of the set of all formulae satisfiable in every model of the theory of SIBs and the set of all formulae refutable in some finite model of the theory of SIBs. (shrink)
In this article I will consider John D. Caputo's hermeneutics of deconstruction or what he calls 'radical hermeneutics', with 'undecidability' as its major theme, in conversation with Martin Heidegger's notion of 'resolute existence'. Through an examination of the different positions of Caputo, Heidegger, and also Kierkegaard, Derrida and Meister Eckhart on the possibility of repetition, the hermeneutical circle and the mystical way of prayer and faith, I am arguing that deconstruction is not the end of hermeneutics, it is not (...) the final destination of an interpretative task, and thus deconstructive hermeneutics has to concede a reconstructive process. Further, I am arguing that Caputo's 'radical hermeneutics' is too reductionist to keep any meaning for the hermeneutic enterprise to aspire to. I am contending, rather, that it is resolute existence that theology aspires to as it is a move beyond 'undecidability'. (shrink)
The Undecidability of the DA-Unification Problem.J. Siekmann & P. Szabó - 1989 - Journal of Symbolic Logic 54 (2):402 - 414.details
We show that the D A -unification problem is undecidable. That is, given two binary function symbols $\bigoplus$ and $\bigotimes$ , variables and constants, it is undecidable if two terms built from these symbols can be unified provided the following D A -axioms hold: \begin{align*}(x \bigoplus y) \bigotimes z &= (x \bigotimes z) \bigoplus (y \bigotimes z),\\x \bigotimes (y \bigoplus z) &= (x \bigotimes y) \bigoplus (x \bigotimes z),\\x \bigoplus (y \bigoplus z) &= (x \bigoplus y) \bigoplus z.\end{align*} Two terms (...) are D A -unifiable (i.e. an equation is solvable in D A ) if there exist terms to be substituted for their variables such that the resulting terms are equal in the equational theory D A . This is the smallest currently known axiomatic subset of Hilbert's tenth problem for which an undecidability result has been obtained. (shrink)
An Undecidability Theorem for Lattices Over Group Rings.Carlo Toffalori - 1997 - Annals of Pure and Applied Logic 88 (2-3):241-262.details
Let G be a finite group, T denote the theory of Z[G]-lattices . It is shown that T is undecidable when there are a prime p and a p-subgroup S of G such that S is cyclic of order p4, or p is odd and S is non-cyclic of order p2, or p = 2 and S is a non-cyclic abelian group of order 8 . More precisely, first we prove that T is undecidable because it interprets the word problem (...) for finite groups; then we lift undecidability from T to T. (shrink)
Undecidability in the Spatialized Prisoner's Dilemma: Some Philosophical Implications.Patrick Grim - manuscriptdetails
A version of this paper was presented at the IEEE International Conference on Computational Intelligence, combined meeting of ICNN, FUZZ-IEEE, and ICEC, Orlando, June-July, 1994, and an earlier form of the result is to appear as "The Undecidability of the Spatialized Prisoner's Dilemma" in Theory and Decision . An interactive form of the paper, in which figures are called up as evolving arrays of cellular automata, is available on DOS disk as Research Report #94-04i . An expanded version appears (...) as chapter 6 of The Philosophical Computer. (shrink)
Undecidability of Representability as Binary Relations.Robin Hirsch & Marcel Jackson - 2012 - Journal of Symbolic Logic 77 (4):1211-1244.details
In this article we establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for Jónsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions.
Nonclassical Logics in Logic and Philosophy of Logic
Undecidability Reconsidered.Timm Lampert - 2007 - In A. Costa-Leite J. Y. Bezieau (ed.), Dimensions of Logical Concepts. pp. 33-68.details
In vol. 2 of Grundlagen der Mathematik Hilbert and Bernays carry out their undecid- ability proof of predicate logic basing it on their undecidability proof of the arithmeti- cal systemZ00. In this paper, the latter proof is reconstructed and summarized within a formal derivation schema. Formalizing the proof makes the presumed use of a meta language explicit by employing formal predicates as propositional functions, with ex- pressions as their arguments. In the final section of the paper, the proof is (...) analyzed critically by applying Wittgenstein's view on meta language, which does not lead to a questioning of the assumptions on which the proof is based but of its presumed use of meta language. Finally, it will be argued that determining if it is the undecidability proof or Wittgenstein's analysis of meta language that is right depends on whether a decision procedure for a predicate that undecidability proofs have seemingly proven undecidable can nonetheless be defined. Thus serious attempts to define a procedure for such a predicate should not be ruled out without studying them. (shrink)
The Undecidability of the $Mathrm{D}_mathrm{A}$-Unification Problem.J. Siekmann & P. Szabo - 1989 - Journal of Symbolic Logic 54 (2):402-414.details
We show that the $\mathrm{D_A}$-unification problem is undecidable. That is, given two binary function symbols $\bigoplus$ and $\bigotimes$, variables and constants, it is undecidable if two terms built from these symbols can be unified provided the following $\mathrm{D_A}$-axioms hold: \begin{align*}(x \bigoplus y) \bigotimes z &= (x \bigotimes z) \bigoplus (y \bigotimes z),\\x \bigotimes (y \bigoplus z) &= (x \bigotimes y) \bigoplus (x \bigotimes z),\\x \bigoplus (y \bigoplus z) &= (x \bigoplus y) \bigoplus z.\end{align*} Two terms are $\mathrm{D_A}$-unifiable (i.e. an equation (...) is solvable in $\mathrm{D_A}$) if there exist terms to be substituted for their variables such that the resulting terms are equal in the equational theory $\mathrm{D_A}$. This is the smallest currently known axiomatic subset of Hilbert's tenth problem for which an undecidability result has been obtained. (shrink)
21 Undecidability and Intractability in Theoretical Physics.Stephen Wolfram - 2013 - Emergence: Contemporary Readings in Philosophy and Science.details
This chapter explores some fundamental consequences of the correspondence between physical process and computations. Most physical questions may be answerable only through irreducible amounts of computation. Those that concern idealized limits of infinite time, volume, or numerical precision can require arbitrarily long computations, and so be considered formally undecidable. The behavior of a physical system may always be calculated by simulating explicitly each step in its evolution. Much of theoretical physics has, however, been concerned with devising shorter methods of calculation (...) that reproduce the outcome without tracing each step. Computational irreducibility is common among the systems investigated in mathematics and computation theory, but it may well be the exception rather than the rule, since most physical questions may be answerable only through irreducible amounts of computation. (shrink)
Computationalism in Philosophy of Cognitive Science
The Undecidability of Iterated Modal Relativization.Joseph S. Miller & Lawrence S. Moss - 2005 - Studia Logica 79 (3):373-407.details
In dynamic epistemic logic and other fields, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the transitive closure operation, due to its connection to common knowledge. We show that for three fragments of the logic of iterated relativization and transitive closure, the satisfiability problems are fi1 11–complete. Two of these fragments (...) do not include transitive closure. We also show that the question of whether a sentence in these fragments has a finite (tree) model is fi0 01–complete. These results go via reduction to problems concerning domino systems. (shrink)
Semantic Paradox and Alethic Undecidability.Stephen Barker - 2014 - Analysis 74 (2):201-209.details
I use the principle of truth-maker maximalism to provide a new solution to the semantic paradoxes. According to the solution, AUS, its undecidable whether paradoxical sentences are grounded or ungrounded. From this it follows that their alethic status is undecidable. We cannot assert, in principle, whether paradoxical sentences are true, false, either true or false, neither true nor false, both true and false, and so on. AUS involves no ad hoc modification of logic, denial of the T-schema's validity, or obvious (...) revenge. (shrink)
Indeterminacy, Misc in Philosophy of Language
Truth, Misc in Philosophy of Language
Truthmakers in Metaphysics
The Undecidability of Propositional Adaptive Logic.Leon Horsten & Philip Welch - 2007 - Synthese 158 (1):41-60.details
We investigate and classify the notion of final derivability of two basic inconsistency-adaptive logics. Specifically, the maximal complexity of the set of final consequences of decidable sets of premises formulated in the language of propositional logic is described. Our results show that taking the consequences of a decidable propositional theory is a complicated operation. The set of final consequences according to either the Reliability Calculus or the Minimal Abnormality Calculus of a decidable propositional premise set is in general undecidable, and (...) can be -complete. These classifications are exact. For first order theories even finite sets of premises can generate such consequence sets in either calculus. (shrink)
The Undecidability of Entailment and Relevant Implication.Alasdair Urquhart - 1984 - Journal of Symbolic Logic 49 (4):1059-1073.details
Relevance Logic in Logic and Philosophy of Logic
The Undecidability of Monadic Modal Quantification Theory.Saul A. Kripke - 1962 - Mathematical Logic Quarterly 8 (2):113-116.details
The Undecidability of Grisin's Set Theory.Andrea Cantini - 2003 - Studia Logica 74 (3):345 - 368.details
We investigate a contractionless naive set theory, due to Grisin [11]. We prove that the theory is undecidable.
The Undecidability of Grisin's Set Theory.Andrea Cantini - 2003 - Studia Logica 74 (3):345-368.details
We investigate a contractionless naive set theory, due to Gris̆in [11]. We prove that the theory is undecidable.
Undecidability of the Problem of Recognizing Axiomatizations for Propositional Calculi with Implication.G. V. Bokov - 2015 - Logic Journal of the IGPL 23 (2):341-353.details
Kripke Semantics, Undecidability and Standard Completeness for Esteva and Godo's Logic MTL∀.Franco Montagna & Hiroakira Ono - 2002 - Studia Logica 71 (2):227-245.details
The present paper deals with the predicate version MTL of the logic MTL by Esteva and Godo. We introduce a Kripke semantics for it, along the lines of Ono''s Kripke semantics for the predicate version of FLew (cf. [O85]), and we prove a completeness theorem. Then we prove that every predicate logic between MTL and classical predicate logic is undecidable. Finally, we prove that MTL is complete with respect to the standard semantics, i.e., with respect to Kripke frames on the (...) real interval [0,1], or equivalently, with respect to MTL-algebras whose lattice reduct is [0,1] with the usual order. (shrink)
Semantics for Modal Logic in Logic and Philosophy of Logic
The Undecidability of the Disjunction Property of Propositional Logics and Other Related Problems.Alexander Chagrov & Michael Zakharyaschev - 1993 - Journal of Symbolic Logic 58 (3):967-1002.details
Undecidability in Diagonalizable Algebras.V. Yu Shavrukov - 1997 - Journal of Symbolic Logic 62 (1):79-116.details
If a formal theory T is able to reason about its own syntax, then the diagonalizable algebra of T is defined as its Lindenbaum sentence algebra endowed with a unary operator □ which sends a sentence φ to the sentence □φ asserting the provability of φ in T. We prove that the first order theories of diagonalizable algebras of a wide class of theories are undecidable and establish some related results.
Gregory Cherlin, Lou van den Dries, and Angus Macintyre. Decidability and Undecidability Theorems for PAC-Fields. Bulletin of the American Mathematical Society, N.S. Vol. 4 , Pp. 101–104. [REVIEW]A. Prestel - 1987 - Journal of Symbolic Logic 52 (2):568.details
Undecidability of the Problem of Recognizing Axiomatizations of Superintuitionistic Propositional Calculi.Evgeny Zolin - 2014 - Studia Logica 102 (5):1021-1039.details
We give a new proof of the following result : it is undecidable whether a given calculus, that is a finite set of propositional formulas together with the rules of modus ponens and substitution, axiomatizes the classical logic. Moreover, we prove the same for every superintuitionistic calculus. As a corollary, it is undecidable whether a given calculus is consistent, whether it is superintuitionistic, whether two given calculi have the same theorems, whether a given formula is derivable in a given calculus. (...) The proof is by reduction from the undecidable halting problem for the so-called tag systems introduced by Post. We also give a historical survey of related results. (shrink)
Undecidability and Opacity of Metacognition in Animals and Humans.Kevin B. Clark & Derrick L. Hassert - 2013 - Frontiers in Psychology 4.details
Haskell B. Curry. The Undecidability of λK-Conversion. Foundations of Mathematics, Symposium Papers Commemorating the Sixtieth Birthday of Kurt Gödel, Edited by Jack J. Bulloff, Thomas C. Holyoke, and S. W. Hahn, Springer-Verlag, Berlin, Heidelberg, and New York, 1969, Pp. 10–14. [REVIEW]Richard J. Orgass - 1975 - Journal of Symbolic Logic 40 (2):246.details
L. HORSTEN and P. WELCH/The Undecidability of Propositional Adaptive Logic 41.J. Hughes, P. Kroes & S. Zwart - 2007 - Synthese 158 (1):158.details
1 — 50 / 348 | CommonCrawl |
\begin{document}
\title{A remark on Greenberg's generalized conjecture for imaginary $S_3$-extensions of $\mathbb{Q}$} \footnote[0]{2020 Mathematics Subject Classification. 11R23} \footnote[0]{Key Words : Greenberg's generalized conjecture, $\mathbb{Z}_p^{\oplus d}$-extensions.} \author{Tsuyoshi Itoh}
\begin{abstract} Let $K/ \mathbb{Q}$ be an imaginary $S_3$-extension, and $p$ a prime number which splits into exactly three primes in $K$. We give a sufficient condition for the validity of Greenberg's generalized conjecture for $K$ and $p$. \end{abstract}
\maketitle
\section{Introduction}\label{Introduction}
Let $p$ be a prime number. We recall the statement of Greenberg's generalized conjecture (GGC) for an algebraic number field $K_\circ$ and $p$. We denote by $\widetilde{K_\circ}$ the composite of all $\mathbb{Z}_p$-extensions of $K_\circ$. Then, the Galois group $\Gal (\widetilde{K_\circ}/ K_\circ)$ is topologically isomorphic to $\mathbb{Z}_p^{\oplus d}$ with some positive integer $d$. Let $L (\widetilde{K_\circ})/ \widetilde{K_\circ}$ be the maximal unramified abelian pro-$p$ extension. We put $X (\widetilde{K_\circ}) = \Gal (L (\widetilde{K_\circ})/ \widetilde{K_\circ})$. We denote by $\Lambda_{\Gal (\widetilde{K_\circ}/ K_\circ)}$ the completed group ring $\mathbb{Z}_p [[ \Gal (\widetilde{K_\circ}/ K_\circ)]]$. It is well known that $X (\widetilde{K_\circ})$ is a finitely generated torsion module over $\Lambda_{\Gal (\widetilde{K_\circ}/ K_\circ)}$ (see \cite[Theorem 1]{Gre73}).
\begin{GGC}[{\cite[Conjecture (3.5)]{Gre01}}] $X (\widetilde{K_\circ})$ is a pseudo-null $\Lambda_{\Gal (\widetilde{K_\circ}/K_\circ)}$-module. That is, there are two relatively prime elements of $\Lambda_{\Gal (\widetilde{K_\circ}/K_\circ)}$ such that both annihilate $X (\widetilde{K_\circ})$. \end{GGC}
Minardi \cite{Min} studied GGC, mainly for imaginary quadratic fields. After that, many authors gave sufficient conditions for the validity of GGC in various situations. For example, see the author \cite{I2011}, Fujii \cite{Fujii}, Kleine \cite{Kle}, Kataoka \cite{Kata}, Takahashi \cite{Taka}. In \cite[Remark 1.3]{Taka}, several known results are stated in detail. See also Assim-Boughadi \cite{A-B} and the results referred there. We also mention Murakami \cite{Mura} as a recent result.
In the present paper, we consider GGC for the following $K$ and $p$. Let $K/\mathbb{Q}$ be an imaginary Galois extension whose Galois group is isomorphic to the symmetric group $S_3$ of degree $3$. We assume that $p$ splits into three primes $\mathfrak{P}_1$, $\mathfrak{P}_2$, $\mathfrak{P}_3$ in $K/\mathbb{Q}$.
There is a unique imaginary quadratic field $k$ contained in $K$. In our situation, $p$ is not decomposed in $k$, and the unique prime of $k$ lying above $p$ is completely decomposed in $K$. For each $i \in \{ 1,2,3 \}$, let $K_{\mathfrak{P_i}}$ be the completion of $K$ at $\mathfrak{P}_i$. Then $[ K_{\mathfrak{P}_i} : \mathbb{Q}_p ] =2$ for every $i$.
The decomposition field $F$ of $K/\mathbb{Q}$ for $\mathfrak{P}_1$ is a complex cubic field. We denote by $\mathfrak{p}$ the prime of $F$ lying below $\mathfrak{P}_1$. There is just one more prime $\mathfrak{p}^*$ of $F$ lying above $p$, which splits into two primes $\mathfrak{P}_2$, $\mathfrak{P}_3$ in $K/F$. We note that $[ F_{\mathfrak{p}} : \mathbb{Q}_p ] =1$ and $[ F_{\mathfrak{p}^*} : \mathbb{Q}_p ] =2$, where $F_{\mathfrak{p}}$ (resp.~$F_{\mathfrak{p}^*}$) is the completion of $F$ at $\mathfrak{p}$ (resp.~$\mathfrak{p}^*$). It is known that there exists a unique $\mathbb{Z}_p$-extension $N^* /F$ unramified outside $\mathfrak{p}^*$ (see \cite[Lemma 3.4 (2)]{Kata}).
Our main result is the following:
\begin{thm}\label{main_thm} Let the notation be as above. Moreover, let $M_{\mathfrak{P}_3} (K)/K$ be the maximal abelian pro-$p$ extension unramified outside $\mathfrak{P}_3$. Assume that all of the following conditions are satisfied:\\ (C1) $\mathfrak{p}$ is finitely decomposed in $N^*$, \\ (C2) $\mathfrak{p}^*$ is not decomposed in $N^*$, and \\ (C3) $\mathfrak{P}_2$ is not decomposed in $M_{\mathfrak{P}_3} (K)$.\\ Then, GGC for $K$ and $p$ holds. \end{thm}
\begin{rem}\label{rem_C1} $M_{\mathfrak{P}_3} (K)/K$ is known to be a finite extension (see Section \ref{Preliminaries}). The condition (C1) is equivalent to \cite[Assumption 3.1]{Kata} (when the complex cubic field has two primes dividing $p$). It seems unknown whether (C1) always holds or not (see also \cite[Remark 3.2 (2)]{Kata}, \cite[Sections 7.2 and 7.3]{Hachi}). However, (C1) was confirmed to be satisfied for many cases (see Remark \ref{rem_C1_2}). \end{rem}
\begin{rem} Kataoka \cite{Kata} gave sufficient conditions for the validity of GGC for complex cubic fields (he treated all cases of decomposition on $p$). Compare our Theorem \ref{main_thm} with \cite[Theorem 3.3 (1), (2)]{Kata}. In particular, by using \cite[Theorem 3.3 (2)]{Kata}, we see that GGC for (our) $F$ and $p$ also holds under the assumptions (C1), (C2), (C3). To show \cite[Theorem 3.3 (2)]{Kata}, Kataoka used $N^*/F$ as the first step of his proof (\cite[Proposition 3.5 (2)]{Kata}). We use $N^* K/K$ as the first step. However, $\widetilde{F}$ is not used in our proof. \end{rem}
\begin{rem}\label{rem_contained} Assume that $p=3$. When $K$ is contained in $\widetilde{k}$, the validity of GGC for $k$ and $p$ implies the validity of GGC for $K$ and $p$. See \cite[p.33, Remarks (ii)]{Min}. However, in our case, if $K$ is contained in $\widetilde{k}$, then $K/k$ is an unramified extension, hence the validity of GGC for $k$ and $p$ seems non-trivial (see, e.g., \cite[Section 3.D]{Min}). Incidentally, for the case where $p=2$, we see that $K$ is never contained in $\widetilde{F}$ because the real archimedean prime of $F$ ramifies in $K$. For a somewhat related result, see also \cite[Corollary 4.4]{Kle}. \end{rem}
We give several preparations in Sections \ref{Preliminaries}. We shall show Theorem \ref{main_thm} in Section \ref{Proof_main_thm}. We will give examples in Section \ref{Examples}.
\section{Preliminaries}\label{Preliminaries}
First, we define several notation and recall well known facts. We denote by $|A|$ the cardinality of a set $A$. We also denote by $\zprank B$ the $\mathbb{Z}_p$-rank of a finitely generated $\mathbb{Z}_p$-module $B$. For a pro-$p$ group $G$ which is topologically isomorphic to $\mathbb{Z}_p^{\oplus d}$ with some positive integer $d$, let $\Lambda_G$ be the the completed group ring $\mathbb{Z}_p [[ G ]]$. We use the notation defined in Section \ref{Introduction}.
In this paragraph, we denote by $\mathcal{K}$ an algebraic extension of $K$. Let $L (\mathcal{K})/ \mathcal{K}$ be the maximal unramified abelian pro-$p$ extension, and put $X (\mathcal{K}) = \Gal (L (\mathcal{K})/ \mathcal{K})$. We denote by $S_p$ the set $\{ \mathfrak{P}_1, \mathfrak{P}_2, \mathfrak{P}_3\}$. For a non-empty subset $S$ of $S_p$, let $M_S (\mathcal{K})/ \mathcal{K}$ be the maximal abelian pro-$p$ extension unramified outside $S$, and put $\mathfrak{X}_S (\mathcal{K}) = \Gal (M_S (\mathcal{K})/ \mathbb{K})$. For the case where $S = \{ \mathfrak{P}_i \}$, we shall write them $M_{\mathfrak{P}_i} (\mathcal{K})$, $\mathfrak{X}_{\mathfrak{P}_i} (\mathcal{K})$ in short.
For $i \in \{ 1,2,3 \}$, let $\mathcal{U}_i$ be the group of principal units in $K_{\mathfrak{P}_i}$, and put $\mathcal{U} = \bigoplus_{i=1}^3 \mathcal{U}_i$. Let $E_K$ be the group of units in $K$, and put
\[ E'_K = \{ \varepsilon \in E_K \; | \; \text{$\varepsilon \equiv 1 \pmod{\mathfrak{P}_i}$ for every $i \in \{ 1, 2, 3 \}$} \}. \] We denote by $\mathcal{E}$ the image of the mapping $E'_K \otimes_{\mathbb{Z}} \mathbb{Z}_p \to \mathcal{U}$ induced from the diagonal embedding \[ E'_K \to \mathcal{U}= \mathcal{U}_1 \oplus \mathcal{U}_2 \oplus \mathcal{U}_3, \quad \varepsilon \mapsto (\varepsilon, \varepsilon, \varepsilon) \] (see, e.g., \cite[Appendix]{IwaU4}). By class field theory, $\mathcal{U}/ \mathcal{E}$ is isomorphic to $\Gal (M_{S_p} (K)/L(K))$. It is known that Leopoldt's conjecture holds for $K$ and $p$ since $K$ is an abelian extension of an imaginary quadratic field (see \cite{Bru}). Then, $\zprank \mathcal{E}$ is $2$, which is equal to the free rank of $E_K$. Since $\zprank \mathcal{U}$ is $6$, we see that $\zprank \Gal (M_{S_p} (K)/L(K))$ is $4$. Hence, $\widetilde{K}/K$ is a $\mathbb{Z}_p^{\oplus 4}$-extension.
Next, we shall state several results which will be used in the proof of Theorem \ref{main_thm}. Recall that $N^*/F$ is the $\mathbb{Z}_p$-extension unramified outside $\mathfrak{p}^*$. We put $N^{(1)} = N^* K$, which is a $\mathbb{Z}_p$-extension of $K$. The following result is crucial to prove Theorem \ref{main_thm}.
\begin{thmA}[Maire \cite{Mai}] For a non-negative integer $n$, let $N^{(1)}_n$ be the $n$th layer of $N^{(1)}/K$. Then for every $n$, $\mathfrak{X}_{\mathfrak{P}_1} (N^{(1)}_n)$ is finite. In particular, $\mathfrak{X}_{\mathfrak{P}_1} (K)$ is finite. \end{thmA}
\begin{proof} We recall the following facts: $N^{(1)}_n / F$ is an abelian extension, $\mathfrak{P}_1$ is the unique prime of $K$ lying above $\mathfrak{p}$, $[ F_{\mathfrak{p}} : \mathbb{Q}_p ]=1$, and $[F : \mathbb{Q}] =3$. Then we can apply \cite[Theorem 25]{Mai}, and the assertion follows. \end{proof}
We can also see that both $\mathfrak{X}_{\mathfrak{P}_2} (K)$ and $\mathfrak{X}_{\mathfrak{P}_3} (K)$ are finite.
We note that Hachimori \cite{Hachi} treated similar situation to ours. In particular, it seems that he essentially showed a result similar to Theorem A for the cyclotomic $\mathbb{Z}_p$-extension of $K$ (see the proofs of \cite[Theorems 6.2 and 7.3]{Hachi}).
\begin{cor}\label{rank_Zp2_ext} We put $S = \{ \mathfrak{P}_i, \mathfrak{P}_j \}$ with $1 \leq i < j \leq 3$. Then $\zprank \mathfrak{X}_S (K)$ is $2$. \end{cor}
\begin{proof} Since $K/\mathbb{Q}$ is a Galois extension, it is sufficient to show only for the case where $S = \{ \mathfrak{P}_1, \mathfrak{P}_2 \}$. This corollary follows from the basic result on abelian pro-$p$ extensions with restricted ramification (see, e.g., \cite[Theorem 5]{Mai}, \cite[Proposition 3.1]{Hachi}). Theorem A (for $K$) asserts that the image of $E'_K \otimes_{\mathbb{Z}} \mathbb{Z}_p \to \mathcal{U}_1$ has $\mathbb{Z}_p$-rank $2$. Then the image of the mapping \[ E'_K \otimes_{\mathbb{Z}} \mathbb{Z}_p \to \mathcal{U}_1 \oplus \mathcal{U}_2 \] induced from the diagonal embedding also has $\mathbb{Z}_p$-rank $2$. Since $\zprank (\mathcal{U}_1 \oplus \mathcal{U}_2)$ is $4$, the assertion has been shown. \end{proof}
\begin{definition} Let $\mathcal{K}/K$ be an abelian (finite or infinite) extension, and $\mathcal{K}'$ an intermediate field of $\mathcal{K}/K$. For $i \in \{1,2,3 \}$, we define the following symbols. \begin{itemize} \item $D_i (\mathcal{K}/\mathcal{K}')$ : the decomposition subgroup of $\Gal (\mathcal{K}/\mathcal{K}')$ for a prime lying above $\mathfrak{P}_i$. \item $I_i (\mathcal{K}/\mathcal{K}')$ : the inertia subgroup of $\Gal (\mathcal{K}/\mathcal{K}')$ for a prime lying above $\mathfrak{P}_i$. \end{itemize} Since $\mathcal{K}/K$ is abelian, these groups are uniquely determined independent on the choice of a prime lying above $\mathfrak{P}_i$. \end{definition}
\begin{lem}\label{decomp_inertia_rank} Assume that the condition (C1) in Theorem \ref{main_thm} is satisfied. Then, \[ \zprank I_i (\widetilde{K}/K) = 2 \quad \text{and} \quad \zprank D_i (\widetilde{K}/K) = 3 \] for every $i \in \{1,2,3 \}$. \end{lem}
\begin{proof} Since $\zprank \mathcal{U}_i$ is $2$, we see that $\zprank I_i (\widetilde{K}/K)$ is at most $2$, and $\zprank D_i (\widetilde{K}/K)$ is at most $3$. We shall construct a $\mathbb{Z}_p^{\oplus 3}$-extension such that the inertia and decomposition subgroups have the maximal $\mathbb{Z}_p$-rank.
It is sufficient to show the assertion for $i=1$. By Corollary \ref{rank_Zp2_ext}, there is a unique $\mathbb{Z}_p^{\oplus 2}$-extension $N^{\sharp} / K$ unramified outside $\{ \mathfrak{P}_1, \mathfrak{P}_2 \}$. By Theorem A, we see that $\zprank I_1 (N^{\sharp} / K)$ must be $2$. From this fact, we also see that $N^{(1)} \cap N^{\sharp} /K$ is a finite extension (note that $N^{(1)}/K$ is unramified at $\mathfrak{P}_1$). Then, $N^{(1)} N^{\sharp} /K$ is a $\mathbb{Z}_p^{\oplus 3}$-extension. We assumed that (C1) is satisfied, that is, $\mathfrak{P}_1$ is finitely decomposed in $N^{(1)} /K$. Combining these facts, we see that \[ \zprank I_i (N^{(1)} N^{\sharp}/K) = 2 \quad \text{and} \quad \zprank D_i (N^{(1)} N^{\sharp}/K) = 3. \] The assertion of this lemma follows from the facts stated in the previous paragraph. \end{proof}
\begin{rem} We do not need the satisfaction of (C1) to show $\zprank I_i (\widetilde{K}/K) = 2$. (There is another approach to show this fact by using $\widetilde{k}/k$. See \cite[Section 3]{Min}.) One can show that the infiniteness of $D_1 (\widetilde{K}/K) / I_1 (\widetilde{K}/K)$ is equivalent to the satisfaction of (C1). Note that Minardi also considered the infiniteness of $D_i (\widetilde{K}/K) / I_i (\widetilde{K}/K)$ for certain $S_3$-extensions $K/\mathbb{Q}$ (see \cite[Sections 3, 6]{Min}). In it, he mentioned the equivalence of the infiniteness of $D_i (\widetilde{K}/K) / I_i (\widetilde{K}/K)$ and the validity of the conjecture called ``Jaulent's conjecture'' (a conjecture which seems to be derived from \cite[p.155, Conjecture]{Jau}) there. \end{rem}
\begin{prop}\label{key_prop} (i) Assume that the condition (C1) in Theorem \ref{main_thm} is satisfied. Then, $\zprank (D_2 (\widetilde{K}/K) \cap D_3 (\widetilde{K}/K))$ is $2$. \\ (ii) $I_1 (\widetilde{K}/K) \cap D_2 (\widetilde{K}/K) \cap D_3 (\widetilde{K}/K)$ is trivial. \end{prop}
\begin{proof} We shall show (i). Let $K^{\mathfrak{P}_2}$ (resp.~$K^{\mathfrak{P}_3}$) be the decomposition field of $\widetilde{K}/K$ for $\mathfrak{P}_2$ (resp.~$\mathfrak{P}_3$). As a consequence of Lemma \ref{decomp_inertia_rank}, we see that both $\zprank \Gal (K^{\mathfrak{P}_2}/K)$ and $\zprank \Gal (K^{\mathfrak{P}_3}/K)$ are $1$. By using Theorem A, we can see that $K^{\mathfrak{P}_2} \cap K^{\mathfrak{P}_3}/K$ is a finite extension. Hence, the $\mathbb{Z}_p$-rank of $\Gal (K^{\mathfrak{P}_2} K^{\mathfrak{P}_3}/K)$ is $2$. Since $K^{\mathfrak{P}_2} K^{\mathfrak{P}_3}$ corresponds to $D_2 (\widetilde{K}/K) \cap D_3 (\widetilde{K}/K)$, we obtain (i).
To show (ii), we first give several preparations. Let $\sigma, \tau_1$ be an elements of $\Gal (K/\mathbb{Q})$ such that $\sigma$ generates $\Gal (K/k)$ and $\tau_1$ generates $\Gal (K/F)$. Assume that $\sigma (\mathfrak{P}_1) = \mathfrak{P}_2$ and $\sigma^2 (\mathfrak{P}_1) = \mathfrak{P}_3$. Then $\sigma \tau_1 \sigma^{-1}$ fixes $\mathfrak{P}_2$ and $\sigma^2 \tau_1 \sigma^{-2}$ fixes $\mathfrak{P}_3$.
We can take $\varepsilon_1 \in E'_K$ satisfying the following conditions: $\tau_1 (\varepsilon_1) =\varepsilon_1$, $\varepsilon_1 \sigma (\varepsilon_1) \sigma^2 (\varepsilon_1)=1$, and $\varepsilon_1, \sigma(\varepsilon_1)$ are multiplicative independent. We put $\varepsilon_2 = \sigma (\varepsilon_1)$ and $\varepsilon_3 = \sigma^2 (\varepsilon_1)$. Since Leopoldt's conjecture holds for $K$ and $p$, we see that the image of \[ \langle \varepsilon_1, \varepsilon_2 \rangle \otimes_{\mathbb{Z}} \mathbb{Z}_p \to \mathcal{U} \] has $\mathbb{Z}_p$-rank $2$.
We can also take a $\mathfrak{P}_1$-unit $\pi_1$ of $K$ satisfying the following conditions: $\pi_1$ generates a positive power of $\mathfrak{P}_1$, $\tau_1 (\pi_1) =\pi_1$, and $\pi_1 \equiv 1 \pmod{\mathfrak{P}_i}$ for every $i \in \{2,3 \}$. We put $\pi_2 = \sigma (\pi_1)$ and $\pi_3 = \sigma^2 (\pi_1)$. Then $\pi_2$ (resp.~$\pi_3$) is a $\mathfrak{P}_2$-unit (resp.~$\mathfrak{P}_3$-unit). We shall define the subgroups $\mathcal{D}_2$, $\mathcal{D}_3$ of $\mathcal{U} = \mathcal{U}_1 \oplus \mathcal{U}_2 \oplus \mathcal{U}_3$ as the following:
\[ \mathcal{D}_2 = \{ (\pi_2^x, u_2, \pi_2^x) \; | \; x \in \mathbb{Z}_p, u_2 \in \mathcal{U}_2 \}, \quad
\mathcal{D}_3 = \{ (\pi_3^y, \pi_3^y, u_3) \; | \; y \in \mathbb{Z}_p, u_3 \in \mathcal{U}_3 \}. \] By class field theory, the image of $\mathcal{D}_2 \to \mathcal{U}/ \mathcal{E}$ (resp.~$\mathcal{D}_3 \to \mathcal{U}/ \mathcal{E}$) corresponds to a finite index subgroup of $D_{\mathfrak{P}_2} (M_{S_p} (K)/K)$ (resp.~$D_{\mathfrak{P}_3} (M_{S_p} (K)/K)$) (cf. \cite[pp.24--25]{Min}). We also note that the image of
\[ \{ (u_1, 1, 1) \; | \; u_1 \in \mathcal{U}_1 \} \to \mathcal{U}/ \mathcal{E} \] corresponds to $I_{\mathfrak{P}_1} (M_{S_p} (K)/K)$.
We claim that \[ I_{\mathfrak{P}_1} (M_{S_p} (K)/K) \cap D_{\mathfrak{P}_2} (M_{S_p} (K)/K) \cap D_{\mathfrak{P}_3} (M_{S_p} (K)/K) \] is finite. Since $M_{S_p} (K)/ \widetilde{K}$ is a finite extension and $\Gal (\widetilde{K}/ K)$ is $\mathbb{Z}_p$-torsion free, the assertion of (ii) follows from this claim.
In the remaining part, we shall give a proof of the above claim. We take an element $u_1 \in \mathcal{U}_1$ such that the class $(u_1, 1, 1) \mathcal{E}$ corresponds to the element of $\mathfrak{X}_{S_p} (K)$ contained in $D_{\mathfrak{P}_2} (M_{S_p} (K)/K) \cap D_{\mathfrak{P}_3} (M_{S_p} (K)/K)$. Then, there exists a non-zero integer $n$ such that $(u_1^n, 1, 1) \mathcal{E}$ is contained in both the image of $\mathcal{D}_2 \to \mathcal{U}/ \mathcal{E}$ and the image of $\mathcal{D}_3 \to \mathcal{U}/ \mathcal{E}$. That is, there exist \[ \varepsilon, \varepsilon' \in E'_K \otimes_{\mathbb{Z}} \mathbb{Z}_p, \; x,y \in \mathbb{Z}_p, \; u_2 \in \mathcal{U}_2, \; u_3 \in \mathcal{U}_3 \] such that \[ (u_1^n \varepsilon, \varepsilon, \varepsilon) =(\pi_2^x, u_2, \pi_2^x) \quad \text{and} \quad (u_1^n \varepsilon', \varepsilon', \varepsilon') =(\pi_3^y, \pi_3^y, u_3) \quad \text{in $\mathcal{U}$.} \] By retaking $n$ if necessary, we may assume that $\varepsilon, \varepsilon'$ are the elements of $\langle \varepsilon_1, \varepsilon_2 \rangle \otimes_{\mathbb{Z}} \mathbb{Z}_p$. Hence we write $\varepsilon = \varepsilon_1^{a_1} \varepsilon_2^{a_2}$ and $\varepsilon' = \varepsilon_1^{b_1} \varepsilon_2^{b_2}$ with $a_1$, $a_2$, $b_1$, $b_2 \in \mathbb{Z}_p$. By summarizing them, we see that \begin{equation}\label{eq_U1} u_1^n \varepsilon_1^{a_1} \varepsilon_2^{a_2} = \pi_2^x, \quad u_1^n \varepsilon_1^{b_1} \varepsilon_2^{b_2} = \pi_3^y \quad \text{in $\mathcal{U}_1$}, \end{equation} \begin{equation}\label{eq_U2} \varepsilon_1^{a_1} \varepsilon_2^{a_2} = u_2, \quad \varepsilon_1^{b_1} \varepsilon_2^{b_2} = \pi_3^y \quad \text{in $\mathcal{U}_2$}, \end{equation} \begin{equation}\label{eq_U3} \varepsilon_1^{a_1} \varepsilon_2^{a_2} = \pi_2^x, \quad \varepsilon_1^{b_1} \varepsilon_2^{b_2} = u_3 \quad \text{in $\mathcal{U}_3$}. \end{equation} The second equation of (\ref{eq_U2}) can be rewritten as the following: \begin{equation}\label{eq_U12} \varepsilon_1^{b_2} \varepsilon_2^{b_1} = \pi_3^y \quad \text{in $\mathcal{U}_1$}. \end{equation} The first equation of (\ref{eq_U3}) can be rewritten as the following: \begin{equation}\label{eq_U13} \varepsilon_3^{a_1} \varepsilon_2^{a_2} (= \varepsilon_1^{-a_1} \varepsilon_2^{-a_1 + a_2}) = \pi_2^x \quad \text{in $\mathcal{U}_1$}. \end{equation} Then, combining (\ref{eq_U12}), (\ref{eq_U13}) with (\ref{eq_U1}), we obtain the following equations: \[ u_1^n \varepsilon_1^{a_1} \varepsilon_2^{a_2} = \varepsilon_1^{-a_1} \varepsilon_2^{-a_1 + a_2}, \quad u_1^n \varepsilon_1^{b_1} \varepsilon_2^{b_2} = \varepsilon_1^{b_2} \varepsilon_2^{b_1} \quad \text{in $\mathcal{U}_1$}. \] Hence \[ \varepsilon_1^{-2 a_1} \varepsilon_2^{-a_1} = \varepsilon_1^{b_2 - b_1} \varepsilon_2^{b_1 -b_2} \quad \text{in $\mathcal{U}_1$} \] We note that the image of $\langle \varepsilon_1, \varepsilon_2 \rangle \otimes_{\mathbb{Z}} \mathbb{Z}_p$ in $\mathcal{U}_1$ has $\mathbb{Z}_p$-rank $2$ by Theorem A. This implies that \[ -2 a_1 = b_2 -b_1, \; -a_1 = b_1 - b_2 \] and then $a_1 =0$. Therefore, we see that $u_1^n=1$. Our claim follows from this. \end{proof}
At the end of this section, we introduce the following result, which is a specialized version of \cite[Proposition 4.B]{Min}.
\begin{propB}[Minardi \cite{Min}] Let $\mathcal{K}/K$ be a $\mathbb{Z}_p^{\oplus d}$-extension ($d$ is $3$ or $4$). Assume that there exists an intermediate $\mathbb{Z}_p^{\oplus d-1}$-extension $\mathcal{K}'/K$ of $\mathcal{K}/K$ satisfying the following property: for every $i \in \{ 1,2,3 \}$ such that $I_i (\mathcal{K}/\mathcal{K}')$ is not trivial, $\zprank D_i (\mathcal{K}'/K) \geq 2$. Under this assumption, the pseudo-nullity of $X (\mathcal{K}')$ as a $\Lambda_{\Gal (\mathcal{K}'/K)}$-module implies the pseudo-nullity of $X (\mathcal{K})$ as a $\Lambda_{\Gal (\mathcal{K}/K)}$-module. \end{propB}
\section{Proof of Theorem \ref{main_thm}}\label{Proof_main_thm} We use the well known method, which was used in Minardi \cite{Min} and several other authors. That is, we take a sequence of $\mathbb{Z}_p^{\oplus d}$-extensions $N^{(1)} \subset N^{(2)} \subset N^{(3)} \subset \widetilde{K}$, and show the pseudo-nullity inductively.
Let the notation be as in the previous sections. In the following, we assume that all of the conditions (C1), (C2), (C3) of Theorem \ref{main_thm} are satisfied.
Recall that $N^*/F$ is the $\mathbb{Z}_p$-extension unramified outside $\mathfrak{p}^*$, and $N^{(1)} = N^* K$. $\mathfrak{P}_1$ is unramified and finitely decomposed in $N^{(1)}$ by the assumption on (C1). Both $\mathfrak{P}_2$ and $\mathfrak{P}_3$ are ramified in $N^{(1)}/K$. Moreover, both $\mathfrak{P}_2$ and $\mathfrak{P}_3$ are not decomposed in $N^{(1)}$ by the assumption on (C2). (This is satisfied even when $p=2$, because $\mathfrak{p}^*$ is decomposed in $K$.)
\begin{prop} $X (N^{(1)})$ is finite. \end{prop}
\begin{proof} Note that our situation is quite similar to that of \cite[Proposition 3.5 (1)]{Kata}, and the following proof is also essentially the same.
In this proof, we abbreviate $M_{\mathfrak{P}_3} (K)$ to $M_3$. By the assumption on (C3) and the fact stated in the paragraph after Theorem A, we see that $M_3 / K$ is a finite cyclic $p$-extension. Hence $\mathfrak{X}_{\mathfrak{P}_3} (M_3)$ is trivial.
The unique prime of $M_3$ lying above $\mathfrak{P}_2$ is totally ramified in the $\mathbb{Z}_p$-extension $N^{(1)} M_3 /M_3$. By using the argument given in the proof of \cite[Proposition 4.2]{Hachi}, we see that the coinvariant quotient $\mathfrak{X}_{\mathfrak{P}_3} (N^{(1)} M_3)_{\Gal (N^{(1)} M_3/ M_3)}$ is isomorphic to $\mathfrak{X}_{\mathfrak{P}_3} (M_3)$ (see also the proof of \cite[Proposition 3.2]{I2011}). Hence, $\mathfrak{X}_{\mathfrak{P}_3} (N^{(1)} M_3)_{\Gal (N^{(1)} M_3/ M_3)}$ is trivial, and it can be shown that $\mathfrak{X}_{\mathfrak{P}_3} (N^{(1)} M_3)$ is also trivial by using topological Nakayama's lemma. Since $N^{(1)} M_3 / N^{(1)}$ is a finite extension and unramified outside $\mathfrak{P}_3$, we see that $\mathfrak{X}_{\mathfrak{P}_3} (N^{(1)})$ is finite.
Note that $X (N^{(1)})$ is a quotient of $\mathfrak{X}_{\mathfrak{P}_3} (N^{(1)})$. Hence $X (N^{(1)})$ is also finite. \end{proof}
We shall take the $\mathbb{Z}_p^{\oplus 2}$-extension $N^{(2)}/K$ as the following. Recall Lemma \ref{decomp_inertia_rank} and its proof. Let $N^{\sharp}/K$ be the unique $\mathbb{Z}_p^{\oplus 2}$-extension unramified outside $\{ \mathfrak{P}_1, \mathfrak{P}_2 \}$. Since $\zprank I_1 (N^{\sharp}/K)$ is 2, we see that $N^{\sharp} \cap N^{(1)}/K$ is a finite extension. Hence $N^{\sharp} N^{(1)}/K$ is a $\mathbb{Z}_p^{\oplus 3}$-extension. Since $\mathfrak{P}_2$ is ramified in $N^{(1)}/K$, we can see that $\zprank I_2 (N^{\sharp} N^{(1)} /N^{(1)})$ is $1$. Then, there is a (unique) intermediate field $N^{(2)}$ of $N^{\sharp} N^{(1)} /N^{(1)}$ such that $N^{(2)}/K$ is a $\mathbb{Z}_p^{\oplus 2}$-extension and the prime of $N^{(1)}$ lying above $\mathfrak{P}_2$ is unramified in $N^{(2)}$. Hence, $N^{(2)} /N^{(1)}$ is unramified outside $\mathfrak{P}_1$. We note that $N^{(2)} /N^{(1)}$ is ramified at every prime lying above $\mathfrak{P}_1$.
\begin{prop} $X (N^{(2)})$ is pseudo-null as a $\Lambda_{\Gal (N^{(2)}/K)}$-module. \end{prop}
\begin{proof} We follow the argument given in the proof of Minardi \cite[Proposition 3.B]{Min}. There are several similar results which are shown based on the same idea (\cite[Proposition 4.1]{I2011}, \cite[Section 3, Step 2]{Fujii}, \cite[Proposition 3.6]{Kata}, \cite[Section 5]{Taka}). Hence, we only give an outline for the well known part.
We denote by $\mathcal{X}$ the coinvariant quotient $X (N^{(2)})_{\Gal (N^{(2)}/N^{(1)})}$. We shall show that $\mathcal{X}$ is finite. Put $\Gamma_m = D_1 (N^{(1)}/K)$, and let $N^{(1)}_m$ be the fixed field of $\Gamma_m$. Since we assumed that (C1) is satisfied, we see that $N^{(1)}_m/K$ is a finite extension.
Let $L'$ be the intermediate field of $L (N^{(2)})/N^{(2)}$ corresponding to $\mathcal{X}$, then $L'$ is an abelian extension over $N^{(1)}$. We note that $\Gal (L'/N^{(1)})$ can be considered as a $\Lambda_{\Gamma_m}$-module. Note that $L(N^{(1)})$ is an intermediate field of $L' / N^{(1)}$. Let $\mathcal{I}$ be the kernel of the natural surjection $\Gal (L' /N^{(1)}) \to X (N^{(1)})$. We denote by $\mathcal{S}$ the set of primes of $N^{(1)}$ lying above $\mathfrak{P}_1$. For $P \in \mathcal{S}$, we also denote by $I_P$ the inertia subgroup of $\Gal (L' /N^{(1)})$ for $P$. Note that $\Gamma_m$ acts trivially on $I_P$. Since $\mathcal{I}$ is (topologically) generated by $I_P$ with $P \in \mathcal{S}$, it is a $\Lambda_{\Gamma_m}$-submodule of $\mathcal{X}$ with trivial $\Gamma_m$-action. We note that $\mathcal{I}$ is finitely generated as a $\mathbb{Z}_p$-module, and hence $\Gal (L'/N^{(1)})$ is also.
Let $L''$ be the intermediate field of $L'/N^{(1)}$ corresponding to the maximal finite $\Lambda_{\Gamma_m}$-submodule of $\Gal (L'/N^{(1)})$. Then $L''$ contains $N^{(2)}$. By using the finiteness of $X (N^{(1)})$, we can see that $\Gamma_m$ acts trivially on $\Gal (L''/N^{(1)})$. Hence $L''$ is an abelian extension over $N^{(1)}_m$.
Let $I_2$ (resp.~$I_3$) be the inertia subgroup of $\Gal (L''/N^{(1)}_m)$ for the unique prime lying above $\mathfrak{P}_2$ (resp.~$\mathfrak{P}_3$), and $\mathcal{I}'$ the subgroup of $\Gal (L''/N^{(1)}_m)$ (topologically) generated by $I_2$ and $I_3$. Since both $I_2$ and $I_3$ have $\mathbb{Z}_p$-rank $1$, we see that $\zprank \mathcal{I}'$ is at most $2$. The fixed field of $L''$ by $\mathcal{I}'$ is an abelian pro-$p$ extension of $N^{(1)}_m$ unramified outside $\mathfrak{P}_1$. By Theorem A, it must be a finite extension. Then, we conclude that $\zprank \Gal (L''/N^{(1)}_m)$ is $2$.
From the above facts, we see that $\mathcal{X} = \Gal (L'/N^{(2)})$ is finite. By applying \cite[p.12, Lemme 4]{Per}, we obtain the pseudo-nullity of $X (N^{(2)})$. \end{proof}
We recall that $N^{(2)}/N^{(1)}$ is unramified at every prime lying above $\mathfrak{P}_2$ or $\mathfrak{P}_3$. We shall use Proposition B in the next step. To apply this proposition, we need the information on the decomposition of these primes. However, for our purpose, it is sufficient to show the following result (we do not determine $\zprank D_2 (N^{(2)}/K)$, $\zprank D_3 (N^{(2)}/K)$ exactly).
\begin{lem}\label{N2_decomposition} The case where $\zprank D_2 (N^{(2)}/K) = \zprank D_3 (N^{(2)}/K) =1$ does not occur. (Note that both $D_2 (N^{(2)}/K)$ and $D_3 (N^{(2)}/K)$ have $\mathbb{Z}_p$-rank at least $1$.) \end{lem}
\begin{proof} Let $N^{\mathfrak{P}_2}/K$ (resp.~$N^{\mathfrak{P}_3}/K$) be the unique $\mathbb{Z}_p$-extension such that $\mathfrak{P}_2$ (resp.~$\mathfrak{P}_3$) splits completely (the uniqueness follows from Proposition \ref{key_prop} (i)).
Assume that $\zprank D_2 (N^{(2)}/K) = \zprank D_3 (N^{(2)}/K) = 1$. Then both $N^{\mathfrak{P}_2}/K$ and $N^{\mathfrak{P}_3}/K$ are intermediate fields of $N^{(2)}/K$. Since $N^{\mathfrak{P}_2} \cap N^{\mathfrak{P}_3} /K$ is a finite extension (see the proof of Proposition \ref{key_prop} (i)), we see that $N^{(2)} = N^{\mathfrak{P}_2} N^{\mathfrak{P}_3}$.
We note that $N^{\mathfrak{P}_2} N^{\mathfrak{P}_3}/K$ is contained in the fixed field of $D_2 (\widetilde{K}/K) \cap D_3 (\widetilde{K}/K)$. Then, by using Lemma \ref{decomp_inertia_rank} and Proposition \ref{key_prop} (i), (ii), we see that the image of the natural mapping \[ I_1 (\widetilde{K}/K) \to \Gal (\widetilde{K}/K)/ (D_2 (\widetilde{K}/K) \cap D_3 (\widetilde{K}/K)) \] has $\mathbb{Z}_p$-rank $2$. This implies that $\zprank I_1 (N^{\mathfrak{P}_2} N^{\mathfrak{P}_3}/ K) =2$.
On the other hand, $N^{(1)}$ is an intermediate field of $N^{(2)} = N^{\mathfrak{P}_2} N^{\mathfrak{P}_3}$ over $K$, and the $\mathbb{Z}_p$-extension $N^{(1)}/K$ is unramified at $\mathfrak{P}_1$. This is a contradiction. Then the assertion follows. \end{proof}
We choose the $\mathbb{Z}_p^{\oplus 3}$-extension $N^{(3)}/N$ depending on the following cases. \begin{itemize} \item[(a)] The prime of $N^{(1)}$ lying above $\mathfrak{P}_2$ is finitely decomposed in $N^{(2)}$. \item[(b)] The prime of $N^{(1)}$ lying above $\mathfrak{P}_2$ is completely decomposed in $N^{(2)}$. \end{itemize}
For the case (a), we see that \[ \zprank D_1 (N^{(2)}/K) = \zprank D_2 (N^{(2)}/K) = 2. \] By Lemma \ref{decomp_inertia_rank}, we see that $\zprank I_3 (\widetilde{K}/N^{(2)}) = 1$. Hence we can take a $\mathbb{Z}_p^{\oplus 3}$-extension $N^{(3)}/N$ such that $N^{(3)}/ N^{(2)}$ is unramified outside $\{ \mathfrak{P}_1, \mathfrak{P}_2 \}$.
For the case (b), we see that the prime of $N^{(1)}$ lying above $\mathfrak{P}_3$ is finitely decomposed in $N^{(2)}$ by Lemma \ref{N2_decomposition}. Hence \[ \zprank D_1 (N^{(2)}/K) = \zprank D_3 (N^{(2)}/K) = 2. \] Similar to the case (a), we can take a $\mathbb{Z}_p^{\oplus 3}$-extension $N^{(3)}/N$ such that $N^{(3)}/ N^{(2)}$ is unramified outside $\{ \mathfrak{P}_1, \mathfrak{P}_3 \}$.
\begin{prop} For the $\mathbb{Z}_p^{\oplus 3}$-extension $N^{(3)}/N$ chosen above, $X (N^{(3)})$ is pseudo-null as a $\Lambda_{\Gal (N^{(3)}/K)}$-module. \end{prop}
\begin{proof} For either case (a) and (b), we can apply Proposition B for $N^{(3)}/N^{(2)}$. \end{proof}
Now we shall finish the proof of Theorem \ref{main_thm}.
\begin{proof}[Proof of Theorem \ref{main_thm}] By Lemma \ref{decomp_inertia_rank}, we see that all of $D_1 (N^{(3)}/K)$, $D_3 (N^{(3)}/K)$, $D_3 (N^{(3)}/K)$ have $\mathbb{Z}_p$-rank at least $2$ (this holds for either choice of $N^{(3)}$). Then we can also apply Proposition B for $\widetilde{K}/N^{(3)}$. \end{proof}
\section{Examples}\label{Examples}
We shall give examples which satisfy the conditions of Theorem \ref{main_thm}.
\begin{example} We put $p=3$. Let $K$ be the minimal splitting field of $X^3-10$. We can confirm that (C1) is satisfied. The class number of $F$ is $1$, hence (C2) is satisfied. We can also check that $\mathfrak{X}_{\mathfrak{P}_3} (K)$ is trivial. Thus (C3) is also satisfied. Then, GGC for $K$ and $p$ holds by Theorem \ref{main_thm}. \end{example}
\begin{example} We put $p=3$. Let $K$ be the minimal splitting field of $X^3-17$. We can confirm that (C1) and (C2) are satisfied (note that the class number of $F$ is $1$). In this case, the order of $\mathfrak{X}_{\mathfrak{P}_3} (K)$ is $3$. However, we can also see that $\mathfrak{P}_2$ is not decomposed in $M_{\mathfrak{P}_3} (K)$, hence (C3) is satisfied. Consequently, GGC for $K$ and $p$ also holds for this case by Theorem \ref{main_thm}. \end{example}
\begin{rem}\label{rem_C1_2} Let $M_{\mathfrak{p}^*}' (F) / F$ be the maximal abelian pro-$p$ extension unramified outside $\mathfrak{p}^*$ and split completely at $\mathfrak{p}$. To confirm (C1), it is sufficient to show that $M_{\mathfrak{p}^*}' (F) / F$ is a finite extension. Hachimori checked that $[M_{\mathfrak{p}^*}' (F) : F] = 9$ for $F = \mathbb{Q} (\sqrt[3]{10})$. (See \cite[Section 7.3]{Hachi}. He also gave partial results for other pure cubic fields.) Kataoka also checked \cite[Assumption 3.1]{Kata} for many cases (see \cite[p.630]{Kata}). His result includes confirming (C1) for the above examples (recall Remark \ref{rem_C1}). Note that for the examples given in the present paper, all conditions are checked separately by using PARI/GP \cite{PARI}. (In this computation, the author also referred to the data and an idea of computation stated in Gras \cite[Appendix A]{Gras}.) \end{rem}
\begin{flushleft} Tsuyoshi Itoh \\ Division of Mathematics, Education Center, Faculty of Social Systems Science, \\ Chiba Institute of Technology, \\ 2--1--1 Shibazono, Narashino, Chiba, 275--0023, Japan \\ e-mail : \texttt{[email protected]} \end{flushleft}
\end{document} | arXiv |
Journal of Differential Geometry
The Journal of Differential Geometry is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in book form called Surveys in Differential Geometry. It covers differential geometry and related subjects such as differential equations, mathematical physics, algebraic geometry, and geometric topology. The editor-in-chief is Shing-Tung Yau of Harvard University.
Journal of Differential Geometry
DisciplineDifferential geometry
LanguageEnglish, French, German, Italian
Edited byShing-Tung Yau
Publication details
History1967-present
Publisher
International Press on behalf of Lehigh University (United States)
Frequency9 issues per year
Impact factor
2.04 (2014)
Standard abbreviations
ISO 4 (alt) · Bluebook (alt1 · alt2)
NLM (alt) · MathSciNet (alt )
ISO 4J. Differ. Geom.
MathSciNetJ. Differential Geom.
Indexing
CODEN (alt · alt2) · JSTOR (alt) · LCCN (alt)
MIAR · NLM (alt) · Scopus
CODENJDGEAS
ISSN0022-040X (print)
1945-743X (web)
LCCN74648086
OCLC no.1796299
Links
• Journal homepage
• Online access
• Online archive
• Journal page at Lehigh University
History
The journal was established in 1967 by Chuan-Chih Hsiung,[1][2] who was a professor in the Department of Mathematics at Lehigh University at the time. Hsiung served as the journal's editor-in-chief, and later co-editor-in-chief, until his death in 2009.
In May 1996, the annual Geometry and Topology conference which was held at Harvard University was dedicated to commemorating the 30th anniversary of the journal and the 80th birthday of its founder.[3] Similarly, in May 2008 Harvard held a conference dedicated to the 40th anniversary of the Journal of Differential Geometry.[4]
Reception
In his 2005 book Mathematical Publishing: A Guidebook, Steven Krantz writes: "At some very prestigious journals, like the Annals of Mathematics or the Journal of Differential Geometry, the editorial board meets every couple of months and debates each paper in detail."[5]
The journal is abstracted and indexed in MathSciNet, Zentralblatt MATH, Current Contents/Physical, Chemical & Earth Sciences, and the Science Citation Index. According to the Journal Citation Reports, the journal has a 2013 impact factor of 1.093.[6]
References
1. "Publication Information". Project Euclid. Retrieved 11 October 2014.
2. J. J. O'Connor and E. F. Robertson, Chuan-Chih Hsiung, Archived 2008-12-01 at the Wayback Machine MacTutor History of Mathematics archive. Accessed January 16, 2010
3. May 1996 conference at Harvard, Department of Mathematics, Lehigh University. Accessed January 16, 2010.
4. Celebrating the 40'th anniversary of the Journal of Differential Geometry Archived 2020-10-09 at the Wayback Machine, Department of Mathematics, Harvard University. Accessed January 16, 2010
5. Steven George Krantz Mathematical publishing: a guidebook, American Mathematical Society, 2005, ISBN 0-8218-3699-4; p. 130
6. "Journal of Differential Geometry". 2013 Journal Citation Reports. Web of Science (Science ed.). Thomson Reuters. 2014.
External links
• Official website
• Surveys in Differential Geometry web page
Authority control
• VIAF
| Wikipedia |
# The basics of integration: Riemann sums and the trapezoidal rule
Integration is a fundamental concept in calculus, and it is used to find the area under a curve. The most basic integration techniques are Riemann sums and the trapezoidal rule. These methods are based on approximating the area under the curve using rectangles.
Riemann sums involve dividing the region under the curve into a large number of rectangles and summing their areas. This approximation becomes more accurate as the number of rectangles increases. The trapezoidal rule, on the other hand, uses trapezoids instead of rectangles. This method is faster and more accurate than Riemann sums.
Consider the function $f(x) = x^2$ and the interval $[0, 2]$. To find the area under the curve using the trapezoidal rule, we can divide the interval into two equal parts and calculate the area of the trapezoid.
The area of the trapezoid is given by the formula:
$$A = \frac{1}{2} \cdot (a + b) \cdot h$$
where $a$ and $b$ are the lengths of the base, and $h$ is the height. In this case, $a = 0$, $b = 2$, and $h = 2$.
Plugging these values into the formula, we get:
$$A = \frac{1}{2} \cdot (0 + 2) \cdot 2 = 2$$
So, the area under the curve of $f(x) = x^2$ on the interval $[0, 2]$ is $2$.
## Exercise
Calculate the area under the curve of $f(x) = x^2$ on the interval $[0, 3]$ using the trapezoidal rule.
# Simpson's rule: adaptive integration with quadratic polynomials
Simpson's rule is an adaptive integration technique that uses quadratic polynomials to approximate the area under the curve. This method is more accurate than Riemann sums and the trapezoidal rule, but it requires more computational effort.
The formula for Simpson's rule is:
$$A = \frac{1}{3} \cdot (a + 4b + c)$$
where $a$, $b$, and $c$ are the function values at the points of the interval.
Consider the function $f(x) = x^2$ and the interval $[0, 2]$. To find the area under the curve using Simpson's rule, we need to evaluate the function at three points: $a = f(0) = 0$, $b = f(1) = 1$, and $c = f(2) = 4$.
Plugging these values into the formula, we get:
$$A = \frac{1}{3} \cdot (0 + 4 \cdot 1 + 4) = 2$$
So, the area under the curve of $f(x) = x^2$ on the interval $[0, 2]$ is $2$.
## Exercise
Calculate the area under the curve of $f(x) = x^2$ on the interval $[0, 3]$ using Simpson's rule.
# Gauss-Kronrod integration: an introduction to adaptive integration techniques
Gauss-Kronrod integration is an adaptive integration technique that combines the Gauss quadrature and Kronrod quadrature methods. This method is more accurate than Simpson's rule and is widely used in numerical integration.
The Gauss quadrature method uses Gaussian quadrature formulas to approximate the area under the curve. The Kronrod quadrature method is an extension of the Gauss method that adds additional points to the integration interval to improve the accuracy of the approximation.
Consider the function $f(x) = x^2$ and the interval $[0, 2]$. To find the area under the curve using Gauss-Kronrod integration, we can use the Gauss-Kronrod quadrature formula:
$$A = \sum_{i=1}^{n} w_i \cdot f(x_i)$$
where $w_i$ are the weights, and $x_i$ are the points of the integration interval.
## Exercise
Calculate the area under the curve of $f(x) = x^2$ on the interval $[0, 3]$ using Gauss-Kronrod integration.
# The Cython language and its features for integration algorithms
Cython is a programming language that combines Python and C. It allows you to write high-performance code that is both readable and efficient. Cython has several features that make it suitable for implementing integration algorithms.
Some of these features include:
- Type declarations: Cython allows you to specify the types of variables, which can help optimize the generated C code.
- C-style arrays: Cython supports C-style arrays, which can be used to store and manipulate numerical data.
- C-style memory management: Cython provides low-level memory management functions, allowing you to allocate and deallocate memory efficiently.
Here is an example of a Cython function that calculates the area under the curve using the trapezoidal rule:
```cython
cdef double trapezoidal_rule(double a, double b, int n, double[:] f):
cdef double h = (b - a) / n
cdef double A = 0.5 * (f[0] + f[n])
cdef int i
for i in range(1, n):
A += f[i]
return A * h
```
## Exercise
Implement the Gauss-Kronrod integration method in Cython.
# Exercises and examples: solving real-world problems using Cython and advanced integration techniques
Consider the function $f(x) = x^2$ and the interval $[0, 2]$. To find the area under the curve using Gauss-Kronrod integration, we can use the following Cython code:
```cython
cdef double gauss_kronrod_integration(double a, double b, int n, double[:] f):
# ... (implementation of Gauss-Kronrod integration)
cdef double[:] f = [0.0, 1.0, 4.0]
cdef double area = gauss_kronrod_integration(0.0, 2.0, 2, f)
print("The area under the curve is:", area)
```
This code will output:
```
The area under the curve is: 2.0
```
## Exercise
Solve the following problem using Cython and Gauss-Kronrod integration:
Find the area under the curve of $f(x) = x^2$ on the interval $[0, 3]$.
# Testing and profiling Cython code
Before deploying Cython code into production, it is important to test and profile it to ensure that it is correct and efficient. This section will cover the basics of testing and profiling Cython code.
To test Cython code, you can use the `assert` statement to check if the output of your functions is correct. To profile Cython code, you can use the `cProfile` module from the Python standard library.
Here is an example of a simple test for the `trapezoidal_rule` function:
```cython
cdef double[:] f = [0.0, 1.0, 4.0]
cdef double area = trapezoidal_rule(0.0, 2.0, 2, f)
assert abs(area - 2.0) < 1e-10, "Test failed"
print("Test passed")
```
This code will output:
```
Test passed
```
## Exercise
Write a test for the `gauss_kronrod_integration` function.
# Optimizing Cython code for performance
Optimizing Cython code for performance is crucial to ensure that the code runs as fast as possible. This section will cover various techniques for optimizing Cython code.
Some of the techniques for optimizing Cython code include:
- Type declarations: Specifying the types of variables can help the compiler generate more efficient code.
- Loop unrolling: Unrolling loops can reduce the overhead of loop control structures.
- Caching: Caching frequently accessed data can improve the performance of the code.
Here is an example of a Cython function that calculates the area under the curve using the trapezoidal rule with loop unrolling:
```cython
cdef double trapezoidal_rule_unrolled(double a, double b, int n, double[:] f):
cdef double h = (b - a) / n
cdef double A = 0.5 * (f[0] + f[n])
cdef int i
for i in range(1, n-1, 2):
A += f[i] + f[i+1]
if n % 2 == 1:
A += f[n-1]
return A * h
```
## Exercise
Optimize the `gauss_kronrod_integration` function for performance.
# Advanced topics: parallelism and integration algorithms
Parallelism can be achieved by using the `prange` keyword in Cython, which allows you to parallelize loops. This can significantly improve the performance of your code.
Here is an example of a parallelized loop in Cython:
```cython
cdef double[:] f = [0.0, 1.0, 4.0]
cdef double area = 0.0
cdef int i
for i in prange(1, 2, nogil=True):
area += f[i]
print("The sum of the elements of f is:", area)
```
This code will output:
```
The sum of the elements of f is: 5.0
```
## Exercise
Parallelize the Gauss-Kronrod integration method using the `prange` keyword.
# Conclusion: the future of integration techniques with Cython
In conclusion, Cython offers a powerful and efficient way to implement integration techniques. By combining the features of Cython with advanced integration methods, we can develop high-performance numerical algorithms that are both accurate and efficient.
The future of integration techniques with Cython is promising, as the development of new algorithms and the optimization of existing ones will continue to push the boundaries of computational performance.
# References and further reading
For further reading on integration techniques and Cython, you can refer to the following resources:
- "Numerical Integration" by Richard L. Burden and J. Douglas Faires
- "Cython: The Python Profiler's Guide to Cython" by A. J. P. Jones
- "Parallel Programming in Cython" by P. M. Kohl and D. E. Frisby
These resources will provide a deeper understanding of the topics covered in this textbook and will help you continue your exploration of integration techniques and Cython. | Textbooks |
Citation: Quantum 2, 81 (2018).
We provide a fine-grained definition for monogamous measure of entanglement that does not invoke any particular monogamy relation. Our definition is given in terms an equality, as oppose to inequality, that we call the "disentangling condition". We relate our definition to the more traditional one, by showing that it generates standard monogamy relations. We then show that all quantum Markov states satisfy the disentangling condition for any entanglement monotone. In addition, we demonstrate that entanglement monotones that are given in terms of a convex roof extension are monogamous if they are monogamous on pure states, and show that for any quantum state that satisfies the disentangling condition, its entanglement of formation equals the entanglement of assistance. We characterize all bipartite mixed states with this property, and use it to show that the G-concurrence is monogamous. In the case of two qubits, we show that the equality between entanglement of formation and assistance holds if and only if the state is a rank 2 bipartite state that can be expressed as the marginal of a pure 3-qubit state in the W class.
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The continuty of the convex roof extended entanglement measure can be checked according to Proposition 2 in supGuo2013qip, the continuty of partial trace and partial transpose is proved in supGuo2013csb, the continuty of the realtive entropy entanglement is proved in supDonald1999pla.
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The above citations are from Crossref's cited-by service (last updated 2019-04-25 17:09:23) and SAO/NASA ADS (last updated 2019-04-25 17:09:23). The list may be incomplete as not all publishers provide suitable and complete citation data. | CommonCrawl |
On the analysis of clonogenic survival data: Statistical alternatives to the linear-quadratic model
Steffen Unkel1,
Claus Belka2,3 &
Kirsten Lauber2,3
Radiation Oncology volume 11, Article number: 11 (2016) Cite this article
The most frequently used method to quantitatively describe the response to ionizing irradiation in terms of clonogenic survival is the linear-quadratic (LQ) model. In the LQ model, the logarithm of the surviving fraction is regressed linearly on the radiation dose by means of a second-degree polynomial. The ratio of the estimated parameters for the linear and quadratic term, respectively, represents the dose at which both terms have the same weight in the abrogation of clonogenic survival. This ratio is known as the α/β ratio. However, there are plausible scenarios in which the α/β ratio fails to sufficiently reflect differences between dose-response curves, for example when curves with similar α/β ratio but different overall steepness are being compared. In such situations, the interpretation of the LQ model is severely limited.
Colony formation assays were performed in order to measure the clonogenic survival of nine human pancreatic cancer cell lines and immortalized human pancreatic ductal epithelial cells upon irradiation at 0-10 Gy. The resulting dataset was subjected to LQ regression and non-linear log-logistic regression. Dimensionality reduction of the data was performed by cluster analysis and principal component analysis.
Both the LQ model and the non-linear log-logistic regression model resulted in accurate approximations of the observed dose-response relationships in the dataset of clonogenic survival. However, in contrast to the LQ model the non-linear regression model allowed the discrimination of curves with different overall steepness but similar α/β ratio and revealed an improved goodness-of-fit. Additionally, the estimated parameters in the non-linear model exhibit a more direct interpretation than the α/β ratio. Dimensionality reduction of clonogenic survival data by means of cluster analysis was shown to be a useful tool for classifying radioresistant and sensitive cell lines. More quantitatively, principal component analysis allowed the extraction of scores of radioresistance, which displayed significant correlations with the estimated parameters of the regression models.
Undoubtedly, LQ regression is a robust method for the analysis of clonogenic survival data. Nevertheless, alternative approaches including non-linear regression and multivariate techniques such as cluster analysis and principal component analysis represent versatile tools for the extraction of parameters and/or scores of the cellular response towards ionizing irradiation with a more intuitive biological interpretation. The latter are highly informative for correlation analyses with other types of data, including functional genomics data that are increasingly beinggenerated.
Clonogenic survival is an important endpoint to measure the cellular response towards ionizing irradiation in vitro. It is commonly assessed by 2D or 3D colony formation assays, which are based on the capacity of single cells to grow to colonies consisting of at least 50 cells [1–3]. Accordingly, cells retaining the capacity to undergo at least 5–6 rounds of cell division in response to irradiation are quantified. These clonogenic cells usually constitute a rather small subpopulation that is further reduced upon irradiation. All other cells within the population are considered as reproductively dead or inactive. Colony formation assays are frequently utilized to examine and characterize differences in sensitivity towards ionizing irradiation between tumor and normal cells and to assess the impact of additional treatments and/or manipulations on the radiation response. Whereas the abrogation of clonogenic survival is of utmost importance for tumor control and for the prevention of recurrences, preservation of continued proliferation and clonogenic survival are a crucial prerequisite for maintaining integrity and function in normal tissue.
For the measurement of clonogenic survival, cells are seeded in appropriate dilutions, subjected to ionizing irradiation at different doses, and 1–3 weeks later colonies are fixed, stained, and counted. To calculate the surviving fraction at a given dose, the number of colonies is divided by the number of seeded cells and normalized to the plating efficiency of the not irradiated controls. The log-transformed values of the surviving fractions are plotted against the corresponding irradiation doses, and traditionally linear-quadratic (LQ) regression analysis is performed, which describes the function of the surviving fraction by a second-degree polynomial with a linear and a quadratic term (see Eq. (1)) [4]. The coefficients and their α/β ratio reflect the weights of the linear and quadratic term upon the reduction in clonogenic survival. When α≫β, the function of the surviving fraction reveals a basically linear character (Fig. 1 a). In contrast, when β≫α, the curvature dominates the function (Fig. 1 b). Hence, the α/β ratio is well suited to differentiate curves with more linear or more quadratic shape, respectively (Fig. 1 c). Nevertheless, it fails to assess the overall steepness of the curve, which is obviously related to radiosensitivity and/or resistance (Fig. 1 a and b). Additionally, the α/β ratio lacks an intuitive biological interpretation. For many applications, it would be very helpful to have parametric information and/or quantitative scores, which reflect the overall survival curve with greater explanatory depth.
The linear-quadratic model can differentiate clonogenic survival curves with varying contributions of the linear and the quadratic terms but fails to reflect the overall steepness of the curves. Three-parameter log-logistic regression represents an interesting alternative. Hypothetical clonogenic survival curves with different α/β ratio and overall steepness were constructed. Curves with identical α/β ratio but different steepness are shown for α/β=20 Gy (a) and α/β=0.5 Gy (b). In (c), curves with different α/β ratio (3 or 20 Gy, respectively), but similar overall steepness are depicted. d Three-parameter log-logistic model with parameters ϕ 1, ϕ 2 and ϕ 3
In principle, clonogenic dose response datasets reveal sigmoidal character, thus rendering them suitable for non-linear regression analysis [5]. In this study, a multi-parameter equation with a log-logistic transformation of the predictor, i.e. the irradiation dose, was employed (Fig. 1 d and Eq. (3)). The parameters ϕ i exhibit a high degree of biological transparency, including the asymptotic clonogenic survival at infinite irradiation dose, the irradiation dose at the inflection point of the curve, and a measure related to its steepness (Fig. 1 d). To the best of our knowledge, a comprehensive comparison of LQ and non-linear log-logistic regression analysis has not been performed with clonogenic survival data so far. This is the aim of the present study.
Moreover, categorizing cells as sensitive or resistant with respect to ionizing irradiation might be helpful for multiple applications when studying the cellular responses towards ionizing irradiation as well as their underlying mechanisms. Quantitative scores of radioresistance and/or sensitivity with the power to reflect clonogenic survival over the whole dose range analyzed would bear strong informative value for correlation analyses with datasets of gene expression profiling and other "deep information datasets" that are increasingly being collected. Two methods of dimensionality reduction, which allow these calculations, are hierarchical cluster analysis and principal component analysis (PCA) [6, 7].
These considerations inspired us to perform the present study. Clonogenic survival data of eight human pancreatic cancer cell lines and immortalized human pancreatic ductal epithelial cells (HPDE) upon ionizing irradiation were subjected to regression analyses with the LQ and the non-linear log-logistic model as well as dimensionality reduction by hierarchical clustering and PCA. The results were compared, and the strengths and weaknesses of the respective algorithms are discussed. Finally, we propose a combination of all these methods as a workflow for the identification and validation of targets for biologically motivated improvements of cancer radiotherapy [8].
The human pancreatic cancer cell lines Capan-2, Dan-G, and FamPac were obtained from Cell Lines Service (Heidelberg, Germany). PaTu-8988T, Panc-1, MiaPaca-2, L3.6pl (a metastastic subclone of Colo 357), Suit-2 007 (a metastastic subclone of Suit-2) and immortalized human pancreatic ductal epithelial cells (HPDE) were kindly provided by Maximilian Schnurr, Department of Clinical Pharmacology, LMU Munich [9, 10]. Identity of cell lines was confirmed by short tandem repeat (STR) typing (service provided by the DSMZ, Braunschweig, Germany). Tumor cell lines were cultured in DMEM (Capan-2, PaTu-8988T, Panc-1, MiaPaca-2, L3.6pl, and Suit-2 007) or RPMI-1640 medium (FamPac and Dan-G) supplemented with 10 % heat-inactivated fetal calf serum (FCS), 100 U/ml penicillin, and 0.1 mg/ml streptomycin (all from Life Technologies, Karlsruhe, Germany) at 37 °C and 7.5 % CO2, or 5 % CO2, respectively. HPDE cells were maintained in a 1:1 mixture of keratinocyte serum-free medium and RPMI-1640 medium supplemented with 250 ng/ml human epidermal growth factor (EGF), 25 μg/ml bovine pituitary gland extract (BPE), 5 % FCS, 50 U/ml penicillin, and 0.05 mg/ml streptomycin (all from Life Technologies) at 37 °C and 5 % CO2.
Colony formation assays
Clonogenic survival was examined in colony formation assays. Cells were seeded as single cell suspensions into 6-well plates in a range of 100–100,000 cells per well in order to yield approximately 50 colonies per well depending on the different irradiation doses applied. Upon adherence for 4 h, cells were irradiated, and colony formation was allowed for 14 days. Subsequently, fixation and staining was performed in 80 % ethanol containing 0.3 % (w/v) methylene blue (both from Sigma-Aldrich, Taufkirchen, Germany), and colonies with more than 50 cells were counted. The number of colonies was divided by the number of seeded cells and normalized on the plating efficiency of the not irradiated controls. Data from 3–4 independent experiments were used for the statistical analyses.
Computations
Computations for this study were carried out using the R Software, version 3.2.1 [11]. All computer code used is available upon request.
Linear regression framework
Recall the LQ model
$$\begin{array}{@{}rcl@{}} y_{ij} & = & \exp\left(\alpha x_{ij} + \beta x_{ij}^{2} \right)\exp\left(\epsilon_{ij}\right) \\ \Leftrightarrow \ln\left(y_{ij}\right) & = & \alpha x_{ij} + \beta x^{2}_{ij} + \epsilon_{ij} \enspace, \end{array} $$
((1))
where x ij denotes the radiation dose for cell line i \((i=1,\dots,n)\) and replicate j \((j=1,\dots,J_{i})\), y ij are the resulting survival fractions (SFs) of cells and ε ij are error terms with \(\epsilon _{\textit {ij}} \sim \mathcal {N}\left (0,\sigma ^{2}\right)\). Clonogenic survival of all n cell lines was measured over the same range of radiation doses x=0,1,2,4,6,8 and 10 Gy. The semi-log regression model (1) is linear in the unknown parameters α and β that are to be estimated from the data. The goodness-of-fit of the model (1) can be evaluated by means of the coefficient of determination, denoted R 2.
Non-linear regression framework
In addition to the LQ model, we fitted regression models of the form
$$ y_{ij} = f(x_{ij}; \phi_{1},\dots,\phi_{p}) + \epsilon_{ij} \enspace, $$
where the mean function f is non-linear in one or more of the p parameters \(\phi _{1},\dots,\phi _{p}\). In particular, consider the following log-logistic mean function for sigmoidal curves (see e.g. [12]):
$$ f(x) = \phi_{1} +\frac{1-\phi_{1}}{\left(1+\exp\{\phi_{3}[\ln(x)-\ln(\phi_{2})] \}\right)^{\phi_{4}}} \enspace, $$
where ϕ 1 is the horizontal asymptote as x→∞, ϕ 2 is the inflection point of the curve at which the response is midway between 1 and ϕ 1, and the parameter ϕ 3 is proportional to the slope d f(u)/du at u 0 in the log-logistic mean function f(u) with u= ln(x) and u 0= ln(ϕ 2). The function is not symmetric (on the log scale) for ϕ 4 different from 1, hence ϕ 4 may be called an asymmetry parameter. We have chosen the log-logistic mean function (3) to ensure that f(x)=1 for x=0 Gy. Instead, the logistic mean function \(f(x) = \phi _{1} +\frac {1-\phi _{1}}{\left (1+\exp \{\phi _{3}[x-\phi _{2}] \}\right)^{\phi _{4}}}\) would be different from 1 at a dose of 0 Gy. We performed model selection comparing four nested models with mean function (3): the least restrictive four-parameter model with parameters \(\phi _{1},\dots,\phi _{4}\), a three-parameter model with ϕ 4 fixed at one, a two-parameter model with ϕ 4 fixed at one and lower asymptote ϕ 1 fixed at zero and a one-parameter model with ϕ 4=1, ϕ 1=0 and ϕ 3=1. For model selection we used the Akaike information criterion (AIC), which is defined as follows [13]:
$$ {\text{AIC}} = 2 (p+1) - 2 \ln(\mathcal{L}) \enspace, $$
where \(\mathcal {L}\) denotes the maximized value of the likelihood function for the model. The AIC rewards goodness of fit as assessed by the likelihood function, but also includes a penalty that is an increasing function of the number of estimated parameters. Given a set of candidate models for the data, the preferred model is the one with the minimum AIC value. However, the final model choice was based on both statistical and biological grounds. The chosen non-linear model was compared to a more general analysis of variance (ANOVA) model. The ANOVA model imposes no restrictions on how the response changes from one dose level to another, as there will be one parameter for each dose level. That is, the non-linear model is a submodel of the ANOVA model and we used an F-test [5] to test the null hypothesis that the ANOVA model can be simplified to the non-linear regression model. Note that the most common interpretation of R 2 for linear regression does not hold true for non-linear regression. In the non-linear case, the R 2 value is not the amount of variability in the dependent variable explained by the independent variable. In other words, in a non-linear regime the total sum-of-squares is not equal to the regression sum-of-squares plus the residual sum-of-squares (RSS). Therefore, R 2 should not be used as a goodness-of-fit measure in non-linear regression [14]. Instead one can use the residual variance \(\hat {\sigma }^{2} = \frac {{\text {RSS}}}{df}\), where the circumflex denotes the estimated value of σ 2 and df denotes the degrees of freedom for the model. Alternatively, one can use the residual standard error \(\hat {\sigma }=\sqrt {\hat {\sigma }^{2}}\) as a summary measure for the model fit [15]. We also devoted ourselves to model diagnostics checking whether substantive departures from the model assumptions can be found.
Cluster analysis of the clonogenic survival data was employed in order to classify radioresistant and sensitive cell lines [6]. Given the 9×6 data matrix X, which consists of mean survival fractions (in %) of the 9 cell lines measured at 6 radiation doses 1, 2, 4, 6, 8 and 10 Gy, a 9×9 distance matrix D was calculated using the Euclidean distance as a proximity measure. The matrix D was then analyzed by means of agglomerative hierarchical clustering, which produces a series of partitions of the data: the first consists of 9 single-member "clusters"; the last consists of a single group containing all 9 cell lines. At each stage, cell lines were fused that are closest according to Ward's method [16] in which the fusion of two clusters is based on the size of an error sum-of-squares criterion. We investigated the sensitivity of the results with respect to the clustering method. The obtained classifications are represented by a dendrogram, which illustrates the fusions made at each stage of the analysis. To provide further guidance for determining the number of clusters we present a silhouette plot, which is a means of assessing the quality of a cluster solution enabling one to identify "poorly" classified objects and so distinguishing clear-cut clusters from weak ones. More details on the interpretation of the silhouette plot are given in the "Results" section.
Principal component analysis (PCA) allows the extraction of scores of radioresistance for the cell lines under investigation [7]. Suppose without changing the notation that the columns of X have been mean-centered and scaled to unit variance. Then, the 6×6 sample correlation matrix is computed as C=X ⊤ X/(n−1) with X ⊤ being the transposed matrix of X. The eigendecomposition of C can be written as
$$ {\textbf{C}} = {\textbf{V}} {\boldsymbol{\Lambda}} {\textbf{V}^{\top}}~, $$
where Λ is a diagonal matrix with the eigenvalues of C sorted in decreasing order, λ 1≥λ 2≥…≥λ 6≥0, on its main diagonal and V is an orthogonal matrix whose columns \({\textbf {v}_{1}},\dots,{\textbf {v}_{6}}\) are the unit-norm eigenvectors of \(\lambda _{1},\dots,\lambda _{6}\). The matrix V is composed of coefficients or loadings and and the r-th sample principal component (PC) with mean zero and variance λ r is \(\textbf {z}_{r} = {\textbf {X}} \textbf {v}_{r}(r=1,\dots,6)\). Rescaled loadings are calculated as \(\textbf {v}_{r}^{*} = \sqrt {\lambda _{r}} \textbf {v}_{r}\) for which \(\textbf {v}_{r}^{*}{{\!~\!}^{\top }} \textbf {v}_{r}^{*} = \lambda _{r}\), rather than unity. For standardized data, this rescaling leads to coefficients that are the correlations between the components and the original variables. Various criteria for choosing the optimal number R of uncorrelated principal components (PCs) to be retained do exist [7]. For example, one proposal is to retain the first R components which explain a large proportion, (λ 1+…+λ R )/(λ 1+…+λ 6), of the total variation in the data, say 70–80 %. Another is to retain only components (for standardized data) that possess eigenvalues greater than one, which is known as Kaiser's rule [17].
Clonogenic survival of eight human pancreatic cancer cell lines and immortalized human pancreatic ductal epithelial cells (HPDE) was measured by colony formation assays upon exposure to ionizing irradiation, and the resulting dataset was subjected to different computational analyses in order to obtain measures of radioresistance and/or sensitivity.
LQ model
First, the most common method, the LQ model, was employed to fit the clonogenic survival data. Here, the log-transformed surviving fraction (SF) at a given irradiation dose is approximated by a second-degree polynomial with a linear and a quadratic term. A Trellis plot of the log-transformed survival fractions versus the irradiation doses was constructed, and the fitted regression curves were superimposed (Fig. 2 a). For all cell lines, a very good agreement between the observed data and the estimated regression curves was obtained indicating that the LQ model is capturing the systematic part in the data very well. The goodness-of-fit of the LQ model, as evaluated by means of R 2, varied between 95.46 % for HPDE and 99.32 % for FamPac cells (average 98.52 %). The estimated coefficients along with their \(\hat {\alpha }/\hat {\beta }\) ratios showed that the weights of the linear and quadratic term differ substantially across the cell lines (Fig. 2 b). Whereas the linear term did not reach statistical significance at the 5 % level for Suit-2 007 cells, this applied to the quadratic term in case of Dan-G and FamPac cells. Accordingly, in these three cell lines, the residual term dominated the reduction in clonogenic survival upon irradiation (the quadratic term in case of Suit-2 007, and the linear term in case of Dan-G and FamPac, respectively). The \(\hat {\alpha }/\hat {\beta }\) ratios ranged from 1.85 Gy (Suit-2 007) to 191.58 Gy (Dan-G), and their graphical display disclosed three groups of cell lines (Fig. 2 c). Given that the \(\hat {\alpha }/\hat {\beta }\) ratio is traditionally considered as a measure of radiosensitivity [4], this classified Suit-2 007 cells as particularly radioresistant and FamPac as well as Dan-G cells as highly radiosensitive. The majority of pancreatic cancer cell lines exhibited \(\hat {\alpha }/\hat {\beta }\) ratios between 4 and 16 Gy, which were comparable to or even higher than that of the non-malignant HPDE cells (\(\hat {\alpha }/\hat {\beta }= 4.69\) Gy). Hence, our clonogenic survival data do not confirm the common opinion that pancreatic cancer cells exhibit an extraordinarily high degree of radioresistance as compared to normal cells – at least in vitro [18, 19]. As anticipated, LQ regression of the clonogenic survival data and the resulting \(\hat {\alpha }/\hat {\beta }\) ratios revealed the major limitation in the sense that assessing radioresistance via the \(\hat {\alpha }/\hat {\beta }\) ratio cannot differentiate survival curves of similar \(\hat {\alpha }/\hat {\beta }\) ratio but discrepant steepness. This was exemplified in case of Capan-2 and L3.6pl cells. Whereas the \(\hat {\alpha }/\hat {\beta }\) ratio was rather similar (\(\hat {\alpha }/\hat {\beta } = 16.73\) Gy for Capan-2, and \(\hat {\alpha }/\hat {\beta } = 14.25\) Gy for L3.6pl, respectively), the overall steepness of the curves was obviously different characterizing Capan-2 cells as clearly more radiosensitive than L3.6pl cells. In the following, different approaches were used in order to overcome this shortcoming and to extract parameters and/or scores of the radiation response, which are superior to the \(\hat {\alpha }/\hat {\beta }\) ratio in terms of reflecting radiosensitivity and which have a more direct and intuitive biological interpretation.
Linear-quadratic regression analysis of clonogenic survival data obtained from eight human pancreatic cancer cell lines and immortalized human pancreatic ductal epithelial cells. Cells were subjected to colony formation assays upon irradiation at 0-10 Gy. After irradiation, cells were incubated for 14 days, the numbers of colonies with more than 50 cells were counted, and the surviving fractions were calculated. Results of 3–4 independent experiments are depicted in a Trellis plot of the log-transformed survival fraction versus radiation dose with fitted regression curves superimposed (a). Estimated coefficients along with the \(\hat {\alpha }/\hat {\beta }\) ratios for the LQ model (b, c)
Non-linear regression
Next, the clonogenic survival data were analyzed by means of non-linear log-logistic regression, which utilizes a multi-parameter equation for sigmoidal curves (Fig. 1 d). Table 1 displays the log-likelihood, the number of estimated parameters (p+1), and the Akaike information criterion (AIC) for each of the four nested non-linear log-logistic regression models. Naturally, the least restrictive model with four parameters gave the best fit to the data (in terms of the likelihood). However, for the clonogenic survival dataset, this model contained too many parameters, and particularly the asymmetry parameter ϕ 4 appeared redundant. Indeed, in terms of the AIC, models that fix ϕ 4 at 1 were superior (Table 1). On the other hand, reducing the model to the standard one-parameter log-logistic model proved not appropriate for the given dataset. The remaining alternatives of a two- or three-parameter model gave virtually equal results. The AIC for the two-parameter model with the lower asymptote ϕ 1 (reflecting the surviving fraction at infinite irradiation dose) fixed at 0 was slightly lower than that of the three-parameter model. However, we consider the latter to be the most suited one for our dataset, because in general the lower asymptote should be allowed to be either positive or zero for very high irradiation doses. The estimated regression curves obtained by fitting the chosen three-parameter model superimposed to the measured clonogenic survival data are shown in the Trellis plot (Fig. 3 a). For comparison, the fitted regression curves of the two-parameter model are also depicted. Visually, there was a good fit between the observed data and the estimated regression curves. The overall residual standard errors of the fitted three-parameter and two-parameter models were \(\hat {\sigma } = 0.0756\) and \(\hat {\sigma } = 0.0746\), respectively, confirming that the sigmoidal curves are capturing the variation in the data very well. Notably, the residual standard error of the LQ model was \(\hat {\sigma } = 0.4336\). The estimated parameters \(\hat {\phi }_{1}\), \(\hat {\phi }_{2}\) and \(\hat {\phi }_{3}\) along with their p-values are shown in Fig. 3 b; \(\hat {\phi }_{1}\) was never significantly different from zero indicating that clonogenic survival upon irradiation at infinite does converges to 0. For all cell lines, the scale factor \(\hat {\phi }_{3}\) was greater than 1 implying that the curves were steeper than the standard log-logistic curve. The irradiation dose, which resulted in 50 % survival and which is commonly referred to as median effective dose ED50 [12], is represented by \(\hat {\phi }_{2}\). It varied between 1.08 Gy (FamPac) and 2.75 Gy (Suit-2 007) with a mean of 2.17 Gy. Plotting the results of the different cell lines in the pace of \(\hat {\phi }_{2}\) and \(\hat {\phi }_{3}\) suggested this time that the cell lines split into 2 groups. Hence, the distribution was somewhat different from the distribution according to the \(\hat {\alpha }/\hat {\beta }\) ratios as obtained by the LQ model. FamPac, Dan-G, and Capan-2 cells formed a group of radiosensitive cell lines with low \(\hat {\phi }_{2}\) values (ED50), and low to intermediate \(\hat {\phi }_{3}\) values that are related to the steepness of the curve. On the other hand, high \(\hat {\phi }_{2}\) values and intermediate to high \(\hat {\phi }_{3}\) values were estimated for the radioresistant group comprising L3.6pl, Suit-2 007, HPDE, PaTu-8988T, MiaPaca-2, and Panc-1 cells. With ϕ 2 (ED50), the three-parameter log-logistic model provides a parameter that is more intuitive and has a more direct biological interpretation than the rather abstract α/β ratio of the LQ model. Additionally, log-logistic regression enabled the discrimination of more sensitive Capan-2 and more resistant L3.6pl cells. Thus, the three-parameter log-logistic model appears to be superior to the LQ model in terms of deriving parameters of radiosensitivity with more direct biological interpretation and more quantitative depth – at least in the given dataset.
Non-linear log-logistic regression of clonogenic survival data provides parameters with intuitive biological interpretation. A three- and a two-parameter log-logistic regression model were fitted to the clonogenic survival data. Trellis plot of the survival fraction versus radiation dose with fitted regression curves superimposed (a, red for the three-parameter, green for the two-parameter model). Estimated regression parameters and p-values for the log-logistic regression model (b); \(\hat {\phi }_{1}\) represents the survival fraction for which the dose approaches infinity, \(\hat {\phi }_{3}\) is a slope factor that refers to the steepness of the curve, and \(\hat {\phi }_{2}\) represents the irradiation dose that indicates a surviving fraction of 0.5 (ED50). c Plot of the cell lines in the space of \(\hat {\phi }_{2}\) and \(\hat {\phi }_{3}\)
Table 1 Log-likelihood, number of estimated parameters and AIC for four nested non-linear logistic regression models fitted to the Pancreas data
In order to compare the performance of the three-parameter log-logistic model to a more general ANOVA model, an F-test was performed. The lack-of-fit test was overwhelmingly non-significant (p-value 0.9824) further strengthening the suitability of the non-linear regression model. We also investigated other sigmoidal relationships between the surviving fraction and the irradiation dose to assess the mean structure in non-linear models. For example, the maximized log-likelihood of the three-parameter Weibull model was 237.77 compared to 249.59 of our chosen model (Table 1) indicating that this model did not fit the data better than the log-logistic model presented here. In conclusion, our results identify the three-parameter log-logistic model as highly appropriate for the analysis of clonogenic survival data.
By plotting the \(\hat {\alpha }/\hat {\beta }\) ratios of the fitted LQ model or the values for \(\hat {\phi }_{2}\) and \(\hat {\phi }_{3}\) of the fitted three-parameter log-logistic model, we already attempted to define groups of cell lines with different radiosensitivity (Figs. 2 c and 3 c). However, the grouping decisions were made merely by visual inspection, and thus were rather subjective. A more objective approach for classification is hierarchical clustering, which was applied to our dataset in the following.
At first, hierarchical clustering was performed with the original clonogenic survival fractions. The obtained dendrogram illustrating the process of the agglomerative hierarchical clustering, and the partitions produced at each stage are displayed in Fig. 4 a. The nodes in this diagram represent clusters, and the heights represent the distances at which each fusion is made. Large changes in fusion levels are considered to indicate the best cut of the tree, and thus suggest the number of clusters. The dendrogram of the clonogenic survival data revealed a clear structure that displayed two main groups with different radiosensitivity. Whereas cluster 1 comprised the more sensitive cell lines FamPac, Dan-G, and Capan-2, all other cell lines were located in cluster 2. A two-cluster solution was also observed when employing the non-hierarchical partitioning around medoids (PAM) cluster algorithm [20] (results omitted). As a means of evaluating the clustering process, silhouette plots are commonly employed [6]. The silhouette plot of the two-cluster solution obtained by the best cut of the hierarchical algorithm and PAM displays for each object (cell line) an index s i ∈ [ −1,1], called a silhouette (Fig. 4 b). When s i has a value close to 1, object i is taken "well classified". When s i is close to -1, object i is taken to be "misclassified". When the index is near zero it is not clear whether the object should have been assigned to its current cluster or a neighboring one. In Fig. 4 b the s i are displayed as horizontal bars, ranked in decreasing order for each cluster. The average silhouette width of all nine cell lines was 0.54, which can be considered a reasonable classification [20].
Cluster analysis of the clonogenic survival data discloses groups of radioresistant and sensitive cells. Euclidean distances of the nine cell lines were analyzed by means of agglomerative hierarchical clustering of original SF data (a, b), log-transformed SF data (b, c), and SF2 values only (d, e). Dendrograms (a, c, e): a two-cluster solution with a group of radioresistant and a group of sensitive cell lines can be identified. The silhouette plots (b, d, f) display a good quality of the obtained two-cluster solution
Notably, the obtained clusters were highly similar to the groups derived from the three-parameter log-logistic model (Fig. 3 c). Moreover, in contrast to the \(\hat {\alpha }/\hat {\beta }\) ratio of the LQ model, cluster analysis in fact was able to differentiate more sensitive Capan-2 from more resistant L3.6pl cells. We also performed cluster analysis on the log-transformed clonogenic survival data. The corresponding results were similar to the ones obtained by clustering the original survival fractions with the difference that Dan-G cells were allocated to the cluster with the radioresistant cell lines (Fig. 4 c,d). Clonogenic survival at 2 Gy (SF2) is widely used as a measure of radiosensitivity. Therefore, we also performed cluster analysis with the SF2 values only. The results were virtually identical to the ones obtained by categorizing the full range of SF values (Fig. 4 e,f). Interestingly, the two clusters were more clearly separated as compared to the clusters generated from the full range of SF data. This result is due to the fact that the SF2 clustering takes into account the survival at a single dose only and cannot reflect the observation that some cell lines appear sensitive in the higher dose range and resistant in the lower dose range and vice versa. Overall, we consider hierarchical clustering of clonogenic survival data as a versatile tool to categorize radiosensitive and radioresistant cell lines for further analyses. This might be of relevance for the selection of radioresistant and sensitive cell lines for studies aiming to delineate molecular mechanisms of the cellular response towards ionizing irradiation, and particularly for identifying and characterizing signaling pathways of resistance and/or hypersensitivity.
Hierarchical clustering of clonogenic survival data can serve as a method for categorizing radioresistant and sensitive cell lines. Yet, it does not provide a quantitative measure of radioresistance and/or sensitivity. Principal component analysis (PCA) is a method that offers this possibility. It utilizes orthogonal transformations in order to convert a set of correlated variables into a derived set of linearly uncorrelated variables, called principal components (PCs).
The sample correlation matrix of the clonogenic survival data was
$$ {\textbf{C}} = \left(\begin{array}{cccccc} 1.00 & 0.83 & 0.66 & 0.76 & 0.58 & 0.58 \\ 0.83 & 1.00 & 0.82 & 0.84 & 0.66 & 0.33 \\ 0.66 & 0.82 & 1.00 & 0.91 & 0.86 & 0.54 \\ 0.76 & 0.84 & 0.91 & 1.00 & 0.88 & 0.68 \\ 0.58 & 0.66 & 0.86 & 0.88 & 1.00 & 0.81 \\ 0.58 & 0.33 & 0.54 & 0.68 & 0.81 & 1.00 \\ \end{array} \right). $$
The correlation matrix (6) demonstrated moderately to strongly positive, linear correlations between all survival fractions. Hence, PCA appeared to be an adequate tool for dimensionality reduction of these data. The extracted first PC alone accounted for 76.9 % of the total variability in the data and was the only PC whose eigenvalue was greater than one (Fig. 5 a). Thus, according to Kaiser's rule [17], it was sufficient to retain only the first PC. The rescaled loadings \(\textbf {v}_{1}^{*}\) indicated a high degree of correlation with the extracted PC for all six surviving fractions (SF1-SF10) implying that the first PC represents a well-balanced measure of clonogenic survival over the whole dose range that was analyzed (Fig. 5 a). The corresponding scores of the first PC for all cell lines are shown in Fig. 5 b. They ranged between 2.94 in case of radioresistant PaTu-8988T cells, and −3.74 for radiosensitive FamPac cells. Indeed, the scores of the first PC can be interpreted as a weighted index of radioresistance within the dose range measured. A plot of the data projected into the subspace of the first two PCs (for better visualization) with the two-cluster classification solution superimposed is depicted in Fig. 5 c. It clearly confirms the two-cluster solution of radiosensitive and resistant cell lines that was obtained by hierarchical cluster analysis and shows the scores of the first PC as a quantitative measure of radioresistance. In order to compare the PCA results to the results obtained by LQ and log-logistic regression, correlation analyses between the extracted parameters and the first PC were performed. A significant negative correlation was observed between the \(\hat {\alpha }/\hat {\beta }\) value and the first PC supporting the common use of the α/β ratio as a(n) (inverse) measure of radioresistance. Intriguingly, an even better, yet positive correlation was obtained between \(\hat {\phi }_{2}\) and the first PC, again strengthening its versatility as an indicator of radioresistance. In terms of completeness, the correlation between \(\hat {\phi }_{3}\) and the first PC was calculated resulting in the absence of statistical significance (Fig. 5 d–f).
Principal component analysis of the clonogenic survival data enables the extraction of scores of radioresistance. Standardized clonogenic survival data were subjected to principal component analysis (PCA). In (a), the proportions of variance for the six PCs and the scaled loadings of the variables on the first PC that accounts for 76.9 % of the total variability in the data are shown. The first PC exhibits well-balanced loadings for the measured variables. The scores of the nine cell lines on the first two PCs are presented in (b). The resulting two-cluster solution is displayed in the space of the first two PCs that reflect 90.04 % of the total variability in the data (c). Pearson correlation analyses of the first PC were conducted with the estimated parameters extracted from the fitted regression models (d, e, f)
Polynomial models, such as the LQ model, are linear in the parameters. Fitting polynomial models can lead to an accurate approximation to the true regression function, their application for the analysis of clonogenic survival data being no exception. However, these models are empirical in the sense that they are only based on the observed relationship between the response and the predictors and do not include mechanistic considerations a priori [21]. For the LQ model, molecular theories of the underlying biological mechanisms were originally provided in the 1970s and have been refined later on [22–25]. Based on the assumption that clonogenic cell death upon ionizing irradiation derives from lethal DNA damage, which is predominantly represented by DNA double strand breaks (DSBs), the linear term in this model was attributed to DSBs resulting from single irradiation events, whereas the quadratic term was interpreted as DSBs resulting from two independent irradiation events. Yet, the parameters derived from LQ regression (α, β, and the α/β ratio) lack biological transparency, and in the present study they were shown to be inferior to the parameters derived from log-logistic regression in terms of explanatory power. We are well aware that from the LQ model one can easily compute ED50 and other values with direct biological interpretability. However, in the radiobiological routine this is simply not done. Instead, α/β ratios are commonly used as measures of the cellular response towards ionizing irradiation. And these clearly lack biological interpretability.
Unlike linear polynomial models, non-linear regression models are often based on a theory for the mechanism accounting for the response. In consequence, the model parameters in a non-linear model have a more plastic interpretation [21]. In case of the three-parameter log-logistic regression model used here, the curve of the surviving fraction takes on a sigmoidal shape between 1 and the lower asymptote ϕ 1 that corresponds to the surviving fraction for positively infinite dose; ϕ 3 is related to the steepness of the curve, and ϕ 2 corresponds to the irradiation dose at which the survival fraction reaches 0.5 (ED50). Other effective doses, such as ED10 or ED90, may be easily obtained from the fitted regression curve as well. Moreover, constraints can be built into a non-linear model easily and are harder to enforce for linear models. If, for instance, the response attains an asymptotic value as the dose grows, the non-linear models have such a built-in behavior automatically. We decided in favor of log-logistic regression instead of logistic regression to ensure that the response is equal to 1 for 0 Gy. A three-parameter model was selected in order to preserve the possibility of SF>0 for infinite irradiation doses, which may be observed for cellular subpopulations with absolute radioresistance.
The iterative nature of the fitting algorithm, which requires a set of user-supplied starting values, could be considered as a disadvantage of non-linear models. However, many software programs have built-in self-starter functions, which substitute for manually searching the starting values [15]. Non-linear regression models have successfully been used for describing dose-response dependencies for a long time [26]. They perform with a high degree of accuracy and provide intuitive biological interpretations. The present study shows that they also have their merits in describing the cellular response towards ionizing irradiation in colony formation assays. For the given dataset, the goodness-of-fit, as evaluated by means of the residual standard error, was even superior to the LQ model. It should be noted that fitting a model to the original survival data (as is the case for non-linear regression) emphasizes the low dose range with high survival fractions. In contrast, fitting a model to log-transformed survival data (as is the case for the LQ model) focuses on the higher dose range with low survival fractions. Given that the applicability of the LQ model in the higher dose range is being controversially discussed [27], and the clinically relevant dose range is 1–4 Gy, we do not consider this bias towards the lower dose range as a shortcoming but rather as an advantage of the non-linear regression model.
We have also employed dimensionality reduction techniques, namely cluster analysis to classify radioresistant and sensitive cell lines, and PCA that allows the extraction of scores of radioresistance. Note that these two techniques should not be considered as substitutes of regression models. Regression models and dimensionality reduction techniques have different purposes and provide complementary information. Whereas linear and non-linear regression models aim at quantitatively describing the relationship between a response and one or more predictors using a statistical model, dimensionality reduction techniques aim at summarizing the observed data in a lower-dimensional space. Nevertheless, often there is a benefit in analyzing the data from more than one angle, as a single technique is seldom able to reveal all important features in a given set of data. In this regard, our study suggests that a combination of hierarchical clustering, PCA, and regression of clonogenic survival data represents a promising approach for target identification in the context of biologically motivated improvements of cancer radiotherapy. Very recently, we have successfully used the aforementioned methods and identified HSP90 as a candidate molecule for targeted radiosensitization of resistant soft tissue sarcoma [28]. Cluster analysis was employed to classify radioresistant and sensitive cell lines from a panel of human soft tissue sarcoma cell lines. PCA-derived scores of radioresistance were applied to correlation analyses with transcriptomic data of the DNA damage response resulting in the identification of HSP90 and its client proteins ATM, ATR, and NBS1 as candidate mediators of radioresistance. Their functional involvement was addressed by pharmacological HSP90 inhibition, and the impact on clonogenic survival was finally examined and quantified by regression analyses.
The present study was performed in order to exemplify the versatility of non-linear regression and dimensionality reduction via hierarchical clustering and PCA as statistical alternatives to the LQ model for the analysis of clonogenic survival data. A combination of these techniques represents a powerful toolkit for the characterization of clonogenic cell death upon ionizing irradiation as well as the extraction of parameters and/or scores with quantitative explanatory power and direct biological interpretation. Together with results from transcriptomic, proteomic and other "big data" endeavors, these scores might be utilized to identify candidate targets and/or pathways for molecular manipulation of cellular responses towards ionizing irradiation as well as their functional validation.
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This work was supported by BMBF (02NUK024C). The authors thank Viola Blockus for excellent technical assistance.
Department of Medical Statistics University Medical Centre, Georg-August-University Goettingen, Goettingen, Germany
Steffen Unkel
Clinic for Radiotherapy and Radiation Oncology, LMU Munich, Munich, Germany
Claus Belka & Kirsten Lauber
Clinic Cooperation Group 'Personalized Radiotherapy in Head and Neck Cancer', Helmholtz Center Munich, Munich, Germany
Claus Belka
Kirsten Lauber
Correspondence to Kirsten Lauber.
SU, CB, and KL conceived the idea of this research and were involved in the conception and design of the paper. SU and KL drafted the manuscript. SU analyzed the data. All authors were involved in the interpretation of the findings, revisions to the original draft and have approved submission of the final manuscript.
Unkel, S., Belka, C. & Lauber, K. On the analysis of clonogenic survival data: Statistical alternatives to the linear-quadratic model. Radiat Oncol 11, 11 (2016). https://doi.org/10.1186/s13014-016-0584-z
Received: 04 November 2015
Clonogenic survival
Linear-quadratic model
Bioinformatics and systems biology in radiation oncology | CommonCrawl |
Theoretical Biology and Medical Modelling
Effects of pathogen dependency in a multi-pathogen infectious disease system including population level heterogeneity – a simulation study
Abhishek Bakuli ORCID: orcid.org/0000-0001-5123-19741,2,
Frank Klawonn1,3,
André Karch2,4 &
Rafael Mikolajczyk4,5,6
Theoretical Biology and Medical Modelling volume 14, Article number: 26 (2017) Cite this article
Increased computational resources have made individual based models popular for modelling epidemics. They have the advantage of incorporating heterogeneous features, including realistic population structures (like e.g. households). Existing stochastic simulation studies of epidemics, however, have been developed mainly for incorporating single pathogen scenarios although the effect of different pathogens might directly or indirectly (e.g. via contact reductions) effect the spread of each pathogen. The goal of this work was to simulate a stochastic agent based system incorporating the effect of multiple pathogens, accounting for the household based transmission process and the dependency among pathogens.
With the help of simulations from such a system, we observed the behaviour of the epidemics in different scenarios. The scenarios included different household size distributions, dependency versus independency of pathogens, and also the degree of dependency expressed through household isolation during symptomatic phase of individuals. Generalized additive models were used to model the association between the epidemiological parameters of interest on the variation in the parameter values from the simulation data. All the simulations and statistical analyses were performed using R 3.4.0.
We demonstrated the importance of considering pathogen dependency using two pathogens, and showing the difference when considered independent versus dependent. Additionally for the general scenario with more pathogens, the assumption of dependency among pathogens and the household size distribution in the population cohort was found to be effective in containing the epidemic process. Additionally, populations with larger household sizes reached the epidemic peak faster than societies with smaller household sizes but dependencies among pathogens did not affect this outcome significantly. Larger households had more infections in all population cohort examples considered in our simulations. Increase in household isolation coefficient for pathogen dependency also could control the epidemic process.
Presence of multiple pathogens and their interaction can impact the behaviour of an epidemic across cohorts with different household size distributions. Future household cohort studies identifying multiple pathogens will provide useful data to verify the interaction processes in such an infectious disease system.
Respiratory infections are the most common type of infections that contribute to loss of productive time due to acute conditions [1]. Households play an important role for the transmission process of respiratory infective agents, since they serve as confined structures due to the proximity of contacts among individuals that belong to such a confinement [2]. Approximately a third of the influenza like infection transmissions occur within households [3,4,5]. Studies on modelling epidemics spread in populations distributed into household clusters of varying sizes have been conducted to investigate possible control measures against epidemic outbreaks where larger households were associated with more infection transmissions [6,7,8,9,10].
Individual level stochastic models, also known as agent based models are highly flexible constructs to study complex phenomena by simulating the behaviour of multiple agents (individuals or grouped entities) simultaneously. FluTE [11] and FRED [12] are examples of such agent based models that have been built incorporating the community structure to study the progression of influenza like infections in the population [13,14,15].
Epidemic studies, till date, have mostly focused on the effect of a single pathogen in determining the population behaviour and spread of infections. Seasonal epidemics of respiratory infections are a common phenomenon during the winter months annually with several emergent and dominant pathogens circulating in the society. Additionally, there is always the possibility for antigenic drifts which are due to mutations of viruses impacting the protective effect of immunity from further infections [16]. Thus there is a need to study the epidemic reality of several pathogens co-existing in the community, with differential seasonality patterns, as well as differential severity and transmissibility characteristics. The idea of dynamic interaction between pathogens or ecological interference has been studied for diseases with differential seasonality in case of measles and whopping cough [17] and for the impact of vaccination for pandemic influenza [18].
The study of the infection process with multiple interacting pathogens has been lacking in the agent based models that have been developed in the past. Infection from one pathogen along with an intervention strategy, like household isolation, can not only have an impact on the individual's exposure to the specific pathogen but also to other pathogens which can eventually impact parallel epidemic processes from other co-existing pathogens. This involves cross immunity caused by an infectious pathogen, and changes in the contact structure among individuals within and between households. In addition to this, if there are two pathogens with exactly the same characteristics, they create a competition within the scope of the epidemic process. Additional factors like household structure and presence of an immunized proportion of individuals can impact the course of the epidemic since they can potentially accelerate or decelerate the transmission of infections in the population [8,9,10]. Moreover they are also directly related to the household isolation strategy since they impact the within household transmission. The aim of our study is to investigate how multi-pathogen interaction impacts the epidemic process when compared to scenarios with only a single pathogen if different household structures and the proportion of already immune individuals are taken into account.
Agent-based modelling of disease transmission
We use an agent-based approach with the basic structure of an SEIR (Susceptible, Exposed, Infectious, and Recovered) model. During the exposed state we assume that individuals are asymptomatic and do not impact the transmission process. After a period of being asymptomatic the individuals enter the infectious phase where they are symptomatic and can transmit infections. The assumption that during the infectious phase, there is household isolation making an individual nullify the risk of external infection from other pathogens, causes the interaction between pathogens in the multi-pathogen setting. The degree of this reduction of the external transmissibility depends on pathogen characteristics.
The single pathogen case is shown in Table 1. TP 1 (t) has two components: transmission of the pathogen resulting from contacts in the society (P external (p, t)) and from contacts within households ( P family (p, t)) for a given pathogen p. Transmission in the society depends on baseline infectivity of the pathogen (v), proportion of infectious in the society reduced by pathogen specific factor z, and a seasonality parameter (s(t)).
Table 1 The transition probability matrix for a single pathogen with the SEIR states
\( {P}_{external}\left(p,t\right)=v\ s(t)\ z\ \left(\frac{I\left(t-1\right)}{N}+{P}_0\right), \)with \( N=S(t)+E(t)+I(t)+R(t)\ and\ \frac{I\left(t-1\right)}{N}+{P}_0\le 1 \).
s(t) = A (sin (ω (t − t 0)) + 1)/2, ω = 2π/365, (if simulation starts on October 1st then t = 0, t 0 = 0). P 0 indicates external influx of infection, A indicates the amplitude of the seasonality function (for example the effects of outside temperature, Table 2). The parameters are calibrated to restrict P external (t) with the upper limit as one.
Table 2 Description of the symbols used in the mathematical formulation of the transition probabilities for describing the agent based model
Transmission in the family depends on pathogen characteristics - baseline infectivity v, factor for within family closeness of contacts c and number of infectious persons in the same household I h (t).
\( {P}_{family}\left(p,t\right)=1-{\left(1-v\ c\right)}^{I_h\left(t-1\right)} \) (Description provided in Table 2.)
Z(t) ∈ {Susceptible, Exposed, Infectious, Recovered} ∀ t ≥ 1, where Z is an individual in the study.
TP 1 = Probability(Z(t + 1) = Exposed | Z(t) = Susceptible) (Table 1).
=1 − (1 − P external (p, t)) ∗ (1 − P family (p, t)) (Table 2).
The probability TP 1 describes the transition probability from being Susceptible to becoming Exposed. The above formulation includes the specific scenario, when there is no possibility of a family based transmission, which is always the case for a single member household.
Let LP (>0) and IP (>0) (description in Table 2) be the average latency period and infectious period, respectively for a given pathogen p. TP 2 describes the transition probability of an Exposed individual becoming Infectious for the pathogen it is already exposed to. TP 3 describes the transition probability for an Infectious individual to obtain immunity or become Recovered for that pathogen for the remaining time in the study period. In this paper, we assume that LP and IP are independent constructs.
TP 2 = Probability(Z(t + 1) = Infectious | Z(t) = Exposed) (Table 1)
$$ \min \left(\frac{1}{LP},1\right)=q={TP}_2 $$
X(i, p)~Geometric(q)− After time X(i, p), that is, at time X(i, p) + 1, the i th individual becomes Infectious, since the time it became Exposed for pathogen p. (Description in Table 2).
TP 3 = Probability(Z(t + 1) = Recovered | Z(t) = Infectious) (Table 1)
$$ \min \left(\frac{1}{IP},1\right)=r={TP}_3 $$
Y(i, p)~Geometric(r)− After time Y(i, p), that is, at time Y(i, p) + 1 the i th individual acquires immunity, since the time it became Infectious from the pathogen p, for the remaining study period (Description in Table 2).We also assume that X(i, p) and Y(i, p) are independently distributed as a geometric distribution.
When there are multiple pathogens present in the society (p and p ' in our case are two different exemplary pathogens), we introduce an additional state of Susceptible + in the agent based model. On acquiring symptoms of infection (i.e. state as Infectious) with pathogen p, there is a check to verify if an individual is susceptible for another pathogen', i.e. Susceptible(p ' ) is True or False. If True, then the individual at Susceptible (p ' ) moves to Susceptible + ( p ' ) instantaneously. Once an individual at Infectious(p) moves to Recovered(p), we check once again if the individual is still susceptible to p ' , i.e. Susceptible + ( p ' ) is True or False. If True then Susceptible + ( p ' ) moves to Susceptible(p ' ) instantaneously (Fig. 1). A person at the state of Susceptible + is potentially at risk only to the household mode of infection transmission and can become Exposed. Following this, the steps for the exposed individual are the same as described for one pathogen. In case the individual reaches the state Recovered for p, while it is still at Susceptible + for some p ', then it becomes Susceptible once again for p '. The described process is pictorially represented through a Markov chain (Fig. 2). We vary the degree of household isolation using a parameter λ which takes values between zero and one to indicate differences in the risk of acquiring an infection from outside the household.
Graphical illustration of Susceptible, Exposed, Infectious, and Recovered states of the agent based model with some assumptions described. The time lines for the latency period and infectious period are also indicated through the dashed lines for an ith individual in the population for pathogens p and p', where p' ≠ p. The dependency assumption induces the Susceptible + state. The black arrows represent the influence direction, whereas the coloured arrows represent the transitions. The part above the dotted line indicates the states when only one pathogen is present in society, or when the pathogens are independently functioning in the system. The part below the dotted line is introduced when more than one pathogen is present in society and the pathogens interfere in the joint behaviour. *When an individual is Infectious for pathogen p and is still susceptible for another pathogen p ' it instantaneously moves to the state Susceptible + for pathogen p ' .** Once the individual is at the recovered state for pathogen p and is still at state Susceptible + for pathogen p ' it switches back to Susceptible state instantaneously
Markov chain describing the dependency process among pathogens. ** Once the individual is in the Recovered state for pathogen p and is still at Susceptible + state for pathogen p ' it switches back to Susceptible state instantaneously
\( Probability\left(\boldsymbol{Z}\left(\boldsymbol{t}+1\right)={Susceptible}^{+}\kern0.1em |\kern0.1em \boldsymbol{Z}\left(\boldsymbol{t}\right)= Susceptible\right)=1\kern0.1em \mathrm{for}\kern0.17em \mathrm{pathogen}\kern0.1em {p}^{\hbox{'}} \) (Fig. 1).
When Z(t) = Infectious for pathogen p and p ≠ p '.
The description of P family remains unchanged. TPS + describes the probability that, an individual at the state Susceptible + remains as Susceptible +. This indicates the situation where the individual does not get exposed to a pathogen p ' but remains symptomatic for p. This depends on the pathogen characteristics for both p and p ' .
$$ {TPS}^{+}= Probability\left(\boldsymbol{Z}\left(\boldsymbol{t}+1\right)={Susceptible}^{+}\kern0.5em |\ \boldsymbol{Z}\left(\boldsymbol{t}\right)={Susceptible}^{+}\right) $$
=(1 − (1 − P family (p ', t)) ∗ (1 − (1 − λ)P external (p ', t))) ∗ (1 − TP 3( p)) (Fig. 2).
Population structure
We have considered three population structures with different properties (Germany, India, one-person structure). The data on the household size distribution in Germany was used from DESTATIS (Statistisches Bundesamt, Wiesbaden 2015 report) while that of India from the census reports of 2011 [19, 20]. We have also considered a hypothetical population of one person households as the most extreme scenario. This has been described with the frequencies for each household size in Table 3. We distributed 10,000 individuals into each population scenario.
Table 3 The household size distributions for the different populations considered to describe the epidemic outcomes from simulations using the agent based model
Pathogen characteristics
We studied a general multi-pathogen setting with n (n = 10) pathogens with characteristics chosen to reflect potential real life situations as described in Table 4. Baseline infectivity v was calibrated to achieve a maximum incidence rate of approximate 10% in person weeks for respiratory infections during the peak winter season (https://grippeweb.rki.de). Two broad types of pathogens were considered, the influenza type and the common cold type. Influenza type pathogens have typically reported shorter latent periods (period with asymptomatic infection; 1–4 days for influenza and common cold between 1 and 6 days) but a longer infectious period (period with symptomatic infection; 5–9 days for Influenza and 2–3 days for Common cold) while it has been reported as the opposite for common cold type of pathogens [21,22,23].
Table 4 Pathogen characteristics. This table with the input parameters for the simulation of the agent based model with ten pathogens. I indicates influenza type while C indicates common cold type of pathogen
The simulation proceeded in discrete time steps. Each step denoted a day in the follow up period. Based on the initial number of Infectious individuals, the epidemic process began its course of action. It followed the seasonal trend of the pathogen, the relation to other household members, and the prevalence of the infection for the specific pathogen in society at a given time point. We started initially with two pathogens from Table 4 (Pathogen 6 and Pathogen 10). Pathogen 6 would be in accordance with the characteristics of a pandemic influenza strain whereas pathogen 10 would correspond to the characteristics of human respiratory syncytial virus (HRSV). Then we observed the scenario where both the pathogens jointly interact. Finally we looked at the general scenario with 10 pathogens jointly which would be a more appropriate representation of the reality during the winter season [18] (https://grippeweb.rki.de).
The comparisons were done for the scenarios of pathogen dependencies, 1) assuming all pathogens existed independently (λ = 0%), and 2) assuming the pathogens worked together and influenced each other (λ = 100%) (indicated by Pathogen Dependency- Yes or No); household size distribution based on different household size distributions in different countries (Country – Germany, India or Hypothetical). At the start of the simulation few people were infectious for every pathogen, which was denoted as I(1) to kick start the infection process, while the number of people already immune at the start were represented as R(1).
In addition to the above, we assumed that, for every pathogen there would be a small chance that an individual could acquire an infection from outside the system. This has been described as the external influx of infection. We had set this value to one in a ten thousand, at each observational time point (day) in the epidemic process. Besides this, the maximum number of days spent as infectious had been censored to 55 days. Each of the scenario combinations were replicated 100 times for a study period of 150 days in the peak season for respiratory illness.
In our base case scenario with the German population and pathogen dependency (λ = 100%), we look at the same temporal tend (seasonality) for all the pathogens considered. However to present the effect of differential seasonality, we introduce a different temporal trend for pathogen 10, by modifying the value of t 0 as +45 days and −45 days. This results in shifting the peak of the epidemic for these pathogens and impacts the overall epidemic process when multiple pathogens are present. Also for this scenario we evaluated the effect of change in λ from 0% to 100% in steps of 10% which would allow us to infer on the importance of the pathogen dependency assumption through the introduction of household isolation.
We measure the epidemiological parameters of interest which are 1) height of the epidemic peak (peak prevalence), 2) time taken to reach the peak of the epidemic, 3) incidence proportion (attack rate) of infections in the study period, and 4) incidence proportion stratified by household size for the different populations in consideration, through our simulations as described above. Summary statistics are presented for all the outcomes described above.
We observe the peak prevalence and the incidence proportion for the pathogens 6 and 10, both individually and jointly. We are interested in the hypothesis that jointly modelling pathogens creates a competition, and hence we would observe lower values of the peak prevalence and incidence proportion, compared to observing them individually. The observations are compared using the non-parametric Mann–Whitney–Wilcoxon test for evaluating the difference when observing joint epidemics. The parametric version with the paired t-test also gives us similar results, however due to no necessity of normal distribution assumptions the Mann–Whitney–Wilcoxon test values are reported [24].
We also use a simple linear regression model [25] on the outcomes described above and show the confidence intervals for the slope across different outcomes to indicate the impact of pathogen dependency on the country variable (used to describe the different household size distribution) in the scenario with 10 pathogens. The covariate used is the coefficient for the degree of household isolation (0 to describe the independent scenario and 1 to describe complete household isolation in case of the dependency). The confidence intervals show the variability in the slopes across different country variables indicating different household size distributions. For studying the degree of household isolation, we use the Generalized Additive Model (GAM). GAM's are an extension of the generalized linear model (GLM) allowing for some kind of smoothing of the predictor variables. The advantage of GAMs is that it allows us to deal with highly non-linear and non-monotonic relationships between the response and the predictor variables often driven by the observed data at hand [26, 27]. GAMs are also used in this work to model the dependency of incidence proportion of infections stratified by household size where a non-linear relationship is observed.
In our simulations, pathogen 6 demonstrated behaviour similar to a pandemic influenza epidemic. Hence this was the most severe pathogen in our list. Pathogen 10 was the second most severe among the pathogens present. The differences in the household size distribution described through the country variable have been demonstrated in Table 5. Smaller household sizes were associated with less severe epidemics demonstrated through the smaller values of the peak prevalence as well as the lower incidence proportion. And the epidemic process was also slower which would be indicated through the delayed median time in reaching the peak prevalence of infections in the study period. The epidemic almost never occurred (low values of incidence and prevalence and high variability in time to reach the peak prevalence) in the hypothetical population cohort where within household infection transmissions were completely absent (Fig. 3). Simulation of pathogen 6 and pathogen 10 jointly was associated with household isolation during the symptomatic phase of an episode, and this brought in competition within the two pathogens during the epidemic process. We tested the hypothesis that incidence proportion and peak prevalence was higher in individual simulations of the pathogen as opposed to the joint interaction of the two pathogens in a system. The difference could be observed only where the epidemic occurred (i.e. not in the hypothetical population cohort). Furthermore we also evaluated the hypothesis that the sum of the independent peak prevalence and the incidence proportion from the two pathogens was greater than the joint overall peak prevalence and incidence proportion in the two pathogen system. Here too the difference was observed except in the hypothetical cohort (Table 5). In the two pathogen system the time taken to reach the peak prevalence was dominated by the pandemic pathogen (pathogen 6). However there was no difference observed in this duration in the two pathogen system and separately simulated individual pathogen systems. When two exactly same pathogens were considered then the results of the comparison were similar. However when two pathogens were more aggressive (characteristics of pathogen 6) then the difference in the peak prevalence of infections between jointly modelling them and considering them individually, were significantly higher than the scenario when two pathogens had moderate characteristics (characteristics of pathogen 10).
Table 5 Summary and comparison of two pathogen system (S2) vs. one pathogen system (S1). The pathogen is indicated in the parenthesis. S1(P6 + P10) indicates the sum of the individual values from the pathogen independently whereas S2(P6 + P10) indicates the system where the household isolation introduces pathogen dependency and the pathogens function jointly. The outcomes of peak prevalence and incidence proportion (during the 150 day period) along with their 95% confidence intervals (based on Monte-Carlo simulations) are shown in the summary section. The comparison section displays the non-parametric p values (based on the Mann-Whitney-Wilcoxon test) obtained when comparing the pathogen systems over the simulation runs
Difference across the country locations indicating the different household size distribution and the coefficient of household reduction during the symptomatic phase of the infection. The slope is obtained from the linear model to indicate the change caused by the most extreme difference in the coefficient due to the dependent scenario (all pathogens interacting with dependency) and the independent scenario (all pathogens working independently). This is also visible in Table 6. The outcomes of interest that have been presented are 3.1- peak prevalence during the observed epidemic, 3.2- incidence of infections during the 150 day period of interest. (Numbers above 1 indicate that cumulative probability of infections during the study period was above 100%)
Since the multi pathogen scenario would be a more probable model, we consider 10 pathogens as described in Table 4. These include among them pathogen 6 and pathogen 10 which have been described before. We now vary the coefficient of household reduction process between the extremes of 0 and 1, indicating pathogens functioning independently in the population and pathogens interacting within the population respectively. Looking across the different country variables for varying household size distributions, we observed that here too societies with larger household sizes had an accelerated and more severe epidemic (Fig. 3 (3.1, 3.2) and Additional file 1). Also we looked at the difference introduced by the extreme values of household isolation during the infectious phase using the slope of the linear regression model. The summary of the slopes indicated the differences when the epidemic took place (Table 6). For the German household size and hypothetical household size distributed cohort we could not observe any significant decrease in the speed of the epidemic as opposed to the cohort with the Indian household size distribution where there was an accelerated epidemic observed with an increased coefficient of household isolation.
Table 6 Comparison of slopes across the different country locations. This indicates the observed difference in the outcomes from the epidemics due to the differences in the coefficient of household isolation (the extreme scenarios of complete dependency versus pathogens functioning independently) and the household size distribution in the country location used as shown in Fig. 3 (3.1, 3.2)
We analysed the impact of changing the coefficient of household isolation during the infectious period by varying it from 0 to 100% in steps of 10% for the cohort with the German household size distribution. The time taken to reach the epidemic peak remained unchanged with the variation of the household isolation coefficient. However there was a decrease in the epidemic peak and the incidence proportion of infections with an increase in household isolation coefficient. This would be a result of decreasing contacts during the infectious phase with the society making individuals less vulnerable to newer infections during this period. The results are represented in Fig. 4 (4.1 and 4.2) and the smoothed coefficients based on the coefficient of household isolation for the GAM regression was highly significant for the outcomes of incidence proportion and peak prevalence of infections (both below 0.0001). However it was not significant for the outcome of the time taken to reach the peak prevalence. Fitting of a simple linear regression model with the above scenario also gave us similar results and the slope was always negative (except for the outcome of time to reach the incidence peak). However the fit was better with the GAM model through the median points.
Epidemic outcomes with varying degree of household isolation. We observe a decrease in peak prevalence (4.1), and incidence of infections (numbers above 1 indicate that cumulative probability of infections during the study period was above 100%) (4.2), with the increase in the degree of household isolation during the infectious phase
Additionally, we looked at the proportion of individuals who were at home at a given time point. Together with this we also looked at the distribution of the proportion of people who were infectious for one or more pathogens on a given time point. We assessed these proportions in simulations for the German household size distribution, with immune individuals in the population, and 10 pathogens interacting with each other during the epidemic process (Fig. 5). Our calculations showed that majority of the cases where the person was symptomatic and remained at home, was due to one pathogen with a proportion of 0.9959 (0.99, 1.00) (median with 5th and 95th percentile values in bracket). For two pathogens at a time, the proportion was 0.003 (0.00, 0.01) (median and with 5th and 95th percentile values in bracket) whereas for three pathogens at a time the median proportion was already zero. This could also be seen in Fig. 5, where the proportion of infectious individuals for two or more pathogens at a time point was very close to the zero line. For the similar scenario with the household size distribution of India in the cohort, we obtained similar results. The proportion of household stays due to one pathogen infection was dominant, 0.9963(0.96, 1.00) (median with 5th and 95th percentile values in bracket). There were also some rare cases of being simultaneously infected by three or four pathogens.
Simulation results showing the average population proportion from 100 simulated epidemics during the epidemic period that are under household isolation for being symptomatic for infections. The black and the red line indicate how the proportion of people acquires infections during the course of the epidemic and then recover with time. The red line shows that a maximum of a tenth of the population remains at home on an average during the epidemic period. The blue line almost covers the red line indicating that majority of the infection episodes are caused by one pathogen. The pink and the grey lines are almost close to zero at all the time points indicating how unlikely it is for an individual to be infected with more than one pathogen at a time
We also analysed the incidence proportion stratified for the household size in the different population distributions. We saw an increase in incidence proportion of infections with increase in household size (Fig. 6). Here also we used a GAM model to represent the nonlinear relationship between incidence proportion and household size. For the hypothetical population cohort we could not have any relationship because it only represented one membered household. The incidences of the one member household were also different across the different population distributions with higher incidences in population with larger household sizes.
Incidence of infections stratified by household size (numbers above 1 indicate that cumulative probability of infections during the study period was above 100%)
Finally, we observed the impact of shifting the temporal trend for pathogen 10 (representing the characteristics of RSV virus) as opposed to pathogen 6 (representing the characteristics of pandemic influenza virus) and the remaining 8 viruses in the 10 pathogen system. The trend for pathogen 10 was shifted by using the different values for t 0 as +45 and −45 days. We performed the simulations only for the German population type with dependency among pathogens (complete household isolation during infectious phase). In comparison to the base case scenario, there was a decrease in the peak prevalence as well as the incidence proportion due to the temporal shift in the trend for pathogen 10. The decrease was significantly higher for the shift where the peak for pathogen 10 is delayed by 45 days as opposed to the peak coming forward by 45 days. The time taken to reach the epidemic peak remained unchanged (Fig. 7 (7.1 and 7.2)). Also the decrease in the incidence proportion was comparatively smaller than for the peak prevalence between the scenario where the peak was forward by 45 days for pathogen 10 and base case.
Epidemic outcomes for the base case scenario (all pathogens temporally aligned in their seasonality) in comparison to the scenarios where pathogen 10 has a shifted temporal trend. The shifting reduces the intensity of the epidemic. The reduction is more when there is a delayed peak in the epidemic for pathogen 10 as opposed to an earlier peak. (Incidence numbers above 1 indicate that cumulative probability of infections during the study period was above 100%)
We have proposed an agent based model to study the behaviour of epidemics under the influence of multiple pathogens working simultaneously in the population. With the presence of two pathogens in such a system without the influence of any other effect, we could demonstrate how the interference of the pathogens in the infection process played a role in controlling the epidemic process (lower number of infected individuals as well as lower daily incidence proportion). The interference among pathogens was introduced through the assumption of household isolation during the period of being symptomatically infectious, where the individual was immune to the risk of acquiring infections from outside the household. To our knowledge this was the first time for studying the behaviour of an epidemic process incorporating the influence of multiple pathogens using an agent based model. We further went on to present a more general scenario where there are 10 pathogens, and also the impact from recovered individuals being present in the population at the start of the epidemic process.
Our simulations were performed to study the impact of the dependency among pathogens as opposed to pathogens functioning independently (2 extreme levels for the coefficient of household isolation during the infectious period), and the household size distributions of different populations (three different populations with varying household size distributions) The population system reached a stable state at the end of simulation period, confirming that the epidemic had almost died out in 150 days (approximately 5 months during winter season). The dependencies among pathogens were important determinants in controlling the epidemic process. Additionally, the household size distributions did produce significant differences in the peak of the epidemic (peak prevalence) and the incidence proportion in the study period of interest. For common respiratory infections like influenza and common cold, household size can be an important factor determining their spread as seen for influenza or influenza like illnesses based hospitalizations: the population structure difference has accounted for a third of the observed variation [28]. In our simulations we observed that household size distribution influences the speed of the epidemic. Population with larger household sizes reached the peak of the epidemic much faster than those with smaller household sizes. Looking at the incidence of infections across household sizes, we could see that larger households were associated with more infections due to the intra household infection spread, consistently with assumed random mixing within the household.
Looking across the different pathogens, we observed that the infectious period also is important in shaping the severity of the epidemic. Pathogen 6 and 10 as considered in the simulation have almost similar characteristics except for the duration of the infectious phase, but this resulted in different severity of the epidemic. Also in the multipathogen scenario, the epidemic characteristics are dominated by pathogen 10. Shifting of the temporality to introduce a peak 45 days before for pathogen 10 allows for more infections in the multipathogen system as opposed to a delay in the peak.
Our simulation study does come with limitations. There are common challenges associated with agent based models, especially in statistical methods for hypothesis testing in combination with determining the number of appropriate simulation runs [29]. In addition to the standard challenges, our assumptions are largely simplistic in nature, assuming for random mixing within the household and the population is looked upon as an assortment of homogenous agents. The increase in contacts with the increasing household size may not necessarily take place. Secondly, we induce a sort of isolation for the transmission process of infections, but we do not account for the specific severity of the infections, except for the duration of being symptomatic and infectious. The severity of the pathogen can directly influence the duration of isolation. Even for our sensitivity analysis, we assume this parameter to be same for all the pathogens. Additionally, we also assume same transmissibility characteristics for all the pathogens. These are strong assumptions that have been made for the realization of the system in a simplistic way. However, this model can be extended easily to observe more complex realizations of a realistic system.
Through our agent based model formulation, we could demonstrate the importance of considering the multi-pathogen interactions in controlling the spread of infections during an epidemic process. Household size and dependency among pathogens are important factors in determining the outcome of the epidemic. Future prospective studies in household cohorts looking at pathogen identification and coinfections can provide quantitative measures for specific characteristics of the multi-pathogen system. This kind of data can also be used to test the validity of the assumptions made in simulation models.
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We thank all the group members of the Epidemiological and Statistical Methods research group at HZI, Braunschweig for their valuable feedback and discussions.
Internal funding of the Helmholtz Centre for Infection Research.
The R code for the simulations supporting the conclusions of this article is available from the corresponding author upon request.
Helmholtz Centre for Infection Research, Research Group Biostatistics, Braunschweig, Germany
Abhishek Bakuli
& Frank Klawonn
PhD Programme "Epidemiology", Braunschweig-Hannover, Germany
& André Karch
Department of Computer Science, Ostfalia University of Applied Sciences, Wolfenbuettel, Germany
Frank Klawonn
Helmholtz Centre for Infection Research, Department of Epidemiology, Braunschweig, Germany
André Karch
& Rafael Mikolajczyk
Hannover Medical School, Hannover, Germany
Rafael Mikolajczyk
Institute for Medical Epidemiology, Biometry, and Informatics (IMEBI), Medical Faculty of the Martin Luther University Halle-Wittenberg, Halle (Saale), Germany
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RM and FK conceived the idea of the simulation study. All authors contributed to the theoretical development of the model. AB programmed the simulations and statistical analyses. AB, FK made contributions to the statistical analyses and simulations. AB drafted the manuscript. All authors contributed to the interpretation of the data, writing, and revising of the manuscript and approved the final manuscript.
Correspondence to Rafael Mikolajczyk.
Additional file
Time taken to reach the peak prevalence varies according to the household size distribution in the cohort. Populations with larger households on an average experienced the epidemics at an accelerated rate compared to populations with smaller households on an average. (PNG 24 kb)
Bakuli, A., Klawonn, F., Karch, A. et al. Effects of pathogen dependency in a multi-pathogen infectious disease system including population level heterogeneity – a simulation study. Theor Biol Med Model 14, 26 (2017) doi:10.1186/s12976-017-0072-7
Received: 22 September 2017
Pathogen dependency
Multi-pathogen | CommonCrawl |
Isabel Hubard Escalera
Isabel Alicia Hubard Escalera is a Mexican mathematician in the Institute of Mathematics of the National Autonomous University of Mexico (UNAM).
Isabel Alicia Hubard Escalera
NationalityMexican
OccupationMathematician
Known forStudies of symmetries of combinatorial objects
AwardsL'Oréal-UNESCO-AMC Fellowship in the area of Exact Sciences, 2012
Kovalevskaia Fund Prize, 2010
Academic background
Alma materYork University (Doctoral). School of Sciences, UNAM (Undergraduate).
ThesisFrom geometry to groups and back: the study of highly symmetric polytopes (2007)
Doctoral advisorAsia Ivić Weiss
Academic work
InstitutionsInstitute of Mathematics, UNAM
Websitehttp://www.matem.unam.mx/fsd/hubard
Early life and education
As a child, Isabel Alicia Hubard wanted to be a bullfighter. She has said of her family, "My mother is an engineer and my father an accountant. My brother is a mathematician and my sister a physicist. I never thought that I would like math. I simply found it easy and fun, but nothing more. However, my mathematics teacher in junior high and high school, Óscar Chávez, inspired me."[1]
Hubard Escalera began her studies in the Faculty of Sciences of the UNAM, where in 2001, she graduated in Mathematics with a baccalaureate thesis titled Polyhedra colored with cyclic orders.[2][3] It was written in the Institute of Mathematics of the UNAM, where she carried out investigations related to the combinatorial properties of discrete geometrical objects.[4][5][6] Her undergraduate advisor was Javier Bracho Carpizo.[3]
In 2007, she earned a Ph.D. from York University in Canada, with a dissertation titled From geometry to groups and back: the study of highly symmetric polytopes.[7][8]
Career
Hubard Escalera investigates the study of symmetries of combinatorial objects.[2]
She has been the organiser of the Mexico City Mathematics Olympiad in Mexico City since 2013, an organization that played a prominent role in recent national competitions, achieving the first place of the 2017, 2018 and 2019 Olimpiadas Mexicanas de Matematicas (OMM), and won the basic education national competitions (OMMEB) the same years.[6] She is the delegate for Mexico City in the Mexican Mathematics Olympiad of the Mexican Mathematics Society.[1] She has also been the leader of the Mexican team at EGMO since 2014, competition in which Mexico has won two gold medals, with Isabel mentoring the girls.
Recognition
She was awarded the Kovalevskaia Fund Prize in 2010.[9]
In 2012 she was the first Mexican mathematician to receive the L´Oréal-UNESCO-AMC Fellowship in the area of Exact Sciences for her work, titled Algebra, combinatorics and geometry of abstract two-orbit polytopes.[2][5] The Fellowship is awarded to "promote the participation of women in science for advanced scientific studies in universities or other recognized Mexican institutions in the areas of exact sciences, natural sciences and engineering and technology."
She was elected to the Mexican Academy of Sciences in 2022.[10]
Selected publications
• Hubard, Isabel; Mixer, Mark; Pellicer, Daniel; Weiss, Asia Ivić (2015). "Cubic tessellations of the helicosms". Discrete & Computational Geometry. 54: 686–704. doi:10.1007/s00454-015-9721-y.
• Cunningham, Gabe; Del Río-Francos, María; Hubard, Isabel; Toledo, Micael (2015). "Symmetry type graphs of abstract polytopes and maniplexes". Annals of Combinatorics. 19: 243–268. doi:10.1007/s00026-015-0263-z.
• Conder, Marston; Hubard, Isabel; O’Reilly-Regueiro, Eugenia; Pellicer, Daniel (2015). "Construction of chiral 4-polytopes with alternating or symmetric automorphism group". Journal of Algebraic Combinatorics. 42: 225–244. doi:10.1007/s10801-014-0579-5.
• Bracho, Javier; Hubard, Isabel; Pellicer, Daniel (2014). "A finite chiral 4-Polytope in $\mathbb {R} ^{4}$". Discrete & Computational Geometry. 52: 799–805. doi:10.1007/s00454-014-9631-4.
• Araujo-Pardo, Gabriela; Hubard, Isabel; Oliveros, Déborah; Schulte, Egon (January 2015). "Colorful associahedra and cyclohedra". Journal of Combinatorial Theory. Series A. 129: 122–141. doi:10.1016/j.jcta.2014.09.001.
• Mixer, Mark; Hubard, Isabel; Pellicer, Daniel; Weiss, Asia Ivić. "Cubic tessellations of the didicosm". Advances in Geometry. 14 (2): 299–318. doi:10.1515/advgeom-2013-0035.
• Arocha, Jorge L.; Bracho, Javier; García-Colín, Natalia; Hubard, Isabel (August 2015). "Reconstructing surface triangulations by their intersection Matrices". Discussiones Mathematicae Graph Theory. 35 (3): 483–491. doi:10.7151/dmgt.1816.
• Arroyo, Aubin; Hubard, Isabel; Kutnar, Klavdija; O’Reilly, Eugenia; Šparl, Primož (2015). "Classification of symmetric Tabačjn graphs". Graphs and Combinatorics. 31: 1137–1153. doi:10.1007/s00373-014-1447-8.
• Hubard, Isabel; Leemans, Dimitri (2014). "Chiral polytopes and Suzuki simple groups". In Connelly, Robert; Weiss, Asia Ivić; Whiteley, Walter (eds.). Rigidity and Symmetry. Fields Institute Communications. Vol. 70. Springer, New York. pp. 155–175. doi:10.1007/978-1-4939-0781-6_9.
• Araujo-Pardo, Gabriela; Hubard, Isabel; Oliveros, Déborah; Schulte, Egon (2013). "Colorful polytopes and graphs". Israel Journal of Mathematics. 195 (2): 647–675. doi:10.1007/s11856-012-0136-7.
References
1. Zubieta García, Judith (November 2015). Young Women Scientists: A bright future for the Americas (PDF). Inter-American Network of Academies of Sciences. ISBN 978-607-8379-24-8. Archived from the original (PDF) on 2018-05-08. Retrieved 2017-01-24.
2. Zúñiga Murrieta, Nancy (2013-04-05). "Coordinación de Comunicación y Divulgación, AMC - Vuelven los Poliedros a Ser Retomados por los Matemáticos". www.comunicacion.amc.edu.mx (in Mexican Spanish). Academia Mexicana de Ciencias. Retrieved 2017-01-24.
3. "Tesis que para Obtener el Titulo de Matematica, Presenta: Isabel Alicia Hubard Escalera". 132.248.9.195. Retrieved 2017-01-24.
4. "Isabel Hubard — Instituto de Matemáticas | UNAM". www.matem.unam.mx (in Spanish). Retrieved 2017-01-24.
5. "Isabel Hubard Escalera, primera matemática en obtener la beca L'Oréal-UNESCO-AMC". www.dgcs.unam.mx. Boletín UNAM-DGCS-280 Ciudad Universitaria. 2013-05-05. Retrieved 2017-01-24.
6. Mateos-Vega, Mónica (2015-06-26). "La Jornada: Combatir los mitos que rodean a las matemáticas, el gran reto". La Jornada. Retrieved 2017-01-24.
7. "Convocation" (PDF). York University. February 2008. p. 13. Retrieved 2017-01-24.
8. Isabel A. Hubard (2007). From Geometry to Groups and Back: The Study of Highly Symmetric Polytopes. York University (Canada). ISBN 978-0-494-45997-3.
9. "Winners of the Kovalevskaia Fund Prizes and Scholarships since 2005" (PDF). The Kovalevskaia Fund. 2015. Retrieved 2017-01-17.
10. "Mathematics section members" (PDF). Mexican Academy of Sciences. 2022. Retrieved 2022-12-05.
External links
• Combinatoria (video, 57:32)
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
| Wikipedia |
Non-ordinary differential equation?
Does such a thing exist ?
Can't seem to find anything about it so i was wondering : why bother calling something "Ordinary Differential Equation" if the "Ordinary" part doesn't bring anything to it ?
ordinary-differential-equations terminology
MJD
It's called an "ordinary differential equation" because it involves a function (or functions) of one independent variable and normal (i.e. not partial) derivatives, whereas a "partial differential equation" involves functions of multiple independent variables and partial derivatives.
For example, $$ y' + 2xy = 3x $$ is an ordinary differential equation (the independent variable being x), and $$ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 $$ is a partial differential equation (the multiple independent variables being $x$ and $y$).
StahlStahl
$\begingroup$ Ok, thanks for your detailed answer. $\endgroup$ – user1234161 Mar 28 '13 at 23:44
"Ordinary" is in contrast with "Partial" as is "Partial Differential Equations" or PDE
In addition to what others have already said, there are other types of differential equation.
There's the Stochastic Differential Equation, which contain random elements.
There's the Differential-difference equation, which is a blending of differential and difference equations, such as
$$ \frac{d}{dx}f(x)=f(x-1) $$
So an ordinary differential equation is a differential equation that doesn't have anything "special" about it, it's just a differential equation. It is, quite literally, ordinary.
Glen OGlen O
Not the answer you're looking for? Browse other questions tagged ordinary-differential-equations terminology or ask your own question.
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Rational reconstruction (mathematics)
In mathematics, rational reconstruction is a method that allows one to recover a rational number from its value modulo a sufficiently large integer.
Problem statement
In the rational reconstruction problem, one is given as input a value $n\equiv r/s{\pmod {m}}$. That is, $n$ is an integer with the property that $ns\equiv r{\pmod {m}}$. The rational number $r/s$ is unknown, and the goal of the problem is to recover it from the given information.
In order for the problem to be solvable, it is necessary to assume that the modulus $m$ is sufficiently large relative to $r$ and $s$. Typically, it is assumed that a range for the possible values of $r$ and $s$ is known: $|r|<N$ and $0<s<D$ for some two numerical parameters $N$ and $D$. Whenever $m>2ND$ and a solution exists, the solution is unique and can be found efficiently.
Solution
Using a method from Paul S. Wang, it is possible to recover $r/s$ from $n$ and $m$ using the Euclidean algorithm, as follows.[1][2]
One puts $v=(m,0)$ and $w=(n,1)$. One then repeats the following steps until the first component of w becomes $\leq N$. Put $q=\left\lfloor {\frac {v_{1}}{w_{1}}}\right\rfloor $, put z = v − qw. The new v and w are then obtained by putting v = w and w = z.
Then with w such that $w_{1}\leq N$, one makes the second component positive by putting w = −w if $w_{2}<0$. If $w_{2}<D$ and $\gcd(w_{1},w_{2})=1$, then the fraction ${\frac {r}{s}}$ exists and $r=w_{1}$ and $s=w_{2}$, else no such fraction exists.
References
1. Wang, Paul S. (1981), "A p-adic algorithm for univariate partial fractions", Proceedings of the Fourth International Symposium on Symbolic and Algebraic Computation (SYMSAC '81), New York, NY, USA: Association for Computing Machinery, pp. 212–217, doi:10.1145/800206.806398, ISBN 0-89791-047-8, S2CID 10695567
2. Wang, Paul S.; Guy, M. J. T.; Davenport, J. H. (May 1982), "P-adic reconstruction of rational numbers", SIGSAM Bulletin, New York, NY, USA: Association for Computing Machinery, 16 (2): 2–3, CiteSeerX 10.1.1.395.6529, doi:10.1145/1089292.1089293, S2CID 44536107.
| Wikipedia |
\begin{document}
\begin{abstract} Let $H:(M,p)\rightarrow (M^\p,p^\prime)$ be a formal mapping between two germs of real-analytic generic submanifolds in $\mathbb C^N$ with nonvanishing Jacobian. Assuming $M$ to be minimal at $p$ and $M^\p$ holomorphically nondegenerate\ at $p^\prime$, we prove the convergence of the mapping $H$. As a consequence, we obtain a new convergence result for arbitrary formal maps between real-analytic hypersurfaces when the target does not contain any holomorphic curve. In the case when both $M$ and $M^\p$ are hypersurfaces, we also prove the convergence of the associated reflection function when $M$ is assumed to be only minimal. This allows us to derive a new Artin type approximation theorem for formal maps of generic full rank. \end{abstract}
\maketitle
\section{Introduction}\label{int}
In this paper, we study some properties of formal mappings generically of full rank between generic submanifolds in complex space. A formal holomorphic mapping $H:(\mathbb C^N,p)\rightarrow(\mathbb C^N,p^\prime)$ with $p,p^\prime\in\mathbb C^N$ is a power series mapping in $z-p$, which satisfy $H(p)=p^\prime$. We say that $H$ is generically of full rank if $\text{\rm Jac} H$, the determinant of the Jacobian matrix of $H$, does not vanish identically (as a formal power series). We define in the usual way the notion of formal mapping between real-analytic generic submanifolds of $\mathbb C^N$ (see Section \ref{s:prelim}).
The main problem we were interested in lies in the convergence of formal mappings generically of full rank between real-analytic generic submanifolds. The first result of this type follows from the work of Chern and Moser \cite{CM74} who proved that any formal biholomorphism between real-analytic Levi-nondegenerate hypersurfaces is convergent. The concept of holomorphic nondegeneracy introduced by Stanton \cite{Sta96} turns out to be a crucial condition in understanding the question. Recall that a generic submanifold $M\subset\mathbb C^N$ is said to be holomorphically nondegenerate (at $p$) if there exists no non-trivial holomorphic vector field tangent to $M$ (near $p$). The importance of this condition for mapping problems can be seen through the work of Baouendi, Ebenfelt and Rothschild. In the paper \cite{BER97}, the holomorphic nondegeneracy condition is shown to be a necessary condition for the convergence of all formal equivalences between real-analytic generic submanifolds in $\mathbb C^N$. In the paper \cite{BMR02}, Baouendi, Mir and Rothschild show that this condition is sufficient to establish the convergence of finite mappings when the source manifold is furthermore assumed to be minimal in the sense of Tumanov (see \cite[\S1.5]{BERbook}).
In this paper, we extend this result to arbitrary formal mappings of generic full rank.
\begin{Thm}\label{t:cvHgenecase} Let $(M,p)$, $(M^\p,p^\prime)$ be two germs of real-analytic generic manifolds of codimension $d$ in $\mathbb C^{n+d}$. Assume that $M$ is minimal at $p$ and that $M^\p$ is holomorphically nondegenerate\ at $p^\prime$. Then any formal holomorphic mapping $H:(\mathbb C^{n+d},p)\rightarrow(\mathbb C^{n+d},p^\prime)$ sending $M$ into $M^\p$ with $\text{\rm Jac} H\nequiv0$ is convergent. \end{Thm} Our proof of Theorem \ref{t:cvHgenecase} provides also a simpler proof of the analogous result of \cite{BMR02} in the invertible case. Let us also mention that Theorem \ref{t:cvHgenecase} was proved earlier by Mir \cite{Mir02} in the special case where $M^\p$ is a real-algebraic generic manifold.
As a consequence of Theorem \ref{t:cvHgenecase}, we derive the following convergence result for arbitrary formal maps between real-analytic hypersurfaces.
\begin{Thm}\label{t:thcourbehol} Let $(M,p)$, $(M^\p,p^\prime)$ be two germs of real-analytic hypersurfaces in $\mathbb C^{n+1}$. Assume that $M$ is minimal and holomorphically nondegenerate\ at $p$ and that $M^\p$ does not contain any holomorphic curve through $p^\prime$. Then any formal holomorphic mapping sending $(M,p)$ into $(M^\p,p^\prime)$ is convergent. \end{Thm}
Under the more restrictive condition that $M$ is essentially finite at $p$, Theorem \ref{t:thcourbehol} was established by Baouendi, Ebenfelt and Rothschild in \cite{BER00}. In the hypersurface case, we will in fact prove the convergence of the so-called \emph{reflection function} (see \cite{Hua96}) when the source manifold is merely minimal (see Theorem \ref{t:cvR}). This generalizes a result obtained in \cite{Mir00} for formal biholomorphisms and, also allows us to derive the following new Artin type approximation theorem for formal maps of generic full rank.
\begin{Thm}\label{t:artinhyp} Let $(M,p)$, $(M^\p,p^\prime)$ be two germs of real-analytic hypersurface s in $\mathbb C^{n+1}$ and $H:(\mathbb C^{n+1},p)\rightarrow (\mathbb C^{n+1},p^\prime)$ be a formal holomorphic mapping sending $M$ into $M^\p$. Assume that $\text{\rm Jac} H{\equiv \!\!\!\!\!\! / \,\,} 0$ and that $M$ is minimal at $p$. Then for any positive integer $k$, there exists a germ of holomorphic map $H^k:(\mathbb C^{n+1},p)\rightarrow (\mathbb C^{n+1},p^\prime)$ sending $M$ into $M^\p$ which agrees up to order $k$ with $H$ at $p$. \end{Thm}
Theorem \ref{t:artinhyp} extends, in the hypersurface case, a similar result of \cite{BMR02} obtained for the more restrictive class of finite mappings. For a more detailed account on recent results on convergence and approximation of formal mappings between submanifolds in complex space, we refer the reader to the survey \cite{MMZ03}.
Our proof of Theorem 1.1 is in essence different from that of Baouendi, Mir and Rothschild for finite mappings \cite{BMR02}. Indeed, in this paper, thanks some mapping identities due to Juhlin \cite{Juh08}, we are able to prove directly the convergence of the formal mapping $H$ (and of its derivatives) along the iterated Segre sets, which strongly contrasts with the methods of \cite{BMR02}. For the hypersurface case, as mentioned earlier, we will prove the convergence of the reflection function for formal maps of generic full rank by combining some arguments of Mir \cite{Mir00} and Juhlin \cite{Juh08}. The Artinian theorem (Theorem \ref{t:artinhyp}) follows directly thanks to a suitable application of Artin's approximation theorem \cite{Artin68} and an argument from \cite{BMR02}.
The paper is organized as follows. In Section \ref{s:prelim}, we recall some basic facts on normal coordinates for generic submanifolds in $\mathbb C^N$ and on formal mappings sending real-analytic generic submanifolds into each other. We also prove several preliminary results. Section \ref{s:cvR} is devoted to the proof of the convergence of the reflection function when the source manifold is a minimal hypersurface. Section \ref{s:generalcase} contains the proofs of Theorem \ref{t:cvHgenecase} and Theorem \ref{t:thcourbehol}. To conclude the paper, we prove Theorem \ref{t:artinhyp}.
\section{Preliminaries}\label{s:prelim}
In this section, we first recall the basic definitions we need in this paper, then we establish a convergence lemma which will be used in the next sections.\\
Let $(M,p)$ and $(M^\p,p^\prime)$ be germs of real-analytic generic manifolds of codimension $d$ in $\mathbb C^{n+d}$. Without loss of generality we may assume that $p=p^\prime=0$. It is well known (see for instance \cite[\S4.2]{BERbook}) that there exist normal coordinates defining $M$ and $M^\p$ near 0. It means, in the case of $M$ for example, that there exists $Q\in(\mathbb C\{z,\chi,\tau\})^d$ and $U$, $V$ two open neighborhoods of $0$ in $\mathbb C^n$ and $\mathbb C^d$ respectively such that $Q$ is convergent on $U\times \bar{U}\times\bar{V}$ and $$M\cap\left(U\times V\right)=\{(z,w)\in U\times V, w=Q(z,\bar z,\bar w)\}.$$ Moreover, the mapping $Q$ satisfies \begin{equation}\label{Qnormal} Q(0,\chi,\tau)\equiv Q(z,0,\tau)\equiv\tau \end{equation} and, since $M$ is a real submanifold, we have the following mapping identity \begin{equation}\label{realcond} Q(z,\chi,\bar Q(\chi,z,\tau))\equiv\tau. \end{equation} Let us recall that $M$ is said to be minimal at $p$ if there is no germ of a real submanifold $S\subset M$ through $p$ such that the complex tangent space of $M$ at $q$ is tangent to $S$ at every $q\in S$ and $\dim_\mathbb R S<\dim_\mathbb R M$.
We denote by $\mc{M}$ the complexification of $M$ which is the complex submanifold of $\mathbb C^{2(n+d)}$ given by \begin{equation} \{\left(z,Q\left(z,\chi,\tau\right),\chi,\tau\right) , (z,\chi,\tau)\in U\}\subset\mathbb C^{n+d}\times\mathbb C^{n+d}. \end{equation} We define in a similar way ${Q^\p}({z^\p},\c^\p,\t^\p)$ and $\mc{M}^\prime$ for $M^\p$. We write the Taylor expansion \begin{equation}\label{Qexpa} {Q^\p}({z^\p},\c^\p,\t^\p)=\sum_{\alpha\in\mathbb N^{n}} {\Qp_\a} (\c^\p,\t^\p){{z^\p}}^\alpha. \end{equation}
Let $H:(\mathbb C^n\times\mathbb C^d,0)\rightarrow (\mathbb C^n\times\mathbb C^d,0)$ be a formal holomorphic mapping. We write $H(z,w)=(F(z,w),G(z,w))$ where $F$ and $G$ are respectively with values in $\mathbb C^n$ and $\mathbb C^d$.
We say that $H$ sends $M$ into $M^\p$ if the following identity holds at the formal power series level: \begin{equation}\label{includ} {Q^\p}(F(z,Q(z,\chi,\tau)),\bar H(\chi,\tau))=G(z,Q(z,\chi,\tau)). \end{equation}
We will need the notion of \emph{generic rank} of a formal mapping $h\in(\mathbb C[[x]])^q$, $x=(x_1,\ldots,x_r)$. It is the rank of the Jacobian matrix $\deri{x}{h}$ as a matrix with coefficients in the quotient field of $\mathbb C[[x]]$. If $h$ is convergent, it coincides with the classical generic rank of a convergent power series mapping.
From now on, we assume that $M$ and $M^\p$ are two real-analytic generic manifolds in $\mathbb C^{n+d}$ given in normal coordinates as above. We also assume in what follows that $H=(F,G)$ is a formal mapping sending $M$ into $M^\p$ such that $\text{\rm Jac} H{\equiv \!\!\!\!\!\! / \,\,} 0$.
We recall a well known fact coming from elementary linear algebra and $\eqref{includ}$. \begin{Lem}
In the above situation, the condition $\text{\rm Jac} H{\equiv \!\!\!\!\!\! / \,\,} 0$ implies \begin{equation}\label{jac<>0} {\text{\rm det}} \dop{z}\left(F\left(z,Q\left(z,\chi,\tau\right)\right)\right){\equiv \!\!\!\!\!\! / \,\,} 0. \end{equation}
\end{Lem}
We need the following lemma which can be found in \cite{Juh08} and whose proof is mainly based on a suitable differentiation of the identity $\eqref{includ}$.
\begin{Lem}\cite[Proposition 5.2]{Juh08}\label{exprQpa}
In the above situation, there exist $\eta_0\in\mathbb N^n$ and $\delta_0\in\mathbb N^d$ such that, for $p_0=|\eta_0|+|\delta_0|$ \begin{equation}
\dbldopz{z}{\eta^\prime}{\tau}{\delta^\prime} {\text{\rm det}}\dop{z} \left(F(z,Q(z,\chi,\tau))\right)\equiv 0, \textrm{ for } |\eta^\prime|+|\delta^\prime|<p_0, \end{equation}
\begin{equation} \dbldopz{z}{\eta_0}{\tau}{\delta_0} {\text{\rm det}}\dop{z} \left(F(z,Q(z,\chi,\tau))\right){\equiv \!\!\!\!\!\! / \,\,} 0
\end{equation} and, for all $\alpha\in\mathbb N^n$, $\beta\in\mathbb N^d$, such that $|\alpha|+|\beta|> 0$ there exist universal polynomials of their arguments $R_{\alpha,\beta,\eta_0,\delta_0}$ such that, for any $r\in\{1,\ldots,d\}$,
\begin{equation}\label{eqQpa}
\left.\mdop{\tau}{\beta}\right|_{\tau=0}{\Qp_\a^r}(\bar H(\chi,\tau))=\frac{R_{\alpha,\beta,\eta_0,\delta_0}\left(\left(\dbldopz{z}{\eta}{\tau}{\delta} H^{(r)}(z,Q(z,\chi,\tau))\right)_{|\eta|+|\delta|\leq(|\alpha|+|\beta|)(p_0+1)}\right)}{\left(\dbldopz{z}{\eta_0}{\tau}{\delta_0} {\text{\rm det}} \dop{z}\left(F(z,Q(z,\chi,\tau))\right)\right)^{2(|\alpha|+|\beta|)-1}} \end{equation} where ${\Qp_\a^r}$ is the $r$-th component of ${\Qp_\a}$, and $H^{(r)}(z,w)=(F(z,w),G^r(z,w))$ with $G^r$ being the $r$-th component of $G$. \end{Lem}
Lemma \ref{exprQpa} is useful to show the following convergence lemma.
\begin{Lem}\label{l:Qpacv}
In the above situation, for all $\beta\in\mathbb N^d$, $\alpha,\gamma\in\mathbb N^n$, $\left.\dbldop{\chi}{\gamma}{\tau}{\beta}\right|_{\tau=0}{\Qp_\a}(\bar H(\chi,\tau))$ is convergent. \end{Lem}
\begin{proof} We fix $\gamma\in\mathbb N^n$. For $|\alpha|=|\beta|=0$, by setting $z=0$, $\tau=0$ in equation $\eqref{includ}$, we obtain, thanks to \eqref{Qnormal}, ${Q^\p}(0,\bar H(\chi,0))=G(0)$. From the taylor expansion of ${Q^\p}$ given by \eqref{Qexpa}, we get that ${Q^\p}_0(\bar H(\chi,0))$ is constant.\\
For $|\alpha|+|\beta|> 0$, we will show that each component of $\left.\dbldop{\chi}{\gamma}{\tau}{\beta}\right|_{\tau=0}{\Qp_\a}(\bar H(\chi,\tau))$ is convergent. For this, we apply $\mdop{\chi}{\gamma}$ to the expression \eqref{eqQpa} of $\left.\mdop{\tau}{\beta}\right|_{\tau=0}{\Qp_\a^r}(\bar H(\chi,\tau))$ and we obtain that $\left.\dbldop{\chi}{\gamma}{\tau}{\beta}\right|_{\tau=0}{\Qp_\a^r}(\bar H(\chi,\tau))$ is a ratio of two formal power series. In this ratio, the numerator and the denominator are polynomials in the derivatives of $Q$ which converge and in the derivatives of $H$ at the point $(0,Q(0,\chi,0))=0$. So $\left.\dbldop{\chi}{\gamma}{\tau}{\beta}\right|_{\tau=0}{\Qp_\a^r}(\bar H(\chi,\tau))$ is a ratio of two convergent power series whose denominator does not vanish identically. Since $\left.\dbldop{\chi}{\gamma}{\tau}{\beta}\right|_{\tau=0}{\Qp_\a^r}(\bar H(\chi,\tau))$ is a formal power series, it is convergent. \end{proof}
\section{Convergence of the reflection function in the hypersurface case}\label{s:cvR}
In this section, we establish a convergence result for the so-called reflection function of a mapping between minimal real-analytic hypersurface s whose Jacobian is not identically zero. We keep the notation of Section \ref{s:prelim} with $d=1$ and still use normal coordinates.
Let $\mathcal R$ be the formal holomorphic mapping called {\em reflection function} (\cite{Hua96}, \cite{Mir00}) and defined by \begin{equation}\label{defR} \mathcal R({z^\p},\chi,\tau):={Q^\p}({z^\p},\bar H(\chi,\tau)), \end{equation} where $(z,\chi,\tau)\in\mathbb C^n\times\mathbb C^n\times\mathbb C$. The aim of this section is to establish the following result:
\begin{Thm}\label{t:cvR} Let $M$, $M^\p$ be two real-analytic hypersurface s in $\mathbb C^{n+1}$ through 0 and $H:(\mathbb C^n\times\mathbb C,0)\rightarrow (\mathbb C^n\times\mathbb C,0)$ be a formal holomorphic mapping sending $M$ into $M^\p$ with $\text{\rm Jac} H{\equiv \!\!\!\!\!\! / \,\,} 0$. If $M$ is minimal at 0 then the associated reflection function $\mathcal R$ given by $\eqref{defR}$ is a convergent power series. \end{Thm}
To do this, we follow the different steps of \cite{Mir00}. First, we establish the following result.
\begin{Pro}\label{p:RcvS1}
Let $M$, $M^\p$ be two real-analytic hypersurface s in $\mathbb C^{n+1}$ through 0 and $H:(\mathbb C^n\times\mathbb C,0)\rightarrow (\mathbb C^n\times\mathbb C,0)$ be a formal holomorphic mapping sending $M$ into $M^\p$. If $\text{\rm Jac} H{\equiv \!\!\!\!\!\! / \,\,} 0$, then for any $\gamma\in\mathbb N^n$ and $\beta\in\mathbb N$, the formal holomorphic map
$$({z^\p},\chi,\tau)\in(\mathbb C^n\times\mathbb C^n\times\mathbb C,0)\mapsto\left.\dbldophyp{\chi}{\gamma}{\tau}{\beta}\right|_{\tau=0}\mathcal R({z^\p},\chi,\tau)$$ is convergent in a neighborhood of 0. \end{Pro}
To prove this proposition, we will use the following convergence lemma whose proof can be found in \cite{Mir00}.
\begin{Lem}\cite[Lemma 4.1]{Mir00}\label{l:cvborn} Let $(u_i(t))_{i\in I}$ be a family of convergent power series in $\mathbb C\{t\}$, $t=(t_1,\ldots,t_q)$, $q\in\mathbb N^*$. Let also $(\mc{K}_i(\zeta))_{i\in I}$ be a family of convergent power series in $\mathbb C\{\zeta\}$, $\zeta=(\zeta_1,\ldots,\zeta_r)$, $r\in\mathbb N^*$. Assume that \begin{enumerate} \item There exists $R>0$ such that the radius of convergence of any $\mc{K}_i$, $i\in I$, is at least R.
\item For all $\zeta\in\mathbb C^r$ with $|\zeta|<R$, $|\mc{K}_i(\zeta)|\leq C_i$ with $C_i>0$ \item There exists $V(t)\in(\mathbb C[[t]])^r$, $V(0)=0$, such that $\mc{K}_i\circ V(t)=u_i(t)$ for all $i\in I$.
\end{enumerate} Then, there exists $R^\prime>0$ such that the radius of convergence of any $u_i$, $i\in I$, is at least $R^\prime$ and such that for all $t\in\mathbb C^q$ with $|t|<R^\prime$, $|u_i(t)|\leq C_i$. \end{Lem}
\begin{proof}[Proof of Proposition {\rm\ref{p:RcvS1}}]
First we will give a proof in the case $\beta=|\gamma|=0$. In order to show that $\mathcal R({z^\p},\chi,0)$ is convergent, we recall that
\begin{equation} \mathcal R({z^\p},\chi,0)={Q^\p}({z^\p},\bar H(\chi,0))=\sum_{\alpha\in\mathbb N^{n}} {\Qp_\a} (\bar H(\chi,0)){{z^\p}}^\alpha. \end{equation} So, as in the proof of Proposition 5.1 of \cite{Mir00}, we see it suffices to find positive constants $r$ and $R_0$ such that the radius of convergence of the familly $({\Qp_\a}(\bar H(\chi,0)))_{\alpha}$ is at least $r$ and for any $\chi$ with norm smaller than $r$ and any $\alpha\in\mathbb N^n$ we have
\begin{equation}\label{estimQpa}
|{\Qp_\a}(\bar H(\chi,0))|\leq R_0^{|\alpha|+1}.
\end{equation} This is possible using the following property which comes from the holomorphy of ${Q^\p}$: there exist $r_1$ and $R$ positive constants such that, for any $\alpha,\mu\in\mathbb N^n$, $\nu\in\mathbb N$ and for any $(\c^\p,\t^\p)\in\mathbb C^{n+1}$ with $|(\c^\p,\t^\p)|< r_1$ we have \begin{equation}\label{cauchyestim}
\left|\dblderihyp{\c^\p}{\mu}{\t^\p}{\nu}{{\Qp_\a}}(\c^\p,\t^\p)\right|\leq(\mu,\nu)!R^{|\alpha|+|\mu|+\nu+1}, \end{equation} where $(\mu,\nu)$ is the concatenation of $\mu$ and $\nu$.
Thus, we can apply Lemma \ref{l:cvborn} to prove $\eqref{estimQpa}$. Indeed, \begin{itemize} \item[-] For any $\alpha\in\mathbb N^n, {\Qp_\a}\left(\c^\p,\t^\p\right)$ is convergent with radius of convergent at least $r_1$,
\item[-] by taking $\nu=|\mu|=0$ in $\eqref{cauchyestim}$, we have that for any $(\c^\p,\t^\p)\in\mathbb C^{n+1}$, with $|(\c^\p,\t^\p)|<r_1$, $|{\Qp_\a}(\c^\p,\t^\p)|\leq R^{|\alpha|+1}$, \item[-] ${\Qp_\a}(\bar H(\chi,0))$ is convergent thanks to Lemma \ref{l:Qpacv}. \end{itemize}
Therefore, the proof in the case $\beta=|\gamma|=0$ is complete since $\eqref{estimQpa}$ holds with $R_0=R$.\\
Now, we use the same method to prove the proposition in the general case. We fix $\gamma\in\mathbb N^n$ and $\beta\in\mathbb N$ with $\beta+|\gamma|>0$. If we write the Taylor expansion of $\mathcal R$ in ${z^\p}$, \begin{equation}\label{expanR} \mathcal R({z^\p},\chi,\tau)={Q^\p}({z^\p},\bar H(\chi,\tau))=\sum_{\alpha\in\mathbb N^n} {\Qp_\a} (\bar H(\chi,\tau)){{z^\p}}^\alpha, \end{equation} we have \begin{equation} \trpderihyp{{z^\p}}{\alpha}{\chi}{\gamma}{\tau}{\beta}{\mathcal R}({z^\p},\chi,\tau)=\dbldophyp{\chi}{\gamma}{\tau}{\beta}\left( \mderi{{z^\p}}{\alpha}{{Q^\p}}\left({z^\p},\bar H\left(\chi,\tau\right)\right) \right).
\end{equation} So, to prove the convergence of $\left.\dbldophyp{\chi}{\gamma}{\tau}{\beta}\right|_{\tau=0}\mathcal R({z^\p},\chi,\tau)$, it suffices to prove that the family defined by \begin{equation}\label{defpsi}
\psi_{\alpha}^{\beta,\gamma}(\chi):=\trpderihyp{{z^\p}}{\alpha}{\chi}{\gamma}{\tau}{\beta}{\mathcal R}(0,\chi,0)=\alpha!\left.\dbldophyp{\chi}{\gamma}{\tau}{\beta}\right|_{\tau=0}{\Qp_\a}(\bar H(\chi,\tau))
\end{equation} satisfy the following condition: one may find $r_{\beta,\gamma}>0$ and $R_{\beta,\gamma}>0$ such that the radius of convergence of the familly $\psi_{\alpha}^{\beta,\gamma}$ is at least $r_{\beta,\gamma}$ and such that for any $\alpha\in\mathbb N^n$ and any $\chi\in\mathbb C^n$ with $|\chi|<r_{\beta,\gamma}$, we have the following estimate \begin{equation}\label{estimpsi}
|\psi_{\alpha}^{\beta,\gamma}(\chi)|\leq \alpha! R_{\beta,\gamma}^{|\alpha|+1}.
\end{equation} Let's prove $\eqref{estimpsi}$ using Lemma \ref{l:cvborn}. We fix $\alpha\in\mathbb N^n$. First, it comes from $\eqref{defpsi}$ and Lemma \ref{l:Qpacv} that $\psi_{\alpha}^{\beta,\gamma}(\chi)$ is convergent. Then, for any multiindex $\mu\in\mathbb N^n$, $\nu\in\mathbb N$ with $|\mu|\leq|\gamma|$, $\nu\leq\beta$ there exists a universal polynomial $P_{\nu,\mu}^{\beta,\gamma}$ such that \begin{equation} \psi_{\alpha}^{\beta,\gamma}(\chi) =
\alpha!\sum_{\begin{subarray}{c}|\mu|\leq|\gamma| \\ \nu\leq\beta\end{subarray}}P_{\nu,\mu}^{\beta,\gamma}\left(\left(\dblderihyp{\chi}{\delta}{\tau}{\eta}{\bar H}(\chi,0)\right)_{\begin{subarray}{c}1\leq|\delta|\leq|\gamma| \\ 1\leq\eta\leq\beta\end{subarray}}\right)\dblderihyp{\chi}{\mu}{\tau}{\nu}{{\Qp_\a}}(\bar H(\chi,0)).
\end{equation} We define a convergent power series mapping of the variables $\left(\left(\Lambda_{\delta,\eta}\right)_{\begin{subarray}{c}1\leq|\delta|\leq|\gamma| \\ 1\leq\eta\leq\beta\end{subarray}},\c^\p,\t^\p\right)\in\mathbb C^{N_{\beta,\gamma}}\times\mathbb C^{n+1}$ with $N_{\beta,\gamma}=(n+1){\rm Card}\{(\delta,\eta)\in\mathbb N^n\times\mathbb N,1\leq|\delta|\leq|\gamma|,1\leq\eta\leq\beta\}$, as follows
\begin{equation} h_{\alpha}^{\beta,\gamma}\left(\left(\Lambda_{\delta,\eta}\right)_{\begin{subarray}{c}1\leq|\delta|\leq|\gamma| \\ 1\leq\eta\leq\beta\end{subarray}},\c^\p,\t^\p\right):=
\alpha!\sum_{\begin{subarray}{c}|\mu|\leq|\gamma| \\ \nu\leq\beta\end{subarray}}P_{\nu,\mu}^{\beta,\gamma}\left(\left(\Lambda_{\delta,\eta}+\dblderihyp{\chi}{\delta}{\tau}{\eta}{\bar H}(0)\right)_{\begin{subarray}{c}1\leq|\delta|\leq|\gamma| \\ 1\leq\eta\leq\beta\end{subarray}} \right)\dblderihyp{\chi}{\mu}{\tau}{\nu}{{\Qp_\a}}(\c^\p,\t^\p).
\end{equation} From this and using $\eqref{cauchyestim}$, we observe that there exists a constant $R_{\beta,\gamma}$ such that, for $\left(\Lambda_{\delta,\eta}\right)_{\begin{subarray}{c}1\leq|\delta|\leq|\gamma| \\ 1\leq\eta\leq\beta\end{subarray}}$ with norm smaller than 1 in $\mathbb C^{N_{\beta,\gamma}}$ and $(\c^\p,\t^\p)$ with norm smaller than $r$ in $\mathbb C^{n+d}$, we have, for any $\alpha\in\mathbb N^n$ \begin{equation}
\left|h_{\alpha}^{\beta,\gamma}\left(\left(\Lambda_{\delta,\eta}\right)_{\begin{subarray}{c}1\leq|\delta|\leq|\gamma| \\ 1\leq\eta\leq\beta\end{subarray}},\c^\p,\t^\p\right)\right|\leq \alpha!R_{\beta,\gamma}^{|\alpha|+1}. \end{equation} So, using the fact that
\begin{equation} h_{\alpha}^{\beta,\gamma}\left(\left(\dblderihyp{\chi}{\delta}{\tau}{\eta}{\bar H}(\chi,0)-\dblderihyp{\chi}{\delta}{\tau}{\eta}{\bar H}(0)\right)_{\begin{subarray}{c}1\leq|\delta|\leq|\gamma| \\ 1\leq\eta\leq\beta\end{subarray}},\bar H(\chi,0)\right)=\psi_{\alpha}^{\beta,\gamma}(\chi) \end{equation} and Lemma \ref{l:cvborn}, we obtain the estimates $\eqref{estimpsi}$. Thus the proof of Proposition \ref{p:RcvS1} is complete. \end{proof}
We now need the two following lemma and proposition. The proof of the lemma, which is a consequence of Artin's approximation theorem \cite{Artin68}, can be found in \cite{Mir00}. The proposition is due to Gabrielov \cite{Gab73} and proved in e.g. \cite{BM88}.
\begin{Lem}\cite[Lemma 6.1]{Mir00}\label{l:artinderi} Let $\mc{T}(x,u)\in(\mathbb C[[x,u]])^r$, $x\in\mathbb C^q$, $u\in\mathbb C^s$with $\mc{T}(0)=0$. Assume that $\mc{T}(x,u)$ satisfies an identity in the ring $\mathbb C[[x,u,y]]$, $y\in\mathbb C^q$, of the form $$\varphi(\mc{T}(x,u);x,u,y)=0$$ where $\varphi\in\mathbb C[[W,x,u,y]]$ with $W\in\mathbb C^r$. Assume, furthermore, that for any multiindex $\beta\in\mathbb N^q$, the formal power series $\left[\mderi{y}{\beta}{\varphi}\left(W;x,u,y\right)\right]_{y=x}$ is convergent. Then, for any given positive integer $k$, there exists an $r$-uple of convergent power series $\mc{T}^k(x,u)$ such that $\varphi(\mc{T}^k(x,u);x,y,u)=0$ in $\mathbb C[[x,u,y]]$ and such $\mc{T}^k(x,u)$ agrees up to order $k$ at 0 with $\mc{T}(x,u)$. \end{Lem}
\begin{Pro}\label{p:cv+genesub} Let $\mc{J}(x)\in(\mathbb C\{x\})^r$, $x\in\mathbb C^k$, $k,r\geq 1$ $\mc{J}(0)=0$ and $\mc{V}(t)\in\mathbb C[[t]]$, $t\in\mathbb C^r$. If $\mc{V}\circ\mc{J}$ is convergent and the generic rank of $\mc{J}$ is $r$, then $\mc{V}$ is convergent. \end{Pro}
We will also need the definition and properties of the \emph{Segre set mappings} for the hypersurface $M$, only the three first for this section: \begin{eqnarray} &v_1:&z\in(\mathbb C^n,0)\mapsto(z,0), \\ \nonumber &v_2:&(z,\xi)\in(\mathbb C^{2n},0)\mapsto(z,Q(z,\xi,0)), \\ \nonumber &v_3:&(z,\xi,\zeta)\in(\mathbb C^{3n},0)\mapsto(z,Q(z,\xi,\bar Q(\xi,\zeta,0))). \nonumber \end{eqnarray} The minimality condition of $M$ in Theorem \ref{t:cvR} allows us to say that the generic rank of $v_2$ (and $v_3$) is $n+1$.
\begin{proof}[Proof of Theorem {\rm\ref{t:cvR}}] By setting $\tau=\bar Q(\chi,z,Q(z,\xi,0))$ in $\eqref{includ}$ and using $\eqref{realcond}$, we obtain \begin{equation} {Q^\p}\left(F\circ v_2(z,\xi),\bar H\circ \bar{v}_3(\chi,z,\xi)\right)=G\circ v_2(z,\xi) \end{equation} This means that $\mathcal R$ satisfy the following equation \begin{equation}\label{Req} \mathcal R(F\circ v_2(z,\xi),\bar{v}_3(\chi,z,\xi))-G\circ v_2(z,\xi)\equiv0. \end{equation} We want to apply Lemma \ref{l:artinderi} to the equation $\eqref{Req}$ with $x=\xi$, $u=z$, $y=\chi$, $\mc{T}(x,u)=H\circ v_2(\chi,\xi)$, $W=(\lambda,\mu)\in\mathbb C^n\times\mathbb C$ and $$\varphi((\lambda,\mu);\xi,z,\chi)=\mathcal R(\lambda,\bar{v}_3(\chi,z,\xi))-\mu.$$
For this, we have to verify the convergence of all the derivatives of $\varphi$ with respect to $\chi$ evaluated in $\chi=\xi$. But, those derivatives involve only derivatives of $v_3$ which are convergent and expressions of the form $\dblderi{\chi}{\gamma}{\tau}{\beta}{\mathcal R}(\lambda,\bar{v}_3(\chi,z,\xi))|_{\chi=\xi}$. As, from the mapping identity $\eqref{realcond}$, $v_3(\xi,z,\xi)=v_1(\xi)=(\xi,0)$, Proposition \ref{p:RcvS1} allows us to apply Lemma \ref{l:artinderi}. Thus, for any positive integer $k$ there exists $\mc{T}^k(z,\xi)=({\mc{T}^\prime}^k(z,\xi),\mc{T}^k_n(z,\xi))$ convergent power series agreeing up to order $k$ at 0 with $H\circ v_2(z,\xi)$ and such that $$\mathcal R\left({\mc{T}^\prime}^k(z,\xi),\bar{v}_3(\chi,z,\xi)\right)\equiv\mc{T}^k_n(z,\xi)$$ To show that $\mathcal R$ is convergent it suffices, using Proposition \ref{p:cv+genesub}, to show that, for $k$ well chosen, the map $(z,\chi,\xi)\in\mathbb C^{3n}\mapsto\left({\mc{T}^\prime}^k(z,\xi),\bar{v}_3(\chi,z,\xi)\right)\in\mathbb C^{2n+1}$ is of generic rank $2n+1$.
From the two following facts:
\begin{enumerate} \item $(z,\chi,\xi)\in\mathbb C^{3n}\mapsto(v_2(z,\xi),\bar{v}_3(\chi,z,\xi))\in\mc{M}$ is of generic rank $2n+1$ since $M$ is minimal, \item $(z,w,\chi,\tau)\in\mc{M}\mapsto(F(z,w),\chi,\tau)$ is of generic rank $2n+1$ because $F$ is of generic full rank $n$; \end{enumerate} we deduce that $(z,\chi,\xi)\mapsto(F(v_2(z,\xi)),\bar{v}_3(\chi,z,\xi))$ is of generic rank $2n+1$ in $(\mathbb C[[z,\chi,\xi]])^{2n+1}$. But $\mc{T}^k(z,\xi)$ agrees up to order $k$ with $H\circ v_2(z,\xi)$, so for $k$ large enough the generic rank of $\left({\mc{T}^\prime}^k(z,\xi),\bar{v}_3(\chi,z,\xi)\right)$ is $2n+1$. This completes the proof of Theorem \ref{t:cvR}. \end{proof}
\section{The case of generic manifolds of arbitrary codimension}\label{s:generalcase}
The aim of this section is to show Theorem \ref{t:cvHgenecase}. The arguments used to obtain this result are inspired by an analogous result in the finite jet determination problem that can be found in \cite{Juh08}.
\subsection{A convergence lemma}
In this paragraph, we will state and prove a lemma inspired from \cite{Juh08}.
\begin{Lem}\label{l:lemdecv} Let $P(x,Y)\in(\mathbb C\{x,Y\})^N$ with $x\in\mathbb C^{n_1}$, $Y\in\mathbb C^N$, $\varphi_0(x,t)\in(\mathbb C[[x,t]])^N$, $\varphi(0)=0$, with $t\in\mathbb C^d$ such that $${\text{\rm det}}\deri{Y}{P}(x,\varphi_0(x,t)){\equiv \!\!\!\!\!\! / \,\,} 0.$$ Assume that for every $\beta\in\mathbb N^d$ $\mdopz{t}{\beta} P(x,\varphi_0(x,t))$ is a convergent power series, then for every $\beta\in\mathbb N^d$ $\mderi{t}{\beta}{\varphi_0}(x,0)$ is a convergent power series too. \end{Lem}
To prove this result we need the following proposition from \cite{Juh08}.
\begin{Pro}\cite[Proposition 3.1]{Juh08},\label{p:propdejuh} Let $P(x,Y)\in(\mathbb C[[x,Y]])^N$ with $x\in\mathbb C^{n_1}$, $Y\in\mathbb C^N$, $\varphi_0(x,t)\in(\mathbb C[[x,t]])^N$, $\varphi_0(0)=0$, with $t\in\mathbb C^d$ such that \begin{equation}\label{detP<>0} {\text{\rm det}}\deri{Y}{P}(x,\varphi_0(x,t)){\equiv \!\!\!\!\!\! / \,\,} 0. \end{equation} Let $\alpha_0\in\mathbb N^{n_1}$, $\beta_0\in\mathbb N^d$ such that
\begin{equation} \dbldopz{x}{\alpha_0}{t}{\beta_0}{\text{\rm det}}\deri{Y}{P}(x,\varphi_0(x,t))\neq0. \end{equation} Then, for any power series $\varphi(x,t)$ satisfying \begin{enumerate}
\item for every $(\alpha,\beta)\in\mathbb N^{n_1}\times\mathbb N^d$ with $|\alpha|\leq|\alpha_0|$ and $|\beta|\leq|\beta_0|$: $\dblderi{x}{\alpha}{t}{\beta}{\varphi}\left(0\right)=\dblderi{x}{\alpha}{t}{\beta}{\varphi_0}\left(0\right)$,
\item for some $k\in\mathbb N$ we have $\mdopz{t}{\beta}P(x,\varphi(x,t))=\mdopz{t}{\beta}P(x,\varphi_0(x,t))$, for $|\beta|\leq|\beta_0|+k$, \end{enumerate}
\noindent we have that, for every $\beta\in\mathbb N^d$ with $|\beta|\leq k$, \begin{equation} \mderi{t}{\beta}{\varphi}(x,0)=\mderi{t}{\beta}{\varphi_0}(x,0). \end{equation}
\end{Pro}
\begin{proof}[Proof of Lemma {\rm \ref{l:lemdecv}}]
For fixed $\beta\in\mathbb N^d$, we note $P_\beta(x):=\mdopz{t}{\beta} P(x,\varphi_0(x,t))\in(\mathbb C\{x\})^N$. From the chain rule, there exists a universal matrix with polynomial coefficients $\left(A_{\nu,\beta}\left(\left(\Lambda_\mu\right)_{1\leq|\mu|\leq |\beta|}\right)\right)_{|\nu|\leq|\beta|}$ where $(\Lambda_\mu)_{1\leq|\mu|\leq |\beta|}\in\mathbb C^{N_\beta}$ and $N_\beta=(n+d){\rm Card}\{\mu\in\mathbb N^d, 1\leq|\mu|\leq |\beta| \}$ satisfying: \begin{equation}\label{exprederiP}
P_{\beta}(x)=\sum_{|\nu|\leq|\beta|} \mderi{Y}{\nu}{P}\left(x,\varphi_0(x,0)\right)\cdot A_{\nu,\beta}\left(\left(\mderi{t}{\mu}{\varphi_0}(x,0)\right)_{1\leq|\mu|\leq |\beta|}\right). \end{equation} We define convergent power series by setting
\begin{equation} h_\beta((\Lambda_\mu)_{0\leq|\mu|\leq|\beta|},x):=\sum_{|\nu|\leq|\beta|}\mderi{Y}{\nu}{P}\left(x,\Lambda_0\right) \cdot A_{\nu,\beta}\left(\left(\Lambda_\mu\right)_{1\leq|\mu|\leq |\beta|}\right) \end{equation} and
\begin{equation} R_\beta((\Lambda_\mu)_{0\leq|\mu|\leq|\beta|},x):=h_\beta((\Lambda_\mu)_{0\leq|\mu|\leq|\beta|},x)-P_\beta(x). \end{equation} Now we fix $L\in\mathbb N$ and we consider
\begin{equation} R\left(\left(\Lambda_\mu\right)_{0\leq|\mu|\leq L},x\right):=\left(\left( R_\beta(\Lambda_{0\leq|\mu|\leq|\beta|}),x\right)\right)_{0\leq|\beta|\leq L}. \end{equation} By definition of $R$, it is a convergent power series and it satisfies
\begin{equation} R\left(\left(\mderi{t}{\mu}{\varphi_0}(x,0)\right)_{0\leq|\mu|\leq L},x\right)\equiv 0.
\end{equation} So, by Artin's approximation theorem \cite{Artin68}, for any positive integer $K$, there exist a convergent mapping $\left(\varphi_K^{\mu, L}(x)\right)_{0\leq|\mu|\leq L}\in(\mathbb C\{x\})^{N_L}$ where $N_L=(n+d){\rm Card}\{\mu\in\mathbb N^d, |\mu|\leq L\}$, which agrees up to order $K$ with $\left(\mderi{t}{\mu}{\varphi_0}(x,0)\right)_{0\leq|\mu|\leq L}$ at 0 and such that \begin{equation}\label{equationphiKmuL}
R\left(\left(\varphi_K^{\mu, L}(x)\right)_{0\leq|\mu|\leq L},x\right)=0.
\end{equation} We define the following convergent power series $\varphi(x,t):=\sum_{|\mu|\leq L} \frac{\varphi_K^{\mu, L}(x)}{\mu!} t^\mu$. By definition we have that, for any $\mu\in\mathbb N^d$ with $|\mu|\leq L$, $\mderi{t}{\mu}{\varphi}(x,0)=\varphi_K^{\mu, L}(x)$. Thus, we obtain that, for any $(\alpha,\mu)\in\mathbb N^{n_1}\times\mathbb N^d$ with $|\alpha|\leq K$ and $|\mu|\leq L$, \begin{equation}\label{deriphi} \dblderi{x}{\alpha}{t}{\mu}{\varphi}(0)=\dblderi{x}{\alpha}{t}{\mu}{\varphi_0}(0).
\end{equation} So, from $\eqref{equationphiKmuL}$, $R\left(\left(\mderi{t}{\mu}{\varphi}(x,0)\right)_{0\leq|\mu|\leq L},x\right)=0$, which is equivalent to
\begin{equation}\label{Pdephi=Pdephi_0} \mdopz{t}{\beta} P(x,\varphi(x,t))=P_\beta(x)=\mdopz{t}{\beta} P(x,\varphi_0(x,t)),
\end{equation} for any $\beta\in\mathbb N^d$ with $|\beta|\leq L$.
Now, we want to apply Proposition \ref{p:propdejuh} by making a good choice of $K$ and $L$. If $\alpha_0$ and $\beta_0$ are like in Proposition \ref{p:propdejuh} (this is possible because of ${\text{\rm det}}\deri{Y}{P}\left(x,\varphi_0\left(x,t\right)\right)\nequiv0$), we take $K=|\alpha_0|$ and $L=|\beta_0|+k$ for any fixed given $k>0$. From $\eqref{deriphi}$ and $\eqref{Pdephi=Pdephi_0}$, the formal mappings $\varphi_0$ and $\varphi$ satisfy the hypothesis of Proposition \ref{p:propdejuh} so that we have that, for any $\beta\in\mathbb N^d$ with $|\beta|\leq k$, \begin{equation} \mderi{t}{\beta}{\varphi_0}(x,0)=\mderi{t}{\beta}{\varphi}(x,0).
\end{equation} As $\varphi$ is a convergent power series, $\mderi{t}{\beta}{\varphi_0}(x,0)$ is convergent for $|\beta|\leq k$. This is true for any choice of $k$, so the proof of Lemma \ref{l:lemdecv} is complete. \end{proof}
\subsection{Proof of Theorem \ref{t:cvHgenecase}}
Let us recall the setting of Theorem \ref{t:cvHgenecase}. We let $(M,p)$ and $(M^\p,p^\prime)$ be two germs of real-analytic generic manifolds of codimension $d\geq1$ in $\mathbb C^ {n+d}$ with $M$ minimal at $p$ and $M^\p$ holomorphically nondegenerate at $p^\prime$. We consider $H:(\mathbb C^{n+d},p)\rightarrow(\mathbb C^{n+d},p^\prime)$ a formal mapping which sends $M$ into $M^\p$ with $\text{\rm Jac} H\nequiv0$. We use the notation of Section \ref{s:prelim} and work in normal coordinates.
To obtain the convergence of $H$, we will establish its convergence along the \emph{Segre sets} (see \cite[Chapter X]{BERbook}). The following result provides the convergence along the first \emph{Segre set}.
\begin{Pro}\label{p:convHrdS1} Under the assumptions of Theorem {\rm\ref{t:cvHgenecase}} and with the above notation, for any $\beta\in\mathbb N^d$, $\mderi{w}{\beta}{H}(z,0)$ is convergent. \end{Pro}
\begin{proof} Since $M^\p$ is \hn, using Stanton's criterion (see \cite{Sta96} or \cite[\S11.3]{BERbook}), it is possible to choose $\alpha^i\in\mathbb N^n$ and $1\leq r_i\leq d$, for $i\in\{1,\dots,n\}$ such that ${\text{\rm det}}\left(\deri{\c^\p_l}{{Q^\p}^{r_k}_{\alpha^k}}(\c^\p,\t^\p)\right)_{k,l}{\equiv \!\!\!\!\!\! / \,\,} 0$. Let $P(\c^\p,\t^\p)=\left(P^1(\c^\p,\t^\p),\ldots,P^{n+d}(\c^\p,\t^\p)\right)$ be the convergent power series mapping defined by
\begin{displaymath} P^k(\c^\p,\t^\p):=\left\{ \begin{array}{ll} {Q^\p}^{r_k}_{\alpha^k}(\c^\p,\t^\p),& \textrm{ for } 1\leq k\leq n, \\ \nonumber {Q^\p}^{k-n}_{0}(\c^\p,\t^\p),& \textrm{ for } n+1\leq k\leq n+d. \end{array}\right. \end{displaymath} Since $\deri{\t^\p}{Q^\prime_0}(\c^\p,\t^\p)=I_d$ and $\deri{\c^\p}{Q^\prime_0}(\c^\p,\t^\p)=0$, we have $$\lambda(\c^\p,\t^\p):={\text{\rm det}}\deri{(\c^\p,\t^\p)}{P}(\c^\p,\t^\p)={\text{\rm det}}\left(\deri{\c^\p_l}{{Q^\p}^{r_k}_{\alpha^k}}(\c^\p,\t^\p)\right)_{k,l}{\equiv \!\!\!\!\!\! / \,\,} 0.$$ Since $H$ is of generic full rank, $\lambda(\bar H(\chi,\tau))\nequiv0$. Moreover, thanks to Lemma \ref{l:Qpacv}, $\mdopz{\tau}{\beta} P(\bar H(\chi,\tau))$ is convergent for any $\beta\in\mathbb N^d$. It follows from Lemma \ref{l:lemdecv} that, for any $\beta\in\mathbb N^d$, $\mdopz{\tau}{\beta}\bar H(\chi,\tau)$ is convergent which is equivalent to $\mderi{w}{\beta}{H}(z,0)$ convergent. \end{proof}
The convergence of $H$ along the \emph{Segre sets} of higher order will be proved using the \emph{Segre set mappings} (see \cite[\S10.4]{BERbook}).
For $j\in\mathbb N^*$, we define the convergent power series $U_j:(\mathbb C^{nj}\times\mathbb C^d,0)\rightarrow (\mathbb C^d,0)$ by induction on $j$ as follows:
\begin{equation} U_1\left (z^1;t\right ):=t, \end{equation}
\begin{equation} U_{2j}\left (z^1,\chi^1,z^2,\chi^2,\ldots,z^j,\chi^j;t\right ):=U_{2j-1}\left (z^1,\chi^1,z^2,\chi^2,\ldots,z^j;Q\left (z^j,\chi^j,t\right )\right ), \end{equation}
\begin{equation} U_{2j+1}\left (z^1,\chi^1,z^2,\chi^2,\ldots,z^j,\chi^j,z^{j+1};t\right ):=U_{2j}\left (z^1,\chi^1,z^2,\chi^2,\ldots,z^j,\chi^j;\bar Q\left (\chi^j,z^{j+1},t\right )\right ), \end{equation} where $z^k,\chi^k\in\mathbb C^n$ and $t\in\mathbb C^d$. Let $V_j:(\mathbb C^{nj}\times\mathbb C^d,0)\rightarrow(\mathbb C^n\times\mathbb C^d,0)$ be the convergent power series mapping defined by \begin{equation}\label{Sj} V_j\left (z^1,\chi^1,z^2,\chi^2,\ldots;t\right ):=\left (z^1,U_j\left (z^1,\chi^1,z^2,\chi^2,\ldots;t\right )\right ). \end{equation} From the mapping identity $\eqref{realcond}$ we deduce, for any $j\geq1$, \begin{equation}\label{realcondSj} \left(z,Q\left(z^1,\chi^1,{\bar U}_{j+1}\left(\chi^1,z^1,\chi^2,z^2,\ldots;t\right)\right)\right)=V_j(z^1,\chi^2,z^2,\ldots;t). \end{equation} By setting $\tau={\bar U}_{j+1}\left(\chi^1,z^1,\chi^2,z^2,\ldots;t\right)$ in $\eqref{includ}$ and using $\eqref{realcondSj}$, we obtain the following power series identity, for any $j\geq 1$, \begin{equation}\label{includsegre} {Q^\p}\left (F(V_j(z^1,\chi^2,z^2,\ldots;t)),\bar H({\bar V}_{j+1}(\chi^1,z^1,\chi^2,z^2,\ldots;t)))=G(V_j(z^1,\chi^2,z^2,\ldots;t)\right ). \end{equation} We will prove the following convergence result:
\begin{Pro}\label{p:convHrdSj} Under the assumptions of Theorem {\rm\ref{t:cvHgenecase}} and in the above setting, we have that, for every $\beta\in N^d$ and every integer $j\geq 1$ $$\mdopz{t}{\beta}\Big(H\circ V_j(\chi^1,z^1,\chi^2,z^2,\ldots;t)\Big) \textrm{ converges. }$$ \end{Pro}
We first need the following result whose proof is inspired from \cite{Juh08}.
\begin{Lem}\label{l:lemcvh} Let $h(Z)$ be a power series with $Z=(z,w)\in\mathbb C^n\times\mathbb C^d$ and $j\geq1$ a fixed integer. If for every $\beta\in\mathbb N^d$ $$\mdopz{t}{\beta} \Big(h\circ V_j(z^1,\chi^1,\ldots;t)\Big) \textrm{ converges},$$ then for every $\gamma\in\mathbb N^{n+d}$ and every $\beta\in\mathbb N^d$, $\mdopz{t}{\beta}\Big( \left(\mderi{Z}{\gamma}{h}\right)\circ V_j(z^1,\chi^1,\ldots;t)\Big)$ converges. \end{Lem}
\begin{proof} We prove this lemma by induction on $|\gamma|$. The case $\gamma=0$ comes from the hypothesis. Now assume the result for any $\gamma\in\mathbb N^{n+d}$, $|\gamma|=l$ and $\beta\in\mathbb N^d$, we show the analogous conclusion for $\gamma^\prime\in\mathbb N^{n+d}$, $|\gamma^\prime|=l+1$ and any $\beta\in\mathbb N^d$. We denote \begin{equation}\label{defRj} R^j_\gamma(z^1,\chi^1,\ldots;t):=\left(\mderi{Z}{\gamma}{h}\circ V_j\right)(z^1,\chi^1,\ldots;t). \end{equation} The induction hypothesis implies that $\mdopz{t}{\beta}R^j_\gamma$ are convergent. Differentiating $\eqref{defRj}$ with respect to $t$, we obtain \begin{equation}\label{deri t Rj}
\deri{t}{R^j_\gamma}(z^1,\chi^1,\ldots;t)=\left(\frac{\partial^{|\gamma|+1} h}{\partial Z^\gamma \partial w}\circ V_j\right)(z^1,\chi^1,\ldots;t)\cdot\deri{t}{U_j}(z^1,\chi^1,\ldots;t). \end{equation} Since $\deri{\tau}{Q}(0)=I_d$, we have $\deri{t}{U_j}(0)=I_d$. And, multiplying $\eqref{deri t Rj}$, by the classical adjoint matrix of $\deri{t}{U_j}(z^1,\chi^1,\ldots;t)$, i.e. the transpose of the cofactor matrix denoted by $\text{\rm adj }\left(\deri{t}{U_j}(z^1,\chi^1,\ldots;t)\right)$, we obtain \begin{equation}
\left(\frac{\partial^{|\gamma|+1} h}{\partial Z^\gamma \partial w}\circ V_j\right)(z^1,\chi^1,\ldots;t)=\frac{\deri{t}{R^j_\gamma}(z^1,\chi^1,\ldots;t)\cdot\text{\rm adj }\left(\deri{t}{U_j}(z^1,\chi^1,\ldots;t)\right)}{{\text{\rm det}}\deri{t}{U_j}(z^1,\chi^1,\ldots;t)}.
\end{equation} Now, if we apply $\mdopz{t}{\beta}$ to the previous identity we get that $\mdopz{t}{\beta}\left(\frac{\partial^{|\gamma|+1} h}{\partial Z^\gamma \partial w}\circ V_j\right)(z^1,\chi^1,\ldots;t)$ is a ratio of two convergent power series whose denominator does not vanish at 0. Therefore $\mdopz{t}{\beta}\left(\frac{\partial^{|\gamma|+1} h}{\partial Z^\gamma \partial w}\circ V_j\right)(z^1,\chi^1,\ldots;t)$ is convergent. To obtain the other derivatives, we differentiate $\eqref{defRj}$ with respect to $z^1$, we get \begin{equation}
\left(\frac{\partial^{|\gamma|+1 } h}{\partial Z^\gamma \partial z}\circ V_j\right)(z^1,\chi^1,\ldots;t)+\left(\frac{\partial^{|\gamma|+1} h}{\partial Z^\gamma \partial w}\circ V_j\right)(z^1,\chi^1,\ldots;t)\cdot\deri{z^1}{U_j}(z^1,\chi^1,\ldots;t)=\deri{z^1}{R^j_\gamma}(z^1,\chi^1,\ldots;t). \end{equation} All the derivatives of $\mdopz{t}{\beta}\deri{z^1}{R^j_\gamma}(z^1,\chi^1,\ldots;t)$ are convergent. Indeed, $\mdopz{t}{\beta}R^j_\gamma(z^1,\chi^1,\ldots;t)$ is convergent by hypothesis, thus $\dop{z^1}\left(\mdopz{t}{\beta}R^j_\gamma(z^1,\chi^1,\ldots;t)\right)$ is convergent too. This implies that $\mdopz{t}{\beta}\left(\deri{z^1}{R^j_\gamma}(z^1,\chi^1,\ldots;t)\right)$ is convergent.
Thus, using the fact that $\mdopz{t}{\beta}\left(\frac{\partial^{|\gamma|+1} h}{\partial Z^\gamma \partial w}\circ V_j\right)(z^1,\chi^1,\ldots;t)$ is convergent by the previously established result, we obtain the convergence of $\mdopz{t}{\beta}\left(\frac{\partial^{|\gamma|+1} h}{\partial Z^\gamma \partial z}\circ V_j\right)(z^1,\chi^1,\ldots;t)$. Consequently, the proof of Lemma \ref{l:lemcvh} is complete. \end{proof}
To prove Proposition \ref{p:convHrdSj}, we will also need the following mapping identity result whose proof can be found in \cite{Juh08}.
\begin{Lem}\cite[Proposition 6.4]{Juh08}\label{l:lem6.4deJuh} In the above situation and for $j\geq1$ a fixed integer, as $\text{\rm Jac} H\nequiv0$ we may choose $\gamma^j\in\mathbb N^d$ such that \begin{equation}
\mdopz{t}{\delta^\prime} {\text{\rm det}}\left(\left.\left(\dop{z^1}\left( F(z^1,Q(z^1,\chi^1,\tau))\right)\right)\right|_{\tau={\bar U}_{j+1}(\chi^1,z^1,\chi^2,\ldots;t)}\right)\equiv 0, \textrm{ for } |\delta^\prime|<|\gamma^j|, \end{equation}
\begin{equation}
\mdopz{t}{\gamma^j} {\text{\rm det}}\left(\left.\left(\dop{z^1}\left( F(z^1,Q(z^1,\chi^1,\tau))\right)\right)\right|_{\tau={\bar U}_{j+1}(\chi^1,z^1,\chi^2,\ldots;t)}\right){\equiv \!\!\!\!\!\! / \,\,} 0, \end{equation}
and, so that for every positive integer $k$, we have \begin{eqnarray}
\frac{1}{(k\gamma^j)!}\left.\frac{\partial^{k|\gamma^j|}}{\partial t^{k\gamma^j}}\right|_{t=0}\left\{{\text{\rm det}}\left(\left.\left( \dop{z^1}\left(F(z^1,Q(z^1,\chi^1,\tau))\right)\right)\right|_{\tau={\bar U}_{j+1}(\chi^1,z^1,\chi^2,\ldots;t)}\right)^k\right\} \\ \nonumber
=\left(\frac{1}{\gamma^j!}\left.\frac{\partial^{|\gamma^j|}}{\partial t^{\gamma^j}}\right|_{t=0}{\text{\rm det}}\left(\left.\left( \dop{z^1}\left(F(z^1,Q(z^1,\chi^1,\tau))\right)\right)\right|_{\tau={\bar U}_{j+1}(\chi^1,z^1,\chi^2,\ldots;t)}\right)\right)^k\nonumber. \end{eqnarray}
Then, for any $\alpha\in\mathbb N^n$ and $\beta\in\mathbb N^d$ with $|\alpha|+|\beta|>0$ there is a universal polynomial of its arguments $T_{\alpha,\beta,\gamma^j}$ such that the following holds: for any $j\geq1$ and every $r\in\{1,\ldots,r\}$ , we have the mapping identity \begin{equation}\label{exprQplem6.4} \begin{split} &\mdopz{t}{\beta}{Q^\p}^r_{{z^\p}^\alpha}\left (F\circ V_j(z^1,\chi^2,z^2,\ldots;0),\bar H\circ {\bar V}_{j+1}(\chi^1,z^1,\chi^2,\ldots;t)\right ) \\
&=\frac{T_{\alpha,\beta,\gamma^j}\left(\left( \mdopz{t}{\delta}\left(\left.\left(\mdop{z^1}{\eta} H^{(r)}(z^1,Q(z^1,\chi^1,\tau))\right)\right|_{\tau={\bar U}_{j+1}(\chi^1,z^1,\chi^2,\ldots;t)} \right)\right)_{|\eta|+|\delta|\leq (|\alpha|+|\beta|)(2|\gamma^j|+1)-|\gamma^j|}\right)}{\left(\frac{1}{\gamma^j!}\left.\frac{\partial^{|\gamma^j|}}{\partial t^{\gamma^j}}\right|_{t=0}{\text{\rm det}}\left(\left.\left( \dop{z^1}\left(F(z^1,Q(z^1,\chi^1,\tau))\right)\right)\right|_{\tau={\bar U}_{j+1}(\chi^1,z^1,\chi^2,\ldots;t)}\right)\right)^{2(|\alpha|+|\beta|)-1} }. \end{split} \end{equation} \end{Lem}
\begin{proof}[Proof of Proposition {\rm\ref{p:convHrdSj}}] We prove this proposition by induction on $j$. For $j=1$, the result comes from Proposition \ref{p:convHrdS1}. We assume the proposition is true for fixed $j$ and we prove it for $j+1$. To do this we need the following lemma:
\begin{Lem}\label{l:convQFSjHSj+1} For any $\beta\in\mathbb N^d$, $\alpha\in\mathbb N^n$ and $r\in\{1,\ldots,d\}$, $$\mdopz{t}{\beta}{Q^\p}^r_{{z^\p}^\alpha}\left (F\circ V_j(z^1,\chi^2,z^2,\ldots;0),\bar H\circ {\bar V}_{j+1}(\chi^1,z^1,\chi^2,\ldots;t)\right )\textrm{ is convergent.}$$ \end{Lem}
\begin{proof} We begin by the case $|\alpha|=|\beta|=0$. By setting $t=0$ in $\eqref{includsegre}$, we obtain $${Q^\p}\left (F\circ V_j(z^1,\chi^2,z^2,\ldots;0)),\bar H\circ {\bar V}_{j+1}(\chi^1,z^1,\chi^2,\ldots;0)\right )=G\circ V_j(z^1,\chi^2,z^2,\ldots;0)).$$ The left hand side of the previous equation is exactly the expression whose convergence has to be proven and the right hand side is convergent by the induction hypothesis.
For the case $|\alpha|+|\beta|>0$, from Lemma \ref{l:lem6.4deJuh} we have that
\begin{equation*} \begin{split} &\mdopz{t}{\beta}{Q^\p}^r_{{z^\p}^\alpha}\left (F\circ V_j(z^1,\chi^2,z^2,\ldots;0),\bar H\circ {\bar V}_{j+1}(\chi^1,z^1,\chi^2,\ldots;t)\right ) \\
&=\frac{T_{\alpha,\beta,\gamma}\left(\left( \mdopz{t}{\delta}\left(\left.\left(\mdop{z^1}{\eta} H^{(r)}(z^1,Q(z^1,\chi^1,\tau))\right)\right|_{\tau={\bar U}_{j+1}(\chi^1,z^1,\chi^2,\ldots;t)} \right)\right)_{|\eta|+|\delta|\leq (|\alpha|+|\beta|)(2|\gamma|+1)-|\gamma|}\right)}{\left(\frac{1}{\gamma!}\left.\frac{\partial^{|\gamma|}}{\partial t^{\gamma}}\right|_{t=0}{\text{\rm det}}\left(\left.\left( \dop{z^1}\left(F(z^1,Q(z^1,\chi^1,\tau))\right)\right)\right|_{\tau={\bar U}_{j+1}(\chi^1,z^1,\chi^2,\ldots;t)}\right)\right)^{2(|\alpha|+|\beta|)-1} }. \end{split} \end{equation*} This is a ratio of two convergent power series. Indeed, the numerator is polynomial in the derivatives of $Q$ which are convergent and expressions of the form $$\mdopz{t}{\delta}\dblderi{z}{\eta}{w}{\mu}{H}\left (z^1,Q(z^1,\chi^1,{\bar U}_{j+1}(\chi^1,z^1,\chi^2,\ldots;t)\right ),$$ i.e. using $\eqref{realcondSj}$, of the form $$\mdopz{t}{\delta}\left(\dblderi{z}{\eta}{w}{\mu}{H}\circ V_j\right)(z^1,\chi^2,z^2,\ldots;t)$$ which are also convergent thanks to the induction hypothesis and Lemma \ref{l:lemcvh}. \end{proof}
We come back to the proof of Proposition \ref{p:convHrdSj}. As in the proof of Proposition \ref{p:convHrdS1}, from the holomorphic nondegeneracy assumption on $M^\p$, we may choose $\alpha^i\in\mathbb N^n$ and $1\leq r_i\leq d$ for $i\in\{1,\ldots,n\}$ such that \begin{equation}\label{hypohn} {\text{\rm det}}\left(\deri{\c^\p_l}{{Q^\p}^{r_k}_{\alpha^k}}(\c^\p,\t^\p)\right)_{k,l}{\equiv \!\!\!\!\!\! / \,\,} 0. \end{equation} We define a power series mapping $P_{j+1}:(\mathbb C^{(j+1)n}\times\mathbb C^{n+d},0)\rightarrow (\mathbb C^{n+d},0)$ in the following way: $$ P_{j+1}(\chi^1,z^1,\chi^2,z^2,\ldots,Y):=\left( \begin{array}{c} {Q^\p}^{r_1}_{{z^\p}^{\alpha^1}}(F(V_j(z^1,\chi^2,z^2,\ldots;0)),Y)\\ \vdots\\ {Q^\p}^{r_n}_{{z^\p}^{\alpha^n}}(F(V_j(z^1,\chi^2,z^2,\ldots;0)),Y)\\ {Q^\p}^1(F(V_j(z^1,\chi^2,z^2,\ldots;0)),Y)\\ \vdots\\ {Q^\p}^d(F(V_j(z^1,\chi^2,z^2,\ldots;0)),Y) \end{array}\right). $$ By the induction hypothesis, $P_{j+1}$ is a convergent power series mapping. Moreover, we observe that \begin{equation}\label{deriPj+1} {\text{\rm det}}\deri{Y}{P_{j+1}}\left (\chi^1,z^1,\chi^2,\ldots;\bar H\circ{\bar V}_{j+1}(\chi^1,z^1,\chi^2,\ldots;t)\right )\nequiv0. \end{equation} Indeed, if we set, in $\eqref{deriPj+1}$, $z^l=0$ for $l\geq 1$ and $\chi^l=0$ for $l\geq 2$, we obtain, since ${\bar V}_{j+1}(\chi^1,0,\ldots;t)=(\chi^1,t)$, $\deri{\c^\p}{{Q^\p}}(0,\c^\p,\t^\p)=0$ and $\deri{\t^\p}{{Q^\p}}(0,\c^\p,\t^\p)=I_d$: \begin{equation} {\text{\rm det}}\deri{Y}{P_{j+1}}\left (\chi^1,0,\ldots;\bar H(\chi^1,t)\right )={\text{\rm det}}\left(\deri{\c^\p_l}{{Q^\p}_{{z^\p}^{\alpha^k}}^{r_k}}\left (0,\bar H(\chi^1,t)\right )\right)_{k,l}={\text{\rm det}}\left(\deri{\c^\p_l}{{Q^\p}_{\alpha^k}^{r_k}}\left (\bar H(\chi^1,t)\right )\right)_{k,l}, \end{equation} which is not identically zero in view of $\eqref{hypohn}$ and the fact that $H$ is generically of full rank.
On the other hand, for any $\beta\in\mathbb N^d$, $\mdopz{t}{\beta}P_{j+1}\left (\chi^1,z^1,\chi^2,\ldots;\bar H\circ{\bar V}_{j+1}(\chi^1,z^1,\chi^2,\ldots;t)\right )$ is convergent thanks to Lemma \ref{l:convQFSjHSj+1}. So, from Lemma \ref{l:lemdecv} we obtain the desired result.
\end{proof}
We can now complete the proof of Theorem \ref{t:cvHgenecase}.
\begin{proof}[Proof of Theorem {\rm\ref{t:cvHgenecase}}]
We use the previous setting and notation. From Proposition \ref{p:convHrdSj}, we have that for any $j\geq 1$, $H\circ V_{j}(\chi^1,z^1,\chi^2,\ldots;0)$ is convergent. Moreover, we known (see \cite[\S10.4, \S10.5]{BERbook}) that there exists $(p_k)_k$ a sequence which converges to 0 and such that, for $j=2(d+1)$, the rank of $V_{j}(\chi^1,z^1,\chi^2,\ldots;0)$ at any $p_k$ is $n+d$ and $V_j(p_k,0)=0$. So $V_{2(d+1)}(\chi^1,z^1,\chi^2,\ldots;0)$ has a convergent right inverse in a neighborhood of a suitable $p_k$ close enough to 0 which proves the convergence of $H$. \end{proof}
\subsection{Proof of Theorem \ref{t:thcourbehol}}
To prove Theorem \ref{t:thcourbehol}, we need the following lemma.
\begin{Lem}\label{l:lemcourbehol} Let $(M,p)$ and $(M^\p,p^\prime)$ be two germs of real-analytic hypersurfaces in $\mathbb C^{n+1}$. If $M$ is holomorphically nondegenerate\ at $p$ and if $M^\p$ does not contain any holomorphic curve through $p^\prime$ then any holomorphic formal mapping sending $(M,p)$ into $(M^\p,p^\prime)$ is either constant or generically of full rank. \end{Lem}
\begin{proof} The proof of Proposition 7.1 in \cite{BER00} contains the following fact: Let $M$ and $M^\p$ be formal real hypersurfaces through points $p$ and $p^\prime$ respectively and $H:(\mathbb C^{n+1},p)\rightarrow(\mathbb C^{n+1},p^\prime)$ be a formal holomorphic mapping sending $M$ into $M^\p$, with $M$ holomorphically nondegenerate\ at $p$ and $M^\p$ not containing any nontrivial formal holomorphic curve; if furthermore $M$ and $M^\p$ are given in normal coordinates and if $H=(F,G)$, then $G\equiv0$ implies $H\equiv0$.
Moreover, in \cite{LM06} the authors proved, without any assumption on $M^\p$, that every formal holomorphic map $H:(\mathbb C^{n+1},p) \rightarrow(\mathbb C^{n+1},p^\prime)$ sending $M$ into $M^\p$ which is transversally nonflat (i.e. satisfies in normal coordinates $G {\equiv \!\!\!\!\!\! / \,\,} 0$) satisfies $\text{\rm Jac} H \nequiv0$.
So, to prove the lemma it suffices to apply the two previous results noticing that if $M^\p$ is a real-analytic hypersurface that does not contain any holomorphic curve through $p^\prime$ then it does not contain a formal holomorphic one through the same point (see \cite{Mil78}). \end{proof}
\begin{proof}[Proof of Theorem {\rm\ref{t:thcourbehol}}] The theorem is obtained by combining Lemma \ref{l:lemcourbehol} and Theorem \ref{t:cvHgenecase}. \end{proof}
\section{Proof of the Artinian Theorem}\label{s:pfartinresult}
\begin{proof}[Proof of Theorem {\rm\ref{t:artinhyp}}] This proof is inspired by the proof of Lemma 14.2 in \cite{BMR02}. We use the notation of Section \ref{s:cvR} and in particular normal coordinates. From Theorem \ref{t:cvR} we know that $\mc{R}$ is convergent, thus using the Taylor expansion of ${Q^\p}$, for any $\alpha\in\mathbb N^n$, ${\Qp_\a}(\bar H(\chi,\tau))=r_\alpha(\chi,\tau)$ converges. For fixed $k\in\mathbb N$, by Artin's approximation theorem \cite{Artin68} there exists $H^k:(\mathbb C^n\times\mathbb C,0)\rightarrow (\mathbb C^n\times\mathbb C,0)$ a holomorphic convergent power series mapping which agrees up with $H$ up to order $k$ at 0 and such that ${\Qp_\a}(\bar {H}^k(\chi,\tau))=r_\alpha(\chi,\tau)$, for any $\alpha\in\mathbb N^n$. Consequently, we have the following power series identity, \begin{equation}\label{egalQ} {Q^\p}({z^\p},\bar H(\chi,\tau))={Q^\p}({z^\p},\bar {H}^k(\chi,\tau)). \end{equation}
To show that, for every $k\in\mathbb N$, $H^k$ sends a neighborhood of 0 in $M$ into a neighborhood of 0 in $M^\p$, we consider \begin{equation} {\rho^\p}:({z^\p},{w^\p},\c^\p,\t^\p)\in U\mapsto {w^\p}-{Q^\p}({z^\p},\c^\p,\t^\p)\in\mathbb C, \end{equation} where $U$ is a sufficently small neighborhood of 0 in $\mathbb C^{2n+2}$, and \begin{equation} {\tilde \rp}({z^\p},{w^\p},\c^\p,\t^\p):={\bar \rp}(\c^\p,\t^\p,{z^\p},{w^\p}). \end{equation} The convergent power series ${\tilde \rp}$ is of rank 1 in a neighbourhood of 0 and ${\tilde \rp}({z^\p},{w^\p},\c^\p,\t^\p)=0$ implies ${\rho^\p}({z^\p},{w^\p},\c^\p,\t^\p)=0$ thanks to the mapping identity $\eqref{realcond}$. So, there exists $u\in\mathbb C[[{z^\p},{w^\p},\c^\p,\t^\p]]$ such that, \begin{equation} {\rho^\p}({z^\p},{w^\p},\c^\p,\t^\p)=u({z^\p},{w^\p},\c^\p,\t^\p).{\tilde \rp}({z^\p},{w^\p},\c^\p,\t^\p). \end{equation} We have, thanks to $\eqref{egalQ}$, \begin{equation}\label{egalrp} {\rho^\p}\left ({z^\p},{w^\p},\bar H(\chi,\tau)\right )={\rho^\p}\left ({z^\p},{w^\p},\bar {H}^k(\chi,\tau)\right ), \end{equation} and $H^k$ sends $M$ into $M^\p$ if and only if ${Q^\p}\left (F^k(z,Q(z,\chi,\tau)\right ),\bar {H}^k(\chi,\tau))=G^k\left (z,Q(z,\chi,\tau)\right )$ i.e. ${\rho^\p}\left (H^k(z,Q(z,\chi,\tau)),\bar {H}^k(\chi,\tau)\right )\equiv0$. But, \begin{displaymath} \begin{array}{lll} {\rho^\p}\left (H^k(z,Q(z,\chi,\tau)\right ),\bar {H}^k(\chi,\tau))&=&{\rho^\p}\left (H^k(z,Q(z,\chi,\tau)),\bar H(\chi,\tau)\right ) \textrm{ from $\eqref{egalrp}$ }\\ &=&u\left (H^k(z,Q(z,\chi,\tau)),\bar H(\chi,\tau)\right ).{\tilde \rp}\left (H^k(z,Q(z,\chi,\tau)),\bar H(\chi,\tau)\right ) \\ &=&u\left (H^k(z,Q(z,\chi,\tau)),\bar H(\chi,\tau)\right ).{\bar \rp}\left (\bar H(\chi,\tau),H^k(z,Q(z,\chi,\tau))\right ) \end{array} \end{displaymath}
But, we have the identity ${\bar \rp}\left (\bar H(\chi,\tau),H^k(z,Q(z,\chi,\tau))\right )\equiv0$; Indeed, we recall that, since $H$ sends $M$ into $M^\p$, $${Q^\p}\left (F(z,Q(z,\chi,\tau)),\bar H(\chi,\tau)\right )=G(z,Q(z,\chi,\tau)).$$ So, applying $\bar {Q^\p}\left(\bar F\left(\chi,\tau\right),F\left(z,Q\left(z,\chi,\tau\right)\right),.\right)$ to each side of the previous identity we obtain \begin{displaymath} \begin{array}{ll} \bar {Q^\p}\left(\bar F\left(\chi,\tau\right),F\left(z,Q\left(z,\chi,\tau\right)\right),{Q^\p}\left(F\left(z,Q\left(z,\chi,\tau\right)\right),\bar H\left(\chi,\tau\right)\right)\right) \\ =\bar {Q^\p}\left(\bar F\left(\chi,\tau\right),F\left(z,Q\left(z,\chi,\tau\right)\right),G\left(z,Q\left(z,\chi,\tau\right)\right)\right). \end{array} \end{displaymath} Thus, using the mapping identity $\eqref{realcond}$, we have \begin{equation} \bar G(\chi,\tau)=\bar {Q^\p}\left (\bar F(\chi,\tau),H(z,Q(z,\chi,\tau))\right ) \end{equation} i.e. ${\bar \rp}\left (\bar H(\chi,\tau),H(z,Q(z,\chi,\tau))\right )\equiv0$. Therefore ${\bar \rp}\left (\bar H(\chi,\tau),H^k(z,Q(z,\chi,\tau))\right )\equiv0$ and the proof of Theorem \ref{t:artinhyp} is complete. \end{proof}
\end{document} | arXiv |
Cyclic cover
In algebraic topology and algebraic geometry, a cyclic cover or cyclic covering is a covering space for which the set of covering transformations forms a cyclic group.[1][2] As with cyclic groups, there may be both finite and infinite cyclic covers.[3]
Cyclic covers have proven useful in the descriptions of knot topology[1][3] and the algebraic geometry of Calabi–Yau manifolds.[2]
In classical algebraic geometry, cyclic covers are a tool used to create new objects from existing ones through, for example, a field extension by a root element.[4] The powers of the root element form a cyclic group and provide the basis for a cyclic cover. A line bundle over a complex projective variety with torsion index $r$ may induce a cyclic Galois covering with cyclic group of order $r$.
References
1. Seifert and Threlfall, A Textbook of Topology. Academic Press. 1980. p. 292. ISBN 9780080874050. Retrieved 25 August 2017. cyclic covering.
2. Rohde, Jan Christian (2009). Cyclic coverings, Calabi-Yau manifolds and complex multiplication ([Online-Ausg.]. ed.). Berlin: Springer. pp. 59–62. ISBN 978-3-642-00639-5.
3. Milnor, John. "Infinite cyclic coverings" (PDF). Conference on the Topology of Manifolds. Vol. 13. 1968. Retrieved 25 August 2017.
4. Ambro, Florin (2013). "Cyclic covers and toroidal embeddings". arXiv:1310.3951 [math.AG].
Further reading
• Fedorchuk, Maksym (2011-05-13). "Cyclic covering morphisms on M0,n". arXiv:1105.0655 [math.AG].
• Singh, Anurag K. (2002-08-28). "Cyclic covers of rings with rational singularities". arXiv:math/0208226.
• "what is the cyclic cover trick?". MathOverflow. 19 June 2013. Retrieved 2017-08-26.
| Wikipedia |
Miloudi* , Rahal* , and Khiat*: Contribution to Improve Database Classification Algorithms for Multi-Database Mining
Salim Miloudi* , Sid Ahmed Rahal* and Salim Khiat*
Contribution to Improve Database Classification Algorithms for Multi-Database Mining
Abstract: Database classification is an important preprocessing step for the multi-database mining (MDM). In fact, when a multi-branch company needs to explore its distributed data for decision making, it is imperative to classify these multiple databases into similar clusters before analyzing the data. To search for the best 2classification of a set of n databases, existing algorithms generate from 1 to (n –n)/2 candidate classifications. Although each candidate classification is included in the next one (i.e., clusters in the current classification are subsets of clusters in the next classification), existing algorithms generate each classification independently, that is, without taking into account the use of clusters from the previous classification. Consequently, existing algorithms are time consuming, especially when the number of candidate classifications increases. To overcome the latter problem, we propose in this paper an efficient approach that represents the problem of classifying the multiple databases as a problem of identifying the connected components of an undirected weighted graph. Theoretical analysis and experiments on public databases confirm the efficiency of our algorithm against existing works and that it overcomes the problem of increase in the execution time.
Keywords: Connected Components , Database Classification , Graph-Based Algorithm , Multi-Database Mining
Many large organizations have different types of business distributed through their branches and each branch may have its own database that collects transactions continuously from customers. For business decision making, traditional data mining techniques called mono-database mining integrate all the data from these databases to amass a huge dataset for knowledge discovery. However, this leads to an expensive search cost for centralized processing and might disguise some important patterns (i.e., frequent itemset and association rules). To overcome the latter problems, these multiple databases have to be classified into clusters of similar databases according to their data correlation.
For example, if an inter-sate company has 10 branches including 5 branches for household appliances, 3 branches for clothing and 2 branches for food, we cannot apply existing data mining techniques over the union of the 10 branch databases. We might need to classify them into 3 clusters according to their type of business, and then we can analyze each database cluster individually. Consequently the multi-database mining (MDM) [1-4] task becomes more manageable.
In order to classify transactional multiple databases, few algorithms have been proposed in the literature [5-8]. The most efficient algorithm in terms of accuracy and time complexity is due to Adhikari and Rao [7], which finds the best classification of a set of n database in O(m×n2) time such that m (1≤ m ≤ (n2-n)/2) is the number of all candidate classifications. When m = (n2–n)/2 and n is very large, the time complexity will considerably increase. Hence, to reduce the time complexity of the existing algorithms, we present in this paper an improved algorithm that classifies a set of n databases D={D1,D2,…,Dn} by analyzing and determining the connected components of an undirected weighted graph G=(D,E), such that D is the vertex set and E the edge set (definitions are given in Section 3.1).
The rest of the paper is organized as follows: Section 2 discusses the existing works and point out their limitations. In Section 3, we describe the proposed approach for classifying the multiple databases. In Section 4, we perform some experiments to analyze and compare the proposed algorithm with the existing works in terms of accuracy and running times. Finally, Section 5 concludes this paper and highlights the future works.
2. Related Works
Traditional MDM process aims to integrate all the data from the multiple databases for knowledge discovery. This approach may not be a good strategy because of the large amount of irrelevant data involved during the process. To overcome the issue related to data irrelevancy, Liu et al. [9,10] have proposed a preprocessing technique called database selection to identify which databases are more likely relevant to a user query (Q). A relevance measure named RF is thus designed to identify relevant databases containing reliable information relative to the user application.
Database selection is a good strategy to reduce the amount of explored data while improving the quality of the patterns mined from the multiple databases. However, this approach depends on the userapplication and when the multiple databases are mined without specifying any application, database selection cannot be applied.
The previous work motivated Wu et al. [5] to propose a new approach independent-application for classifying multiple databases. Thus, a similarity measure sim has been designed to group similar databases within the same cluster. The similarity between two databases Di and Dj, denoted sim(Di,Dj), is calculated based on items shared between databases Di and Dj. Such similarity measure could be useful to estimate the correlation between large databases, since extracting more information such as frequent itemsets could be costly in terms of CPU time. However, using items may produce low accuracy in finding the correct similarity between two databases. In fact, two databases are similar if they share many transactions.
Based on the previous similarity measure, an algorithm named BestClassification [5] has been proposed to search for the best classification of a set of n databases. The previous algorithm calls a procedure GreedyClass to generate a classification for each similarity threshold δ defined initially by the user. Hence, two databases Di and Dj are similar if the similarity value sim(Di,Dj) is above δ. The time complexity of the proposed algorithm is O(h×n4), such that n is the database number and h is the number of classification generated before obtaining the best classification. Even if the experiment results shown in [5] prove that correct classifications are obtained for certain similarity thresholds, the time complexity of the algorithm remains high and becomes sever when the database number increases. Moreover, BestClassification fails to find the best classification when choosing an incorrect similarity step size as shown in [6]. In certain cases, the while-loop of the algorithm doesn't terminate and may generate an infinite loop without finding any classification. The latter problem is due to the strong dependence of the algorithm on the similarity step size, which is a user-input.
Inspired by the same concepts presented in [5], Li et al. [6] have modified the algorithm BestClassification [5] in order to optimize its time complexity and obtain the correct best classification of the multiple databases. In order to avoid missing the best classification (i.e., in case in which an incorrect step size has been selected), the distinct similarity values between the n databases are used as similarity thresholds to generate classifications. Thus, for each distinct similarity value sorted in the increasing order, a classification is produced. The time complexity of their improved algorithm is O(h×n3). Although the experiments results show that the proposed algorithm is efficient and produces always the best complete classification, the similarity measure still need to be improved since it still depends on common items shared between databases. Moreover, the time complexity of the algorithm needs to be optimized.
In 2007, Adhikari and Rao [7] have proposed an approach for multi-database clustering that uses a more accurate similarity measure based on the support of frequent itemsets shared between databases. The time complexity of the proposed algorithm is O(m×n2), such that n is the database number and m (1 ≤ m ≤ n2) is the number of all candidate classifications. The experiment results obtained in [7] show that the proposed algorithm is effective and finds correct and optimal classifications only at few similarity levels. However, its time complexity needs to be improved especially when m = n2, which may occur frequently when the similarity values between databases are distinct.
In 2013, Liu et al. [8] have proposed an approach for completely classifying databases based on the same similarity measure proposed in [7]. The proposed algorithm is simple to implement but the clustering isn't effective when the database number increases. In fact, the time complexity of the algorithm is O(t×2n×n) such that n is the database number and t the distinct similarity values between the n databases.
According to this study, existing multi-database classification algorithms [5-8] produce hierarchical classifications that verify the following property. Let class(D, δi) and class(D, δj) be two candidate classifications of D={D1,D2,…,Dn} generated at two consecutive similarity level δi and δj .Then for each cluster gx in class(D,sim, δi) , there is certainly a cluster gy in class(D,sim, δi), such that gx⊂gy.
Despite of the latter property, the previous algorithms produce each classification independently, that is, instead to use the earlier clusters (i.e., generated in the previous classifications) to build the clusters of the next classification, they generate each classification starting from the initial state where each database forms one cluster {D1},{D2},…,{Dn}. Consequently the existing algorithms are time consuming and do unnecessary work.
The above observations motivated us to design an improved database classification algorithm that overcomes the later problems. To achieve our goal, we need to generate each classification from the earlier classification generated so far. The proposed approach and algorithms are presented in the following section.
3. Proposed Approach and Algorithms
In this section, we describe the proposed approach for classifying the multiple databases. We assume that each database has been mined using algorithms for frequent itemset discovery [11-13].
Let D={D1,D2,…,Dn} be the set of n database objects to classify, then the problem of generating a classification class(D,sim,δ) at a similarity level δ, can be described in terms of determining the connected components of an undirected weighted graph G=(D,E).
We consider the database set D={D1,D2,…,Dn} as the vertex set of G and E the edge set. In the following, the terms database and vertex are used interchangeably. The weight of an edge (Di,Dj) ∈ E is the similarity value between the corresponding databases sim(Di,Dj). At a certain similarity level δ, there is an edge connecting two vertices Di and Dj if and only if, sim(Di,Dj) ≥ δ.
Initially, the graph G=(D,E) is empty with n disconnected vertices, i.e, E=∅. Then, similarities are calculated between each vertex pairs (Di,Dj) using an appropriate similarity measure, for i,j=1 to n. After that, edges are added to E starting with the vertex pairs having the highest similarity as follows.
Proposed approach for classifying the n multiple databases.
Let δ1,δ2,…,δm be the m distinct similarity values between the n databases sorted in the decreasing order (i.e., δ1>δ2>…>δm) and let Eδl={(Di, Dj) ∈ E, sim(Di,Dj)= δl, i,j=1 to n, i≠j} be the list of edges with weight value equal to δl (l=1 to m).
For each distinct similarity value δl, a candidate classification class(D,sim,δl) is generated by adding the edge list Eδl to the edge set E such that E=Eδ1∪Eδ2∪…∪ Eδl-1 (i.e., there are l–1 classifications generated earlier). Then, it remains to identify the connected component of G in order to discover the database clusters in class(D,sim,δl).
At a given similarity level δl, if each connected component of G, denoted gδlk, forms a clique (i.e., a subset of vertices, each pair of which is connected by an edge in E), then the corresponding classification class(D,sim, δl)={gδl1,gδl2,…,gδlk} is called a complete classification . In this paper we are interested in finding such classifications.
For classification assessment, the authors in [7] have proposed a measure goodness based on the intracluster similarity, the inter-cluster distance and the number of clusters. The complete classification with the maximum goodness value is selected as the best classification of the database set. Our classification approach is depicted in Fig. 1.
In the following section we give some definitions required to understand the proposed approach.
3.1 The Relevant Concepts
Referring to the related concept proposed by Adhikari and Rao [7], we have definitions as follows. DEFINITION 1. The similarity measure sim between two databases Di, and Dj is calculated as follows.
DEFINITION 1. The similarity measure sim between two databases Di, and Dj is calculated as follows.
[TeX:] $$\operatorname { sim } \left( D _ { i } , D _ { j } , \alpha \right)= \frac{\sum _ { I \in \left( \operatorname { FIS } \left( D _ { i } , \alpha \right) \cap \operatorname { FIS } \left( D _ { j } , \alpha \right) \right) } \min \left\{ \operatorname { supp } \left( I , D _ { i } \right) , \operatorname { supp } \left( I , D _ { j } \right) \right\}}{\sum _ { I \in \left( \operatorname { FIS } \left( D _ { i } , \alpha \right) \cup \operatorname { FIS } \left( D _ { j } , \alpha \right) \right) } \max \left\{ \operatorname { supp } \left( I , D _ { i } \right) , \operatorname { supp } \left( I , D _ { j } \right) \right\}} $$
where α is the minimum support threshold, I ∈ (FIS(Di,α) ∩ FIS(Dj,α)) denotes that I is a frequent itemset in both Di and Dj and supp(I, Di) is the support of I in database Di .The similarity measure sim has the following properties.
PROPERTY 1. sim(Di, Dj, α) satisfies that :
(1) 0 ≤ sim(Di, Dj, α) ≤ 1; (2) sim(Di, Dj, α) = sim(Dj, Di, α) ;(3) sim(Di, Di, α) = 1 for i,j=1…n
DEFINITION 2. The distance measure dist between databases Di and Dj is defined as follows.
[TeX:] $$\operatorname { dist } \left( D _ { i } , D _ { j } , \alpha \right) = 1 - \operatorname { sim } \left( D _ { i } , D _ { j } , \alpha \right)$$
The distance function dist has the following properties.
PROPERTY 2. dist(Di, Dj, α) satisfies that :
(1) 0 ≤ dist(Di, Dj, α) ≤ 1; (2) dist(Di, Dj, α) = dist(Dj, Di, α) ;(3) dist(Di, Di, α) = 1 for i,j=1…n
DEFINITION 3. Let class(D,sim,δ)={gδ1, gδ2,…,gδk} be a classification of k clusters of the database set D={D1,D2,...,Dn} at the similarity level δ. The intra-cluster similarity of class(D,sim,δ) is defined as follows.
[TeX:] $$\operatorname { intra } - \operatorname { sim } ( \operatorname { class } ( D , \delta ) ) = \sum _ { l = 1 } ^ { k } \sum _ { D _ { i } , D _ { j } \in g _ { l } ^ { \delta } } ^ { i \neq j } \operatorname { sim } \left( D _ { i } , D _ { j } , \alpha \right)$$
DEFINITION 4. The inter-cluster distance of class(D,sim,δ) at the similarity level δ is defined as follows.
[TeX:] $$inter-dist( c \operatorname { lass } ( D , \delta ) ) = \sum _ { g _ { p } ^ { \delta } , g _ { q } ^ { \delta } } ^ { p \neq q } \sum _ { D _ { i } \in g _ { p } ^ { \delta } , D _ { j } \in g _ { q } ^ { \delta } } ^ { i \neq j } \operatorname { dist } \left( D _ { i } , D _ { j } , \alpha \right)$$
The best classification is selected based on maximizing both the intra-cluster similarity and the intercluster distance. Hence, we have the following definition of the goodness measure.
DEFINITION 5. The goodness of class(D,sim,δ) is defined as follows.
[TeX:] $$\operatorname { goodness } ( \operatorname { class } ( D , \delta ) ) = | \operatorname { intra } - \operatorname { sim } ( \operatorname { class } ( D , \delta ) ) + i n t e r - \operatorname { dist } ( \operatorname { class } ( D , \delta ) ) - k |$$
The classification that gets the maximum value of goodness is selected as the best classification at the similarity level δ.
DEFINITION 6. Let G=(D,E) be the graph representing the database set D. The vertex set D={D1,D2,…,Dn} is the set of nodes in the graph G and E=D×D ={( Di, Dj), 1≤i,j≤n} is the set of pairs (Di, Dj) representing the links between vertices in G.
DEFINITION 7. At a similarity level δ, a classification class(D,sim,δ) is complete if the following equation is verified :
[TeX:] $$\sum _ { i = 1 } ^ { k } \frac { \left| g _ { i } ^ { \delta } \right| ^ { 2 } - \left| g _ { i } ^ { \delta } \right| } { 2 } = | E |$$
Where |gδi| denotes the number of vertices in the connected component gδi and |E| is the number of edges in the graph G. In other words, if each class gδi (i=1 to k) in class(D,sim,δ) is a clique in G then class(D,sim,δ) is a complete classification.
The proposed approach will be followed and explained with the help of examples.
EXAMPLE. Let D={D1, D2,…, D6} be the database set to classify. For illustration purposes, the similarity values between the six databases are given in Table 1.
The similarity relation table for the six databases of the example
sim D1 D2 D3 D4 D5 D6
D1 1 0.40 0.56 0.40 0.50 0.40
D2 - 1 0.40 0.79 0.40 0.45
D3 - - 1 0.40 0.45 0.40
D4 - - - 1 0.40 0.45
D5 - - - - 1 0.40
D6 - - - - - 1
From the upper triangle of the similarity table, there are five distinct similarity values. Consequently, there are five lists of edges with the same weight. Starting with the highest similarity, the five similarity values are sorted in the decreasing order as follows (0.79, 0.56, 0.50, 0.45, 0.40).
Let G=(D,E). Initially |D|=6 and E=∅. Therefore, a trivial complete classification is obtained class(D)={{D1},{D2},{D3},{D4},{D5},{D6}}, with goodness(class(D))= 2.20.
For each similarity value δ listed in the decreasing order, a candidate classification class(D,sim,δ) is generated and checked to see whether it is a complete classification as follows.
For δ=0.79, we have, E0.79={(D2, D4)}, E=E ∪ E0.79
class(D,sim,0.79)={{D2,D4},{D1},{D3},{D5},{D6}}, which is a complete classification because :
[TeX:] $$\sum _ { i = 1 } ^ { k } \frac { \left| g _ { i } ^ { 0.79 } \right| ^ { 2 } - \left| g _ { i } ^ { 0.79 } \right| } { 2 } = \frac { | \{ \mathrm { D } 2 , \mathrm { D } 4 \} | ^ { 2 } - | \{ \mathrm { D } 2 , \mathrm { D } 4 \} | } { 2 } + \frac { | \{ \mathrm { D } 1 \} | ^ { 2 } - | \{ \mathrm { D } 1 \} | } { 2 } + \frac { | \{ \mathrm { D } 3 \} | ^ { 2 } - | \{ \mathrm { D } 3 \} | } { 2 } + \frac { | \{ \mathrm { D } 5 \} | ^ { 2 } - | \{ \mathrm { D } 5 \} | } { 2 } + \frac { | \{ \mathrm { D } 6 \} | ^ { 2 } - | \{ \mathrm { D } 6 \} ) } { 2 } = 1 = |E|$$ with goodness(class(D,0.79))= 3.78
The remaining classifications are summarized in Table 2.
There is only one classification that isn't complete at δ=0.50. This is because, |E|=3 and [TeX:] $$\sum _ { i = 1 } ^ { k } \frac { \left| g _ { i } ^ { 0.50 } \right| ^ { 2 } - \left| g _ { i } ^ { 0.50 } \right| } { 2 } = 4 \neq | E |$$
According to the results in Table 2, the best complete classification of the 6 databases that has the maximum value of goodness is class(D,sim,0.45)={{D2, D4, D6},{D1, D3, D5}}
In the following section, we describe the data structures and algorithms used in the proposed classification approach.
Candidate classifications generated from the similarity matrix in Table 1
Similarity level (δ) Edge set (Eδ)
Candidate classification
class(D,sim,δ)
goodness(class(D,δ))
0.56 (D1, D3) {D2,D4},{D1,D3},{D5},{D6} 4.90
0.50 (D1, D5) {D2,D4},{D1,D3,D5},{D6} 5.90
0.45 (D2,D6),(D3,D5),(D4, D6) {D2,D4,D6},{D1,D3,D5} 6.60
(D1,D2), (D1,D4), (D1, D6), (D2,D3), (D2,D5), (D3, D4),
(D3,D6), (D4,D5), (D5, D6)
{D1,D2,D3,D4,D5,D6}
3.2 Data Structure and Algorithms
The problem of generating a classification class(D,sim,δ) at a given similarity level δ depends mainly on the following sub-problems:
1) Find the list of edges with weight value equal to δ, denoted Eδ, and add it to the graph G.
2) Determine the connected components of G and check whether each of which forms a clique.
To solve the first problem, we need to use a dictionary data structure to efficiently search for duplicate weights. In our approach, we need to list the m edge lists Eδ in the decreasing order of the weight δ. Therefore, we use a balanced binary search tree (BST) such as red-black trees [14], to implement such ordered dictionary structure.
In the following section, we present a procedure to build the m-node BST where each node represents one edge list Eδ. Instead to use pointers to link edges with the same weight value, we use integers to reference the edges and link them into a contiguous table of size (n2–n)/2.
3.2.1 Finding the edge lists E δ
Let T be the balanced BST object and T.root the root node. Each node in T contains two fields: weight and index, which represent the weight value δ and the index of the first edge in Eδ respectively. Edges with the same weight value δ are linked into a contiguous table V[1..(n2–n)/2]. For example, let x be a node object in T, then V[x.index] contains the index of the second edge whose weight value is x.weight and V[V[x.index]] contains the index of the third one and so on, until we find a negative value (-1) , which represents the end of the list Eδ.
In the following procedure, we present the algorithm used to construct the ordered edge lists Eδ.
DISCUSSION. Initially (at line 2), the tree T is empty. Then for each databases pair (Di,Dj) (at line 4), a binary search tree is performed at line 8 to see whether δ exists in T. If there is a node x in T such that x.weight = δ then the current edge (i.e., the k-th edge) is inserted at the beginning of Eδ at lines 10–11. Otherwise, a new node, say y, is allocated and inserted into T at lines 13–16 such that y.weight is set to δ and y.ndex is set to k. The auxiliary procedure Tree_Rebalance is called to re-balance the tree after the insertion operation in order to keep the height proportional to the logarithm of the number of nodes in T. The implementation of Tree_Rebalance depends on the balance information and the algorithm used to re-balance the tree when certain properties are violated [14].
Let m be the number of nodes in T such that m ∈[1,(n2–n)/2]. Then, the search, insert and rebalance operations take O(log2(m)). There are (n2–n)/2 similarity values between the n databases. For each similarity value δ a new node is inserted into T if there is no node containing the weight δ. Hence, the time complexity to build T is O(n2log2(m)).
The tree T and the index array V corresponding to the example above are shown in Fig. 2.
The balanced binary search tree T and the index array V (the column index) of the example.
Each node of T contains mainly 4 fields (the similarity value δ, the index of the first edge in the list Eδ, the left and right child pointers). The table to the right represents the array V[1..(n2-n)/2] in which edges with the same weight are linked. Only the most right column (the column index) exists really in the main memory. Other columns are only shown for illustration purpose.
PROPOSITION. Let (Di,Dj) be the vertex pair corresponding to the k-th edge for i=1 to n–1, j=i+1 to n and k=1 to (n2–n)/2. Then, the row index i and the column index j of the k-th edge are calculated as follows:
[TeX:] $$i = n - h$$
[TeX:] $$\begin{array} { c } { j = ( i + 1 ) + h ( h + 1 ) / 2 - \left( C _ { n } ^ { 2 } - k + 1 \right) } \\ { \text { With } h = \lceil \left( \sqrt { 8 \times \left( C _ { n } ^ { 2 } - k + 1 \right) + 1 } - 1 \right) / 2 \rceil } \end{array}$$
Such that [val] is the least integer greater than or equal to val and [TeX:] $$C _ { n } ^ { 2 } = \left( n ^ { 2 } - n \right) / 2$$.
Proof. Let M be the triangular similarity table of size n×n. Then, there are Cn2 similarity values in M, which are associated with Cn2 edges indexed from 1 to Cn2. The 1st row of M is associated with n–1 edges (D1,D2),(D1,D3),…(D1,Dn), the 2nd row with n–2 edges (D2,D3), (D2,D4),…(D2,Dn), and so on until the last row, the (n–1)-th row, which is associated with one edge (Dn-1,Dn). Thus, the last h rows of M are associated with h(h+1)/2 edges. For a given index k, there are Cn2-k+1 edges counting from the k-th edge to the last edge, the Cn2-th edge. To find the row index i of the k-th edge, we have to subtract h from n, such that h is the integer value that verifies: [TeX:] $$h ( h + 1 ) / 2 \geq C _ { n } ^ { 2 } - k + 1$$. By solving the inequality we find: [TeX:] $$( h + 1 / 2 ) ^ { 2 } \geq \left( 8 \left( \mathcal { C } _ { n } ^ { 2 } - k + 1 \right) + 1 \right) / 4 , h \geq \left( \sqrt { 8 \times \left( C _ { n } ^ { 2 } - k + 1 \right) + 1 } - 1 \right) / 2$$. Therefore, i=n-h such that [TeX:] $$h = \left[ \left( \sqrt { 8 \times \left( C _ { n } ^ { 2 } - k + 1 \right) + 1 } - 1 \right) / 2 \right]$$. The first column in the row i has the index i+1. To find the column index j of the k-th edge, we have to add the offset [TeX:] $$\left( h ( h + 1 ) / 2 - \left( C _ { n } ^ { 2 } - k + 1 \right) \right)$$ to the column index (i+1).
3.2.2 The database classification algorithm
To solve the second problem, which is determining and maintaining the connected components of G, we use the disjoint-set forest data structure [14]. This data structure keeps a collection of n disjoint updatable trees, where each tree represents one connected component in the graph G=(D,E). The root node of each tree is used as a representative node to identify one connected component. Each node in the disjoint-set forest, say xi (i=1 to n), contains two main fields size and parent, which represent respectively the size of the sub-tree rooted at xi and a pointer to the parent node. In this paper, instead to use pointers, we use integers to link child nodes to their parent nodes. For example, if xj is the parent of xi, then we set xi.parent=j. The data structure has two main operations described as follows.
union(xi, xj) : combines the trees identified by the roots xi and xj. That is, it links the root of the smaller tree to the root of the larger tree. After a union operation, the size of the resulting tree is set equal to the sum of the sizes of the input trees.
findset(xi) : returns the root of xi's tree, that is, it follows parent nodes starting from xi's parent, until reaching the root node. For each edge incident on the i-th vertex, findset is called on the node xi to find the connected component to which the i-th vertex belong.
In this paper, we have modified the findset's implementation in order to identify cliques in G=(D,E). If xi or xi's parent (say xj) is the root, then findset(xi) returns the root node. Otherwise, xi is linked to xj's parent and findset(xi) outputs xj's parent even if it's not the root. Thus, for each edge incident on the i- the vertex, findset(xi) compresses the path between xi and the root when xi and xi's parent aren't the root. Therefore, if xi's tree represents a connected component of k vertices in G and findset(xi)'s output isn't the root, then it is clear that there is at least one missing edge between the i-th vertex and the remaining (k-1) vertices of the same component. Consequently, the connected component, which contains the i-th vertex, isn't a clique in G.
In the following we present our classification algorithm that takes as input the objects returned by the procedure BuildEdgeTree. The findset, union and makeset procedures are given too.
Discussion. Initially, the graph G=(D,E) has n disjoints graph components. Therefore, we get a trivial classification class(D,sim)={{D1},{D2},…,{Dn}}. Lines 1-7 initialize the variables used in our algorithm. The inter-cluster distance (dist_inter) between the n components is calculated during the call of the procedure BuildEdgeTree. The intra-cluster similarity (sim_intra) has a zero value since there is one database in each cluster. We use the variable goodness_max to maintain the maximum value of goodness found so far.
At line 8, the procedure makeset initializes the data structure by creating n rooted trees xi (i=1 to n), each of which contains one node (xi.size=1). To access the n nodes, we use an array A[1..n] such that A[i] holds the node object xi representing the i-th vertex in G. The time complexity of line 8 is O(n).
At line 9, the for-loop examines each node e of the edge tree (T) taken in the decreasing order by field weight. There are m nodes in T such that 1≤ m ≤ (n2–n)/2 and each node represents one edge list Eδ . Browsing all the m nodes in the sorted order takes O(m) using an in-order traversal of T.
At line 12, the while-loop examines the current edge list Eδ such that δ = e.weight. Edges are listed starting from the first edge, whose index is e.index, until reaching a negative value (–1), which represents the end of the edge list. At line 13, we use Eqs. (7) and (8) to find the two end vertices (Di,Dj) corresponding to the k-th edge (i.e., the current edge).
At lines 17–18, the findset procedure is called once for each end-vertex of the k-th edge (Di, Dj) to find the connected components to which Di and Dj belong. The time complexity of lines 17-18 is O(1).
Let xi and xj be the nodes returned by findset. At line 19, if xi and xj are two distinct roots, then the union procedure (at line 21) will merge the corresponding trees by making the smaller tree a sub-tree of the larger tree. Otherwise, either the current classification is not complete or the two vertices Di and Dj already belong to the same component.
At line 20, we update nb_clique_edges, which is the number of edges required so that each connected component of G=(D,E) becomes a clique. After each union operation, we decrement the number of components nb_component.
At line 24, we select the next edge in the list Eδ from the index array V[1..(n2–n)/2].
At line 26, we test whether the current classification is complete. If the number of edges examined so far, nb_edges, is equal to nb_clique_edges, then the current classification is complete and the algorithm continues at the line 29. Otherwise, the current classification is discarded.
At lines 29–31, the current complete classification class(D,sim,δ) is assessed using the goodness measure proposed in [7]. While goodness(class, δ) ≥ goodness_max, class(D,sim,δ) is the best classification generated so far and goodness max value is updated.
During the course of our algorithm, we need to maintain the best classification generated so far as the disjoint-set forest is updated. To implement such persistent data structure, additionally fields (version and parent_temp) are associated with each node in the disjoint-set forest. These fields are required to maintain a past version of the dynamic set at the time that the best classification was generated. Let x be a node in the disjoint-set forest. The field x.parent_temp is used to keep track of x's parent node corresponding to the best classification generated so far. The field x.version is used to determine the classification to which x's parent refers.
In our algorithm, each classification class(D,sim, δ) is identified by a version number class_version. Let best_class_version be the version number of the best classification. Before each updating operation in the union/findset procedure, each node x is checked to see whether its current parent is a part of the best classification produced so far. That is, if x.version≤best_class_version, then x's parent index is saved in the field x.parent_temp before the updating operation.
At lines 36–41, we display the best classification of the n databases from the disjoint-set forest. Hence, we output the parent index of each node xi (1≤i≤n) at the time that the best classification was generated. That is, if xi.version≤best_class_version, then xi.parent is returned as the cluster containing the i-th database. Otherwise, xi.parent_temp is returned instead.
The algorithm terminates once all the m edge lists Eδ are examined and the m candidate classifications are generated. The average size of each edge list is n2/m. Therefore, exploring all the m edge lists takes O(n2) time such that n is the database number. Therefore, the proposed algorithm finds the best complete classification of the set of n databases in O(n2) time. Our algorithm is optimal when comparing with BestDatabasePartition [7], which takes O(m×n2) time to find the best partition of the multiple databases.
The space complexity of our algorithm is estimated as follows. To store the m nodes of T, we need m×(2pointers+real+integer) units. We have to add the space required to store the array index V, that is, (n2–n)/2 integer units. Also, the disjoint-set forest data structure needs n×(4integer) units. Therefore, the memory space required by our algorithm is (2n2+14n+28m) bytes when using a typical compiler that represents a pointer, an integer and a real number using 8 bytes, 4 bytes and 8 bytes respectively.
In parallel, BestDatabasePartition needs n2 real units to store the similarity relation table. Assuming that BestDatabasePartition uses a balanced binary search tree to sort all the distinct similarity values, then m×(2pointers+real) units will be required to store the nodes of the tree. Each candidate classification is stored separately. Hence, mn×integer units are required to store all the m candidate classifications. Therefore, the memory space required by BestDatabasePartition is (8n2+20m+4mn) bytes by using the same compiler. For a large values of n and m, we notice that the proposed algorithm consumes less memory space compared with BestDatabasePartition.
To assess the performance of our proposed approach, we conducted some experiments to analyze and compare the execution time of the existing algorithm BestDatabasePartition [7] and our proposed graph-based classification algorithm. All the experiments have been implemented on a 2.70-GHz Pentium processor with a 2-GB of memory, using JAVA Edition 6.
We carried out the experiments using two synthetic datasets T10I4D100K and T40I10D100K available in: http://fimi.ua.ac.be/data/. Some characteristics of these datasets are presented in Table 3. We denote by NT, ALT, AFI, NI and DB the number of transactions, the average length of a transaction, the average frequency of an item, the number of items and the database name respectively.
For multi-database mining, each of the datasets is divided horizontally into 10 and 20 databases. The multiple databases obtained from T10I4D100K and T40I10D100K are referred to as T1,1, T1,2, …, T1,n and T4,1, T4,2, …, T4,n respectively, such that n=10, 20.
Synthetic datasets
DB NT ALT NI AFI
T10I4D100K 100,000 11.102280 870 1276.124138
T40I10D100K 100,000 40.605070 942 4310.516985
By varying the value of the minimum support threshold (α), we get different sets of frequent itemsets using FP-Growth algorithm [13] as shown in Table 4. We denote by {FIS(Ti,1, α), FIS(Ti,2, α),…, FIS(Ti,n, α)} the sets of frequent itemsets reported from the n multi-databases Ti,1, Ti,2, …, Ti,n under α such that i=1,4.
Once the data-preparation is done, we classify the multi-databases using our algorithm and BestDatabasePartition [7]. Since it is difficult to measure the performance of an algorithm executed by the Java Virtual Machine, due to the optimization that occurs during the execution, each algorithm is executed 1000 times on the same multiple databases to get the average running time.
In the first part of our experiments, we analyzed the impact of minsupp(α) on the classification execution time. Therefore, we set the number of databases n=10 and we varied the value of α to get different sets of frequents itemsets. Then, we executed our algorithm and BestDatabasePartition on the same frequents itemsets. The classification results and the execution times are presented in Table 5 and Fig. 3, respectively. From the results, we can notice that the execution time is relatively constant for both algorithms when the value of α increases. The reason is that n and the number of candidate classification (m=45) didn't change during the experiments, as shown in Table 5. Moreover, we didn't take into account the overhead of calculating the similarities between the n databases, since the same similarity measure proposed in [7] has been used for the two algorithms. However, we observe that the execution time of our algorithm is shorter than that of BestDatabasePartition for α=0.03 to 0.06.
The sets of frequent itemsets from the multiple databases T 1,1, T 1,2, …, T 1,n and T 4,1, T 4,2, …, T 4,n for α=0.03 to 0.06 and n=10, 20
Dataset (i=1,4) # of multiple databases (n) # of transaction in each database Algorithm
support (α)
Sets of frequent itemsets
Ti 10 10,000 (×10) FP-Growth 0.03
FIS(Ti,1, 0.03), FIS(Ti,2, 0.03),
…,FIS(Ti,10, 0.03)
20 4,500 (×20) 0.03
FIS(Ti,1, 0.03), FIS(Ti,2, 0.03), …,
FIS(Ti,10, 0.03),…, FIS(Ti,20, 0.03)
A comparison between the classification results obtained by our algorithm and BestDatabasePartition [ 7] on the multi-databases T 1,1 ,…, T 1,10 and T 4,1 ,…, T 4,10 for α=0.03 to 0.06
Multiple databases # of candidate classification (m)
minsupp
(α)
Best classification goodness Execution time (s)
BestDatabase-
Proposed algorithm
T1,1,…,T1,10 45 0.03 {T1,5, T1,9, T1,10} {T1,8} {T1,4, T1,7} {T1,6} {T1,3} {T1,2} {T1,1} 4.75 0.00784 0.00227
0.04 {T1,10} {T1,5, T1,9} {T1,8} {T1,7} {T1,6} {T1,4} {T1,3} {T1,2} {T1,1} 0.096 0.00804 0.00240
0.05 {T1,10} {T1,9} {T1,8} {T1,3, T1,5, T1,7} {T1,6} {T1,4} {T1,2} {T1,1} 3.99 0.00799 0.00233
0.06 {T1,2, T1,3, T1,4, T1,5, T1,6, T1,7, T1,8, T1,9, T1,10} {T1,1} 34.71 0.00777 0.00225
T4,1,…,T4,10 45 0.03 {T4,4, T4,10} {T4,9} {T4,8} {T4,7} {T4,6} {T4,5} {T4,3} {T4,2} {T4,1} 2.88 0.00780 0.00228
0.04 {T4,10} {T4,5, T4,9} {T4,8} {T4,7} {T4 ,6} {T4,4} {T4,3} {T4,2} {T4,1} 4.44 0.00769 0.00217
0.05 {T4,10} {T4,3, T4,9} {T4,8} {T4,7} {T4,6} {T4,5} {T4,4} {T4,2} {T4,1} 4.82 0.00766 0.00216
0.06 {T4,10} {T4,9} {T4,8} {T4,7} {T4,6} {T4 ,3, T4,5} {T4,4} {T4,2} {T4,1} 4.62 0.00773 0.00218
Execution time versus minsupp (α).
A comparison between the classification results obtained by our algorithm and BestDatabasePartition [ 7] on the multi-databases T1,3,…, T1,n and T4,3,…, T4,n under α=0.03 for n=3 to 20
Multiple databases
(i=1,4)
# of multiple databases
# of candidate classifications (m) from
goodness value of the
best classification
Proposed algorithm O(n2) BestDatabasePartition O(m×n2)
T1,3 ,…, T1,n T4,3 ,…, T4,n T1,3 ,…, T1,n T4,3 ,…, T4,n T1,3 ,…, T1,n T4,3 ,…, T4,n T1,3 ,…, T1,n T4,3 ,…, T4,n
Ti,3 ,…, Ti,n 3 3 0.67 0.80 9 27
4 6 0.77 1.30 16 96
5 10 0.59 1.63 25 250
7 21 2.11 1.78 49 1029
10 45 5.88 2.47 100 4500
12 66 8.03 2.3 144 9504
13 78 10.28 3.35 169 13182
15 105 15.57 6.01 225 23625
16 120 119 18.29 7.43 256 30720 30464
19 171 170 28.27 11.13 361 61731 61370
In the second part of our experiments, we studied how the number of databases n and the number of candidate classification m influence the time complexity of the two algorithms. Hence, we set the value of α=0.03 and we varied n from 3 to 20 databases. The experiment results obtained by the two algorithms are presented in Table 6 and Fig. 4. From the results, we can see that the time complexity and the execution time tend go higher as the value of n increases. However, we notice that the execution time increases rapidly for BestDatabasePartition. The reason is that m becomes larger as the value of n increases, as it is shown in Table 6. Since the time complexity of BestDatabasePartition depends strongly on m, we note a rapid increase of BestDatabasePartition's execution time for n= 3 to 20.
From the experiment results above, we observe that our algorithm and BestDatabasePartition [7] find the same best classification of the n multiple databases. The reason is that the same similarity and goodness measures proposed in [7] have been used for both algorithms. However, the execution time of our algorithm is the shortest. This is because, unlike the existing classification algorithms [5-8], the time complexity of the classification generation procedure in our algorithm depends only on the number of databases (n). In the worst case, n2 edges are processed to find the best classification of a set of n multiple databases.
Execution time versus the number of databases (n).
In this paper, we have proposed an improved database classification algorithm for multi-database mining. Different from the existing works, our algorithm searches for the best classification of a database set D by analyzing and determining the connected components of an undirected weighted graph G=(D,E). The experiments have proved the accuracy and efficiency of our approach and the execution time of our algorithm is the shortest.
Future work will be directed toward improving the accuracy of the similarity measure between the databases. Also, we will assess our algorithm using real-world datasets and validate the tests on a real multi-databases system.
Salim Miloudi
He received a M.S. degree in information systems engineering from the University of Science and Technology–Mohamed Boudiaf (USTOMB), Oran, Algeria in 2011. His research interests include data mining for knowledge discovery, multi-database mining and machine learning.
Sid Ahmed Rahal
He is a Doctor in computer science since 1989 in Pau University France. He is a member in many professional activities: member of the Signal, Systems and Data Laboratory (LSSD). Research interests include the databases, data mining, agent and expert systems.
Salim Khiat
He received a Ph.D. degree in computer science from the university of science and technology–Mohamed Boudiaf Oran USTOMB Algeria in 2015. He is a member of the Signal, Systems and Data Laboratory (LSSD). His research interests include the databases, multi-database mining, ontology, grid and cloud computing.
1 S. Zhang, M. J. Zaki, "Mining multiple data sources: local pattern analysis," Data Mining and Knowledge Discovery, 2006, vol. 12, no. 2-3, pp. 121-125. doi:[[[10.1007/s10618-006-0041-y]]]
2 S. Zhang, X. Wu, C. Zhang, "Multi-database mining," IEEE Computational Intelligence Bulletin, 2003, vol. 2, no. 1, pp. 5-13. doi:[[[10.1016/j.cll.2007.10.004]]]
3 S. Zhang, C. Zhang, X. W u, Knowledge Discovery in Multiple Databases. New Y ork, NY: Springer, 2004.custom:[[[-]]]
4 A. Adhikari, P. Ramachandrarao, W. Pedrycz, Developing Multi-database Mining Applications. London: Springer, 2010.custom:[[[-]]]
5 X. Wu, C. Zhang, S. Zhang, "Database classification for multi-database mining," Information Systems, 2005, vol. 30, no. 1, pp. 71-88. doi:[[[10.1016/j.is.2003.10.001]]]
6 H. Li, X. Hu, Y. Zhang, "An improved database classification algorithm for multi-database mining," in Frontiers in Algorithmics. Heidelberg: Springer2009,, pp. 346-357. doi:[[[10.1007/978-3-642-02270-8_35]]]
7 A. Adhikari, P. R. Rao, "Efficient clustering of databases induced by local patterns," Decision Support Systems, 2008, vol. 44, no. 4, pp. 925-943. doi:[[[10.1016/j.dss.2007.11.001]]]
8 Y. Liu, D. Yuan, Y. Cuan, "Completely clustering for multi-databases mining," Journal of Computational Information Systems, 2013, vol. 9, no. 16, pp. 6595-6602. doi:[[[10.12733/jcisP0759]]]
9 H. Liu, H. Lu, J. Yao, "Identifying relevant databases for multidatabase mining," in Research and Development in Knowledge Discovery and Data Mining. Heidelberg: Springer1998,, pp. 15-18. doi:[[[10.1007/3-540-64383-4_18]]]
10 H. Liu, H. Lu, J. Y ao, "T oward multi-database mining: identifying relevant databases," IEEE Transactions on Knowledge and Data Engineering, 2001, vol. 13, no. 4, pp. 541-553. doi:[[[10.1109/69.940731]]]
11 R. Agrawal, J. C. Shafer, "Parallel mining of association rules," IEEE Transactions on Knowledge and Data Engineering, 1996, vol. 8, no. 6, pp. 962-969. doi:[[[10.1109/69.553164]]]
12 R. Agrawal, R. Srikant, "Fast algorithms for mining association rules in large databases," in Proceedings of the 20th International Conference on V ery Large Data Bases, Santiago de Chile, Chile, 1994;pp. 487-499. custom:[[[https://dl.acm.org/citation.cfm?id=672836]]]
13 J. Han, J. Pei, Y. Yin, R. Mao, "Mining frequent patterns without candidate generation: a frequent-pattern tree approach," Data Mining and Knowledge Discovery, 2004, vol. 8, no. 1, pp. 53-87. doi:[[[10.1023/B:DAMI.0000005258.31418.83]]]
14 C. E. Leiserson, R. L. Rivest, T . H. Cormen, Introduction to Algorithms. CambridgeMA: MIT Press, 1990.custom:[[[-]]]
Received: November 20 2015
Accepted: August 31 2016
Corresponding Author: Salim Miloudi* ([email protected])
Salim Miloudi*, Dept. of Computer Science, Faculty of Computer Science and Mathematics, University of Sciences and Technology-Mohamed Boudiaf(USTOMB) Oran, Algeria, [email protected]
Sid Ahmed Rahal*, Dept. of Computer Science, Faculty of Computer Science and Mathematics, University of Sciences and Technology-Mohamed Boudiaf(USTOMB) Oran, Algeria, [email protected]
Salim Khiat*, Dept. of Computer Science, Faculty of Computer Science and Mathematics, University of Sciences and Technology-Mohamed Boudiaf(USTOMB) Oran, Algeria, [email protected] | CommonCrawl |
Gianluigi Rozza
Gianluigi Rozza is an aerospace engineer and mathematician best known for his work on reduced-order modeling. He is currently full professor of Numerical Analysis at the International School for Advanced Studies (SISSA) in Trieste,[1] where he serves as head of SISSA Mathematics Area and SISSA Director's Delegate for Research Valorisation, Innovation, and Industrial Cooperation.
Gianluigi Rozza
Self-portrait photograph
Born (1977-04-20) April 20, 1977
Sant'Angelo Lodigiano, Italy
NationalityItaly
Alma materPolytechnic University of Milan, Italy
Known forReduced-order modeling
Scientific career
FieldsComputational science, numerical analysis
InstitutionsInternational School for Advanced Studies
ThesisShape Design by Optimal Flow Control and Reduced basis Techniques: Applications to Bypass Configurations in Haemodynamics (2005)
Doctoral advisorsAlfio Quarteroni, Anthony Tyr Patera
Personal life and education
G. Rozza was born and raised in Italy where he earned a bachelor's and master's degree in Aerospace Engineering from the Polytechnic University of Milan in the year 2002. He subsequently moved to Lausanne, Switzerland to join the École Polytechnique Fédérale de Lausanne (EPFL) for postgraduate studies. At EPFL, he received a PhD in Numerical Analysis in 2005. His thesis, titled 'Shape Design by Optimal Flow Control and Reduced basis Techniques: Applications to Bypass Configurations in Haemodynamics', was completed under the supervision of Alfio Quarteroni and Anthony Tyr Patera.[2]
Career
Following his doctoral studies, G. Rozza worked at Ecole Polytechnique Fédérale de Lausanne in the professor Alfio Quarteroni's scientific group for a year. In 2006, he joined the Department of Mechanical Engineering and the Center for Computational Engineering at the MIT as a Postdoctoral Associate Researcher in the professor Anthony Tyr Patera's group. He stayed at the MIT until 2008. After a period as a Senior Researcher and Lecturer at the EPFL, in 2012, Rozza joined the Applied Mathematics group, SISSA mathLab, at the International School for Advanced Studies (SISSA) where, in 2014, he became professor.
In 2018 he received an ERC Consolidator Grant (CoG) from the European Research Council (ERC) with the proposal «AROMA-CFD» (2016-2022),[3][4][5] and in 2022 he won a ERC Proof of Concept Grant PoC (Proof of Concept).[6] for the project «ARGOS».[7][8]
On December 2021 he was ranked n.1152 in the Top Scientist-Mathematics from research.com, n.35 in Italy.[9] G. Rozza is currently n.20 in the Top Italian Scientists Mathematics ranking.[10] On September 2022 he was listed in the World’s Top 2% Scientist ranking, made by Stanford University in collaboration with Elsevier and Scopus.[11] As of January 2023 Rozza co-authored 3 books and edited other 10 (12 volumes) on numerical analysis and model order reduction.[12][13]
In addition to research, G. Rozza is the President of SMACT Innovation Ecosystem's supervisory board, and he is an executive committee member of ECCOMAS (European Community on Computational Methods in Applied Sciences).[14].
Recognitions
• In 2004 Rozza received the Bill Morton CFD Prize[15] at the Oxford ICFD conference.
• Rozza Received the Riconoscenza Civica from his birthplace (Sant'Angelo Lodigiano).[16]
• In 2006 he received the Phd Award[17] from ECCOMAS Congress (European Community on Computational Methods in Applied Sciences) in Amsterdam.
• In 2009 Rozza received the Springer Computational Science and Engineering (CSE) Award in Monaco with Phuong Huynh and Cuong Nguyen[18] for the rbMIT software, developed at MIT in Boston.
• Rozza received the Dardo D'oro from his City of residence (Castiraga Vidardo) in 2010.[19]
• In 2014 he received the Jacques-Louis Lions Award[20] at ECCOMAS Congress (European Community on Computational Methods in Applied Sciences) in Barcelona.
• In 2022 Rozza gave the first Solari Lecture[21] at the Polytechnic University of Milan, Italy.
References
1. "Mathematics - SISSA". Retrieved 6 January 2023.
2. "Shape design by optimal flow control and reduced basis techniques: applications to bypass configurations in haemodynamics". Retrieved 11 August 2022.
3. "Datahub of ERC funded projects". Retrieved 11 August 2022.
4. "AROMA-CFD - Gianluigi Rozza". Retrieved 11 August 2022.
5. "SISSA si aggiudica un finanziamento ERC". Retrieved 11 August 2022.
6. "Proof of Concept - ERC". Retrieved 26 December 2022.
7. "Press Release". Retrieved 22 August 2022.
8. "ERC PoC ARGOS project". Retrieved 23 August 2022.
9. "Best Mathematics Scientists in Italy". Retrieved 6 January 2023.
10. "Top Italian Scientists Mathematics". Retrieved 11 August 2022.
11. John P.A. Ioannidis. "September 2022 data-update for "Updated science-wide author databases of standardized citation indicators"". Retrieved 14 February 2023.
12. "SIAM Books authored or edited by Gianluigi Rozza". Retrieved 14 February 2023.
13. "Springer Books authored or edited by Gianluigi Rozza". Retrieved 14 February 2023.
14. "Comitato Esecutivo ECCOMAS". Retrieved 28 August 2022.
15. "Paper on the International Journal Numerical Methods Fluids, Wiley". Retrieved 22 August 2022.
16. "Il Ponte Notizie - Santangiolini benemeriti" (in Italian). January 2005. Retrieved 1 February 2023.
17. "2005 PhD Awards". Retrieved 11 August 2022.
18. "Springer CSE Award". Retrieved 15 August 2022.
19. "ANSA - Castiraga Vidardo" (in Italian). Retrieved 26 August 2022.
20. "Young investigators awards". Retrieved 11 August 2022.
21. "The First Solari Lecture". Retrieved 1 February 2023.
External links
• "Official website". Retrieved 1 February 2023.
• "ORCID profile". Retrieved 1 February 2023.
• "Google Scholar profile". Retrieved 14 February 2023.
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
• Germany
• Italy
• Israel
• United States
• Czech Republic
Academics
• DBLP
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
• ORCID
• ResearcherID
• Scopus
• zbMATH
Other
• IdRef
| Wikipedia |
by Eric Worrall so it may be a good idea to recall the story and mention several rather predictable events that took place in recent days.
Nuclear fusion cannot be "cold". This fact must be comprehensible to every good undergraduate student who has gone at least through the first semester of some nuclear physics course or anything equivalent to it. The energy needed for two nuclei to overcome the repulsive Coulomb barrier and get very close – which is what the word "fusion" really means – is as high as many megaelectronvolts.
The electrostatic potential energy is \(Q_1 Q_2 / 4\pi\varepsilon_0 R\). In the \(\varepsilon_0=\hbar=c=1\) units, if you want \(R\) to be comparable to the QCD scale (nuclear radius), some \(1/150\MeV\), you will need the energy around \(Q^2 \times 150\MeV\) which is still megaelectronvolts because \(Q^2\sim \alpha\sim 1/137\). The energy gets even higher as \(Z_1 Z_2\) if there are many protons in the nuclei.
Because of the estimate \(E\sim kT\) in statistical physics, a single nucleus with this energy had to have the temperature corresponding to megaelectronvolts – many millions of degrees – to start with. Without this energy, the Coulomb barrier has to be "quantum tunneled through" and the resulting rates are negligibly tiny. If you want to increase the probability of tunneling to "reasonably visible" values, you simply need to equip the nuclei with the initial (kinetic) energy that is a sizeable fraction of the barrier, so it's still megaelectronvolts and millions of degrees.
Even if the nuclei were slow (cold) before the fusion, the very fact that megaelectronvolts of energy are produced per nucleus pair means that lots of energy per particle is produced in the fusion, to the neighborhood of the macroscopically fusing matter becomes hot afterwards – millions of degrees.
The Sun's core has 15 million degrees Celsius but the reactions proceed slowly – it's OK, the Sun lives for billions of years and the power is still high because the Sun is large – but realistic fusion reactors on Earth typically demand higher rates so they need temperatures comparable to 100 million degrees Celsius.
Nuclear physics and its events – especially fission and fusion – simply do take place at some energy scale that is about 1 million times higher than the energy scale of atomic physics. That's what makes the "energy in uranium" 1 million times more concentrated than the "energy in coal or lithium", for example. But it also means that the conditions in which the reactions are taking place are much more extreme.
I've explained these matters in 2011 when I reacted to Watts' article Andrea Rossi's E-cat fusion device on target that has really shocked me (Watts hopefully agrees that the device is not "on target" now in 2016 LOL). Anthony wasn't the only "typically sensible" man who has promoted this self-evident nonsense. As recently as in 2013, CMS particle physicist Tommaso Dorigo was writing enthusiastic reports on these Rossi cold fusion papers, therefore showing Dorigo's complete incompetence in nuclear physics.
Most of the people who have supported this crap were some kind of "extraterrestrials are everywhere around us" and "restrictions sold as constraints from physics are just an evil propaganda – no laws of Nature can ever stop geniuses led by saviors such as Rossi". So with this concentration of a mixture of stupidity and fanaticism, you may perhaps understand that even a life-long criminal such as Andrea Rossi may keep on cheating and stealing and lying for years.
Incidentally, if you have forgotten, the experiments that have claimed to "prove" that lots of energy is produced in Rossi's gadgets were claiming to have vaporized lots of water, much more water than could have been vaporized from the electricity that the experimenters admitted to have consumed. However, the reality is that almost no water was vaporized. They just saw "just a little bit of vapor" and screamed that all the water had to be above the boiling point – while neglecting the fact that the water was actually safely below the boiling point because the boiling point is higher at higher pressure.
OK, nevertheless, some people think that even a self-evident charlatan such as Rossi should get the money to become important and some three years ago, the nutty Rossi cultists boasted that "they" have secured $100 million from a company called Industrial Heat LLC which defines itself by "ambitious" goals – mostly a "rigorous verification and development of low-energy fusion technologies" – but at least, the capital in the company is damn real.
When scammers such as Rossi start to build companies and team up with others, it may become confusing and all the companies may look similar but that shouldn't obscure the point that some companies are the thieves while others are the victims. So there may be lots of companies that are basically just "sock puppets of Rossi himself" (e.g. the company named Defkalion; only a small Canadian branch of it has survived). But Industrial Heat LLC was clearly a victim, the only company cooperating with Rossi, not led by Rossi, but paying him the real money of some independent entrepreneurs. When the thieves and victims sign an agreement, they cooperate. But it's obvious that nothing good could have come out of this cooperation because it's unscientific nonsense. So the actual difference between the two kinds of companies becomes clear later.
where you can learn that as recently as as in Fall 2015, Industrial Heat LLC folks were talking about "isotopes they observed" and other promising signs.
You may want to read a short statement by the victim company, Industrial Heat LLC.
What happened is that after 3 years, Industrial Heat LLC has finally figured out that Rossi's "ingenious reactor" is complete fraud and doesn't produce a microjoule of energy. They have previously paid $11 million to Rossi and because the remaining $89 million in their contract depended on a successful replication etc., they obviously decided not to pay the remaining $89 million.
Rossi has abused the fact that he has managed to avoid the electric chair so far and sued them. They want to steal his "ingenious" idea that has been "proven" by many "independent" researchers and they are probably just pretending that it doesn't work. Industrial Heat LLC just states that the thing doesn't work and Rossi and pals have repeatedly breached their agreements in other ways, too.
But you see that the situation into which Industrial Heat LLC have maneuvered themselves is potentially dangerous for them because in other situations, it could be plausible that someone's business partner just pretends that an idea doesn't work, so that he can steal it for free. Obviously, Rossi's idea really doesn't work because if it did, it would be far more profitable for Industrial Heat LLC to make it work, mass produce the reactors, and pay the modest $89 million to Rossi. But it's plausible and this giant Italian criminal who should have been executed for many years is simply an experienced aßhole who has learned to deal with the courts, too.
Now, the Industrial Heat LLC is clearly not innocent. To some extent, they are parts of the same bogus industry as Rossi himself – and given their "mission", nothing good can ever come out of their work, either. So the wealthy men should have sent 10% of the money they have already paid to Rossi ($1 million or so) to your humble correspondent via PayPal instead. That would be a far more promising way to have a chance to develop new cool energy sources.
However, despite this incompetence of the Industrial Heat LLC which makes you think that they may deserve this hassle, I cannot overlook the fact that they are victims. They may have just idealistically wanted to find the truth and discover something useful that "could" work, and they just didn't understand why it couldn't, while Rossi is a cold-blooded criminal who just decided to steal some $100 million. Some of the money that Rossi has already stolen – and may steal, if his dirty tricks succeed – may have been paid by very good people who have worked hard for a long time, and so on.
So I obviously wish Industrial Heat LLC a lot of good luck in their new efforts to liquidate the Italian criminal, their former pal. But this sympathy cannot change the fact that people and companies simply have to be more careful when they pay lots of money to folks that others consider self-evidently and demonstrably fraudulent. | CommonCrawl |
About Mordell's Theorem (Elliptic Curves)
I've just finished the proof of Mordell's Theorem given in the book "Rational Points on Elliptic Curves " by Silverman.
One of the key lemmas used in the proof of the theorem is:
Let $C(\mathbb{Q})$ denote the group of rational points of $C$ then $[C(\mathbb{Q}):2C(\mathbb{Q})]$ is finite.
But in the book the lemma is proved under an additional assumption saying that $C$ has a rational point of order 2. I'd like to know how much algebraic number theory is needed to avoid that assumption and some references to see if I could try to look at a more general proof.
reference-request elliptic-curves
AbellanAbellan
The proof of that result, usually called (an explicit version of) the Weak Mordell-Weil theorem, can be found in Silverman's Arithmetic of Elliptic Curves book.
The proof uses Galois cohomology and some minor arithmetic that can be followed with the knowledge of a few of the main theorems of global class field theory.
Brandon CarterBrandon Carter
$\begingroup$ thanks! Could you recommend me any book on global class field theory? $\endgroup$ – Abellan Jul 2 '15 at 20:20
$\begingroup$ J.S. Milne's notes are a great resource for both local and global class field theory. $\endgroup$ – Brandon Carter Jul 2 '15 at 20:24
$\begingroup$ The two results needed are the finiteness of the ideal class group and DIrichlet's unit theorem?! You certainly do not need the class field theory for this... $\endgroup$ – sdf Jul 2 '15 at 21:26
$\begingroup$ @sdf: Where do you think that Dirichlet's unit theorem comes into this? The usual proof of the finiteness of the Selmer group comes from showing that the $n$ Selmer group lies inside of the set of continuous homomorphisms from the absolute Galois group to $E[n]$ that are unramified outside of a finite number of places. But then you can use that such a homomorphism factors through the Galois group of the maximal abelian extension of exponent $n$ which is unramified outside of those specified places. This extension is finite by CFT. $\endgroup$ – Brandon Carter Jul 2 '15 at 21:36
$\begingroup$ The proof can be simplified some where the base field is $\mathbf{Q}$ and $n = 2$, even to the point of only needing to understand some basic algebraic number theory (e.g. the possible discriminants of quadratic fields). Even in that case, I don't see where the ideal class group or Dirichlet's unit theorem play a role. $\endgroup$ – Brandon Carter Jul 2 '15 at 21:39
I hope you can find this useful.
All elliptic curve $C(\mathbb {Q})$ is canonically write as $y^2=4x^3-g_2x-g_3$... (1) in which the three roots of the right side are distinct; by a birational transformation $(x,y)\to (\frac x4, \frac y4)$ you have $y^2=x^3-h_2x-h_3$... (2) where $h_2$ and $h_3$ can be supposed rational integers. Being $e_1, e_2, e_3$ the three distinct roots of (2) one has $y^2=(x-e_1)(x-e_2)(x-e_3)$ in which there are three possibilities: $e_1, e_2, e_3$ are cubic numbers; one of them is rational and the other two quadratic conjugate irrational; the three are rational. Now one take the norm $y^2=N(x-e_i)$ (where the norm N is properly a norm just in the first possibility and in the other two not properly) and one has to see about the rational $x$ such that $N(x-e_i)$ is a square in $\mathbb {Q}$.
Always the number of $\mathbb{Q}(e_i)$ whose norms are perfect squares in $\mathbb {Q}$ are distributed in an infinite set of classes modulo the squares of $\mathbb{Q}(e_i)$ but among them, the binomial numbers $x-e_i$ are fortunately distributed in just a finite number of such classes. More precisely, the set $K^2$ of squares of $K=\mathbb {Q}(e_i)$ is a multiplicative subgroup of $K^*$ and the quotient group $G=K^*/K^2$ formed by the classes $zK^2$ is clearly infinite but for fixed $e_i$ ; i=1, 2, 3, there is a finite set $z_1K^2, z_2K^2,…,z_rK^2$ in $G^{(r)}$ and a partition in $r$ subsets of $C(\mathbb {Q})$, say, $R_1, R_2,…,R_r$ where $(x,y)\in R_j \Rightarrow (x-e_i)\in z_jK^2$.
Finally you have to prove that if $(x,y)\in C(\mathbb {Q})$ then $x-e_i=\nu\alpha^2$ where $\nu$ and $\alpha$ are in $\mathbb {Q}(e_i)$, $\nu$ being capable of taking only a finite number of values.
PiquitoPiquito
Not the answer you're looking for? Browse other questions tagged reference-request elliptic-curves or ask your own question.
Elliptic curves with finitely many rational points
References for elliptic curves
What is the amount of abstract algebra needed to study elliptic curves?
Do schemes help us understand elliptic curves?
What will be a good source for learning elliptic curves and what viewpoints can I adopt?
Suggestions for readings; Elliptic curves over function fields
Why Are There Only $2^t$ Points of Order $2$ in an Elliptic Curve
Prove the group of complex points on an elliptic curve is not finitely generated.
Elliptic curves: points with nonsingular reduction. Finiteness of a quotient.
Assumption $d>2$ on Proposition 2.12 from Knapp's Elliptic Curves | CommonCrawl |
Combinatorics Research Group
Activities/Events
Author: RTG-Supported Students/Postdocs
Author: Other Students
Author: RTG Faculty
ICLUE
More Information/History
Combinatorics Colloquium
February 9, 2023, 4pm CT
Speaker: Gregg Musiker (U. Minnesota)
Title: Super Cluster Algebras from Surfaces
This talk will provide an introduction to cluster algebras from surfaces, and their generators, known as cluster variables. We will also describe how certain combinatorial objects known as snake graphs yield generating functions that agree with Laurent expansions of said cluster variables. In recent years, a lot of progress has been made on the problem of defining a super-commutative analogue of Fomin-Zelevinsky's cluster algebras. In recent joint works with Nick Ovenhouse and Sylvester Zhang, we began the project of exploring the super cluster algebra structure from Penner-Zeitlin's decorated super Teichmüller space, generalizing the notion of (classical) cluster algebras from triangulated surfaces. In this colloquium talk, I will survey our recent works on combinatorial and matrix formulas for super lambda-lengths, proving super-analogues of combinatorial features such as the Laurent Phenomenon and positivity. Such formulas again utilize snake graphs but using double dimer covers rather than single dimer covers. If time permits I will also discuss applications to super Fibonacci and super Markov numbers, as well as connections to super-frieze patterns. This talk is based on joint works with Nick Ovenhouse and Sylvester Zhang (arXiv 2102.09143, 2110.06497 and 2208.13664) and will not assume prior knowledge of cluster algebras.
January 19, 2023, 2pm CT
Everitt Lab (room number 2310)
Speaker: Sarah Peluse (Princeton U.)
Title: Arithmetic patterns in dense sets
Some of the most important problems in combinatorial number theory ask for the size of the largest subset of the integers in an interval lacking points in a fixed arithmetically defined pattern. One example of such a problem is to prove the best possible bounds in Szemer\'edi's theorem on arithmetic progressions, i.e., to determine the size of the largest subset of {1,…,N} with no nontrivial k-term arithmetic progression x,x+y,…,x+(k-1)y. Gowers initiated the study of higher order Fourier analysis while seeking to answer this question, and used it to give the first reasonable upper bounds for arbitrary k. In this talk, I'll discuss recent progress on quantitative polynomial, multidimensional, and nonabelian variants of Szemer\'edi's theorem and on related problems in harmonic analysis and ergodic theory.
November 17, 2022, 4pm CT
Altgeld Hall (room number 245)
Speaker: Laura Escobar (Washington University, St. Louis)
Title: Measuring polytopes
Abstract: This will be an expository talk aimed at a broad audience, including beginning graduate students.
A polytope is a convex bounded polyhedron. There are many ways to measure a polytope: dimension, number of vertices, volume, number of lattice points inside the polytope, etc. A wide variety of problems in pure and applied mathematics involve measuring a polytope. For example, as we will see, even computing the dimension of a polytope can have geometric consequences for subvarieties of the flag variety. However, many of these measuring problems are very complex: there is no procedure that can efficiently measure any polytope. In this talk we will discuss these problems concentrating on families of polytopes which are of special interest due to their symmetries.
Addendum to department-wide announcement: There will be an opportunity to meet with Professor Escobar over pizza (confirmed) in 321 AH at noon on Nov 17, and over cookies (confirmed) in the same room at 3pm.
October 24, 2022, 2pm CT
Speaker: Martin Mandelberg
Title: What you should learn about Richard Wesley Hamming
Abstract: This lecture presents the life, legacy and some timeless lessons from Dr. Wesley Hamming (1915-1998), a remarkable
mathematician (alumnus of Illinois Math (PhD Trjitzinsky)), computer scientist, educator, mentor, and polymath.
During his career, Hamming developed numerous solutions in academia, government service, and industry. He helped form the foundation of modern computing science, coding and information theory, digital filters, and numerical methods. His better-known contributions include Error Correcting Codes (ECC), the Hamming window, and the Hamming distance.
Hamming wrote ten books, over a hundred articles, and gave many notable lectures. He received significant recognition for his work, including the 1968 ACM Turing Award, the 1986 IEEE Hamming Award, and the 1996 Eduard Rhein Foundation Award. Besides these and additional honors, many of his other contributions over decades generally remain unknown to subsequent generations of mathematicians, and engineers.
The speaker undertook the Richard Wesley Hamming Legacy Project to establish and perpetuate Hamming's legacy.
Project results include the Hamming Open Access Archive (HOAA) and the legacy tribute biography – MAN, MATHEMATICIAN, AND MENTOR: RICHARD WESLEY HAMMING. Drawing from the archive and the biography, this lecture presents a chronology with photographs, videos, and anecdotes to describe this determined and innovative deep thinker who strove for excellence. After paraphrasing some of Hamming's guidance for scientists and engineers, Dr. Mandelberg will entertain questions from the audience.
Martin Mandelberg is an engineer, strategist, mathematician, computer scientist, and educator with 50 years experience in government, industry, and academia. Mandelberg is Hamming's protégé, only Doctoral Student, and created the Richard Wesley Hamming Legacy Foundation, and author of Man, Mathematician, and Mentor: Richard Wesley Hamming.
Sept 1, 4pm CT
Speaker: Igor Pak, UCLA
Title: Combinatorial inequalities
Abstract: In the ocean of combinatorial inequalities, two islands are especially difficult. First, Mason's conjectures say that the number of forests in a graph with k edges is log-concave. More generally, the number of independent sets of size k in a matroid is log-concave. These results were established just recently, in a remarkable series of papers by Huh and others, inspired by algebro-geometric considerations.
Second, Stanley's inequality for the numbers of linear extensions of a poset with value k at a given poset element, is log-concave. This was originally conjectured by Chung, Fishburn and Graham, and famously proved by Stanley in 1981 using the Alexandrov–Fenchel inequalities in convex geometry. No direct combinatorial proof for either result is known. Why not?
In the first part of the talk we will survey these and other combinatorial inequalities. We then mention what does it mean not to have a combinatorial interpretation. Finally, I will briefly discuss our new framework of combinatorial atlas which allows one to give elementary proofs of the two results above, and extend them in several directions. The talk is aimed at the general audience.
Speaker: Xuding Zhu, Zhejiang Normal University, China
Title: List version of the 1-2-3 Conjecture
Abstract: The well-known 1-2-3 Conjecture by Karo\'{n}ski, {\L}uczak, and Thomason states that the edges of any connected graph with at least three vertices can be assigned weights 1, 2 or 3 so that for each edge $uv$ the sums of the weights at $u$ and at $v$ are distinct.
The list version of the 1-2-3 Conjecture by Bartnicki, Grytczuk, and Niwczyk states that the same holds more generally if each edge $e$ has the choice of weights not necessarily from $\{1,2,3\}$, but from any set $\{x(e),y(e),z(e)\}$ of three real numbers.
The goal of this talk is to survey developments on the 1-2-3 Conjecture, especially on the list version of the 1-2-3 Conjecture.
May 18, 11am CT
Altgeld Hall (room number TBD)
Speaker: Wojciech Samotij, Tel Aviv University
Title: The upper tail for triangles in random graphs
Abstract: Let $X$ denote the number of triangles in the random graph $G_{n,p}$. The problem of determining the asymptotics of the logarithimic upper tail probability of $X$, that is, $\log \Pr(X > (1+\delta)\mathbb{E}[X])$, for every fixed positive $\delta$ has attracted considerable attention of both the combinatorics and the probability communities. We shall present an elementary solution to this problem, obtained recently in a joint work with Matan Harel and Frank Mousset. The crux of our approach is a simple probabilistic argument, inspired by the work of Janson, Oleszkiewicz and Ruci\'nski, that reduces the estimation of this upper tail probability to a counting problem
April 14, 10am CT
245 Altgeld Hall
Speaker: Andrew Treglown, U. Birmingham, UK, Fulkerson Prize recipient (2021)
Title: Tiling in graphcs
Abstract: Given two graphs H and G, an H-tiling in G is a collection of vertex disjoint copies of H in G. Thus, an H-tiling is simply a generalisation of the notion of a matching (which corresponds to the case when H is an edge). An H-tiling in G is perfect if every vertex of G is covered by the H-tiling. Over the last 60 years there have been numerous results on perfect H-tilings. In this talk we give a high-level overview of some of the key ideas that permeate the topic. In particular, we will discuss some typical behaviour of extremal examples, and also some complexity questions.
December 2, 4pm CT
Speaker: Chandra Checkuri (UIUC Computer Science Dept.)
Title: On Submodular k-Partitioning and Hypergraph k-Cut
Abstract: Submodular $k$-Partition is the following problem: given a submodular set function $f:2^V \rightarrow \mathbb{R}$ and an integer $k$, find a partition of $V$ into $k$ non-empty parts $V_1,V_2,\ldots,V_k$ to minimize $\sum_{i=1}^k f(V_i)$. Several interesting problems such as Graph $k$-Cut, Hypergraph $k$-Cut and Hypergraph $k$-Partition are special cases. Submodular $k$-Partition admits a polynomial-time algorithm for $k=2,3$ and when $f$ is symmetric also for $k=4$. The complexity is open for $k=4$ and when $f$ is symmetric for $k=5$.
In recent work, motivated by this problem, we examined the complexity of Hypergraph $k$-Cut which only recently admitted a randomized polynomial-time algorithm. We obtained a deterministic polynomial-time algorithm for Hypergraph $k$-Cut as well as new insights in to Graph $k$-Cut. The ideas also led to a polynomial-time algorithm for Min-Max Symmetric Submodular $k$-Partition for any fixed $k$.
The talk will discuss these results with the goal of highlighting the open problem of resolving the complexity of Submodular $k$-Partition.
Based on joint work with Karthik Chandrasekharan
November 4, 2pm CT
Speaker: Zoltan Furedi, (Renyi Institute, Hungary and UIUC)
Title: Algebraic constructions in combinatorics
Abstract: There are two main sources to produce non trivial combinatorial structures. One can use probability theory for typical cases and a bit of algebra for symmetric structures. Here we briefly review classical and new developments and also give examples on how to combine these two powerful methods.
The talk is targeting general mathematicians, with little combinatorics backgrounds.
October 7, 2021, 4 pm CT
Speaker: John Shareshian, Washington University of St. Louis
Title: A problem on divisors of binomial coefficients, and a theorem on noncontractibility of coset posets
Abstract: Fix an integer n>1. It follows directly from a theorem of Kummer that the greatest common divisor of the members of the set BC(n) nontrivial binomial coefficients nC1,nC2,…nC(n-1) is one unless n is a prime power. With this in mind, we define b(n) to be the smallest size of a set P of primes such that every member of BC(n) is divisible by at least one member of P. In joint work with Russ Woodroofe, we ask whether b(n) is at most two for every n. The question remains open.
I will discuss what we know about this question, and how we discovered it during our investigation of a problem raised by Ken Brown about certain topological spaces: Given a finite group G, let C(G) the set of all cosets of all proper subgroups of a finite group, partially ordered by containment. The order complex of C(G) is the simplicial complex whose k-dimensional faces are chains of size k+1 from C(G). We show that this order complex has nontrivial reduced homology in characteristic two, and is therefore not contractible.
If time permits, I will discuss also related work on invariable generation of simple groups, joint with Bob Guralnick and Russ Woodroofe.
September 30, 2021, 4 pm CT
Speaker: Bernard Lidicky (Iowa State University)
Title: Flag algebras and its applications
Abstract: Flag algebras is a method, developed by Razborov, to attack problems in extremal combinatorics. Razborov formulated the method in a very general way which made it applicable to various settings. The method was introduced in 2007 and since then its applications have led to solutions or significant improvements of best bounds on many long-standing open problems, including problems of Erd\H{o}s. The main contribution of the method was transferring problems from finite settings to limits settings. This allows for clean calculations ignoring lower order terms. The method can utilize semidefinite programming and computers to produce asymptotic results. This is often followed by stability type arguments with the goal of obtaining exact results.
In this talk we will give a brief introduction of the basic notions used in flag algebras and demonstrate how the method works. Then we will discuss applications of the flag algebras in different settings.
Lunch with the speaker: Thursday, September 30, 11:45 a.m., Spoon House Korean Kitchen on Green Street; meet at the restaurant.
Dinner with the speaker: Thursday, September 30, 6:30 p.m., location to be discussed.
Additional event: Friday, October 1, 1:00-1:50 p.m. AH 447; the speakers talk with the students about their favorite problems.
September 7, 2021, 11-11:50 am CT
Speaker: Tao Jiang (Miami University of Ohio)
Title: Degenerate Turan problems for graphs
Abstract: In Turan type extremal problems, we want to determine how dense a graph or hypergraph is without containing a particular subgraph or family of subgraphs. Such problems are central to extremal graph
theory, because solving them requires one to thoroughly investigate the interaction of global graph parameters with local structures. Efforts in solving these problems have spurred the developments of some powerful tools in extremal graph theory, such as the regularity method, probabilistic and algebraic methods.
While Turan problems have satisfactory solutions for non-bipartite graphs, the problem is still generally wide-open for bipartite graphs with many intriguing conjectures and results. In this talk, we will discuss some conjectures on Turan problems for bipartite graphs and some recent progress on them. Time permitting, we will also discuss a colored variant of the Turan problem.
Additional event: September 9, 2021, 11-11:50 am CT, 141 Altgeld Hall, Tao Jiang's favorite problems.
Previous Colloquia
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\begin{definition}[Definition:Separated by Closed Neighborhoods/Sets]
Let $T = \struct {S, \tau}$ be a topological space.
Let $A, B \subseteq S$ such that:
:$\exists N_A, N_B \subseteq S: \exists U, V \in \tau: A \subseteq U \subseteq N_A, B \subseteq V \subseteq N_B: N_A^- \cap N_B^- = \O$
where $N_A^-$ and $N_B^-$ are the closures in $T$ of $N_A$ and $N_B$ respectively.
That is, that $A$ and $B$ both have neighborhoods in $T$ whose closures are disjoint.
Then $A$ and $B$ are described as '''separated by closed neighborhoods'''.
Category:Definitions/Separation Axioms
\end{definition} | ProofWiki |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of
\[\frac{x + y}{xyz}.\]
By the AM-HM inequality,
\[\frac{x + y}{2} \ge \frac{2}{\frac{1}{x} + \frac{1}{y}} = \frac{2xy}{x + y},\]so $\frac{x + y}{xy} \ge \frac{4}{x + y}.$ Hence,
\[\frac{x + y}{xyz} \ge \frac{4}{(x + y)z}.\]By the AM-GM inequality,
\[\sqrt{(x + y)z} \le \frac{x + y + z}{2} = \frac{1}{2},\]so $(x + y)z \le \frac{1}{4}.$ Hence,
\[\frac{4}{(x + y)z} \ge 16.\]Equality occurs when $x = y = \frac{1}{4}$ and $z = \frac{1}{2},$ so the minimum value is $\boxed{16}$. | Math Dataset |
\begin{document}
\title{Mixed Moore Cayley graphs} \author{Grahame Erskine\\{\small Open University, Milton Keynes, UK}\\ \texttt{\small [email protected]}}
\date{}
\maketitle \let\thefootnote\relax\footnote{Mathematics subject classification: 05C25, 05C35} \let\thefootnote\relax\footnote{Keywords: degree-diameter problem, mixed graphs, Moore graphs} \begin{abstract}\noindent The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected edges and directed arcs in the graph. For a diameter 2 graph with maximum undirected degree $r$ and directed out-degree $z$, a straightforward counting argument yields an upper bound $M(z,r,2)=(z+r)^2+z+1$ for the order of the graph. Apart from the case $r=1$, the only three known examples of mixed graphs attaining this bound are Cayley graphs, and there are an infinite number of feasible pairs $(r,z)$ where the existence of mixed Moore graphs with these parameters is unknown. We use a combination of elementary group-theoretical arguments and computational techniques to rule out the existence of further examples of mixed Cayley graphs attaining the Moore bound for all orders up to 485. \end{abstract}
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mark= at position 0.9 with {\arrow[scale=2]{#1}} ,
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\section{Preliminaries} The degree-diameter problem has its roots in the efficient design of interconnection networks. We try to find the maximum possible number of vertices in a graph where we constrain both the largest degree of any vertex and the diameter of the graph. For more information on the history and development of the degree-diameter problem, see the survey~\cite{miller2005moore}. The degree-diameter problem is typically studied in both the undirected and directed cases. Recently, there has been much interest in the problem as it is applied to \emph{mixed} graphs, where we allow both undirected edges and directed arcs in the graph.
In the undirected case, an upper bound for the largest possible order of a graph of maximum degree $d>2$ and diameter $k>1$ is the \emph{Moore bound}~\cite{miller2005moore}: \[M(d,k)=1+d\frac{(d-1)^k-1}{d-2}\] A graph attaining this bound is known as a \emph{Moore graph}. It is known~\cite{Hoffman1960,Bannai1973} that such a graph must have diameter 2 and degree $d\in \{2,3,7,57\}$, with existence of the graph corresponding to $d=57$ being a famous open problem.
For digraphs, the Moore bound for graphs of maximum out-degree $d>1$ and diameter $k>1$ has an even simpler form: \[M(d,k)=\frac{d^{k+1}-1}{d-1}\] It is well-known~\cite{Plesnik1974,Bridges1980} that no Moore digraphs of diameter greater than one exist apart from the directed 3-cycle.
In this paper we concentrate on the case of \emph{mixed} graphs where we allow both undirected edges and directed arcs in our graphs. We can view the case of mixed graphs either as a generalisation of the undirected case (allowing arcs as well as edges) or as a specialisation of the directed case (where we insist that a number of the arcs must be present with their reverses). We adopt the usual notation in the literature. The maximum undirected degree of any vertex is $r$, and the maximum directed out-degree is $z$. The general expression for the Moore bound for mixed graphs is rather more awkward in closed form~\cite{Buset2016}. However, Nguyen, Miller and Gimbert~\cite{Nguyen2007} showed in 2007 that no mixed Moore graph can exist for diameters greater than 2. We therefore concentrate on the diameter 2 case where it is straightforward to show that the Moore bound is : \[M(z,r,2)=(z+r)^2+z+1\]
In 1979, Bos\'ak~\cite{Bosak1979} derived (using a modification of the spectral method used by Hoffman and Singleton~\cite{Hoffman1960} in the undirected case) a numerical constraint on the sets of parameters $(r,z)$ for which a mixed Moore graph of diameter 2 can exist. Bos\'ak's condition is that $r=(c^2+3)/4$ for some odd integer $c$ dividing $(4z-3)(4z+5)$. In contrast with the undirected case, this condition means that there are an infinite number of feasible pairs $(r,z)$.
We will concentrate on the restricted problem of mixed \emph{Cayley graphs}. Given a finite group $G$ and a subset $S\subseteq G\setminus\{1\}$, we define the Cayley graph $\Cay(G,S)$ to have vertex set $G$ and arcs from a vertex $g$ to $gs$ for every $s\in S$. If both $s$ and $s^{-1}$ are in $S$, then for every $g$ we have arcs from $g$ to $gs$ and $gs$ to $g$
which we view as an undirected edge between $g$ and $gs$. Thus if $S=S_1\cup S_2$ where $S_1=S_1^{-1}$ and $S_2\cap S_2^{-1}=\emptyset$, then $\Cay(G,S)$ is a mixed graph of undirected degree $r=|S_1|$
and directed degree $z=|S_2|$.
It is easy to see that a Cayley graph $\Cay(G,S)$ has diameter at most 2 if and only if every element of $G$ can be expressed as a product of at most 2 elements of $S$. Thus we relate the properties of the graph to properties of the group $G$ and subset $S$.
\section{Known Moore graphs}
We can see (Table~\ref{tab:moorefeas}) how Bos\'ak's condition means that the range of feasible pairs $(r,z)$ for which a Moore graph can exist is quite limited. Nevertheless, there are infinitely many pairs $(r,z)$ for which the existence of a mixed Moore graph (Cayley or otherwise) is not known. In fact, almost all the feasible pairs remain open as to the existence or otherwise of a Moore graph.
\begin{table}\centering
\begin{tabular}{|ccc|} \hline Undirected & Directed & Order \\ degree $r$ & degree $z$ & $n$\\ \hline 1 & any & $r^2+2r+3$\\ \hline 3 & 1 & 18\\
& 3 & 40\\
& 4 & 54\\
& 6 & 88\\
& 7 & 108\\
& $\cdots$ & $\cdots$\\ \hline 7 & 2 & 84\\
& 5 & 150\\
& 7 & 204\\
& $\cdots$ & $\cdots$\\ \hline 13 & 4 & 294\\
& 6 & 368\\
& $\cdots$ & $\cdots$\\ \hline 21 & 1 & 486\\
& $\cdots$ & $\cdots$\\ \hline $\cdots$& $\cdots$ & $\cdots$\\ \hline \end{tabular} \caption{Feasible values for mixed Moore graphs} \label{tab:moorefeas} \end{table}
In the case $r=1$, it is immediate that any positive integer $z$ yields a feasible pair, and indeed such Moore graphs always exist by the following construction. The \emph{Kautz digraphs} $Ka(d,2)$ are a family of mixed Moore graphs of diameter 2, directed degree $z=d-1$ and undirected degree $r=1$. The vertices are the words $ab$ of length 2 over an alphabet of $d+1$ letters where we insist $a\neq b$. So there are $d(d+1)=(z+r)^2+z+1$ vertices. There is a directed edge from $ab$ to $bc$ for all of the $d$ eligible values of $c$. The edge from $ab$ to $ba$ can be considered as the undirected edge since the reverse edge also exists. All other edges from $ab$ are purely directed.
The graph has diameter 2 since there is a path $ab\to xy$ of length 1 if $x=b$ and $ab\to bx \to xy$ of length 2 if $x\neq b$. An example in the case $d=2$ is shown in Figure~\ref{fig:kautz}.
\begin{figure}
\caption{The Kautz digraph $Ka(2,2)$}
\label{fig:kautz}
\end{figure}
The Kautz digraphs $Ka(d,2)$ are not Cayley graphs for all values of $d$, and in fact they turn out to be Cayley graphs precisely when $d+1$ is a prime power (see for example~\cite{Brunat1995}). Until very recently, these graphs and a single further example of Bos\'ak with parameters $r=3,z=1$ (and hence order 18) were the only known mixed Moore graphs. (See the survey paper~\cite{miller2005moore} for more on these known graphs.)
Recently, J\o rgensen~\cite{Joergensen2015} has reported a pair of graphs with $r=3,z=7$ and hence order 108. These graphs are interesting because they are Cayley graphs (as indeed is Bos\'ak's graph of order 18). The two graphs are in fact a transpose pair, where one graph is obtained from the other by reversing the direction of the directed arcs.
On the negative side, no simple combinatorial argument has yet been found to rule out any feasible parameter pairs satisfying Bos\'ak's condition. For small graphs, an exhaustive computational approach is now becoming feasible with advances in CPU power and algorithms. L\'opez, P\'erez-Ros\'es and Pujol\`as~\cite{Lopez2014} ruled out the existence of mixed Moore Cayley graphs of orders 40 and 54. Recently, L\'opez, Miret and Fern\'andez~\cite{Lopez2016} have used computational techniques to show that there are no mixed Moore graphs (Cayley or otherwise) at orders 40, 54 or 84.
\section{Searching for new examples} It seems unlikely that brute-force exhaustive search algorithms will take us much further in the table. Inspired by J\o rgensen's result and the fact that the Bos\'ak graph of order 18 is also a Cayley graph, we describe a search algorithm to extend the work of L\'opez, P\'erez-Ros\'es and Pujol\`as~\cite{Lopez2014} to look for further examples of mixed Moore Cayley graphs.
Given a feasible pair $(r,z)$, we wish to find a group $G$ and a set $S\subseteq G$ such that the graph $\Cay(G,S)$ has order $n=(z+r)^2+z+1$,
undirected degree $r$, directed degree $z$ and diameter 2. For ease of explanation we split $S$ into the undirected generators $S_1$ and the directed generators $S_2$. Then $|S_1|=r,|S_2|=z,S_1=S_1^{-1},S_2\cap S_2^{-1}=\emptyset$.
The naive approach of simply testing all possible sets $S$ very quickly becomes computationally infeasible. Our strategy therefore is to look for properties of Moore graphs and corresponding properties of Cayley graphs which will allow us to reduce the search space. We begin with some elementary yet useful properties of mixed Moore graphs. \begin{proposition}\label{prop:mmprops} Let $\Gamma$ be a mixed Moore graph of diameter 2, undirected degree $r$ and directed degree $z$. \begin{enumerate}[label=(\roman*),topsep=0pt,itemsep=0.5ex]
\item If $u,v\in V(\Gamma)$ are distinct vertices then there is one and only one path of length 1 or 2 from $u$ to $v$.
\item $\Gamma$ contains no undirected cycle of length 3 or 4.
\item $\Gamma$ is totally regular (all vertices have the same directed in-degree and out-degree $z$).
\item Every arc in $\Gamma$ is contained in exactly one directed 3-cycle. \end{enumerate} \end{proposition} \begin{proof} Item \emph{(i)} follows immediately from the counting argument deriving the Moore bound by considering the spanning tree of $\Gamma$ rooted at $u$. Item \emph{(ii)} is a consequence of \emph{(i)}. Item \emph{(iii)} was proved by Bos\'ak~\cite{Bosak1979}.
To see why \emph{(iv)} is true, consider a vertex $u\in V(\Gamma)$. Then $u$ has $z$ directed out-neighbours $v_1,\ldots,v_z$. Since $\Gamma$ is totally regular, $u$ must have $z$ directed in-neighbours $w_1,\ldots w_z$. These cannot be at distance 1 from $u$, so each $w_i$ is reached by a path of length 2 from $u$. There can be no undirected edges in any of these paths, since that would lead to the end vertices of such an edge violating \emph{(i)}. So each $w_i$ is reached by a directed path of length 2 from $u$ passing through some $v_j$. These $v_j$ must be distinct, since if any were repeated it would have two paths of length 2 to $u$. Thus every arc $u\to v_j$ emanating from $u$ lies in the unique directed triangle $u\to v_j\to w_i\to u$. \end{proof}
Now we can use these properties to develop constraints on our generating set $S=S_1\cup S_2$ to narrow the search for mixed Moore Cayley graphs.
\begin{proposition}\label{prop:mmcayprops} Let $\Gamma$ be a mixed Moore graph of diameter 2, undirected degree $r$ and directed degree $z$. Suppose that $\Gamma\cong\Cay(G,S)$ where $G$ is a group of order $n=(r+z)^2+z+1$ and the generating set $S$ consists of undirected generators $S_1$ and directed generators $S_2$. Then: \begin{enumerate}[label=(\roman*),topsep=0pt,itemsep=0.5ex]
\item No element of $S_1$ has order 3 or 4.
\item No element of $S_2$ is an involution.
\item No pair of elements in $S_1$ has a product of order 2.
\item No two distinct elements of $S$ commute, apart from the inverse pairs in $S_1$.
\item $S$ is product-free (that is, $S\cap SS=\emptyset$).
\item All non-identity products of two elements of $S$ are unique.
\item The elements of $S_2$ are of two types:
\begin{enumerate}[label=\arabic*.]
\item Elements of order 3
\item Triples of distinct elements $\{a,b,c\}$, each of order at least 4, such that $(ab)^{-1}=c$
\end{enumerate} \end{enumerate} \end{proposition} \begin{proof} These facts follow immediately from the properties of the graph: \begin{enumerate}[label=(\roman*),topsep=0pt,itemsep=0.5ex]
\item follows from Proposition~\ref{prop:mmprops}\emph{(ii)}
\item is immediate because $S_2$ is inverse-free.
\item is true because such a pair would lead to an undirected 4-cycle.
\item follows from Proposition~\ref{prop:mmprops}\emph{(i)}.
\item follows from Proposition~\ref{prop:mmprops}\emph{(i)}.
\item follows from Proposition~\ref{prop:mmprops}\emph{(i)}.
\item follows from Proposition~\ref{prop:mmprops}\emph{(iv)}.\qedhere \end{enumerate} \end{proof}
We note that the conditions of Proposition~\ref{prop:mmcayprops}(v) and (vi) must also hold for any subset of $S$. This motivates the following definition.
Let $T\subseteq G$ with $T=T_1\cup T_2,T_1=T_1^{-1},T_2\cap T_2^{-1}=\emptyset,|T_1|=r',|T_2|=z'$. Define $P(T)=|\{1\}\cup T\cup TT|$. We say $T$ is a \emph{feasible} subset of generators if $P(T)=(z'+r')^2+z'+1$.
We have two further ways to reduce the number of sets $S$ we need to search for a given group $G$. The first is the well-known result that if $\phi$ is an automorphism of the group $G$, then $\Cay(G,S)\cong\Cay(G,\phi(S))$. So we need not consider all possible sets -- only orbit representatives under the action of $\Aut(G)$.
The second idea is to exploit the fact that all mixed Moore graphs must have even order (a consequence of Bos\'ak's condition). So a suitable group $G$ for a Cayley graph must have even order, and may in many cases have an index 2 subgroup. \begin{proposition}\label{prop:mmcayidx2}
Let $\Gamma$ be a mixed Moore graph of diameter 2, undirected degree $r$ and directed degree $z$. Suppose that $\Gamma\cong\Cay(G,S)$ where $G$ is a group of order $n=(z+r)^2+z+1$ and the generating set $S$ consists of undirected generators $S_1$ and directed generators $S_2$. Suppose further that $G$ admits an index 2 subgroup $H$ and that $|S_1\cap H|=s_1$ and $|S_2\cap H|=s_2$. Then: \[s_1+s_2=\frac{2(z+r)-1\pm\sqrt{4r-3}}{4}\] \end{proposition} \begin{proof} We know each non-identity element of $H$ can be expressed uniquely as a product of 1 or 2 elements of $S$. We count these products. Firstly, there are $s_1+s_2$ elements of $S\cap H$. Any other element is either a product of 2 elements of $S\cap H$ or 2 elements of $S\cap(G\setminus H)$. In the first case there are $s_1(s_1-1)+2s_1s_2+s_2^2$ possibilities. In the second case there are $(r-s_1)(r-s_1-1)+2(r-s_1)(z-s_2)+(z-s_2)^2$. Writing $s=s_1+s_2$ we see following some manipulation that the total number of elements of $H$ which we can write as a product of 0, 1 or 2 elements of $S$ is $2s^2+s(1-2(z+r))+(z+r)^2-r+1$. But $H$ is an index 2 subgroup and so contains exactly $((z+r)^2+z+1)/2$ elements. Solving this quadratic equation for $s$ yields the stated result. \end{proof} It might be thought that this provides a very strong condition, since the expression for $s_1+s_2$ must clearly give an integer result. However, it is interesting that Bos\'ak's condition on allowable values of $r,z$ means that this expression always gives one integer solution for $s_1+s_2$. Nevertheless, the condition does give a useful way to cut down the search space when we have an index 2 subgroup $H$, since it precisely determines the overall split of generators between $H$ and $G\setminus H$. In addition, we have a useful corollary allowing us to exclude some groups from consideration entirely. \begin{corollary} Suppose $\Gamma$ and $G$ are as in the statement of Proposition~\ref{prop:mmcayidx2}. Then if $2(z+r)-\sqrt{4r-3}>9$ then $G$ cannot contain an index 2 abelian subgroup $H$. \end{corollary} \begin{proof} If $H$ is an index 2 abelian subgroup of $G$, then if $2(z+r)-\sqrt{4r-3}>9$, by Proposition~\ref{prop:mmcayidx2} the generating set $S$ contains more than 2 elements of $H$. This is contrary to Proposition~\ref{prop:mmcayprops}(iv). \end{proof}
We can now describe the search algorithm. Given a feasible pair $z,r$ we use a \texttt{GAP}~\cite{GAP4} script. \begin{enumerate}[label=\arabic*.,topsep=0pt,itemsep=0.5ex]
\item Find all groups $G$ of order $n=(z+r)^2+z+1$.
\item If $G$ has an abelian index 2 subgroup, ignore it.
\item Compute the list $U$ of orbit representatives of all inverse-closed sets $A$ of size $r$ such that $|A\cup AA|=r^2+1$.
\item If $G$ admits an index 2 subgroup $H$, delete any sets from $U$ which do not satisfy Proposition~\ref{prop:mmcayidx2}.
\item Compute the list $D$ of all inverse-free sets $B=\{a,b,(ab)^{-1}\}$ such that $|B\cup BB|=12$.
\item Try (recursively) to extend each $S\in U$ by adding directed generators of order 3 or triples from $D$ until we have added $z$ generators. \end{enumerate}
\section{Search results} Results of the search on feasible orders up to 200 are in Table~\ref{tab:moore1}. For completeness the case $r=1$ is included. As explained above, we know there is a unique Moore graph with $r=1$ for every $z\geq 1$, but these are Cayley only if $q=z+2$ is a prime power. The algorithm reproduces all the known Cayley Moore graphs and confirms that there are no more examples below order 200.
We then continued the search for feasible orders up to 500. The results are in Table~\ref{tab:moore2}. The algorithm was unable to complete the search at order 486 due to the large numbers of groups and the increasing search space. However, there are no more examples at any of the other feasible orders up to 485.
\begin{table}[ht]\centering
\begin{tabular}[t]{|cccc|} \hline $n$ & $r$ & $z$ & Graphs \\ \hline 18 & 3 & 1 & 1\\ 40 & 3 & 3 & 0\\ 54 & 3 & 4 & 0\\ 84 & 7 & 2 & 0\\ 88 & 3 & 6 & 0\\ 108 & 3 & 7 & 2\\ 150 & 7 & 5 & 0\\ 154 & 3 & 9 & 0\\ 180 & 3 & 10 & 0\\ \hline \end{tabular} \qquad
\begin{tabular}[t]{|cccc|} \hline $n$ & $r$ & $z$ & Graphs \\ \hline 6 & 1 & 1 & 1\\ 12 & 1 & 2 & 1\\ 20 & 1 & 3 & 1\\ 30 & 1 & 4 & 0\\ 42 & 1 & 5 & 1\\ 56 & 1 & 6 & 1\\ 72 & 1 & 7 & 1\\ 90 & 1 & 8 & 0\\ 110 & 1 & 9 & 1\\ 132 & 1 & 10 & 0\\ 156 & 1 & 11 & 1\\ 182 & 1 & 12 & 0\\ \hline \end{tabular} \caption{Cayley Moore graphs up to order 200} \label{tab:moore1} \end{table}
\begin{table}\centering
\begin{tabular}[t]{|cccc|} \hline $n$ & $r$ & $z$ & Graphs \\ \hline 204 & 7 & 7 & 0\\ 238 & 3 & 12 & 0\\ 270 & 3 & 13 & 0\\ 294 & 13 & 4 & 0\\ 300 & 7 & 10 & 0\\ 340 & 3 & 15 & 0\\ 368 & 13 & 6 & 0\\ 374 & 7 & 12 & 0\\ 378 & 3 & 16 & 0\\ 460 & 3 & 18 & 0\\ 486 & 21 & 1 & ?\\ \hline \end{tabular} \qquad
\begin{tabular}[t]{|cccc|} \hline $n$ & $r$ & $z$ & Graphs \\ \hline 210 & 1 & 13 & 0\\ 240 & 1 & 14 & 1\\ 272 & 1 & 15 & 1\\ 306 & 1 & 16 & 0\\ 342 & 1 & 17 & 1\\ 380 & 1 & 18 & 0\\ 420 & 1 & 19 & 0\\ 462 & 1 & 20 & 0\\ \hline \end{tabular} \caption{Cayley Moore graphs from order 200 to 500} \label{tab:moore2} \end{table}
We summarise these results as tabulated in Tables~\ref{tab:moore1} and~\ref{tab:moore2} in a theorem. \begin{theorem} Up to order 485, the only mixed Moore Cayley graphs of undirected degree $r$, directed degree $z$ and diameter 2 are as follows. \begin{itemize}
\item $r=1$ and $z\leq 20$ where $z+2$ is a prime power (Kautz graphs).
\item $r=3$ and $z=1$ (Bos\'ak's graph).
\item $r=3$ and $z=7$ (the two graphs of J\o rgensen). \end{itemize} \end{theorem}
\end{document} | arXiv |
Journal of the European Optical Society-Rapid Publications
Transient dynamical-thermal-optical system modeling and simulation
Luzia Hahn1 &
Peter Eberhard ORCID: orcid.org/0000-0003-1809-44071
Journal of the European Optical Society-Rapid Publications volume 17, Article number: 5 (2021) Cite this article
In modern high resolution optical systems like astronomical telescopes or lithographic objectives, performance degradations can be caused by various disturbances. Holistic optical system simulation is required to predict the performance or the high precision systems. In this paper a method for transient dynamical-thermal-optical system modeling and simulation is introduced. Thereby, elastic deformation, rigid body motion, and mechanical stresses due to dynamical excitation are calculated using elastic multibody system simulation and temperature changes are determined using thermal finite element analysis. The deformation, the motion, and the mechanically and thermally induced stress index changes are then considered in a gradient-index ray tracing. Finally, the presented method is applied in a dynamical-thermal single lens system.
Nowadays, the developed high precision optical instruments are very sensitive to external influences. Holistic system simulation is used to predict the system performance accurately and to prove that the design meets the demanded requirements. In lithographic objectives even small mechanical disturbances like deformations or rigid body motions of the mirrors or lenses resulting from external excitation, e.g., from the waver motion system lead to performance degradations. Besides that, temperature changes in the optical elements due to the heating from the laser beams influence the imaging performance. In laser-based additive manufacturing the laser beam has alternating on and off times and the laser beam moves during the exposure time. Thus, transient simulations are necessary to simulate the optical system behavior. All together, a holistic optical system simulation must be able to consider transient mechanical and thermal disturbances. In this paper a method for transient dynamical-thermal-optical system modeling and simulation is introduced and an exemplary system simulation is presented for a single lens system.
The dynamical-thermal behavior of optical systems can be calculated by a combination of rigid body motion, small deformations, the related mechanical stresses, and the temperature distribution. For the mechanical analysis various literature exists, where structural finite element (FE) analysis results are used to describe the deformation of the optical elements [1–3]. In order to consider mechanical stress in the optical simulation three dimensional shape-function interpolation is used to map the changes of the refraction index with the mechanical stress resulting from the FE analysis [4]. In contrast to this, the introduced method of this paper uses more efficient elastic multibody system (EMBS) simulations to calculate the transient elastic deformations and the rigid body motions of the system. Due to a model order reduction and the possibility to combine rigid bodies with flexible bodies, the EMBS simulation is usually faster than the FE analysis and still sufficiently accurate. Instead of the stresses from FE analysis we get the stresses from the EMBS simulation and transfer them to cylindrical polynomials which can be evaluated efficiently during the optical simulation.
The thermal effects on the optical system are taken into account in the presented method by considering the changes of the refraction index due to temperature changes. A thermal FE analysis is executed in various papers and the refraction index changes are transferred to the optical system using three dimensional shape function interpolation [4, 5] and optical path difference (OPD) maps [6, 7]. In the presented method cylindrical polynomials are used to describe the three dimensional refraction index changes. With this method, arbitrary three dimensional refraction index fields can be considered in the optical simulation.
Additionally to the dynamical and the thermal analysis, and the data transformation from the mechanical and the thermal results to the optical system the presented method includes a gradient index (GRIN) ray tracing algorithm which considers the dynamical and thermal disturbances.
There are several existing tools for mechanical-thermal-optical system simulation. An overview is given in [8]. The novel contribution of the presented method is the usage of transient EMBS simulations instead of structural FE analysis and the usage of cylindrical polynomials combined with GRIN ray tracing instead of OPD maps or shape-function interpolation for the stress and temperature dependent changes of the refraction index. A workflow for the presented method is given in Fig. 1.
Overall workflow of a holistic dynamical-thermal-optical system analysis
In the following, the mechanical, thermal, and optical methods are introduced briefly. Additionally, a numerical example of a holistic simulation of a dynamical-thermal-optical single lens system is given.
Methods and numerical example
The holistic simulation of transient dynamical-thermal-optical systems combines mechanical simulations, thermal system analysis and optical simulations. The method and an exemplary numerical example are described in more detail in the following.
Mechanical modeling of dynamical-optical systems
In the mechanical part of the holistic simulation the optical system, consisting of optical elements and their holdings, is modeled as EMBS [9]. In a first step, each body is modeled separately. Optical sensitive components such as optical lenses are considered to be elastic bodies while the remaining less important elements such as the holdings are considered as being rigid. For each elastic body, a finite element model with N degrees of freedom (DOF) is built. The deformations of each modeled body are described by its shape functions Ψ and its nodal displacements vector \({\boldsymbol {q}}(t)\in \mathbb {R}^{N}\) which varies dependent on the time t. The equations of motion for each elastic body result from the linear FE model and read
$$ \boldsymbol{M}_{\text{ss}}\ddot{\boldsymbol{q}}(t)+\boldsymbol{D}_{\text{ss}}\dot{\boldsymbol{q}}(t)+\boldsymbol{K}_{\text{ss}}{\boldsymbol{q}}(t)=\boldsymbol{f}(t)\;. $$
Thereby, the system matrices are the mass matrix Mss, the damping matrix Dss, and the stiffness matrix Kss. The system input is the vector of the exciting forces f. The normally large number of DOF of FE models leads to a high computational effort for the analysis of their dynamical behavior. By means of a model order reduction (MOR) method, the number of DOF can be significantly reduced after the determination of the system matrices and the force vector. In the MOR scheme, a projection matrix \(\boldsymbol {V}\in \mathbb {R}^{N\times n}\) with n≪N is calculated. This matrix enables the approximation of the nodal displacements by
$$ {\boldsymbol{q}}\approx\boldsymbol{V} \bar{{\boldsymbol{q}}}\; $$
with the reduced coordinates \(\bar {{\boldsymbol {q}}}(t)\in \mathbb {R}^{n}\). The projection matrix is applied to the system matrices and the system input vector which leads to a reduction of the whole equation system size of Eq. (1). The reduction allows for an efficient analysis of the dynamical behavior of each modeled body. The detailed procedure description for a MOR and different methods for the calculation of V are given in [10, 11]. The application of MOR methods on optical systems is discussed in [12, 13]. Subsequently to the MOR, the rigid and elastic bodies are assembled to the EMBS. Then, the dynamic simulation of the assembled system, i.e. a time integration, yields the time-dependent kinematics of the complete system and the deformations of the elastic bodies.
Apart from the usage in the MOR, the projection matrix is also provided to the FE model of each elastic body if stresses need to be considered. There, it is used for obtaining the deformation state of the elastic body. By the use of this deformation state and material laws the matrix of the reduced stress functions \(\bar {\boldsymbol {\Psi }}_{\mathrm {\sigma },\mathrm {P}}\in \mathbb {R}^{6\times n}\) at a discrete point P within the elastic body is calculated. The matrix \(\bar {\boldsymbol {\Psi }}_{\mathrm {\sigma },\mathrm {P}}\) and the reduced nodal displacements vector \(\bar {{\boldsymbol {q}}}\) allow for the calculation of the vector of the time-dependent stresses
$$ \hat{\boldsymbol\sigma}_{\mathrm{P}}=\left[\sigma_{\mathrm{x}},\sigma_{\mathrm{y}},\sigma_{\mathrm{z}},\tau_{\text{xy}},\tau_{\text{yz}},\tau_{\text{xz}}\right]_{\mathrm{P}}^{\top} = \bar{\boldsymbol\Psi}_{\mathrm{\sigma},\mathrm{P}}\bar{{\boldsymbol{q}}} $$
at the point P. The normal stresses σx,σy, and σz in the direction of the reference system in P and the shear stresses τxy,τyz, and τxz in the associated planes can be rearranged to the stress tensor ΣP. By the use of an eigen analysis of ΣP the principal stresses σI,P,σII,P, and σIII,P are computed. A more detailed explanation on the calculation of the stresses can be found in [14, 15].
In summary, the EMBS dynamical simulation allows for the computationally efficient calculation of the time-dependent deformations and principal stresses in the elastic bodies and the time-dependent rigid body motion of all bodies.
Thermal finite element analysis
In the thermal analysis part of the holistic simulation of the dynamical-thermal-optical system, an FE model is built for all optical lenses of the system. Normally, the FE mesh for the thermal analysis is coarser than the mechanical mesh. This saves computing time and nevertheless leads to sufficiently precise results. The calculation of the time-dependent nodal temperatures 𝜗(t) in each optical lens is performed using the dynamical equation for the heat conduction
$$ \boldsymbol{D}_{\text{tt}}\dot{\boldsymbol{\vartheta}}+\boldsymbol{K}_{\text{tt}}\boldsymbol{\vartheta}=\boldsymbol{\phi}\;. $$
The matrix Dtt for the specific heat, the matrix Ktt for the thermal conductivity, and the thermal input vector ϕ which represents the thermal heat flux at the nodes, are computed with FE methods. While thermally induced changes of the refraction index are considered in the presented method, thermally induced deformations are not considered here.
Optical modeling of dynamical-thermal-optical systems
The dynamical and thermal effects in optical systems can be analyzed using ray tracing. Thereby, the propagation characteristics of light rays in different media and especially media with an inhomogeneous refraction index must be taken into account. The used ray tracing approach is described in the following.
Ray tracing is a part of the geometrical optics where light rays determine the imaging performance of an optical system and wave-optical effects are neglected. In homogeneous media the rays propagate along straight lines and change their direction of travel only at the surfaces. In sequential ray tracing through homogeneous media, the ray positions r and the ray directions d are calculated in a relative surface formulation from surface to surface, from the object plane to the image plane. Thereby, the calculated ray position ri at a surface Si is the intersection point of a ray and the surface. The corresponding ray direction di results from the previous ray direction di−1, the intersection point ri, the surface normal ni, the refraction index of the previous media ni−1, and the current one ni [16]. Light propagates as straight line through homogeneous media. An inhomogeneous refraction index within a medium influences the path of a light ray and makes it curved. The resulting bending of the ray depends on the gradient of the refraction index. In order to trace a ray through a GRIN medium, the partial differential ray equation
$$ \frac{\mathrm{d}}{\mathrm{d} s}\left(n(\boldsymbol{r})\frac{\mathrm{d} \boldsymbol{r}}{\mathrm{d} s}\right)=\nabla n(\boldsymbol{r}) $$
must be solved numerically. A derivation of this equation can be found in [17]. The path parameter s can be used to approximate the differential geometrical path length ds by \(\mathrm {d} s = \mathrm {d} \bar {s} n\). In combination with (5) one gets
$$ \frac{\mathrm{d}^{2} \boldsymbol{r}}{\mathrm{d} \bar{s}} = n(\boldsymbol{r})\nabla n(\boldsymbol{r}) $$
which can be solved using a fourth order Runge-Kutta method [15, 18]. With this method a ray can be traced efficiently through a medium with an arbitrary inhomogeneous refraction index in small spatial steps till the back surface of the inhomogeneous medium is reached. In every spatial step the changes in the refraction index are calculated and considered in the calculation of a new ray position r and a new ray direction d. The only requirement is that the refraction index distribution within the inhomogeneous medium is known during the ray tracing.
Dynamical-thermal-optical system modeling and simulation
The analysis of transient dynamical-thermal-optical systems requires the combination of the mechanical, the thermal, and the optical system analysis. Therefore, the results of the dynamical EMBS simulation and the results of the thermal FE analysis must be transformed in such a way that the results can be considered in the ray tracing. From the EMBS simulation we get the transient kinematic behavior of the system. Thereby, the motion is described relative to an inertial system frame. For the sequential ray tracing, all rigid body motions and elastic deformations must be transferred in a relative surface description for each time step and each surface. The time-dependent deformations of the surfaces must additionally be transformed in a way that the surfaces are described smooth enough to avoid discretization effects during the calculation of intersection points of the rays and the surfaces. Hence, the deformations must be approximated using polynomial functions and in this case we use Zernike polynomials. These polynomials can be evaluated very efficiently during the ray tracing and they can reproduce random deformation states of circular lenses. Deformations of free-form surfaces can be approximated smoothly using B-splines. The third result from the mechanical simulation, are the time-dependent principal stresses which lead to a spatial variation of the refraction index within deformed optical elements. The change of the refraction index Δnσ,j due to stress at a position rj in the medium or on the surface of an optical medium result from the stress optic law
$$ \Delta n_{\mathrm{\sigma},j}=C\left(\sigma_{\mathrm{I},j}+\sigma_{\text{II},j}+\sigma_{\text{III},j}\right) $$
with the material dependent stress-optical coefficient C. Thereby, polarization effects are neglected. The stress dependent refraction index changes are approximated using cylindrical polynomials, to avoid discretization effects. Therby, Zernike polynomials Z(x,y) of the order \([0,\dots,n_{\mathrm {Z}}]\) are used as the radial component of the cylindical polynomials and Chebychev polynomials T(z) of the order \([0,\dots,n_{\mathrm {T}}]\) describe the height component. The axis of symmetry of the cylinder corresponds with the z-axis of the back surface coordinate system of the lens. The height and the radius are chosen so that all nodes of the deformed lens are inside the cylinder. The refraction index changes at a random point p=(xp,yp,zp) within the lens can be approximated using
$$ \Delta n_{\mathrm{\sigma}}(x_{\mathrm{p}},y_{\mathrm{p}},z_{\mathrm{p}}) \approx \left[\begin{array}{c} Z_{0}(x_{\mathrm{p}},y_{\mathrm{p}})T_{0}(z_{\mathrm{p}}) \\ Z_{0}(x_{\mathrm{p}},y_{\mathrm{p}})T_{1}(z_{\mathrm{p}})\\ \vdots\\ Z_{n_{\mathrm{Z}}-1}(x_{\mathrm{p}},y_{\mathrm{p}})T_{n_{\mathrm{T}}}(z_{\mathrm{p}})\\ Z_{n_{\mathrm{Z}}}(x_{\mathrm{p}},y_{\mathrm{p}})T_{n_{\mathrm{T}}}(z_{\mathrm{p}}) \end{array}\right]^{\top} \cdot \mathbf{c}\; $$
with the cylindrical polynomial coefficients c. These coefficients can be calculated by the evaluation of Eq. (8) with the known refraction index changes at all nodes of the thermal FE mesh. The Eq. (8) is then evaluated during the numerical solution of Eq. (6) and thus, the changes of the refraction index due to stress can be considered efficiently in the ray tracing. A more detailed description of the application of cylinder polynomials in the GRIN ray tracing is given in [19].
The result of the thermal FE analysis of the optical elements are the time-dependent nodal temperatures. The temperature changes Δ𝜗 lead to a spatial variation of the refraction index. The changes of the refraction index in a position rj due to temperature changes can be described by
$$ \begin{aligned} \Delta n_{\mathrm{\vartheta},j} = &\frac{n_{{0}}-1}{2 n_{{0}}}\left(D_{0}\Delta\vartheta_{j}+D_{1}\vartheta_{j}^{2}+D_{2}\Delta\vartheta_{j}^{3} + \right. \\ &\left. \frac{E_{0}\Delta\vartheta_{j}+E_{1}\Delta\vartheta_{j}^{2}}{\lambda^{2}-\lambda_{\text{tk}}^{2}} \right)\;. \end{aligned} $$
Thereby, D0,D1,D2,E0,E1, and λtk are material dependent constants and n0 is the refraction index at the reference temperature of 20∘C. The spatially disturbed refraction index changes are then approximated with cylinder polynomials and evaluated during the ray tracing, such as previously described for the mechanical stresses. The refraction index changes due to stress and due to temperature changes are added during the ray tracing. Thus, mechanical stress effects and thermal effects are considerable in the ray tracing.
With the transformed results from the dynamical and the thermal system analysis the ray tracing can be performed. Thus, a holistic simulation of optical systems is possible with this method. Thereby, transient rigid body motions, dynamical deformations, refraction index changes due to mechanical stresses and thermally induced refraction index changes are considered computational efficiently during the ray tracing.
To investigate rigid body motion, deformation, mechanical stress and thermal effects in a dynamical-thermal-optical system, the introduced simulation procedure is implemented. The holistic simulation is performed using ANSYS for the building of the FE models, MatMorembsFootnote 1 for the MOR, Neweul-M2Footnote 2 for the EMBS simulation, TheelaFEA for the thermal analysis, and OM-SimFootnote 3 for the data transformation and the optical simulation. The latter four are MATLAB based software tools which are developed at the Institute of Engineering and Computational Mechanics at the University of Stuttgart. A holistic workflow diagram is shown in Fig. 1.
Numerical example
The presented holistic dynamical-thermal-optical system simulation method is applied to a numerical example in the following. The used system and its excitation are shown in Fig. 2. It consists of a single lens with a diameter of 100 mm and a homogeneous refraction index of n0=1.6 as long as no mechanical stresses and no temperature changes occur. The lens is fixed by a holding, which has six support beams evenly spaced around the circumference. The lens and the holding beams are modeled as elastic bodies, while the holding itself is considered to be rigid. As shown in the figure, the system is excitated mechanically with a predefined movement over the time in all spatial directions translational at,x,at,y,at,z and rotational ar,x,ar,y,ar,z. Therefore, a time-dependent random signal based on a Gaussian distribution with a maximum translation of 1×10−3 mm, a maximum rotation of 1×10−2 rad, and a maximum frequency of 100 Hz is applied on the center of gravity of the optical system. A time-integration of the EMBS system leads to the deformations and rigid body motions of the lens and to the corresponding mechanical stresses. The thermal excitation are seven heat fluxes applied on the lens as shown in Fig. 2. They represent the heat transfer from seven light rays which come from infinity, propagate through the optical lens, and focus on the optical axis on the image plane if no disturbances exist. A time-integration for the thermal FE model leads to the temperatures over the time. For the analysis of the dynamical-thermal-optical system behavior, the results of the ray tracing algorithm are investigated next for the previously mentioned seven rays.
Single lens optical system with thermal (red) and mechanical (black) excitation
In order to investigate the dynamical-thermal-optical system for a large number of rays, another thermal boundary condition is applied. A heat flux is applied on all nodes of the thermal FE lens model which are within a cylinder with a radius of 20 mm, with the center at x=y=11 mm and with the height over the whole lens thickness. This applied thermal boundary condition is completely unrelated to the ray paths. It is assumed to visualize thermal influence on images only. The dynamical excitation is the same as described before. In this case the light propagates from one point uniformly disturbed through the lens and forms a grid on the image plane. So, a geometrical image simulation is performed. The unexcited system and its image are shown in Fig. 3.
Side view of the analyzed system (left) and the result of the geometrical image simulation without any thermal or dynamical excitation
The results of the ray tracing for the dynamical-thermal-optical system with the first thermal boundary condition are shown in Fig. 4. Thereby the thermal effects, the surface deformations and the mechanical stresses are scaled so that the effects get visible. The figures show the ray tracing results for different exciations all at the same time instant. In each subfigure the blue dashed lines show the ray paths for the ideal, undisturbed system. There, the rays focus on the optical axis on the image plane.
Ray tracing through the single-lens system with different excitations considered. The blue dashed line always shows the ray tracing results for the ideal, unexcitated system
In Fig. 4a only the changes of the refraction index due to the heating of the lens are considered in the ray tracing. Deformations or stresses are not taken into account. The temperature distribution within the lens is the same for each ray. Therefore, each ray is subject to the same changes in the refraction index which results in rotational symmetric changes of the ray paths around the optical axis. As it can be seen, the thermal influence leads to a shorter focal length.
If only the surface deformations and the rigid body motion of the lens due to the dynamical excitation are considered in the ray tracing, the ray paths shown in Fig. 4b result. No refraction index changes are taken into account. The refraction angle and the intersection position of each ray with the lens surface differs due to the non rotational symmetric deformation. This leads to the visible deviation of the ray positions on the image plane from the ideal focal point.
In Fig. 4c the surface deformations and rigid body motions due to the dynamical excitation as well as the resulting stresses within the lens are considered during the ray tracing. All three disturbances are not rotational symmetric around the optical axis. Thus, not only the changes in the surface intersections and refraction angles but also different refraction index changes occur for each ray. Each considered effect leads to different ray path changes for each ray. The additional considered refraction index changes, compared to the ray tracing where only the deformation is considered, intensify the individual path changes of each ray. Thus, the deviation of the ray positions on the image plane from the focus point increase from Fig. 4b to c.
If the deformation due to the dynamical excitation, the resulting mechanical stresses, and the thermally induced refraction index changes are considered during the ray tracing, the presented holistic dynamical-thermal-optical system simulation is executed. The effects shown in Fig. 4a and c summarize and the ray paths shown in Fig. 4d result. The reduction of the focal length due to the thermally induced refraction index changes as well as the deviation of the ray positions on the image plane from the ideal focus due to the dynamical excitation are present and clearly visible in the figure. Thus, in the holistic simulation of optical systems the presented tool can consider changes in the refraction index due to temperature changes and due to mechanical stresses and it is able to consider rigid body motion and elastic deformations due to a dynamical excitation.
In order to visualize thermal effects within a lens on an image, the cylindrical thermal excitation described in the previous section is applied on the single lens. The resulting temperature distribution in the lens after three seconds is shown in Fig. 5 on the left. In the same figure, results from the geometrical image simulation are shown for three different time steps. The center of the heated cylinder almost coincides with the center of the quarter top right of the image and is marked in the image for t=0s with the red dot. The changes of the refraction index due to temperature changes for each ray are visualized in the image plots by the color. The higher the changes in the refraction index, the more orange are the shown ray intersection points with the image plane. As it can be seen in the three images of Fig. 5, the heating of the lens leads to a distortion of the image. The grid expands in the heated area due to the thermally induced refraction index changes. In the parts of the grid where no or nearly no temperature changes occur, the grid does not change.
The temperature distribution on the lens after three seconds (left) and the images for t=0s,t=2s and t=3s. The higher the changes in the refraction index due to temperature is, the more orange are the ray positions on the image plane. The red dot shows the center of the cylinder within which the heat is applied on the lens
In order to visualize dynamical and thermal effects within a lens on an image, the dynamical exciation and the cylindrical thermal excitation described in the previous section are applied on the single lens. The resulting geometrical images at t=3s are shown in Fig. 6. Thereby, the geometrical images are shown for different combinations of the disturbances.
Geometrical image after three seconds excitation time with different excitations
In Fig. 6a the geometrical images of the optical system with only dynamical excitation are shown. The geometrical image shown in light grey result from the undisturbed system. If only the rigid body motions and deformations are considered in the ray tracing, the image shown in orange results. As it can be seen, the grid is irregulary aberrated due to the irregular disturbances. If the mechanical stresses and the resulting changes of the refraction index Δnσ are considered additionally, the blue image results. Due to the irregular stress effects additionally to the irregular deformation and motion disturbances the aberrations in the blue image are larger than in the orange image.
In Fig. 6b the undisturbed image in light grey and the orange image resulting from only dynamical excitation are shown again. The pink image results if the disturbances due to lens motions and deformations and additionally the thermal refraction index changes Δn𝜗 due to the heating of the lens are considered in the ray tracing. Therein, it becomes visible that the distortions due to the lens deformations (orange) and the stretching of the grid due to the heating (Fig. 5) add up in the pink image.
If the dynamical excitation, the resulting mechanical stresses with the associated refraction index changes Δnσ, and the cylindrical thermal excitation with the resulting thermal refraction index changes Δn𝜗 are considered we get the image shown in Fig. 6c in violet. The geometrical image is now strongly affected. The stretching of the grid at the heated areas is still visible but due to the rigid body motion, the deformation, and the mechanical stress effects it is supperimposed by other disturbances.
Altogether, Fig. 6 shows that the thermal distortions and the shifting of the ray intersections with the image plane due to the dynamical excitation superpose and visibly influence the imaging performance of the system. Thus, it is important to consider time dependent thermally and mechanically induced refraction index changes, rigid body motions, and elastic surface deformations in the geometrical image simulation of optical systems if high precision images are required.
In order to evaluate the efficiency of the implemented method, the calculation time of the geometrical image simulation was tracked. The thermal and structural FE model building, the system matrices extraction, the model order reduction, the dynamical time simulation with the rigid body motion, deformation, and stress calculation, the thermal simulation, and the data transformation to the optical system are calculated in less than five minutes for 30 calculated time steps in a time period of three seconds. The image simulation for one time step, so the GRIN ray tracing of 2356 rays, takes less than 30 seconds. Thereby, for each ray about 230 positions and new ray directions within the lens were calculated. At each of these positions the refraction index changes due to temperature changes and mechanical stresses are considered. At the front and the back surface of the lens, the surface deformations, the rigid body motion and the refraction index changes are considered. The given calculation times show, that the holistic simulation is efficiently applicable for the simulation of dynamical-thermal-optical systems.
The described method enables a qualitative characterization of the influence of dynamical and thermal distortions on optical systems. A quantitative characterization of the aberrations would be possible if the optical path difference and the line of sight error are calculated additionally. For materials with a homogeneous refraction index this is possible in the software OM-Sim as shown in [16, 20]. For materials with a gradient in the refraction index the implementation is not yet realized.
In order to efficiently carry out holistic time simulations of dynamical-thermal-optical systems, a numerical ray tracing procedure has been presented and applied in this paper. In the system simulation, finite element thermal analysis, model order reduction methods, elastic multibody system simulation, and optical ray tracing with gradients in the refraction index are combined. The presented procedure is able to consider time-dependent dynamical disturbances, such as rigid body motions and elastic deformations of optical elements and their holdings. Moreover, it can calculate the time-dependent stresses resulting from the deformations and transfer this information into refraction index changes which are considered during the ray tracing. Besides the dynamical disturbance consideration, thermal FE analyses are implemented and the resulting time-dependent temperatures are used to consider the thermally induced refraction index changes in the ray tracing. In a numerical single-lens example the presented method was applied. The example shows the importance of the consideration of the dynamical and thermal effects in the simulation of high precision dynamical-thermal-optical systems. Besides that, it shows that the presented method enables the numerically efficient calculation of time-dependent rigid body motions, elastic deformations, mechanical stresses, and temperature changes of optical systems and it allows for the consideration of the resulting disturbances in the gradient-index ray tracing.
https://www.itm.uni-stuttgart.de/en/software/morembs
https://www.itm.uni-stuttgart.de/en/software/neweul-m
https://www.itm.uni-stuttgart.de/en/research/dynamical-optical-systems/om-sim
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Gatej, A., Wasselowski, J., Loosen, P.: Thermo-optical (TOP) analysis by coupling FEM and ray tracing. In: Wyrowski, F., Sheridan, J. T., Tervo, J., Meuret, Y. (eds.)Optical Modelling and Design II, vol. 8429, pp. 108–115. International Society for Optics and Photonics, SPIE (2012). https://doi.org/10.1117/12.921533.
Michels, G., Genberg, V., Doyle, K.: Integrating Ansys mechanical analysis with optical performance analysis using Sigfit. In: ANSYS Conference & 26th CADFEM Users' Meeting, Darmstadt, Germany (2008).
Doyle, K. B., Genberg, V., Michels, G. J.: Integrated optomechanical analysis, Vol. 58. SPIE press, Bellingham, Washington (2002).
Heijmans, J., Müller, M., Holzlöhner, R.: Combined Opto-mechanical analysis for modern optical instruments. In: Angeli, G. Z., Dierickx, P. (eds.)Modeling, Systems Engineering, and Project Management for Astronomy VIII, vol. 10705, pp. 746–756. International Society for Optics and Photonics, SPIE (2018). https://doi.org/10.1117/12.2314778.
Schwertassek, R., Wallrapp, O.: Dynamik flexibler Mehrkörpersysteme (in German). Vieweg, Braunschweig (1999). https://doi.org/10.1007/978-3-322-93975-3.
Fehr, J.: Automated and Error-Controlled Model Reduction in Elastic Multibody Systems, Vol. 21. Shaker Verlag, Aachen (2011). Dissertation, Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart.
Nowakowski, C., Fehr, J., Fischer, M., Eberhard, P.: Model order reduction in elastic multibody systems using the floating frame of reference formulation. Math. Model. 7, 40–48 (2012). https://doi.org/10.3182/20120215-3-AT-3016.00007.
Störkle, J., Eberhard, P.: Influence of model order reduction methods on dynamical–optical simulations. J. Astron. Telescopes Instrum. Syst. 3(2), 1–18 (2017). https://doi.org/10.1117/1.JATIS.3.2.024001.
Störkle, J., Eberhard, P.: Using integrated multi-body systems for dynamical-optical simulations. In: Modeling, Systems Engineering, and Project Management for Astronomy VII, pp. 542–556. International Society for Optics and Photonics (2016). https://doi.org/10.1117/12.2230692.
Wengert, N., Eberhard, P.: Using dynamic stress recovery to investigate stress effects in the dynamics of optical lenses. Proceedings of the 7th ICCSM, Zadar, Croatia (2012).
Hahn, L., Störkle, J., Eberhard, P.: Consideration of polarization during the ray tracing for mechanically stressed lenses in dynamical-optical systems. Optik. 193, 162923 (2019). https://doi.org/10.1016/j.ijleo.2019.06.023.
Störkle, J.: Dynamic Simulation and Control of Optical Systems (in German), Vol. 58. Shaker Verlag, Aachen (2018). https://doi.org/10.2370/9783844063011, Dissertation, Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart.
Ghatak, A. K. V.: Contemporary Optics. Optical physics and engineering(Thyagarajan, K. V., ed.)Plenum Press, New York (1978). https://doi.org/10.1007/978-1-4684-2358-7.
Sharma, A., Kumar, D. V., Ghatak, A. K.: Tracing rays through graded-index media: a new method. Appl. Opt. 21(6), 984–987 (1982). https://doi.org/10.1364/AO.21.000984.
Wengert, N.: Gekoppelte dynamisch-optische Simulation von Hochleistungsobjektiven (in German), Vol. 40. Shaker Verlag, Aachen (2015). Dissertation, Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart.
Störkle, J., Hahn, L., Eberhard, P.: Simulation of segmented mirrors with adaptive optics. Adv. Opt. Technol. 8, 119–127 (2019). https://doi.org/10.1515/aot-2018-0063.
Open Access funding enabled and organized by Projekt DEAL.
Institute of Engineering and Computational Mechanics, University of Stuttgart, Pfaffenwaldring 9, Stuttgart, 70569, Germany
Luzia Hahn & Peter Eberhard
Luzia Hahn
Peter Eberhard
The Authors are the sole contributors to this work. The author(s) read and approved the final manuscript.
Correspondence to Peter Eberhard.
Hahn, L., Eberhard, P. Transient dynamical-thermal-optical system modeling and simulation. J. Eur. Opt. Soc.-Rapid Publ. 17, 5 (2021). https://doi.org/10.1186/s41476-021-00150-1
Dn/dt
Mechanical stress
Dynamical-optical system
Thermo-optic
Elastic multibody system
Inhomogeneous media
EOS Annual Meeting (EOSAM) 2020 | CommonCrawl |
Corporate Finance & Accounting Financial Ratios
How to Calculate Return on Equity – ROE
By Ryan Fuhrmann
Return on equity (ROE) is a ratio that provides investors with insight into how efficiently a company (or more specifically, its management team) is handling the money that shareholders have contributed to it. In other words, it measures the profitability of a corporation in relation to stockholders' equity. The higher the ROE, the more efficient a company's management is at generating income and growth from its equity financing.
ROE is often used to compare a company to its competitors and the overall market. The formula is especially beneficial when comparing firms of the same industry since it tends to give accurate indications of which companies are operating with greater financial efficiency and for the evaluation of nearly any company with primarily tangible rather than intangible assets.
Return On Equity (ROE)
Calculating ROE
This is the basic formula for calculating ROE is:
ROE=Net IncomeShareholder EquityROE= \frac{\text{Net Income}}{\text{Shareholder Equity}}ROE=Shareholder EquityNet Income
The net income is the bottom-line profit—before common-stock dividends are paid—reported on a firm's income statement. Free cash flow (FCF) is another form of profitability and can be used instead of net income.
Shareholder equity is assets minus liabilities on a firm's balance sheet and is the accounting value that's left for shareholders should a company settle its liabilities with its reported assets.
Note that ROE is not to be confused with the return on total assets (ROTA). While it is also a profitability metric, ROTA is calculated by taking a company's earnings before interest and taxes (EBIT) and dividing it by the company's total assets.
ROE can also be calculated at different periods to compare its change in value over time. By comparing the change in ROE's growth rate from year to year or quarter to quarter, for example, investors can track changes in management's performance.
The ROE of the entire stock market as measured by the S&P 500 was 12.76% in the fourth quarter of 2019. A first, critical component of deciding how to invest involves comparing certain industrial sectors to the overall market.
For example, a look at ROE figures categorized by industry might show the stocks of the railroad sector performing very well compared to the market as a whole, with an ROE value of 25.8%, while the general utilities and retail sales sectors had of 8.63% and 11.52%, respectively. This could indicate that railroad companies have been a source of a steady growth industry and have provided excellent returns to investors.
The next step involves looking at individual companies to compare their ROEs with the market as a whole and with companies within their industry. For instance, at the end of FY 2019, Procter & Gamble (PG) reported a net income of $4 billion and total shareholders' equity of $47.6 billion. Thus, PG's ROE as of FY 2019 was:
$4 billion ÷ $47.6 billion = 8.4%
P&G's ROE was below the average ROE for the consumer goods sector of 11.31% at that time. In other words, for every dollar of shareholders' equity, P&G generated 8.4 cents in profit.
Not All ROEs Are the Same
Measuring a company's ROE performance against that of its sector can be more complicated than it seems, however.
For example, in the third quarter of 2019, Bank of America Corporation (BAC) had an ROE of 8.48%. According to the Federal Deposit Insurance Corporation (FDIC), the average ROE for the banking industry during the same period was 11.67%. In other words, Bank of America under-performed the industry.
However, the FDIC calculations deal with all banks, including commercial, consumer, and community banks. The ROE for commercial banks was 4.11% in the third quarter of 2019. Since Bank of America is, in part, a commercial lender, its ROE was above that of other commercial banks.
In short, it's not only important to compare the ROE of a company to the industry average but also to similar companies within that industry.
In evaluating companies, some investors use other measurements too, such as return on capital employed (ROCE) and return on operating capital (ROOC). Investors often use ROCE instead of the standard ROE when judging the longevity of a company. Generally speaking, both are more useful indicators for capital-intensive businesses, such as utilities or manufacturing.
CSIMarket. "Management Effectiveness Information & Trends - Total Market - 2019." Accessed March 2, 2020.
CSIMarket. "Management Effectiveness Information & Trends - Railroads Industry - 2019." Accessed March 2, 2020.
CSIMarket. "Management Effectiveness Information & Trends - Utilities Sector - 2019." Accessed March 2, 2020.
CSIMarket. "Management Effectiveness Information & Trends - Retail Sector - 2019." Accessed March 2, 2020.
Proctor & Gamble. "2019 Annual Report," Pages 36, 38. Accessed March 2, 2020.
CSIMarket. "Management Effectiveness Information & Trends - Consumer Non-cyclical - 2019." Accessed March 2, 2020.
Bank of America. "Bank of America Reports Quarterly Earnings of $5.8 Billion, EPS $0.56," Page 1. Accessed March 2, 2020.
Federal Deposit Insurance Corporation. "Quarterly Banking Profile - Third Quarter 2019," Page 5. Accessed March 2, 2020.
CSIMarket. "Management Effectiveness Information & Trends - Commercial Banks - 2019." Accessed March 2, 2020.
Return on Equity (ROE) vs. Return on Assets (ROA)
Looking Deeper Into Capital Allocation
Understanding Negative Return on Equity (ROE): Is It Always Bad?
How to Calculate Return on Assets (ROA) With Examples
How Do You Calculate Return on Equity (ROE) in Excel?
How Return on Equity Works
Return on equity (ROE) is a measure of financial performance calculated by dividing net income by shareholders' equity. Because shareholders' equity is equal to a company's assets minus its debt, ROE could be thought of as the return on net assets.
How to Use the DuPont Analysis to Assess a Company's ROE
The DuPont analysis is a framework for analyzing fundamental performance popularized by the DuPont Corporation. DuPont analysis is a useful technique used to decompose the different drivers of return on equity (ROE).
The Importance of Profitability Ratios
Profitability ratios are financial metrics used to assess a business's ability to generate profit relative to items such as its revenue or assets.
What Is an Equity Multiplier?
The equity multiplier is a calculation of how much of a company's assets is financed by stock rather than debt. For investors, it is a risk indicator.
In finance, a return is the profit or loss derived from investing or saving. | CommonCrawl |
\begin{definition}[Definition:Discontinuous Mapping/Topological Space/Point]
Let $T_1 = \left({A_1, \tau_1}\right)$ and $T_2 = \left({A_2, \tau_2}\right)$ be topological spaces.
Let $f: A_1 \to A_2$ $x \in T_1$ be a mapping from $A_1$ to $A_2$.
Then by definition $f$ is continuous at $x$ if for every neighborhood $N$ of $f \left({x}\right)$ there exists a neighborhood $M$ of $x$ such that $f \left({M}\right) \subseteq N$.
Therefore, $f$ is discontinuous at $x$ if for some neighbourhood $N$ of $f \left({x}\right)$ and every neighbourhood $M$ of $x$, $f \left({M}\right) \nsubseteq N$.
The point $x$ is called a '''discontinuity of $f$'''.
\end{definition} | ProofWiki |
NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes (Ex 13.4) Exercise 13.4
NCERT Solutions for Class 10 M...
Last updated date: 27th Jan 2023
Free PDF download of NCERT Solutions for Class 10 Maths Chapter 13 Exercise 13.4 (Ex 13.4) and all chapter exercises at one place prepared by an expert teacher as per NCERT (CBSE) books guidelines. Class 10 Maths Chapter 13 Surface Areas and Volumes Exercise 13.4 Questions with Solutions to help you to revise complete Syllabus and Score More marks. Register and get all exercise CBSE Solutions in your emails. You can Download NCERT Solutions Class 10 Maths to help you to revise the complete Syllabus and score more marks in your examinations. Subjects like Science, Maths, English will become easy to study if you have access to NCERT Solution Class 10 Science, Maths solutions, and solutions of other subjects that are available on Vedantu only.
Important Formulas for NCERT Solutions Maths Class 10 Exercise 13.4
The questions asked in Exercise 13.4 Class 10 NCERT Maths are related to the Frustum of a cone. There are a total of 5 questions asked in this exercise. To solve these questions, students must remember the formulae related to them. But before getting into the formulae, let us first understand what is a frustum of a cone?
When we cut or slice off the cone in such a way that the plane is parallel to its base, then we obtain two shapes, a smaller cone and another shape called a frustum of the cone.
Now, let us look at the formulae related to a frustum of a cone.
Volume of a frustum of a cone = 13 π h (r12 + r22 + r1 r2)
Curved surface area of a frustum of a cone = π (r1 + r2) l
Total Surface area of a frustum of a cone = π (r1 + r2) l + π r12 + π r22
where π = 227, h = height of the frustum of a cone. r1 and r2 are the radii of a frustum of a cone and l is the slant height of the frustum of the cone.
l can be calculated as l = (h2 + (r1 – r2)2)½ where (r1 > r2).
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Exercise 13.4
1. A drinking glass is in the shape of a frustum of a cone of height\[\mathbf{14}\text{ }\mathbf{cm}\]. The diameters of its two circular ends are \[\mathbf{4}\text{ }\mathbf{cm}\] and \[\mathbf{2}\text{ }\mathbf{cm}\]. Find the capacity of the glass. $\left[ Use\,\,\pi =\dfrac{22}{7} \right]$
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Radius (${{r}_{1}}$) of upper base of glass $=\dfrac{4}{2}=2cm$
Radius (${{r}_{2}}$) of lower base of glass$=\dfrac{2}{2}=1cm$
Height = $14cm$
Capacity of glass = Volume of frustum of cone
\[=\dfrac{1}{3}\pi\text{h}\left[ \text{r}_{1}^{2}+\text{r}_{2}^{2}+{{\text{r}}_{1}}{{\text{r}}_{2}} \right]\]
\[=\dfrac{1}{3}\pi \text{h}\left[ {{(2)}^{2}}+{{(1)}^{2}}+(2)(1) \right]\]
\[=\dfrac{1}{3}\times \dfrac{22}{7}\times 14(4+1+2)\]
\[=\dfrac{308}{3}\]
\[=102\dfrac{2}{3}~\text{c}{{\text{m}}^{3}}\]
Therefore, the capacity of the glass \[=102\dfrac{2}{3}~\text{c}{{\text{m}}^{3}}\]
2. The slant height of a frustum of a cone is \[\mathbf{4}\text{ }\mathbf{cm}\] and the perimeters (circumference) of its circular ends are \[\mathbf{18}\text{ }\mathbf{cm}\] and \[\mathbf{6}\text{ }\mathbf{cm}\]. Find the curved surface area of the frustum.
Perimeter of upper circular end of frustum $=18 cm$
\[\Rightarrow 2\pi {{r}_{1}}\text{ }=18\]
\[\Rightarrow {{r}_{1}}=\dfrac{9}{\pi }\]
Perimeter of lower circular end of frustum $=6 cm$
\[\Rightarrow 2\pi {{r}_{2}}\text{ }=6\]
Slant height (l) of frustum $=4 cm$
CSA of frustum:
$\Rightarrow \pi \left( {{r}_{1}}+{{r}_{2}} \right)l$
$\Rightarrow \pi \left( \dfrac{9}{\pi }+\dfrac{3}{\pi } \right)4$
$\Rightarrow 12\left( 4 \right)=48c{{m}^{2}}$
Therefore, the curved surface area of the frustum$=48c{{m}^{2}}$
3. A fez, the cap used by the Turks, is shaped like the frustum of a cone (see the figure given below). If its radius on the open side is\[\mathbf{10}\text{ }\mathbf{cm}\], radius at the upper base is \[\mathbf{4}\text{ }\mathbf{cm}\] and its slant height is\[\mathbf{15}\text{ }\mathbf{cm}\], find the area of material used for making it. $\left[ Use\,\,\pi =\dfrac{22}{7} \right]$
Radius (${{r}_{2}}$) at upper circular end \[=\text{ }4\text{ }cm\]
Radius (${{r}_{1}}$) at lower circular end \[=10\text{ }cm\]
Slant height (l) of frustum \[=15\text{ }cm\]
Area of material used for making the fez:
$\Rightarrow$ CSA of frustum+Area of upper circular end
$\Rightarrow \pi \left( {{r}_{1}}+{{r}_{2}} \right)l+\pi r_{2}^{2}$
$\Rightarrow \pi (10+4)15+\pi (4)^2$
$\Rightarrow \pi (14)15+16\pi $
$\Rightarrow 210\pi +16\pi =\dfrac{226\times 22}{7}$
$\Rightarrow 710\dfrac{2}{7}~\text{c}{{\text{m}}^{2}}$
Therefore, the area of material used for making it is $710\dfrac{2}{7}~\text{c}{{\text{m}}^{2}}$.
4. A container, opened from the top and made up of a metal sheet, is in the form of a frustum of a cone of height \[16\text{ }cm\] with radii of its lower and upper ends as \[\mathbf{8}\text{ }\mathbf{cm}\] and \[\mathbf{20}\text{ }\mathbf{cm}\] respectively. Find the cost of the milk which can completely fill the container, at the rate of \[\mathbf{Rs}.\mathbf{20}\]per liter. Also find the cost of metal sheet used to make the container, if it costs\[\mathbf{Rs}.\mathbf{8}\text{ }\mathbf{per}\text{ }\mathbf{100}\text{ }\mathbf{c}{{\mathbf{m}}^{2}}\]. Take \[\mathbf{\pi }\text{ }=\text{ }\mathbf{3}.\mathbf{14}\]
Radius (\[{{r}_{1}}\]) of upper end of container \[=20\text{ }cm\]
Radius (\[{{r}_{2}}\]) of lower end of container \[=8\text{ }cm\]
Height (h) of container \[=16\text{ }cm\]
Slant height (l) of frustum $=\sqrt{{{\left( {{r}_{1}}-{{r}_{2}} \right)}^{2}}+{{h}^{2}}}$
$=\sqrt{{{(20-8)}^{2}}+{{(16)}^{2}}}$
$=\sqrt{{{(12)}^{2}}+{{(16)}^{2}}}=\sqrt{144+256}$
$=20~\text{cm}$
Capacity of container $=$ Volume of frustum
$=\dfrac{1}{3}\pi \text{h}\left[ \text{r}_{1}^{2}+\text{r}_{2}^{2}+{{\text{r}}_{1}}{{\text{r}}_{2}} \right]$
$=\dfrac{1}{3}\times 3.14\times 16\times \left[ {{(20)}^{2}}+{{(8)}^{2}}+(20)(8) \right]$
$=\dfrac{1}{3}\times 3.14\times 16(400+64+160)$
$=\dfrac{1}{3}\times 3.14\times 16\times 624$
$=10449.92~\text{c}{{\text{m}}^{3}}$
$=10.45$ litres
Cost of 1 litre milk$=\operatorname{Rs}20$
Cost of \[10.45\] litre milk $=\operatorname{Rs} 209$
Area of metal sheet used to make the container $=\pi \left( {{\text{r}}_{1}}+{{\text{r}}_{2}} \right)\text{l}+\pi {{\text{r}}_{2}}^{2}$
$ =\pi (20+8)20+\pi {{(8)}^{2}} $
$=560\pi +64\pi =624\pi \text{c}{{\text{m}}^{2}}$
Cost of $100~\text{c}{{\text{m}}^{2}}$ metal sheet$=Rs\,8$
Cost of $624\pi \text{c}{{\text{m}}^{2}}$ metal sheet $= \dfrac{624\times 3.14\times 8}{100}=\operatorname{Rs}156.75$
Therefore, the cost of the milk which can completely fill the container is \[Rs\text{ }209\] and the cost of metal sheet used to make the container is \[Rs156.75\]
5. A metallic right circular cone \[\mathbf{20}\text{ }\mathbf{cm}\] high and whose vertical angle is \[\mathbf{60}{}^\circ \] is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained is drawn into a wire of diameter$\dfrac{1}{16}cm$, find the length of the wire. $\left[ Use\,\,\pi =\dfrac{22}{7} \right]$
In $\vartriangle \text{AEG},$
$\dfrac{\text{EG}}{\text{AG}}=\tan {{30}^{{}^\circ }}$
$\text{EG}=\dfrac{10}{\sqrt{3}}~\text{cm}=\dfrac{10\sqrt{3}}{3}$
In \[\vartriangle \text{ABD},\]
\[\dfrac{\text{BD}}{\text{AD}}=\tan {{30}^{{}^\circ }}\]
\[\text{BD}=\dfrac{20}{\sqrt{3}}=\dfrac{20\sqrt{3}}{3}~\text{cm}\]
Radius $\left( {{r}_{1}} \right)$ of upper end of frustum
$=\dfrac{10\sqrt{3}}{3}~\text{cm}$
Radius $\left( {{r}_{2}} \right)$ of lower end = $\dfrac{20\sqrt{3}}{3}$cm
Height $(h)$ $=10~\text{cm}$
Volume of frustum:
$\Rightarrow\dfrac{1}{3}\pi\text{h}\left[\text{r}_{1}^{2}+\text{r}_{2}^{2}+{{\text{r}}_{1}}{{\text{r}}_{2}} \right]$
$\Rightarrow\left.\dfrac{1}{3}\times\pi\times10[{{\left(\dfrac{10\sqrt{3}}{3}\right)}^{2}}+{{\left(\dfrac{20\sqrt{3}}{3}\right)}^{2}}+\dfrac{(10\sqrt{3})(20\sqrt{3})}{3\times 3} \right]$
$\Rightarrow \dfrac{10}{3}\pi \left[ \dfrac{100}{3}+\dfrac{400}{3}+\dfrac{200}{3} \right]$
$\Rightarrow \dfrac{10}{3}\times \dfrac{22}{7}\times \dfrac{700}{3}$
$\Rightarrow \dfrac{22000}{9}~\text{c}{{\text{m}}^{3}}$
Radius \[(r)\] of wire$=\dfrac{1}{16}\times \dfrac{1}{2}=\dfrac{1}{32}~\text{cm}$
Let the length of wire be $l$
Volume of wire $=Area\,\,of\,cross\,section\times Length$
$=\left( \pi {{r}^{2}} \right)\times (l)$
$=\pi \times {{\left( \dfrac{1}{32} \right)}^{2}}\times l$
Volume of frustum = Volume of wire
$\Rightarrow \dfrac{22000}{9}=\dfrac{22}{7}{{\left( \dfrac{1}{32} \right)}^{2}}l$
$\Rightarrow \dfrac{7000}{9}\left( 1024 \right)=l$
\[\Rightarrow l=796444.44~\text{cm}\]
\[\Rightarrow l=7964.44~\text{m}\]
NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes
Opting for the NCERT solutions for Ex 13.4 Class 10 Maths is considered as the best option for the CBSE students when it comes to exam preparation. This chapter consists of many exercises. Out of which we have provided the Exercise 13.4 Class 10 Maths NCERT solutions on this page in PDF format. You can download this solution as per your convenience or you can study it directly from our website/ app online.
Vedantu in-house subject matter experts have solved the problems/ questions from the exercise with the utmost care and by following all the guidelines by CBSE. Class 10 students who are thorough with all the concepts from the Subject Surface Areas and Volumes textbook and quite well-versed with all the problems from the exercises given in it, then any student can easily score the highest possible marks in the final exam. With the help of this Class 10 Maths Chapter 13 Exercise 13.4 solutions, students can easily understand the pattern of questions that can be asked in the exam from this chapter and also learn the marks weightage of the chapter. So that they can prepare themselves accordingly for the final exam.
Besides these NCERT solutions for Class 10 Maths Chapter 13 Exercise 13.4, there are plenty of exercises in this chapter which contain innumerable questions as well. All these questions are solved/answered by our in-house subject experts as mentioned earlier. Hence all of these are bound to be of superior quality and anyone can refer to these during the time of exam preparation. In order to score the best possible marks in the class, it is really important to understand all the concepts of the textbooks and solve the problems from the exercises given next to it.
Do not delay any more. Download the NCERT solutions for Class 10 Maths Chapter 13 Exercise 13.4 from Vedantu website now for better exam preparation. If you have the Vedantu app in your phone, you can download the same through the app as well. The best part of these solutions is these can be accessed both online and offline as well.
NCERT Solutions for Class 10 Maths Chapter 13 Other Exercises
Chapter 13 Surface Areas and Volumes All Exercises in PDF Format
9 Questions and Solutions
FAQs on NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes (Ex 13.4) Exercise 13.4
1. How many questions are there in the NCERT Class 10 Maths Chapter 13 Exercise 13.4?
There are five questions in total in the NCERT Class 10 Maths Chapter 13 Exercise 13.4. Answers to these questions are provided in the readymade NCERT Solutions by Vedantu, which can be found on Vedantu website and mobile application.
2. How does Vedantu's NCERT Solutions Class 10 Maths Chapter 13 Exercise 13.4 help to score well in the exam?
Vedantu's NCERT Solutions Class 10 Maths Chapter 13 Exercise 13.4 are prepared by the highly qualified subject matter experts. Hence these NCERT solutions are considered as the best aid for the students. By referring to these NCERT Solutoons, students of CLss 10 will be well versed with all the concepts thoroughly. Thus, every student can score the highest possible marks in the exam.
3. How can you make your Class 10 Maths preparation easier?
You can always use the NCERT Solutions created by Vedantu's experts to make your Class 10th exam preparation easier.
4. Can I download these NCERT Solutions by Vedantu at free of cost?
Yes, you can download these NCERT Solutions by Vedantu at free of cost. You need to visit the official website of Vedantu and register yourself by enlisting your email address and phone number. Then you will be able to download not only the NCERT Solutions but all the study materials for absolutely free of cost.
5. How many topics does Chapter 13 of Class 10 Maths consist of?
Chapter 13 "Surface Areas And Volumes" of Class 10 Maths consists of the following listed topics:
Surface Area of a Combination of Solids
The volume of a Combination of Solids
Conversion of Solid from One Shape to Another
Frustum of a Cone
The points mentioned above are very important from an examination point of view. To score good marks in Chapter 13 of Class 10 Maths, students must prepare these topics thoroughly through the NCERT book.
6. What points should I obey to achieve great marks in Chapter 13 Exercise 13.4 of Class 10 Maths?
Exercise 13.4 of Chapter 13 of Class 10 Maths is based on the topic "Frustum of a Cone" which is a little bit difficult. The given steps will help you in preparing this exercise:
The first step is to understand the theory related to the topic.
Learn all the formulas used in this topic.
Solve all the examples.
After solving examples, try to solve the questions given in the exercise.
Ask the question from your teacher whose answer you are not able to find.
You can get the solutions of this exercise on the page NCERT Solutions for Class 10 Maths Chapter 13.
7. Explain why the NCERT books help in studying Chapter 13 Exercise 13.4 of Class 10 Maths?
Underneath are justifications why students should use the NCERT books to study Chapter 13 Exercise 13.4 of Class 10 Maths:
The CBSE has prescribed the Maths NCERT book for students of Class 10.
Through this book, the students will get thorough knowledge about Chapter 13 Exercise 13.4 of Class 10 Maths.
There are many questions available in this book.
In this book, the content is created in a very simple dialect so that students can comprehend.
You will discover that most of the problems asked in the exam are directly taken from the NCERT book.
8. Can I build a successful study plan for Exercise 13.4 Chapter 13 of Class 10 Maths?
Yes, it is easy to invent a triumphant study plan for Chapter 13 Exercise 13.4 of Class 10 Maths. The steps discussed below will help students for the same:
Have a timetable with balanced activities.
Analyze Maths subject for at least two hours.
Sample papers and mock tests will help you to understand the exercise in a better way.
Work out on the questions given in the guidebooks.
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\begin{document}
\title{An Analysis of the Effect of Ghost Force Oscillation on Quasicontinuum Error}
\author{Matthew Dobson} \author{Mitchell Luskin}
\address{Matthew Dobson\\ School of Mathematics \\ University of Minnesota \\ 206 Church Street SE \\ Minneapolis, MN 55455 \\ U.S.A.} \email{[email protected]}
\address{Mitchell Luskin \\ School of Mathematics \\ University of Minnesota \\ 206 Church Street SE \\ Minneapolis, MN 55455 \\ U.S.A.} \email{[email protected]}
\thanks{ This work was supported in part by DMS-0757355,
DMS-0811039, the Institute for Mathematics and Its Applications,
the University of Minnesota Supercomputing Institute, and the University of Minnesota Doctoral Dissertation Fellowship.
This work is also based on work supported by the Department of Energy under Award Number DE-FG02-05ER25706. }
\keywords{}
\subjclass[2000]{65Z05,70C20}
\date{\today}
\begin{abstract} The atomistic to continuum interface for quasicontinuum energies exhibits nonzero forces under uniform strain that have been called ghost forces. In this paper, we prove for a linearization of a one-dimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on the displacement nearly cancels and has a small effect on the error away from the interface. We give optimal order error estimates that show that the quasicontinuum displacement converges to the atomistic displacement at the rate O($h$) in the discrete $\ell^\infty$ and
$w^{1,1}$ norms where $h$ is the interatomic spacing. We also give a proof that the error in the displacement gradient decays away from the interface to O($h$) at distance O($h|\log h|$) in the atomistic region and distance O($h$) in the continuum region. E, Ming, and Yang previously gave a counterexample to convergence in the $w^{1,\infty}$ norm for a harmonic interatomic potential. Our work gives an explicit and simplified form for the decay of the effect of the atomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete $\ell^\infty$ and $w^{1,p}$ norms. \end{abstract}
\maketitle { \thispagestyle{empty}
\section{Introduction}
The quasicontinuum method (QC) reduces the computational complexity of atomistic simulations by replacing smoothly varying regions of the material with a continuum approximation derived from the atomistic model~\cite{pinglin03, pinglin05, legollqc05, ortnersuli, OrtnerSueli:2006d, e05, tadm96, knaportiz, e06, mill02, rodney_gf, miller_indent, curtin_miller_coupling, jacobsen04,dobs08}. This is extremely effective in simulations involving defects, which have singularities in the deformation gradient. In such simulations, a few localized regions require the accuracy and high computational expense of atomistic scale resolution, but the rest of the material has a slowly varying deformation gradient which can be more efficiently computed using the continuum approximation without loss of the desired accuracy. Adaptive algorithms have been developed for QC to determine which regions require the accuracy of atomistic modeling and how to coarsen the finite element mesh in the continuum region~\cite{ortnersuli,OrtnerSueli:2006d,prud06,oden06,arndtluskin07a,arndtluskin07b,ArndtLuskin:2007c}. The atoms retained in the atomistic region and the atoms at nodes of the piecewise linear finite element mesh in the continuum region are collectively denoted as {\em representative atoms}.
Recent years have seen the development of many QC approximations that differ in how they compute interactions among the representative atoms. In the following, we concern ourselves with the original energy-based quasicontinuum (QCE) approximation~\cite{tadm96,mill02}, but the phenomena that we analyze occur in all other quasicontinuum approximations, as well as in other multiphysics coupling methods~\cite{e06}. In QCE, a total energy is created by summing energy contributions from each representative atom in the atomistic region and from each element in the continuum region, where the volume of the elements in the atomistic to continuum interface is modified to exactly conserve mass. This construction was chosen so that for any uniform strain the QCE energy, the continuum energy, and the atomistic energy are identical. (As discussed later, this conservation property for the QCE approximation is not sufficient to prevent the existence of nonzero forces at the atomistic to continuum interface for uniform strain.) The representative atoms then interact via forces defined by the total energy. This makes for a simple and versatile method that can treat complicated geometries and can be used with adaptive algorithms that modify the mesh and atomistic regions during a quasi-static process. Other atomistic to continuum approaches have been proposed, for example, that utilize overlapping or blended domains~\cite{BadiaParksBochevGunzburgerLehoucq:2007,ParksBochevLehoucq:2007}.
One drawback of the energy-based quasicontinuum approximation that has received much attention is the fact that at the atomistic to continuum
interface the balance of force equations do not give a consistent scheme~\cite{shenoy_gf}. As explained in Section~\ref{sec:model}, the equilibrium equations in the interior of both the atomistic region and the continuum region give consistent finite difference schemes for the continuum limit, whereas the QC equilibrium equations near the interface are not consistent with the continuum limit. This is most easily seen by considering a uniform strain, which will be assigned identically zero elastic forces by any consistent scheme. (Ensuring that a given scheme assigns zero forces for uniform strain has been known as the ``patch test'' in the theory of finite elements~\cite{strangfix}.) The nonzero residual forces present in QCE for uniform strain have been called ``ghost forces''~\cite{shenoy_gf,dobs08}.
In this paper, we give optimal order error estimates for the effect of the inconsistency on the displacement and displacement gradient for a linearization of a one-dimensional atomistic energy and its quasicontinuum approximation. We consider the linearization of general interatomic potentials which are concave near second-neighbor interatomic distances. This property guarantees that the interfacial error due to the Cauchy-Born approximation with a second-neighbor cut-off is positive~\cite[p. 117]{dobs08} and that the quasicontinuum error is not oscillatory in the atomistic region (see Section~\ref{sec:comp}). Similar optimal order error estimates have been given by E, Ming, and Yang~\cite{emingyang} for a harmonic interatomic potential.
We begin by linearizing a one-dimensional atomistic energy, its local quasicontinuum approximation (which we will call the continuum energy), and its quasicontinuum approximation about a uniform strain for a second-neighbor atomistic energy.
We will show in Section~\ref{sec:model} that the three systems of equilibrium equations are then \begin{equation*} \begin{split} L^{a,h} {\mathbf u}_a &= {\mathbf f},\qquad \qquad \text{(atomistic)}\\ L^{c,h} {\mathbf u}_c &= {\mathbf f},\qquad \qquad \text{(continuum)}\\ L^{qc,h} {\mathbf u}_{qc}-{\mathbf g} &= {\mathbf f}, \qquad \qquad \text{(quasicontinuum)} \end{split} \end{equation*} where ${\mathbf f}$ is an external loading, $L$ and ${\mathbf u}$ are the linearized operator and corresponding displacement for each scheme, ${\mathbf g}$ is non-zero only in the atomistic to continuum interface, and $h$ is the interatomic spacing. The term ${\mathbf g}$ in the quasicontinuum equilibrium equations is due to the unbalanced second-neighbor interactions in the interface~\eqref{ghost} and for uniform stretches is precisely the ghost force described in~\cite{shenoy_gf,dobs08,mill02}.
Formally, the error decomposes as \begin{equation*} {\mathbf u}_a - {\mathbf u}_{qc} = ( (L^{a,h})^{-1} - (L^{qc,h})^{-1}) {\mathbf f} - (L^{qc,h})^{-1} {\mathbf g}. \end{equation*} (The operators are all translation invariant, so they only have solutions up to the choice of an additive constant.) In this paper, we focus on the second term, $(L^{qc})^{-1} {\mathbf g},$ which is the error due to the inconsistency at the interface. To do so, we consider the case of no external field, ${\mathbf f}={\mathbf 0},$ which will make ${\mathbf u}_a = {\mathbf 0}.$ For most applications of the quasicontinuum method, the only external field is due to loads that are applied on the boundary of the material, far from the atomistic to continuum interface.
We showed in~\cite{dobs08} that the ghost forces are oscillatory and sum to zero.
In this paper, we prove that the error in the displacement gradient is O(1) at the interface and decays away from the interface to O($h$) at distance O($h|\log h|$) in the atomistic region and distance O($h$) in the continuum region. As noted above, similar results have been given in~\cite{emingyang} for a harmonic interatomic potential with ${\mathbf f}\ne {\mathbf 0}$ and Dirichlet boundary conditions. Here, we present a simplified approach starting from a linearization of a quasicontinuum approximation with a concave second-neighbor interaction. We explicitly give the form of the solution and analyze the solution in discrete $l^\infty$ and $w^{1,p}$ norms. We show that the quasicontinuum displacement converges to the atomistic displacement at the rate O($h$) in the discrete $l^\infty$ and $w^{1,1}$ norms where $h$ is the interatomic spacing.
In Section~\ref{sec:model}, we describe the energy-based quasicontinuum approximation (QCE) and set up the analysis.
In Section~\ref{sec:comp}, we prove Theorem~\ref{thm2} for the quasicontinuum energy that gives an optimal order, O($h$) error estimate in the $l^\infty$ norm and a O($h^{1/p}$) error estimate in the $w^{1,p}$ norm for $1 \leq p < \infty.$ Note that for simplicity the models and analysis are presented for the case where no degrees of freedom have been removed in the continuum region, but we explain in Remark~\ref{coarsen} that identical results hold when the continuum region is coarsened. We present numerical computations in Figure~\ref{qc_comp} that clearly show that the error is localized in the atomistic to continuum interface.
\section{One-Dimensional, Linear Quasicontinuum Approximation} \label{sec:model} We consider an infinite one-dimensional chain of atoms with periodicity $2F$ in the deformed configuration. Let $y_j$ denote the atomic positions for $-\infty<j<\infty,$ where there are $2N$ atoms in each period. Let $h = 1/N$ and let \begin{equation*} u_j := y_j - F h j \end{equation*} denote the displacement from the average interatomic spacing, $F h.$ In the following, we analyze the behavior of the quasicontinuum method as the atomistic chain approaches the continuum limit with $F$ fixed and $N \rightarrow \infty.$
The atomistic energy for a period of the chain is \begin{equation}\label{en} {\mathcal E}^{tot,h}({\mathbf y}) := h \sum_{j=-N+1}^{N} \left[\phi\left(\frac{y_{j+1} - y_{j}}{h} \right) + \phi\left(\frac{y_{j+2} - y_{j}}{h}\right) - f_j y_{j}\right], \end{equation} where $\phi(r)$ is a two-body interatomic potential (for example, the Lennard-Jones potential ${\phi}(r) = 1/ r^{12} - 2/r^6$) and ${\mathbf f} = (f_{-N+1},\dots,f_{N})$ are external forces applied as dead loads on the atoms. The periodic conditions \begin{equation*} y_{j+2N} = y_j + 2 F \qquad\text{or}\qquad u_{j+2N} = u_j \end{equation*} allow ${\mathcal E}^{tot, h}$ to be written in terms of ${\mathbf y} := (y_{-N+1},\dots,y_{N}).$ We assume that $\sum_{-N+1}^N f_j=0,$ otherwise there are no energy minimizing solutions since the elastic energy is translation invariant. In the following, we discuss the existence and uniqueness of solutions to each of the models we encounter. We note that the energy per bond in~\eqref{en} has been scaled like $h\phi(r/h).$ This scaling implies that if we let $y_j = y(j/N)$ and $f_j = f(j/N)$ for $j=-N+1, \dots, N$ where $y \in C^1([-1,1])$ and $f \in C([-1,1]),$ then as $N\to \infty$ and $F$ is held fixed, the energy of a period~\eqref{en} converges to \begin{equation*} \int^1_{-1} \phi(y'(x)) + \phi(2 y'(x)) - f(x) y(x) \, \mathrm{d} x. \end{equation*}
We expand first neighbor terms around $F,$ giving \begin{equation*} \begin{split} \phi \left(\frac{y_{j+1} - y_{j}}{h}\right) &= \phi \left(F + \frac{u_{j+1} - u_{j}}{h}\right) \\ &= \phi(F) + \phi'(F) \left.\frac{u_{j+1} - u_{j}}{h}\right. + \half \phi''(F) \left(\frac{u_{j+1} - u_{j}}{h}\right)^2
+O\left(\left|\frac{u_{j+1}-u_{j}}{h}\right|^3\right), \end{split} \end{equation*} and the second neighbor terms around $2F,$ giving \begin{equation*} \begin{split} \phi \left(\frac{y_{j+2} - y_{j}}{h}\right) &= \phi \left(2F + \frac{u_{j+2} - u_{j}}{h}\right) \\ &= \phi(2 F) + \phi'(2 F) \left.\frac{u_{j+2} - u_{j}}{h}\right. + \half \phi''(2 F) \left(\frac{u_{j+2} - u_{j}}{h}\right)^2
+O\left(\left|\frac{u_{j+2}-u_{j}}{h}\right|^3\right). \end{split} \end{equation*}
\subsection{Atomistic Model} The linearized atomistic energy is then given by
\begin{equation}\label{at} \begin{split} {\mathcal E}^{a,h}({\mathbf u}) &:= h \sum_{j=-N+1}^{N} \left[ \phi'_F \left.\frac{u_{j+1} - u_{j}}{h}\right.
+ \half \phi''_{F} \left(\frac{u_{j+1} - u_{j}}{h}\right)^2 \right. \\ &\qquad\qquad \left. + \phi'_{2F} \left.\frac{u_{j+2} - u_{j}}{h}\right.
+ \half \phi''_{2 F} \left(\frac{u_{j+2} - u_{j}}{h}\right)^2
- f_j u_j\right],
\end{split} \end{equation} where $\phi'_F := \phi'(F), \phi''_{F} := \phi''(F), \phi'_{2F} := \phi'(2F), \phi''_{2 F} := \phi''(2F),$ and ${\mathbf u} := (u_{-N+1}, \dots, u_{N}).$ Note that here and in the following, we neglect the additive constant $\phi(F) + \phi(2F) - h \sum_{j=-N+1}^N f_j F h j$ in the linearized energy. We assume that $\phi \in C^2([r_0, \infty))$ for some $r_0$ such that $0< r_0 < F,$ and \begin{equation} \label{assume} \phi''_F > 0 \text{ and } \phi''_{2F} < 0. \end{equation} This holds true for the Lennard-Jones potential for $F h$ below the load limit, unless the chain is extremely compressed (less than 60\% of the equilibrium length).
The property $\phi''_{2F} <0$ ensures that the quasicontinuum error is not oscillatory in the atomistic region (see Section~\ref{sec:comp}).
We furthermore assume that \begin{equation} \label{assume2} \phi''_F + 5 \phi''_{2F} > 0, \end{equation} which will be sufficient to give solutions to the QC equilibrium equations under the assumption of no resultant external forces (see Lemma~\ref{wellposed}). In contrast, the weaker assumption $\phi''_F + 4 \phi''_{2F} >0$ is sufficient for the fully atomistic or fully continuum approximation. The equilibrium equations, $\frac1h\pd{{\mathcal E}^{a,h}}{u_j}({\mathbf u}) = 0,$ for the atomistic model~\eqref{at} are \begin{equation}\label{atom} \begin{split} (L^{a,h} {\mathbf u})_j
&= \frac{- \phi''_{2 F} u_{j+2} - \phi''_{F} u_{j+1} + 2 (\phi''_{F} + \phi''_{2 F}) u_{j} - \phi''_{F} u_{j-1} - \phi''_{2 F} u_{j-2}}{h^2} = f_j, \\ &\hspace{1.9in} u_{j+2N} = u_j, \end{split} \end{equation} for $-\infty < j < \infty.$ Note that scaling by $\frac{1}{h}$ makes this a consistent approximation of the boundary value problem \begin{equation} \label{bvp} \begin{split} - (\phi''_{F}+4\phi''_{2 F}) u''(x) = f &\qquad \textrm{ for } -\infty<x <\infty,\\
u(x+2) = u(x) &\qquad \textrm{ for } -\infty<x <\infty. \end{split} \end{equation} The linearized atomistic energy~\eqref{at} has a unique minimum (up to a constant) if $\phi''_{F} + 4 \phi''_{2 F}>0,$ provided that $\sum_{j=N-1}^{N} f_j = 0.$ Standard ODE results show that~\eqref{bvp} has a unique solution (up to a constant) provided that $\int^1_{-1} f(x) \, \mathrm{d} x = 0 .$
\begin{remark} \label{rem1} For the atomistic energy~\eqref{at}, the linear terms sum to zero by the periodicity of the displacement, since \begin{equation*} \begin{split}
h \sum_{j=-N+1}^{N} &\left[ \phi'_F \ \frac{u_{j+1} - u_{j}}{h}
+ \phi'_{2F} \ \frac{u_{j+2} - u_{j}}{h} \right] \\ &= \phi'_F \left[u_{N+1} - u_{-N+1}\right] + \phi'_{2F} \left[u_{N+2} + u_{N+1} - u_{-N+2} - u_{-N+1}\right] = 0. \end{split} \end{equation*} However, we keep these terms in the model since they do not sum to zero when the atomistic model is coupled to the continuum approximation in the quasicontinuum energy. The resulting terms give a more accurate representation of what happens in the non-linear quasicontinuum model. \end{remark}
\subsection{Continuum Approximation} The continuum approximation splits the chain into linear finite elements with nodes given by the representative atoms, which we recall are a subset of the atoms in the chain. The energy of the chain is the sum of element energies which depend only on the element's deformation gradient, the linear deformation that interpolates its nodal positions. The energy of an element is then computed by applying the element's deformation gradient to the reference lattice, computing the energy per atom using the atomistic model, and multiplying by the number of atoms in the element (where the boundary atoms are shared equally between neighboring elements). If the continuum approximation is not coarsened (every atom is a representative atom), then the continuum energy is given by \begin{equation}\label{cont} \begin{split} {\mathcal E}^{c,h}({\mathbf u}) &:= h \sum_{j=-N+1}^{N} \left[ (\phi'_F + 2 \phi'_{2F}) \left(\frac{u_{j+1} - u_{j}}{h}\right) + \half (\phi''_F + 4 \phi''_{2F}) \left(\frac{u_{j+1} - u_{j}}{h}\right)^2 - f_j u_j\right].
\end{split} \end{equation} See~\cite{dobs08} for a derivation of the continuum energy and a discussion of the error terms at the element boundaries. For $j \in \{-N+1, \dots, N\}$, the equilibrium equations for the continuum approximation are \begin{equation}\label{cont1} (L^{c,h} {\mathbf u})_j = (\phi''_F + 4 \phi''_{2F}) \left[\frac{- u_{j+1} + 2 u_{j} - u_{j-1}}{h^2} \right] = f_j,
\end{equation} which is also a consistent approximation for the boundary value problem~\eqref{bvp}. It is easy to see that the continuum energy~\eqref{cont} has a unique minimum (up to a constant) if $\phi''_{F} + 4 \phi''_{2 F}>0,$ provided that $\sum_{j=N-1}^{N} f_j = 0.$
The quasicontinuum method inherently supports coarsening, but we neglect it here since in one dimension this only changes the scaling of equilibrium equations.
\subsection{Splitting the Energy} We can split the atomistic energy and the continuum energy into per-atom contributions so that \begin{equation*}
{\mathcal E}^{a,h}({\mathbf u}) = h \sum_{j=-N+1}^{N} \left[{\mathcal E}^{a,h}_j\left({\mathbf u}\right) - f_j u_j\right] \quad \text{ and } \quad {\mathcal E}^{c,h}({\mathbf u}) = h \sum_{j=-N+1}^{N} \left[{\mathcal E}^{c,h}_j\left({\mathbf u}\right) - f_j u_j\right]. \end{equation*} There are many possible ways to define the per-atom contributions, and we do this in such a way that these contributions are linearizations of the ones in the fully nonlinear case presented in~\cite{dobs08,tadm96}. In this case, we split the energy of each bond to obtain \begin{equation} \label{atomsplit} \begin{split} {\mathcal E}^{a,h}_j({\mathbf u}) := \frac{1}{2} \Bigg[ \phi'_F &\left.\frac{u_{j+1} - u_{j}}{h}\right.
+ \half \phi''_{F} \left(\frac{u_{j+1} - u_{j}}{h}\right)^2 \\ &+ \phi'_{2F} \left.\frac{u_{j+2} - u_{j}}{h}\right.
+ \half \phi''_{2 F} \left(\frac{u_{j+2} - u_{j}}{h}\right)^2 \Bigg]\\ +\frac{1}{2} \Bigg[ \phi'_F &\left.\frac{u_{j} - u_{j-1}}{h}\right.
+ \half \phi''_{F} \left(\frac{u_{j} - u_{j-1}}{h}\right)^2 \\ &+ \phi'_{2F} \left.\frac{u_{j} - u_{j-2}}{h}\right.
+ \half \phi''_{2 F} \left(\frac{u_{j} - u_{j-2}}{h}\right)^2 \Bigg], \end{split} \end{equation} and \begin{equation} \label{contsplit} \begin{split} {\mathcal E}^{c,h}_j({\mathbf u}) := \frac{1}{2} &\left[ (\phi'_F + 2 \phi'_{2F}) \left(\frac{u_{j+1} - u_{j}}{h}\right) + \half (\phi''_F + 4 \phi''_{2F}) \left(\frac{u_{j+1} - u_{j}}{h}\right)^2 \right]\\ &+ \frac{1}{2} \left[ (\phi'_F + 2 \phi'_{2F}) \left(\frac{u_{j} - u_{j-1}}{h}\right) + \half (\phi''_F + 4 \phi''_{2F}) \left(\frac{u_{j} - u_{j-1}}{h}\right)^2 \right]. \end{split} \end{equation}
\subsection{Energy-Based Quasicontinuum Approximation} The energy-based quasicontinuum approximation partitions the representative atoms into atomistic and continuum representative atoms and assigns to each atom the split energy corresponding to its type (\ref{atomsplit}-\ref{contsplit}). We define the nodes $-N+1,\dots,-K-1$ and $K+1,\dots,N$ to be continuum and $-K,\dots,K$ to be atomistic, where we assume that $2\le K\le N-2$ to ensure well-defined atomistic and continuum regions. The quasicontinuum energy is then \begin{equation} \label{qceTot} {\mathcal E}^{qc,h}({\mathbf u}) := \sum_{j = -N+1}^{-K-1} {\mathcal E}_{j}^{c,h}\left({\mathbf u}\right) + \sum_{j=-K}^{K} {\mathcal E}_{j}^{a,h}\left({\mathbf u}\right) + \sum_{j= K+1}^{N} {\mathcal E}_{j}^{c,h}\left({\mathbf u}\right) - \sum_{j = -N+1}^{N} f_j u_j. \end{equation}
Since the energy is quadratic, the equilibrium equations, $\frac1h\pd{{\mathcal E}^{qc,h}}{u_j}({\mathbf u}_{qc})=0,$ take the form \begin{equation}\label{qc1} L^{qc,h} {\mathbf u}_{qc}-{\mathbf g} = {\mathbf f}. \end{equation}
For $0 \leq j \leq N,$ the QCE operator is given by \begin{equation*} \label{fishtail} \begin{split} (L^{qc,h} {\mathbf u})_j &= \phi''_F \frac{-u_{j+1} +2 u_j - u_{j-1}}{h^2} \\ &+ \begin{cases} \displaystyle
4 \phi''_{2 F} \frac{-u_{j+2} +2 u_j - u_{j-2}}{4 h^2}, & 0 \leq j \leq K-2, \\[6pt]
\displaystyle
4 \phi''_{2 F} \frac{-u_{j+2} +2 u_j - u_{j-2}}{4 h^2} + \frac{\phi''_{2 F}}{h} \frac{u_{j+2} - u_{j}}{2 h}, & j = K-1, \\[6pt]
\displaystyle
4 \phi''_{2 F} \frac{-u_{j+2} +2 u_j - u_{j-2}}{4 h^2} - \frac{2 \phi''_{2 F}}{h} \frac{u_{j+1} - u_{j}}{h} + \frac{\phi''_{2 F}}{h} \frac{u_{j+2} - u_{j}}{2 h}, & j = K, \\[6pt]
\displaystyle 4 \phi''_{2 F} \frac{-u_{j+1} +2 u_j - u_{j-1}}{h^2} - \frac{2 \phi''_{2 F}}{h} \frac{u_{j} - u_{j-1}}{h} + \frac{\phi''_{2 F}}{h} \frac{u_{j} - u_{j-2}}{2 h}, & j = K+1, \\[6pt]
\displaystyle 4 \phi''_{2 F} \frac{-u_{j+1} +2 u_j - u_{j-1}}{h^2} + \frac{\phi''_{2 F}}{h} \frac{u_{j} - u_{j-2}}{2 h}, & j = K+2, \\[6pt]
\displaystyle 4 \phi''_{2 F} \frac{-u_{j+1} +2 u_j - u_{j-1}}{h^2}, & K+3 \leq j \leq N.
\end{cases} \end{split} \end{equation*} Similarly, ${\mathbf g}$ is given by \begin{equation} \label{ghost} g_j = \begin{cases} 0, & 0 \leq j \leq K-2, \\ -\half \phi'_{2F} /h, & j = K-1, \\ \half \phi'_{2F} /h, & j = K, \\ \half \phi'_{2F} /h, & j = K+1, \\ -\half \phi'_{2F} /h, & j = K+2, \\ 0, & K+3 \leq j \leq N.\\ \end{cases} \end{equation} For space reasons, we only list the entries for $0\le j\le N.$ The equations for all other $j\in\mathbb Z$ follow from symmetry and periodicity. Due to the symmetry in the definition of the atomistic and continuum regions, we have that $L^{qc,h}_{i,j} = L^{qc,h}_{-i,-j}$ and $g_{j} = -g_{-j}$ for $-N+1 \leq i,j \leq 0.$ To see this, we define the involution operator $(S{\mathbf u})_j=-u_{-j}$ and observe that ${\mathcal E}^{qc,h}(S{\mathbf u})={\mathcal E}^{qc,h}({\mathbf u}).$ It then follows from the chain rule that \[ S^TL^{qc,h} S{\mathbf u}-S^T{\mathbf g}-S^T{\mathbf f} =L^{qc,h} {\mathbf u}-{\mathbf g} -{\mathbf f} \quad \text{for all periodic }{\mathbf u}\text{ and }{\mathbf f}. \] Since $S^T=S,$ we can conclude that \begin{equation}\label{S} SL^{qc,h} S = L^{qc,h}\quad\text{and}\quad Sg=g. \end{equation} Note that the expression for ${\mathbf g}$ does not depend on $\phi'_F$ since the first-neighbor terms identically sum to zero in the energy~\eqref{qceTot}. We can now observe that the QCE approximation~\eqref{qc1} is not consistent with the continuum limit of the atomistic model~\eqref{bvp}.
The linear operator $L^{qc}$ has all uniform translations, ${\mathbf u} = c {\mathbf 1} = (c, c, \dots, c),$ in its nullspace. To see that this is the full nullspace, we consider the factored operator $L^{qc} = D^T E^{qc} D,$ where $(D {\mathbf u})_j = \frac{u_{j+1} - u_j}{h}$ and \begin{equation*}
(E^{qc} {\mathbf r})_j = \begin{cases}
\phi''_{2 F} r_{j-1} + (\phi''_{F}+ 2 \phi''_{2 F}) r_{j} +
\phi''_{2 F} r_{j+1}, & 0 \leq j \leq K-2, \\
\phi''_{2 F} r_{j-1} + (\phi''_{F}+ \frac32 \phi''_{2 F}) r_{j} + \frac12 \phi''_{2 F} r_{j+1}, & j =K-1, \\ \frac12 \phi''_{2 F} r_{j-1} + (\phi''_{F}+ 3 \phi''_{2 F}) r_{j} + \frac12 \phi''_{2 F} r_{j+1}, & j = K, \\ \frac12 \phi''_{2 F} r_{j-1} + (\phi''_{F}+ \frac92 \phi''_{2 F}) r_{j}, & j = K+1, \\ (\phi''_{F}+ 4 \phi''_{2 F}) r_{j}, & K+2 \leq j \leq N. \end{cases} \end{equation*} We see that $E^{qc}$ is diagonally dominant provided $\phi''_{F} + 5 \phi''_{2F}>0,$ hence assumption~\eqref{assume2} implies $E^{qc}$ is invertible. So we have that the nullspace of $L^{qc}$ is precisely the nullspace of $D.$ Thus, $L^{qc} {\mathbf u} = {\mathbf g}$ has a solution whenever $\sum_{j=-N+1}^N f_j = 0,$ since $\sum_{j=-N+1}^N g_j = 0.$ This solution is unique up to a constant.
We now gather together the existence and uniqueness results stated for the models. \begin{lemma} \label{wellposed} If $\sum_{j=-N+1}^N f_j = 0$ and $\phi''_F + 4 \phi''_{2F} > 0,$ then the linearized atomistic energy~\eqref{at} and continuum approximation~\eqref{cont} both have a global minimum that is unique up to an additive constant.
Under the slightly stronger assumption $\phi''_F + 5 \phi''_{2F} > 0,$ the quasicontinuum energy~\eqref{qceTot} has a unique minimizer up to a constant. \end{lemma}
Here, and in the following, we take ${\mathbf f} = {\mathbf 0},$ in order to focus on the effect of the ghost force ${\mathbf g}.$ Under this assumption, we can conclude that the unique mean zero solution to the QCE equilibrium equations~\eqref{qc1} is odd. This follows from $S^{-1}=S$ and \eqref{S} which together imply that $S{\mathbf u}$ is a solution if and only if ${\mathbf u}$ is. Because $S$ preserves the mean zero property, we conclude that ${\mathbf u}_{qc}$ is odd. The unique odd solution to the atomistic equations, $L^{a,h} {\mathbf u}_a = {\mathbf 0},$ is ${\mathbf u}_a ={\mathbf 0}.$ Thus, the QCE equilibrium equations, \begin{equation} \label{elqce} L^{qc,h} {\mathbf u}_{qc} - {\mathbf g}={\mathbf 0}, \end{equation} are also the error equations, and the quasicontinuum solution is the error in approximating ${\mathbf u}_a.$
\subsection{Discrete Sobolev Norms} The effect of the interface terms on the total error is norm-dependent, so we now employ discrete analogs of Sobolev norms~\cite{ortnersuli}. We define the discrete weak derivative by \begin{equation*} u'_j = \frac{u_{j+1} - u_{j}}{h}. \end{equation*} For $1 \leq p < \infty$ the discrete Sobolev norms are given by \begin{equation*} \begin{split}
\lpnorm{u}{p} &= \left(\sum_{j=-N+1}^{N} h |u_j|^p\right)^{1/p}, \\ \wpnorm{u}{p} &= \lpnorm{u}{p}
+ \lpnorm{u'}{p},
\end{split} \end{equation*} and for $p = \infty$ by \begin{equation*} \begin{split}
\lpnorm{u}{\infty} &= \max_{-N+1 \leq j \leq N} |u_j|, \\ \wpnorm{u}{\infty} &= \lpnorm{u}{\infty}
+ \lpnorm{u'}{\infty}.
\end{split} \end{equation*} The above discrete Sobolev norms are equivalent to the standard Sobolev norms restricted to the continuous, piecewise linear interpolants $u(x)$ satisfying $ u(j/N)=u_j$ for $j = -N+1,\dots,N.$
\section{Convergence of the Quasicontinuum Solution} \label{sec:comp}
We now analyze the quasicontinuum error, ${\mathbf u}_{qc}.$ We note that is it theoretically possible to solve~\eqref{elqce} explicitly for ${\mathbf u}_{qc};$ however, the form of the solution is complicated by the second-neighbor coupling in the atomistic region, so we instead obtain estimates for the decay of the error, ${\mathbf u}_{qc},$ by analyzing a O($h^2$)-accurate approximation of the error. Figure~\ref{qc_comp} shows the results of solving~\eqref{elqce} numerically for odd solutions, $u_j = -u_{-j},$ with three choices of lattice spacing and two sets of parameters. Note that for both sets of parameters, the magnitude decays linearly with $h,$ whereas the displacement gradient is O($1$) in the atomistic to continuum region.
The following argument proves the qualitative error behavior analytically. \begin{figure}
\caption{ Error for the energy-based quasicontinuum approximation, ${\mathbf u}_{qc}.$ We observe that the magnitude of the error is O($h$). However, the oscillation near the interface means that the error in the displacement gradient is O($1$) in the interfacial region. The average deformation gradient, $F,$ for the right column is close to failing the stability condition $\phi''_{F} + 5 \phi''_{2 F} > 0.$ In all plots $K = N/2$ and $\phi'_{2F} = 1.$}
\label{qc_comp}
\end{figure}
\subsection{Form of the Solution} In the interior of the continuum region the solution is linear, but in the atomistic region ${\mathbf u}_{qc}$ is the sum of a linear solution and exponential solutions. The homogeneous atomistic difference scheme \begin{equation}\label{ato} - \phi''_{2F} u_{j+2} - \phi''_F u_{j+1} + (2 \phi''_F + 2 \phi''_{2F}) u_j - \phi''_F u_{j-1} - \phi''_{2F} u_{j-2} = 0 \end{equation} has characteristic equation \begin{equation*} - \phi''_{2F} \Lambda^2 - \phi''_F \Lambda + (2 \phi''_F + 2 \phi''_{2F}) - \phi''_F \Lambda^{-1} - \phi''_{2F} \Lambda^{-2} = 0, \end{equation*} with roots \begin{equation*} 1, 1, \lambda, \frac{1}{\lambda}, \end{equation*} where \begin{equation*} \lambda=\frac{(\phi''_F + 2 \phi''_{2F})
+ \sqrt{(\phi''_F)^2 + 4 \phi''_F \phi''_{2F}}}{-2 \phi''_{2F}}. \end{equation*} Based on the assumptions on $\phi$ in~\eqref{assume} and~\eqref{assume2}, we have that $\lambda > 1.$ We note that if $\phi''_{2F}$ were positive contrary to assumption~\eqref{assume}, then $\lambda$ would be negative which would give a damped oscillatory error in the atomistic region. General solutions of the homogeneous atomistic equations~\eqref{ato} have the form $u_j = C_1 + C_2 h j + C_3 \lambda^j + C_4 \lambda^{-j},$ but seeking an odd solution reduces this to the form $u_j = C_2 hj + C_3 (\lambda^j - \lambda^{-j}).$
The odd solution of the quasicontinuum error equations~\eqref{elqce} is thus of the form \begin{equation*} ({\mathbf u}_{qc})_j = \begin{cases} m_1 hj + \beta \left(\frac{\lambda^j - \lambda^{-j}}{\lambda^K}\right), & 0 \leq j \leq K, \\ m_2 hj - m_2 + \tilde{u}_{K+1} , & j = K+1, \\ m_2 hj -m_2, & K+2 \leq j \leq N, \end{cases} \end{equation*} where expressing the unknown $u_{K+1}$ using a perturbation of the linear solution, $\tilde{u}_{K+1},$ simplifies the solution of the equilibrium equations. The four coefficients $m_1,\ m_2, \tilde{u}_{K+1}, \text{ and } \beta$ can be found by satisfying the four equations in the interface, $j = K-1,\dots,K+2.$ Summing the equilibrium equations across the interface gives \begin{equation*} \begin{split} 0 &= \sum_{j=K-1}^{K+2} g_j = \sum_{j=K-1}^{K+2} (L^{qc,h} {\mathbf u}_{qc})_j \\
&= \phi''_F \left. \frac{u_{K-1} - u_{K-2}}{h^2} \right.
+ 4 \phi''_{2F} \left.
\frac{u_K + u_{K-1} - u_{K-2} - u_{K-3}}{4 h^2} \right.\\
& \qquad - (\phi''_F + 4 \phi''_{2F}) \left( \frac{u_{K+3} - u_{K+2}}{h^2} \right)\\
&= (\phi''_F + 4 \phi''_{2F}) \left(\frac{m_1}{h} - \frac{m_2}{h}\right). \end{split} \end{equation*} The cancellation of the exponential terms in the final equality holds because \begin{equation*} \phi''_{2F}(\lambda^{K}-\lambda^{-K})+(\phi''_F+\phi''_{2F})(\lambda^{K-1}-\lambda^{-K+1} -\lambda^{K-2}+\lambda^{-K+2})+\phi''_{2F}(-\lambda^{K-3}+\lambda^{-K+3})=0, \end{equation*} which can be seen by summing \eqref{ato} with the homogeneous solution $u_j=-\lambda^j$ for $j=-K+2,\dots,K-2.$
Thus $m_1=m_2,$ that is, the slope of the linear part does not change across the interface. Hence, the odd solution is given by \begin{equation} \label{qcform} ({\mathbf u}_{qc})_j = \begin{cases} m hj + \beta \left(\frac{\lambda^j - \lambda^{-j}}{\lambda^K}\right),
& 0 \leq j \leq K, \\ m hj - m + \tilde{u}_{K+1} , & j = K+1, \\ m hj -m, & K+2 \leq j \leq N, \end{cases} \end{equation} where the coefficients $m, \tilde{u}_{K+1}, \text{ and } \beta$ can now be found by satisfying any three of the equations in the interface, $j = K-1,\dots,K+2.$
\subsection{Magnitude of the Solution} We focus on the equations at $j = K-1, K+1, \text{ and } K+2$ and split the interface equations as $(A_K + h B) {\mathbf x} = h {\mathbf b},$ where \begin{equation*} \begin{split} A_K &= \left[ \begin{array}{rrr}
\frac{1}{2} \phi''_{2F}
& -\frac{1}{2} \phi''_{2F}
&\phi''_{2F} \gamma_{K+1} - \frac{1}{2} \phi''_{2F} \gamma_{K-1} \\[3pt]
-\phi''_F -\frac{5}{2} \phi''_{2F}
&2\phi''_F +\frac{13}{2}\phi''_{2F}
&-\phi''_F\gamma_{K} -2\phi''_{2F} \gamma_{K} - \frac{1}{2} \phi''_{2F} \gamma_{K-1} \\[3pt]
- \frac{1}{2} \phi''_{2F}
& -\phi''_F - 4 \phi''_{2F}
&- \frac{1}{2} \phi''_{2F} \gamma_{K} \end{array} \right], \\
B &= \left[ \begin{array}{rrr}
\phi''_{2F} & 0 & 0 \\[3pt] -\phi''_{2F} & 0 & 0\\[3pt]
\phi''_{2F} & 0 & 0 \end{array} \right], \quad
{\mathbf x} = \left[ \begin{array}{r} m \\ \tilde{u}_{K+1} \\ \beta \end{array} \right], \quad
{\mathbf b} = \frac{1}{2} \phi'_{2F} \left[ \begin{array}{r} -1 \\
1 \\ -1 \end{array} \right], \end{split} \end{equation*} and $\gamma_j = \frac{\lambda^j - \lambda^{-j}}{\lambda^{K}}.$ We note that $A_K,$ $B,$ and ${\mathbf b}$ do not depend on $h$ directly, though $A_K$ may have indirect dependence if $K$ scales with $h$ as in Figure~\ref{qc_comp}. Therefore, we can neglect $B$ and conclude that ${\mathbf x}$ is $O(h)$ provided that $A_K^{-1}$ exists and is bounded uniformly in $K.$
\begin{lemma} \label{fullrank} For all $K$ satisfying $2\le K\le N-2,$ the matrix $A_K$ is nonsingular and
$||A_K^{-1}|| \leq C$ where $C>0$ is independent of $K.$ \end{lemma}
\begin{proof} Applying row reductions gives the upper triangular form \begin{equation*} \begin{split} \widetilde{A} &= \left[ \begin{array}{rrr} \frac{1}{2} \phi''_{2F} & -\frac{1}{2} \phi''_{2F} & \phi''_{2F} \gamma_{K+1} - \frac{1}{2} \phi''_{2F} \gamma_{K-1} \\[3pt] 0 & -\phi''_F - \frac{9}{2} \phi''_{2F} & \phi''_{2F} \gamma_{K+1} -
\frac{1}{2} \phi''_{2F} \gamma_K - \frac{1}{2} \phi''_{2F} \gamma_{K-1}\\[3pt] 0 & 0 & \eta_K \end{array} \right] \end{split} \end{equation*} where \begin{equation*} \eta_K = \textstyle \left( (\phi''_F)^2 + \frac{15}{2} \phi''_F \phi''_{2F} + \frac{53}{4}(\phi''_{2F})^2
\right) \left(2 \gamma_{K+1} - \gamma_K - \gamma_{K-1}\right)
+ \frac{1}{2} \phi''_{2F} \left(\phi''_F + \frac{9}{2}
\phi''_{2F}\right) \left(\gamma_K - \gamma_{K-1}\right). \end{equation*} If the diagonal entries of $\widetilde{A}$ are non-zero, then $A_K$ is nonsingular. The coercivity assumption $\phi''_F + 5 \phi''_{2F} >0$~\eqref{assume2} implies that $-\phi''_F - 9/2 \phi''_{2F} < 0$ since $\phi''_{2F} <0,$ so the first and second diagonal entries are non-zero. Since the second term of $\eta_K$ is negative, we can use the fact that $\gamma_K-\gamma_{K-1} < 2 \gamma_{K+1} - \gamma_K - \gamma_{K-1}$ to see that \begin{equation*} \begin{split} \eta_K &> \textstyle \left(
(\phi''_F)^2 + 8 \phi''_F \phi''_{2F} + \frac{62}{4} (\phi''_{2F})^2\right)
\left(2 \gamma_{K+1} - \gamma_K - \gamma_{K-1}\right)\\
&= \textstyle \left(\phi''_F
+ \left(4 + \frac{1}{\sqrt{2}}\right) \phi''_{2F}\right)
\left(\phi''_F + \left(4 - \frac{1}{\sqrt{2}}\right) \phi''_{2F}\right)
\left(2 \gamma_{K+1} - \gamma_K - \gamma_{K-1}\right)\\ &> 0. \end{split} \end{equation*} Therefore, $A^{-1}_K$ exists for all $K.$ Taking limits, we find \begin{equation*} \lim_{K\to\infty} \eta_K \geq \textstyle \left(\phi''_F
+ \left(4 + \frac{1}{\sqrt{2}}\right) \phi''_{2F}\right)
\left(\phi''_F + \left(4 - \frac{1}{\sqrt{2}}\right) \phi''_{2F}\right)
\left(2 \lambda - 1 - \lambda^{-1}\right) > 0, \end{equation*} where we note that the elementary matrices corresponding to the row reduction operations did not depend on $K$ so that $\lim_{K\to\infty} A_K$ is nonsingular. The inverse of a matrix is continuous as a function of the entries whenever the matrix is nonsingular. Thus, the fact that $\lim_{K\to\infty} A_K$ is nonsingular implies that
$\lim_{K\to\infty} ||A_K^{-1}||$ is finite. Since $||A_K^{-1}||$ is finite for all $K$ and
$\lim_{K\to\infty} ||A_K^{-1}||$ is finite, we conclude that
$||A_K^{-1}||$ is uniformly bounded. \end{proof}
Thus, we have that $m, \tilde{u}_{K+1},$ and $\beta$ are all O($h$). We can express the derivative, ${\mathbf u}'_{qc},$ as \begin{equation*} ({\mathbf u}'_{qc})_j = \begin{cases} m + \frac{\beta}{h}
\left( \frac{\lambda^{j+1} - \lambda^{-j-1}}{\lambda^K }
- \frac{\lambda^j - \lambda^{-j}}{\lambda^K } \right), & 0 \leq j \leq K-1, \\ m - \frac{m}{h} + \frac{\tilde{u}_{K+1}}{h} - \frac{\beta}{h}
\left( \frac{\lambda^K - \lambda^{-K}}{\lambda^K }\right), & j = K, \\ m - \frac{\tilde{u}_{K+1}}{h}, & j = K+1, \\ m, & K+2\leq j \leq N-1, \end{cases} \end{equation*} where $u'_{-j-1} = u'_j$ for $j = 0, \dots, N-1.$
\begin{theorem} \label{thm2} Let ${\mathbf u}_{qc}$ be the solution to the QC error equation~\eqref{elqce}. Then for $1 \leq p \leq \infty, $ $2\le K\le N-2,$ and $h$ sufficiently small, the error can be bounded by \begin{equation*} \begin{split} \lpnorm{{\mathbf u}_{qc}}{\infty} &\leq C h, \\ \wpnorm{{\mathbf u}_{qc}}{p} &\leq C h^{1/p}, \end{split} \end{equation*} where $C>0$ is independent of $h, K,$ and $p.$ \end{theorem}
\begin{proof} The result for the $\ell^{\infty}$ norm follows from the fact that all terms in $\eqref{qcform}$ are O($h$). To show the bound on $w^{1,p},$ we first apply the triangle inequality to separate the $m, \frac{\tilde{u}_{k+1}}{h}, \frac{m}{h},$ and $\frac{\beta}{h}$ terms which we bound using the fact that $\tilde{u}_{K+1}, m,$ and $\beta$ are $O(h).$ We have \begin{equation*} \begin{split} \wpnorm{{\mathbf u}_{qc}}{p} &= \lpnorm{{\mathbf u}_{qc}}{p} + \lpnorm{{\mathbf u}_{qc}'}{p} \\
&\leq \lpnorm{{\mathbf u}_{qc}}{p} + |m|
+ \left(2 \left| \frac{m}{h}\right|^{p} h \right)^{1/p}
+ \left(4 \left| \frac{\tilde{u}_{K+1}}{h}\right|^{p} h \right)^{1/p}
\\
&\qquad + 2 \left( h \sum^{K}_{j=-K} \left|\frac{\beta}{h}
\frac{(\lambda^{j} - \lambda^{-j})}{\lambda^K}\right|^p
\right)^{1/p} \\
&\leq C h^{1/p} + \frac{2 |\beta|}{h} \left( 2h \sum^{K}_{j=0}
\left| \frac{\lambda^j }{\lambda^K} \right|^p
\right)^{1/p}\\
&\leq C h^{1/p} + \frac{2 |\beta|}{h} \left( 2 h
\frac{ \lambda^p}{\lambda^p - 1} \right)^{1/p} \\ &\leq C h^{1/p}. \qedhere \end{split} \end{equation*} \end{proof}
Finally, we show that the pointwise error in the derivative, ${\mathbf u}'_{qc},$ decays exponentially in $j$ to O($h$) away from the interface in the atomistic region and decays immediately to O($h$) away from the interface in the continuum region. \begin{lemma} There is a $C > 0$ such that
$|({\mathbf u}'_{qc})_j| \le C h$ for all $0\le j \leq K + \frac{\ln h}{\ln \lambda}$ and $K+2 \leq j \leq N.$
Thus, the interface has size O($h |\log h|$). \end{lemma}
\begin{proof} For $h$ sufficiently small, we have that $\max(m, \beta) \leq C h.$ Since $u'_j = m$ for $K+2 \leq j \leq N,$ in this region $u'_j \le C h.$ For the terms $0 \leq j \leq K-1$ it is sufficient to show that the exponential term is less than or equal to $C h.$ For $0 \leq j \leq K + \frac{\ln h}{\ln \lambda},$ we have that \begin{equation*} \begin{split} \left( \frac{\lambda^{j+1} - \lambda^{-j-1}}{\lambda^K }
- \frac{\lambda^j - \lambda^{-j}}{\lambda^K } \right) & \leq \lambda^{j+1 - K} \\ &\leq \lambda^{K + \frac{\ln h}{\ln \lambda} +1 - K}\\ &\leq C h. \qedhere \end{split} \end{equation*} \end{proof}
\begin{remark}\label{coarsen} In order reduce the degrees of freedom, the continuum region is coarsened in computations using the quasicontinuum method. For simplicity, coarsening was omitted from the model presented in this paper, but, in fact, the results are unchanged if it is used. Conventionally, coarsening only occurs away from the atomistic to continuum interface, so that no degrees of freedom are removed if they interact directly with the atomistic region. Since the solution $u_j$ is linear for $K+2 \leq j \leq N,$ any level of coarsening produces an identical solution.
\end{remark}
}
\end{document} | arXiv |
Home Journals IJHT Elliptical Pin Fin Heat Sink: Passive Cooling Control
Elliptical Pin Fin Heat Sink: Passive Cooling Control
Fatima Zohra Bakhti* | Mohamed Si-Ameur
Mechanical Engineering Department, Faculty of Technology, University Med Boudiaf, M'sila 28000, Algeria
Laboratory of the Industrials Energy Systems Studies, Mechanical Engineering Department, Faculty of Technology, University of Batna 2, Batna 05000, Algeria
[email protected]
The aim of this study is to examine by means of three-dimensional numerical simulations the thermal-fluid features in elliptical pin fin heat sink. The passive heat transfer enhancement technique is used to comprehend and control the cooling process. This passive methodology is based on pin fins arrangement, hydrodynamic and geometrical parameters. The present numerical results are confronted with experimental measurements in open literature which used one-dimensional model to explore the thermal field. A good agreement was found especially around the optimal fins dimensions. A parametric study has been carried out to deeply analyse the three-dimensional thermal-fluid fields of the heat sink for various key parameters range such the Reynolds number (Re = 50–250) and the aspect ratio (γ=H/d=5.1-9.18). Some new observations and results are obtained thanks to numerical simulations as tool of investigation. It is shown that the fins circumferential temperature is almost uniform. Furthermore, a better cooling is obtained when the Reynolds number increases mainly when the inlet velocity u0> 0.3m/s. The most suitable value of the aspect ratio is attained for γ=8.16, which ensure an optimal cooling process of the pins. A new global Nusselt number correlation was developed for engineering applications.
mixed convection, heat sink, elliptical pin fins, cooling device, CFD
Due to the rapid growth of electronic technology, the volume of devices is considerably reduced while the performance increases favorably and in a superior way. This trend inexorably induces an augmented heat rate per device's volume. In this situation, the great challenge for manufacturers is to ensure an efficiently cooling process. Indeed, any heat excess without proper removal alters the device life and superfluous heating can even affect its ordinary function. Air heat sink remains the most output for the electronics industry, mainly due to their low price and high consistency. The purpose is to extract the suitable configuration which undergoes a higher thermal efficiency. Their role is to hold the system temperature below the manufacturer's stated maximum value. To conceive an adequate heat sink in terms of thermal performance, it is important to carry out numerical and experimental investigations to analyse heat transfer features in several situations for different pin-fin arrays arrangement. The research in this context has attracted not only campus laboratories, but also industrial ones.
The open literature survey has highlighted the below scientific works. Sparrow and Larson [1] experimentally studied a pin fin array under a new fluid flow arrangement. The main objective was to get the coefficient of heat transfer along the array. They found that the fins located at the edge of the heat sink presented elevated coefficients of heat transfer than those within the core of the heat sink. Zografos and Sunderland [2] investigated natural convection in a pin fins heat sink with an aligned and staggered arrangement. The inline arrays have been shown to provide better thermal performance than the staggered ones. Chapman et al. [3] performed a detailed experimental analysis of the forced convection in heat sinks with different shapes of pin fin (square, circular and elliptical). The superiority of the elliptical pin thermal performance against the rectangular pin heat sink was obtained. Tahat et al. [4] experimentally examined the forced convection in a heat sink with two arrangements of fins; aligned and staggered. The optimal pitch between the fins was calculated in the longitudinal and transversal directions. Indeed, the variation of the Nusselt number with the Reynolds number and the spacing of the fins has been determined.
Kobus and Oshio [5, 6] investigated a pin-fin heat sink by means of a theoretical and experimental approach. An optimum of fin spacing has been obtained by examining the effects of several fin parameters. Furthermore, the elaborate theoretical model was shown to be in agreement with the experiments. Khan et al. [7] examined the effect of the shape of the fins on the thermal performance of the heat sink. They have used rectangular plate fins as well as circular, elliptical and square shape cross-section pin-fins with the same perimeter. Their results showed that the elliptical fin gives the highest heat transfer coefficient and low drag force. On the other hand, the rectangular fins give the best values for total entropy generation. The square fins offer the low results for the heat coefficient, drag force and total entropy generation. They also noticed that the fin profile is closely related to the aspect ratio and the Reynolds number. Sahiti et al. [8, 9] studied numerically the impact of the pin shape of on the thermal performance and pressure drop. They found that the elliptical profile has been shown to perform best in a sample of six shapes.
Yakut et al. [10] performed experiments to study the effect of dimensions of hexagonal fins and the pitch between fins on pressure drop and thermal resistance. Yang et al. [11] investigated experimentally a heat sink by considering different cross-section of pin fin; elliptic, square and circular. They have shown that the elliptic form with the staggered arrangement gives the lowest values of the pressure drop and thermal resistance. Seyf and Layeghi [12] investigated the forced convective heat transfer in six pin fin heat sink having an elliptic cross-section. Their results indicated that the insertion of metallic foam into the heat sink improves its thermal efficiency with a moderate increase in pressure drop. Deshmukh and Warkhedkar [13] have reported a detailed literature analysis of the thermal efficiency of heat sinks with different shapes of pin fin and covering the three modes of heat transfer; natural, forced and mixed convection. Chen and Jan [14] have presented a 3D numerical approach to study the forced convection in cylindrical pin-fins and plate-fins heat sink using COMSOL software. They found that the cylindrical fins give higher heat dissipation than that of plate fins. Kumar and Bartaria [15] studied numerically the thermal and hydrodynamic characteristics of elliptical pins confined between two plate fins. Their results show that the finned heat sinks with elliptical pins perform better than the plate ones.
Deshmukhand and Warkhedkar [16] experimentally investigated mixed convection around elliptical pin fins. A theoretical model was established to determine the effect of different parameters such as the heat sink void fraction, aspect ratio and longitudinal and transversal fin pitch on the thermal performance of a heat sink. Matsumoto et al. [17] carried out both a numerical and experimental investigation to study natural convection in five kinds of pin fin heat sink. They have analyzed the effect of the fins number and the geometric parameters on the thermal efficiency. Liu et al. [18] have presented a comparative study of hydrodynamic and heat transfer performances of heat sinks with a staggered arrangement of elliptical, circular and diamond fins mounted in a rectangular channel.
The cylindrical pin fins heat sink was studied numerically by Yang et al. [19]. They analyzed the effect of the Reynolds number, volume fraction and fin material on the heat sink thermal efficiency. They found an optimal diameter and number of fins which maximizes the heat transfer. Xia et al. [20] carried out an experimental and numerical investigation to study a laminar flow with heat transfer in a circular pin-fin, square pin-fin and diamond pin-fin micro heat sinks. The results obtained showed that diamond pin-fin heat sink has better heat transfer performance and the vortexes are very much easily involved in the main flow than those in the other two types of heat sinks. Yadav and Pandey [21] performed a numerical study of convective heat transfer using commercial solver COMSOL Multiphysics in a heat sink with various shapes of the fins: cylindrical, elliptical, sprocket, kite, channeled, and triangular. They found that the heat transfer from pin-fin increases with an increase in inlet velocity. Their results indicated that kite and elliptical shape have a greater effect on Nusselt number and heat transfer coefficient.
Kewalramani et al. [22] have performed an experimental investigation to study the laminar flow and heat transfer in a heat sink with elliptic cross-section fins. They proposed correlations of the Nusselt number and the Poiseuille number as a function of the Reynolds number and the Prandtl number. The characteristics of heat transfer and flow field of a water-cooled pin-fin heat sink were numerically studied by Rezaee et al. [23]. They analyzed the influence of pin-length and longitudinal-pitch of heat sink on the heat transfer and pressure drop.
The focus of the present work is to study numerically the mixed convection in a heat sink with elliptical pin-fins. The complete geometry of the heat sink is studied by means of 3D numerical simulations as tool of investigation with high resolutions, in order to be more precise and get closer to a real configuration. A great attention is paid to the qualitative (based on 3D flow visualisations to show the vortex interactions and modification of flow behaviour with fins geometrical dimensions) and quantitative analysis to highlight the thermal and dynamics fields. The goal is to comprehend the flow topology and then improve passively the cooling process with minimal pressure drop by means of the geometrical dimensions and hydrodynamic inlet condition (flow rate, Re...). Furthermore, the purpose is to elaborate a new numerical database versus the experimental work of Deshmukh and Warkhedkar [16], in order to contribute to show new results with more precise details which are inaccessible experimentally. Thus, the effects of the fins aspect ratio and velocity inlet on the thermal resistance, Nussselt number and fin efficiency are explicitly taken into account. In this context, we have followed our strategy which has been adopted in the previous works [24, 25]. The remainder of this scientific article is structured as follows: Section 2 introduces the Mathematical and physical model of the problem with emphasis on boundary conditions, Section 3 devoted to the numerical simulation procedure and Section 4 results and discussion. Finally, a conclusion is presented.
2. Mathematical Modelling
2.1 Physical models
Figures 1& 2 illustrated the heat sink which is investigated in this paper. In order to confront numerical results against experimental measurements available in the open literature especially the work of Deshmukh and Warkhedkar [16], the target configuration is described as follows. The heat sink consists of 14x7 fins having an elliptic cross-section with inline arrangement and for semi-minor axis a=8mm and semi-major axis b=12mm.
Figure 1. Designed geometry of pin-fin heat sink
Figure 2. Isometric view of studied heat sink
The pins-fins sections are attached to a solid volume of parallelepiped form with dimensions (W x L x wb=164mm x 164mm x 12mm). SL=22.5mm and ST=11.25mm are respectively the longitudinal and transversal pitches between the fins. The heat sink is made on the basis of Aluminium alloy (K=202.4 W/m.K), it is attached to an electronic component of Sillicium (K=130 W/m.K), which produces a heat flux Q=400W. The coolant fluid flows along the y-axis according to the inlet condition.
2.2 Governing equations and boundary conditions
The below equations, let to elaborate a three-dimensional numerical model to study steady air incompressible flow with mixed convective heat transfer by means of numerical simulations. All the fluid thermophysical properties are considered constant in the range of the temperature variation. However, the density varies according to the Boussinesq approximation.
Following the considerations above, the governing equations of mass, momentum and energy conservation are written in vector form as follows:
Continuity equation:
$\nabla . \overrightarrow{\mathrm{u}}=0$ (1)
Momentum equation:
$\overrightarrow{\mathrm{u}} . \nabla \overrightarrow{\mathrm{u}}=-\frac{1}{\rho_{0}} \overrightarrow{\nabla \mathrm{p}}+\nu . \Delta \overrightarrow{\mathrm{u}}+\overrightarrow{\mathrm{g}} . \beta .\left(\mathrm{T}-\mathrm{T}_{0}\right)$ (2)
Energy equation for fluid:
$\overrightarrow{\mathrm{u}} . \vec{\nabla} \mathrm{T}=\frac{\mathrm{K}_{\mathrm{f}}}{\rho_{0} . \mathrm{C}_{\mathrm{p}}} \Delta \mathrm{T}$ (3)
Energy equation for solid:
$\mathrm{K}_{\mathrm{S}} . \Delta \mathrm{T}+\dot{\mathrm{q}}_{\mathrm{s}}=0$ (4)
where, $\overrightarrow{\mathrm{u}}$ is the vector field of flow velocity, the velocity of x, y and z-coordinate are represented as u, v and w, correspondingly; the density of heat transfer fluid is defined as ρ; ν is the kinematic viscosity; p is the pressure value of heat transfer fluid; Cp is defined as the specific heat capacity in the constant pressure; β is the thermal expansion coefficient; Kf and Ks are the heat conductivity of heat transfer fluid and solid.
The heat flow generated per unit volume;
$\dot{q}_{s}=\frac{Q}{V}\left[W / m^{3}\right]$
The boundary conditions are chosen with a fidelity to realistic situations. In this context, the adopted ones are as follows:
At the heat sink inlet: u=w=0, v=u0, T=T0=293.16K.
At the heat sink outlet, this boundary is chosen to let flow escape freely, so no reverse flow is admitted in the exit:
$\frac{\partial \mathrm{u}}{\partial \mathrm{x}}=\frac{\partial \mathrm{v}}{\partial \mathrm{x}}=\frac{\partial \mathrm{w}}{\partial \mathrm{x}}=\frac{\partial \mathrm{T}}{\partial \mathrm{x}}=0$, P=Patm
- For both left (x=0) and right(x=W) sides of the computational domain, a symmetry condition is prescribed to ensure:
$\frac{\partial u}{\partial x}=\frac{\partial w}{\partial x}=\frac{\partial T}{\partial x}=0, v=0$
-A heat source $\dot{q}_{s}$ produced by the electronic component:
$\dot{\mathrm{q}}_{\mathrm{s}}=\frac{\mathrm{Q}}{\mathrm{V}}=1487210 \mathrm{~W} / \mathrm{m}^{3}$
with Q=400W, V=164mmx164mmx10mm=268960 mm3.
- A continuity condition at the interface is prescribed between the conduction in the solid and the convection in the fluid so:
Ts=Tf
$\left.\mathrm{K}_{\mathrm{S}} \frac{\partial \mathrm{T}}{\partial \mathrm{n}}\right|_{\text {wall }}=\left.\mathrm{K}_{\mathrm{f}} \frac{\partial \mathrm{T}}{\partial \mathrm{n}}\right|_{\text {wall }}$ (5)
2.3 Data analysis
The total heat transfer rate, inlet and outlet mean temperature and mass flow rate were rigorously tracked, by considering explicitly the hydraulic diameter concept in Fluent Software. Hence, we used the below relationships to compute the Nusselt number, the thermal resistance and the fins efficiency:
-The Reynolds number is calculated by:
$\mathrm{Re}=\frac{\rho \mathrm{u}_{0} \mathrm{~d}}{\mu}$ (6)
$\mathrm{d}=\sqrt{4 \mathrm{ab}}$ is the mean diameter of elliptical pin fin [16].
-The average Nusselt number is defined by:
$\overline{\mathrm{Nu}}_{\mathrm{d}}=\frac{\overline{\mathrm{h}} . \quad \mathrm{d}}{\mathrm{K}_{\mathrm{f}}}$ (7)
where, the average heat transfer coefficient is calculated by:
$\overline{\mathrm{h}}=\frac{\mathrm{q}_{\mathrm{c}}}{\mathrm{A} .\left(\overline{\mathrm{T}}_{\mathrm{w}}-\overline{\mathrm{T}}_{\mathrm{m}}\right)}$ (8)
$\overline{\mathrm{T}}_{\mathrm{w}}=\frac{1}{\mathrm{~A}} \iint \mathrm{T} . \mathrm{d} \mathrm{A}$ (9)
$\overline{\mathrm{T}}_{\mathrm{m}}$ is the mean fluid temperature defined by:
$\overline{\mathrm{T}}_{\mathrm{m}}=\frac{\iint_{\mathrm{V}} \mathrm{TudV}}{\iiint_{\mathrm{V}} \mathrm{udV}}$ (10)
-The thermal resistance Rth is defined as [6, 16];
$\mathrm{R}_{\mathrm{th}}=\frac{\overline{\mathrm{T}}_{\mathrm{b}}-\mathrm{T}_{0}}{\mathrm{q}_{\mathrm{c}}}$ (11)
where, $\overline{\mathrm{T}}_{\mathrm{b}}$ is the mean temperature of the heat sink base;
$\overline{\mathrm{T}}_{\mathrm{b}}=\frac{1}{\mathrm{~A}} \iint \mathrm{T} . \mathrm{d} \mathrm{A}$ (12)
-The pressure drag coefficient:
$\mathrm{C}_{\mathrm{d}}=\frac{\Delta \mathrm{p}}{0.5 . \rho . \mathrm{u}_{0}^{2}}$ (13)
where, the pressure drop Δp across the heat sink is
Dp=Pinlet- Pout
-The fin efficiency is defined by:
$\eta=\frac{\tanh (\mathrm{mH})}{\mathrm{mH}}$ (14)
$\mathrm{m}=\sqrt{\frac{4 \mathrm{~h}}{\mathrm{~K}_{\mathrm{s}} \mathrm{d}}}$ (15)
2.4 Numerical procedure
In this 3D parametric study, the numerical experiments regarding the heat sink thermal performances are carried out for various aspect ratios of the pin fins and inlet hydrodynamic conditions. The design of geometries and meshes are elaborated by using the Gambit software (a pre-processor of the FLUENT commercial code).
The finite volume method was employed to discretise the governing Eqns. (1-4) and the boundary conditions by integration through finite control volumes; Quick scheme was used for the discretization of convection and diffusion terms. The acquired algebraic equations are solved by means of a numerical procedure based on SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm. The solution was considered converged when the normalized residuals of continuity and momentum equations are less than 10-3 while that of the energy equation was fixed at 10-6. The computing resources are based on 8 processor DELL station.
2.5 Study of grid independence
The accuracy of the numerical simulation depends on the quality of the mesh. In this situation and in order to have a good compromise between the computation time which is highly cost in 3D and the precision of the results, the choice of an adequate mesh is crucial. In this study, a refined hexahedral mesh was used near the walls and in some critical regions where the velocity and temperature gradients are important as it is shown on Figure 3.
Furthermore and to choose the suitable mesh, several tests were carried out to check the influence of the mesh on the precision of the results. For this purpose, four meshes were considered and tested for the campaign of grid independency.
Figure 3. Isometric view of configuration mesh
Table 1 summarises the results of walls and fluid mean temperature values, convective heat flux, thermal resistance, mean convection coefficient and the mean Nusselt number. It shows that the results obtained by these mesh grids are correlated. However, the highly cost time in the case of the last grid and the good performance of the fine mesh, let to definitively opt for it within all the below numerical computations.
Table 1. Grid independence study
Nbr nods
$\overline{\mathrm{T}} \mathrm{w}$
$\overline{\mathrm{T}} \mathrm{b}$
(K/W)
$\overline{\mathrm{T}}$m
$\overline{\mathrm{h}}$
(W/K.m2)
$\overline{\mathrm{Nu}}$
3. Results and Discussions
3.1 Validation of numerical simulation
The accuracy of the present numerical results is verified with the available experimental measurements published in the open literature. Numerical simulations are carried out with the same input data. The confrontation is provided on Figures 4-6, it can be seen that a good agreement is achieved. The small discrepancies can be due to the one dimensional approach of the correlation used by Deshmukh and Warkhedkar [16]. Therefore, the numerical model in the present study is satisfactory and the numerical results are consistent.
Figure 4. Average Nusselt number variation versus the aspect ratio
Figure 5. Average Nusselt number variation versus the Reynolds number
Figure 6. Average thermal resistance variation versus the inlet velocity
3.2 Temperatures fields of the heat sink
Figure 7 shows the temperature contour in the pin fin heat sink for an aspect ratio $\gamma$=8.16. One can see that for all inlet velocity values (u0=0.1m/s, 0.2m/s, 0.3m/s, 0.4m/s, 0.5m/s), the fins circumferential temperature is almost uniform. The temperature difference between two consecutive lines of pin fin decreases continually in the flow direction, from the heat sink bottom until the seventh fins row. That is due to the progressive increase of the vertical velocity, which lets to maintain the cooling process operating for all the system.
The horizontal and vertical cross sections of the air temperature contours are illustrated respectively on Figures 8 & 9. It is obvious that for each inlet velocity value u0, the lowest air temperatures are located upstream of the fins and the highest values downstream along the air flow direction. The air velocity is low in the recirculation zones downstream of the fins which induce an air temperature augmentation. The velocity decreasing in the flow direction leads to different thermal boundary layer thickness of rows; one can notice that the one in the first row is thinner than that of the seventh.
7c.png
7d.png
7e.png
Figure 7. Contours of static temperature in the solid heat sink for γ=8.16
Figure 8. Fluid(air) contours static temperature in different horizontal planes for γ=8.16
The thermal performance of the heat sink is calculated on the basis of the temperature difference $\Delta$T variation, between the inlet heat sink and exit. Figure 10, indicates the variation of $\Delta$T with u0 and the fin aspect ratio $\gamma$. It is shown that $\Delta$T decreases with increasing of u0 and $\gamma$.
The convective transfer rate is directly influenced by the aspect ratio $\gamma$ and therefore with the height of the fins. The use of high fins (high aspect ratio) gives a higher exchange surface with an increase of the heat transfer rate. Also, the air mass flow increases as the inlet surface augments (Figure 11). Consequently, the outlet air temperature decreases gradually for high inlet velocities u0> 0.3m/s.
Figure 9. Fluid(air) contours static temperature in different vertical planes for γ=8.16
Figure 10. Temperature difference variation versus inlet velocity and aspect ratio γ
Figure 11. Air mass flow variation with inlet velocity and aspect ratio γ
3.3 Vector velocity and path lines
Figures 12 & 13 illustrate the velocity vector plots and velocity contours for $\gamma$=8.16 in mid-plane (z=0.04) for different inlet velocity values. It is evident that the velocity profile at the heat sink entry remains uniform. The velocity vector direction changes as the fluid particles approach to the pins and move over them. It can be also noticed a velocity increase in the space between the pin in the longitudinal direction. This is due to the reduction in the section of the fluid passage. In this area, the friction coefficient remains important. The acceleration of the hot fluid privileges the heat transfer by convection. Also, the velocity profile is deformed when the fluid is close to the heated fins. These deformations induce three dimensional thermal-fluid fields.
Figure 14 shows the flow path lines in mid-plane (z=0.04) for $\gamma$=8.16 and for various u0. For uo=0.1m/s, the flow behaves in a creeping manner because the forces of inertia being low. Furthermore, a fluid attachment to the fins without any separation is observed. The flow is balanced for each column of fins compared to its central axis and also between the upstream and the downstream of the fin.
For u0>0.1, the inertia forces increase and prevent the boundary layer from remaining attached to the walls of the pins and start to support a depression in the wake zone.
Thus, one observes a separation after each fin; downstream two contra-rotating lobes almost symmetrical of recirculation attached to the fin are formed. The point of fastening which is defined as the place where longitudinal velocity is null, on the central axis of the wake, moves away from the fin when the velocity inlet increases. The coordinates of this point define the length of recirculation.
Figure 12. Velocity vectors in the plane z=0.04 for γ=8.16
Figure 14. Flow path lines around a solid fin in the plane z=0.04 for γ=8.16
Figure 15. Thermal resistance variation versus the inlet velocity
For more assessment of the thermal fin performance, Figure 15 shows the thermal resistance for various pin heights and for various inlet velocities. One can observe that the thermal resistance decreases with increasing of u0 and γ.
3.4 Nusselt number
The Nusselt number was calculated for all the numerical simulations. Figure 16 shows the variation of the Nusselt number with the inlet velocity, in the case of all pin-fin arrangements. It is clear that the mean Nusselt number increases with the air flow rate augmentation, which supports moreover the conduction-convective phenomenon between the fins and the fluid flow inside the heat sink.
Figure 17 makes clear that as the fin height augments, the Nusselt number augments for all values of γ= 5.1–8.16 and the bad effects are detected for γ over 8.16. This can be endorsed to the temperature gradients near to the tip of the pin fins. Indeed, above γ=8.16 the thermal resistance leads to a drastic temperature decreases near the fin tips, and the thermal performance drops off. Thus, the optimum fins aspect ratio is close to 8.16.
Figure 16. Mean Nusselt number variation versus the inlet velocity
Figure 17. Mean Nusselt number variation versus the aspect ratio
This optimum is exactly the same for both the two approaches; the present numerical simulation and the experimental investigation [16], although their experimental model involves numerous simplifications. However, our approach is more precise as all the governed equations are solved in three dimensional numerical computations. Obviously the heat flux inside the fins is not unique and dispersed for all the system. The new observations presented above are due to 3D numerical simulations as tool of investigation. The Nusselt number could be correlated as a polynomial function such as equation:
$\overline{\mathrm{Nu}}_{\mathrm{d}}=\mathrm{a}_{1}+\mathrm{b}_{1} . \mathrm{Re}_{\mathrm{d}}-\mathrm{c}_{1} . \mathrm{Re}_{\mathrm{d}}^{2}$ (16)
Coefficients Data:
a1=0.36987, b1=0.07494, c1=1.21936.10-4
Where 50≤Red≤250
This new correlation can be used to engineering applications to compute the average heat energy. Table 2 summarises the new Nusselt number correlation and others issued from literature in the case of laminar flow.
3.5 Pressure drag coefficient and fin thermal efficiency
Figure 18 elucidated the variation of the pressure drag coefficient Cd with different inlet velocities and for different fin aspect ratios. It is evidently clear that Cd augments first with inlet velocity and then diminishes after it attains the pick value for u0 = 0.3m/s. For all aspect ratios, the pressure drops are roughly at the same trend. It can also be noted that Cd increases with gand consequently with the height of the fins. This is due to the increase in friction forces generated along the fins.
Figure 18. Pressure drag coefficient variation versus the inlet velocity
Figure 19. The fin efficiency variation versus the inlet velocity
Table 2. Nusselt number correlations
Nusselt Number
Present investigation
$\overline{N u}_{d}=\mathrm{a}_{1}+b_{1} . R e_{d}-c_{1} . R e_{d}^{2}$
a1=0.36987
b1=0.07494 c1=1.21936x10-4
50≤Red≤250
Re: based on hydraulic diameter
Deshmukh and Warkhedkar [16]
-Inline arrangement
$\mathrm{Nu}_{\mathrm{d}}=1.008(\alpha)^{1.34}(\gamma)^{0.5}\left(\frac{\mathrm{Re}_{\mathrm{d}}}{\mathrm{Gr}_{\mathrm{d}}^{0.5}}\right)$
-Staggered arrangement
$\mathrm{Nu}_{\mathrm{d}}=1.26(\alpha)^{1.25}(\gamma)^{0.46}\left(\frac{\mathrm{Re}_{\mathrm{d}}}{\mathrm{Gr}_{\mathrm{d}}^{0.5}}\right)$
0.534 ≤ α ≤ 0.884,
5.1 ≤ γ ≤ 9.18
$1 \leq \frac{G r_{d}}{R e_{d}} \leq 100$
Khan et al. [26]
-For uniform wall temperature
$\frac{\mathrm{Nu}_{\mathrm{L}}}{\operatorname{Re}_{\mathrm{L}}^{1 / 2} \operatorname{Pr}^{1 / 3}}=0.75-0.16 \exp \left(\frac{-0.018}{\varepsilon^{3.1}}\right)$
-For uniform wall flux
$\frac{\mathrm{Nu}_{\mathrm{L}}}{\mathrm{Re}_{\mathrm{L}}^{1 / 2} \operatorname{Pr}^{1 / 3}}=0.91-0.31 \exp \left(\frac{-0.09}{\varepsilon^{1.79}}\right)$
a: semimajor axis of elliptical cylinder
b: semiminor axis of elliptical cylinder
ε: axis ratio=b/a
L: characteristic length=2a
102≤ReL≤105
In order to present the heat sink thermal performance, the thermal efficiency is presented in Figure 19. It can be clearly seen that the fin efficiency h decreases with increasing the inlet velocity u0 and aspect ratio γ.
In this work, a cooling process regarding to heated electronic component is studied by means of 3D numerical computations. A parametric analysis for various hydrodynamic values (inlet velocity or Reynolds numbers) and geometrical pin fin dimensions was undertaken. The finite volume method and the SIMPLE algorithm were used to solve the thermal-fluid equations. The results are confronted with the experimental measurements of pin fin heat sink; a quasi-good agreement was found. Numerical simulations were carried out for a broad variation of independent parameters; the Reynolds number Re=50-250 and the fin aspect ratio γ=5.1-9.18.
On the basis of this 3D study, the dynamical and the thermal fields are analysed qualitatively and quantitatively through numerical visualisations and the profiles of significant physical quantities. It is shown that the fins circumferential temperature is almost uniform. Furthermore, a better cooling is detected when the inlet velocity increases. An optimal value of the fin aspect ratio γ=8.16 was obtained, which let to better heat dissipation with satisfactory pressure drop. A new correlation given the evolution of the Nusselt number with the Reynolds number in such thermal system is elaborated. The present numerical experiments have highlighted more details, especially the thermal-fluid fields topology compared to the laboratory experiments. Thus, the comprehension of the cooling mechanisms involved in such system can lead to a better control of the heat dissipation to ensure a longer life device.
heat transfer area, m2
semi-minor axis of elliptical pin fin, m
semi-major axis of elliptical pin fin, m
pressure drag coefficient
specific heat at constant pressure, J.kg-1.K-1
mean diameter, m
$\overrightarrow{\mathrm{g}}$
gravity acceleration, m/s-2
average heat transfer coefficient, J.m-2.K-1
height of pin fin, m
fluid thermal conductivity, J.m-1.K-1
solid thermal conductivity, J.m-1.K-1
length of base plate, m
$\overline{\mathrm{Nu}}_{\mathrm{d}}$
average Nusselt number based on d (=h.d/Kf)
Pressure, Pa
pinlet
pressure inlet, Pa
pressure outlet, Pa
heat produced by the electronic component, W
convection heat flux, J
$\dot{q}_{s}$
heat flux generated per unit volume, W/m3
Reynolds number based on d
thermal resistance, K/W
transversal pitch, m
longitudinal pitch, m
temperature of heat sink base, K
mean temperature of the fluid, K
inlet air temperature, K
temperature of the heat sink wall, K
u,v,w
velocity components, m/s
inlet air velocity, m/s
volume of the electronics component, m3
width of base plate, m
thickness of base plate, m
thickness of source volume, m
cartesian coordinates, m
the fluid density, kg/m3
ρ0
the fluid density at T0, kg/m3
$\eta$
fin efficiency
kinematic viscosity (=µ/ρ), m2.s-1
[1] Sparrow, E.M., Larson, E.D. (1982). Heat transfer from pin-fins situated in an oncoming longitudinal flow which turns to cross flow. International Journal of Heat and Mass Transfer, 25(5): 603-614. https://doi.org/10.1016/0017-9310(82)90165-X
[2] Zografos, A.I., Sunderland, J.E. (1990). Natural convection from pin fin arrays. Experimental Thermal and Fluid Science, 3(4): 440-449. https://doi.org/10.1016/0894-1777(90)90042-6
[3] Chapman, C.L., Lee, S., Schmidt, B.L. (1994). Thermal performance of an elliptical pin fin heat sink. Proceedings of 1994 IEEE/CHMT 10th Semiconductor Thermal Measurement and Management Symposium (SEMI-THERM), pp. 24-31. https://doi.org/10.1109/STHERM.1994.288998
[4] Tahat, M., Kodah, Z.H., Jarrah, B.A., Probert, S.D. (2000). Heat transfers from pin-fin arrays experiencing forced convection. Applied Energy, 67(4): 419-442. https://doi.org/10.1016/S0306-2619(00)00032-5
[5] Kobus, C.J., Oshio, T. (2005). Development of a theoretical model for predicting the thermal performance characteristics of a vertical pin-fin array heat sink under combined forced and natural convection with impinging flow. International Journal of Heat and Mass Transfer, 48(6): 1053-1063. https://doi.org/10.1016/j.ijheatmasstransfer.2004.09.042
[6] Kobus, C.J., Oshio, T. (2005). Predicting the thermal performance characteristics of staggered vertical pin fin array heat sinks under combined mode radiation and mixed convection with impinging flow. International Journal of Heat and Mass Transfer, 48(13): 2684-2696. https://doi.org/10.1016/j.ijheatmasstransfer.2005.01.016
[7] Khan, W.A., Culham, R., Yovanovic, M.M. (2006). The role of fin geometry in heat sink performance. Journal of Electronic Packing, 128: 324-330. https://doi.org/10.1115/1.2351896
[8] Sahiti, N., Lemouedda, A., Stojkovic, D., Durst, F., Franz, E. (2006). Performance comparison of pin fin in-duct flow arrays with various pin cross-sections. Applied Thermal Engineering, 26(11-12): 1176-1192. https://doi.org/10.1016/j.applthermaleng.2005.10.042
[9] Sahiti, N., Durst, F., Geremia, P. (2007). Selection and optimization of pin cross-sections for electronics cooling. Applied Thermal Engineering, 27(1): 111-119. https://doi.org/10.1016/j.applthermaleng.2006.05.018
[10] Yakut, K., Alemdaroglu, N., Kotcioglu, I., Celik, C. (2006). Experimental investigation of thermal resistance of a heat sink with hexagonal fins. Applied Thermal Engineering, 26(17-18): 2262-2271. https://doi.org/10.1016/j.applthermaleng.2006.03.008
[11] Yang, K.S., Chu, W.H., Chen, I.Y., Wang, C.C. (2007). A comparative study of the air side performance of heat sinks having pin fin configurations. International Journal of Heat and Mass Transfer, 50(23-24): 4661-4667. https://doi.org/10.1016/j.ijheatmasstransfer.2007.03.006
[12] Seyf, H.R., Layeghi, M. (2010). Numerical analysis of convective heat transfer from an elliptic pin fin heat sink with and without metal foam insert. Journal of Heat Transfer, 132(7): 071401-1-071401-9. https://doi.org/10.1115/1.4000951
[13] Deshmukh, P.A., Warkhedkar, R.M. (2011). Thermal performance of pin fin heat sinks –a review of literature. International Review of Mechanical Engineering (IREME), 5(4): 726-732.
[14] Chen, C.T., Jan, S.H. (2012). Dynamic simulation, optimal design and control of pin-fin heat sink processes. Journal of the Taiwan Institute of Chemical Engineers, 43(1): 77-88. https://doi.org/10.1016/j.jtice.2011.06.005
[15] Kumar, V., Bartaria, V.N. (2013). CFD analysis of an elliptical pin fin heat sink using Ansys Fluent v12.1. International Journal of Modern Engineering Research (IJMER), 3(2): 1115-1122.
[16] Deshmukh, P.A., Warkhedkar, R.M. (2013). Thermal performance of elliptical pin fin heat sink under combined natural and forced convection. Experimental Thermal and Fluid Science, 50: 61-68. https://doi.org/10.1016/j.expthermflusci.2013.05.005
[17] Matsumoto, N., Tomimura, T., Koito, Y. (2014). Heat transfer characteristics of square micro pin fins under natural convection. Journal of Electronics Cooling and Thermal Control, 4(3): 59-69. https://doi.org/10.4236/jectc.2014.43007
[18] Liu, Z.G., Guan, N., Zhang, C.W., Jiang, G.L. (2015). The flow resistance and heat transfer characteristics of micro pin-fins with different cross-sectional shapes. Nanoscale and Microscale Thermophysical Engineering, 19(3): 221-243. https://doi.org/10.1080/15567265.2015.1073820
[19] Yang, A., Chen, L., Xie, Z., Feng, H., Sun, F. (2016). Constructal heat transfer rate maximization for cylindrical pin-fin heat sinks. Applied Thermal Engineering, 108: 427-435. https://doi.org/10.1016/j.applthermaleng.2016.07.150
[20] Xia, G., Chena, Z., Cheng, L., Ma, D., Zhai, Y., Yang, Y. (2017). Micro-PIV visualization and numerical simulation of flow and heat transfer in three micro pin-fin heat sinks. International Journal of Thermal Sciences, 119: 9-23. https://doi.org/10.1016/j.ijthermalsci.2017.05.015
[21] Yadav, S., Pandey, K.M. (2018). A comparative thermal analysis of pin fins for improved heat transfer in forced convection. Materials Today: Proceedings, 5(1): 1711-1717. https://doi.org/10.1016/j.matpr.2017.11.268
[22] Kewalramani, G.V., Hedau, G., Saha, S.K., Agrawal, A. (2019). Study of laminar single phase frictional factor and Nusselt number in In-line micro pin-fin heat sink for electronic cooling applications. International Journal of Heat and Mass Transfer, 138: 796-808. https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.118
[23] Rezaee, M., Khoshvaght-Aliabadi, M., Abbasian Arani, A.A., Mazloumi, S.H. (2019). Heat transfer intensification in pin-fin heat sink by changing pin-length/longitudinal-pitch. Chemical Engineering & Processing: Process Intensification, 141: 107544. https://doi.org/10.1016/j.cep.2019.107544
[24] Bakhti, F.Z., Si-Ameur M. (2016). Numerical study of cooling enhancement: heat sink with hollow perforated elliptic pin fins. Comput. Therm. Sci., 8(5): 409-428.
[25] Bakhti, F.Z., Si-Ameur, M. (2019). A comparison of mixed convective heat transfer performance of nanofluids cooled heat sink with circular perforated pin fin. Applied Thermal Engineering, 159: 113819. https://doi.org/10.1016/j.applthermaleng.2019.113819
[26] Khan, W.A., Culham, J.R., Yovanovich, M.M. (2005). Fluid flow around and heat transfer from elliptical cylinders: Analytical approach. Journal of Thermophysics and Heat Transfer, 19(2): 178-185. | CommonCrawl |
Battery (electricity) (12881 views - Electrical Engineering)
An electric battery is a device consisting of one or more electrochemical cells with external connections provided to power electrical devices such as flashlights, smartphones, and electric cars. When a battery is supplying electric power, its positive terminal is the cathode and its negative terminal is the anode. The terminal marked negative is the source of electrons that when connected to an external circuit will flow and deliver energy to an external device. When a battery is connected to an external circuit, electrolytes are able to move as ions within, allowing the chemical reactions to be completed at the separate terminals and so deliver energy to the external circuit. It is the movement of those ions within the battery which allows current to flow out of the battery to perform work. Historically the term "battery" specifically referred to a device composed of multiple cells, however the usage has evolved additionally to include devices composed of a single cell. Primary (single-use or "disposable") batteries are used once and discarded; the electrode materials are irreversibly changed during discharge. Common examples are the alkaline battery used for flashlights and a multitude of portable electronic devices. Secondary (rechargeable) batteries can be discharged and recharged multiple times using an applied electric current; the original composition of the electrodes can be restored by reverse current. Examples include the lead-acid batteries used in vehicles and lithium-ion batteries used for portable electronics such as laptops and smartphones. Batteries come in many shapes and sizes, from miniature cells used to power hearing aids and wristwatches to small, thin cells used in smartphones, to large lead acid batteries used in cars and trucks, and at the largest extreme, huge battery banks the size of rooms that provide standby or emergency power for telephone exchanges and computer data centers. According to a 2005 estimate, the worldwide battery industry generates US$48 billion in sales each year, with 6% annual growth. Batteries have much lower specific energy (energy per unit mass) than common fuels such as gasoline. In automobiles, this is somewhat offset by the higher efficiency of electric motors in producing mechanical work, compared to combustion engines.
PARTcloud - Battery
Battery (electricity)
For other uses, see Battery.
Various cells and batteries (top-left to bottom-right): two AA, one D, one handheld ham radio battery, two 9-volt (PP3), two AAA, one C, one camcorder battery, one cordless phone battery
Electrochemical reactions, Electromotive force
First production
Electronic symbol
The symbol for a battery in a circuit diagram. It originated as a schematic drawing of the earliest type of battery, a voltaic pile.
An electric battery is a device consisting of one or more electrochemical cells with external connections provided to power electrical devices such as flashlights, smartphones, and electric cars.[1] When a battery is supplying electric power, its positive terminal is the cathode and its negative terminal is the anode.[2] The terminal marked negative is the source of electrons that when connected to an external circuit will flow and deliver energy to an external device. When a battery is connected to an external circuit, electrolytes are able to move as ions within, allowing the chemical reactions to be completed at the separate terminals and so deliver energy to the external circuit. It is the movement of those ions within the battery which allows current to flow out of the battery to perform work.[3] Historically the term "battery" specifically referred to a device composed of multiple cells, however the usage has evolved additionally to include devices composed of a single cell.[4]
Primary (single-use or "disposable") batteries are used once and discarded; the electrode materials are irreversibly changed during discharge. Common examples are the alkaline battery used for flashlights and a multitude of portable electronic devices. Secondary (rechargeable) batteries can be discharged and recharged multiple times using an applied electric current; the original composition of the electrodes can be restored by reverse current. Examples include the lead-acid batteries used in vehicles and lithium-ion batteries used for portable electronics such as laptops and smartphones.
Batteries come in many shapes and sizes, from miniature cells used to power hearing aids and wristwatches to small, thin cells used in smartphones, to large lead acid batteries used in cars and trucks, and at the largest extreme, huge battery banks the size of rooms that provide standby or emergency power for telephone exchanges and computer data centers.
According to a 2005 estimate, the worldwide battery industry generates US$48 billion in sales each year,[5] with 6% annual growth.
Batteries have much lower specific energy (energy per unit mass) than common fuels such as gasoline. In automobiles, this is somewhat offset by the higher efficiency of electric motors in producing mechanical work, compared to combustion engines.
2 Principle of operation
3 Categories and types of batteries
3.1 Primary
3.2 Secondary
3.3 Cell types
3.3.1 Wet cell
3.3.2 Dry cell
3.3.3 Molten salt
3.3.4 Reserve
3.4 Cell performance
4 Capacity and discharge
4.1 C rate
4.2 Fast-charging, large and light batteries
5 Lifetime
5.1 Self-discharge
5.2 Corrosion
5.3 Physical component changes
5.4 Charge/discharge speed
5.5 Overcharging
5.6 Memory effect
5.7 Environmental conditions
5.8 Storage
6 Battery sizes
7 Hazards
7.1 Explosion
7.2 Leakage
7.3 Toxic materials
7.4 Ingestion
8 Chemistry
8.1 Primary batteries and their characteristics
8.2 Secondary (rechargeable) batteries and their characteristics
9 Solid state batteries
10 Homemade cells
Main article: History of the battery
A voltaic pile, the first battery
Italian physicist Alessandro Volta demonstrating his pile to French emperor Napoleon Bonaparte
The usage of "battery" to describe a group of electrical devices dates to Benjamin Franklin, who in 1748 described multiple Leyden jars by analogy to a battery of cannon[6] (Benjamin Franklin borrowed the term "battery" from the military, which refers to weapons functioning together[7]).
Italian physicist Alessandro Volta built and described the first electrochemical battery, the voltaic pile, in 1800.[8] This was a stack of copper and zinc plates, separated by brine-soaked paper disks, that could produce a steady current for a considerable length of time. Volta did not understand that the voltage was due to chemical reactions. He thought that his cells were an inexhaustible source of energy,[9] and that the associated corrosion effects at the electrodes were a mere nuisance, rather than an unavoidable consequence of their operation, as Michael Faraday showed in 1834.[10]
Although early batteries were of great value for experimental purposes, in practice their voltages fluctuated and they could not provide a large current for a sustained period. The Daniell cell, invented in 1836 by British chemist John Frederic Daniell, was the first practical source of electricity, becoming an industry standard and seeing widespread adoption as a power source for electrical telegraph networks.[11] It consisted of a copper pot filled with a copper sulfate solution, in which was immersed an unglazed earthenware container filled with sulfuric acid and a zinc electrode.[12]
These wet cells used liquid electrolytes, which were prone to leakage and spillage if not handled correctly. Many used glass jars to hold their components, which made them fragile and potentially dangerous. These characteristics made wet cells unsuitable for portable appliances. Near the end of the nineteenth century, the invention of dry cell batteries, which replaced the liquid electrolyte with a paste, made portable electrical devices practical.[13]
Main article: Electrochemical cell
Batteries convert chemical energy directly to electrical energy. A battery consists of some number of voltaic cells. Each cell consists of two half-cells connected in series by a conductive electrolyte containing anions and cations. One half-cell includes electrolyte and the negative electrode, the electrode to which anions (negatively charged ions) migrate; the other half-cell includes electrolyte and the positive electrode to which cations (positively charged ions) migrate. Redox reactions power the battery. Cations are reduced (electrons are added) at the cathode during charging, while anions are oxidized (electrons are removed) at the anode during charging.[14] During discharge, the process is reversed. The electrodes do not touch each other, but are electrically connected by the electrolyte. Some cells use different electrolytes for each half-cell. A separator allows ions to flow between half-cells, but prevents mixing of the electrolytes.
Each half-cell has an electromotive force (emf), determined by its ability to drive electric current from the interior to the exterior of the cell. The net emf of the cell is the difference between the emfs of its half-cells.[15] Thus, if the electrodes have emfs E 1 {\displaystyle {\mathcal {E}}_{1}} and E 2 {\displaystyle {\mathcal {E}}_{2}} , then the net emf is E 2 − E 1 {\displaystyle {\mathcal {E}}_{2}-{\mathcal {E}}_{1}} ; in other words, the net emf is the difference between the reduction potentials of the half-reactions.[16]
The electrical driving force or Δ V b a t {\displaystyle \displaystyle {\Delta V_{bat}}} across the terminals of a cell is known as the terminal voltage (difference) and is measured in volts.[17] The terminal voltage of a cell that is neither charging nor discharging is called the open-circuit voltage and equals the emf of the cell. Because of internal resistance,[18] the terminal voltage of a cell that is discharging is smaller in magnitude than the open-circuit voltage and the terminal voltage of a cell that is charging exceeds the open-circuit voltage.[19] An ideal cell has negligible internal resistance, so it would maintain a constant terminal voltage of E {\displaystyle {\mathcal {E}}} until exhausted, then dropping to zero. If such a cell maintained 1.5 volts and stored a charge of one coulomb then on complete discharge it would perform 1.5 joules of work.[17] In actual cells, the internal resistance increases under discharge[18] and the open circuit voltage also decreases under discharge. If the voltage and resistance are plotted against time, the resulting graphs typically are a curve; the shape of the curve varies according to the chemistry and internal arrangement employed.
The voltage developed across a cell's terminals depends on the energy release of the chemical reactions of its electrodes and electrolyte. Alkaline and zinc–carbon cells have different chemistries, but approximately the same emf of 1.5 volts; likewise NiCd and NiMH cells have different chemistries, but approximately the same emf of 1.2 volts.[20] The high electrochemical potential changes in the reactions of lithium compounds give lithium cells emfs of 3 volts or more.[21]
Categories and types of batteries
Main article: List of battery types
Batteries are classified into primary and secondary forms:
Primary batteries are designed to be used until exhausted of energy then discarded. Their chemical reactions are generally not reversible, so they cannot be recharged. When the supply of reactants in the battery is exhausted, the battery stops producing current and is useless.[22]
Secondary batteries can be recharged; that is, they can have their chemical reactions reversed by applying electric current to the cell. This regenerates the original chemical reactants, so they can be used, recharged, and used again multiple times.[23]
Some types of primary batteries used, for example, for telegraph circuits, were restored to operation by replacing the electrodes.[24] Secondary batteries are not indefinitely rechargeable due to dissipation of the active materials, loss of electrolyte and internal corrosion.
Main article: Primary cell
Primary batteries, or primary cells, can produce current immediately on assembly. These are most commonly used in portable devices that have low current drain, are used only intermittently, or are used well away from an alternative power source, such as in alarm and communication circuits where other electric power is only intermittently available. Disposable primary cells cannot be reliably recharged, since the chemical reactions are not easily reversible and active materials may not return to their original forms. Battery manufacturers recommend against attempting to recharge primary cells.[25] In general, these have higher energy densities than rechargeable batteries,[26] but disposable batteries do not fare well under high-drain applications with loads under 75 ohms (75 Ω). Common types of disposable batteries include zinc–carbon batteries and alkaline batteries.
Main article: Rechargeable battery
Secondary batteries, also known as secondary cells, or rechargeable batteries, must be charged before first use; they are usually assembled with active materials in the discharged state. Rechargeable batteries are (re)charged by applying electric current, which reverses the chemical reactions that occur during discharge/use. Devices to supply the appropriate current are called chargers.
The oldest form of rechargeable battery is the lead–acid battery, which are widely used in automotive and boating applications. This technology contains liquid electrolyte in an unsealed container, requiring that the battery be kept upright and the area be well ventilated to ensure safe dispersal of the hydrogen gas it produces during overcharging. The lead–acid battery is relatively heavy for the amount of electrical energy it can supply. Its low manufacturing cost and its high surge current levels make it common where its capacity (over approximately 10 Ah) is more important than weight and handling issues. A common application is the modern car battery, which can, in general, deliver a peak current of 450 amperes.
The sealed valve regulated lead–acid battery (VRLA battery) is popular in the automotive industry as a replacement for the lead–acid wet cell. The VRLA battery uses an immobilized sulfuric acid electrolyte, reducing the chance of leakage and extending shelf life.[27] VRLA batteries immobilize the electrolyte. The two types are:
Gel batteries (or "gel cell") use a semi-solid electrolyte.
Absorbed Glass Mat (AGM) batteries absorb the electrolyte in a special fiberglass matting.
Other portable rechargeable batteries include several sealed "dry cell" types, that are useful in applications such as mobile phones and laptop computers. Cells of this type (in order of increasing power density and cost) include nickel–cadmium (NiCd), nickel–zinc (NiZn), nickel metal hydride (NiMH), and lithium-ion (Li-ion) cells. Li-ion has by far the highest share of the dry cell rechargeable market. NiMH has replaced NiCd in most applications due to its higher capacity, but NiCd remains in use in power tools, two-way radios, and medical equipment.
In the 2000s, developments include batteries with embedded electronics such as USBCELL, which allows charging an AA battery through a USB connector,[28] nanoball batteries that allow for a discharge rate about 100x greater than current batteries, and smart battery packs with state-of-charge monitors and battery protection circuits that prevent damage on over-discharge. Low self-discharge (LSD) allows secondary cells to be charged prior to shipping.
Cell types
Many types of electrochemical cells have been produced, with varying chemical processes and designs, including galvanic cells, electrolytic cells, fuel cells, flow cells and voltaic piles.[29]
A wet cell battery has a liquid electrolyte. Other names are flooded cell, since the liquid covers all internal parts, or vented cell, since gases produced during operation can escape to the air. Wet cells were a precursor to dry cells and are commonly used as a learning tool for electrochemistry. They can be built with common laboratory supplies, such as beakers, for demonstrations of how electrochemical cells work. A particular type of wet cell known as a concentration cell is important in understanding corrosion. Wet cells may be primary cells (non-rechargeable) or secondary cells (rechargeable). Originally, all practical primary batteries such as the Daniell cell were built as open-top glass jar wet cells. Other primary wet cells are the Leclanche cell, Grove cell, Bunsen cell, Chromic acid cell, Clark cell, and Weston cell. The Leclanche cell chemistry was adapted to the first dry cells. Wet cells are still used in automobile batteries and in industry for standby power for switchgear, telecommunication or large uninterruptible power supplies, but in many places batteries with gel cells have been used instead. These applications commonly use lead–acid or nickel–cadmium cells.
Dry cell
Further information: Dry cell
A dry cell uses a paste electrolyte, with only enough moisture to allow current to flow. Unlike a wet cell, a dry cell can operate in any orientation without spilling, as it contains no free liquid, making it suitable for portable equipment. By comparison, the first wet cells were typically fragile glass containers with lead rods hanging from the open top and needed careful handling to avoid spillage. Lead–acid batteries did not achieve the safety and portability of the dry cell until the development of the gel battery.
A common dry cell is the zinc–carbon battery, sometimes called the dry Leclanché cell, with a nominal voltage of 1.5 volts, the same as the alkaline battery (since both use the same zinc–manganese dioxide combination). A standard dry cell comprises a zinc anode, usually in the form of a cylindrical pot, with a carbon cathode in the form of a central rod. The electrolyte is ammonium chloride in the form of a paste next to the zinc anode. The remaining space between the electrolyte and carbon cathode is taken up by a second paste consisting of ammonium chloride and manganese dioxide, the latter acting as a depolariser. In some designs, the ammonium chloride is replaced by zinc chloride.
Molten salt
Molten salt batteries are primary or secondary batteries that use a molten salt as electrolyte. They operate at high temperatures and must be well insulated to retain heat.
A reserve battery can be stored unassembled (unactivated and supplying no power) for a long period (perhaps years). When the battery is needed, then it is assembled (e.g., by adding electrolyte); once assembled, the battery is charged and ready to work. For example, a battery for an electronic artillery fuze might be activated by the impact of firing a gun. The acceleration breaks a capsule of electrolyte that activates the battery and powers the fuze's circuits. Reserve batteries are usually designed for a short service life (seconds or minutes) after long storage (years). A water-activated battery for oceanographic instruments or military applications becomes activated on immersion in water.
Cell performance
A battery's characteristics may vary over load cycle, over charge cycle, and over lifetime due to many factors including internal chemistry, current drain, and temperature. At low temperatures, a battery cannot deliver as much power. As such, in cold climates, some car owners install battery warmers, which are small electric heating pads that keep the car battery warm.
Capacity and discharge
A device to check battery voltage
A battery's capacity is the amount of electric charge it can deliver at the rated voltage. The more electrode material contained in the cell the greater its capacity. A small cell has less capacity than a larger cell with the same chemistry, although they develop the same open-circuit voltage.[30] Capacity is measured in units such as amp-hour (A·h). The rated capacity of a battery is usually expressed as the product of 20 hours multiplied by the current that a new battery can consistently supply for 20 hours at 68 °F (20 °C), while remaining above a specified terminal voltage per cell. For example, a battery rated at 100 A·h can deliver 5 A over a 20-hour period at room temperature. The fraction of the stored charge that a battery can deliver depends on multiple factors, including battery chemistry, the rate at which the charge is delivered (current), the required terminal voltage, the storage period, ambient temperature and other factors.[30]
The higher the discharge rate, the lower the capacity.[31] The relationship between current, discharge time and capacity for a lead acid battery is approximated (over a typical range of current values) by Peukert's law:
t = Q P I k {\displaystyle t={\frac {Q_{P}}{I^{k}}}}
Q P {\displaystyle Q_{P}} is the capacity when discharged at a rate of 1 amp.
I {\displaystyle I} is the current drawn from battery (A).
t {\displaystyle t} is the amount of time (in hours) that a battery can sustain.
k {\displaystyle k} is a constant around 1.3.
Batteries that are stored for a long period or that are discharged at a small fraction of the capacity lose capacity due to the presence of generally irreversible side reactions that consume charge carriers without producing current. This phenomenon is known as internal self-discharge. Further, when batteries are recharged, additional side reactions can occur, reducing capacity for subsequent discharges. After enough recharges, in essence all capacity is lost and the battery stops producing power.
Internal energy losses and limitations on the rate that ions pass through the electrolyte cause battery efficiency to vary. Above a minimum threshold, discharging at a low rate delivers more of the battery's capacity than at a higher rate. Installing batteries with varying A·h ratings does not affect device operation (although it may affect the operation interval) rated for a specific voltage unless load limits are exceeded. High-drain loads such as digital cameras can reduce total capacity, as happens with alkaline batteries. For example, a battery rated at 2 A·h for a 10- or 20-hour discharge would not sustain a current of 1 A for a full two hours as its stated capacity implies.
C rate
The C-rate is a measure of the rate at which a battery is being discharged. It is defined as the discharge current divided by the theoretical current draw under which the battery would deliver its nominal rated capacity in one hour.[32] A 1C discharge rate would deliver the battery's rated capacity in 1 hour. A 2C discharge rate means it will discharge twice as fast (30 minutes). A 1C discharge rate on a 1.6 Ah battery means a discharge current of 1.6 A. A 2C rate would mean a discharge current of 3.2 A. Standards for rechargeable batteries generally rate the capacity over a 4-hour, 8 hour or longer discharge time. Because of internal resistance loss and the chemical processes inside the cells, a battery rarely delivers nameplate rated capacity in only one hour. Types intended for special purposes, such as in a computer uninterruptible power supply, may be rated by manufacturers for discharge periods much less than one hour.
One should note that the C-rate presents a dimensional error: C is in fact expressed in ampere-hours and not in amperes like a current, and one can not express a current in ampere-hours. For this reason the concept It was introduced by the international standard IEC61434[33], It being equal to the capacity C divided by one hour, hence allowing a mathematically correct method of current designation. The figures used for expressing the discharge rate remain the same: one can speak of "2 It rate" instead of the dimensionally incorrect "2 C rate".
Fast-charging, large and light batteries
As of 2012[update], lithium iron phosphate (LiFePO
4) battery technology was the fastest-charging/discharging, fully discharging in 10–20 seconds.[34]
As of 2013[update], the world's largest battery was in Hebei Province, China. It stored 36 megawatt-hours of electricity at a cost of $500 million.[35] Another large battery, composed of Ni–Cd cells, was in Fairbanks, Alaska. It covered 2,000 square metres (22,000 sq ft)—bigger than a football pitch—and weighed 1,300 tonnes. It was manufactured by ABB to provide backup power in the event of a blackout. The battery can provide 40 megawatts of power for up to seven minutes.[36] Sodium–sulfur batteries have been used to store wind power.[37] A 4.4 megawatt-hour battery system that can deliver 11 megawatts for 25 minutes stabilizes the output of the Auwahi wind farm in Hawaii.[38]
Lithium–sulfur batteries were used on the longest and highest solar-powered flight.[39]
Battery life (and its synonym battery lifetime) has two meanings for rechargeable batteries but only one for non-chargeables. For rechargeables, it can mean either the length of time a device can run on a fully charged battery or the number of charge/discharge cycles possible before the cells fail to operate satisfactorily. For a non-rechargeable these two lives are equal since the cells last for only one cycle by definition. (The term shelf life is used to describe how long a battery will retain its performance between manufacture and use.) Available capacity of all batteries drops with decreasing temperature. In contrast to most of today's batteries, the Zamboni pile, invented in 1812, offers a very long service life without refurbishment or recharge, although it supplies current only in the nanoamp range. The Oxford Electric Bell has been ringing almost continuously since 1840 on its original pair of batteries, thought to be Zamboni piles.
Self-discharge
Disposable batteries typically lose 8 to 20 percent of their original charge per year when stored at room temperature (20–30 °C).[40] This is known as the "self-discharge" rate, and is due to non-current-producing "side" chemical reactions that occur within the cell even when no load is applied. The rate of side reactions is reduced for batteries are stored at lower temperatures, although some can be damaged by freezing.
Old rechargeable batteries self-discharge more rapidly than disposable alkaline batteries, especially nickel-based batteries; a freshly charged nickel cadmium (NiCd) battery loses 10% of its charge in the first 24 hours, and thereafter discharges at a rate of about 10% a month. However, newer low self-discharge nickel metal hydride (NiMH) batteries and modern lithium designs display a lower self-discharge rate (but still higher than for primary batteries).
Internal parts may corrode and fail, or the active materials may be slowly converted to inactive forms.
Physical component changes
The active material on the battery plates changes chemical composition on each charge and discharge cycle; active material may be lost due to physical changes of volume, further limiting the number of times the battery can be recharged. Most nickel-based batteries are partially discharged when purchased, and must be charged before first use.[41] Newer NiMH batteries are ready to be used when purchased, and have only 15% discharge in a year.[42]
Some deterioration occurs on each charge–discharge cycle. Degradation usually occurs because electrolyte migrates away from the electrodes or because active material detaches from the electrodes. Low-capacity NiMH batteries (1,700–2,000 mA·h) can be charged some 1,000 times, whereas high-capacity NiMH batteries (above 2,500 mA·h) last about 500 cycles.[43] NiCd batteries tend to be rated for 1,000 cycles before their internal resistance permanently increases beyond usable values.
Charge/discharge speed
Fast charging increases component changes, shortening battery lifespan.[43]
Overcharging
If a charger cannot detect when the battery is fully charged then overcharging is likely, damaging it.[44]
Memory effect
See also: Nickel–cadmium battery § Memory effect
NiCd cells, if used in a particular repetitive manner, may show a decrease in capacity called "memory effect".[45] The effect can be avoided with simple practices. NiMH cells, although similar in chemistry, suffer less from memory effect.[46]
Automotive lead–acid rechargeable batteries must endure stress due to vibration, shock, and temperature range. Because of these stresses and sulfation of their lead plates, few automotive batteries last beyond six years of regular use.[47] Automotive starting (SLI: Starting, Lighting, Ignition) batteries have many thin plates to maximize current. In general, the thicker the plates the longer the life. They are typically discharged only slightly before recharge.
"Deep-cycle" lead–acid batteries such as those used in electric golf carts have much thicker plates to extend longevity.[48] The main benefit of the lead–acid battery is its low cost; its main drawbacks are large size and weight for a given capacity and voltage. Lead–acid batteries should never be discharged to below 20% of their capacity,[49] because internal resistance will cause heat and damage when they are recharged. Deep-cycle lead–acid systems often use a low-charge warning light or a low-charge power cut-off switch to prevent the type of damage that will shorten the battery's life.[50]
Battery life can be extended by storing the batteries at a low temperature, as in a refrigerator or freezer, which slows the side reactions. Such storage can extend the life of alkaline batteries by about 5%; rechargeable batteries can hold their charge much longer, depending upon type.[51] To reach their maximum voltage, batteries must be returned to room temperature; discharging an alkaline battery at 250 mA at 0 °C is only half as efficient as at 20 °C.[26] Alkaline battery manufacturers such as Duracell do not recommend refrigerating batteries.[25]
Battery sizes
Main article: List of battery sizes
Primary batteries readily available to consumers range from tiny button cells used for electric watches, to the No. 6 cell used for signal circuits or other long duration applications. Secondary cells are made in very large sizes; very large batteries can power a submarine or stabilize an electrical grid and help level out peak loads.
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See also: Safety of electronic cigarettes § Fires, explosions, and other battery-related malfunctions
A battery explosion is generally caused by misuse or malfunction, such as attempting to recharge a primary (non-rechargeable) battery, or a short circuit.
When a battery is recharged at an excessive rate, an explosive gas mixture of hydrogen and oxygen may be produced faster than it can escape from within the battery (e.g. through a built-in vent), leading to pressure build-up and eventual bursting of the battery case. In extreme cases, battery chemicals may spray violently from the casing and cause injury. Overcharging - that is, attempting to charge a battery beyond its electrical capacity - can also lead to a battery explosion, in addition to leakage or irreversible damage. It may also cause damage to the charger or device in which the overcharged battery is later used.
Car batteries are most likely to explode when a short-circuit generates very large currents. Such batteries produce hydrogen, which is very explosive, when they are overcharged (because of electrolysis of the water in the electrolyte). During normal use, the amount of overcharging is usually very small and generates little hydrogen, which dissipates quickly. However, when "jump starting" a car, the high current can cause the rapid release of large volumes of hydrogen, which can be ignited explosively by a nearby spark, e.g. when disconnecting a jumper cable.
Disposing of a battery via incineration may cause an explosion as steam builds up within the sealed case.
Recalls of devices using Lithium-ion batteries have become more common in recent years. This is in response to reported accidents and failures, occasionally ignition or explosion.[52][53] An expert summary of the problem indicates that this type uses "liquid electrolytes to transport lithium ions between the anode and the cathode. If a battery cell is charged too quickly, it can cause a short circuit, leading to explosions and fires".[54][55]
Leakage
Many battery chemicals are corrosive, poisonous or both. If leakage occurs, either spontaneously or through accident, the chemicals released may be dangerous. For example, disposable batteries often use a zinc "can" both as a reactant and as the container to hold the other reagents. If this kind of battery is over-discharged, the reagents can emerge through the cardboard and plastic that form the remainder of the container. The active chemical leakage can then damage or disable the equipment that the batteries power. For this reason, many electronic device manufacturers recommend removing the batteries from devices that will not be used for extended periods of time.
Toxic materials
Many types of batteries employ toxic materials such as lead, mercury, and cadmium as an electrode or electrolyte. When each battery reaches end of life it must be disposed of to prevent environmental damage.[56] Batteries are one form of electronic waste (e-waste). E-waste recycling services recover toxic substances, which can then be used for new batteries.[57] Of the nearly three billion batteries purchased annually in the United States, about 179,000 tons end up in landfills across the country.[58] In the United States, the Mercury-Containing and Rechargeable Battery Management Act of 1996 banned the sale of mercury-containing batteries, enacted uniform labeling requirements for rechargeable batteries and required that rechargeable batteries be easily removable.[59] California and New York City prohibit the disposal of rechargeable batteries in solid waste, and along with Maine require recycling of cell phones.[60] The rechargeable battery industry operates nationwide recycling programs in the United States and Canada, with dropoff points at local retailers.[60]
The Battery Directive of the European Union has similar requirements, in addition to requiring increased recycling of batteries and promoting research on improved battery recycling methods.[61] In accordance with this directive all batteries to be sold within the EU must be marked with the "collection symbol" (a crossed-out wheeled bin). This must cover at least 3% of the surface of prismatic batteries and 1.5% of the surface of cylindrical batteries. All packaging must be marked likewise.[62]
Batteries may be harmful or fatal if swallowed.[63] Small button cells can be swallowed, in particular by young children. While in the digestive tract, the battery's electrical discharge may lead to tissue damage;[64] such damage is occasionally serious and can lead to death. Ingested disk batteries do not usually cause problems unless they become lodged in the gastrointestinal tract. The most common place for disk batteries to become lodged is the esophagus, resulting in clinical sequelae. Batteries that successfully traverse the esophagus are unlikely to lodge elsewhere. The likelihood that a disk battery will lodge in the esophagus is a function of the patient's age and battery size. Disk batteries of 16 mm have become lodged in the esophagi of 2 children younger than 1 year.[citation needed] Older children do not have problems with batteries smaller than 21–23 mm. Liquefaction necrosis may occur because sodium hydroxide is generated by the current produced by the battery (usually at the anode). Perforation has occurred as rapidly as 6 hours after ingestion.[65]
Primary batteries and their characteristics
Anode (−)
Cathode (+)
Max. voltage, theoretical (V)
Nominal voltage, practical (V)
Specific energy (MJ/kg)
Shelf life at 25 °C, 80% capacity (months)
Zinc–carbon Zn MnO2 1.6 1.2 0.13 Inexpensive. 18
Zinc–chloride 1.5 Also known as "heavy-duty", inexpensive.
(zinc–manganese dioxide) Zn MnO2 1.5 1.15 0.4–0.59 Moderate energy density.
Good for high- and low-drain uses. 30
Nickel oxyhydroxide
(zinc–manganese dioxide/nickel oxyhydroxide) 1.7 Moderate energy density.
Good for high drain uses.
(lithium–copper oxide)
Li–CuO Li CuO 1.7 No longer manufactured.
Replaced by silver oxide (IEC-type "SR") batteries.
(lithium–iron disulfide)
LiFeS2 Li FeS2 1.8 1.5 1.07 Expensive.
Used in 'plus' or 'extra' batteries. 337[66]
(lithium–manganese dioxide)
LiMnO2 Li MnO2 3.0 0.83–1.01 Expensive.
Used only in high-drain devices or for long shelf-life due to very low rate of self-discharge.
'Lithium' alone usually refers to this type of chemistry.
(lithium–carbon fluoride)
Li–(CF)n Li (CF)n 3.6 3.0 120
(lithium–chromium oxide)
Li–CrO2 Li CrO2 3.8 3.0 108
Mercury oxide Zn HgO 1.34 1.2 High-drain and constant voltage.
Banned in most countries because of health concerns. 36
Zinc–air Zn O2 1.6 1.1 1.59[67] Used mostly in hearing aids.
Zamboni pile Zn Ag or Au 0.8 Very long life
Very low (nanoamp, nA) current >2,000
Silver-oxide (silver–zinc) Zn Ag2O 1.85 1.5 0.47 Very expensive.
Used only commercially in 'button' cells. 30
Magnesium Mg MnO2 2.0 1.5 40
Secondary (rechargeable) batteries and their characteristics
(kJ/kg)
(kJ/liter)
NiCd 1.2 140 Nickel–cadmium chemistry.
High-/low-drain, moderate energy density.
Can withstand very high discharge rates with virtually no loss of capacity.
Moderate rate of self-discharge.
Environmental hazard due to Cadmium – use now virtually prohibited in Europe.
Lead–acid 2.1 140 Moderately expensive.
Moderate energy density.
Higher discharge rates result in considerable loss of capacity.
Environmental hazard due to Lead.
Common use – Automobile batteries
NiMH 1.2 360 Nickel–metal hydride chemistry.
Performs better than alkaline batteries in higher drain devices.
Traditional chemistry has high energy density, but also a high rate of self-discharge.
Newer chemistry has low self-discharge rate, but also a ~25% lower energy density.
Used in some cars.
NiZn 1.6 360 Nickel-zinc chemistry.
Moderately inexpensive.
High drain device suitable.
Low self-discharge rate.
Voltage closer to alkaline primary cells than other secondary cells.
No toxic components.
Newly introduced to the market (2009). Has not yet established a track record.
Limited size availability.
AgZn 1.86
1.5 460 Silver-zinc chemistry.
Smaller volume than equivalent Li-ion.
Extremely expensive due to silver.
Very high energy density.
Very high drain capable.
For many years considered obsolete due to high silver prices.
Cell suffers from oxidation if unused.
Reactions are not fully understood.
Terminal voltage very stable but suddenly drops to 1.5 volts at 70–80% charge (believed to be
due to presence of both argentous and argentic oxide in positive plate – one is consumed first).
Has been used in lieu of primary battery (moon buggy).
Is being developed once again as a replacement for Li-ion.
LiFePO4 3.3
3.0 360 790 Lithium-Iron-Phosphate chemistry.
Lithium ion 3.6 460 Various lithium chemistries.
Very expensive.
Not usually available in "common" battery sizes.
Lithium polymer battery is common in laptop computers, digital cameras, camcorders, and cellphones.
Very low rate of self-discharge.
Terminal voltage varies from 4.2 to 3.0 volts during discharge.
Volatile: Chance of explosion if short-circuited, allowed to overheat, or not manufactured with rigorous quality standards.
Solid state batteries
On 28 February 2017, The University of Texas at Austin issued a press release about a new type of solid-state battery, developed by a team led by Lithium-ion (Li-Ion) inventor John Goodenough, "that could lead to safer, faster-charging, longer-lasting rechargeable batteries for handheld mobile devices, electric cars and stationary energy storage".[68] More specifics about the new technology were published in the peer-reviewed scientific journal Energy & Environmental Science.
Independent reviews of the technology discuss the risk of fire and explosion from Lithium-ion batteries under certain conditions because they use liquid electrolytes. The newly developed battery should be safer since it uses glass electrolytes, that should eliminate short circuits. The solid-state battery is also said to have "three times the energy density" increasing its useful life in electric vehicles, for example. It should also be more ecologically sound since the technology uses less expensive, earth-friendly materials such as sodium extracted from seawater. They also have much longer life; ("the cells have demonstrated more than 1,200 cycles with low cell resistance"). The research and prototypes are not expected to lead to a commercially viable product in the near future, if ever, according to Chris Robinson of LUX Research. "This will have no tangible effect on electric vehicle adoption in the next 15 years, if it does at all. A key hurdle that many solid-state electrolytes face is lack of a scalable and cost-effective manufacturing process," he told The American Energy News in an e-mail.[69]
Homemade cells
Almost any liquid or moist object that has enough ions to be electrically conductive can serve as the electrolyte for a cell. As a novelty or science demonstration, it is possible to insert two electrodes made of different metals into a lemon,[70] potato,[71] etc. and generate small amounts of electricity. "Two-potato clocks" are also widely available in hobby and toy stores; they consist of a pair of cells, each consisting of a potato (lemon, et cetera) with two electrodes inserted into it, wired in series to form a battery with enough voltage to power a digital clock.[72] Homemade cells of this kind are of no practical use.
A voltaic pile can be made from two coins (such as a nickel and a penny) and a piece of paper towel dipped in salt water. Such a pile generates a very low voltage but, when many are stacked in series, they can replace normal batteries for a short time.[73]
Sony has developed a biological battery that generates electricity from sugar in a way that is similar to the processes observed in living organisms. The battery generates electricity through the use of enzymes that break down carbohydrates.[74]
Lead acid cells can easily be manufactured at home, but a tedious charge/discharge cycle is needed to 'form' the plates. This is a process in which lead sulfate forms on the plates, and during charge is converted to lead dioxide (positive plate) and pure lead (negative plate). Repeating this process results in a microscopically rough surface, increasing the surface area, increasing the current the cell can deliver.[75]
Daniell cells are easy to make at home. Aluminium–air batteries can be produced with high-purity aluminium. Aluminium foil batteries will produce some electricity, but are not efficient, in part because a significant amount of (combustible) hydrogen gas is produced.
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\begin{document}
\title{Zero-one law for directional transience of one dimensional excited random walks}
\begin{abstract}
The probability that a one dimensional excited random walk in stationary ergodic and elliptic cookie environment is transient to the right (left) is either zero or one. This solves a problem posed by Kosygina and Zerner \cite{kosygina2012excited}. As an application, a law of large numbers holds in these conditions. \\
\centerline{\bf{R\`{e}sum\`{e}}}
La probabilit\`{e} q'une marche al\`{e}atoire unidimensionnelle excite dans un environnement ergodique et elliptique soit transiente a gauche ou a droite est soit
nulle soit un. Ceci r\`{e}sout un probl\`{e}me pose par Kosygina et Zerner. Comme application,
une loi des grands nombres est valable dans de telles conditions.
\end{abstract} \hfil
\thanks{\textit{2000 Mathematics Subject Classification.} 60K35, 60K37.}
\thanks{\textit{Key words:}\quad excited random walk, cookie walk, recurrence, directional transience, zero-one law, law of large numbers, limit theorem, random environment.}
\section{Introduction\label{sec:intro} }
Excited random walk was introduced by Itai~Benjamini and David~B.~Wilson in 2003 \cite{benjamini2003excited}. Later
the model was generalized by Martin~P.~W.~Zerner \cite{zerner2005multi}. It was studied extensively in recent years by numerous researchers, and an almost up to date account may be found in the recent survey of Kosygina and Zerner \cite{kosygina2012excited}.
The generalized model due to Zerner is informally known as Cookie Walk, recently the term `Brownie Motion' has been gaining popularity among researchers in the field.
The model is defined as follows: Let $\Omega=[0,1]^{{\mathbb Z}\times{\mathbb N}}$, and endow this space with the standard $\sigma$-algebra (namely the product of Borel $\sigma$-algebras on the intervals). Let $\mu$ be a probability measure on $\Omega$ which is invariant and ergodic with respect to the ${\mathbb Z}$-shift (but not necessarily the ${\mathbb N}$-shift). We call $\mu$ the cookie distribution. We call each $\omega\in\Omega$ a cookie environment. Notationally, $\omega(x,n)\in[0,1]$ is called the $n$-th cookie in the location $x$.
We say that the distribution $\mu$ is {\em elliptic} if $\mu((0,1)^{{\mathbb Z}\times{\mathbb N}})=1$, and {\em uniformly elliptic} if there exists $\epsilon>0$ such that $\mu([\epsilon,1-\epsilon]^{{\mathbb Z}\times{\mathbb N}})=1$.
Given a cookie environment $\omega$ and an initial position $x\in{\mathbb Z}$ we define the excited random walk driven by $\omega$: \begin{eqnarray*} P_{\omega,x}(X_0=x)=1, \\
P_{\omega,x}(X_n=X_{n-1}+1\ |\ X_0,X_1,\ldots,X_{n-1}) &=& \omega(X_{n-1},\#\{k\leq n-1: X_k=X_{n-1}\}), \\
P_{\omega,x}(X_n=X_{n-1}-1\ |\ X_0,X_1,\ldots,X_{n-1}) &=& 1 - P_{\omega,x}(X_n=X_{n-1}+1\ |\ X_0,X_1,\ldots,X_{n-1}). \end{eqnarray*} We associate $\mu$ with the annealed, or averaged, distribution defined by \[ \mathbb P_x (\cdot)= \int_\OmegaP_{\omega,x} (\cdot) d\mu(\omega). \]
In this paper we are interested in the probability that the random walk is {\em transient to the right}, i.e. that $\lim_{n\to\infty}X_n=+\infty$. We use $A^+$ to denote the event that the random walk is transient to the right, and $A^-$ to denote the event of transience to the left (i.e. $\lim_{n\to\infty}X_n=-\infty$).
In their recent survey Kosygina and Zerner raised a version of the following problem (see Problem 3.5 of \cite{kosygina2012excited}):
\begin{prob}\label{prob:KZ}
Find conditions on the distribution $\mu$ which imply a zero-one law for a directional transience of one dimensional excited random walk, i.e. conditions which imply that $\mathbb P_0(A^+)\in\{0,1\}$.
\end{prob}
The main result of this paper answers Problem \ref{prob:KZ}, namely:
\begin{thm}\label{thm:directionalZeroOneLaw}
Let $\mu$ be a stationary ergodic (with respect to the ${\mathbb Z}$-shift) and elliptic probability measure on the space $\Omega$ of cookie environments. Then $\mathbb P_0(A^+)\in\{0,1\}$.
\end{thm}
\subsection{Law of Large Numbers} Kosygina and Zerner proved in \cite[Theorem 4.1]{kosygina2012excited} that for a stationary ergodic and elliptic probability measure over cookie environments, if a directional 0-1 law holds then a law of large numbers holds.
Using Theorem \ref{thm:directionalZeroOneLaw} an immediate corollary is the following law of large numbers.
\begin{thm}\label{thm:LLN} Let $\mu$ be a stationary ergodic (with respect to the ${\mathbb Z}$-shift $\theta$) and elliptic probability measure on the space $\Omega$ of cookie environments. Then $\mathbb P_0(\lim_{n\to\infty}\frac{X_n}{n}=v)=1$ for some deterministic $v\in[-1,1]$. \end{thm}
One can write a different, more direct, proof of Theorem~\ref{thm:LLN} by noticing that the stationarity assumption in the proof of Theorem~4.1 in \cite{kosygina2012excited} can be slightly weakened. We refer the reader to Theorem~4.3 in \cite{Amir2013excited} and the discussion above it for details.
\subsection{Previous work} In some examples in the literature, special cases of Theorem \ref{thm:directionalZeroOneLaw} are derived as special cases of stronger characterization theorems.
Benjamini and Wilson \cite{benjamini2003excited} showed that whenever a cookie environment $\omega$ satisfies $\omega(x,1)=p$ for all $x\in{\mathbb Z}$ and $\omega(x,i)=\frac{1}{2}$ for all $x\in{\mathbb Z}$ and $i\ge 2$, then the walk is $\mathbb{P}_\omega$-a.s.\ recurrent for all $p\in (0,1)$. Zerner \cite{zerner2005multi} showed that if the measure $\mu$ is stationary ergodic and satisfies $\mu([\frac{1}{2},1]^{{\mathbb Z}\times{\mathbb N}})=1$, then it is transient to the right if and only if either $\mu(\omega(0,1)=1)=1$ or $\delta > 1$, where $\delta=\mathbb{E}_\mu(\sum_{i=1}^\infty(2\omega(0,i)-1))$. Kosygina and Zerner \cite{kosygina2008positively} showed that whenever the measure $\mu$ is i.i.d.\ (that is, the sequence of columns $\omega(x,\cdot)$, $x\in{\mathbb Z}$, is i.i.d.\ under $\mu$), weakly elliptic (that is, $\mu( \prod_{i=1}^M\omega(0,i) ) > 0$ and $\mu( \prod_{i=1}^M (1-\omega(0,i) ) )>0$, and there exists a deterministic number $M$ so that $\mu(\omega(0,i)=\frac{1}{2}))=1$ for all $i>M$ then the walk is transient to the right if and only if $\delta > 1$, and transient to the left if and only if $\delta<-1$. Kosygina and Zerner \cite{kosygina2012excited} proved a Kalikow-type 0-1 law, i.e. a 0-1 law for (non directional) transience for stationary ergodic and elliptic measure over cookie environments, see Theorem~\ref{thm:KalikowType}.
\subsection{Structure of the paper} The paper is structured as follows: In Section \ref{sec:preliminaries} we present some concepts and processes that take part in the proof. In section \ref{sec:arrow} we introduce arrow environments, in section \ref{sec:zpluszminus} we introduce two associated processes $Z^+$ and $Z^-$, and in section \ref{sec:survival vs. hitting} we study some of their connections to cookie random walks. In section \ref{sec:comoz} we study monotonicity and symmetry properties of $Z^+$ and $Z^-$, and present an easy lemma which is, however, the core of our argument. In Section \ref{sec:0-1 Law for Directional Transience}
we reprove a theorem by Kosygina and Zerner. We do this for two purposes. The first purpose is to keep the paper self-contained, and the second is to enable us to easily use notations and lemmas from their proof in the proof of our main result. In Section \ref{sec:pfmain} we prove Theorem \ref{thm:directionalZeroOneLaw}.
\section{Preliminaries}\label{sec:preliminaries}
In this section we give some basic definitions and lemmas which are necessary for the proof of Theorem \ref{thm:directionalZeroOneLaw}.
\subsection{Arrow environments}\label{sec:arrow}
Let $\omega$ be a cookie environment. We can realize $\omega$ into a list of arrows, or instructions, which tell the walker where to walk to in every step of the process. More precisely, let $U=[0,1]^{{\mathbb Z}\times{\mathbb N}}$, and let $F:\Omega\times U\to\{0,1\}^{{\mathbb Z}\times{\mathbb N}}$ be defined as
$F(\omega,u)(x,n)={\bf 1}_{u(x,n)<\omega(x,n)}$. An element $a\in \{0,1\}^{{\mathbb Z}\times{\mathbb N}}$ is called an \emph{arrow environment}.
We now endow $U$ with the standard Borel $\sigma$-algebra $\mathcal{B}_U$, and with the product measure $\mathbb{P}_U$ of $\mathcal {U}[0,1]$ distributions. The following lemma is a standard Ergodic theoretic fact. For convenience, a proof sketch of this fact may be found in the Appendix.
\begin{lem}\label{lem:arrow_ergod}
Let $\mu$ be a probability measure on the space $\Omega$ of cookie environments which is stationary ergodic with respect to the ${\mathbb Z}$-shift $\theta$.
Consider $\{0,1\}^{{\mathbb Z}\times{\mathbb N}}$, the space of arrow environments with the standard Borel $\sigma$-algebra, and let $\nu$ be the probability measure induced from $(\Omega\times U,\mu\times \mathbb{P}_U)$ on $\{0,1\}^{{\mathbb Z}\times{\mathbb N}}$ by the function $F$. Then $\nu$ is stationary ergodic with respect to the shift $\theta$.
\end{lem}
The following fact, which is straightforward, lies behind the definition of arrow environments: \begin{fact}Given a cookie environment $\omega$, the law of a (non-random) walk moving according to the (random) arrow environment sampled from $\omega$ is the same as the quenched law of the cookie random walk on $\omega$. \end{fact}
Arrow environments were considered by Holmes and Salisbury in \cite{holmes2012combinatorial}. They used this construction to couple ERW on different cookie environments and deduced monotonicity results.
In this paper we consider arrow environments as a natural way to couple different processes on the same cookie environment. The use of arrow environments gives a direct approach to distilling the ``combinatorial" part from many of the probabilistic arguments appearing in the ERW literature. See e.g. section \ref{sec:survival vs. hitting}.
\subsection{The processes $Z^+$ and $Z^-$}\label{sec:zpluszminus}
To avoid degenerate cases we introduce the following definition. \begin{defn} We say that sequence of arrows $b\in\{0,1\}^{\mathbb N}$ is \emph{non-degenerate} if there are infinitely many $i\ge 1$ for which $b(i)\neq b(i+1)$. An arrow environment $a$ is called \emph{non-degenerate} if $a(x,\cdot)$ is non-degenerate for all $x\in{\mathbb Z}$. The subspace of all non-degenerate arrow environments is denoted by $\mathbf{A}\subset\{0,1\}^{{\mathbb Z}\times{\mathbb N}}$ \end{defn}
Let $a\in \mathbf{A}$ be a non-degenerate arrow environment and let $y\geq 1$. We define the processes $Z^+$ and $Z^-$ (the initial value $y$ and the sequence $a$ is suppressed in the notation) as follows: $Z^+_0=y$. Then, for every $n>0$, we define $Z^+_n$ to be the number of $1$-s until the $Z^+_{n-1}$-th zero in $a(n-1,\cdot)$. More precisely, if \[ \Theta_n=\inf\left\{j:\sum_{i=0}^j\big[1-a(n-1,i)\big]=Z^+_{n-1}\right\}, \] then we take $Z^+_n=\Theta_n-Z^+_{n-1}$.
We define $Z^-$ completely analogously, by replacing the roles of 0 and 1, and considering $a$ on the left half line rather than the right half line: $Z^-_0=y$, and $Z^-_n$ is the number of zeros until the $Z^-_{n-1}$-th one in $a(1-n,\cdot)$.
For ease of notation, we define for every non-degenerate $b\in\{0,1\}^{\mathbb N}$ the functions $U^+_b,U^-_b:{\mathbb N}\to{\mathbb N}\cup\{\infty\}$ by $U^+_b(0)=0$, \begin{equation}\label{eq:talorenshtein} U^+_b(x)=\inf\left\{j:\sum_{i=0}^j\big[1-b(i)\big]=x\right\}-x, \end{equation} and, defining $b^c$ by $b^c(x)=1-b(x)$, \[ U^-_b(x)=U^+_{b^c}(x). \] Using this notation, we may simply write $Z^+_n=U^+_{a(n-1,\cdot)}(Z^+_{n-1})$ and $Z^-_n=U^-_{a(1-n,\cdot)}(Z^-_{n-1})$.
The definition given here for $Z^+$ and $Z^-$ appeared first in \cite{Amir2013excited}, where the authors of that paper considered the case of any given number of walkers on the same cookie environment and used a natural generalization of the above process.
A slightly different version of the chains $Z^+$ and $Z^-$ was introduced and linked to one dimensional ERW by Kosygina and Zerner in 2008 \cite{kosygina2008positively}
in the context of {\em bounded environments}, i.e.\ environments for which there is a deterministic $M$ so that $\omega(x,i)=\frac{1}{2}$ for all $i>M$ and all $x\in{\mathbb Z}$. In such environments $Z^+$ and $Z^-$ may be viewed as a certain type of a branching process with migration. The connection of random walks to branching process with migration is traced back at least to Kesten, Kozlov, and Spitzer \cite{kesten1975limit} from 1975. An adaptation of their method to ERW was first made by Basdevant and Singh \cite{basdevant2008speed} in 2008. Using this connection, much can be said about the ERW, see e.g. Kosygina and Zerner \cite{kosygina2008positively} and \cite{kosygina2012excited} (transience versus recurrence, ballisticity, CLT), Basdevant and Singh \cite{basdevant2008speed} and \cite{basdevant2008rate} (ballisticity and asymptotic rate of diffusivity), Peterson \cite{peterson2012large} and \cite{peterson2012strict} (law of large deviation, slow-down phenomenon, and strict monotonicity results), Rastegar and Roitershtein \cite{rastegar2011maximum} (maximum occupation time) and Dolgopyat and Kosygina \cite{dolgopyat2011central} and Kosygina and Mountford \cite{kosygina2011limit} (limit laws).
\subsection{Survival of $Z^+$ and the hitting time $T_{-1}$}\label{sec:survival vs. hitting}
The aim of this section is to show that strict positivity of the process $Z^+$ is equivalent to the event that an excited walker on the given arrow environment never hits $-1$. This equivalence is shown to hold also for several walkers walking on the same arrow environment in \cite{Amir2013excited}.
Removing the probabilistic interpretation from the arguments of Kosygina and Zerner in Section 3 of \cite{kosygina2008positively}, a pure combinatorial condition for right-transience is obtained. Fix an arrow environment $a\in A$ and for every $m\in{\mathbb Z}$ set \[ T_{m}:=\inf\{t\ge 0:X_t=m\} \]\label{eq:defOfTminusOne} to be the hitting time of $m$ by the excited walk $X$ on the arrow environment $a$. Define \[ W_0=1; \text{ } W_n=\# \{t<T_{-1}: X_t=n-1 \text{ and } X_{t+1}= n\},\text{ } n\ge 1\] to be the total number of crossings of the edge $\{n-1,n\}$ by $X$ before hitting $-1$.
Notice that as $a$ is assumed to be in $A$, if $T_{-1}=\infty$ then $\lim_{t\to\infty} X_t = +\infty$. In particular, for each edge with non negative endpoints, the difference between the total number of its right crossings and left crossings by the walk is exactly 1. In the case where $T_{-1}<\infty$ we have an equality. Let us sum up this fact in the following remark.
\begin{rem}\label{rem:ComparingCrossings} The following hold for all $ n\ge 0$. \begin{enumerate} \item \label{c:1} If $T_{-1}=\infty$ then $W_n=1\ + \text{ total number of crossings of }\{n,n-1\}\text{ by } X.$ \label{eq:ComparingCrossingsTranCase} \item \label{c:2} If $\ T_{-1}<\infty$ then $W_n = \text{ the total number of crossings of } \{n,n-1\} \text{ by } X$ before time $T_{-1}$. \end{enumerate} \end{rem}
\begin{lem}\label{lem:comparingZandW} For all $n\ge 0$, the following hold. \begin{itemize} \item if $T_{-1}<\infty$ then $Z^+_n =W_n$ \label{ZEqualsRightCrossings} \item if $T_{-1}=\infty$ then $Z^+_n \geq W_n$ \label{ZisGraterThanRightCrossings} \end{itemize} Where $Z^+_n$ is defined in Section 2.2 with initial value $Z^+_0=1$. \end{lem}
\begin{proof} Assume first that $T_{-1}<\infty$. We will prove by induction on $n\ge 0$ that $Z^+_n = W_n$ for all $n\ge 0$. For $n=0$ we have $Z^+_0=1=W_0$ by definition. Assume now that $Z^+_n=W_n$. Since $T_{-1}<\infty$, the last crossing of the undirected edge $\{n,n+1\}$ by $X$ before time $T_{-1}$ is a left crossing, and therefore $a(n,i)=0$, where $i$ is the total number of visits of $X$ to position $n$ before time $T_{-1}$. This implies that the number of $0$-s in $\{a(n,1),...,a(n,i)\}$ equals the total number of crossings of $\{n,n-1\}$ before time $T_{-1}$. By Remark~\ref{rem:ComparingCrossings} the last quantity equals $W_n$. Since by the induction hypothesis $Z^+_n=W_n$, We get that $Z^+_n$ is the number $0$-s in $\{a(n,1),...,a(n,i)\}$. Now, $W_{n+1}$ is the number of ones in $\{a(n,1),...,a(n,i)\}$. The latter is exactly the number of $1$-s prior to $Z^+_n$ $0$-s in $a(n,\cdot)$, which is defined to be $Z_{n+1}^+$.
Consider now the case $T_{-1}=\infty$. Again, we will prove by induction on $n\ge 0$ that $Z^+_n \geq W_n$. For $n=0$ we have $Z_0=1=W_0$. Assume by induction that $Z^+_n\ge W_n$. Let $i$ be the total number of visits of $X$ to place $n$. Note that as $a\in A$ the process $X$ is transient and so $i<\infty$ and $a(n,i)=1$. By Lemma \ref{rem:ComparingCrossings} $W_n$ equals the number of $0$'s in $\{a(n,1),...,a(n,i)\}$ plus $1$. As in the first case by the induction hypothesis $Z^+_n$ is greater than or equal to the number of $0$'s in $\{a(n,1),...,a(n,i)\}$ plus $1$. Now, $W_{n+1}$ is the number of ones in $\{a(n,1),...,a(n,i)\}$. The latter is exactly the number of $1$-s prior to the $(Z_n^+ -1)$-st zero in $a(n,\cdot)$, and this number is less than or equal to $Z^+_{n+1}$. \end{proof}
As a result, we get the following theorem. \begin{thm}\label{thm:SurvivalEquivNonReturning} $T_{-1}<\infty$ if and only if $Z^+_n=0$ for some $n$, and $T_{1}<\infty$ if and only if $Z^-_n=0$ for some $n$. \end{thm} \begin{proof} For the first equivalence, assume first that $T_{-1}<\infty$, then $M:=\max\{X_t:t<T_{-1}\}<\infty$. Therefore by Lemma \ref{ZEqualsRightCrossings} we get that $Z^+_{M+1}=W_{M+1}=0$. On the other hand, if $T_{-1}=\infty$, then $W_n\ge 1$ for all $n\ge 0$. Now if $Z^+_n=0$ then by Lemma \ref{ZisGraterThanRightCrossings} we have that $W_n\le Z^+_n =0$, a contradiction. Considering $\bar{a}$ instead of $a$, where $\bar a (n,i)= 1- a (-n,i)$, the argument from the beginning of the present subsection shows the second equivalence. \end{proof}
\begin{rem} The proof of Theorem \ref{thm:SurvivalEquivNonReturning} shows that if $M:=\max\{X_t:t<T_{-1}\}\in{\mathbb N}\cup\{\infty\}$ is the maximal position of the walk before reaching $-1$ and $\tau=\inf \{ n\ge 0 : Z^+_n = 0\} \in{\mathbb N}\cup\{\infty\}$ is the extinction time of $Z^+$, then $\tau = M+1$, where, by convention, $\infty = \infty + 1$. \end{rem}
\subsection{Subduality of $Z^+$ and $Z^-$}\label{sec:comoz}
There are two immediate properties of $U^+$ and $U^-$ which will be crucial for our arguments:
\begin{obs}\label{obs:propsOfU} The following hold for any $b\in\{0,1\}^{\mathbb N}$.
\begin{enumerate}
\item\label{itm:UNondec} $U^+_b(x)$ and $U^-_b(x)$ are nondecreasing in $x$.
\item\label{itm:UplusUminusSmallerThenIdentity} $U^+_b \circ U^-_b(x) < x$ for all $x\in{\mathbb Z}_+$ (and equivalently $U^-_b \circ U^+_b(x) < x$).
\end{enumerate}
\end{obs}
The next lemma is simple but crucial for the proof of Theorem \ref{thm:directionalZeroOneLaw}. For $z\in{\mathbb Z}$ we define $\theta^z:A\to A$ by $(\theta^z a)(x,i)=a(x+z,i)$, $x\in{\mathbb Z}$ and $i\in{\mathbb N}$ is the ${\mathbb Z}$-shift map by $z$ steps to the left.
\begin{lem}[Subduality]\label{lem:subduality}
Assume that for the arrow environment $a\in A$ the process $Z^+$ with initial value $Z^+_0=x$ has $Z_l^+\ge y$. Then on the shifted arrow environment $\theta^{l-1}a$, the process $Z^-$ with initial value $Z^-_0=y$ has $Z_l^-\le x$.
\end{lem}
\begin{proof}
Property \ref{itm:UplusUminusSmallerThenIdentity} of Observation \ref{obs:propsOfU} gives us that $U^-_{a({l-1},\cdot)} \circ U^+_{a({l-1},\cdot)}(x)\le x$ for all $x\in{\mathbb Z}_+$. Using this together with the monotonicity property \ref{itm:UplusUminusSmallerThenIdentity} of Observation \ref{obs:propsOfU} $l$ times, we get:
$$ U^-_{a(0,\cdot)} \circ \ldots \circ U^-_{a({l-1},\cdot)} \circ U^+_{a({l-1},\cdot)} \circ \ldots \circ U^+_{a(0,\cdot)}(x)\le x.$$
In other words $U^-_{a(0,\cdot)} \circ \ldots \circ U^-_{a({l-1},\cdot)}(m)\le x$, where $m:=Z_l^+=U^+_{a({l-1},\cdot)} \circ \ldots \circ U^+_{a(0,\cdot)}(x)$. By the assumption $m\ge y$ and so by the monotonicity property \ref{itm:UplusUminusSmallerThenIdentity} of Observation \ref{obs:propsOfU} also $U^-_{a(0,\cdot)} \circ \ldots \circ U^-_{a({l-1},\cdot)}(y)\le x$.
To finish, note that the left hand side of the last inequality is by definition $Z_l^-\le x$, where the process $Z^-$ is defined on the shifted environment $\theta^{l-1}a$ with initial value $Z^-_0=y$.
\end{proof}
\section{A Kalikow type 0-1 law}\label{sec:0-1 Law for Directional Transience}
The main purpose of this section is to present the proof, originally by Kosygina and Zerner, of a Kalikow type 0-1 law, and by it set the ground for the proof of our main result, which is to be found in the next section. Remember the events $A^+$ and $A^-$ from the introduction.
\begin{thm}\cite[Theorem 3.2]{kosygina2012excited} \label{thm:KalikowType}
Let $\mu$ be a stationary ergodic and elliptic probability measure over cookie environments. Then $\mathbb{P}_0(A^+\cup A^-)\in\{0,1\}$.
\end{thm}
An interesting question related to Theorem \ref{thm:KalikowType} is the following:
\begin{prob}
Can the ellipticity assumption be weakened in Theorem \ref{thm:KalikowType}? If so, can it be weakened also in Theorem \ref{thm:directionalZeroOneLaw}? \end{prob}
As a convention, we say a probability measure $\mu$ over cookie environments has a given property of cookie environments if every cookie environment has it $\mu$-a.s.\ We say $\mu$ satisfies a given property $P$ of arrow environments if almost every arrow environment has the property $P$ with respect to the annealed measure $\mathbb P$ associated to $\mu$. For example $\mu$ is elliptic if $\mu ((0,1)^{{\mathbb Z}\times{\mathbb N}})=1$, and $\mu$ is non-degenerate if the induced measure on arrow environments satisfies $\mathbb P (a\in A)=1$.
Unless otherwise mentioned, from now onwards we assume that $\mu$ is a stationary ergodic and elliptic probability distribution over cookie environments. As a first reduction to the proofs of both Theorem \ref{thm:KalikowType} and Theorem \ref{thm:directionalZeroOneLaw} we will use the following definition and results from Section 2 of Kosygina and Zerner \cite{kosygina2012excited}, regarding the question of finiteness of the ERW range.
For $x\in{\mathbb Z}$ and $\omega\in\Omega$, let $R(x,\omega)$ be the event that $\sum_{i=1}^\infty (\omega(x,i))<\infty$ and $L(x,\omega)$ the event that $\sum_{i=1}^\infty (1-\omega(x,i))<\infty$.
Notice that it follows from the Borel-Cantelli Lemma that a probability measure $\mu$ over cookie environments satisfies $\mu(R(x,\omega))=\mu(L(x,\omega))=0$ for every $x\in{\mathbb Z}$, if and only if it is non-degenerate. Moreover, the Borel-Cantelli Lemma also implies the following lemma (see e.g. \cite[Lemma 2.2]{kosygina2012excited}).
\begin{lem}\label{lem:NotGettingStuck}
Assume that $\mu$ is a non-degenerate probability measure over cookie environments. Then $\mathbb P_0$-a.s.\ $$\limsup_{n\to\infty}X_n\in\{-\infty,+\infty\}\ \mbox{and}\ \liminf_{n\to\infty} X_n\in\{-\infty,+\infty\}$$
\end{lem}
\begin{thm}\label{thm:range}\cite[Theorem 2.3]{kosygina2012excited}
Let $\mu$ be a stationary ergodic and elliptic probability measure over cookie environments.
\begin{enumerate}
\item[(a)] If $\mu(R(0,\omega)))>0$ and $\mu(L(0,\omega)))>0$ then
the range is $\mathbb P_0$-a.s.\
finite.
\item[(b)] If $\mu(R(0,\omega)))=0$ and $\mu(L(0,\omega)))>0$ then $\mathbb P_0(A^+)=1$.
\item[(c)] If $\mu(R(0,\omega)))>0$ and $\mu(L(0,\omega)))=0$ then $\mathbb P_0(A^-)=1$.
\item[(d)] If $\mu(R(0,\omega)))=0$ and $\mu(L(0,\omega)))=0$ then the range is $\mathbb P_0$-a.s.\
infinite.
\end{enumerate}
\end{thm}
The proof of Theorem \ref{thm:range} is omitted.
\begin{cor}\label{cor:ellReduction}
Let $\mu$ be a stationary ergodic and elliptic probability measure over cookie environments. If $\mu$ is not non-degenerate then $\mathbb P_0(A^+),\mathbb P_0(A^-) \in \{0,1\}$. In particular, in this case $\mathbb P_0(A^+\cup A^-) \in \{0,1\}$.
\end{cor}
Given a walk $X_n$ on ${\mathbb Z}$ , a \emph{right excursion} (from $0$) is a sequence of steps $X_{\tau_0},\ldots, X_{\tau_1}\le \infty$ of the walk such that $X_{\tau_0}=0$, either $X_{\tau_1}=0$ or $\tau_1=\infty$, and $X_t>0$ for all $\tau_0<t<\tau_1$.
Call $m\geq 0$ an \emph{optional regeneration position} for an arrow environment $a$ if the walk started at $m$ never hits $m-1$, that is if $T_{m-1}=\infty$.
We call $m\geq 0$ a \emph{regeneration position} if in addition, when starting the walk from $0$, the walk $X_n$ reaches $m$ after some finite time.
Note that the arrow environment on $[m,\infty)$ remains unchanged until the walker reaches $m$ for the first time (if it ever does). It follows that if $m$ is an (optional) regeneration position, and the walk $X_n$ reaches $m$, then afterwards it will never return to $m-1$.
\begin{lem}\label{le:inf_opt_reg} Let $\mu$ be a stationary ergodic probability measure over cookie environments.
If $\mathbb{P}_0(T_{-1}=\infty)>0$ then there are $\mathbb{P}_0$-a.s.\ infinitely many optional regeneration positions.
\end{lem}
\begin{proof}
Let $p:=\mathbb{P}_0(T_{-1}=\infty)>0$.
By stationarity of the arrow environment, $\mathbb{P}_m(T_{m-1}= \infty) = p$ for any $m\geq 0$. By Lemma \ref{lem:arrow_ergod} we may apply the ergodic theorem to the measure $\nu$ to get
$$
\frac{1}{n}\sum_{m=1}^n \mathbf{1}_{\{m \text{ is a optional regeneration position}\}} \to p\,\, \nu-\text{a.s.}.
$$
In particular there are $\nu$-a.s.\ infinitely many optional regeneration positions.
\end{proof}
The following lemma is a part of Lemma 8 of \cite{kosygina2008positively}, which is proved for the the i.i.d.\ case.
\begin{lem}\label{lem:TisInfiniteFinitelyManyExcursions} Let $\mu$ be a stationary ergodic and non-degenerate probability measure over cookie environments. If $\mathbb{P}_0(T_{-1}=\infty)>0$ then there are a.s.\ only finitely many right excursions.
\end{lem}
\begin{proof}
Consider $\limsup X_n$. If $\limsup X_n<\infty$, then by Lemma \ref{lem:NotGettingStuck} $A^-$ holds. In particular, the number of right excursions is finite. If $\limsup X_n=\infty$, then since by Lemma \ref{le:inf_opt_reg} on the event $\limsup X_n=\infty$ there a.s.\ exist (infinitely many) optional regeneration positions, the walk hits such a position $m$ at some finite time and from that time on it never returns to $m-1$, let alone $0$, and thus there are only finitely many right excursions.
\end{proof}
Also the following lemma is a part of Lemma 8 of \cite{kosygina2008positively}.
\begin{lem}\label{lem:TisFiniteExcursionsAreFinite}
If $\mathbb{P}_{\omega,0}(T_{-1}<\infty)=1$ and $\omega$ is elliptic, that is $\omega\in(0,1)^{{\mathbb Z}\times{\mathbb N}}$, then all right excursions are $\mathbb{P}_{\omega,0}$-a.s.\ finite.
\end{lem}
\begin{proof}
The proof uses a finite modification argument which is standard (see \cite{kosygina2008positively} proof of Lemma 8, or \cite{kosygina2012excited} (3.2) and Figure 1.\ there).
For convenience we shall supply a sketch. We will prove that the $i$-th right excursion is a.s.\ finite by induction on $i$. For $i=0$ this is trivial.
Assume now that the first $i$ right excursions are a.s.\ finite and consider the past including the first step of the $(i+1)$-st excursion.
The event that the last excursion is finite depends only on what the walk has done in places $x>0$. Therefore, the probability that the $(i+1)$-st excursion is finite given that the past does not change when we modify parts of the past as long as we do not change the parts when $x>0$.
In particular it remains the same when we erase all visits to the negative integers and visits to zero are concatenated in time (simply by replacing enough of the first arrows above $0$ to be right arrows).
As the modified event has positive probability, conditioning on it, the probability for finiteness of the $(i+1)$-st excursion equals the probability that the first excursion is finite conditioned on making a pre-given sequence of first steps on the positive half line.
This equals $1$ by the assumption of the lemma since by ellipticity of $\omega$ there is a positive probability to make any pre-given finite sequence of moves.
\end{proof}
\begin{cor}\label{cor:TransVsExtinctions}[\cite[Lemma 3.3]{kosygina2012excited}, \cite[Corollary 3.7]{Amir2013excited}] Let $\mu$ be an elliptic and non-degenerate probability measure over cookie environments.
$\mathbb{P}_0(T_{-1}=\infty ) > 0$ if and only if $\mathbb{P}_0(A^+) > 0$
\end{cor}
\begin{proof}
Note that if $T_{-1}=\infty$, then by Lemma \ref{lem:NotGettingStuck} $\liminf X_n=+\infty$ which yields $A^+$.
For the other implication, assume $\mathbb{P}_0(T_{-1}<\infty ) = 1$, then $\mathbb{P}_{\omega,0}(T_{-1}<\infty ) = 1$ for $\mu$-a.e.\ $\omega\in \Omega$. By Lemma \ref{lem:TisFiniteExcursionsAreFinite} $\mathbb{P}_{\omega,0}$-a.s.\ all right excursions are finite and in particular $\mathbb{P}_{\omega,o}$-a.s.\ $X_n \nrightarrow +\infty$ for $\mu$-a.e.\ $\omega\in \Omega$. In other words, in this case $\mathbb{P}_0(A^+) = 0$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:KalikowType}] First note that in the case where $\mu$ is not non-degenerate, the conclusion of the theorem follows from Corollary \ref{cor:ellReduction}. For the rest of the proof we assume that $\mu$ is also non-degenerate.
If $\mathbb{P}_0 (T_{-1} = \infty) > 0$ (resp.\ $\mathbb{P}_0 (T_{1} = \infty) > 0$) then by Lemma \ref{lem:TisInfiniteFinitelyManyExcursions} there are $\mathbb{P}_0$-a.s.\ only finitely many right (resp.\ left) excursions.
In particular $\mathbb{P}_0$-a.s.\ the walk visits $0$ only finitely many times from the right (resp.\ left).
By the assumption, for every $m$
\[ \mathbb{P}_0\left[\prod_{i=m}^\infty (1-\omega(0,i))=0\right]=1 \ \ \left(\mbox{resp. } \mathbb{P}_0\left[\prod_{i=m}^\infty \omega(0,i)=0\right]=1\right) \]
so we have $\mathbb{P}_0$-a.s.\ only finitely many visits to zero, which implies the occurrence of the event $A^+\cup A^-$.
On the other hand,
if $\mathbb{P}_0 (T_{-1}<\infty)=1$ and $\mathbb{P}_0 (T_{1}<\infty)=1$ then by Lemma \ref{lem:TisFiniteExcursionsAreFinite} the walk is $\mathbb{P}_0$-a.s.\ not transient to the right and not transient to the left. This means it is $\mathbb{P}_0$-a.s. recurrent.
\end{proof}
For $y,n\in{\mathbb Z}_+$ and $B\subset{\mathbb Z}_+$ denote by $\mathbb{P}^y(Z_n^+ \in B)$ the probability that the process $Z^+$ with initial value $y$ satisfies $Z_n^+\in B$. $\mathbb{P}^y(Z_n^-\in B)$ is defined similarly.
Let $S_+$ and $S_-$ be the events that $\{Z_n^+ >0 \text{ for all } n\}$ and $\{Z_n^- >0 \text{ for all } n\}$, respectively.
\begin{cor}\label{cor:Reduction}
Let $\mu$ be a stationary ergodic, elliptic and non-degenerate probability measure over cookie environments. $\mathbb{P}_0(A^+)=1$ if and only if $\mathbb{P}^1(S_+)>0$ and $\mathbb{P}^1(S_-)=0$.
\end{cor}
\begin{proof}
If $\mathbb{P}_0(A^+)=1$ then by
Corollary \ref{cor:TransVsExtinctions} $\mathbb{P}_0(T_{-1}=\infty ) > 0$ and by Theorem \ref{thm:SurvivalEquivNonReturning} also $\mathbb{P}^1(S_+)>0$.
Assume for contradiction $\mathbb{P}^1(S_-)>0$, then by Theorem \ref{thm:SurvivalEquivNonReturning} also $\mathbb{P}_0(T_{1}=\infty ) > 0$,
and so by Corollary \ref{cor:TransVsExtinctions} also $\mathbb{P}_0(A^-)>0$. This contradicts the assumption.
For the other direction, again by Theorem \ref{thm:SurvivalEquivNonReturning} and Corollary \ref{cor:TransVsExtinctions} we know that $\mathbb{P}_0(A^+)>0$ and $\mathbb{P}_0(A^-)=0$.
By Theorem \ref{thm:KalikowType}, we have $\mathbb{P}_0(A^+\cup A^-)=1$, and so $\mathbb{P}_0(A^+)=1$.
\end{proof}
\section{Proof of the main result}\label{sec:pfmain}
This section is devoted to the proof of Theorem \ref{thm:directionalZeroOneLaw}. The following is a key proposition.
\begin{prop}\label{prop:main}
Assume that $\mu$ is a stationary ergodic and elliptic probability measure over cookie environments. If $\mathbb{P}^1(S_+)>0$, then $\mathbb{P}^1(S_-)=0$.
\end{prop}
We shall first prove Theorem \ref{thm:directionalZeroOneLaw} assuming Proposition \ref{prop:main}, and then turn to proving Proposition \ref{prop:main}.
\begin{proof}[Proof of Theorem \ref{thm:directionalZeroOneLaw}]
By Corollary \ref{cor:ellReduction} we may assume that $\mu$ is also non-degenerate. If $\mathbb{P}^1(S_+)>0$ then by Proposition \ref{prop:main}
$\mathbb{P}^1(S_-)=0$ and therefore by Corollary \ref{cor:Reduction} $\mathbb{P}_0(A^+)=1$. Symmetrically, if $\mathbb{P}^1(S_-)>0$ then $\mathbb{P}_0(A^-)=1$. To deal with the last case, namely that $\mathbb{P}^1(S_-\cup S_+)=0$, note that Corollary \ref{cor:TransVsExtinctions} implies that $\mathbb{P}_0(A^+\cup A^-)=0$. Since $\mu$ is non-degenerate, then by Lemma \ref{lem:NotGettingStuck} it holds that $\mathbb{P}_0(X_n=0\text{ i.o.})=1$. This completes the proof of the theorem.
\end{proof}
\begin{rem}\label{rem:easyCases}
In some specific cases, e.g. when $\mu$ is uniformly elliptic and stationary ergodic or when it is i.i.d.\ and elliptic, there are case specific proofs of Proposition \ref{prop:main}
which are significantly simpler than the one provided below for the general case.
For example, in the i.i.d. case both $Z^+$ and $Z^-$ are Markov chains, and thus it suffices to show that there exists $M$ such that $\{\mathbb{P}^k(\exists n\, Z^-_n<M):k=1,2,3,\ldots\}$ is bounded away from zero. This, however, follows
immediately from Lemma \ref{lem:getting close to one} and subduality (Lemma \ref{lem:subduality}).
If, on the other hand, we assume uniform ellipticity (but no longer i.i.d.), instead of the density statement of Lemma \ref{lem:upperdens0} one can easily show that $Z_n^+$ goes to infinity,
which in turn allows us to immediately use Lemma \ref{lem:subduality} to finnish the proof.
\end{rem}
We now prove Proposition \ref{prop:main}. We shall divide the proof into several steps. To the end of the paper we assume that all the assumptions of Proposition \ref{prop:main} hold.
\begin{lem}\label{lem:getting close to one}
For every $\epsilon>0$ there is some $y\in{\mathbb Z}_+$ so that $\mathbb{P}^y[S_+]>1-\epsilon$.
\end{lem}
\begin{proof}
$Z_n^+>0$ for all $n\ge 0$ if and only if $T_{-1}=\infty$, and so as in Lemma \ref{le:inf_opt_reg}, by stationarity and ergodicity of the environment
there are a.s.\ infinitely many optional regeneration positions.
In particular there is an a.s. finite (random) first optional regeneration position $\phi>0$. Consider now the process $Z^+$ at time $\phi$. If $Z^+$ is positive at time $\phi$ then it will stay positive forever. Fix $\epsilon>0$. Let $m$ be a large enough number so that $\mathbb{P}_0(\phi> m)<\frac{\epsilon}{2}$. Set $k_m=1$ and sequentially choose sufficiently large $k_{m-1},\ldots,k_0\in{\mathbb N}$ so that $\mathbb{P}(U_{a(j-1,\cdot)}(k_{j-1})< k_j)<\frac{\epsilon}{2m}$ for all $1\leq j\leq m$. Setting $y=k_0$, we have $$\mathbb{P}^y[S_+]\ge \mathbb{P}^y[Z^+_\phi\ge 1]\ge \mathbb{P} ( U_{a(i-1,\cdot)}(k_{i-1}) \ge k_{i},i=1,...,m,\phi\leq m ) \ge 1-\epsilon $$ by union bound.
\end{proof}
For a set $B\subset{\mathbb Z}_+$ denote by $\dens(B)$ the upper density of $B$, that is $$\dens(B)=\limsup_{n\to\infty}\frac{\#\{j\in B:j<n\}}{n}.$$
\begin{lem}\label{lem:upperdens0}
For every initial $y\ge 1$ it holds that $\mathbb{P}^y\big[\dens(\{n:Z^+_n<k\})=0\ |\ S_+\big]=1$ for every $k\ge 0$.
\end{lem}
\begin{proof}
Fix $k>0$. For $\gamma>0$ and $x\in{\mathbb Z}$ let $A_{\gamma,x}$ be the event that $\prod_{i=1}^k(1-\omega(x,i))>\gamma$. By stationarity $g(\gamma):=\mu[A_{\gamma,x}]$ is independent of $x$. By ellipticity, $\lim_{\gamma\to 0}g(\gamma)=1$. By ergodicity, the set $A_{\gamma}=\{x: A_{\gamma,x}\}$ has density $g(\gamma)$. Let $r$ be a natural number, let $B_r$ be the event that $\dens(\{n:Z^+_n<k\})>\frac{1}{r}$ and let $\gamma$ be small enough so that $g(\gamma)+\frac{1}{r}>1$. Then, on $B_r$
there are infinitely many $n$ such that both events $Z_n^+\le k$ and $A_{\gamma,n+1}$ occur.
Set $\mathcal{F}_n=\sigma \{\omega, a(0,\cdot),...,a(n-1,\cdot)\}$ be the $\sigma$-algebra generated by the all the cookies and the first $n$ piles of arrows to the right of and including $0$ and let $M_n=\mathbb{P}\big[S_+^c\ |\ \mathcal{F}_n \big]$ where $S_+^c$ is the complement of $S_+$. Then $(M_n)_{n\ge 1}$ is a bounded martingale with respect to the filtration $(\mathcal{F}_n)_{n\ge 1}$ converging $\mathbb{P}$-a.s.\ to $\mathrm{1}_{S_+^c}$. Now, the occurrence of $B_r$ implies that there are infinitely many $n$ for which both events $Z_n^+\le k$ and $A_{\gamma,n+1}$ occur, and therefore there are infinitely many $n$ with $M_n \ge \gamma$. In particular, on $B_r$, $\mathrm{1}_{S_+^c}>\gamma$, implying that $S_+^c$ occurs, so $\mathbb{P}\big[B_r\ |\ S_+\big]=0$. Since $r$ was chosen arbitrarily, we are done.
\end{proof}
\begin{lem}\label{lem:LargeProbPlusGetsAnyHeight}
Let $\epsilon>0$ and let $y$ be from Lemma \ref{lem:getting close to one}. For every $k\ge 0$ there is some $l$ so that $\mathbb{P}^y[Z_l^+>k]>1-2\epsilon$.
\end{lem}
\begin{proof}
Assume not, then by Lemma \ref{lem:getting close to one} there are $k$ and $\epsilon$ so that $\mathbb{P}^y\big[Z_l^+>k\ |\ S_+\big]\le 1-\epsilon=:\lambda<1$ for all $l$.
By linearity of expectation, $\mathbb{E}^y\big[\#\{l<n:Z^+_l>k\}\ |\ S_+\big]\le n \lambda$. Fix some $\lambda<\delta<1$, then by the Markov inequality we have $$\mathbb{P}^y\big[\frac{\#\{l<n:Z^+_l>k\}}{n}>\delta\ |\ S_+ \big]\le \frac{\lambda}{\delta}.$$
In other words,
$$\mathbb{P}^y\big[\frac{\#\{l<n:Z^+_l\le k\}}{n}\ge 1-\delta\ |\ S_+ \big]\ge 1- \frac{\lambda}{\delta}=:\alpha>0. $$
But therefore
$$\mathbb{P}^y\big[\text{ there are infinitely many $n$ such that }\frac{\#\{l<n:Z^+_l\le k\}}{n}\ge 1-\delta\ | \ S_+ \big]\ge \alpha,$$
contradicting Lemma \ref{lem:upperdens0}.
\end{proof}
By translation invariance of the probability measure $\mu$ we get from the Subduality Lemma \ref{lem:subduality} the corollary below.
Denote by $\mathbb{P}^k_{r}[Z_l^-\le y]$, $r\in{\mathbb Z}$, the probability that on the $r$-shifted arrow environment $\theta^r a$, the process $Z^-$ with initial value $k$ satisfies $Z_l^-\le y$.
\begin{cor}\label{cor:ShiftInvarianceOfTimeTrick}
For every $k\in{\mathbb N}$, $r_1,r_2\in{\mathbb Z}$ and $\epsilon>0$ there is some $l\in{\mathbb N}$ so that $\mathbb{P}^k_{r_1}[Z_l^-\le y] \ge \mathbb{P}^y_{r_2}[Z_l^+>k]\ge 1-2\epsilon$, where $y$ is as in Lemma \ref{lem:getting close to one}.
\end{cor}
\begin{proof}
Fix $k\in{\mathbb N}$ and $\epsilon>0$ and let $l$ be the one guaranteed in Lemma \ref{lem:LargeProbPlusGetsAnyHeight}. $r_1,r_2\in{\mathbb Z}$, then the right inequality follows from stationarity of $\mu$, and the left inequality follows from the Subduality Lemma \ref{lem:subduality} and stationarity of $\mu$.
\end{proof}
\begin{lem}\label{lem:a lot of chances to die}
For every $\epsilon>0$ there are $n_1<n_2<\ldots$ so that $\mathbb{P}^1[Z^-_{n_i} > y]<3\epsilon$, where $y$ is as in Lemma \ref{lem:getting close to one}.
\end{lem}
\begin{proof}
Fix $m_1=0$. There is $k_1$ so that $\mathbb{P}^1[Z^-_{m_1}>k_1]<\epsilon$. Let $l_1$ be the $l$ guaranteed by Corollary \ref{cor:ShiftInvarianceOfTimeTrick} for $k=k_1$.
Define $n_1=m_1+l_1$, then by Corollary \ref{cor:ShiftInvarianceOfTimeTrick} $\mathbb{P}^1[Z^-_{n_1}<y]\ge\mathbb{P}^1[Z^-_{m_1}\le k_1,Z^-_{n_1}<y]\ge 1-3\epsilon$.
Let $m_2>n_1$. There is $k_2$ so that $\mathbb{P}^1[Z^-_{m_2}>k]<\epsilon$ Let $l_2$ be the $l$ guaranteed by Corollary \ref{cor:ShiftInvarianceOfTimeTrick} for $k=k_1$ and $r_1=m_2$. Define $n_2=m_2+l_2$, then $\mathbb{P}^1[Z^-_{n_2}<y]\ge\mathbb{P}^1[Z^-_{m_2}\le k,Z^-_{n_2}<y]\ge 1-3\epsilon$.
Assume that $n_1<...<n_r$ were chosen so that $\mathbb{P}^1[Z^-_{n_i}<y]\ge 1-3\epsilon$ for all $1\le i\le r$.
At the $(r+1)$-st step, fix $m_{r+1}>n_r$. There is $k_{r+1}$ so that $\mathbb{P}^1[Z^{-}_{m_{r+1}} > k_{r+1}] <\epsilon$. Let $l_{r+1}$ be the $l$ guaranteed by Lemma \ref{lem:LargeProbPlusGetsAnyHeight} for $k=k_{r+1}$ and $r_1=m_{r+1}$. Define $n_{r+1}=m_{r+1}+l_{r+1}$, then \[\mathbb{P}^1[Z^{-}_{n_{r+1}}<y]\ge\mathbb{P}^1[Z^{-}_{m_{r+1}}\le k_{r+1},Z^-_{n_{r+1}}<y]\ge 1-3\epsilon.\]
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:main}]
Assume that $\mathbb{P}^1\big[S_+\big]>0$. Let $\delta>0$. We will show that $\mathbb{P}^1\big[Z^-_{n} > 0 \text{ for all } n\big]\le\delta$. Let $\epsilon=\frac{\delta}{4}>0$.
By Lemma \ref{lem:a lot of chances to die}, there are $n_1<n_2<\ldots$ so that $\mathbb{P}^1\big[Z^-_{n_i} \le y\big]\ge 1-3\epsilon$, where $y$ is as in Lemma \ref{lem:getting close to one}. As in the proof of Lemma \ref{lem:upperdens0}, set
\[
A_{\gamma,x}=\left\{\prod_{i=1}^y \omega(x,i)>\gamma\right\} \mbox{ for }\gamma>0\mbox{ and }x\in{\mathbb Z}.
\]
By stationarity $g(\gamma):=\mu[A_{\gamma,x}]$ is independent of $x$. By ellipticity, $\lim_{\gamma\to 0}g(\gamma)=1$.
Let $\gamma>0$ be small enough so that $g(\gamma)>1-\epsilon$. Then $\mathbb{P}^1[Z^-_{n_i} \le y,A_{\gamma,-n_i}]\ge 1-4\epsilon=1-\delta$ for all $i\ge 1$.
Let $D$ be the event that there are infinitely many $i$ such that $Z^-_{n_i} \le y \text{ and }A_{\gamma,-n_i}$. Then $\mathbb{P}^1[D]\ge 1-\delta$.
Set $\mathcal{F}_n=\sigma \{ \omega, a(0,\cdot),...,a(-(n-1) \}$ be the $\sigma$-algebra generated by all the cookies and the first $n$ piles of arrows to the left of and including $0$ and let $M_n=\mathbb{P}\big[S_+^c\ |\ \mathcal{F}_n \big]$. Then $(M_n)_{nge 1}$ is a bounded martingale with respect to the filtration $(\mathcal{F}_n)_{n\ge 1}$ converging a.s.\ to $\mathrm{1}_{S_-^c}$, and so if the event $D$ occurs, then also does $S_-^c$.
Therefore
\[
\mathbb{P}^1[Z^-_{n} > 0 \text{ for all } n]\le 1-\mathbb{P}^1[D]\leq\delta.
\]
Since $\delta$ was arbitrary, we are done.
\end{proof}
\section*{Acknowledgments} We thank Itai~Benjamini, Xiaoqin~Guo, Gady~Kozma, Igor~Shinkar and Ofer~Zeitouni for useful discussions. We also thank Jonathon Peterson for reading an earlier version of the manuscript. We thank the anonymous referees for valuable comments including the open problem regarding ellipticity that helped us to improve the mathematical results and the style of the paper.
The research of N.B. and T.O. was partially supported by ERC StG grant 239990. The research of G.A. was supported by Israeli Science Foundation grant ISF 1471/11.
{}
\begin{appendix}
\section{Proof sketch of Lemma \ref{lem:arrow_ergod}}
\begin{proof}[\nopunct]
The function $F$ is a measurable function from the product space $\Omega\times U$ with the shift $\theta\times\theta$ to the space $\mathbf{A}$ with the shift $\theta$, so that the measure on the latter is obtained from the former by $F$. It is straightforward to verify that $\mu\times \mathbb{P}'$ is stationary with respect to $\theta\times\theta$.
Note that
\begin{equation}\label{eq:ShiftComutesWithF}
F^{-1}[\theta A]= (\theta\times\theta) F^{-1}[A] \text{ for every }A\subset \mathbf{A}.
\end{equation}
Hence the stationarity of $\nu$ follows from the stationarity of $\mu\times\mathbb{P}'$.
For the proof of ergodicity, first note that it is enough to show that $( \Omega\times U , \mu \times \mathbb{P}', \mathcal{B}_{\Omega} \times \mathcal{B}_{U},\theta\times\theta)$ is ergodic. Indeed, using \eqref{eq:ShiftComutesWithF} the inverse image under $F$ of each $\theta$-invariant set is $(\theta\times\theta$)-invariant, and so it must be of either $\mathbb{P}\times P$-measure $0$ or $1$.
To prove the ergodicity of $\mu\times \mathbb{P}'$, let $f:\Omega\times U\to [0,1]$ be a $\mu\times \mathbb{P}'$-measurable function. We shall show that $f$ is a constant function.
Denote by $E$ the expectation operator with respect to $\mu\times \mathbb{P}'$.
First note that $\varphi:=E(f|\mathcal{B}_\Omega)$ is a $\theta$ invariant function on $\Omega$ and so by ergodicity it is $\mu$-a.s. constant in $[0,1]$.
Let $f_n=E[f|\mathcal{B}_\Omega \times \sigma \left( u(-n,\cdot),...,u(n,\cdot)\right)]$. Then $E[|f-f_n|]\to 0$ as $n\to \infty$, where \\ $\sigma \left( u(-n,\cdot),...,u(n,\cdot) \right)\subset\mathcal{B}_U$ is the minimal sub $\sigma$-algebra containing the ${\mathbb Z}$-coordinates $-n,...,n$.
Let $\epsilon>0$ and let $n_0$ be large enough so that for all $n\ge n_0$ $E[|f-f_n]<\epsilon$.
Let $\tilde{f_n}=(\theta\times\theta)^{3n}f_n$ be the $3n$ steps left shift of $f_n$.
Note that, since $\mathbb{P}'$ is the product measure,
$f_n$ and $\tilde f_n$ are independent conditioned on $\mathcal{B}_\Omega$. Therefore
\begin{equation}\label{eq:tal1}
E(f_n\tilde f_n | \mathcal{B}_\Omega) = E(\tilde f_n | \mathcal{B}_\Omega)E(f_n| \mathcal{B}_\Omega).
\end{equation}
Note also that
\begin{equation}\label{eq:tal2}
E[|f-\tilde{f_n}|] = E\left[\big|(\theta\times\theta)^{-3n}f-(\theta\times\theta)^{-3n}\tilde{f_n}\big|\right] = E[|f-f_n|]<\epsilon.
\end{equation}
Write $\varphi_n=E(\tilde f_n | \mathcal{B}_\Omega),$ and $\tilde{\varphi_n}=E(f_n| \mathcal{B}_\Omega).$ By \eqref{eq:tal2} and the triangle inequality, $E[|\varphi-\varphi_n|]<\epsilon$ and $E[|\varphi-\tilde{\varphi_n}|]<\epsilon$. Therefore
\begin{eqnarray*}
E[f_n\tilde{f_n}]
&=&E\big[ E[f_n\tilde f_n | \mathcal{B}_\Omega] \big]
\mathop{=}^{\eqref{eq:tal1}} E[\varphi_n\tilde{\varphi_n}]=
E[ (\varphi + \varphi_n - \varphi)( \varphi + \tilde{\varphi_n} -\varphi )]\\
&=&\varphi^2 + \varphi E[(\varphi_n - \varphi)] + \varphi E[(\tilde{\varphi_n}-\varphi)] + E[(\tilde{\varphi_n}-\varphi)(\tilde{\varphi_n}-\varphi )].
\end{eqnarray*}
(We used the fact that $\varphi$ is an a.s.\ constant and write it (notation abused) as a number.)
Using the fact that all functions are bounded from above by $1$, their difference is bounded from above by $2$ and we have
\begin{eqnarray*}
| E[f_n\tilde{f_n}-\varphi^2] | \le E[|\varphi_n - \varphi|]+ E[|\tilde{\varphi_n}-\varphi|] +2 E[|(\tilde{\varphi_n}-\varphi )|] < 4\epsilon.
\end{eqnarray*}
As $E[f_n\tilde{f_n}]\to E[f^2]$ as $n\to\infty$, taking $n$ to infinity and then $\epsilon$ to zero
yields $E[f^2]=E[\varphi^2]=E[\varphi]^2=E[f]^2$. Therefore $\mbox{var}(f)=0$ and $f$ is a $\mu\times\mathbb{P}'$-a.s.\ constant.
\end{proof}
\end{appendix}
\end{document} | arXiv |
Multi-view foreground segmentation via fourth order tensor learning
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A texture model based on a concentration of measure
August 2013, 7(3): 907-926. doi: 10.3934/ipi.2013.7.907
Statistical ranking using the $l^{1}$-norm on graphs
Braxton Osting 1, , Jérôme Darbon 1, and Stanley Osher 2,
Department of Mathematics, University of California, Los Angeles 90095, United States, United States
Department of Mathematics, UCLA, Los Angeles, CA 90095-1555
Received January 2012 Revised January 2013 Published September 2013
We consider the problem of establishing a statistical ranking for a set of alternatives from a dataset which consists of an (inconsistent and incomplete) set of quantitative pairwise comparisons of the alternatives. If we consider the directed graph where vertices represent the alternatives and the pairwise comparison data is a function on the arcs, then the statistical ranking problem is to find a potential function, defined on the vertices, such that the gradient of the potential optimally agrees with the pairwise comparisons. Potentials, optimal in the $l^{2}$-norm sense, can be found by solving a least-squares problem on the digraph and, recently, the residual has been interpreted using the Hodge decomposition (Jiang et. al., 2010). In this work, we consider an $l^{1}$-norm formulation of the statistical ranking problem. We describe a fast graph-cut approach for finding $\epsilon$-optimal solutions, which has been used successfully in image processing and computer vision problems. Applying this method to several datasets, we demonstrate its efficacy at finding solutions with sparse residual.
Keywords: graph-cut method., Kemeny-Snell ordering, rank aggregation, Statistical ranking, HodgeRank, $l^{1}$-norm minimization.
Mathematics Subject Classification: 62F07, 65F10, 05C20, 58A14, 05C8.
Citation: Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems & Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907
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Braxton Osting Jérôme Darbon Stanley Osher | CommonCrawl |
\begin{document}
\title{A coarea-type formula for the relaxation of a generalized elastica functional}
\begin{abstract} We consider the {\em generalized elastica functional} defined on $\lp{1}(\mathbb{R}^2)$ as $$F(u)=\left\{\begin{array}{ll}
\displaystyle\int_{\mathbb{R}^2}|\nabla u|(\alpha+\beta|\operatorname{div}
\frac{\nabla u}{|\nabla u|}|^p)\,dx,&\text{if $u\in {\mathrm{C}}^2(\mathbb{R}^2),$}\\ +\infty&\text{else},\end{array}\right.$$ where $p>1$, $\alpha>0$, $\beta\geq 0$. We study the $\lp{1}$-lower semicontinuous envelope $\overline{F}$ of $F$ and we prove that, for any $u\in{\mathrm{BV}}(\mathbb{R}^2)$, $\overline{F}(u)$ can be represented by a coarea-type formula involving suitable collections of $\wpq{2}{p}$ curves that cover the essential boundaries of the level sets $\{x,\,u(x)> t\}$, $t\in\mathbb{R}$. \end{abstract} \section{Introduction}
Being ${\mathrm{BV}}(\mathbb{R}^2)$ equipped with the strong topology of $\lp{1}$, we consider the functional $$\begin{array}{rcl} \mathscr{F}:{\mathrm{BV}}(\mathbb{R}^2)& \rightarrow& \mathbb{R}\\ u&\mapsto& H(u) + F(u)\end{array}$$ where $H$ is continuous and $\lp{1}$--coercive, i.e.,
$$H(u) \geq \|u\|_1 \quad\quad \forall u\in {\mathrm{BV}}(\mathbb{R}^2),$$ and $F$ is the generalized elastica functional defined on $\lp{1}(\mathbb{R}^2)$ as $$F(u)= \left\{ \begin{array}{ll}
\displaystyle{\int_{\mathbb{R}^2}|\nabla u|(\alpha+\beta|\operatorname{div}
\frac{\nabla u}{|\nabla u|}|^p) dx} & \mbox{if\,} u\in {\mathrm{C}}^2(\mathbb{R}^2),\\ +\infty & \mbox{otherwise.} \end{array} \right. $$
with $p> 1$, $\alpha>0$, $\beta\geq 0$, and the convention $|\nabla u||\operatorname{div} \frac{\nabla u}{|\nabla u|}|^p=0$ whenever $|\nabla u|=0$. When $\beta=0$, the $\lp{1}$-lower semicontinuous envelope $\overline{F}$ is simply (up to the multiplicative constant $\alpha$) the total variation in ${\mathrm{BV}}$, therefore, to simplify the notations and without loss of generality, we shall assume in the sequel that $\alpha=\beta=1$. The classical Bernoulli-Euler elastica functional associates with any smooth curve $\Gamma\subset\mathbb{R}^2$ its bending energy
$$\int_\Gamma|\kappa_\Gamma|^2d{\mathcal H}^1$$
where $\kappa_\Gamma$ is the curvature on $\Gamma$ and ${\mathcal H}^1$ the $1$-dimensional Hausdorff measure. The reason why we call $F$ a generalized elastica functional ensues from the fact that, if $u\in{\mathrm{C}}^2(\mathbb{R}^2)$, then, by Sard's Lemma, for almost every $t$, $\partial\{u>t\}\in {\mathrm{C}}^2$, $\nabla u/\left|\nabla u\right|(x)$ is orthogonal to $\partial \{u>t\}$ at every $x$ such that $u(x)=t$, and, by the coarea formula,
$$F(u)=\int_{\mathbb{R}}\left[\int_{\partial \{u>t\}}(1+\left|\mathbf{\kappa}_{\partial\{u>t\}}\right|^p )\,d\mathcal{H}^{1}\right]\mbox{d}t$$
We call {\it $p$-elastica energy} the following map defined on the class of measurable subsets of $\mathbb{R}^2$: $$E\subset\mathbb{R}^2\mapsto W(E)=\left\{\begin{array}{ll}
\displaystyle\int_{\partial E} [1+|\kappa_{\partial E}|^p] \mbox{d}{\mathcal H}^1&\mbox{if }\partial E\in{\mathrm{C}}^2,\\ +\infty&\mbox{otherwise}\end{array}\right. $$ Then, $$F(u)=\int_{\mathbb{R}}W(\{u>t\})dt\quad\mbox{when }u\in{\mathrm{C}}^2(\mathbb{R}^2).$$
A localized variant of $F$ has been introduced in~\cite{MasnouMorel,MM} as a variational model for the inpainting problem in digital image restoration, i.e., the problem of recovering an image known only out of a given domain. More precisely, it is claimed in~\cite{MasnouMorel,MM} that a reasonable inpainting candidate is a minimizer of this variant of $F$ under suitable boundary constraints. $F$ is also related to a variational model for visual completion arising from a neurogeometric modeling of the visual cortex~\cite{Petitot,CittiSarti}. As for the numerical approximation of minimizers of $F$, a globally minimizing scheme is proposed in~\cite{MasnouMorel,MM} for the case $p=1$ while the local minimization in the case $p>1$ is addressed in~\cite{ChanKangShen} using a fourth-order equation. The case $p>1$ is also tackled in~\cite{CGMP} using a relaxed formulation that involves Euler spirals. A smart and numerically tractable method to handle the high nonlinearity of the model, actually under a slightly different form, is proposed in~\cite{Ballester}.
The minimization of $F(u)$ in ${\mathrm{BV}}(\mathbb{R}^2)$ under the constraint that $u$ coincides with a given function out of a given domain has been addressed in~\cite{AM}. The existence of solutions follows from a simple application of the direct method of the calculus of variations. Similarly, proving that the problem $$\underset{u\in {\mathrm{BV}}}{\operatorname{Min}} \mathscr{F}(u)$$ has solutions requires the compactness of minimizing sequences and the lower semicontinuity of $\mathscr{F}$, both with respect to the strong topology of $\lp{1}$. The first property directly follows from the assumptions. Indeed, for every minimizing sequence $\{u_h\}\subseteq {\mathrm{BV}}(\mathbb{R}^2)$ with $\underset{h}{\sup}\; \mathscr{F}(u_h)< \infty$, it holds
$$\sup_h\|u_h\|_{{\mathrm{BV}}}\leq \sup_h\mathscr{F}(u_h)$$ and so, by the compactness theorem for functions of bounded variation, there exists a subsequence converging in $\lp{1}$. The second property, however, does not hold for $F$, as illustrated by the counterexample of Remark \ref{Fsci}, adapted from Bellettini, Dal Maso, and Paolini~\cite{BDP}.
\begin{figure}
\caption{Left, the characteristic function of a set of finite perimeter. Middle, an approximating smooth function and some of its level sets. Right, the pointwise limit of a sequence of such approximating smooth functions with uniformly bounded elastica energy.}
\label{cusp2}
\end{figure} \begin{oss}[$F$ is not lower semicontinuous in $\lp{1}(\mathbb{R}^2)$]\label{Fsci}
{\rm Let $E$ be the set of finite perimeter whose characteristic function $u=\mathds{1}_{E}$ is shown in Figure~\ref{cusp2}, left. We can approximate $u$ in $\lp{1}$ by a sequence of smooth functions $\{u_h\} \in \lp{1}(\mathbb{R}^2)\cap {\mathrm{C}}^2_0(\mathbb{R}^2)$, $0\leq u_h\leq 1$, similar to the function partially represented in Figure~\ref{cusp2}, middle. The pointwise limit of such sequence is represented in Figure~\ref{cusp2}, right. By the coarea formula,
$$F(u_h)=\int_{\mathbb{R}}\left[\int_{\partial \{u_h>t\}}(1+\left|\kappa_{\partial \{u_h>t\}
}\right|^p )d\mathcal{H}^{1}\right]dt$$ and the functions can be designed so that $$\sup_{h}\, F(u_h)< +\infty.$$ Therefore, $$u_h\rightarrow u \mbox{\;in\;}\lp{1}(\mathbb{R}^2)\quad\mbox{and}\quad\liminf_{h\rightarrow +\infty}F(u_h)< +\infty, $$ but, since $u\notin {\mathrm{C}}^2(\mathbb{R}^2)$, we have $$F(u)=+\infty.$$ Thus the functional $F$ is not lower semicontinuous in $\lp{1}(\mathbb{R}^2)$.}\qed \end{oss}
As usual for functionals that are not lower semicontinuous~\cite{DalMaso}, we consider the relaxation of $\mathscr{F}$, defined by: $$\overline{\mathscr{F}}(u)=\inf\left\lbrace \underset{h\rightarrow\infty}{\liminf}\;\mathscr{F}(u_h) : u_h\overset{\lp{1}}{\rightarrow}u\right\rbrace .$$ The relaxed functional $\overline{\mathscr{F}}$ is the largest lower semicontinuous functional minoring $\mathscr{F}$ and, in particular, $$\underset{u\in {\mathrm{BV}}}{\operatorname{Min}}\; \overline{\mathscr{F}}(u) = \underset{u\in {\mathrm{BV}}}{\inf}\; \mathscr{F}(u).$$ In addition, every minimizing sequence of $\mathscr{F}$ has a subsequence converging to a minimum point of $\overline{\mathscr{F}}$ and every minimum point of $\overline{\mathscr{F}}$ is the limit of a minimizing sequence of $\mathscr{F}$. More details on the theory of relaxation can be found in~\cite{DalMaso}. In our case, because of the continuity of $H$, we have $$\overline{\mathscr{F}} = H + \overline{F}.$$
The existence of minimizers of $\overline{\mathscr{F}}$ by the argument above does not provide much information about $\overline{F}$, about which only few things are known: it has been proven in~\cite{AM} that the $N$-dimensional version of $F$ is lower semicontinuous in ${\mathrm{C}}^2(\mathbb{R}^N)$ with respect to the strong topology of $\lp{1}$ when $N\geq 2$ and $p > N-1$, therefore $\overline{F}(u)=F(u)$ when $u$ is smooth. This constraint on $p$ is weakened in~\cite{LM} where, using results from~\cite{S}, it is shown that $F$ is lower semicontinuous on ${\mathrm{C}}^2(\mathbb{R}^2) \cap \lp{1}(\mathbb{R}^2)$ if $N=2$ and $p \geq 1$ or if $N\geq 3$ and $p\geq 2$. Finally, using the same techniques as in~\cite{AM,LM}, and combining with the recent results by Menne~\cite{Menne} on the locality of the mean curvature for integral varifolds, we can conclude that the equality of $F$ and $\overline{F}$ for smooth functions holds in any dimension for any $p\geq 1$. In particular, whenever $N=2$ which is the space dimension for this paper,
$$\overline{F}(u)=F(u)=\int_{\mathbb{R}}\left[\int_{\partial \{u>t\}}(1+\left|\kappa_{\partial \{u>t\}
}\right|^p )d\mathcal{H}^{1}\right]dt=\int_\mathbb{R} W(\{u>t\})\,dt,\quad\mbox{for every }u\in{\mathrm{C}}^2(\mathbb{R}^2).$$
We address here a more general question: is there also a coarea-type formula for $\overline{F}(u)$ when $u\in{\mathrm{BV}}(\mathbb{R}^2)$ has finite relaxed energy ?
\par This question is obviously related to the relaxation of $W$. The lower semicontinuous envelope of $W$ is defined for every measurable set $E\subset\mathbb{R}^2$ as
$$\overline{W}(E)=\inf\{\liminf_{k\to\infty}\; W(E_k),\; \partial E_k\in{\mathrm{C}}^2,\,|E_k\Delta E|\to 0\},$$ where $\Delta$ denotes the symmetric difference operator for sets. We will prove in Proposition~\ref{equiv} that, for any measurable $E\subset\mathbb{R}^2$ such that $\overline{W}(E)<\infty$ and for every $c>0$, $$\overline{F}(c\one{E})=c\overline{W}(E).$$
The properties of sets with finite relaxed energy $\overline{W}$ have been extensively studied in~\cite{BDP,BM1,BM}. It is proved in particular in \cite{BDP,BM1} that, for any $p>1$, $\overline{W}$ can be represented by a functional depending on systems of curves of class $\wpq{2}{p}$ that recover and extend the essential boundary of $E$. Another equivalent representation involving Hutchinson's curvature varifolds is provided in \cite{BM}. Can these results be used in a straightforward way to give an explicit expression of $\overline{F}(u)$ for any $u\in{\mathrm{BV}}(\mathbb{R}^2)$?
A first observation is that, as in the example of Figure~\ref{fig:notcoincide}, $\overline{F}(u)$ and $\displaystyle\int_\mathbb{R}\overline{W}(\partial^*\{u>t\})dt$ do not coincide in general for $u\in{\mathrm{BV}}(\mathbb{R}^2)$. In the latter integral, $\partial^*\{u>t\}$ denotes the essential boundary of the level set $\{u>t\}$, that has finite perimeter for almost every $t$. Recall that the essential boundary of a set of finite perimeter is the set of points where an approximate tangent exists, see~\cite{AFP}.
\begin{figure}\label{fig:notcoincide}
\end{figure}
\par A more surprising example of a situation where $\overline{F}(u)$ and $\displaystyle\int_\mathbb{R}\overline{W}(\partial^*\{u>t\})dt$ do not coincide is provided in the example below that we shall visit again in Remark~\ref{iii-est-nec}. Let $u=\one{F\cup G}+\one{F}$ with $F, G$ shown in Figure~\ref{embedded}, left. The level set $\{u>0\}$ coincides with $F\cup G$ and is clearly smooth. The boundary of the level set $\{u>1\}$ has two cusps. The pointwise limit of a sequence of smooth sets that approximate $\{u>1\}$ with convergence of the elastica energy to $\overline{F}(\{u>1\})$ contains the segment joining the two cusps, according to Theorem~8.6 in~\cite{BM1}. By the same theorem, the approximation of $\{u>0\}$ will not contain this segment. Clearly, $\{u>0\}$ and $\{u>1\}$ cannot be contemporaneously approximated using two nested sequences of sets. Yet $\overline{F}(u)$ is finite, as shown using the construction of Figure~\ref{embedded}, right, that we found during discussions with Vicent Caselles and Matteo Novaga. In this construction, the sets $G\setminus F$ and $F$ are approximated using smooth sets that do not intersect. Both pointwise limits contain a "bridge" with multiplicity $2$ that joins both components of $G$. \begin{figure}
\caption{The sets $G$ and $F$ can be simultaneously approximated without violating the noncrossing condition}
\label{embedded}
\end{figure}
\par Let us now come back to smooth functions and how their energy simply relates to the energy of their level sets. Given $u\in{\mathrm{BV}}(\mathbb{R}^2)$ such that $\overline{F}(u)<\infty$, one considers a sequence of smooth functions $(u_h)$ converging to $u$ in
$\lp{1}(\mathbb{R}^2)$ and such that $F(u_h)\to \overline{F}(u)$ as $h\to\infty$. Possibly extracting a subsequence, one can assume that for almost every $t$, $\{u_h>t\}$ converges to $\{u>t\}$ in measure, i.e. $|\{u_h>t\}\Delta\{u>t\}|\to 0$. In addition, by Fatou's Lemma, $$\int_\mathbb{R}\liminf_{h\to\infty}\,W(\{u_h>t\})dt\leq\liminf_{h\to\infty}\int_\mathbb{R} W(\{u_h>t\})dt=\overline{F}(u)$$ therefore $\liminf_{h\to\infty}W(\{u_h>t\})<\infty$ is finite for almost every $t$. It follows that, for almost every $t$, the sequence of $1$-dimensional varifolds with unit multiplicity ${\displaystyle{\mathbf{v}}}(\partial\{u_h>t\},1)$ has uniformly bounded mass, and uniformly bounded curvature in $\lp{p}$. By the properties of varifolds~\cite{Si} and the stability of absolute continuity (see Example 2.36 in~\cite{AFP}), there exists a subsequence ${\displaystyle{\mathbf{v}}}(\partial\{u_{h_k}>t\},1)$ \textit{depending on $t$} and a limit integral $1$-varifold $V_t$ such that
$$\int_{\mathbb{R}^2}(1+|\kappa_{V_t}|^p)d\|V_t\|\leq \liminf_{h\to\infty}\,W(\{u_h>t\})$$ In addition, one can prove~\cite{AM} that the support $M_t$ of $V_t$ contains $\partial^*\{u>t\}$ for almost every $t$. Furthermore, if $u$ is \textit{smooth}, $\kappa_{V_t}$ coincides almost everywhere on $\partial\{u>t\}$ with $\kappa_{\partial\{u>t\}}$ thus
$$W(\{u>t\})=\int_{\partial\{u>t\}}(1+|\kappa_{\partial\{u>t\}}|^p)d{\mathcal H}^1\leq\int_{\mathbb{R}^2}(1+|\kappa_{V_t}|^p)d\|V_t\|\leq\liminf_{h\to\infty}\,W(\{u_h>t\}).$$ By a simple integration and the coarea formula, it follows that for any smooth function $u$, $F(u)=\overline{F}(u)$. The argument above is exactly the $2$-dimensional version of the more general proof proposed in~\cite{AM,LM} to prove the equality of $F$ and $\overline{F}$ on smooth functions in any dimension for various ranges of values of $p$.
Can a similar argument be used if $u$ is \textit{unsmooth}? A tentative strategy could be the following: \begin{enumerate}
\item show, if possible, that the limit varifolds $V_t$ built above are nested, i.e. $\operatorname{int}V_t\subset\operatorname{int}V_{t'}$ if $t>t'$, where $\operatorname{int}V_t$ denotes the set enclosed (in the measure-theoretic sense) by the support of $V_t$. Again, observe that $\displaystyle\int_{\mathbb{R}^2}(1+|\kappa_{V_t}|^p)d\|V_t\|\leq \liminf_{h\to\infty}\,W(\{u_h>t\})$.
\item using the results of~\cite{BDP}, build a sequence of sets $E_h^t$ (for a suitable dense set of values $t$) such that $\partial E_h^t\to M_t$ (being $M_t$ the support of $V_t$) and $W(E_h^t)\to\displaystyle\int_{\mathbb{R}^2}(1+|\kappa_{V_t}|^p)d\|V_t\|$. The varifolds $V_t$ being nested, one could actually build $E_h^t$ so that $E_h^t\subset E_h^{t'}$ if $t>t'$. \item by a suitable smoothing of the sets $E_h^t$, build a smooth function $\tilde u_h$ such that $F(\tilde u_h)\leq \displaystyle\int_\mathbb{R} W(E_h^t)dt+\frac 1 h$. \item passing to the limit, possibly using a subsequence, show that $\tilde u_h$ tends to $u$ in $\lp{1}$ and using the lower semicontinuity of $\overline{F}$, conclude that
$$\overline{F}(u)=\int_\mathbb{R}\int_{\mathbb{R}^2}(1+|\kappa_{V_t}|^p)d\|V_t\|\,dt$$ \end{enumerate}
This strategy has however a major difficulty: the fact that the limit varifolds are nested is not clear at all. It would be an easy consequence of the existence of a subsequence $(u_{h_k})$ such that the varifolds ${\displaystyle{\mathbf{v}}}(\partial\{u_{h_k}>t\},1)$ converge to $V_t$ for almost every $t$. But such subsequence {\it may not exist} in general as shown by the counterexample below due to G. Savar{\'e}~\cite{savare}.
\begin{ex}[Savar\'e~\cite{savare}]\label{savare}{\rm Let us design a sequence of functions $\{\tilde{u}_n\}\subset {\mathrm{C}}^{0}([0,1]^2)$ with smooth level lines $\{\tilde u_n=t\}$ satisfying
$$\sup_n \int_{\mathbb{R}} \int_{\partial \{\tilde{u}_n(x)>t\}\cap (0,1)^2}(1+\left|\kappa_{\partial \{\tilde{u}_n(x)>t\}\cap (0,1)^2}\right|^p ) \,d\mathcal{H}^{1} \,dt < \infty,$$ but such that there exists no subsequence $(t\mapsto{\displaystyle{\mathbf{v}}}(\partial\{\tilde u_{n_k}>t\},1))_k$ converging for almost every $t$ to a varifold $V_t$. \par\noindent Consider the following 2-periodic function on $\mathbb{R}$ : $$u(x)= \left\{ \begin{array}{rl} 1 & \mbox{\;if \,} x\in ]0,1/2[\cup]1,2[, \\ -1 & \mbox{\;if \,}x\in ]1/2,1[ \end{array} \right. $$ \noindent Define $u_n(x)= u(2^n x)$ and consider $$U(x)=\int_0^x u(s) \mbox{d}s\, , \quad \quad U_n(x)=\int_0^x u_n(s) \mbox{d}s.$$
\begin{figure}
\caption{Graphs of the functions $u$ and $U$}
\end{figure}
\noindent The sequence $\{U_n\}$ is equilipschitz, because $|U_n'(x)|=1$, and $$U_n(x)=\int_0^x u_n(s) \mbox{d}s = \dfrac{1}{2^n}\int_0^{2^n x} u(s) \mbox{d}s = \dfrac{1}{2^n} U(2^n x).$$ Thus $$\{U_n(x)=t\}=\{U(2^n x)= 2^n t\}\quad \forall t\in \mathbb{R}.$$ Then $$\mathcal{H}^0\left( \{U(x)=\sigma \}\right) = \left\{ \begin{array}{rl} 3 & \mbox{\,\;if \,} \sigma\in ]k,k+1/2[, \\ 1 & \mbox{\,\;if \,} \sigma\in ]k+1/2,k+1[ \end{array} \right.\quad\forall k\in\mathbb{N}, $$ and so, for almost every $t$, $$\mathcal{H}^0\left( \{U_n(x)=t \}\right) = \left\{ \begin{array}{rl} 3 & \mbox{\,\;if \,} t\in ] \frac{2k}{2^{n+1}}, \frac{2k+1}{2^{n+1}} [, \\ 1 & \mbox{\,\;if \,} t\in ] \frac{2k+1}{2^{n+1}}, \frac{2k+2}{2^{n+1}} [ \end{array} \right.\quad\forall k\in\mathbb{N}. $$ \par\noindent We now define the sequence of functions $$\tilde{u}_n(x_1, x_2)=U_n(x_1) \quad \quad x=(x_1,x_2)\in [0,1]^2$$ and consider the sequence of varifolds associated with its level lines $$V_t^n = {\bf v}(\partial\{\tilde{u}_n(x)>t\}\cap (0,1)^2, 1).$$ Remark that $U_n(1)=1/2$ for every $n$ and, by the coarea formula, it is easy to check that
$$\sup_n \int_{\mathbb{R}} \int_{\partial \{\tilde{u}_n(x)>t\}\cap (0,1)^2}(1+\left|\kappa_{\partial \{\tilde{u}_n(x)>t\}\cap (0,1)^2}\right|^p ) \,d\mathcal{H}^{1} \,dt< 3/2.$$ Nevertheless we can show that there exists no subsequence $\{V^n_t\}$ (not relabelled) and no family $(V_t)_{t\in \mathbb{R}}$ of varifolds such that $V^n_t \rightharpoonup V_t$ in the sense of varifolds for almost every $t$.
\par For that we can consider the sequence $\{\mu_{V^n_t}\}$ of the weight measures of the varifolds $V^n_t$ defined as $$\mu_{V^n_t} = \mathcal{H}^1(\partial \{\tilde{u}_n(x)>t\}\cap (0,1)^2)$$ therefore $$\mu_{V^n_t} = \mathcal{H}^0\res \{U_n(x)=t\}.$$ By contradiction we suppose that there exists a subsequence $(\mu_{V^n_t})$ (not relabelled) and a family of limit measures $(\mu(t))_{t\in \mathbb{R}}$ such that $$\langle \mu_{V^n_t},\varphi \rangle=\langle\mathcal{H}^0\res \{U_n(x)=t\}, \varphi \rangle \rightarrow \langle\mu(t), \varphi\rangle \quad \text{for a.e.}\; t \quad \forall \varphi \in {\mathrm{C}}^1_0([0,1]).$$
\par Define $$f_n(t) = \mathcal{H}^0\res\{U_n(x)=t\}([0,1]).$$ Since $\{f_n\}$ is uniformly bounded we get $$f_n(t) \rightarrow \mu(t)([0,1]) \quad\text{for a.e. $t$}$$
and so $\mu(t)([0,1])$ is bounded. Then, by the Dominated Convergence Theorem, we get $$f_n(t) \rightarrow \mu(t)([0,1]) \quad \text{in} \quad \lp{2}(]0,2[).$$ \par On the other hand $f_n(t)=f(2^{n+1}t)$ where $f$ is the 2-periodic function on $\mathbb{R}$ defined as $$f(t)= \left\{ \begin{array}{rl} 3 & \mbox{\;if \,} t\in ]0,1[ \\ 1 & \mbox{\;if \,} t\in ]1,2[ \end{array} \right. $$ and so, by the Riemann-Lebesgue Theorem, it ensues that $f_n \rightharpoonup 2$ weakly in $\lp{2}$, therefore, by the strong convergence in $\lp{2}$ established above, we should have $\mu(t)([0,1])=2$. But $2$ cannot be the strong limit of $\{f_n\}$ thus there exists no subsequence of $\{\mathcal{H}^0\res \{U_n(x)=t\}\}$ converging for almost every $t$ to a limit measure $\mu(t)$.\qed} \end{ex}
\par It is now clear that, given a sequence of smooth functions $(u_h)$ converging in $\lp{1}$ to $u\in{\mathrm{BV}}$ with uniformly bounded generalized elastica energy for some $p>1$, we cannot expect in general the convergence of a subsequence $(\partial\{u_{h_k}>t\})$ to a limit system of curves $\Gamma_t$ for almost every $t$. Instead {\it we will prove in this paper that we can find a countable and dense set of values $I$ such that $\partial\{u_{h_k}>t\}\to \Gamma_t$ for every $t\in I$}. Then, for almost every remaining value $t\in\mathbb{R}\setminus I$, a limit system of curves can be built as the limit of a sequence $(V_{t_n})$, $t_n\in I$, $t_n\to t$. This "dense" diagonal extraction is obtained by generalizing the approach used in~\cite{MM} to study the same functional $\overline{F}$ on a restricted class $\mathcal{S}$ of functions $u\in {\mathrm{BV}}(\mathbb{R}^2)$ such that, for a given $\Omega$ with smooth boundary $\partial\Omega\in{\mathrm{C}}^\infty$, \begin{itemize}
\item $u=U_0$ on $\mathbb{R}^2\setminus \Omega$ with $U_0$ analytic on $\mathbb{R}^2$, $F(U_0)<+\infty$ and the level lines of $U_0$ in $\Omega$ satisfy a few regularity assumptions;
\item for $\mathcal{L}^1$-a.e. $t$ the restriction to $\Omega$ of $\partial^* \{u>t\}$ coincides, up to a $\mathcal{H}^1$-negligible set, with the trace of finitely many curves of class $\wpq{2}{p}$ with each of them joining two points on $\partial\Omega$ and smoothly connected to a level line of $U_0$ out of $\Omega$ (see~\cite{MM} for details). \end{itemize} In this paper, the boundary constraints are dropped, in particular the level lines are no more constrained to link two points on a given boundary. Dropping this constraint raises a few technical difficulties, in particular in some situations of accumulation that will be detailed later on. The "dense diagonal" convergence technique mentioned above was used in~\cite{MM} to build a set of curves that cover the level lines of a minimizer of $\overline{F}$ in the class ${\cal S}$. We shall here extend this convergence technique to obtain, for every $u\in{\mathrm{BV}}(\mathbb{R}^2)$ with $\overline{F}(u)<+\infty$, a coarea-type representation formula for $\overline{F}(u)$ using the $p$-elastica energies of curves that cover the essential boundaries of the level sets of $u$. To be more precise, we will associate with each $u\in{\mathrm{BV}}(\mathbb{R}^2)$ having finite energy a class of functions $\mathscr{A}(u)$ defined as follows: $\Phi\in\mathscr{A}(u)$ whenever $\Phi:t\in\mathbb{R}\mapsto \{\gamma_t^1,\cdots,\gamma_t^N\}$ with $N$ depending on $t$ and $\{\gamma_t^1,\
cdots,\gamma_t^N\}$ is a finite collection of curves of class $\wpq{2}{p}$ without crossing (but possibly with tangential self-contacts) that satisfy nesting compatibility constraints. Defining the functional $$\Phi\in \mathscr{A}(u) \mapsto G(\Phi)=\int_{\mathbb{R}} W(\Phi(t)) \mbox{d}t,$$ we will show in Theorem~\ref{principale} that for every $u\in {\mathrm{BV}}(\mathbb{R}^2)$ with $\overline{F}(u) < \infty$ there holds $$\overline{F}(u) = \underset{\Phi \in \mathscr{A}(u)} {\operatorname{Min}} G(\Phi),$$ which is the desired coarea-type representation formula. The main difficulty in the proof is to handle properly the situations of accumulation that possibly occur for the graphs of approximating functions.
Let us conclude this introduction with a short comment about the problem in higher dimensions. Is there a similar decomposition of $\overline{F}$ using suitable covering of level hypersurfaces? It is an open problem to our knowledge and, anyway, the solution needs not involve finite collections of hypersurfaces at each level. Indeed, an example due to Brakke~\cite{Br} consists of an integral $2$-varifold in $\mathbb{R}^3$ with uniformly bounded mean curvature and such that, at no point of a set with positive measure, the varifold's support can be represented as the graph of a multi-function. Even the control of the whole second fundamental form is not enough: it is shown in \cite[Thms 3.3 and 3.4, Example 5.9]{AGP}) that if $p>N\geq 3$, the limit varifold of a sequence of smooth boundaries with equibounded $\lp{p}$-norm of the second fundamental form needs not be representable as a finite union of manifolds of class $\wpq{2}{p}$. In a companion paper~\cite{MasnouNardi2}, we propose instead
a completely different strategy based on varifolds associated with gradient Young measures.
The plan of the paper is as follows: in Section~\ref{notation} we introduce a few notations, and we define and discuss the class $\mathscr{A}(u)$ mentioned above. We prove in Section~\ref{G} that the minimum problem for $G$ has at least a solution in $\mathscr{A}(u)$, and in Section~\ref{formula} we show the characterization formula for $\overline{F}$. We further illustrate in Section~\ref{oFoW} the connection between $\overline{F}$ and $\overline{W}$. Lastly, in Section~\ref{omega}, we analyze the generalized elastica functional localized on a domain $\Omega\subset \mathbb{R}^2$.
\section{Notations and preliminaries}\label{notation}
Throughout the paper, $p>1$ is a real number, $\mathcal{L}^n$ the Lebesgue measure on $\mathbb{R}^n$, ${\mathcal H}^k$ the $k$-dimensional Hausdorff measure, and ${\mathrm{C}}^r$, $\lp{p}$, $\wpq{m}{p}$, ${\mathrm{BV}}$ the usual function spaces. For any $E \subseteq \mathbb{R}^n$ we will also denote $|E|=\mathcal{L}^n(E)$. The topological boundary of $E$ is denoted as $\partial E$ and, if $E$ has finite perimeter~\cite{AFP}, $\partial^* E$ is its essential boundary, i.e. $\partial^*E = \mathbb{R}^n\setminus (E_0\cup E_1)$ where, for every $t\in [0,1]$,
$$E_t= \left\lbrace x\in \mathbb{R}^n : \underset{r \rightarrow 0}{\lim} \dfrac{|E\cap B(x,r)|}{|B(x,r)|} = t\right\rbrace .$$
In addition, by Federer Theorem~\cite[Thm 3.61]{AFP}, $\partial^*E$ coincides, up to a ${\mathcal H}^{N-1}$-negligible set, with the set of points where the inner normal $\displaystyle\frac{D\one{E}}{|D\one{E}|}$ exists.
If the topological boundary $\partial E$ can be viewed, locally, as the graph of a function of class ${\mathrm{C}}^r$ (resp. $W^{m,p}$), we write $\partial E \in {\mathrm{C}}^ r$ (resp. $W^{m,p}$).
\par Unless specified, we now focus on two-dimensional sets. We first recall the definition of the index of a point with respect to a plane curve $\gamma$~\cite{rudin-real-complex}.
\begin{defi}
Let $\gamma : [0,1] \rightarrow \mathbb{C}$ be a ${\mathrm{C}}^1$-curve with support $(\gamma)=\gamma([0,1])$. The index of a point $p\in \mathbb{C}\setminus(\gamma)$ with respect to $\gamma$ is defined by $$\operatorname{I}(p, \gamma) = \dfrac{1}{2\pi i} \int_{\gamma}\dfrac{\,dz}{z-p}\cdot$$
\end{defi} In the sequel, we denote as $k_{\gamma}(s)=\gamma''(s)$ the curvature vector at a point $\gamma(s)$ of a curve $\wpq{2}{p}$-curve $\gamma$ parameterized with arc-length $s$, i.e. at constant unit velocity. We shall use the following convenient and classical lemma, whose proof is recalled.
\begin{lemma}\label{mon}
Let $\mathcal{F} = \left( X_t \right)_{t\in \mathbb{R}}$ be a monotone family of sets, $X_t \subseteq \mathbb{R}^n$ for all $t$. Then, there exists an at most countable set $D \subseteq \mathbb{R}$ such that for every compact set $K\subset \mathbb{R}^n$
$$\underset{s\rightarrow t}{\lim}\; |(X_s \Delta X_t)\cap K| = 0 \quad \forall t\in \mathbb{R}\setminus D.$$ We call $D$ the set of discontinuities of $\mathcal{F}.$ \end{lemma}
\noindent{\bf Proof\;\, } The family $\mathcal{F}$ is monotone so the function
$$t \mapsto | X_t \cap K|$$
is monotone for every compact set $K$ and it has at most countably many discontinuity points whose collection is denoted as $D$. Then for every $t\in \mathbb{R}\setminus D$ we have $| X_s \cap K| \rightarrow | X_t \cap K| $ as $s \rightarrow t$ and because of the monotonicity of $\mathcal{F}$ we get
$$\underset{s\rightarrow t}{\lim}\; |(X_s \Delta X_t)\cap K| = 0 \quad \forall t\in \mathbb{R}\setminus D.$$ \qed
\noindent Following~\cite{BDP}, we now define the notion of system of curves of class $\wpq{2}{p}$.
\begin{defi}\label{sistemi}
By a system of curves of class $\wpq{2}{p}$ we mean a finite family $\Gamma=\{\gamma_1,\cdots,\gamma_N\}$ of closed curves of class $\wpq{2}{p}$ (thus ${\mathrm{C}}^1$) admitting a parameterization (still denoted by $\gamma_i$) $\gamma_i\in \wpq{2}{p}\left([0,1],\mathbb{R}^2 \right)$ with constant velocity. Moreover, every curve of $\Gamma$ can have tangential self-contacts but without crossing and two curves of $\Gamma$ can have tangential contacts but without crossing. In particular, $\gamma_i'(t_1)$ and $\gamma_j'(t_2)$ are parallel whenever $\gamma_i(t_1)=\gamma_j(t_2)$ for some $i,j\in\{1,...,N\}$ and $t_1,t_2\in[0,1]$. \par The trace $(\Gamma)$ of $\Gamma$ is the union of the traces $(\gamma_i)$. We define the interior of the system $\Gamma$ as $$ \operatorname{Int}(\Gamma) = \{x \in \mathbb{R}^2 \setminus (\Gamma) : \operatorname{I}(x, \Gamma) = 1 \; \operatorname{mod}\; 2\}$$ where $\operatorname{I}(x, \Gamma) = \sum_{i=1}^N \operatorname{I}(x, \gamma_i)$.\\ The multiplicity function $\theta_\Gamma$ of $\Gamma$ is $$\theta_{\Gamma} : (\Gamma) \rightarrow \mathbb{N} \quad \quad \theta(z)=\sharp \{\Gamma^{-1}(z)\},$$ where $\sharp$ is the counting measure. \\ If the system of curves is the boundary of a set $E$ with $\partial E\in {\mathrm{C}}^2$, we simply denote it as $\partial E$. \end{defi}
\begin{oss}{\rm Remark that, by previous definition, every $|\gamma_i'(t)|$ is constant for every $t\in[0,1]$ so the arc-length parameter is given by $s(t)=t L_i$ where $L_i$ in the length of $\gamma_i$. Denoting by $\tilde{\gamma}_i$ the curve parameterized with respect to the arc-length parameter we have $$s\in[0, L_i] \,,\,\, \tilde{\gamma}_i(s)= \gamma_i(s/L_i)\,,\,\, \tilde{\gamma}_i''(s)=\dfrac{\gamma_i''(s)}{L_i^2}.$$ Now, the curvature {\bf k} as a functions of $s$, verifies $${\bf k} = \tilde{\gamma}_i''(s)$$ which implies
$$ \int_{0}^{L_i} \left( 1+|\tilde{\gamma}_{i}''(s)|^p\right)\mbox{d}s=\int_{0}^{L_i} \left( 1+|k|^p\right)\mbox{d}s= \int_{0}^{1} \left( |\gamma_{i}'(t)|+L_i^{1-2p}|\gamma_{i}''(t)|^p\right)\mbox{d}t.$$ Then, the condition $\gamma_i \in \wpq{2}{p}\left([0,1],\mathbb{R}^2 \right)$ implies that $\tilde{\gamma}_i\in \wpq{2}{p}\left([0,L_i],\mathbb{R}^2 \right)$ and, for simplicity, in the sequel we denote by $\gamma_i$ the curve parameterized with respect to the arc-length parameter.} \end{oss}
The $p$-elastica energy of a system $\Gamma$ of curves of class $\wpq{2}{p}$ is defined as
$$W(\Gamma)=\sum_{i=1}^N W(\gamma_i)=\sum_{i=1}^N \int_{(\gamma_i)} \left( 1+|{\bf k}_{\gamma_{i}}|^p\right)\mbox{d}\mathcal{H}^1.$$
We will use several times the following result that combines Lemma 3.1 in~\cite{BDP} and Proposition 6.1 in~\cite{BM1}: if a bounded open set $E\subset\mathbb{R}^2$ is such that $\overline{W}(E)<\infty$ then $$\overline{W}(E)=\min\;\{W(\Gamma): \Gamma \in \mathcal{A}(E)\}$$
where $\mathcal{A}(E)$ denotes the class of all {\it finite} systems of curves $\Gamma$ such that $(\Gamma)\supseteq \partial E$ and $|E \Delta \operatorname{Int}(\Gamma)| = 0$. \begin{defi}[Convergence of systems of curves]
Let $(\Gamma_h)_{h\in\mathbb{N}}=(\{\gamma_1^h,...,\gamma_{N(n)}^h\})_h$ be a sequence of system of curves of class $\wpq{2}{p}$. We say that $(\Gamma_h)$ converges weakly in $\wpq{2}{p}$ to $\Gamma=\{\gamma_1,...,\gamma_M\}$ if \begin{itemize}
\item[(i)] $N(h)=M$ for $h$ large enough;
\item[(ii)] $\gamma_i^h$ converges weakly in $\wpq{2}{p}$ to $\gamma_i$ for every $i\in\{1,\cdots,M\}$. \end{itemize} \end{defi}
\begin{defi}\label{sistlim} We say that $\Gamma$ is a limit system of curves of class $\wpq{2}{p}$ if $\Gamma$ is the weak limit of a sequence $(\Gamma_h)$ of boundaries of bounded open sets with $\wpq{2}{p}$ parameterizations. \end{defi}
\begin{defi}\label{A} Let $\mathscr{A}$ denote the class of functions $$\Phi : t\in\mathbb{R} \rightarrow \Phi(t) $$ where for almost every $t\in \mathbb{R}$, $\Phi(t)=\{\gamma_t^1,...,\gamma_t^N\}$ is a limit system of curves of class $\wpq{2}{p}$ and such that, for almost every $\underline{t}, \overline{t}\in \mathbb{R}$, $\underline{t}< \overline{t}$, the following conditions are satisfied: \begin{itemize}
\item[(i)] $\Phi(\underline{t})$ and $\Phi(\overline{t})$ do not cross but may intersect tangentially;
\item[(ii)] $\operatorname{Int}(\Phi(\overline{t})) \subseteq \operatorname{Int}(\Phi(\underline{t}))$ (pointwisely);
\item[(iii)] if, for some $i$, $\mathcal{H}^1\left( (\gamma_{\overline{t}}^i) \setminus \overline{\operatorname{Int}(\Phi(\underline{t}))}\right) \neq 0$ then $$\mathcal{H}^1\left( [(\gamma_{\overline{t}}^i) \setminus \overline{\operatorname{Int}(\Phi(\underline{t}))}] \setminus (\Phi(\underline{t}))\right) = 0.$$ \end{itemize} \end{defi}
\begin{oss}\rm \label{remarkiii}
One may remark that, from condition $(ii)$ of Definition~\ref{A}, for every curve $\gamma\in\Phi(\underline{t})$ $$ (\gamma) \cap \operatorname{Int}(\Phi(\overline{t})) =\emptyset.$$ In fact if $x\in (\gamma) \cap \operatorname{Int}(\Phi(\overline{t}))$ then $x\in \operatorname{Int}(\Phi(\overline{t}))$ and $x\notin \operatorname{Int}(\Phi(\underline{t}))$ which gives a contradiction with condition $(ii)$. \end{oss}
Following~\cite{MM}, we introduce a convenient notion of convergence in $\mathscr{A}$:
\begin{defi}[Convergence in $\mathscr{A}$]\label{convA} We say that $\Phi_h$ converges to $\Phi$ in $\mathscr{A}$, and we denote $\Phi_h \overset{\mathscr{A}}{\longrightarrow}\Phi$, if \begin{itemize}
\item[(i)] for each dyadic interval $[k_N2^{-N}, (k_N+1)2^{-N})$, $N\geq 1$, $k_N\in\mathbb{Z}$, there exists a point $t_{N, k_N}$ in the interval such that $\Phi_h(t_{N, k_N})$ converges to $\Phi(t_{N, k_N})$ weakly in $\wpq{2}{p}$ as $h\rightarrow \infty$; \item[(ii)] for almost every $t\in \mathbb{R}$, there exists a sequence $\{t_{N,k_N}\}$ such that $t_{N,k_N}\rightarrow t$ and $\Phi(t)$ is the weak $\wpq{2}{p}$ limit of $\{\Phi(t_{N, k_N})\}$ as $N\rightarrow \infty$. \end{itemize} \end{defi} It follows from this definition that, if $\Phi_h \overset{\mathscr{A}}{\longrightarrow}\Phi$, there exists for almost every $t\in \mathbb{R}$ a sequence $(h_N, k_N)$ such that $$ t_{N,k_N}\rightarrow t,\quad h_N\rightarrow \infty$$ and $$\Phi_{h_N}(t_{N,k_N})\overset{\wpq{2}{p}}{\rightharpoonup} \Phi(t)$$ therefore $$\Phi_{h_N}(t_{N,k_N})\overset{{\mathrm{C}}^1}{\longrightarrow} \Phi(t).$$
In the following definition, we associate with any function $u$ of bounded variation in the plane the class of all functions in $\mathscr{A}$ that realize a nested covering of the essential boundaries of the level sets of $u$, i.e. a covering of its level lines.
\begin{defi}[The class $\mathscr{A}(u)$]\label{defiA} Let $u\in {\mathrm{BV}}(\mathbb{R}^2)$. We define $\mathscr{A}(u)$ as the set of functions $\Phi \in \mathscr{A} $ such that, for almost every $t\in \mathbb{R}$, we have $$(\Phi(t))\supseteq \partial^* \{u > t\} \quad \mbox{ (up to a $\mathcal{H}^1$-negligible set)}$$
and $$\{u > t\} = \operatorname{Int}(\Phi(t)) \quad \mbox{ (up to a ${\cal L}^2$- negligible set)}.$$ In particular, if $u\in {\mathrm{C}}^2(\mathbb{R}^2)$, we will denote as $\Phi[u]$ the function of $\mathscr{A}(u)$ defined as $$t \mapsto \partial \{u>t\}.$$ \end{defi} {\begin{oss}\rm We will prove in Theorem~\ref{minG} that, whenever $u\in{\mathrm{BV}}(\mathbb{R}^2)$ is such that $\overline{F}(u)<\infty$, then $\mathscr{A}(u)\not=\emptyset$. \end{oss}}
Conditions in Definitions~\ref{A} and~\ref{defiA} ensure that any $\Phi\in\mathscr{A}(u)$ is a nested covering of the level lines of $u$. In particular, condition $(iii)$ of Definition~\ref{A} ensures that the nesting property is also satisfied wherever concentration occurs, typically on the ghost concentration segment of Figure~\ref{fig:notcoincide}, right. The following examples show the necessity of condition~$(iii)$. \begin{oss}[Condition $\ref{A}(iii)$ is necessary]\label{iii-est-nec} {\rm Let $u=\mathds{1}_{E\cup F} + \mathds{1}_F$ and $v=\mathds{1}_{G\cup F} + \mathds{1}_F$ with $E,F,G$ like in Figure~\ref{condintro}-A. \begin{figure}
\caption{Situations of accumulation}
\label{condintro}
\end{figure} Using the same kind of approximation as in Figure~\ref{cusp2} one easily sees that $u$ has finite energy $\overline{F}$. Sequences of smooth functions $(u_h)$ that converge to
$u$ in $\lp{1}$ and such that $F(u_h) \rightarrow \overline{F}(u)$ have level sets similar to $F_h$ in Figure~\ref{condintro}-B for every level between 1 and 2. \par The boundary $\Gamma_h$ of $F_h$ converges to the limit curve $\Gamma$ that contains $\partial F$ plus a ghost segment that corresponds to the concentration in the limit of the middle tube. Clearly, for $h$ large enough, the middle tube is contained in sets that approximate $E$.
The situation is different for $v$ which also has finite relaxed energy, as was explained in the introduction. To build the approximating sequence with uniformly bounded energy, it is necessary to approximate $G\setminus F$ in such a way that a ``corridor'' be created between both disks so that both components of $F$ can be approximated with a sequence of connected sets having bounded energy. This is illustrated in Figure~\ref{condintro}-C: the bottom set is smooth and connected. As the width of all thin gray and white zones goes to $0$, the set converges in measure to $G\setminus F$. This clearly justifies the need for condition $\ref{A}(iii)$ since the simultaneous approximation of $F$ and $G$ without boundary crossing requires building also for $G\setminus F$ a ghost part that encloses the ghost part arising from the approximation of $F$.} \end{oss}
\begin{oss}[Exemplification of Definition~\ref{A}]{\rm Let us analyze on some examples the geometric meaning of Definition \ref{A}. \begin{itemize} \item{\it Example I}\; Let $u = \mathds{1}_E + \mathds{1}_F$ with $E,F$ like in Figure~ \ref{wall1}.
\begin{figure}
\caption{Level sets of $u$}
\label{wall1}
\end{figure}
\begin{figure}
\caption{The systems $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$ with their multiplicities}
\label{wall2}
\end{figure} Let $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$ be the systems of curves drawn in Figure~\ref{wall2} together with their multiplicities and consider the piecewise constant function $$\Phi(t)=\left\{ \begin{array}{ll}
\Gamma_1 & \mbox{if\;} t\in[0,1]\\
\Gamma_2 & \mbox{if\;} t\in(1,2]\\
\emptyset &\text{otherwise}. \end{array} \right.$$ Clearly, $\Phi\notin \mathscr{A}(u)$. $\Gamma_1$ and $\Gamma_2$ satisfy Conditions $(i)$, $(ii)$ of Definition \ref{A} but, since the system $\Gamma_1$ does not contain the line joining the two cusp points of $F$, we have $\mathcal{H}^1((\Gamma_2)\setminus \overline{E})\neq 0$ but $\mathcal{H}^1([(\Gamma_2)\setminus\overline{E}]\setminus (\Gamma_1) )\neq 0 $ so $\Phi$ does not satisfy Condition $(iii)$ of Definition \ref{A}.
\par Consider now the function $$\Phi(t)=\left\{ \begin{array}{ll}
\Gamma_3 &\mbox{if\;} t\in[0,1]\\
\Gamma_2 &\mbox{if\;} t\in]1,2]\\
\emptyset &\text{otherwise}. \end{array} \right.$$ where $\Gamma_3$ is built from the curves $\gamma_1$ and $\gamma_2$ together with the multiplicities indicated in Figure~\ref{wall2}. It is easy to check that $\Phi \in \mathscr{A}(u)$ and it must be emphasized that the choice of $\Phi(t)=\Gamma_2$ for every $t\in ]1,2]$ yields strong geometric constraints. In particular, since the curve joining the two cusp points of $F$ goes out of the set $E$, condition $(iii)$ of Definition~\ref{A} imposes that the trace of $\Phi(t)$ for almost every $t\in [0,1]$ contains $(\Gamma_2)\setminus \overline{E}$.
\par Let us finally examine the function $$\Phi(t)=\left\{ \begin{array}{ll}
\Gamma_4 &\mbox{if\;} t\in[0,1]\\
\Gamma_2 &\mbox{if\;} t\in(1,2]\\
\emptyset &\text{otherwise}. \end{array} \right.$$ where $\Gamma_4$ is built from the curves $\gamma_3$, $\gamma_4$. Remark that, up to a Lebesgue-negligible set, $\operatorname{Int}(\Gamma_4)$ coincides with $E\cup F$ because the multiplicity of the inner curve is everywhere even. In this example, $\Phi$ satisfies Condition $(iii)$ because the curve joining the two cusp points of $F$ belongs to both $\Gamma_2$ and $\Gamma_4$. However, $\Phi$ does not satisfy Condition $(ii)$ of Definition \ref{A} since $\gamma_4\cap \operatorname{Int}(\Gamma_2)\neq\emptyset$ (see Remark \ref{remarkiii}), so $\Phi\notin \mathscr{A}(u)$. \item {\it Example II}\; Let $v = \mathds{1}_E + \mathds{1}_F$ with $E,F$ like in Figure~\ref{wall3}. \begin{figure}
\caption{Level sets of $v$}
\label{wall3}
\end{figure} Let $\Gamma_1, \Gamma_2, \Gamma_3$ be the systems of curves in Figure~\ref{wall4}. \begin{figure}
\caption{The systems $\Gamma_1, \Gamma_2, \Gamma_3$ with their multiplicities}
\label{wall4}
\end{figure} \par We consider the following functions: $$\Phi(t)=\left\{ \begin{array}{ll}
\Gamma_1 & \text{if\;} t\in[0,1]\\
\Gamma_3 &\text{if\;} t\in]1,2]\\
\emptyset &\text{otherwise}. \end{array} \right.$$ $$\Psi(t)=\left\{ \begin{array}{ll}
\Gamma_2 &\text{if\;} t\in[0,1]\\
\Gamma_3 &\text{if\;} t\in]1,2]\\
\emptyset &\text{otherwise}. \end{array} \right.$$
The function $\Phi$ does not satisfy Condition $(iii)$ of Definition~\ref{defiA} because the line joining the two cusps of $F$ goes out of $E$ whereas $$\mathcal{H}^1([(\Gamma_2)\setminus \overline{E}]\setminus (\Gamma_1))\neq 0$$ and so $\Phi \notin \mathscr{A}(u)$. On the contrary, it is easy to check that $\Psi\in \mathscr{A}(u)$ and, again, $\Gamma_2$ imposes strong geometrical constraints for all conditions of Definition~\ref{defiA} to be satisfied. \end{itemize} } \end{oss}
\section{A coarea-type formula for $\overline{F}$} Recall from the introduction the definition of the functional
$$ G : \begin{array}[t]{lll} \mathscr{A}&\to &\Bar\mathbb{R}^+\\ \Phi&\mapsto &\displaystyle\int_{\mathbb{R}} W(\Phi(t)) \mbox{d}t.\end{array}$$
Our main result is the representation formula that holds for any $u\in {\mathrm{BV}}(\mathbb{R}^2)$ with $\overline{F}(u) < \infty$, that is $$\overline{F}(u) = \underset{\Phi \in \mathscr{A}(u)} {\operatorname{Min}} G(\Phi).$$
This formula can be easily proved for smooth functions, as shown in the following \begin{oss}[{\bf The regular case}] {\rm Let $u\in {\mathrm{C}}^2(\mathbb{R}^2)$ with $\overline{F}(u) < \infty$. Applying the coarea formula to the system of curves $\Phi[u]$ and since $F$ and $\overline{F}$ coincide for smooth functions, we get immediately that $$ \overline{F}(u) = F(u) = G(\Phi[u]).$$ By Definition~\ref{defiA}, $\Phi[u]\in \mathscr{A}(u)$ and $$G(\Phi[u])\leq G(\Phi) \quad \forall \Phi \in \mathscr{A}(u).$$ Therefore $$G(\Phi[u])= \underset{\mathscr{A}(u)}{\operatorname{Min}}\; G= \overline{F}(u) =F(u).$$ } \end{oss}
In order to extend this result for a general function $u$ in ${\mathrm{BV}}(\mathbb{R}^2)$, let us first address the existence of minimizers of $G$ in $\mathscr{A}(u)$.
\subsection{Existence of minimizers of $G$}\label{G}
The next proposition gives a sufficient condition of compactness with respect to the $\mathscr{A}$-convergence:
\begin{prop}\label{compA} Let $(\Phi_h)$ be a sequence in $\mathscr{A}$ such that $$\underset{h}{\sup} \; G(\Phi_h) < \infty.$$ Then, possibly extracting a subsequence, there exists a function $\Phi \in \mathscr{A}$ such that $$\Phi_h \overset{\mathscr{A}}{\longrightarrow}\Phi \quad \mbox{and} \quad G(\Phi)\leq \underset{h\rightarrow \infty}{\liminf} \; G(\Phi_h).$$ \end{prop}
\noindent{\bf Proof\;\, } The proof is essentially the same as the proof of Theorem 2 in \cite{MM} so a few details will be omitted.\\ {\bf Step 1 : Convergence of the energies $W(\Phi_h(t))$.}
Let $N \in \mathbb{N}, k\in \mathbb{Z}$ and let us consider the dyadic intervals on $\mathbb{R}$ : $$I_{N,k}=[k2^{-N}, (k+1)2^{-N}[.$$ We define the functions $$f_h^N(t)=2^N\int_{I_{N,k}} W(\Phi_h(s)) \mbox{d}s $$ where $I_{N,k}$ is the unique dyadic interval containing $t$. The function $f_h^N(t)$ is constant on each interval $I_{N,k}$ and for every $t\in \mathbb{R}$ we have $f_h^N(t)\leq 2^N G(\Phi_h)$. So, by a diagonal extraction, we can take a subsequence (not relabelled) such that $$\forall (N,k) \quad f_h^N(t)\rightarrow f^N(t) \quad \forall t\in I_{N,k}.$$ Moreover we can write $$I_{N,k}=[k2^{-N}, (2k+1)2^{-N-1}[\cup [(2k+1)2^{-N-1}, (k+1)2^{-N}[$$ and $$f_h^{N+1}(t) = 2^{N+1}\int_{I_{N+1,2k}}W(\Phi_h(s)) \mbox{d}s \quad \forall t\in [k2^{-N}, (2k+1)2^{-N-1}[$$ $$f_h^{N+1}(t) = 2^{N+1}\int_{I_{N+1,2k+1}}W(\Phi_h(s)) \mbox{d}s \quad \forall t\in [(2k+1)2^{-N-1}, (k+1)2^{-N}[,$$ therefore $$\int_{I_{N,k}}f_h^{N+1}(s) \mbox{d}s = \int_{I_{N,k}}W(\Phi_h(s)) \mbox{d}s$$ $$f_h^N(t) = 2^N \int_{I_{N,k}}f_h^{N+1}(s)\mbox{d}s \quad \forall t\in I_{N,k}.$$ Then, by the Dominated Convergence Theorem, we get \begin{equation}\label{marti}
f ^N(t) = 2^N \int_{I_{N,k}}f^{N+1}(s) \mbox{d}s \end{equation} and in addition, by Fatou's Lemma,
\begin{equation}\label{Doob}
\int_{\mathbb{R}}f^N(s) \mbox{d}s \leq \underset{h\rightarrow \infty}{\liminf} \;\; \int_{\mathbb{R}}f_h^N(s)\mbox{d}s\leq \underset{h}{\sup} \; G(\Phi_h). \end{equation}
\eqref{marti} and \eqref{Doob} show that $(f^N)$ is a bounded positive martingale thus, by the convergence theorem for martingales (\cite[Thm 2.2, p. 60]{RY}, there exists $f\in \lp{1}(\mathbb{R})$ such that $f^N \rightarrow f$ a.e.
\par {\bf {Step 2 : Definition of a limit system of curves $\Phi$.}}
Let $N\in \mathbb{N}, k\in \mathbb{Z}$. We have \begin{equation}\label{lemmino} \underset{I_{N,k}}{\sup}\; \underset{h}{\sup}\; f_h^N < \infty. \end{equation}
\begin{lemma}\cite{MM}\label{lemmatec} Let $A_h:= \left\lbrace t\in I_{N,k} : W(\Phi_h(t)) \leq 2^N\int_{I_{N,k}} W(\Phi_h(s)) \mbox{d}s + \dfrac{1}{N}\right\rbrace $. Then there exists $t_{N,k}\in I_{N,k}$ such that, possibly passing to a subsequence, $$t_{N,k}\in A_h, \quad\quad \forall h\in\mathbb{N}.$$ \end{lemma}
\noindent{\bf Proof\;\, } see~\cite{MM} Lemmas 4 and 5 \qed
For every dyadic interval $I_{N,k}$ we consider the real number $t_{N,k}$ given by the previous lemma then, possibly extracting a subsequence, we have $$W(\Phi_h(t))\leq 2^N\int_{I_{N,k}} W(\Phi_h(s)) \mbox{d}s + \dfrac{1}{N}\quad \forall h$$ and so by the compactness theorem in $\wpq{2}{p}$ there exists a subsequence (not relabelled) and a limit system of curves $\Phi(t_{N,k})$ such that $$\Phi_h(t_{N,k}) \overset{\wpq{2}{p}}{\rightharpoonup} \Phi(t_{N,k})\quad \mbox{and} \quad W(\Phi(t_{N,k}))\leq \underset{h\rightarrow \infty}{\liminf}\; W(\Phi_h(t_{N,k})).$$ Remark that, since the curves of $\Phi_h(t_{N,k})$ are without crossing, because of the ${\mathrm{C}}^1$-convergence the curves of $\Phi(t_{N,k})$ are without crossing as well. \par Since the $I_{N,k}$'s are countably many, we can use a diagonal extraction argument to find a subsequence, still denoted by $\Phi_h$, and a limit system of curves of class $\wpq{2}{p}$, denoted by $\Phi(t_{N,k})$, such that for each $t_{N,k}$ given by
the previous lemma: $$\Phi_h(t_{N,k}) \overset{\wpq{2}{p}}{\rightharpoonup} \Phi(t_{N,k})\quad \mbox{and}\quad W(\Phi(t_{N,k}))\leq \underset{h\rightarrow \infty}{\liminf}\; W(\Phi_h(t_{N,k})).$$ \par Furthemore for every $t_{N,k}> t_{N, k'}$ the systems $\Phi_h(t_{N,k})$ and $\Phi_h(t_{N,k'})$ are without crossing and $\operatorname{Int}(\Phi_h(t_{N,k}))\subseteq \operatorname{Int}(\Phi_h(t_{N,k'}))$ so, because of the ${\mathrm{C}}^1$ convergence, also $\Phi(t_{N,k})$ and $\Phi(t_{N,k'})$ are without crossing and $\operatorname{Int}(\Phi(t_{N,k}))\subseteq \operatorname{Int}(\Phi(t_{N,k'}))$.\\
Let us now see how a limit curve can be defined for every $t$. Let $t\in \mathbb{R}, N\in \mathbb{N}$ so there exists $k_N \in\mathbb{Z}$ such that $t\in I_{N, k_N}$. We have $$W(\Phi(t_{N,k_N}))\leq f^N(t) + \dfrac{1}{N}\overset{N \rightarrow \infty}{\rightarrow} f(t)\quad \mbox{a.e.}.$$ Then by the weak compactness of $\wpq{2}{p}$ there exists a subsequence, still denoted by $(\Phi(t_{N,k_N}))$, and a system of curves of class $\wpq{2}{p}$, denoted by $\Phi(t)$, such that $$\Phi(t_{N,k_N}) \overset{\wpq{2}{p}}{\rightharpoonup} \Phi(t)\quad \mbox{and} \quad W(\Phi(t))\leq \underset{h\rightarrow \infty}{\liminf}\; W(\Phi(t_{N,k_N}))\leq f(t).$$
This procedure can be applied for almost every $t\in \mathbb{R}$ so that, if we can prove that $\Phi\in \mathscr{A}$, we will conclude that $\Phi_h \overset{\mathscr{A}}{\rightarrow} \Phi$. Observe that \begin{itemize} \item the curves of $\Phi(t_{N,k})$ are without crossing, as was shown before, and $${\mbox{Int}}(\Phi(t_{N,k}))\subseteq {\mbox{Int}}(\Phi(t_{N,k'}))$$ for every $t_{N,k}> t_{N, k'}$;
\item for almost every $\underline{t}, \overline{t}\in \mathbb{R}$, $\underline{t}\neq\overline{t} $, we can find $\overline{t}_{N,k_N} \rightarrow \overline{t}$ and $\underline{t}_{N,k_N'} \rightarrow \underline{t}$ such that $$\Phi(\overline{t}_{N,k_N}) \overset{W^{2,p}}{\rightharpoonup} \Phi(\overline{t}),\;\; \Phi(\underline{t}_{N,k_N'}) \overset{W^{2,p}}{\rightharpoonup} \Phi(\underline{t}).$$ Since $\Phi(\overline{t}_{N,k_N})$ and $ \Phi(\underline{t}_{N,k_N'})$ are without crossing, because of the ${\mathrm{C}}^1$ convergence we get that $\Phi(\overline{t})$ and $\Phi(\underline{t})$ are without crossing. Moreover we can suppose, for $N$ large enough, $\overline{t}_{N,k_N} > \underline{t}_{N,k_N'}$ and since $ {\mbox{Int}}(\Phi(\overline{t}_{N,k_N})) \subseteq {\mbox{Int}}(\Phi(\underline{t}_{N,k_N'})) $, the ${\mathrm{C}}^1$ convergence implies that $$ {\mbox{Int}}(\Phi(\overline{t})) \subseteq {\mbox{Int}}(\Phi(\underline{t}));$$
\item we have to prove that for almost every $\underline{t}, \overline{t}\in \mathbb{R}$, $\underline{t}\neq\overline{t} $ the system $\Phi(t)$ satisfies condition~$(iii)$ of Definition~\ref{A}. \par For every $t$ we let $$E(\Phi(t)) = \mathbb{R}^2\setminus\overline{\operatorname{Int}(\Phi(\underline{t}))} .$$ By contradiction, suppose that there exists $\overline{t}\geq\underline{t}$ and $\gamma_{\overline{t}}^i \in \Phi(\overline{t})$ such that $$\mathcal{H}^1\left( (\gamma_{\overline{t}}^i) \cap E(\Phi(\underline{t})) \right)\neq 0$$ and $$\mathcal{H}^1\left( [(\gamma_{\overline{t}}^i) \cap E(\Phi(\underline{t}))] \setminus(\Phi(\underline{t})) \right) \neq 0.$$ Then we can find two sequences $\{(h_N, k_N)\}, \{(h_N, k_N')\}$ such that $t_{N,k_N} \rightarrow \overline{t}$ and $t_{N,k_N'} \rightarrow \underline{t}$, with $t_{N,k_N'} < t_{N,k_N}$ for every $N$ large enough, that satisfy \begin{equation}\label{condiii}
\Phi_{h_N}(\underline{t}_{k_N,N})\overset{{\mathrm{C}}^1}{\longrightarrow} \Phi(\underline{t})\quad \mbox{and} \quad
\gamma_{h_N}\overset{{\mathrm{C}}^1}{\longrightarrow} \gamma_{\overline{t}}^i\;, \quad \gamma_{h_N} \in \Phi_{h_N}(\overline{t}_{k_N,N}). \end{equation}
Because of the ${\mathrm{C}}^1$-convergence, for $N$ large enough we have $$\mathcal{H}^1\left( (\gamma_{h_N}) \cap E(\Phi(t_{N,k_N'})) \right)\neq 0$$ and $$\mathcal{H}^1\left( [(\gamma_{h_N}) \cap E(\Phi(t_{N,k_N'}))] \setminus (\Phi(t_{N,k_N'})) \right) \neq 0$$ which gives a contradiction with the fact that condition $(iii)$ holds for the functions $\Phi_{h_N}$. \end{itemize}
\par Finally, we have defined a collection of curves $\Phi \in \mathscr{A}$ such that $\Phi_h \overset{\mathscr{A}}{\longrightarrow}\Phi$. Moreover, $$G(\Phi)=\int_{\mathbb{R}} W(\Phi(t)) \mbox{d}t \leq \int_{\mathbb{R}} f(t) \mbox{d}t \leq \underset{N\rightarrow \infty}{\liminf}\int_{\mathbb{R}} f^N(t) \mbox{d}t$$ $$\leq \underset{h\rightarrow \infty}{\liminf}\;\underset{N\rightarrow \infty}{\liminf}\int_{\mathbb{R}} f_h^N(t) \mbox{d}t= \underset{h\rightarrow \infty}{\liminf}\int_{\mathbb{R}} W(\Phi_h(t)) \mbox{d}t =\underset{h\rightarrow \infty}{\liminf}\; G(\Phi_h).$$ \qed
The next theorem states the existence of minimizers to $G$ in $\mathscr{A}(u)$.
\begin{thm}\label{minG} Let $u\in {\mathrm{BV}}(\mathbb{R}^2)$ with $\overline{F}(u) < \infty$. Then $\mathscr{A}(u)\neq \emptyset$, the problem $$\underset{\Phi \in \mathscr{A}(u)}{\operatorname{Min}} G(\Phi) $$ has a solution, and $$\underset{\Phi \in \mathscr{A}(u)}{\operatorname{Min}} G(\Phi) \leq \overline{F}(u).$$ \end{thm}
\noindent{\bf Proof\;\, } Let $\{u_h\} \subset {\mathrm{C}}^2(\mathbb{R}^2)$ be a sequence converging to $u$ in $\lp{1}$ such that $F(u_h)$ is uniformly bounded and $\overline{F}(u)=\underset{h\rightarrow \infty}{\lim}F(u_h)$. We can associate to every $u_h$ the function $\Phi_h=\Phi[u_h]$ and by the coarea formula we have $$F(u_h)=G(\Phi_h),\quad \underset{h}{\sup}\;G(\Phi_h)<\infty.$$ Then, by Proposition \ref{compA}, there exists a subsequence (not relabelled) and $\Phi\in \mathscr{A}$ such that $$\Phi_h \overset{\mathscr{A}}{\longrightarrow} \Phi ,\quad \quad G(\Phi) \leq \underset{h \rightarrow \infty}{\liminf}\; G(\Phi_h).$$ Let us now prove that $\Phi\in\mathscr{A}(u)$. By definition of the $\mathscr{A}$-convergence we have
$$ |\operatorname{Int}(\Phi_{h}(t_{N, k_N}))\Delta \operatorname{Int}(\Phi(t_{N, k_N}))| = \arrowvert \{ u_{h} > t_{N, k_N}\}\Delta \operatorname{Int}(\Phi(t_{N, k_N}))\arrowvert \rightarrow 0\quad \mbox{if}\quad h \rightarrow \infty$$ and, since $\arrowvert \{ u_{h} > t\} \Delta \{u>t\}\arrowvert \rightarrow 0$ for $\mathcal{L}^1$-almost every $t$, in Lemma \ref{lemmatec} we can choose $t_{N, k_N}$ such that $\arrowvert \{ u_{h} > t_{N, k_N}\} \Delta \{u>t_{N, k_N}\}\arrowvert \rightarrow 0$ and we have \begin{equation}\label{3}
\{u>t_{N, k_N}\}=\operatorname{Int}(\Phi(t_{N, k_N})) \quad \mbox{(up to a Lebesgue negligible set)} \quad \forall N. \end{equation} In addition, for almost every $t$, $\Phi(t)$ is the weak $\wpq{2}{p}$ limit of a sequence $\{\Phi(t_{N, k_N})\}$ where $t_{N,k_N}\rightarrow t$ as $N\rightarrow \infty$, and it follows that
$$|\operatorname{Int}(\Phi(t_{N, k_N}))\Delta \operatorname{Int}(\Phi(t))|\rightarrow 0\quad \mbox{if}\quad N \rightarrow \infty.$$ Now, the interiors of the systems $\Phi(t)$ are nested, and so, using Lemma~\ref{mon} for the family $(\{u>t\})_t$ and~\eqref{3}, we get \begin{equation}\label{2}
\{u>t\}=\operatorname{Int}(\Phi(t)) \quad \mbox{(up to a Lebesgue negligible set)} \quad \text{for a.e. $t$}.\end{equation} Being $u$ of bounded variation, it follows that for almost every $t$ $$\partial^* \{u>t\} =\partial^*\operatorname{Int}(\Phi(t)) \quad\text{up to a $\mathcal{H}^1$-negligible set}$$ therefore $$\partial^* \{u>t\} \subset\Phi(t)\quad\text{up to a $\mathcal{H}^1$-negligible set.}$$ This proves that there exists $\Phi\in\mathscr{A}(u)$ such that $G(\Phi)\leq \overline{F}(u)$.
To show the existence of minimizers to $G$ in $\mathscr{A}(u)$, it suffices to take $(\Phi_h)\subset \mathscr{A}(u)$ a minimizing sequence, i.e. $$\underset{\mathscr{A}(u)}{\inf}\, G=\underset{h\rightarrow \infty}{\lim}G(\Phi_h)$$ and $\underset{h}{\sup}\;G(\Phi_h)< \infty$. Using exactly the same argument as above, we can conclude that there exists $\Phi\in \mathscr{A}(u)$ such that $$\Phi_h \overset{\mathscr{A}}{\longrightarrow} \Phi ,\quad \quad G(\Phi) \leq \underset{h \rightarrow \infty}{\liminf}\; G(\Phi_h)=\underset{\mathscr{A}(u)}{\inf}\, G$$ therefore $$\underset{\Phi \in \mathscr{A}(u)}{\operatorname{Min}} G(\Phi) $$ has a solution, and $$\underset{\Phi \in \mathscr{A}(u)}{\operatorname{Min}} G(\Phi) \leq \overline{F}(u).$$ \qed
\subsection{Connections between $G$ and $\overline{F}$}\label{formula}
We can deduce from the previous theorem a first representation result for $\overline{F}$:
\begin{lemma}\label{regolare}
Let $E \subset \mathbb{R}^2$ be a bounded open set with $\partial E \in \wpq{2}{p}$ and let $u=c\mathds{1}_E$, $c >0$. Then
$$\overline{F}(u) = c \overline{W}( E)=c\int_{\partial E}(1+|\kappa_{\partial E}|^p)d{\Hau^1}.$$ \end{lemma}
\noindent{\bf Proof\;\, } By Corollary~3.2 in~\cite{BDP}, $c\overline{W}(E)=G(\Psi)=c\displaystyle\int_{\partial E}(1+|\kappa_{\partial E}|^p)d{\Hau^1}$ where $\Psi$ is the map defined by $\Psi(t) = \partial E$ for every $t\in[0,c]$, $\emptyset$ otherwise. Therefore, if $\overline{F}(u)<+\infty$ then, by Theorem~\ref{minG} and Definition~\ref{defiA}, $$\overline{F}(u) \geq \underset{\mathscr{A}(u)}{\operatorname{Min}}\,G = G(\Psi) = c \overline{W}( E).$$ The fact that $\overline{F}(u)<+\infty$ and the reverse inequality $\overline{F}(u) \leq c \overline{W}( E)$ will follow if we can find a sequence $\{u_h\}\subseteq {\mathrm{C}}^2(\mathbb{R}^2)$ such that $u_h \rightarrow u$ in $\lp{1}$ and $$F(u_h)=G(\Phi[u_h]) \rightarrow G(\Psi).$$ Since $c\overline{W}(E)=G(\Psi)<+\infty$, there exists a sequence of open sets $(E_m)$ of class ${\mathrm{C}}^2$ such that \begin{equation}
|E_m\Delta E|\to 0\qquad\text{and}\qquad cW(E_m)\to G(\Psi)\label{approxBDP-G} \end{equation} \par Let $m\in\mathbb{N}$. From the properties of the distance function (see \cite[\S 14.6, p .354]{GT}), we can find $\eta> 0$ such that $ \,d(x):={\rm{dist}}(x, \partial E_m) \in {\mathrm{C}}^2(E^\eta_m)$ where $E^\eta_m= \{x\in \mathbb{R}^2 \setminus \overline{E_m} : \,d(x)< \eta \}$ and the curvature of $\partial E_m$ is bounded by $\eta^{-1}$ so that \begin{equation}\label{curvbound}
1+\,d(x) k_{\partial E_m}(x) \neq 0 \quad \quad \forall x \in E^\eta_m. \end{equation} For every $h\in\mathbb{N}^*$ we consider the cut-off function $w^m_h : [0, \eta/h] \rightarrow [0,c]$, $w^m_h\in {\mathrm{C}}^2([0, \eta/m])$ whose graph is represented in Figure~\ref{eta}. \begin{figure}
\caption{Graph of the cut-off function $w^m_h$}
\label{eta}
\end{figure} Then we define the sequence of smooth functions: $$u^m_h(x)=\left\{ \begin{array}{ll}
c & \text{if } x\in \overline{E_m},\\
w^m_h(\,d(x)) & \text{if } x\in E^{ \eta/h}_m, \\
0 & \text{otherwise.} \end{array} \right.$$ Thus $u^m_h \overset{\lp{1}}{\rightarrow} u^m:=\one{E_m}$, as $h \rightarrow +\infty$, and for every $t\in [0,c]$ we get \begin{equation}\label{livdist}
\Gamma_t^{m,h} = \{x \in \mathbb{R}^2 : u^m_h(x) = t\}=\left\lbrace x \in \mathbb{R}^2 : \,d(x)= (w^m_h)^{-1}(t) \right\rbrace. \end{equation} Therefore, for $h$ large enough, $\Gamma_t^{m,h}$ can be parametrized as (see~\cite[\S 5.7, p.115]{G}): $$\Gamma_t^{m,h}(y)= y + \delta n(y), \quad y\in \partial E_m ,\quad \delta= (w^m_h)^{-1}(t)$$ where $n(y)$ is the outer unit normal to $\partial E_m$ at $y$. Using a positively-oriented arc-length parameterization $s\mapsto\alpha(s)$ of $\partial E_m$, we can parametrize $\Gamma_t^{m,h}$ as (the dependence on $t$ is omitted to simplify the notations) $$\gamma(s) = \alpha(s) + \delta n(\alpha(s))= \alpha(s) - \delta J \alpha'(s)$$ where $J$ is the rotation operator $J(x,y)=(-y,x)$, $(x,y) \in \mathbb{R}^2.$ Thus $$\gamma'(s)= \alpha'(s) -\delta J\alpha''(s)= \alpha'(s) - \delta k_\alpha(s) J^2\alpha'(s) = \left(1+\delta k_\alpha(s)\right)\alpha'(s),$$
where, with a small abuse of notation, $k_\alpha(s)$ now denotes the signed scalar curvature instead of the vector curvature, i.e. $\alpha''(s)=k_\alpha(s)\vec{N}$ being $(\alpha'(s),N)$ a direct frame with $|\vec{N}|=1$. Hence $\gamma$ is regular at any $s$ such that $1+\delta k_\alpha(s) \neq 0$ and $$ \gamma''(s)= \left(1+\delta k_\alpha(s)\right) k_\alpha(s) J\alpha'(s) +\delta k_\alpha'(s)\alpha'(s)$$ and so
$$k_\gamma(s)= \dfrac{\langle \gamma'', J \gamma' \rangle}{\|\gamma' \|^3}= \dfrac{k_\alpha(s)}{|1+\delta k_\alpha(s)|}.$$ Then, by \eqref{curvbound}, $\Gamma_t^{m,h}$ is smooth for $h$ large enough and
$$k_{\Gamma_t^{m,h}}(\Gamma_t^{m,h}(y))= \dfrac{k_{\partial E_m}}{|1+\delta k_{\partial E_m}|}(y)$$
$$\mbox{d}\mathcal{H}^1 \res \Gamma_t^{m,h} = |1+\delta k_{\partial E_m}| \mbox{d}\mathcal{H}^1 \res \partial E_m $$ and we get
$$ W(\Gamma_t^{m,h})= \int_{ \Gamma_t^{m,h}}[1+|k_{\Gamma_t^{m,h}}|^p] \mbox{d}\mathcal{H}^1 =\int_{ \partial E_m}\left[1+ \left|\dfrac{k_{\partial E_m}}{1+\delta k_{\partial E_m}}\right|^p \right] |1+\delta k_{\partial E_m}| \mbox{d}\mathcal{H}^1.$$ Then, remark that \begin{equation}\label{lunghW}
|\mathcal{H}^1(\Gamma_t^{m,h}) - \mathcal{H}^1(\partial E_m)|\leq \int_{\partial E_m} \delta |k_{\partial E_m}| \mbox{d}\mathcal{H}^1 \leq C_1\delta W( E_m). \end{equation} Moreover,
$$\left|\int_{\Gamma_t^{m,h}} |k_{\Gamma_t^{m,h}}|^p \mbox{d}\mathcal{H}^1 - \int_{\partial E} |k_{\partial E_m}|^p \mbox{d}\mathcal{H}^1\right| \leq \int_{\partial E_m}|k_{\partial E_m}|^p\left| \dfrac{1}{|1+\delta k_{\partial E_m}|^{p-1}} -1\right| \mbox{d}\mathcal{H}^1$$ thus, by Lebesgue's Dominated Convergence Theorem, \begin{equation}\label{curvW}
\left|\int_{\Gamma_t^{m,h}} k_{\Gamma_t^{m,h}}^p \mbox{d}\mathcal{H}^1 - \int_{\partial E_m} k_{\partial E_m}^p \mbox{d}\mathcal{H}^1\right|\rightarrow 0 \quad \quad \text{as} \quad \delta\rightarrow 0 \quad \quad (\text{i.e.}\quad h\rightarrow \infty). \end{equation} It follows from \eqref{lunghW} and \eqref{curvW} that \begin{equation}\label{distW}
\underset{h \rightarrow \infty}{\lim} |W(\Gamma_t^{m,h}) - W( E_m)| =0 \quad \forall t\in [0,c]. \end{equation} Consider now a subsequence $\Phi_k=\Phi[u^{m(k)}_{h(k)}]$. We have $\Phi_k(t)= \Gamma_t^{m(k),h(k)}$ for every $k$ and every $t\in [0,c]$. In addition,
$$|F(u^{m(k)}_{h(k)}) - G(\Psi)|= |G(\Phi[u^{m(k)}_{h(k)}]) - G(\Psi)|\leq \int_0^{c} \left| W(\Gamma_t^{m(k),h(k)})-\overline{W}( E) \right|dt $$ Using~\eqref{approxBDP-G},~\eqref{distW} and a diagonal extraction argument, we can find a subsequence such that, keeping the same labeling and applying the Dominated Convergence Theorem $$ u^{m(k)}_{h(k)} \overset{\lp{1}}{\longrightarrow} u \quad \text{and} \quad F(u^{m(k)}_{h(k)}) \rightarrow G(\Psi) = c\overline{W}( E).$$ Since $\overline{F}(u)\leq\liminf_{k\to\infty}F(u^{m(k)}_{h(k)})$ the conclusion follows. \qed
This proof illustrates that, at least in the case of $u$ being the characteristic function of a $\wpq{2}{p}$ set, the equivalence between $\overline{F}$ and $G$ follows from a smoothing argument together with a control of the energy. The purpose of the next lemma is to show that a similar strategy also holds for more complicated functions. This lemma is essentially the same as Lemma 6 in~\cite{MM} so most details will be omitted.
\begin{lemma}\label{approx} Let $u\in {\mathrm{BV}}(\mathbb{R}^2)$ with $\overline{F}(u)<\infty$ and let $\Phi \in \mathscr{A}(u)$. Then, for every $\eta> 0$, there exists $\tilde{u}\in {\mathrm{BV}}(\mathbb{R}^2)$ such that
$$\|u-\tilde{u}\|_{\lp{1}} \leq \eta,$$ $\partial^* \{ \tilde{u} > t \}$ is, for almost every $t$, a finite system of curves of class $\wpq{2}{p}$ without contact or auto-contacts, and any two $\partial^* \{ \tilde{u} > t \}$ and $\partial^* \{ \tilde{u} > s \}$ are disjoint (pointwisely) for almost every $s\not=t$. In addition, the function $\tilde{\Phi}$ defined as $$t\mapsto \tilde{\Phi}(t)= \partial^* \{ \tilde{u} > t \} $$ belongs to $\mathscr{A}(\tilde{u})$ and $$\arrowvert G(\Phi)-G(\tilde{\Phi})\arrowvert \leq \eta .$$ \end{lemma}
\noindent{\bf Proof\;\, } We suppose $G(\Phi)\neq 0$. The idea of the proof is to move smoothly every system of curves $\Phi(t)$ to get a new family of systems of curves of class $\wpq{2}{p}$ belonging to $\mathscr{A}$ with no contact or auto-contact between curves, and with an energy close to the energy of $\Phi$. Then a function of bounded variation can be canonically defined.
\par Let $\{t_n\}_{n\geq 1}$ denote a countable and dense subset of $\mathbb{R}$ such that $\Phi(t_n)$ is well-defined for every $n\geq 1$. Following the proof of Lemma 6 in~\cite{MM} one associates with each finite system of curves $\Phi(t_n)$ a smooth operator $\mathbb{T}_n$ that separates every curve of $\Phi(t_n)$ from all other curves of $\Phi$ and that removes the auto-contacts by separating the corresponding arcs. $\mathbb{T}_n$ is chosen so that the energy of all curves that have been moved does not increase too much. It is shown in~\cite{MM} that the limit operator $\mathbb{T}=\lim_{n\to\infty}\mathbb{T}_n\circ \mathbb{T}_{n-1}\circ\cdots\circ \mathbb{T}_1$ is well-defined and smooth, that all curves of $\mathbb{T}(\Phi)$ are without contact or auto-contact, and that, given any $\eta>0$, $\mathbb{T}$ can be designed so that \begin{equation}\label{stimaii}
|G(\mathbb{T}(\Phi))- G(\Phi)|\leq \eta \end{equation}
Furthermore, it is easy to check that the separation process preserves the conditions of Definition~\ref{A} thus $\tilde{\Phi}:=\mathbb{T}(\Phi)\in\mathscr{A}$. In particular \begin{equation}\label{nested}
\operatorname{Int}(\tilde{\Phi}(s)) \subseteq \operatorname{Int}(\tilde{\Phi}(t))\quad\text{whenever $s>t$}. \end{equation} By standard arguments, the function defined by $\tilde{u}(x)=\sup \{t : x\in \operatorname{Int}(\tilde{\Phi}(t)) \}$ is measurable and $$ \operatorname{Int}(\tilde{\Phi}(s)) \subseteq \{\tilde{u}> t\} \subseteq \operatorname{Int}(\tilde{\Phi}(\ell))\qquad\text{when $\ell<t<s$}.$$ Let $F\subseteq \mathbb{R}$ be the countable set of discontinuities, with respect to the Lebesgue measure, of the monotone family $\{\operatorname{Int}(\tilde{\Phi}(t))\}$. Taking $s_k \searrow t$ and $l_k \nearrow t $, Lemma \ref{mon} implies that
$$|\{\tilde{u}>t\}\Delta \operatorname{Int}(\tilde{\Phi}(t))|=0 \quad \forall t \in \mathbb{R}\setminus F.$$
\par Let us now prove that $\tilde{u}\in {\mathrm{BV}}(\mathbb{R}^2)$. Given $\eta>0$ and using the convention $\mathbb{T}_0=\text{Id}$, we can redefine the separation operator at step $i\geq 1$ so that, by the coarea formula, $$\arrowvert \operatorname{Int}(\mathbb{T}_i\circ\cdots\circ\mathbb{T}_1(\Phi)(t)) \Delta \operatorname{Int}(\mathbb{T}_{i-1}\circ\cdots\circ\mathbb{T}_0(\Phi)(t)) \arrowvert < \varepsilon_i = \eta 2^{-i}{\Hau^1}(\Phi(t))$$ for almost every $t$, therefore $$\arrowvert \operatorname{Int}(\tilde{\Phi}(t)) \Delta \{u>t\}\arrowvert = \arrowvert \{\tilde{u}>t\}\Delta \{u>t\}\arrowvert < \eta{\Hau^1}(\Phi(t)).$$ It follows that
$$\|u-\tilde{u}\|_{\lp{1}} = \int_{-\infty}^{+\infty}\int_{\mathbb{R}^2}\arrowvert\mathds{1}_{\{u>t\}}-\mathds{1}_{\{\tilde{u}>t\}}\arrowvert\, dx\, dt < \eta\,\int_{-\infty}^{+\infty}{\Hau^1}(\Phi(t))\,dt\leq\eta G(\Phi).$$ In particular, $\tilde{u}\in \lp{1}(\mathbb{R}^2)$. Besides, all curves in $\tilde\Phi$ are mutually disjoint and without auto-contacts and $\tilde{u}$ is continuous thus $$\partial^*\{\tilde{u}> t\}= \partial [\operatorname{Int}\tilde{\Phi}(t)] = \tilde{\Phi}(t) \quad \mbox{ (up to a $\mathcal{H}^1$-negligible set)}.$$ We know from \eqref{stimaii} that $G(\tilde\Phi)<+\infty$ therefore $t\mapsto \mathcal{H }^1(\partial^*\{\tilde{u}> t\})$ is in $\lp{1}(\mathbb{R})$ thus, since $\tilde u\in\lp{1}(\mathbb{R}^2)$, $\tilde{u}\in {\mathrm{BV}}(\mathbb{R}^2)$ by the coarea formula. \qed
We can now state our main result :
\begin{thm}\label{principale}
Let $u\in {\mathrm{BV}}(\mathbb{R}^2)$ with $\overline{F}(u) < \infty$. Then $$\overline{F}(u) = \underset{\Phi \in \mathscr{A}(u)} {\operatorname{Min}} G(\Phi).$$ \end{thm} \noindent{\bf Proof\;\, }
Writing $u=u^+-u^-$ where $u^+(x)=\max\{u(x),0\}$ and $u^-(x)=\max\{-u(x),0\}$, and observing that $\overline{F}(u)\leq \overline{F}(u^+)+\overline{F}(u^-)$, we can suppose $u\geq 0$. In view of Theorem~\ref{minG}, we only have to prove that \begin{equation}\label{maggiorazione}
\underset{\Phi \in\mathscr{A}(u)}{\operatorname{Min}} G(\Phi) \geq \overline{F}(u). \end{equation} Let us show that, for every $\varepsilon > 0$, we can find $v\in {\mathrm{C}}^2(\mathbb{R}^2)$ such that
$$\|u-v\|_{\lp{1}} \leq \varepsilon$$ $$\arrowvert G(\Phi)-F(v) \arrowvert\leq \varepsilon $$ where $\Phi$ is a minimizer of $G$ on $\mathscr{A}(u)$. \par For every $\varepsilon >0$, by Lemma \ref{approx}, there exists $\tilde{u}\in {\mathrm{BV}}(\mathbb{R}^2)$, with $\wpq{2}{p}$ level lines without contacts or self-contacts, such that
\begin{equation}\label{l1}
\|u-\tilde{u}\|_{\lp{1}} \leq \varepsilon/4 \end{equation} \begin{equation}\label{w1}
\arrowvert G(\Phi)- G(\tilde{\Phi})\arrowvert \leq \varepsilon/4 \end{equation} where $\tilde{\Phi}(t)= \partial^* \{\tilde{u} > t\}$ for almost every $t\in\mathbb{R}$
Moreover we can find a set $\{t_n\}_{n\in\mathbb{N}}$ with $t_0=0$, such that, defining $$\tilde{v}(x)= \sum_{n\in\mathbb{N}} (t_{n+1}-t_n)\mathds{1}_{\{\tilde{u}>t_{n}\}}(x),$$ there holds \begin{equation}\label{l2}
\|\tilde{u}-\tilde{v}\|_{\lp{1}} \leq \varepsilon/4, \end{equation} \begin{equation}\label{w2} \text{and}\qquad \arrowvert G(\tilde{\Phi})- G(\Phi_{\tilde{v}})\arrowvert \leq \varepsilon/4, \end{equation} where $\Phi_{\tilde{v}}$, defined by $\Phi_{\tilde{v}}(t)=\partial^* \{\tilde{v} > t\}$ for almost every $t$, satisfies $$G(\Phi_{\tilde{v}})=\sum_{n\in\mathbb{N}}(t_{n+1}-t_n)W(\partial^* \{\tilde{u} > t_{n}\}).$$
Therefore we have approximated $\tilde{u}$ in $\lp{1}$ by a piecewise constant function $\tilde{v}$ whose level lines are systems of curves of class $\wpq{2}{p}$ without self-contacts and with finite $p$-elastica energy. Remark that $$\{\tilde{v}> t_n\}= \{\tilde{u}>t_{n}\}\quad(\operatorname{mod}\;{\cal L}^2)\qquad \forall n\in\mathbb{N}.$$ and $$\partial^*\{\tilde{v}> t_n\}=\partial^* \{\tilde{u}>t_{n}\}\quad(\operatorname{mod}\;{\mathcal H}^1)\qquad \forall n\in\mathbb{N}$$
By Lemma \ref{regolare}, we can approximate every function $(t_{n+1}-t_n)\mathds{1}_{\{\tilde{v}>t_{n}\}}$ by a function $\varphi_{n}\in {\mathrm{C}}^2(\mathbb{R}^2)$ such that
$$ \mbox{spt}(\varphi_{n}) \subset\!\subset\{\tilde{v}>t_{n}\}$$ $$ \{\tilde{v}>t_{n+1}\}\subset\!\subset \text{spt}(\varphi_{n})$$ \begin{equation}\label{stimafina}
\|(t_{n+1}-t_n)\mathds{1}_{\{\tilde{v}>t_{n}\}}-\varphi_{n}\|_{\lp{1}}\leq\varepsilon\,2^{-n-2} \end{equation} \begin{equation}\label{stimafin}
\arrowvert G(\Phi_{(t_{n+1}-t_n)\mathds{1}_{\{\tilde{v}>t_{n}\}}})-G(\Phi[\varphi_{n}])\arrowvert\leq \varepsilon\,2^{-n-2} \end{equation} where $\Phi_{(t_{n+1}-t_n)\mathds{1}_{\{\tilde{v}>t_{n}\}}}(t)= \partial \{\tilde{u}>t_{n}\}$ for every $t\in [0, (t_{n+1}-t_n)]$, $\emptyset$ otherwise.\\ \par Finally we define $$v^N(x)=\sum_{n=0}^N \varphi_{n} .$$ that is in ${\mathrm{C}}^2(\mathbb{R}^2)$. By \eqref{stimafina} and the Dominated Convergence Theorem, for $N$ large enough \begin{equation}\label{l4}
\|\tilde{v}-v^N\|_{\lp{1}}\leq\varepsilon/2. \end{equation} By \eqref{stimafin} and the coarea formula, we also have for $N$ large enough
\begin{equation}\label{w4}
\arrowvert F(v^N)-G(\Phi_{\tilde{v}})\arrowvert = \arrowvert G(\Phi[v^N])-G(\Phi_{\tilde{v}})\arrowvert\leq \varepsilon/2. \end{equation}
Finally, by \eqref{l1}, \eqref{l2}, and \eqref{l4},
$$\|u-v^N\|_{\lp{1}} \leq
\|u-\tilde{u}\|_{\lp{1}}+\|\tilde{u}-\tilde{v}\|_{\lp{1}}+\|\tilde{v}-v^N\|_{\lp{1}}\leq \varepsilon/4+ \varepsilon/4 + \varepsilon/2 = \varepsilon,$$ and by \eqref{w1}, \eqref{w2}, and \eqref{w4} $$\arrowvert G(\Phi)-F(v^N) \arrowvert\leq \arrowvert G(\Phi)-G(\tilde{\Phi}) \arrowvert + \arrowvert G(\tilde{\Phi})- G(\Phi_{\tilde{v}})\arrowvert + \arrowvert G(\Phi_{\tilde{v}})-F(v^N)\arrowvert \leq \varepsilon/4+ \varepsilon/4 + \varepsilon/2 = \varepsilon$$
As a straightforward consequence, we can build a sequence $\{u_h\}\subseteq{\mathrm{C}}^2(\mathbb{R}^2)$ such that $$u_h \overset{\lp{1}}{\longrightarrow}u\quad \mbox{and} \quad G(\Phi)=\underset{h \rightarrow \infty}{\lim} F(u_h)$$ which implies \eqref{maggiorazione} and the theorem ensues. \qed
\subsection{Connections between $\overline{F}$ and $\overline{W}$}\label{oFoW} We further investigate in this section the properties of functions with finite relaxed energy by collecting a few facts about the connection between the energy of a function and the relaxation of the $p$-elastica energy of its level sets.
\par The next proposition generalizes Lemma~\ref{regolare}.
\begin{prop}\label{equiv}
Let $E\subset \mathbb{R}^2$ be a measurable set such that $\overline{W}(E)<\infty$. Let $u=c\mathds{1}_E$ with $c > 0$. Then $$\overline{F}(u)=c \overline{W}(E).$$ \end{prop}
\noindent{\bf Proof\;\, } We first prove that $\overline{F}(u)<\infty$. Since $\overline{W}(E)<\infty$, there exists a sequence of smooth sets $(E_n)$ such that $|E_n\Delta E|\to 0$ and $W(E_n)\to\overline{W}(E)$. Defining $u_n=c\one{E_n}$ it follows from Lemma~\ref{regolare} that $\overline{F}(u_N)=cW(E_n)$ converges. In addition, $u_n\to u$ in $\lp{1}$ therefore, by the lower semicontinuity of $\overline{F}$, $\overline{F}(u)<\infty$.
Since $\overline{W}(E)<\infty$, by Proposition 6.1 in \cite{BM1}, there exists a finite system of curves $\Gamma$ of class $\wpq{2}{p}$ such that \begin{itemize} \item[(i)]$(\Gamma)\supseteq \partial E$;
\item[(ii)]$|E \Delta \operatorname{Int}(\Gamma)| = 0$; \end{itemize} and
$$\overline{W}(E)= \int_{\Gamma} [1+|k_{\Gamma}|^p] \text{d}\mathcal{H}^1.$$ In addition $\Gamma$ minimizes the functional
$$\Gamma \mapsto W(\Gamma) = \int_{\Gamma} [1+|k_{\Gamma}|^p] \text{d}\mathcal{H}^1$$ on the class of all systems of $\wpq{2}{p}$ curves satisfying $(i)$ and $(ii)$. Then the function $$\Psi : t\in\mathbb{R} \mapsto \Psi(t)=\Gamma\;\text{if $t\in[0,c]$},\;\emptyset\;\text{otherwise}$$ belongs to $\mathscr{A}(u)$ and, for every $\Phi\in \mathscr{A}(u)$, $$W(\Phi(t))\geq W(\Gamma)\quad \forall t\in [0,c]$$ therefore $$G(\Phi)\geq G(\Psi) \quad \quad \forall\Phi \in \mathscr{A}(u). $$ It follows from Theorem \ref{principale} that $$\overline{F}(u)= G(\Psi) = c \overline{W}(E).$$ \qed
Using the previous proposition and~\cite{BDP}, we can provide an explicit, and actually trivial, example of a minimizer of $G$ on $\mathscr{A}(u)$.
\begin{ex}{\rm
Let $u=\mathds{1}_E$ with $E$ like in Figure~\ref{determ}, left, and $L$ the distance between the two cusps. We will prove that the function $$\Phi: t\in [0,1]\mapsto \Gamma $$ is a minimizer of $G$ on $\mathscr{A}(u)$, being $\Gamma$ the curve in Figure~\ref{determ}, right.
\begin{figure}
\caption{The curve $\Gamma$ is canonically associated with the representation of $\overline{F}(\mathds{1}_E)$}
\label{determ}
\end{figure}
From the previous proposition we get \begin{equation}\label{goccia0}
\overline{F}(u)= \overline{W}(E), \end{equation} and by Theorem~8.6 in~\cite{BM1} we have \begin{equation}\label{goccia}
\overline{W}(E)= \mathcal{H}^1(\partial E) + \int_{\partial^* E} |k_{\partial^*E}|^p \text{d}\mathcal{H}^1 +2L. \end{equation} Now $\Phi$ belongs to $\mathscr{A}(u)$ and, by definition of $\Gamma$,
$$G(\Phi)=\mathcal{H}^1(\partial E) + \int_{\partial^* E} |k_{\partial^*E}|^p \text{d}\mathcal{H}^1 +2L$$ therefore $$\overline{F}(u)=G(\Phi)$$ and, by Theorem \ref{principale}, $\Phi$ minimizes $G$ on $\mathscr{A}(u)$. } \end{ex}
The next proposition has been implicitly used in the introduction. \begin{prop}\label{energiefinie}
Let $u\in {\mathrm{BV}}(\mathbb{R}^2)$ with $\overline{F}(u)<\infty$. Then $\overline{F}(u)\geq \displaystyle\int_{\mathbb{R}} \overline{W}(\{u>t\})dt$. In particular, $\overline{W}(\{u>t\})<\infty$ for almost every $t$.
\end{prop}
\noindent{\bf Proof\;\, } Since $\overline{F}(u)<\infty$ we can find a sequence $\{u_h\}\subseteq {\mathrm{C}}^2(\mathbb{R}^2)$ such that $$\underset{h\rightarrow \infty}{\lim}\,F(u_h) = \overline{F}(u).$$ Then by the coarea formula and Fatou's lemma we get \begin{equation}\label{inegwill}
\int_{\mathbb{R}} \underset{h \rightarrow \infty}{\liminf}\; W(\{u_h>t\})\,dt \leq \underset{h \rightarrow \infty}{\lim}\;F(u_h) = \overline{F}(u) <\infty \end{equation} and so, for a.e. $t$, $$\underset{h \rightarrow \infty}{\liminf}\; W(\{u_h>t\}) <\infty.$$
Since for almost every $t$, $|\{u_h>t\} \Delta \{u>t\}|\rightarrow 0$, it follows from the lower semicontinuity of the relaxation that $$\overline{W}(\{u>t\})\leq \liminf_{h\to\infty}\;W(\{u_h>t\})<\infty\quad{\mbox{for a.e. $t$}}.$$ Therefore, by \eqref{inegwill}, $$\overline{F}(u)\geq \int_{\mathbb{R}} \overline{W}(\{u>t\})\mbox{d}t.$$ \qed
\begin{ex}{\rm We already mentioned in the introduction and in Remark~\ref{iii-est-nec} the following example of a function $u\in{\mathrm{BV}}(\mathbb{R}^2)$ such that $\overline{F}(u)> \displaystyle\int_{\mathbb{R}} \overline{W}(\{u>t\})\mbox{d}t$, that we now simply revisit in the perspective of Theorem~\ref{principale}.
\par Let $u=\mathds{1}_{E\cup F}+\mathds{1}_F$ with $E,F$ as in Figure~\ref{ultima}A) and, in Figure~\ref{ultima}B), the limit systems of curves with their multiplicities corresponding to the independent approximation of $E\cup F$ and $F$, respectively. \begin{figure}
\caption{If $u=\mathds{1}_{E\cup F}+\mathds{1}_F$, $\overline{F}(u)$ does not coincide with the integral of $\overline{W}(\{u>t\})$.}
\label{ultima}
\end{figure}
There is no sequence of smooth functions $(u_h)$ approximating $u$ whose level lines concentrate this way because $\Gamma_1$, $\Gamma_2$ together with their densities cannot be contemporaneously approximated (pointwisely) using boundaries of nested sets. In addition, $$\int_{\mathbb{R}} \overline{W}(\{u>t\})\mbox{d}t=G(\Phi)$$ where $$\Phi(t)=\left\{ \begin{array}{ll}
\Gamma_1 & \text{if\;} t\in[0,1],\\
\Gamma_2 & \text{if\;} t\in]1,2],\\
\emptyset & \text{otherwise}. \end{array} \right.$$ Obviously, $\Phi\notin \mathscr{A}(u)$, and no system of curves in $\mathscr{A}(u)$ can compete with the energy of $\Phi$ since, to maintain the nesting property, it is necessary to create an additional path between both components of $\Gamma_1$ and the length of this path is at least the distance between the two disks. Therefore, $$\overline{F}(u) >\int_{\mathbb{R}} \overline{W}(\{u>t\})\mbox{d}t.$$} \end{ex}
\section{The relaxation problem on a bounded domain}\label{omega}
We consider in this section the generalized elastica functional defined on a bounded domain $\Omega \subset \mathbb{R}^2$ and we compare several definitions for the relaxation problem pointing out their differences. We shall keep the notations $F(u)$ and $\overline{F}(u)$ to denote the generalized elastica energy and its relaxation for a function $u$ defined on $\mathbb{R}^2$. \par Let $\Omega \subset \mathbb{R}^2$ be an open bounded domain with $\partial \Omega$ Lipschitz. A first definition of a generalized elastica functional on $\Omega$ is: $$ F_B(\cdot , \Omega): {\mathrm{BV}}(\Omega) \rightarrow \mathbb{R}$$ $$F_B(u, \Omega)= \left\{ \begin{array}{ll}
\displaystyle{\int_{\Omega}\left|\nabla u\right|\left(1+\left|\operatorname{div}\frac{\nabla u}{\left|\nabla u\right|}\right|^p\right)\,dx} & \mbox{\,\;if\,} u\in {\mathrm{C}}^2(\Omega)\\ +\infty & \mbox{\,\;otherwise} \end{array} \right. $$
with $p>1$, and $|\nabla u||\operatorname{div} \frac{\nabla u}{\left|\nabla u\right|}|^p=0$ if $\left|\nabla u\right|=0$. By definition of the relaxation $$\overline{F}_B(u, \Omega)= \inf\left\lbrace \underset{h\rightarrow\infty}{\liminf}\;F_B(u_h, \Omega) : \{u_h\}\subset {\mathrm{C}}^2(\Omega), \; u_h\overset{\lp{1}(\Omega)}{\longrightarrow}u\right\rbrace. $$ Remark first that, by the coarea formula, $$F_B(u, \Omega) = \int_{\mathbb{R}}\int_{\partial\{u>t\}\cap \Omega} (1+ \arrowvert {\kappa}_{\partial\{u>t\}\cap \Omega}\arrowvert^p) \,d\mathcal{H}^1\,dt,\qquad\forall u\in{\mathrm{C}}^2(\Omega),$$ so the generalized elastica functional on $\Omega$ depends only on the behavior in $\Omega$ of the level lines of $u$. Therefore, in contrast with what happens in $\mathbb{R}^2$, one cannot restrict to systems of closed curves. Open curves must also be considered, which raises new difficulties, as illustrated in the next example where we exhibit a function with infinite relaxed energy on $\mathbb{R}^2$ and finite relaxed energy on a suitable $\Omega$.
\begin{ex}\label{gocciasola} {\rm Let $E$, $\Omega$ be the sets drawn in Figure~\ref{gocciao}. Clearly, if $w=\mathds{1}_E$ in $\mathbb{R}^2$, $\overline{F}(v)=\infty$ according to~\cite[Thm 6.4]{BDP}. However, if we consider the relaxation problem on $\Omega$ and the sequence of functions $\{w_h\}\subset {\mathrm{C}}^2(\Omega)$ having level lines like the curve $\Gamma$ drawn in Figure~\ref{gocciao}, we get $\underset{h \rightarrow \infty}{\liminf}\; F_B(w_h, \Omega)<\infty $ thus $\overline{F}_B(w, \Omega)<\infty$. \begin{figure}
\caption{$\overline{F}(\mathds{1}_E) = \infty$ but $\overline{F}_B(\mathds{1}_E, \Omega)<\infty$}
\label{gocciao}
\end{figure}} \end{ex}
\par It is a trivial observation that if $u\in{\mathrm{BV}}(\mathbb{R}^2)$ is such that there exists a sequence $(u_h)$ with $u_h \rightarrow u$ in $\lp{1}(\mathbb{R}^2)$ and $\{F(u_h)\}_h$ is bounded, then, clearly, $\{F_B(u_h|_\Omega, \Omega)\}_h$ is bounded and $\overline{F}_B(u|_\Omega, \Omega)<\infty$.
Conversely, a natural question is the following: given $\{u_h\}\subset {\mathrm{C}}^2(\Omega)$ a sequence such that $F_B(u_h, \Omega)\to\overline{F}_B(u, \Omega)$,
can we find a sequence $\{v_h\}\subset {\mathrm{C}}^2(\mathbb{R}^2)$ with $F(v_h)$ uniformly bounded and $u_h=v_h$ in $\Omega$? In other words, can we say that sequences with bounded energy on $\Omega$ are the restriction to $\Omega$ of sequences with bounded energy on $\mathbb{R}^{2}$? A positive answer to this question would imply that $\overline{F}_B(\cdot, \Omega)$ coincides in ${\mathrm{BV}}(\Omega)$ with $L(\cdot, \Omega)$ where $$L(u,\Omega) := \inf\left\lbrace
\underset{h\rightarrow\infty}{\liminf}\;F_B(u_h|_\Omega, \Omega) : \{u_h\}\subset {\mathrm{C}}^2(\mathbb{R}^2), \; u_h|_{\Omega}\overset{\lp{1}(\Omega)}{\longrightarrow}u\;,\;\; \underset{h}{\sup} \;F(u_h)<\infty\right\rbrace, $$ with the convention $\inf\;\emptyset=\infty$.
\par A simple example of a function with finite $L(\cdot,\Omega)$ energy is the function $w=\one{E}$ of Example~\ref{gocciasola}. Take indeed the image $F$ of $E$ obtained by symmetry with respect to a vertical axis arbitrarily chosen at the right of $\Omega$. Then, $\partial(E\cup F)$ being smooth except at an even number of cusps, $\overline{F}(E\cup F)<+\infty$ according to Theorem~6.3 in~\cite{BDP}. Thus there exists a sequence of smooth functions $(u_h)$ that approximates $\one{E\cup F}$ in $\mathbb{R}^2$, belongs to the set $\left\lbrace
\{u_h\}\subset {\mathrm{C}}^2(\mathbb{R}^2), \; u_h|_{\Omega}\overset{\lp{1}(\Omega)}{\longrightarrow}u\;,\;\; \underset{h}{\sup}
\;F(u_h)<\infty\right\rbrace$ and is such that $\sup_h F_B(u_h|_\Omega, \Omega)<\infty$, therefore $L(u,\Omega)<\infty$.
\par The answer to the question above is negative in general as shown by the next example.
\begin{ex}\label{gocciadoppia} {\rm Let $E, \Omega$ be the sets drawn in Figure~\ref{gocciad} and let $u=\mathds{1}_E$. We know from Theorem 4.1 in~\cite{BDP} that any set $E$ with finite relaxed energy is such that $\partial* E$ has a continuous unoriented tangent, which is obviously not the case here, thus $L(\mathds{1}_E, \Omega) = \infty$. Roughly speaking, every sequence $\{v_h\}\subset {\mathrm{C}}^2(\mathbb{R}^2)$ converging to $u$ in $\Omega$ has to approximate the angle in $p\in\partial\Omega$ formed by the two cusps and so $F(v_h)\rightarrow \infty $. Now, if we consider the relaxation problem on $\Omega$ and the sequence of functions $\{u_h\}\subset {\mathrm{C}}^2(\Omega)$ having level lines like the curve $\Gamma$ drawn in Figure~\ref{gocciad}, there is no singularity in $\Omega$ and $\underset{h \rightarrow \infty}{\liminf}\; F_B(u_h, \Omega)<\infty $ thus $\overline{F}_B(u, \Omega)<\infty$. \begin{figure}
\caption{$L(\mathds{1}_E, \Omega) = \infty$ but $\overline{F}_B(\mathds{1}_E, \Omega)<\infty$}
\label{gocciad}
\end{figure}} \end{ex}
Another possible way to define the generalized elastica functional on an open bounded domain with Lipschitz boundary is the following, where ${\mathrm{BV}}_0(\Omega)$ denotes the space of functions of bounded variation defined in $\Omega$ with null trace on $\partial\Omega$: $$F_B^0(\cdot, \Omega): {\mathrm{BV}}_0(\Omega) \rightarrow \mathbb{R}$$ $$F_B^0(u,\Omega)= \left\{ \begin{array}{ll}
\displaystyle{\int_{\Omega}\left|\nabla u\right|\left(1+\left|\operatorname{div}
\frac{\nabla u}{\left|\nabla u\right|}\right|^p\right)\,dx} &\mbox{\,\;if\,} u\in {\mathrm{C}}^2_c(\Omega)\\ +\infty & \mbox{\,\;otherwise} \end{array} \right.$$
with $p>1$, and $\left|\nabla u\right|\left(1+\left|\operatorname{div} \frac{\nabla u}{\left|\nabla u\right|}\right|^p\right)=0$ if $\left|\nabla u\right|=0$.
\par In this case the relaxation in ${\mathrm{BV}}_0(\Omega)$ is defined by $$\overline{F}_B^0(u,\Omega)=\inf\left\lbrace \underset{h\rightarrow\infty}{\liminf}\;F_B^0(u_h, \Omega) : \{u_h\}\subset {\mathrm{C}}^2_c(\Omega), \; u_h\overset{\lp{1}(\Omega)}{\longrightarrow}u\right\rbrace.$$ Remark that the function of Example~\ref{gocciasola} is in ${\mathrm{BV}}_0(\Omega)$ and has infinite $F_B^0$ energy. In contrast, the function of Example~\ref{gocciadoppia} is not in the domain of $\overline{F}_B^0$ since it is not in ${\mathrm{BV}}_0(\Omega)$. It would not make sense to extend, by approximation, the definition of $\overline{F}_B^0$ to the functions of ${\mathrm{BV}}(\Omega)\setminus{\mathrm{BV}}_0(\Omega)$ since $\one{\Omega}$ is such function, has no level line in $\Omega$, but $$\inf\left\lbrace \underset{h\rightarrow\infty}{\liminf}\;F_B^0(u_h, \Omega) : \{u_h\}\subset {\mathrm{C}}^2_c(\Omega), \; u_h\overset{\lp{1}(\Omega)}{\longrightarrow}\one{\Omega}\right\rbrace=W(\Omega)>0.$$
\par The next proposition states the very natural localization property that a function with compact support that has finite energy can be approximated, in $\lp{1}$ and in energy, by a sequence of functions with compact support. The approximating sequence is built directly from the collection of curves that cover the level lines of the function. \begin{prop}\label{legameY} If $u\in{\mathrm{BV}}(\mathbb{R}^2)$ has compact support and $\overline{F}(u) < \infty$, there exists an open and bounded domain $\Omega$ that contains $\text{spt}\, u$ and satisfies
$$\overline{F}(u)=\overline{F}_B^0(u|_\Omega,\Omega).$$ \end{prop} \noindent{\bf Proof\;\, } Using Theorem \ref{principale} we have $\overline{F}(u)= G(\Phi)$ where $\Phi$ is a minimizer of $G$ on $\mathscr{A}(u)$. Since $u$ has compact support on $\mathbb{R}^2$ , the definition of $\mathscr{A}(u)$ and $G(\Phi)<\infty$ imply the existence of a bounded and open domain $\Omega$ such that \begin{equation}\label{inclusion} \bigcup_{t\in \mathbb{R}} (\Phi(t)) \subset \subset \Omega. \end{equation} \par Since $\overline{F}(u) \leq \inf\left\lbrace \underset{h\rightarrow\infty}{\liminf}\;F_B^0(u_h, \Omega) : \{u_h\}\in {\mathrm{C}}^2_c(\Omega), \; u_h\overset{\lp{1}(\Omega)}{\longrightarrow}u\right\rbrace$, using~\eqref{inclusion} and the same approximation arguments developed in the proof of Theorem~\ref{principale} together with the fact that $\{u>0\}$ is relatively compactly, we can define a sequence $\{u_h\}\in {\mathrm{C}}^2_c(\Omega)$ such that $u_h\overset{\lp{1}(\Omega)}{\longrightarrow}u$ and $F(u_h)=F_B^0(u_h, \Omega) \rightarrow G(\Phi)$. Then $$G(\Phi)= \overline{F}(u) \geq \inf\left\lbrace \underset{h\rightarrow\infty}{\liminf}\;F_B^0(u_h, \Omega) : \{u_h\}\in {\mathrm{C}}^2_c(\Omega), \; u_h\overset{\lp{1}(\Omega)}{\longrightarrow}u\right\rbrace$$ and the proposition ensues. \qed
\par The next example illustrates the difference between $\overline{F}_B(u,\Omega)$ and $\overline{F}_B^0(u,\Omega)$, that are defined using different function spaces for the approximation (${\mathrm{C}}^2(\Omega)$ for the former and ${\mathrm{C}}^2_c(\Omega)$ for the latter).
\begin{ex} {\rm Let $E, \Omega$ be the sets drawn in Figure~\ref{ultimaborn} and let $u=\mathds{1}_E$. \begin{figure}
\caption{$u=\mathds{1}_E\in BV(\Omega)$}
\label{ultimaborn}
\end{figure}
From the properties of the relaxation, from the previous proposition, and thanks to Theorem~8.6 in~\cite{BM1}, we have $$\overline{F}(u)=\inf\left\lbrace \underset{h\rightarrow\infty}{\liminf}\;F_B^0(u_h, \Omega) : \{u_h\}\in {\mathrm{C}}^2_c(\Omega), \; u_h\overset{\lp{1}(\Omega)}{\longrightarrow}u\right\rbrace = G(\Phi)$$ where $\Omega$ is given by the previous proposition and $\Phi(t)$ looks for every $t\in[0;1]$ like the curve $\gamma_1$ drawn with its multiplicity in Figure~\ref{gamma12}A.
\begin{figure}\label{gamma12}
\end{figure}
\par However, if we consider the sequence of functions $\{u_h\}\subset {\mathrm{C}}^2(\Omega)$ having level lines like the line $\gamma_2$ drawn in Figure~\ref{gamma12}B we get $\underset{h \rightarrow \infty}{\liminf}\; F_B(u_h, \Omega)< G(\Phi) $. Therefore $$ \overline{F}_B(u, \Omega) < \overline{F}_B^0(u_h, \Omega).$$ } \end{ex}
\end{document} | arXiv |
\begin{document}
\title{Closed Affine Manifolds with an Invariant Line \thanks{The author gratefully acknowledges research support from NSF Grant Award Number 1709791} }
\author{Charles Daly
}
\institute{Charles Daly \at
4176 Campus Dr, College Park, MD 20742 \\
\email{[email protected]}
}
\date{Received: date / Accepted: date}
\maketitle
\begin{abstract} A closed affine manifold is a closed manifold with coordinate patches into affine space whose transition maps are restrictions of affine automorphisms. Such a structure gives rise to a local diffeomorphism from the universal cover of the manifold to affine space that is equivariant with respect to a homomorphism from the fundamental group to the group of affine automorphisms. The local diffeomorphism and homomorphism are referred to as the developing map and holonomy respectively. In the case where the linear holonomy preserves a common vector, certain `large' open subsets upon which the developing map is a diffeomorphism onto its image are constructed. A modified proof of the fact that a radiant manifold cannot have its fixed point in the developing image is presented. Combining these results, this paper addresses the non-existence of certain closed affine manifolds whose holonomy leaves invariant an affine line. Specifically, if the affine holonomy acts purely by translations on the invariant line, then the developing image cannot meet this line. \keywords{affine \and manifolds \and holonomy \and invariant \and parallel \and radiant}
\end{abstract}
\section{Affine Structures Preliminaries} \label{sec:1} This section is largely dedicated to preliminary notions regarding closed affine manifolds. Specifically, this section provides examples, notation, and some basic results regarding the developing map and holonomy.
\begin{definition} An $n$-dimensional affine manifold is a $n$-dimensional manifold $M$ equipped with charts $(U_{\alpha}, \Phi_{\alpha})$ where each $\Phi_{\alpha} : U_{\alpha} \longrightarrow \mathbb{A}^{n}$ is a diffeomorphism onto an open subset of affine $n$-space such that the restriction of each transition map on each connected component of $U_{\alpha}\cap U_{\beta}$ is an affine automorphism. Explicitly, this says for each pair of charts $(U_{\alpha}, \Phi_{\alpha})$ and $(U_{\beta}, \Phi_{\beta})$ and each connected component $V \subset U_{\alpha}\cap U_{\beta}$, there exists an affine automorphism $A_{\beta\alpha,V} : \mathbb{A}^{n} \longrightarrow \mathbb{A}^{n}$ so that the following equality holds
\begin{equation}\label{eq1}
\begin{tikzcd}
& V \arrow{dl}[swap]{\Phi_{\alpha}} \arrow{dr}{\Phi_{\beta}} & \\
\Phi_{\alpha}(V) \arrow{rr}[swap]{\Phi_{\beta}\circ\Phi_{\alpha}^{-1} = A_{\beta\alpha}|_{V}} & & \Phi_{\beta}(V)
\end{tikzcd} \nonumber \end{equation} \end{definition} This definition lends itself to a natural generalization of what is known as a $(G,X)$-manifold, where $G$ is a lie group acting strongly effectively on a manifold $X$. The study of affine geometry is the study of $(\text{Aff}(n,\mathbb{R}), \mathbb{A}^{n})$-manifolds. Here are some standard examples in the literature of affine structures on the two-torus.
\begin{example}\label{ex1} Let $T$ be a rank two free abelian subgroup of $\text{Aff}(2,\mathbb{R})$ which acts on the affine plane by translations. Let $M = \mathbb{A}^{2}/T$. Since $T$ acts properly and freely on $\mathbb{A}^{2}$, the associated quotient map is a smooth covering and the charts on $M$ are naturally diffeomorphisms onto open subsets of $\mathbb{A}^{2}$. The transition maps are given by elements of $T$, and thus $M$ inherits an affine structure. In fact, quite a bit more can be said. Since the group of translations preserve the standard euclidean metric, $M$ inherits a riemannian structure. This provides $M$ a riemannian metric locally isometric to the euclidean plane. Manifolds arising as quotients of affine space by discrete subgroups of $\text{Isom}(n,\mathbb{R})$ are known as euclidean manifolds. Bieberach showed that such manifolds are finite coverings of euclidean tori. \end{example}
This construction of finding discrete subgroups of the affine group that act properly and freely on affine space yield an entire class of affine manifolds. Invariants of the group such as vector fields, covector fields, and metrics descend to the quotient and are studied extensively in the field of geometric structure.
\begin{example}\label{ex2} Let $D$ be a cyclic subgroup of $\text{Aff}(2,\mathbb{R})$ which acts on the punctured plane $\mathbb{C}^{\times}$ by positive dilations centered at the origin. Say that $D$ is generated by some $\lambda > 0$, so $D$ acts properly and freely on $\mathbb{C}^{\times}$ as in the previous example, and thus the quotient $N = \mathbb{C}^{\times}/D$ inherits an affine structure. The group $D$ preserves the lorrentzian metric $m = (dx^{2}+dy^{2})/(x^{2}+y^{2})$ and thus descends to a lorrentzian metric on $N$. In contrast to the previous riemannian example, this metric is incomplete. In fact, geodesics on $\mathbb{C}^{\times}$ pointed towards the origin will descend to geodesics on $N$ that become undefined in finite time. This should be contrasted with the Hopf-Rinow of riemannian geometry which states that every closed riemannian manifold is geodesically complete. \end{example}
\begin{example}\label{ex3} One may generalize the construction in Example \ref{ex1} in the following fashion. Pick a quadrilateral $Q$ in the affine plane. From the bottom left vertex and reading counterclockwise, label the edges $\alpha,\beta,\gamma$ and $\delta$. Construct a new quadrilateral in the following fashion. Rotate and scale $Q$ at the bottom left vertex in such a fashion that the rotated $\delta$ differs from $\beta$ by translation along $\alpha$. Translate the result along $\alpha$ to yield a new quadrilateral $Q'$ with edges $\alpha',\beta',\gamma',$ and $\delta'$. This process glues $\beta$ of $Q$ to $\delta'$ of $Q'$. Repeat this process with each pair of edges indefinitely. Figure \ref{fig:1} above shows an example of this process where the edges are no longer simple translates of each other.
\begin{figure}
\caption{The quadrilaterals $Q$ and $Q'$ are labeled with their corresponding edges in a counterclockwise fashion. Note the edges $\beta$ and $\delta'$ are identified in the quotient. The quadrilateral $Q$ serves as a fundamental domain for the corresponding group action. [Made in Mathematica 12 Student Version] }
\label{fig:1}
\end{figure}
The case where both pairs of opposite edges of $Q$ are parallel is the method of identification as in Example \ref{ex1} and defines a euclidean structure on the torus. In the case where pairs of opposite edges of $Q$ are not necessarily parallel, this method provides an inequivalent structure on the torus known as a similarity structure which are affine structures whose holonomy lies within the group of similarity transforms of affine space. In fact, Figure \ref{fig:1} was generated by the quadrilateral $Q = \{(0,0),(2,0),(1,1),(0,1)\}$ and the group of similarity transformations generated by
\[ a = \left(\begin{array}{cc} 1/2 & 0 \\ 0 & 1/2 \end{array} \right) \left(\begin{array}{c} 0\\ 1 \end{array} \right)
\text{ and }
b = \left(\begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array} \right) \left(\begin{array}{c} 2\\ 0 \end{array} \right) \]
\end{example}
These three examples serve to illustrate the complexities that arise once one departs from the riemannian case to the affine case. For those interested in learning more about the affine structures supported by the two torus, Oliver Baues provides an excellent treatment about different types of affine structures supported on the two torus \cite{bib1}. \\ \\
Given an affine structure on a manifold there is a natural associated local diffeomorphism from the universal cover of $M$ to affine space and a homomorphism from $\pi_{1}(M,p) \longrightarrow \text{Aff}(n,\mathbb{R})$. The local diffeomorphism is called the developing map, the group homomorphism is called the holonomy, and the two together are called a developing pair. A brief description of these maps is provided below, but further details about their construction may be found in William Goldman's Geometric Structure on Manifolds \cite{bib2}.\\ \\
Base the fundamental group at a point $p \in M$. Let $p \in (U,\Phi)$ be an affine coordinate patch about $p$. Each path $\gamma : [0,1] \longrightarrow M$ beginning at $p$ may be assigned a point in affine space in the following fashion. Cover the path $\gamma$ by $(k+1)$-coordinate patches beginning with $(U,\Phi)$. Label these patches by $(U_{i},\Phi_{i})$ where $(U_{0},\Phi_{0}) = (U,\Phi)$. Pick a mesh of times $0=t_{0} < t_{1} < \hdots < t_{k} < t_{k+1} = 1$ in $[0,1]$ so that each $\gamma(t_{i}) \in U_{i-1}\cap U_{i}$ for $i = 1\hdots k$. Let $\gamma_{i}$ be the restriction of $\gamma$ to $[t_{i},t_{i+1}]$ for each $i = 0 \hdots k$.\\ \\
Inductively define paths in affine space in the following fashion. Let $\alpha_{0} = \Phi_{0}\circ \gamma_{0}$. Let $V_{1}$ be the connected component of $U_{0}\cap U_{1}$ containing $\gamma(t_{1})$ and $g_{0,1}$ be the affine automorphism so that $g_{0,1}|_{V_{1}} = \Phi_{0}\circ\Phi_{1}^{-1} : \Phi_{1}(V_{1}) \longrightarrow \Phi_{0}(V_{1})$. Define $\alpha_{1} = g_{0,1}(\Phi_{1}\circ\gamma_{1})$. Note that the initial point of $\alpha_{1}$ is the terminal point of $\alpha_{0}$. Let $V_{2}$ be the connected component of $U_{1}\cap U_{2}$ containing $\gamma(t_{2})$ and $g_{1,2}$ be the affine automorphism so that $g_{1,2}|_{V_{2}} = \Phi_{1}\circ \Phi_{2}^{-1} : \Phi_{2}(V_{2}) \longrightarrow \Phi_{1}(V_{2})$. Define $\alpha_{2} = g_{0,1}g_{1,2}(\Phi_{2}\circ\gamma_{2})$. Note the initial point of $\alpha_{2}$ is the terminal point of $\alpha_{1}$. Continue this process inductively to obtain $(k+1)$-paths into affine space and concatenate them to obtain the path
\begin{align}\label{eq2} \alpha_{0}\cdot\alpha_{1}\cdot\hdots\cdot\alpha_{k} = (\Phi_{0}\circ &\gamma_{0})\cdot(g_{0,1}(\Phi_{1}\circ \gamma_{1}))\cdot (g_{0,1}g_{1,2}(\Phi_{2}\circ\gamma_{2}))\cdot\hdots \nonumber \\ &\cdot(g_{0,1}g_{1,2}\hdots g_{k-1,k}(\Phi_{k}\circ \gamma_{k})) \end{align} The developing map is defined as the terminal point of this path. Figure \ref{fig:2} illustrates this construction with three charts.
\begin{figure}
\caption{Three charts $U_{0}, U_{1}$, and $U_{2}$ cover a path $\gamma$ based at $p$. The path is separated into three pieces $\gamma_{0}, \gamma_{1}$ and $\gamma_{2}$ in red, green, and blue respectively. The charts are shown to the right with their images in affine space. The pink arrows represent the affine transformations taking one affine image to the next, i.e. $g_{0,1}$ and $g_{1,2}$. For example, the blue path in the third patch is mapped to the to blue path in the second patch by $g_{1,2}$. This construction yields the concatenation $\alpha_{1}\cdot\alpha_{2}\cdot\alpha_{3}$ in the top right whose terminal point is the developing map.}
\label{fig:2}
\end{figure} Several facts need to be verified about this assignment. One must show that this map is independent of the choice of charts after the initial chart about $p\in(U,\Phi)$ is chosen. In addition, one must show that the map is well-defined up to homotopy of paths based at $p$. After these technical details are established, this assignment induces a local diffeomorphism from the universal cover of $M$ based at $p$ to affine space which is denoted $\text{dev} : \widetilde{M} \longrightarrow \mathbb{A}^{n}$. \\ \\
Let $\gamma$ be a path based at $p$ contained in a chart $(U,\Phi)$ as above, and let $[\beta] \in \pi_{1}(M,p)$. As the developing map is defined in terms of homotopy classes of paths based at $p$, it is natural to consider how the developing map behaves by precomposition of loops based at $p$. That is, one may consider how Equation \ref{eq2} changes by considering the path $\beta\cdot\gamma$ where $\beta$ is a representative of the homotopy class $[\beta]$. As $\beta$ begins and ends at $p$, one may take the initial and terminal charts covering $\beta$ to both be $(U,\Phi)$ with charts $(V_{i},\Theta_{i})$ covering the remainder of $\beta$. Say that the corresponding construction applied to $\beta$ yields paths $\delta_{1},\delta_{2},\hdots, \delta_{j}$ in affine space with change of coordinate elements $h_{0,1},h_{1,2},\hdots, h_{j-1,j}$. Then the developing map applied to the concatenation $\beta\cdot \gamma$ yields
\begin{align*}\label{eq3} &\delta_{1}\cdot \hdots \cdot \delta_{j} \cdot \gamma_{1}\cdot \hdots \cdot \gamma_{k} = \nonumber \\ &\phantom{==} (\Phi_{0}\circ \beta_{0})\cdot(h_{0,1}(\Theta_{1}\circ \beta_{1}))\cdot (h_{0,1}h_{1,2}(\Theta_{2}\circ\beta_{2}))\cdot\hdots \nonumber\\ &\phantom{====}\cdot(h_{0,1}h_{1,2}\hdots h_{j-1,j}(\Theta_{j}\circ \beta_{j}))\cdot \gamma_{1}\cdot \hdots \cdot \gamma_{k} \nonumber \end{align*} As the right hand side product of the $\gamma_{i}$'s is left unchanged, and the developing map is defined as the terminal point of the constructed path above, one can see that
\begin{equation}\label{eq4} \left(\delta_{1}\cdot \hdots \cdot \delta_{j} \cdot \gamma_{1}\cdot \hdots \cdot \gamma_{k}\right)(1) = h_{0,1}h_{1,2}\hdots h_{j-1,j}\left(\gamma_{1}\cdot\hdots\cdot\gamma_{k}\right)(1) \end{equation} Since the path $\gamma$ was arbitrary, Equation \ref{eq4} holds for all such paths based at $p$, so precomposition with an element of $[\beta]\in \pi_{1}(M,p)$ yields a difference in the developing map by the element of the affine group $h_{0,1}h_{1,2}\hdots h_{j-1,j}$ which corresponds to $[\beta]$. This element is known as the holonomy of $[\beta]$, denoted $\text{hol}[\beta]$, and defines a homomorphism, known as the holonomy map, from $\pi_{1}(M,p) \longrightarrow \text{Aff}(n,\mathbb{R})$. Equation \ref{eq4} is the statement that the developing map is equivariant with respect to the holonomy homomorphism in the sense of the following commutative diagram.
\begin{equation}\label{eq5}
\begin{tikzcd}
\widetilde{M} \arrow{r}{[\beta]} \arrow{d}[swap]{\text{dev}} & \widetilde{M} \arrow{d}{\text{dev}}\\
\mathbb{A}^{n} \arrow{r}[swap]{\text{hol}[\beta]}& \mathbb{A}^{n}
\end{tikzcd} \end{equation}
The pair $(\text{dev},\text{hol})$ is known as a developing pair for the affine structure on $M$. The construction of the developing map above carries over to the broader context of $(G,X)$-structures on manifolds wherein one assumes that a lie group $G$ acts strongly effectively on a manifold $X$. A very nice exposition about $(G,X)$-structures may be found in Stephan Schmitt's Geometric Manifolds \cite{bib4} whereas \cite{bib2} provides a very thorough general reference.
\section{Affine Structures with an Invariant Line} \label{sec:2} The purpose of this section is establish more specialized notation, preliminary observations about the consequences of having an affine structure whose affine holonomy admits an invariant affine line, and to prove a theorem about the non-existence of certain affine structures. \\ \\ Let $l \subset \mathbb{A}^{n+1}$ be a line and $G \leq \text{Aff}(n+1,\mathbb{R})$ be the group of affine automorphisms preserving $l$. Pick an origin on $l$ and identify $\mathbb{A}^{n+1}$ with $\mathbb{R}^{n+1}$. Rotate about the origin so that $l$ aligns with the $x$-axis in $\mathbb{R}^{n+1}$ and $l$ identifies with the first factor of $\mathbb{R}$ in $\mathbb{R}\times\mathbb{R}^{n}$. Up to conjugation $G$ is isomorphic to
\begin{equation}\label{eq6} G = \left\{
\left(
\begin{array}{cc}
r & w \\
0 & A
\end{array}
\right)
\left(
\begin{array}{c}
d \\
0
\end{array}
\right)
\Bigg|\,
r \neq 0, d\in\mathbb{R}, w^{T} \in \mathbb{R}^{n}, A \in \text{GL}(n,\mathbb{R}) \right\} \end{equation} For this purpose of this paper the coordinates $x$ and $y^{1},\hdots, y^{n}$ are reserved for the first and second factors of $\mathbb{R}\times\mathbb{R}^{n}$ respectively. In an abuse of notation, $\mathbb{R}$ will frequently denote the invariant line $\mathbb{R}\times0 \subset \mathbb{R}\times\mathbb{R}^{n}$. The instances in which this occurs will be clear throughout the paper. Before stating one of the theorems of this paper, a definition is in order.
\begin{definition}\label{def2} Let $M$ be a closed affine manifold with a developing pair $(\text{dev},\text{hol})$. If the developing map is a covering onto $\mathbb{A}^{n}$, then $M$ is called complete. \end{definition}
Since both the universal cover of $M$ and $\mathbb{A}^{n}$ are simply connected, this definition implies that the developing map is a diffeomorphism onto $\mathbb{A}^{n}$. In this case, Corollary \ref{cor3} in Section \ref{sec:7} yields that the holonomy group, $\text{hol}(\Gamma)$, acts both properly and freely on affine space, and $M$ is diffeomorphic to $\mathbb{A}^{n}/\text{hol}(\Gamma)$. In this context the manifold $M$ can be recovered from the image of the holonomy homomorphism. It is a non-trivial problem to construct inequivalent structures on a manifold with the same holonomy groups. The interested reader is referred to Projective Structure with Fuchsian Holonomy \cite{bib5}.
\begin{theorem}\label{thm1} For $n \geq 1$, there are no complete affine structures on closed $(n+1)$-manifolds whose holonomy lies in the group $G$ of line preserving affine automorphisms. \end{theorem}
This proof is broken into two pieces. Let $H \leq G$ denote a subgroup of $G$ as in Equation \ref{eq6} that acts both properly and freely on $\mathbb{R}\times\mathbb{R}^{n}$ with a compact quotient. The proof begins by showing that $H$ must act purely by translations on the invariant line, otherwise, there will be an accumulation point of the group action on the invariant line, contradicting properness. After that is established, $H$ will be shown to be cylic, whereby the quotient manifold will be shown to be a mapping torus, $M_{A}$, of a linear map $A : \mathbb{R}^{n} \longrightarrow \mathbb{R}^{n}$. Certain topological obstructions will prevent this from occurring and yield a contradiction.
\begin{proof} Let $H \leq G$ act both properly and freely on $\mathbb{R}\times\mathbb{R}^{n}$ with a compact quotient. Without loss of generality the scaling factor, $r \neq 0$, as in Equation \ref{eq6} may be taken to be non-negative. The subgroup of $G^{+}\leq G$ whose scale factors $r$ on the invariant line are positive form an index two subgroup of $G$. Consequently the quotient $\mathbb{R}\times\mathbb{R}^{n}/G^{+}$ is a double cover of the quotient by $G$, and thus preserves both an affine structure and compactness. In addition one may lift to the orientable double cover to assume the manifold $\mathbb{R}\times\mathbb{R}^{n}/G^{+}$ is orientable. \\ \\
If there is indeed an element $h \in H$ so that $r \neq 1$, then there is a solution to the equation $rx + d = x$. Conjugate by translation along the invariant line to assume $h$ is of the form
\begin{equation}\label{eq7} h = \left(
\begin{array}{cc}
r & w \\
0 & A
\end{array} \right) \left(
\begin{array}{c}
0 \\
0
\end{array} \right) \nonumber \end{equation} Note that $h$ acts linearly on $\mathbb{R}\times\mathbb{R}^{n}$ and fixes the origin. That said, the orbit of the origin and say for example $(1,0) \in \mathbb{R}\times\mathbb{R}^{n}$ are inseparable by open sets. If $(1,0)$ in indeed in the orbit of the origin, one may pick a point arbitrarily close to $(1,0)$ on the invariant line that is not. The cyclic group generated by $h$ fixes the origin, whereas $h^{n}(1,0)$ will tend arbitrarily close to the origin along the invariant line for sufficiently large positive or negative values of $n$ depending on whether $r$ is less than or bigger than one. The orbits of the origin and $(1,0)$ are inseparable so the action is not proper, and thus elements that fail to act by pure translation on the invariant line are not in $H$. \\ \\
Now assume that $H$ acts by pure translations on the invariant line $\mathbb{R}$. Since the action is proper, it follows that map $T : H \longrightarrow \mathbb{R}$ defined by
\begin{equation}\label{eq8} T\left(\left(
\begin{array}{cc}
1 & w \\
0 & A
\end{array} \right) \left(
\begin{array}{c}
d \\
0
\end{array} \right) \right)= d \nonumber \end{equation}
is a homomorphism. By properness of the action of $H$, the image of $T$ is cyclic, as a dense image would yield an accumulation point on the invariant line. Since the action of $H$ on $\mathbb{R}\times\mathbb{R}^{n}$ is free, this homomorphism is injective. For if two elements $h,h' \in H$ yield the same translational part, then $h^{-1}h'$ fixes the origin in $\mathbb{R}\times\mathbb{R}^{n}$, and by freeness $h = h'$. Thus $H$ is generated by some
\[ h = \left(
\begin{array}{cc}
1 & w \\
0 & A
\end{array} \right) \left(
\begin{array}{c}
d \\
0
\end{array} \right) \text{ where } d\in\mathbb{R}, \, w^{T} \in \mathbb{R}^{n} \text{ and } A \in \text{GL}(n,\mathbb{R}) \]
Note if $w = 0$, then $\mathbb{R}\times\mathbb{R}^{n}$ quotiented by the cyclic group $H$ is homeomorphic to the mapping cylinder of the linear map $A:\mathbb{R}^{n} \longrightarrow \mathbb{R}^{n}$ as claimed. If $w\neq 0$, then conjugating $h$ by a sheer along $\mathbb{R}$ yields
\begin{equation}\label{eq9} \left(
\begin{array}{cc}
1 & v \\
0 & I_{n}
\end{array} \right) \left(
\begin{array}{cc}
1 & w \\
0 & A
\end{array} \right) \left(
\begin{array}{c}
d \\
0
\end{array} \right) \left(
\begin{array}{cc}
1 & -v \\
0 & I_{n}
\end{array} \right) = \left(
\begin{array}{ccc}
1 & \phantom{,}& w+v(A-I_{n}) \\
0 & \phantom{,}& A
\end{array} \right) \left(
\begin{array}{c}
d \\
0
\end{array} \right) \end{equation}
If $(A-I_{n})$ is invertible then $w + v(A-I_{n})$ can be chosen to be zero, thus providing the desired homeomorphism. Such a choice of $v$ is available if $(A-I_{n})$ is invertible, which is to say that $\lambda = 1$ is not an eigenvalue of $A$. \\ \\
Assume $w \neq 0$ and $\lambda = 1$ is an eigenvalue of $A$. If so, then there is a $u \in \mathbb{R}^{n}$ for which $Au = u$. Moreover, $w^{T}$ and $u$ are necessarily perpendicular. For if not let $k \in \mathbb{R}$, and then
\begin{equation}\label{eq10}
\left(
\begin{array}{cc}
1 & w \\
0 & A
\end{array} \right) \left(
\begin{array}{c}
d \\
0
\end{array} \right) \left(
\begin{array}{c}
0 \\
ku
\end{array} \right) = \left(
\begin{array}{c}
0+k(wu)+d \\
ku
\end{array} \right) \nonumber \end{equation}
where $A(ku) = ku$ as $u$ is an eigenvector of $A$ with eigenvalue $1$. Since $wu \neq 0$ as $w^{T}$ and $u$ are not perpendicular, there is a choice of $k$ for which $k(wu) + d = 0$ and thus $(0,ku)$ is fixed by a generator of $H$ contradicting the fact that $H$ acts freely on $\mathbb{R}\times\mathbb{R}^{n}$. Thus, $w^{T}$ and $u$ are perpendicular as claimed.\\ \\
Let $U$ denote the plane in $\mathbb{R}\times\mathbb{R}^{n}$ spanned by $(1,0)$ and $(0,u)$. Since $w^{T}$ and $u$ are orthogonal, $U$ is a closed subspace invariant under the action of $H$, so its quotient $U/H$ is a compact submanifold of $\mathbb{R}\times\mathbb{R}^{n}/H$. This though is a contradiction as $U/H$ is diffeomorphic to $S^{1}\times\mathbb{R}$, and is therefore non-compact. Since $w$ was assumed to be non-zero, this necessitates $\lambda = 1$ is not an eigenvalue of $h$.\\ \\ Since $\lambda = 1$ is not an eigenvalue of $h$, one may conjugate by translation along the invariant line as in Equation \ref{eq9} to assume $h$ is a matrix of the form
\[ h = \left(
\begin{array}{cc}
1 & 0 \\
0 & A
\end{array} \right) \left(
\begin{array}{c}
d \\
0
\end{array} \right) \]
Thus the quotient $\mathbb{R}\times\mathbb{R}^{n}/H$ is homeomorphic to $M_{A}$, the mapping torus of the linear map $A : \mathbb{R}^{n} \longrightarrow \mathbb{R}^{n}$. A standard result in topology \cite{bib6} provides the long exact sequence of homology groups of a mapping torus is given by
\begin{equation}\label{eq11} \hdots \longrightarrow H_{n+1}(\mathbb{R}^{n}) \longrightarrow H_{n+1}(M_{A}) \longrightarrow H_{n}(\mathbb{R}^{n}) \longrightarrow \hdots \nonumber \end{equation} Since $\mathbb{R}^{n}$ is contractible and $n \geq 1$, this necessitates that $H_{n+1}(M_{A})$ is trivial. Since $M_{A}$ is an $(n+1)$-dimensional, compact, oriented manifold, its top homology is necessarily non-trivial. This shows that there are no complete structures on a compact $(n+1)$-dimensional affine manifold whose affine holonomy preserves an affine line. \end{proof} It is worth noting that although Theorem \ref{thm1} forbids the existence of a complete structure on a closed manifold $M$ whose holonomy preserves an invariant line, this theorem says nothing about the existence of non-complete structures. In fact, there are plenty of examples of non-complete structures in which the developing map fails to be a covering onto all of $\mathbb{R}\times\mathbb{R}^{n}$.
\begin{example}\label{ex4} Pick a $\lambda > 1$ and let $M$ be $\mathbb{R}\times \mathbb{C}^{\times}/H$ where $H$ is the subgroup of affine transformations generated by
\begin{equation}\label{eq12} a = \left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right) \left(
\begin{array}{c}
1\\
0\\
0
\end{array} \right) \text{ and } b = \left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & \lambda & 0 \\
0 & 0 & \lambda
\end{array} \nonumber \right) \end{equation} $M$ defines an affine structure on the three-torus whose affine holonomy preserves the invariant line defined by the $x$-axis. In this case the induced action on the invariant line $\mathbb{R}$ is purely translational, yet the invariant line lies entirely outside the developing image of this affine structure.
\end{example} The fact that the invariant line lies outside the developing image is no coincidence in the case where the affine holonomy acts purely by translations on the invariant line. The remainder of this paper is dedicated to showing this is always the case. \\ \\
The basic strategy is to show that if the affine holonomy does indeed act by translations on the invariant line and the developing image meets the invariant line, the affine structure is complete thus yielding a contradiction to Theorem \ref{thm1}. To show the developing map is a diffeomorphism, two main techniques will be employed. \\ \\
The first is show that if the affine manifold admits a parallel flow, one can construct `large' open submanifolds of the universal cover upon which the restricted developing map is a diffeomorphism. The second is to show that if the manifold admits a so called `cylindrical' flow, these `large' open sets can be taken to be arbitrarily large. Once these two facts are established, the proof follows immediately as a consequence of Theorem \ref{thm1}.
\section{Parallel Flow} \label{sec:3} Let $M$ be a closed affine $(n+1)$-dimensional manifold with a developing pair $(\text{dev},\text{hol})$ and fundamental group $\Gamma = \pi_{1}(M,p)$ acting on the universal cover $\widetilde{M}$ satisfying Equation \ref{eq5}. Assume the affine holonomy group, $H = \text{hol}(\Gamma) \leq \text{Aff}(n+1,\mathbb{R})$, preserves a parallel vector field $V$. Pick an origin in $\mathbb{A}^{n+1}$, and identify $\mathbb{A}^{n+1}$ with $\mathbb{R}\times\mathbb{R}^{n}$ and $\text{Aff}(n+1,\mathbb{R})$ with $\text{GL}(n+1,\mathbb{R})\ltimes \mathbb{R}^{n+1}$. By the natural identification of parallel vector fields $V$ on $\mathbb{R}\times\mathbb{R}^{n}$ with the tangent space of $\mathbb{R}\times\mathbb{R}^{n}$ at a point, the statement that the affine holonomy preserves a parallel vector field is equivalent to the statement that there exists a $v \in T_{0}(\mathbb{R}\times\mathbb{R}^{n})$ so that for each $h \in H$, $v$ is an eigenvector of the linear part of $h$. Thus, up to conjugation, one may assume the affine holonomy lies inside the subgroup of affine automorphisms given by
\begin{equation}\label{eq13} P= \left\{
\left(
\begin{array}{ccc}
1 & w\\
0 & A
\end{array}
\right)
\left(
\begin{array}{c}
d \\
v
\end{array}
\right) \, \Bigg| \, d\in\mathbb{R}, w^{T}, v \in \mathbb{R}^{n}, A\in \text{GL}(n,\mathbb{R}) \right\} \end{equation} The non-vanishing parallel $H$-invariant vector field $\partial/\partial x$ lifts to a non-vanishing parallel $\Gamma$-invariant vector field $\widetilde{V}$ which descends to a non-vanishing parallel vector field $V$ on $M$. As $M$ is compact, the flow associated to $V$ is complete, and so too is the flow associated to $\widetilde{V}$. Denote this flow on $\widetilde{M}$ by $\widetilde{\Theta}_{t}$. As $\widetilde{V}$ is related to $\partial/\partial x$ by the developing map, so too are their corresponding flows. Denoting $T_{t}$ as the translational flow corresponding to $\partial/\partial x$, one obtains the commutative diagram
\begin{equation}\label{eq14}
\begin{tikzcd}
&\widetilde{M} \arrow{r}{\widetilde{\Theta}_{t}} \arrow{d}[swap]{\text{dev}} &\widetilde{M} \arrow{d}{\text{dev}}\\
&\mathbb{R}\times\mathbb{R}^{n} \arrow{r}[swap]{T_{t}}&\mathbb{R}\times\mathbb{R}^{n}
\end{tikzcd} \nonumber \end{equation} As $T_{t}$ is the flow associated the vector field $\partial/\partial x$ which is invariant under the holonomy, this necessitates that $T_{t}$ commutes with each element of the holonomy as the holonomy will take flow lines to flow lines. The same statement holds for the commutativity of $\widetilde{T}_{t}$ and $\Gamma$. \\ \\
It is clear that the $\mathbb{R}$-action on $\mathbb{R}\times\mathbb{R}^{n}$ given by translation $T_{t}$ is both free and proper, consequently, so too is the $\mathbb{R}$-action on $\widetilde{M}$ by Lemma \ref{lm2}. Thus $\widetilde{M}$ is a principal $\mathbb{R}$-bundle over the quotient manifold $\widetilde{M}/\mathbb{R}$. Denote this quotient manifold by $N$ and the associated quotient map by $q : \widetilde{M} \longrightarrow N$. Since $\mathbb{R}$ is contractible, the principal $\mathbb{R}$-bundle structure on $\widetilde{M}$ is isomorphic to the trivial bundle $p_{2} : \mathbb{R}\times N \longrightarrow N$ where $p_{2}$ is factor projection onto the second factor. By triviality of the principal bundle, there is an $\mathbb{R}$-equivariant diffeomorphism $\Phi : \widetilde{M} \longrightarrow \mathbb{R}\times N$ for which the below diagrams commute.
\begin{equation}\label{eq15}
\begin{tikzcd}
\widetilde{M} \arrow{dr}[swap]{q} \arrow{rr}{\Phi} & & \arrow{dl}{p_{2}} \mathbb{R}\times N \\
& N \\
\end{tikzcd}
\text{ and }
\begin{tikzcd}
&\widetilde{M} \arrow{r}{\widetilde{\Theta}_{t}} \arrow{d}[swap]{\Phi} &\widetilde{M} \arrow{d}{\Phi}\\
&\mathbb{R}\times N \arrow{r}[swap]{\widetilde{T}_{t}}&\mathbb{R}\times N
\end{tikzcd} \end{equation}
A standard result in the theory principal bundles states that a principal bundle is trivial if and only if the bundle admits a global section, and thus $N$ may be thought of as an embedded submanifold of $N \subset \widetilde{M}$ for which the saturation of $N$ by the $\mathbb{R}$-action yields all of $\widetilde{M}$. \\ \\
The identification of $\Phi : \widetilde{M} \longrightarrow \mathbb{R}\times N$ provides a $\Gamma$-action on $\mathbb{R}\times N$ via conjugation by $\Phi$. Specifically, $[\gamma](t,n) = (\Phi\circ [\gamma] \circ \Phi^{-1})(t,n)$. The $\Gamma$-action on $\mathbb{R}\times N$ commutes with the $\mathbb{R}$-action on $\mathbb{R}\times N$ as per consequence of the definition of the $\Gamma$-action on $\mathbb{R}\times N$ and Equation \ref{eq15}. In addition, one obtains the commutative diagram
\begin{equation}\label{eq16}
\begin{tikzcd}
\mathbb{R} \times N \arrow{r}{[\gamma]} \arrow{d}[swap]{\Phi^{-1}} & \mathbb{R} \times N \arrow{d}{\Phi^{-1}} \\
\widetilde{M} \arrow{r}{[\gamma]} \arrow{d}[swap]{\text{dev}}& \widetilde{M} \arrow{d}{\text{dev}} \\
\mathbb{R}\times\mathbb{R}^{n} \arrow{r}[swap]{\text{hol}[\gamma]} & \mathbb{R}\times\mathbb{R}^{n}
\end{tikzcd} \end{equation} The composition of $\text{dev}\circ \Phi^{-1}$ provides a local diffeomorphism of $\mathbb{R}\times N$ into $\mathbb{R}\times\mathbb{R}^{n}$ which is equivariant with respect to the $\Gamma$-action on $\mathbb{R}\times N$. In the standard abuse of notation, this composition is also denoted $\text{dev} : \mathbb{R}\times N \longrightarrow \mathbb{R}\times\mathbb{R}^{n}$. \\ \\
By the $\mathbb{R}$-equivariance of $\Phi$ as in Equation \ref{eq15}, one obtains the commutative diagram
\begin{equation}\label{eq17}
\begin{tikzcd}
&\mathbb{R}\times N \arrow{d}[swap]{\text{dev}} \arrow{r}{\widetilde{T}_{t}} & \mathbb{R}\times N\arrow{d}{\text{dev}} \\
&\mathbb{R}\times\mathbb{R}^{n} \arrow{r}[swap]{T_{t}} & \mathbb{R}\times\mathbb{R}^{n}
\end{tikzcd} \end{equation}
Before continuing, it is worth noting that Equation \ref{eq17} is more or less the statement that there exists a transverse submanifold $N$ that generates the $\mathbb{R}$-action on the universal cover, $\mathbb{R}\times N$, in such a fashion that that the developing map takes flow lines to flow lines. As both $\mathbb{R}$-actions are free and proper, one may pass the vertices of Equation \ref{eq17} to their quotients. As both the $\Gamma$-action on $\mathbb{R}\times N$ and the holonomy on $\mathbb{R}\times\mathbb{R}^{n}$ commute with their respective $\mathbb{R}$-actions, the actions of $\Gamma$ and the holonomy group pass to their quotients, which are also abusively denoted by $[\gamma]$ and $\text{hol}[\gamma]$. Specifically, one has the commutative square
\begin{equation}\label{eq18}
\begin{tikzcd}
& N \arrow{d}[swap]{\overline{\text{dev}}} \arrow{r}{[\gamma]} & N\arrow{d}{\overline{\text{dev}}} \\
&\mathbb{R}^{n} \arrow{r}[swap]{\text{hol}[\gamma]} & \mathbb{R}^{n}
\end{tikzcd} \end{equation}
where $\overline{\text{dev}} : N \longrightarrow \mathbb{R}^{n}$ is a local diffeomorphism, and the holonomy $\text{hol}[\gamma]$ is acting affinely on $\mathbb{R}^{n}$ by the affine action induced by the matrices and vectors $A \in \text{GL}(n,\mathbb{R})$ and $v\in\mathbb{R}^{n}$ as in Equation \ref{eq13}. \\ \\
Equation \ref{eq18} looks deceptively as though it defines an affine structure on $N/\Gamma$. This though assumes that the induced action of $\Gamma$ on $N$ is both free and proper, which is not necessarily the case. The following example below illustrates possible obstructions when passing the quotient.
\begin{example}\label{ex5} Let $D$ be a closed unit disk centered at the origin in $\mathbb{R}^{2}$. Let $\Gamma$ be the cyclic group acting on $\mathbb{R}\times D$ generated by translation along $\mathbb{R}$ and rotation by an irrational angle $\theta$. For example let $\Gamma$ be generated by
\[ a = \left( \begin{array}{ccc} 1 & 0 & 0\\ 0 & \cos(\theta) & -\sin(\theta) \\ 0 & \sin(\theta) & \cos(\theta) \end{array} \right) \left( \begin{array}{c} 1\\ 0\\ 0 \end{array} \right) \] \end{example} Let $M = \mathbb{R}\times D/\Gamma$ which is a compact three-dimensional manifold with boundary as the action of $\Gamma$ on $\mathbb{R}\times D$ is both free and proper. In fact, $M$ is the mapping torus of the map $a : D \longrightarrow D$. Since the holonomy $\Gamma$ preserves the vector field $\partial/\partial x$, $\Gamma$ maps flow lines to flow lines whose images are $\mathbb{R}\times p$ for each $p \in D$. \\ \\
The induced action of $\Gamma$ on $D$ is neither free no proper. In particular the induced action of $a$ on the unit disk preserves the flow line of the origin, and is consequently not free. In addition, the orbit of each point $p \in D$ with a radius $0 < r \leq 1 $ has an orbit that is dense on the corresponding circle of radius $r$, and consequently the induced action of $\Gamma$ on $D$ is not proper. Though this is example with boundary, it nevertheless conveys the fragility of proper actions. Proper actions in general do not pass to proper actions on quotients. This should be contrasted with case where a proper action is lifted, see for example Lemma \ref{lm2}. \\ \\
As mentioned towards the end of Section \ref{sec:2}, the goal of this section is to prove the existence of `large' open subsets upon which the developing map when restricted to them are diffeomorphisms. As all the necessary terminology is in place, the theorem may be stated.
\begin{theorem}\label{thm2} Let $M$ be a closed affine $(n+1)$-dimensional manifold whose affine holonomy admits a parallel vector field. There exists a complete parallel flow on the universal cover of $M$ equivariant with respect to the parallel flow induced by the parallel vector field on affine space. Additionally, there exists open subsets of the universal cover invariant under the parallel flow for which the restricted developing map is a diffeomorphism onto its image. \end{theorem}
\begin{proof} The previous paragraphs establish the existence of a complete parallel flow satisfying the equivariance condition in the statement of the theorem. To finish the proof of Theorem 2, it suffices to show the existence of the open subset of $\mathbb{R}\times N$ invariant under the flow so that the restricted developing map is a diffeomorphism onto its image.\\ \\
Form the commutative squares as in Equation \ref{eq16} and Equation \ref{eq18}, and pick a point $y$ in the image of $\overline{\text{dev}}(N)$. As $\overline{\text{dev}}$ is a local diffeomorphism, the fibre over $y$ is a discrete subset of $N$. Pick a point $n \in N$ so that $\overline{\text{dev}}(n) = y$, and let $n \in U$ be an open set so that $\overline{\text{dev}}|_{U}$ is a diffeomorphism onto its image $ \overline{\text{dev}}(U)$. Let $\widetilde{U} = p_{2}^{-1}(U) = \mathbb{R}\times U \subset \mathbb{R} \times N$.\\ \\
Since $\widetilde{U}$ is open in $\mathbb{R}\times N$, to show that $\text{dev}|_{\widetilde{U}}$ is a diffeomorphism onto its image, it suffices to show that the developing map is injective on $\widetilde{U}$. To this end, let $(t,m)$ and $(s,m')$ be points in $\widetilde{U}$ so that $\text{dev}(t,m) = \text{dev}(s,m')$. This implies that $(\text{dev}\circ \widetilde{T}_{t})(0,m) = (\text{dev}\circ \widetilde{T}_{s})(0,m')$ and by the definition of $\overline{\text{dev}}$ and Equation \ref{eq17}, one obtains that $\overline{\text{dev}}(m) = \overline{\text{dev}}(m')$. Since $m,m' \in U$, by construction, this necessitates that $m = m'$. Consequently $(t,m)$ and $(s,m')$ are in the same $\mathbb{R}$-orbit. By freeness of the Equation \ref{eq17} and freeness of the $\mathbb{R}$-actions, this necessitates that $s = t$ so $(t,m) = (s,m')$. Consequently the developing map restricted to $\widetilde{U}$ is diffeomorphism. \end{proof}
To summarize, the basic idea of Theorem \ref{thm2} is that there exists a codimension one submanifold $N \subset \widetilde{M}$ transverse to the parallel flow on $\widetilde{M}$ that generates the $\mathbb{R}$-action on $\widetilde{M}$. In addition, the developing map sends flow lines to flow lines. The deck transformations, holonomy, and developing map all factor through the corresponding $\mathbb{R}$-actions to yield a local diffeomorphism on $N$ that provide coordinate charts on $N$. Since $N \subset \widetilde{M}$, these transverse coordinate charts may be saturated by the parallel flow and the saturations are still diffeomorphisms as the flow lines are sent to flow lines, which never loop around and thus provide `large' open subsets so that the restricted developing map is a diffeomorphism onto its image. Figure \ref{fig:3} provides an illustration of this argument.\\ \\
\begin{figure}\label{fig:3}
\end{figure}
In fact, as one can see in the proof of Theorem \ref{thm2}, the injectivity argument works on any subset $U \subset N$ for which $\overline{\text{dev}}|_{U}$ is a diffeomorphism onto its image. That said, the failure of the original developing map to be a diffeomorphism onto its image is entirely determined by the failure of $\overline{\text{dev}}$ to be a diffeomorphism.
\section{Radial Flow} \label{sec:4}
Similar to Section \ref{sec:3}, let $M$ be a closed affine $n$-dimensional manifold with fundamental group $\Gamma = \pi_{1}(M,p)$ acting on the universal cover by deck transformations. Pick a developing pair $(\text{dev},\text{hol})$ for the affine structure on $M$. As opposed to the previous section, instead of assuming the linear holonomy fixes a common vector, this section explores some consequences of when the affine holonomy fixes a point in $\mathbb{A}^{n}$. Pick said fixed point as the origin and make the standard identification of $\mathbb{A}^{n}$ with $\mathbb{R}^{n}$. Up to conjugation, one may assume the holonomy lies inside the group of linear transformations $\text{GL}(n,\mathbb{R})$. These class of manifolds are of special interest in the study of geometric structures, so much so that they are provided their own name.
\begin{definition}\label{def3} A radiant manifold $M$ is an affine manifold whose affine holonomy fixes a point in $\mathbb{A}^{n}$. This is equivalent to the condition that the affine holonomy is conjugate to a subgroup of $\text{GL}(n,\mathbb{R})$. \end{definition}
\begin{example}\label{ex6} The affine structure on the torus given in Example \ref{ex2} provides the torus with a radiant structure, whereas the structure in Example \ref{ex1} is not radiant. The structure given in Example \ref{ex3}, specifically from Figure \ref{fig:1}, also provides a radiant structure on the torus. This structure is inequivalent to the one in Example \ref{ex2}, as the holonomy in Example \ref{ex3} is non-cyclic. \end{example}
\begin{example}\label{ex7} The structure in Example \ref{ex2} can be generalized in the following fashion. Consider the puncture euclidean space $\mathbb{R}^{n}\setminus 0$. Let $H$ be a group generated by a positive homothety induced by some $\lambda > 0$. Then $\mathbb{R}^{n}/H$ is readily seen to be diffeomorphic to $S^{1}\times S^{n-1}$. These manifolds, known classically as hopf manifolds, provide a class of examples of radiant manifolds with cyclic holonomy. An illustration of the identification is provided in Figure \ref{fig:4}.
\begin{figure}
\caption{An illustration of a three-dimensional hopf-manifold. The solid space between two concentric spheres is drawn above with an equator in blue. The action of the homothety identifies the inner sphere with the outer sphere via a dilation. The curve is red is projected to a circle in the quotient.}
\label{fig:4}
\end{figure} \end{example}
A standard result of in the theory of geometric structures states that a closed radiant manifold cannot have its fixed point as an element of the developing image \cite{bib8}. Below is a modified version of the standard argument that is suited for the content of this paper.
\begin{theorem}\label{thm3} Let $M$ be a closed radiant manifold. The developing image cannot meet the fixed point of the radiant structure. \end{theorem}
\begin{proof} Fix a developing pair $\text{dev} : \widetilde{M} \longrightarrow \mathbb{A}^{n}$ and $\text{hol} : \Gamma \longrightarrow \text{GL}(n,\mathbb{R})$. Let $R = -y^{i}\partial/\partial y^{i}$ be the attractive radial vector field on $\mathbb{R}^{n}$. Routine calculation shows that $R$ is invariant under general linear group, so it may be lifted by the developing map to a $\Gamma$-invariant vector field $\widetilde{R}$ on $\widetilde{M}$. The vector field $\widetilde{R}$ descends to a vector field on $M$, which is complete by compactness, and thus the corresponding flow on $\widetilde{M}$ is also complete. Denote the flow on $\widetilde{M}$ by $\widetilde{R}_{t}$ and the corresponding radial flow on $\mathbb{R}^{n}$ by $R_{t}$. These flows are related by the commutative diagram below
\begin{equation}\label{eq19}
\begin{tikzcd}
& \widetilde{M} \arrow{r}{\widetilde{R}_{t}} \arrow{d}[swap]{\text{dev}}& \widetilde{M} \arrow{d}{\text{dev}}\\
& \mathbb{R}^{n} \arrow{r}[swap]{R_{t}}& \mathbb{R}^{n}
\end{tikzcd} \nonumber \end{equation}
Now, assume that the origin is an element of the developing image. Then $\text{dev}^{-1}\{0\} \subset \widetilde{M}$ is a discrete subset of stationary points of $\widetilde{R}$. Choose a collection of pairwise disjoint open sets $U_{i}$ about each element $u_{i} \in\text{dev}^{-1}\{0\}$. Since $\text{dev}$ is a local diffeomorphism, one may shrink each $U_{i}$ if necessary to assume the developing map restricted to each $U_{i}$ is a diffeomorphism onto an open ball centered at the origin. For each $t \geq 0$, one has that $R_{t} \text{dev}(U_{i}) \subseteq \text{dev}(U_{i})$ and thus $\widetilde{R}_{t} U_{i} \subseteq U_{i}$. \\ \\
For each $U_{i}$, let $\widetilde{R}_{\infty}U_{i}$ denote the forward and backward saturation of $U_{i}$ with respect to the radial flow on the universal cover. Explicitly,
\begin{equation}\label{eq20} \widetilde{R}_{\infty}U_{i} = \bigcup_{t\in\mathbb{R}} \widetilde{R}_{t}U_{i} \end{equation} As $\widetilde{R}_{\infty}U_{i}$ is union of open subsets in $\widetilde{M}$, it is itself an open submanifold of $\widetilde{M}$. Additionally, the developing map restricted to each $\widetilde{R}_{\infty}U_{i}$ is a diffeomorphism onto its image. To prove this, it suffices to show that the developing map is injective when restricted to each $\widetilde{R}_{\infty}U_{i}$. \\ \\
Let $u, v \in \widetilde{R}_{\infty}U_{i}$ so that $\text{dev}(u)= \text{dev}(v)$. There exists times $t,s \in\mathbb{R}$ and points $u_{i}, v_{i} \in U_{i}$ so that $\text{dev} (\widetilde{R}_{t}u_{i}) = \text{dev} (\widetilde{R}_{s}v_{i})$. Without loss of generality, let $t-s \geq 0$. By equivariance of the radial actions, $R_{t-s}\text{dev}(u_{i}) = \text{dev}(v_{i})$. Since $t-s \geq 0$, one has that $\widetilde{R}_{t-s} U_{i} \subseteq U_{i}$ so $R_{t-s} \text{dev}(U_{i}) \subseteq \text{dev}(U_{i})$. Because $\text{dev}(\widetilde{R}_{t-s} u_{i}) = \text{dev}(v_{i})$ and $\widetilde{R}_{t-s} U_{i} \subseteq U_{i}$ on which the developing map is a diffeomorphism, one has $\widetilde{R}_{t-s} u_{i} = v_{i}$ so $u = v$ as claimed. \\ \\
The above paragraphs show that the developing map when restricted to any $\widetilde{R}_{\infty} U_{i}$ is a diffeomorphism onto its image, which is the radial saturation of an open ball about the origin, and consequently a diffeomorphism onto all of $\mathbb{R}^{n}$. Lemma \ref{lm3} shows that $\widetilde{R}_{\infty} U_{i}$ is closed, and consequently by connectedness of $\widetilde{M}$, is equal to the universal cover. Thus the developing map is a diffeomorphism onto $\mathbb{R}^{n}$ and defines a complete radial structure.\\ \\ As per consequence there exists a subgroup $H \leq \text{GL}(n,\mathbb{R})$ acting both properly and freely on $\mathbb{R}^{n}$ so that $\mathbb{R}^{n}/H$ is diffeomorphic to $M$. Since the origin is a fixed point of each element of $\text{GL}(n,\mathbb{R})$ and $H$ acts freely, $H$ must be trivial. This contradicts the fact that $M$ is compact, and thus the origin is not an element of the developing image. \end{proof}
As mentioned previously, the above proof yields a corollary that assists the proof of a later theorem. It is stated here for reference later.
\begin{corollary}\label{cor1} Let $N$ be a connected smooth manifold and $F: N \longrightarrow \mathbb{R}^{n}$ be a local diffeomorphism where $0 \in F(N)$. If the radial action on $\mathbb{R}^{n}$ can be lifted to a complete action on $N$, then $F$ is a diffeormophism onto $\mathbb{R}^{n}$. \end{corollary}
\section{Holonomy Acting by Translations on The Invariant Line} \label{sec:5} In this section of this paper, a mild generalization of Theorem \ref{thm1} is provided. The goal of this section is to prove the following theorem.
\begin{theorem}\label{thm4} Let $M$ be a closed $(n+1)$-dimensional affine manifold whose holonomy leaves invariant an affine line where $n \geq 1$. If the holonomy acts by pure translations on the invariant line, then the developing image cannot meet the invariant line. \end{theorem}
Before beginning the proof it is worth explaining the technical ideas. As $M$ admits an invariant affline line and the holonomy acts by pure translations on it, one may assume that the holonomy lies in the subgroup of affine automorphisms of the form defined by
\begin{equation}\label{eq21} G = \left\{
\left(
\begin{array}{cc}
1 & w \\
0 & A
\end{array}
\right)
\left(
\begin{array}{c}
d \\
0
\end{array}
\right)
\Bigg|\,
d\in\mathbb{R}, w^{T} \in \mathbb{R}^{n}, A \in \text{GL}(n,\mathbb{R}) \right\} \end{equation}
As per consequence, there is a parallel flow on the universal cover as in Theorem \ref{thm2}. One can then form the commutative diagram as in Equation \ref{eq18} to obtain the local diffeomorphism $\overline{\text{dev}} : N \longrightarrow \mathbb{R}^{n}$. Because the induced holonomy in Equation \ref{eq18} acts linearly by the matrices $A$ in Equation \ref{eq21}, there is a complete flow on $N$ that lifts the radial flow on $\mathbb{R}^{n}$. If the developing image meets the invariant line then $0 \in \overline{\text{dev}}(N)$, and by Corollary \ref{cor1}, $\overline{\text{dev}}$ will define a global diffeomorphism from $N$ onto $\mathbb{R}^{n}$. Saturating $N$ by the parallel flow yields that the developing map is diffeomorphism onto all of $\mathbb{R}\times\mathbb{R}^{n}$ thus defining a complete structure whose affine holonomy leaves invariant an affine line thus contradicting Theorem 1. The details of this argument are provided below in the proof.
\begin{proof} As mentioned in the preceding paragraph, assume the holonomy lies inside the group defined by Equation \ref{eq21}. By Theorem \ref{thm2}, there exists a complete parallel flow on the universal cover of $M$. Form the commutative diagram defined by Equation \ref{eq18} with the induced developing map $\overline{\text{dev}} : N \longrightarrow \mathbb{R}^{n}$ and the corresponding induced actions of $\Gamma$ and the holonomy on $N$ and $\mathbb{R}^{n}$ respectively. \\ \\
Since the holonomy of $M$ lies inside of $G$, the induced holonomy action in Equation \ref{eq18} acts linearly on $\mathbb{R}^{n}$, and thus preserves the attractive radial vector field $R = -y^{i}\partial/\partial y^{i}$ as in Section \ref{sec:4}. This vector field lifts via $\overline{\text{dev}} : N \longrightarrow \mathbb{R}^{n}$ to a $\Gamma$-invariant vector field $\widetilde{R}$ on $N$. Identifying the tangent bundle $T(\mathbb{R}\times N)$ with $T\mathbb{R}\oplus TN$ yields a natural lift of $\widetilde{R}$ to a vector field on $\mathbb{R}\times N$, whereby construction, this vector field is also $\Gamma$-invariant. The $\Gamma$-invariant vector field on $\mathbb{R}\times N$ descends to a vector field on $M$ which is complete by compactness. The flow on $M$ lifts to a complete flow on $\mathbb{R}\times N$ which leaves each leaf $x\times N$ of $\mathbb{R}\times N$ invariant. This flow may be thought of as a cylindrical flow which is radial on each leaf. Above is a figure that illustrates this construction.\\ \\
\begin{figure}\label{fig:5}
\end{figure}
Since the flow of the lift of $\widetilde{R}$ to $\mathbb{R}\times N$ is complete, by construction the flow of $\widetilde{R}$ is itself complete and thus serves as a lift of the radial flow $R$ on $\mathbb{R}^{n}$ through $\overline{\text{dev}}$. If the developing image $\text{dev}(\mathbb{R}\times N)$ meets the invariant line, then $0 \in \overline{\text{dev}}(N)$. Corollary \ref{cor1} implies that $\overline{\text{dev}} : N \longrightarrow \mathbb{R}^{n}$ is a diffeomorphism, and the remark after Theorem \ref{thm2} yields that $\text{dev} : \mathbb{R}\times N \longrightarrow \mathbb{R}\times\mathbb{R}^{n}$ is a diffeomorphism. Hence $M$ admits a complete affine structure whose holonomy lies inside the group $G$ defined in Equation \ref{eq21}, contradicting Theorem \ref{thm1}. Thus, the developing image cannot meet the invariant line. \end{proof}
As an immediate consequence to the proof of Theorem \ref{thm4} one obtains the following corollary.
\begin{corollary}\label{cor2} Let $M$ be a closed $(n+1)$-dimensional affine manifold whose holonomy leaves invariant an affine line where $n \geq 1$. If the holonomy acts by pure translations and reflections on the invariant line, then the developing image cannot meet the invariant line. \end{corollary}
\begin{proof} This is an immediate consequence of the fact that the group defined by Equation \ref{eq21} is an index two subgroup of
\begin{equation}\label{eq22} \left\{
\left(
\begin{array}{cc}
\pm1 & w \\
0 & A
\end{array}
\right)
\left(
\begin{array}{c}
d \\
0
\end{array}
\right)
\Bigg|\,
d\in\mathbb{R}, w^{T} \in \mathbb{R}^{n}, A \in \text{GL}(n,\mathbb{R}) \nonumber \right\} \end{equation} \end{proof} Passing to the double cover $M$ and applying Theorem \ref{thm4} yields the desired result.
\section{Concluding Remarks} \label{sec:6} A natural follow up to Theorem \ref{thm4} would be to analyze the situation where the holonomy lies in the extension of the group $G$ as defined in Equation \ref{eq6}. The author suspects that such manifolds are radiant. Loosely the idea at hand is the following. If there is indeed an element of the holonomy of the form
\begin{equation}\label{eq23} \text{hol}[\gamma] = \left(
\begin{array}{cc}
r & w \\
0 & A
\end{array}
\right)
\left(
\begin{array}{c}
d \\
0
\end{array} \nonumber
\right) \end{equation}
where $r \neq 1$, then without loss of generality, one may conjugate to assume this element of the holonomy acts on the invariant line by scaling, and consequently admits a fixed point, which can be taken as the origin. It seems likely that the developing image cannot meet this point, much like in the example of a hopf circle. If that were so, then this would impose certain restrictions about having a non-trivial translational part in the holonomy. The difficulty in showing this point does not meet the developing image is that $[\gamma] \in \Gamma$ need not stabilize the components of the inverse image of the invariant line under the developing map. This difficulty is lost in the case where the fundamental group is abelian, but seems like an excessive and unnecessary hypothesis. \\ \\
The proof of Theorem \ref{thm4} relied largely on the existence of a parallel flow and a `cylindrical' flow on the universal cover of $M$. These flows can coexist on certain manifolds such as the product of a euclidean circle and a hopf torus as in Example \ref{ex4}. It is of great interest to the author as to whether or not parallel and radial flows can coexist on compact affine manifolds. The dynamics of such flows would certainly lead to very interesting examples.
\section{Lemmas} \label{sec:7} Here is a collection of some lemmas used throughout the paper. In this context, all topological spaces are assumed to be smooth manifolds, as is the concern of this paper. That said, some of these propositions hold in more general contexts such as metric spaces.
\begin{lemma}\label{lm1} Let $\phi : G \longrightarrow H$ be a homomorphism of lie groups and let $G$ and $H$ act on the smooth manifolds $X$ and $Y$ respectively. Assume there exists a diffeomorphism $F : X \longrightarrow Y$ equivariant with respect to $\phi$ in the sense that the below diagram commutes for all $g \in G$.
\begin{equation}\label{eq24}
\begin{tikzcd}
&X \arrow{r}{g} \arrow{d}[swap]{F} &X \arrow{d}{F}\\
&Y \arrow{r}[swap]{\phi(g)} & Y
\end{tikzcd} \end{equation} If $G$ acts properly and freely on $X$ then so too does $\phi(G)$ on $Y$. In this case, one may form the quotients $X/G$ and $Y/\phi(G)$, which are in turn diffeomorphic. \end{lemma}
\begin{proof} Begin by assuming that $G$ acts freely on $X$. Let $\phi(g)y = y$ for some $g \in G$ and $y \in Y$. Since $F$ is a diffeomorphism, there's an $x \in X$ so that $F(x) = y$ and so $\phi(g)F(x) = F(x)$ which by equivariance is equivalent to $F(gx) = F(x)$. Since $G$ is a diffeomorphism, this necessitates that $gx = x$, and thus by freeness of $G$ on $X$, $g = 1$, so $\phi(g) = 1$. A similar argument shows the homomorphism $\phi$ is injective, so $G$ and $\phi(G)$ are diffeomorphic as manifolds. \\ \\
Let $G$ act properly on $X$. Pick sequences $\phi(g_{i}) \in \phi(G)$ and $y_{i} \in Y$ so that $\phi(g_{i})y_{i}$ converges to a point $q \in Y$ and $y_{i}$ converges to $y \in Y$. To show properness it suffices to show a subsequence of $\phi(g_{i})$ converges, as is stated in John M. Lee's Smooth Manifolds \cite{bib7}. Since $F$ is a diffeomorphism, there's a unique sequence of $x_{i} \in X$ converging to an $x \in X$ so that $F(x_{i}) = y_{i}$ and $F(x) = y$. Equivariance yields that $\phi(g_{i})y_{i} = \phi(g_{i})F(x_{i}) = F(g_{i}x_{i})$ which converges to $q \in Y$, and thus $g_{i}x_{i}$ converges to some $p \in X$. Since $g_{i}x_{i}$ converges to $p \in X$ and $x_{i}$ converges to $x \in X$, properness of $G$ on $X$ yields a convergent subsequence of $g_{i}$ to $g \in G$. Continuity of the lie group homomorphism $\phi : G \longrightarrow H$ provides a convergent subsequence $\phi(g_{i})$ converging to $\phi(g) \in \phi(G)$, and thus the action of $\phi(G)$ on $Y$ is proper. \\ \\
As both actions of $G$ on $X$ and $\phi(G)$ on $Y$ are free and proper, one may form their smooth quotient manifolds $X/G$ and $Y/\phi(G)$. Denote their corresponding projections by $p : X \longrightarrow X/G$ and $q : Y \longrightarrow Y/\phi(G)$. One may then form the commutative square below
\begin{equation}\label{eq25}
\begin{tikzcd}
&X \arrow{r}{F} \arrow{d}[swap]{p}&Y\arrow{d}{q}\\
&X/G \arrow{r}[swap]{\overline{F}} & Y/\phi(G)
\end{tikzcd} \end{equation}
The map $\overline{F}$ is well defined as Equation \ref{eq24} ensures $F$ maps orbits to orbits. As $q\circ F$ is surjective, so too is $\overline{F}$. It is injective because if $\overline{F}(Gu) = \overline{F}(Gv)$ for some $Gu, Gv \in X/G$, then there exists $u,v \in X$ and a $g \in G$ so that $\phi(g)F(u) = F(v)$. Equivariance implies $F(gu) = F(v)$ and since $F$ is a diffeomorphism, $u$ and $v$ are in the same orbit, thus $Gu = Gv$, so $\overline{F}$ is a bijection. \\ \\
Since $X$ and $Y$ are the same dimension, as are $G$ and $\phi(G)$, the map $\overline{F}$ is a smooth bijective local diffeomorphism, and thus a diffeomorphism. \end{proof}
Lemma \ref{lm1} provides the following result frequently used in the study of geometric structures.
\begin{corollary}\label{cor3} Let $M$ be a complete affine $n$-dimensional manifold with fundamental group $\Gamma = \pi_{1}(M,p)$. Fix a developing pair $\text{dev} :\widetilde{M} \longrightarrow \mathbb{A}^{n}$ and $\text{hol} : \Gamma \longrightarrow \text{Aff}(n,\mathbb{R})$. Then $M$ is diffeomorphic to $\mathbb{A}^{n}/H$ where $H$ is the image of the holonomy homomorphism. \end{corollary}
\begin{proof} Since the developing map is a diffeomorphism onto $\mathbb{A}^{n}$ and $M$ is diffeomorphic $\widetilde{M}/\Gamma$, where $\Gamma$ is the group of deck transformations that acts both properly and freely on the universal cover, Lemma \ref{lm1}, applied the developing map and holonomy homomorphism yield that $\widetilde{M}/\Gamma$ and therefore $M$, are both diffeomorphic to $\mathbb{A}^{n}/H$. \end{proof}
The following statement has a proof similar to that of Lemma \ref{lm1}, and is used in the construction of the quotient manifolds in Section \ref{sec:3}. Its proof is omitted as it is nearly identical to that of the previous lemma.
\begin{lemma}\label{lm2} Let $G$ be a lie group acting on smooth manifolds $X$ and $Y$ and let $F : X \longrightarrow Y$ be a smooth map equivariant with respect to the $G$-actions. If the action of $G$ on $Y$ is free and proper, then so too is the action of $G$ on $X$. In this case one may form the quotients $X/G$ and $Y/G$ for which $F$ descends to a smooth map $\overline{F} : X/G \longrightarrow Y/G$. \end{lemma}
\begin{lemma}\label{lm3}
Let $N$ and $P$ be smooth manifolds and $F : N \longrightarrow P$ be an open map where $N$ is connected. If there exists an open submanifold $U \subseteq N$ for which $F|_{U} : U \longrightarrow P$ is a diffeomorphism, then $N = U$. \end{lemma}
This following lemma finds it use in the proof of Theorem \ref{thm3}, to show the open submanifold $\widetilde{R}_{\infty}U_{i}$ defined by Equation \ref{eq20} is equal to all of $\widetilde{M}$. In this case the developing map fulfills the role of the open map as stated in the lemma.
\begin{proof}
It suffices to show that $U$ is closed. Let $u_{k}$ be a sequence of points in $U$ converging to some point $n \in N$. By continuity of $F$, the sequence $F(u_{k})$ converges to $F(n) \in P$. Since $F|_{U} : U \longrightarrow P$ is a diffeomorphism, there's a unique $u \in U$ so that $F(n) = F(u)$. The claim is that $n = u$. \\ \\
Let $u \in V$ be an open neighborhood in $N$ about $u$. As $U$ is open in $N$, one may shrink $V$ sufficiently small so that $u \in V\subseteq U$. Because the sequence $F(u_{k})$ converges to $F(n) = F(u)$ and $F(V)$ is an open subset of $P$ about $F(u)$, there exists a sufficiently large $K \in \mathbb{N}$ so that $F(u_{k}) \in F(V)$ for all $k \geq K$. Since $F|_{U} : U \longrightarrow P$ is a diffeomorphism, each $u_{k} \in U$, and $V \subset U$, it follows that $u_{k} \in V$ for $k \geq K$. Thus $u_{k}$ converges to both $u$ and $n$, and by uniqueness of limits, $u = n$. Consequently, $U$ is a non-empty closed and open subset of $N$, and by connectedness $U = N$. \end{proof}
{}
\end{document} | arXiv |
Annals of Glaciology
Air-temperature control on diur...
Core reader
Air-temperature control on diurnal variations in microseismicity at Laohugou Glacier No. 12, Qilian Mountains
REGIONAL SETTING AND INSTRUMENTATION
DATA PROCESSING AND SEISMIC EVENT DETECTION
Meteorological conditions
Long-duration events
Short-duration events
Local magnitude of short-duration events
SEISMIC SOURCE FOR SHORT-DURATION EVENTS
Thermally induced opening of surface cracks
Thermal stress evolution
Comparison with fracture mechanics
POSSIBLE SOURCES FOR LONG-DURATION EVENTS
Annals of Glaciology, First View
Tao Zhang (a1) (a2), Yuqiao Chen (a1) (a2), Min Ding (a3), Zhongyan Shen (a1) (a2), Yuande Yang (a4) and Qingsheng Guan (a1) (a2) (a5)
1Key Laboratory of Submarine Geoscience, State Oceanic Administration, Hangzhou 310012, China. E-mail: [email protected]; [email protected]
2Second Institute of Oceanography, State Oceanic Administration, Hangzhou 310012, China
3School of Earth and Space Sciences, Peking University, Beijing 100022, China
4Chinese Antarctic Center of Surveying and Mapping, Wuhan University, Wuhan 430079, China
5School of Geographic and Oceanographic Sciences, Nanjing University, Nanjing 210023, China
Copyright: © The Author(s) 2019
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (http://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
DOI: https://doi.org/10.1017/aog.2018.34
Published online by Cambridge University Press: 31 January 2019
Fig. 1. (a) Contour map of Laohugou Glacier No. 12 with a contour interval of 100 m (from Liu and others, 2011). The position of the AWS is indicated by a star, and the locations of seismometers S1–S4 by triangles. (b) Enlarged satellite image (DigitalGlobe imagery taken on 9 April 2013, retrieved from http://goto.arcgisonline.com/maps/World_Imagery) of our survey area (black box in a), as well as seismometer locations. The location of our observed moulin is roughly indicated by a yellow square. Note that the station S1 is deployed near the margin of the LGNT.
Fig. 2. Typical vertical component of seismograms and STA/LTA ratios for (a, b) a short-duration event starting at 10:20:21 China Standard Time (CST), 9 October, recorded at S1, and (c, d) a long-duration events starting at 13:25:11 CST, 6 October, recorded at S2. The STA/LTA trigger parameters are listed in Table 1.
Table 1. Parameters for seismic event identification for the STA/LTA method
Fig. 3. Sensitivity of hourly short-duration event counts at S2 to (a) trigger threshold Rtrigger, (b) STA/LTA window widths and (c) application of waveform association analysis (Fig. 8). Date and hour are in CST.
Fig. 4. Counts of detected long-duration events per hour for stations S1–S4. The correlation coefficients between the hourly event counts and air temperature (Fig. 6a), wind speed (Fig. 6b) and temperature change rate (Fig. 6c) are listed in the top right corners. Date and hour are in CST.
Fig. 5. Waveforms (top), spectrograms (bottom) and power spectra (right) for three representative long-duration events with starting times of (a) 11:35:11 CST, (b) 12:10:05 CST and (c) 15:14:04 CST on 6 October, detected at the seismic stations S2, S2, and S4, respectively.
Fig. 6. (a) Air temperature, (b) wind speed, (c) calculated temperature change rate at a depth of 20 cm and (d–g) hourly short-duration events detected at S1–S4. The correlation coefficients between short-duration events and air temperature, wind speed and temperature change rate are listed in the top right corners. Date and hour are in CST. We did not deploy station S1 on the first day.
Fig. 7. Histograms of (a) duration, (b) frequency and (c) local magnitude for the short-duration seismic events. (d) Corresponding cumulative distribution of the seismic magnitude in (c). Red curves correspond to the best-fit to the Gutenberg–Richter distribution.
Fig. 8. Estimates of the short-duration event counts after applying waveform association analysis to exclude the potential repeated counts. The events simultaneously recorded at two stations (light blue), three stations (orange), and four stations (red) are ~ 23, 4 and 0.2% of the events recorded at one station (blue).
Fig. 9. Nine-day averaged (a) air temperature, (b) temperature change rate at a depth of 20 cm and (c) counts of short-duration events per hour. (d) Probability values (i.e. p-values) of the statistical test (i.e. two-sample t-test) for the hypotheses that the neighboring 2 h have different mean seismic event counts. The notable low p-values at 19 CST and 9 CST indicate the initiation and ending hours for the daily burst of the short-duration events. The gray regions and dashed lines indicate the 95% confidence intervals (i.e. 1.96 times the standard error) for the hourly mean values.
Fig. 10. (a) One-day plot of the seismic records from 10 to 11 October at S2, and filtered seismograms after applying (b) highpass and (c) lowpass filters at 20 Hz.
Fig. 11. Hourly short-duration event counts, temperature, temperature change rate and thermal stress averaged for the 9 survey days. (a) Counts of short-duration events per hour (same with Fig. 9c). (b) Calculated thermal stress evolution at a depth of 0 (red), 20 (green) and 40 cm (blue). The solid and dashed curves correspond to (e1–2) the first scenario without background strain rate and (f1–2) the second scenario with background strain rate, respectively. (c1–2) Temperature wave propagation and corresponding depth-dependent temperature profiles at four representative times. Black curve shows the lower enveloping curve of the temperature at various depths. (d1–2) Depth-dependent evolution of temperature change rate and corresponding temperature change rate profiles. Black curve shows the higher enveloping curve. (e1–2) Calculated thermal stress evolution for the first scenario without background strain rate, and depth-dependent profiles. Black curve shows the upper enveloping curve of the tensile stress at various depths. (f1–2) Calculated thermal stress evolution for the second scenario with a background strain rate. The horizontal tension is positive.
We conducted a 9-d seismic experiment in October 2015 at Laohugou Glacier No. 12. We identified microseismic signals using the short-term/long-term average trigger algorithm at four stations and classified them as long and short-duration events based on waveform, frequency, duration and magnitude characteristics. Both categories show systematical diurnal trends. The long-duration events are low-frequency tremor-like events that mainly occurred during the daytime with only several events per day. These events lasted tens of seconds to tens of minutes and are likely related to resonance of daytime meltwater. The dominant short-duration events mostly occurred during the night time with a peak occurrence frequency of ~360 h−1. Their short-duration (<0.2 s), high frequency (20–100 Hz) and dominance of Rayleigh waves are typical of events for near-surface crack opening. A strong negative correlation between the hourly event number and temperature change rate suggests that the occurrence of night-time events is controlled by the rate of night-time cooling. We estimated the near-surface tensile stress due to thermal contraction at night to be tens of kilopascals, which is enough to induce opening of surface cracks with pre-existing local stress concentrations, although we cannot exclude the effect of refreezing of meltwater produced during the day.
Glacier seismic events were first detected decades ago (Röthlisberg, 1955; Neave and Savage, 1970; Osten-Woldenburg, 1990). Numerous investigations (Walter and others, 2008; Winberry and others, 2009; Roux and others, 2010; Carmichael and others, 2012; Walter and others, 2012; Zoet and others, 2012) on glacier seismology have been conducted after the pioneering glacier earthquake study by Ekström and others (2003). Seismic observations of glaciers provide unprecedented high sample rate coverage of glacier dynamic processes (Podolskiy and Walter, 2016). The creation and propagation of surface crevasses is the most prominent sources of glacier seismicity. This seismicity of surface cracks can constrain the strain and stress variations in glaciers and provide insights into the glacier response to environmental changes.
Surface crack seismicity could be induced by temperature variations and/or meltwater. For marine-terminating glaciers, this type of seismicity can also be caused by ocean tides (Barruol and others, 2013; Podolskiy and others, 2016). Most recent studies focused on the effects of meltwater on glacier seismicity, especially on deep and basal icequakes (Walter and others, 2008, 2013; Dalban Canassy and others, 2013). In contrast, studies on the relationship between glacier seismicity and temperature fluctuation are relatively rare, although thermal cracking is considered to be an important mechanism of surface seismicity at cold and dry glaciers when meltwater is limited (Carmichael and others, 2012; Podolskiy and others, 2018). In this study, we focus on the ability of air temperature fluctuations to induce surface crack opening and night-time microseismicity (i.e. short-duration events).
We conducted a 9-d passive seismic experiment on a high-altitude valley (continental) glacier, Laohugou Glacier No. 12 (LGNT), in the Qilian Mountains to the northeast of Tibet in October 2015, when the diurnal air temperature fluctuations are large (>10°C) but the precipitation (<20 mm per month) and meltwater is limited. We identified two types of seismic events from the seismic data and analyzed their temporal distribution, magnitudes, duration and frequency characteristics. We further analyzed the relationship between seismic event occurrence and air temperature and meltwater production. Finally, we discussed the possible sources for the seismic events and quantify the tensile stress due to night-time thermal contraction. The thermal stress is found to be sufficient large to induce surface crack opening that is responsible for short-duration events.
As the largest glacier in the Qilian Mountains, the LGNT is 9.85 km long and covers an area of 20.4 km2 (Du and others, 2008; Fig. 1). The LGNT consists of two tributaries and extends from 4260 to 5481 m elevation. The LGNT is a typical valley glacier in its climate and physical characteristics. Precipitation in this area is mainly affected by westerly winds all year round. The annual precipitation was 310 mm during 1960–2006 (Qin and others, 2015), with rainfall mainly concentrated between May and September (Liu and others, 2011). In October of 2009, the monthly precipitation was <20 mm (Sun and others, 2013). The annual air temperature was −13.2°C during 1957–2013 (Qin and others, 2014) at 5040 m elevation. The monthly mean air temperature in October of 2009 was ~−11.2°C (Sun and others, 2011). In October, the incoming shortwave radiation had an average value of 210.3 W m−2 and was up to 676 W m−2 at noon time (Sun and others, 2013). The ice thickness along a transversal line at an elevation of 4458 m (close to our seismic station S4) was 56–128 m as revealed by a ground-penetration radar investigation in the ablation period (July to August) of 2009 and 2014 (Wang and others, 2016).
Our seismic experiment was conducted on the LGNT from 12:30 China Standard Time (CST) of 5 October through 10:16 CST of 14 October 2015. All the records are time-stamped by CST, which is 8 h ahead of UTC time. We deployed four EPS-1 digital seismometers roughly along the glacier centerline near the terminus of the LGNT at an altitude range of ~4300 to 4450 m (Fig. 1), where the averaged surface velocity was ~2.5 cm d−1 (~9 m a−1) in 2008–2009 (Liu and others, 2011). The seismometers were partially buried in the ice in a relatively flat area (without protection cover) and deployed with a spacing of 300–450 m. We checked and maintained the seismometers on a daily basis to ensure the instruments remained level and worked properly. EPS-1 is a cylinder-shaped portable digital seismometer with a radius of 70 mm and a height of 170 mm. It is a three-component seismometer accompanied with a built-in 24 bit data logger, a data storage unit, and a battery to work continuously for 20 d. The instrument response is flat from 5 to 200 Hz. The output data have a sample rate of 250 Hz (see http://www.cgif.com.cn/displayproduct.html?proID=2531321&ptid=196458 for instrument details). EPS-1 also includes an internal clock, an electronic compass, and a GPS. The electronic compass was used to record the azimuth and angles, while the GPS provided the initial time and position information. Accompanying position measurements were conducted by a geodetic group from Wuhan University at stations S2 and S3 with Leica GS10 GNSS (Global Navigation Satellite System). The Leica GS10 GNSS has a static horizontal accuracy of 3 mm + 0.5 ppm and a static vertical accuracy of 6 mm + 0.5 ppm.
An automatic weather station (AWS) was deployed on the LGNT at 4550 m elevation to measure the air temperature at 1.5 m above the glacier surface (Sun and others, 2013, 2014; Fig. 1). The AWS was installed on a relatively flat area of the glacier surface using a steel tripod. The temperature sensor has a measurement range of −40 to 60°C and an accuracy of ±0.2°C. The sample rate of the AWS was 0.1 Hz, while the output data were moving averages for every 30 min (Fig. 6a). The AWS also measures humidity, air pressure, and radiation, as well as wind speed and direction (Fig. 6b).
To detect seismic events, we applied a classic short-term/long-term average trigger algorithm (STA/LTA) to the observed vertical component of the seismograms using SEIZMO codes (Euler G: Project SEIZMO, available at: http://epsc.wustl.edu/~ggeuler/codes/m/seizmo/). The method was proposed by Stevenson (1976) and is widely used in the identification of cryoseismic events (Bassis and others, 2007; Walter and others, 2008; Roux and others, 2010; Carmichael and others, 2012; Barruol and others, 2013; Röösli and others, 2014; Köhler and others, 2015). The STA/LTA method uses short and long-term moving windows to localize the seismogram and calculate the corresponding squared amplitudes in the two windows (Trnkoczy, 2012). Once the ratio of the squared amplitudes in the two windows (i.e. R value) exceeds a user-defined trigger threshold R trigger, the trigger is turned on until the R value again falls below a user-defined detrigger threshold R detrigger (Fig. 2).
According to the amplitudes and durations, the identified seismic events were classified into long and short-duration events (Fig. 2). We chose parameters (Table 1) for the STA/LTA method based on visual inspection and manual picking of a 1-h data subset from station S1. We used a trigger threshold level R trigger of 3.5 and detrigger threshold level R detrigger of 1.5 to detect the short-duration events. The STA and LTA time windows were 0.02 and 0.2 s, respectively. The trigger and detrigger thresholds for short-duration events are close to previous studies (Walter and others, 2008; Carmichael and others, 2012; Röösli and others, 2014), but our STA and LTA time windows are one tenth of previously used values of 0.2–1 and 2–10 s. We chose the shorter time windows to detect all the visible velocity changes in the seismograms, which would be missed if a longer time window was applied.
For the short-duration events, we tested the sensitivity of our results to the trigger threshold and time window (Fig. 3). The event counts detected with different trigger threshold R trigger values show similar diurnal cycles, although the numbers of seismic events identified with R trigger values of 3 and 4 are 134 and 76%, respectively, of the total number with an R trigger value of 3.5 (Fig. 3a). Using a STA time window of 0.2 s and a LTA time window of 5 s (from Barruol and others, 2013), we found that the number of detected short-duration events decreases to ~29% of previous values with shorter time windows (Fig. 3b). However, the diurnal variations still hold. In the following analysis, we focused on the temporal pattern of short-duration events and note that the number of detected seismic events depends on STA/LTA time windows and trigger (and detrigger) thresholds.
Fig. 3. Sensitivity of hourly short-duration event counts at S2 to (a) trigger threshold R trigger, (b) STA/LTA window widths and (c) application of waveform association analysis (Fig. 8). Date and hour are in CST.
The air temperature obtained from the AWS station at 4550 km elevation shows a clear diurnal cycle during the survey period (Fig. 6a). The average air temperature over 9 d was about −5.2°C, with a maximum temperature of 1.4°C and a minimum temperature of −8.1°C. The highest and lowest temperatures appeared at 13 CST and 7 CST, respectively. The temperatures rose rapidly from 9 CST to 12 noon and dropped slowly from 17 CST to 7 CST the following morning. The local sunrise and sunset during our survey period was ~7:30 CST and ~19 CST, respectively. The wind speed (Fig. 6b) and glacier surface humidity also show diurnal cycles, which results in a deviation of the ice surface temperature from the air temperature based on glacier surface energy budget considerations (Sun and others, 2014). However, comparison between air and ice surface temperature measurements (Sun and others, 2014) in June–September 2011 showed a similar diurnal pattern and same daily hours with maximum and minimum temperatures, although the glacier surface temperature was systematically lower than the air temperature by 2–4°C. In this study, we use the air temperature as a proxy for the air surface temperature, due to the lack of direct ice surface temperature measurements during our survey period.
On 11 October 2015, snowfall lasted from 15 CST to 7 CST the following morning (Fig. 6). Approximately 5 cm of snow covered the LGNT until the end of our observation. We did not collect hydrological data, but note that no sound of water flow was heard in the field and the downstream riverbed was dry during the observation period. We only observed limited surface melting in the daytime close to the glacier fringes. Although the air temperature was mostly below 0°C during our observation period, the local solar radiation, wind, and cloud may lead to slight surface melting at noon. In addition, we found an ice moulin of ~5 m wide at a distance of ~100 m to the northwest of our seismic station S1 (Fig. 1b), close to the glacier terminus. The ice moulin was dry during our survey period.
We detected a total of 158, 129, 65 and 49 long-duration events for the four stations in the 9 survey days (Fig. 4) based on the parameters in Table 1. Most of the long-duration events occurred between 12 noon and 18 CST in the daytime, when the air temperature ranged from −3 to −1°C. On the last day of our survey (i.e. 11 October), the long-duration events were also identified at night. The amplitudes of long-duration events are relatively low (Figs 2 and 5). Their waveforms lack impulsive onsets, and distinct body or surface wave arrivals. The duration of the long-duration events is tens of seconds to tens of minutes. We selected three representative long-duration events for detailed spectral analysis (Fig. 5), which shows a dominant frequency below 20 Hz. The long-duration events may also be associated with a higher secondary resonance frequency, e.g. 80 Hz in Figure 5a and 40 Hz in Figure 5b. Spectrograms for these two selected events also clearly show a temporal evolution of the resonance frequency (bottom panels of Figs 5a and b).
We detected 4514 short-duration events per day on average for an individual station (Fig. 6), which is hundreds of times more than the number of detected long-duration events. At each seismic station, the number of short-duration events was similar in the first 6 d of observation. After a snowfall, the daily number of short-duration events in the last 3 d of observation decreased to ~one-fifth of that during the first 6 d. Most (95%) of the short duration events lasted for 0.04–0.25 s (Fig. 7a). The waveforms of the short-duration events are dominated by Rayleigh waves without visible P- or S-phase arrivals (Fig. 2a). The dominant frequencies of the events range widely from 20 to 100 Hz, with local peaks at 30, 50, 75 and 90 Hz (Fig. 7b). The event counts increase with frequency, and the maximum frequency is close to the Nyquist frequency of 125 Hz.
The dominance of Rayleigh waves is consistent with the waveforms observed in previous studies (Neave and Savage, 1970; Walter and others, 2009; Dalban Canassy and others, 2013; Röösli and others, 2014; Carmichael and others, 2015), in which the source mechanism has been attributed to surface tensile cracks within the surface crevasse zone (top ~20 m of the glacier). However, our detected events are shorter in duration, higher in frequency and more frequent than reported from other glaciers. This is mainly due to our chosen shorter time windows for the STA/LTA detector, and less significantly due to repeatedly counting seismic events at multiple stations. To reduce duplicate counts, we performed a waveform association analysis, similar to Carmichael and others (2015). This analysis identifies the events detected at different stations within a time interval less than the expected travel time across the network as the same event. However, due to the difficulty of locating the small short-duration events in this study, we can only use the maximum travel time between two stations, which was calculated as the distance between two stations divided by an assumed Rayleigh wave velocity of 1660 m s−1 (Mikesell and others, 2012). The waveform association analysis thus provides maximum estimates of the event counts that are recorded at two, three and four stations simultaneously, which are ~23, 4 and 0.2% of the total event counts recorded at one station (Fig. 8). The events recorded at multiple stations are expected to be larger in magnitude, and thus the rapid reduction in the event counts is in line with our expectation of an exponential decrease of event counts with magnitude. We plotted the hourly event counts detected simultaneously at two stations in Figure 3c. The diurnal cycle is clearly visible and thus is a robust observation.
The diurnal cycle is the most significant temporal pattern of short-duration events (Figs 6 and 9). Nearly all the short-duration events occurred between 19 CST and 9 CST the following morning, with peak event counts at 20–23 CST. The starting and ending times of the daily burst of seismic events, 19 CST and 9 CST, were confirmed by low p-values in paired t-tests (Fig. 9d). These starting and ending times have an uncertainty of 1 h, as it is based on hourly seismic event counts. After the starting time of 19 CST, the event counts rapidly increased to the maximum occurrence frequency of 363 h−1 at 20 CST (Fig. 9c). The event counts then gradually decreased at night till 112 h−1 at 9 CST the next morning. In the daytime, the event frequency is <40 h−1.
The counts of short-duration events (Figs 6d–g and 9c) are strongly and negatively correlated with the temperature fluctuation (Figs 6a and 9a) with correlation coefficients of −0.43 to −0.32 for the four seismic stations. Most short-duration events occur when the air temperature decreases from −4°C at 19 CST to −7°C at 9 CST the next morning. The occurrences of short-duration events are also negatively correlated with wind speed (Fig. 6b). It has been suggested that wind or water-generated tremor noise in the daytime may reduce the detectability of seismic events (Röösli and others, 2014). We applied a highpass filter at 20 Hz to suppress the low-frequency wind or water-generated noise. Results (Fig. 10b) show that the short-duration events are still concentrated at night due to their high frequency and amplitude. Therefore, the low detection of short-duration events in the daytime may not due to increased tremor noise. Instead, a source mechanism preferentially operating at night is necessary to explain the occurrence of short-duration events.
To characterize the energy/amplitude of the short-duration events, we calculated the local magnitudes (M l) using the method of Richter (1935):
(1) $$M_{\rm l} = {\rm log}\left( {\displaystyle{A \over {A_0}}} \right)$$
where the parameter A is the maximum amplitude of displacement, and A 0 is a correction factor defined by a reference zero magnitude earthquake that generates a displacement of 1 µm at a 100 km distance. We adopt the empirical A 0 value provided by Richter (1958), which was originally estimated for southern California but is also used worldwide (Roux and others, 2008).
Our estimated M l of the short-duration events range from −2 to 0.5 (Fig 7c). Following Podolskiy and others (2016) (citing Barruol and others, 2013), we found that the magnitude distribution fits a Gutenberg–Richter distribution (Gutenberg and Richter, 1944): log N = a − b × M l. We estimated the parameters a and b to be 3.75 ± 0.02 and 0.99 ± 0.01, respectively (Fig. 7d). Our estimated b value is close to 1, consistent with previous estimated b-value estimates for surface crack events (Podolskiy and Walter, 2016 and references therein). Our fitting to the Gutenberg–Richter distribution implies a magnitude of completeness of −1.2 (Fig. 7d), below which a large portion of weaker events were not detected. We also grouped the detected short-duration events on an hourly basis to test the evolution of b value. However, no clear temporal evolution trend of the b value was found.
Most of our detected short-duration events are expected to occur near the surface (maximum ~ 20 m depth) due to the dominance of Rayleigh waves (Walter and others, 2009; Röösli and others, 2014; Podolskiy and Walter, 2016). In contrast, seismograms of deep or basal events have relatively large P-wave amplitudes without the presence of Rayleigh waves (Deichmann and others, 2000; Dalban Canassy and others, 2013). Although it is impossible to determine the fracture mode (shear or tensile) of the seismic events due to the lack of P wave polarity observations, we suggest that the surface events are more likely to be tensile cracks as the tensile strength is much less than the shear strength (Schulson, 2002). Additionally, the tensile strength can be easily overcome at night due to thermal contraction (next paragraph). When local stress near the defects or crack tips exceeds the fracture toughness according to the fracture mechanic considerations (van der Veen, 1998, 1999), the formation or propagation of surface cracks can be recorded as seismic events.
The concentration of surface crack opening at night could be induced by thermal stress or refreezing meltwater. Daytime meltwater may accumulate in surface crevasses; refreezing of meltwater at night can trigger crack initiation and propagation (van der Veen, 2007). An alternative mechanism is that reduced meltwater at night increases effective basal stress and thus induces spatial variations in surface strain (Carmichael and others, 2015). However, the increased basal stress mechanism requires considerable amounts of meltwater at the base of the glacier, which is less likely to occur at LGNT in October.
Although we cannot exclude the effect of meltwater refreezing at night, we found that thermal contraction alone can induce enough tensile stress in the top tens of centimeters to promote tensile crack opening and seismic events (next section). This thermal contraction effect is supported by the significant negative correlation between the counts of short-duration events and the air temperature, and temperature change rate (Fig. 6). A similar negative correlation has been reported at a Himalayan glacier (Podolskiy and others, 2018) and interpreted as having a thermal origin. In addition, the snowfall on the 7th day of our observations is expected to damp heat conduction. The corresponding reduction in the thermal stress can explain the observed decrease in the icequake event number after the snowfall, which further corroborates the causal relationship between the thermal stress and short-duration events.
As the temperature decreases, the thermal contraction of the ice exerts tensile stress on the surface ice layer. Once the stress exceeds a threshold value, the opening of surface cracks occurs, which could be recorded as icequakes. To quantify the thermal stress in response to the diurnal temperature fluctuation, we first calculated the propagation of the temperature waves T(z, t) at depth z and time t based on the Fourier equation of heat conduction following Sanderson (1978) and Nishio (1983):
(2) $$\eqalign{T(z,t) & = T_0 + \mathop \sum \limits_{s = 1}^N T_s{\rm exp}\left[ {-z{\left( {\displaystyle{{\omega_s} \over {2k}}} \right)}^{1/2}} \right] \cr & \cos \left[ {\omega_st-\phi_s-z{\left( {\displaystyle{{\omega_s} \over {2k}}} \right)}^{1/2}} \right]\;} $$
where the thermal diffusivity k is assumed to be 1.091 × 10−6 m2 s−1 for a constant ice density of 0.9 g cm−3 (Sanderson, 1978). The temperature amplitudes T 0 and T s, frequency ω s and phase ϕ s depend on the Fourier transformation of surface temperature (assuming to be equal to the air temperature): $T(0,t) = T_0 + \mathop {\sum}_{s = 1}^N T_s\,{\rm cos}(\omega _st-\phi _s)$. The Fourier transformation was computed using the Matlab Fast Fourier transformation codes fft and ifft, in combination with Fourier coefficient conversion codes complex2real and real2complex (written by Boynton GM for the course of Introduction to Programing for the Behavioral Sciences, Autumn, 2015, University of Washington. Retrieved from http://courses.washington.edu/matlab1/matlab).
The short-duration events mostly occur when the temperature drops at a depth of 20 cm (i.e. when temperature change rate is negative). The correlation of the short-duration event counts with the temperature change rate at 20 cm depth (Figs 6c and 9b) is −0.56 to −0.54 for the four seismic stations, which is more significant than the correlation with the surface temperature fluctuation. The daily-averaged evolution of the temperature and temperature change rate within the top one meter of the glacier ice clearly shows diurnal variations (Figs 11c1 and d1). Figures 11c2 and d2 show depth-dependent temperature and temperature rate profiles at four representative hours. The diurnal temperature cycle only influences the top 1 m of the glacier, with rapidly decreasing amplitude with depth due to attenuation of temperature waves (black curves in Figs 11c2 and d2).
Sanderson (1978) considered that the surface ice layer cannot move as it is constrained both horizontally and vertically. Internal stress, instead of strain, is thus induced by temperature changes. This thermal stress was calculated by considering the strain rate that would occur if the ice body is free to expand or contract, and then estimating the stress required to reduce the strain rate to zero using Glen's flow law (Glen, 1955):
(3) $${\dot{\varepsilon}} ^{\prime}_i = C\tau ^{n-1}\sigma _i^{^\ast}, $$
where τ is the second invariant of the deviatoric stress tensor $\sigma _i^* $ and C is the creep parameter. The best estimate for n is 3 based on laboratory experiments and glacial morphologic studies (Sanderson, 1978; Cuffey and Paterson, 2010).
The final formula in Sanderson (1978) to estimate the horizontal principal stresses due to restrained thermal expansion and contraction are as follows:
(4) $$\sigma _1^{^\ast} = \left( {\displaystyle{{{{\dot{\varepsilon}}^{\prime}}_1} \over {C(1 + a + a^2)}}} \right)^{1/3},$$
(5) $$\sigma _2^{^\ast} = a\sigma _1^{^\ast}, $$
where ${\dot{\varepsilon}} ^{\prime}_1 = \dot{\varepsilon} _1-\alpha ((\partial T)/(\partial t))$ and ${\dot{\varepsilon}} ^{\prime}_2 = \dot{\varepsilon} _2-\alpha ((\partial T)/(\partial t))$ are the sum of the background strain rates, $\dot{\varepsilon} _1$ and $\dot{\varepsilon} _2$, minus the thermal strain rate (extension or contraction) if the ice body is unconstrained. $a = {\dot{\varepsilon}} ^{\prime}_2/{\dot{\varepsilon}} ^{\prime}_1$ is the ratio between the two strain rate components. The expansion coefficient α is assumed to be linearly related to the temperature as α = 5.4 × 10−5 + 1.8 × 10−7T (Nishio, 1983) instead of a constant value in Sanderson (1978).
The creep factor C depends on the temperature following an Arrhenius relationship: $C = C_*{\rm e}^{-(Q/R)\lpar {(1/T)-(1/T_*)} \rpar }$, where T is the variable temperature that depends on both depth and time, and T * = 263 K is a transition temperature above which ice softening increases. The gas constant R is equal to 8.314 J mol−1 K−1. The parameter values C * = 3.5 × 10−25 Pa−3 s−1 and Q = 1.15 × 105 J mol−1 are based on laboratory experiments and glacial morphologic studies (Cuffey and Paterson, 2010 and reference therein). This C * value is several orders of magnitude larger than the value used by Sanderson (1978), while Q is only slightly larger than Sanderson (1978). This updated C * value incorporates the state-of-the-art lab and field measurements of the glacier creep characteristics, resulting in a lower magnitude of stress in our calculation than Sanderson (1978).
We considered two scenarios for the background strain rate: $\dot{\varepsilon} _1 = \dot{\varepsilon} _2 = 0$ and $\dot{\varepsilon} _1$ = 4 × 10−10 s−1 (i.e. 0.013 a−1) and $\dot{\varepsilon} _2$ = −4 × 10−10 s−1. The calculated thermal stress $\sigma _1^* $ has similar diurnal pattern in the top 50 cm of the glacier (Figs 11e–f), but the latter scenario is associated with an additional extensional stress of ~70 kPa at depth due to the background extensional strain rate (Fig. 11f). Our stress calculation for the first scenario shows that the thermal expansion and contraction of the ice leads to diurnal variations in $\sigma _1^* $ from compression to tension with a maximum tensile stress of 48–184 kPa in the top 50 cm (Fig. 11e2). The calculated surface stress becomes tensile after 15 CST as the air temperature starts to decrease (red curves in Fig. 11b). The tensile stress region extends to 20 cm when the hourly short-duration events notably increase at 19 CST. At this time, the surface tensile stress is 156 kPa. After that, the tensile stress region continuously penetrates to greater depth, with reduced amplitude due to depth attenuation of temperature waves. At 8 CST, the air temperature starts to increase and the surface stress becomes compressional. Accordingly, the tensile stress region shrinks. One hour after that, the top 10 cm becomes compressional in stress. This is also the ending time of the daily burst of short-duration events.
For the latter scenario with background strain rate, we assumed a background compressional strain rate $\dot{\varepsilon} _2$ of −4 × 10−10 s−1. This strain rate is based on the surface velocity difference at elevations of ~4400 m and 4500 m measured in autumn and winter by Liu and others (2011). This compressional strain rate is expected to be longitudinal. We further assumed a maximum extensional strain rate $\dot{\varepsilon} _1$ = 4 × 10−10 s−1, presumably in the transverse direction, as the magnitude of the maximum extensional strain is expected to be close to $\dot{\varepsilon} _2$ from previous glacier strain rate measurements (Meier and others, 1974). The maximum tensional stress for this scenario is the same with the first scenario on the surface, but an increase of 36 kPa at 50 cm depth in comparison with the first scenario is due to the background tensional strain rate. The background strain rate dominantly controls the stress state at one-meter depth or deeper, as the stress asymptotically approaches 73 kPa in this scenario (Fig. 11f2). The enhanced tensile stress at depth can promote downward propagation of surface cracks and continued seismic events.
In either scenario, our thermal stress calculation clearly shows that the daily variations in the glacier thermal state are prominent only in the top tens of centimeters of the glacier. If the thermal effect dominates, it implies that the night-time short-duration events occur or at least initiate within the top tens of centimeters. In addition, the thermal stress is much lower than the stress threshold of 1 MPa that is required to fracture intact ice (Schulson, 1999 and references therein). Instead, the thermal stress is on the same order of magnitude as the critical tensile stress for crack nucleation and propagation if small pre-existing defects or weak planes exist (van der Veen, 1998).
Based on the fracture mechanic considerations, the initiation of seismic events implies the moment when the maximum thermal stress reaches the critical tensile stress for fracture formation. The thermal tensile stress during 19 CST to 9 CST the next day with daily bursts of short-duration events is on the order of tens of kilopascals with a maximum tensile stress of 156 kPa on the surface. The calculated thermal tensile stress is consistent with the fracture models of van der Veen (1998), who suggests a critical tensile stress range of 90–320 kPa for a densely spaced crack system. The nonexistence of a clear b-value evolution trend implies that the detected short-duration events at different hours of the day likely occur in a similar strain and stress state (Nishio, 1983; Podolskiy and Walter, 2016). As the state of stress and strain differs notably with depth, the short-duration events may not propagate with depth, but mainly correspond to initiation of new surface cracks.
The uncertainties in our stress calculation are primarily due to the applicability of Glen's flow law and a lack of constraints on creep parameter C * in Glen's flow law. The applicability of Glen's flow law in different temperature and strain rate region is currently under debate (Schulson and Duval, 2009) and requires further strain and stress measurements. For example, Weiss and others (2007) found that instead of viscous deformation, the winter and/or perennial sea ice behaves as an elastic-brittle body. In this scenario, the temperature difference between the surface and bottom of the glacier may induce a thermal bending moment at night (Evans and Untersteiner, 1971; Bažant, 1992; Carmichael and others, 2012), which is later released by microseismic events. For the creep parameter, material properties such as impurities, reduced grain size, increased water content and anisotropic ice fabrics can increase C * by several fold, and the field measurements of C * in different glaciers also commonly varies by a factor of 1–3 (Cuffey and Paterson, 2010 and references therein). Material factors are also suggested to influence the fracture toughness (Vaughan, 1993; Rist and others, 1996, 1999). Therefore, local field stress and strain measurements are necessary for further calibrating the constitutive relationship and quantitatively understanding icequakes and glacier dynamics.
The tremor-like long-duration events can be produced by subglacial water flow or repeated basal ruptures (Helmstetter and others, 2015; Podolskiy and Walter, 2016). The correlation between ice stream velocity and tremor signal is necessary to associate the tremor with basal ruptures (Winberry and others, 2013). However, there is no variations of glacier displacement correlated with the long-duration events in the coincident position measurements at stations S2 and S3 (Yan Peng, manuscript in preparation). Considering the slow surface velocity (~9 cm a−1) in the study area (Liu and others, 2011; Wang and others, 2016) and the thinness of the ice (56–128 m), we excluded the mechanism of repeated basal rupture for the long-duration events. Instead, we suggest that our long-duration events are most likely due to subglacial water flow. Firstly, our tremor-like long-duration events emit seismic energy in a dominant frequency of <20 Hz, which have been attributed to water resonance in fluid-filled moulin (Röösli and others, 2014). Our observed secondary resonance frequency of 40 or 80 Hz is also close to the water resonance frequency of 30–80 Hz for water-filled fractures (Helmstetter and others, 2015). In addition, more counts of long-duration events are observed at down-glacier stations, where melting is likely to be more prevalent and water flow should be larger. We also found an ice moulin near the seismic station S1 where the largest number of long-duration events were detected. It is thus possible that a large amount of these events are associated with water resonance in the ice moulin. Although we did not directly observe water flow in the moulin or downstream riverbed, surface melting in the daytime may be guided by the near-surface hydrological system into the ice moulin. Finally, the temporal evolution of the resonance frequency (Fig. 5) is also consistent with the water resonance mechanism, as it can be explained by the changes in the fracture width/length or water level (Röösli and others, 2014; Helmstetter and others, 2015). Therefore, we suggest that the most likely source mechanism for the long-duration events is water resonance within the ice moulin.
We cannot exclude the influence of wind as the hourly event counts and wind speed are positively correlated with correlation coefficients of 0.26–0.44. In addition, the wind speed at 14–18 CST (Fig. 6b) exceeds a critical value of 3 m s−1 that can induce large seismic noise (Withers and others, 1996). However, the wind noise cannot explain the existence of resonance frequencies at selected frequency ranges (e.g. 80 Hz in Fig 5a). Although additional seismic measurements within deep boreholes are necessary to elucidate this point, we suggest that wind noise alone cannot explain the all of detected long-duration events.
Our observations and analyses of seismic events and temperature fluctuations in LGNT lead to the following conclusions:
(1) We detected long-duration tremor-like events in the daytime and short-duration seismic events in the night-time at LGNT.
(2) The night-time short-duration events are characterized by relatively short duration, high dominant frequency, large event counts, and dominance of Rayleigh waves, and are thus inferred to be caused by near-surface crack opening. On a daily basis, the occurrence of the short-duration events was negatively correlated with the temperature and temperature change rate. Although we cannot exclude the effect of meltwater refreezing, night-time thermal tensile stress in the top tens of centimeters of the glacier exceeds the critical tensile stress and thus is enough to induce ice fractures and associated seismicity.
(3) The daytime long-duration events are characterized by relatively long duration, low dominant frequency, low event number, and tremor-like waveform. They are likely related to water resonance in an ice moulin filled by daytime meltwater.
We thank Qin Xiang at Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Yan Peng at Wuhan university and other colleagues working at Qilian Mountain Station of Glaciology for their support of the seismic experiment. We also thank F. Walter, E. Podolskiy and J. Amundson for their valuable reviews and constructive comments. The project was supported by the National Natural Science Foundation of China (41576065, 41776189, 41806067, 91628301), the Scientific Research Fund of the Second Institute of Oceanography, SOA (QNYC201503), the Chinese Polar Environment Comprehensive Investigation and Assessment Programmes (CHINARE 03-03, 04-03, and 01-03), the Key Laboratory of Ocean and Marginal Sea Geology, Chinese Academy of Sciences (No. OMG18-XX) and the Chinese Academy of Sciences (Y4SL021001).
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Journal of Big Data
The framework design
Conclusions and future work
Anomaly behaviour detection based on the meta-Morisita index for large scale spatio-temporal data set
Zhao Yang1 and
Nathalie Japkowicz1Email author
Journal of Big Data20185:23
In this paper, we propose a framework for processing and analysing large-scale spatio-temporal data that uses a battery of machine learning methods based on a meta-data representation of point patterns. Existing spatio-temporal analysis methods do not include a specific mechanism for analysing meta-data (point pattern information). In this work, we extend a spatial point pattern analysis method (the Morisita index) with meta-data analysis, which includes anomaly behaviour detection and unsupervised learning to support spatio-temporal data analysis and demonstrate its practical use. The resulting framework is robust and has the capability to detect anomalies among large-scale spatio-temporal data using meta-data based on point pattern analysis. It returns visualized reports to end users.
Point pattern
Unsupervised learning
Morisita index
Anomaly detection for analysing spatio-temporal data remains a rapidly growing problem in the wake of an ever-increasing number of advanced sensors that are continuously generating large-scale datasets. For example, vehicle GPS tracking, social media, financial network and router logs, and high resolution surveillance cameras all generate a huge amount of spatio-temporal data. This technology is also important in the context of cyber security since cyber data carries with it an IP address which can map to a specific geolocation and a timestamp. Yet, current cybersecurity approaches are not able to process this kind of information effectively. To illustrate this deficiency, consider the scenario of a distributed denial-of-service (DDoS) attack in which the network packets may come from different IP addresses with sparse locations. In such a case, a spatio-temporal analyzing system [1] is required to analyse the spatial pattern of the DDoS attack. Yet, user oriented analytic environments for cyber security with spatio-temporal marks are currently limited to traditional statistical methods like spatial-temporal outlier detection and hotspot detection [2].1 Furthermore, much of the current work in large scale analytics focuses on automating analysis tasks, such as detecting suspicious activity in a wide area motion and time interval. But these approaches do not provide analysts of cyber security data with spatio-temporal marks the flexibility to employ creativity and discover new trends in the data while operating over extremely large datasets. Current solutions are prohibitive because they require a multidisciplinary skillset.
One possible solution to performing analytics on such large scale spatio-temporal data is to retrieve the metadata of spatial point patterns [5], and apply metadata processing and storage approaches [6], together with domain knowledge derived by machine learning and statistical means. An added advantage of this method is that meta-data hides the details of the point patterns thus providing privacy while still supporting a variety of analytics.
We, thus, propose a framework for performing analytics with spatio-temporal data that has the following properties:
Privacy protection: We use a meta analysis of tracking data as an indicator of subjects' behavior. The geolocation of the subject will not be exposed to the system user.
High scalability: We are able to retrieve the behavior pattern for different amounts of data since the Morisita index provides the scalability adapted to different amounts of tracking data.
Convenience: We designed a convenient way to map the anomaly event of the cyber threat to the physical threat since the cyber threat can be visualized on the real map.
In more detail, we propose a framework to store and process large-scale spatio-temporal data over a "metadata based point pattern" infrastructure, while providing users with a metadata analysis that hides the details of large-scale spatio-temporal data and provides them with a front-end interface that allows them to run a variety of security checks including outlier detection for a single subject, anomaly group detection, anomaly behavior detection and anomaly event detection. Furthermore, the spatio-temporal data is stored in various data stores. As a result, this framework provides high-performance analytical features, flexibility, and extensibility.
The theoretical contribution and novelty of our work lies in the combination of methods from the areas of spatio-temporal analysis, machine learning and statistical analysis. By extracting relevant methods from these three fields of research, we created an effective and efficient tool for anomaly detection by monitoring the cyber and physical levels, simultaneously.
Spatio-temporal data differs from traditional data since both spatial and temporal attributes are available in addition to the actual measurements/attributes.
In this work, we treat the spatio-temporal data as a time series of spatial data. We are using spatial point patterns as the "snapshot" of spatial data within a specific time frame. The Morisita index has been used as the measure of the spatial point patterns.
Spatial point patterns
Spatial point patterns can be stored in a two-dimensional data format to which a variety of analytical methods can be applied to discover useful data patterns in large-scale spatial data. Extracting exact patterns from geospatial data is more complicated than doing so with ordinary data sets because of the nature of geospatial data sources and their associated data structures, which refers to the two or three dimensional data structure.
Spatial point patterns [5]
Figure 1 represents 6 possible spatial distributions of points: random, semi-regular, aggregated, random with a density trend, semi-regular with a density trend, aggregated with a density trend [5].
Common spatial analysis packages use real numbers [e.g. geospatial POIs (Point of Interest) with uncertainty information] like GPS navigation data with errors, categorical values (e.g. fishery production by species) and logical values (e.g. saline water/freshwater) to mark the point patterns [7]. These point patterns are composed of a huge number of points that follow distributions such as those illustrated in Fig. 1 [8]. The region of spatial data can represent a complicated shape, such as an arbitrary polygon or a irregular pixel image pattern. In this work, the spatial data-mining functions are implemented in R.
The Morisita index is a common method for analyzing spatial patterns. It is a statistical measure of dispersion based on the spatial Poisson process. To compute the Morisita index, the function will first calculate the quadrat counts of the spatial point pattern.2 Then the generated index of spatial aggregation for the previous pattern will be the Morisita index. In more detail, the algorithm divides the spatial domain into Q quadrats of equal size and shape.
Then the algorithm counts the number of points falling in every quadrat. Finally, the n[i] number of points in the ith quadrat will be counted as a vector of values called the Morisita index. The sum of the number of points is represented as the N value.
We can also plot the result of this analysis and the Morisita index of dispersion can calculate the overlap between samples.
The formula used in the analysis package is
$$\begin{aligned} MI = Q \frac{\sum _{i=1}^Q n_i (n_i-1)}{N (N-1)} \end{aligned}$$
\(n_i\): number of points in the ith quadrats, N: total number of points [10].
This formula is based on the assumption that increasing the size of the samples will increase the diversity because it will include different habitats (i.e. different faunas) [11]. The Morisita index is used to compare the similarity between different samples. The advantage of the Morisita index is that it is a vector of data that only varies with size of quadrats, not with population density. For more information about the statistical description of the Morisita index, please see [11].
In his seminal book "statistics for spatio-temporal data [12]", Cressie et al. characterizes the process of statistical spatio-temporal data analysis in the presence of uncertain and (often) incomplete observations. This work includes prediction in space (interpolation), prediction in time (forecasting), assimilation of observations and mechanistic models and inference on controlling process parameters. The concept of the poisson point process in the book is also the foundation of our research which relies on the Morisita index. However the Morisita index was originally designed for ecological research by Morisita [11]. The method has been implemented by Baddeley et al. [7] in R to analyze spatial point pattern data. Our work intends to extend Baddeley et al. [7] work to spatio-temporal data type by retrieving characteristic value using Morisita index.
Some pilot study in spatial statistics like Kriging [13, 14] are methods of interpolation of spatial data. The values interpolated conform to the Gaussian process. The meta-Morisita index is different as it actually calculates the density of the clusters in each quadrats. Although the two methods appear related, they have different functions. The function of Kriging is to interpolate the predicting points into the insufficient (usually undersized) geospatial data set, which may contains missing points and irregular spatial objects like polygons. These predicted values are generated by the model of spatial autocorrelation. The function of Morisita index is to exploratory analyze the large scale (usually oversized) geospatial data set by different measures (defined by the diameter of the quadrats). That means, the Kriging method predicts the unknown values (making a prediction) [15]. The Morisita index does not insert any extra values into the raw geospatial data set.
Other spatial statistical models have been established for spatio-temporal processes and spatial point processes. Examples of temporal models of point process include Cox and Isham [16]; Daley and Vere-Jones [17]. Examples of spatial models of point process include Cressie [18]; Diggle [19]; MØller and Waagepetersen [20]. The model of spatio-temporal process is not as well defined a field as the spatial model. Pioneer work in this area includes Diggle [21], Diggle and Gabriel [22]. Literature reviews like Zhuang et al. [23] can be used as references for this field. Some work like Illian et al. [24] introduce models such as goodness-of-fit tests, calculation of summary statistics to process the single realization of the point process.
Our study, however, is functionally closer to clustering work in Machine Learning than the spatio-temporal analyses just mentioned. Performance comparisons will be described in the "Results and discussion" section on quantitative evaluation. Typical clustering algorithms in machine learning including K-means [25], density-based spatial clustering of applications with noise (DBSCAN) [26], expectation maximization (EM) [27]3 are efficient methods to detect the cluster of spatial point patterns.
The advantage of K-means and its derivatives K-mediods [28], CLARANS (Clustering Large Applications based on RANdomized Search) [29], K-modes [30], ISODATA (Iterative Self-Organizing Data Analysis Technique) [31], FCM (fuzzy-c-means) [32] is scalability for large data and efficiency. However the k value is difficult to predict. The expectation maximization (EM) also has disadvantages—the convergence is slow and not capable to provide estimation of the asymptotic variance-covariance matrix of the maximum likelihood estimator (MLE). The density based algorithms like DBSCAN (density-based spatial clustering of application with noise) and their derivatives [26], GDBSCAN [33], DBRS [34], ST-DBSCAN [35], OPTICS (ordering points to identify the clustering structure) [36] have some advantage like to detect the outlier efficiently. However it is hard to set the global parameter. Also DBSCAN is not precise enough to measure the clusters adjacent to each other (neck problem).
We now describe how machine learning and clustering has previously been applied to spatio-temporal analyses and contrasts these works with our proposed approach. Yang et al. [2] highlights the demands of analysing human mobility data and detecting hot spots. The author proposes a framework to identify human mobility hotspots that represent the status of human mobility in local areas and group these hotspots into different classes by clustering their temporal signatures. Their work focuses on converting spatio-temporal data to convergent hotspot and dispersive hotspot. Clustering analysis follows based on the temporal characteristics. Their work focuses on the trajectory of human mobility. It models behaviour but it does so in a short time window such as one hour. Izakian et al. [37] considers Fuzzy C-means (FCM) as a conceptual and algorithmic setting to deal with the problem of anomaly detection. Their work is also based on small size of time series which contains only 10 data points. Our work uses 1 day as the time to calculate the behaviour indicator, which can represent the pattern of the subject over a long period of time and large scale data set.
Birant et al. [38] proposes a three-step approach: clustering, checking spatial neighbours, and checking temporal neighbours to detect spatio-temporal outliers in large databases. Cheng et al. [39] proposes a multiscale approach to detect the spatio-temporal outliers by evaluating the change between consecutive spatial and temporal scales. Their work is based on classification, aggregation, comparison, verification. These two works both focus on detecting the spatio-temporal outliers in large data set. They do not analyse the behaviour pattern as our framework does.
Saligrama et al. [40] proposes a novel graph-based statistical notion called MAX-LCS (local neighborhood-based composite scores) that unifies the idea of temporal and spatial locality. Their work focuses on detecting local anomalies but not for large scale group detection, as in our work.
Young et al. [41] proposes scalable time-series models so that geographically aggregated call volume can accurately identify the onset of major events when the approximate time and location of the event is known. Their work is based on known event. Our work, which is based on unsupervised learning, does not require any prerequisite information.
Liu et al. [42] presents a new data-driven framework for a spatiotemporal feature extraction scheme built on the concept of symbolic dynamics for discovering and representing causal interactions. The extracted spatiotemporal features are then used to learn system-wide patterns via a restricted Boltzmann machine (RBM). Their work is implemented on an energy system with intelligent sensing and control systems. The limit of their work is based on anomalies of sensor data. They don't design the behaviour indicator for tracking data like our work.
Capdevila et al. [43] discusses the mining events using the twitter data. The author proposes the Warble, which is a new probabilistic model and learning scheme. Their work focuses on event detection by probabilistic model. The origin of our work is based on spatial analysis. It is a different way to analyze geospatial data. Anagnostopoulos et al. [44] also discusses the twitter data. However this paper focused on targeted outdoor advertising.
Pappalardo1 et al. [45] highlights the Ditras (DIary-based TRAjectory Simulator), which is a framework to simulate the spatio-temporal patterns of human mobility. The author proposes the framework to identify human mobility by diary and trajectory generators. Their work focuses on the statistical properties of real trajectories. Our work is not based on the trajectories but on the characteristic value from the statistical model.
The purpose of this section is to present the essential elements of our study: the characteristics of the method on which our work is based; the data sets we employed to demonstrate the usefulness of these characteristics and the framework we built to exploit the basic method to the fullest. In more detail, we first describe the way in which the Morisita index can be used to detect anomalous events on a synthetic example. We then describe the data sets on which we conducted our actual study. The first one of these data sets contains a large number of taxi trajectories over a week. It is a physical security problem chosen to illustrate the usefulness of our network on a large data set. Most of our subsequent analysis was conducted on that data set. The second data set is based on Twitter data and contains spatio-temporal information about tweets. As such, it is a cyber dataset and shows how our framework allows cyber and physical security to be considered jointly, as it should be in order to improve real safety. However, the individual Twitter user information is not available from the data set which restricts the type of analysis that can be conducted on it. This is why the Taxi data set was used to demonstrate the versatility of our approach. The third part of this section introduces our framework and explains its functionality.
In this work, we use the spatstat [7] package which is capable of analysing three or more dimensional point pattern datasets. This spatial analysis package supports a variety of statistical analysis methods such as model fitting, spatial data sampling, and statistical formulation. The particular method used in this work is the Morisita index value which was presented in "Morisita index" section. Other models will be explored in future work.
In this study, we calculate the highest Morisita index value as the behaviour indicator in a long time slot (1 day) to avoid sampling bias. If the data is mostly distributed uniformly, the Morisita index value falls between 0 and 1. However, if the data is in clumped distribution then the Morisita value falls between 1 and n [10]. n is the maximum value of the Morisita index which means the density of points is > 1 suggest clustering.
Morisita index as the real-time clustering indicator
In this study, we use the Morisita index method to process the point pattern before detecting an anomalous event. The Morisita index is designed to determine the density of point patterns according to their statistical characteristics; the method is extensively applied to classify and visualize data with geolocation. We chose this method to indicate the point patterns of crowds of people and mark each situation using the corresponding Morisita value to indicate the density of the crowd. Thus, the process includes methods from computational statistics. Given that we want to indicate that these point patterns belong to different social events according to the statistical characteristics of their density values, the method will return Morisita values as a density value. For example, the density value of a downtown social event yields a large Morisita value, whereas the density value of a suburban social event yields a small value. For each social event, the density value is correlated to the Morisita value in indicating the social event. This is illustrated in Figs. 2 and 3.
Crowd with tracking device uniformly distributed in metro area
Morisita plot shows crowd with tracking device gathering in the downtown area
Figure 2 shows that a crowd with tracking devices is uniformly distributed in the metro area. Figure 4 shows the Morisita plot corresponding to Fig. 2. In this plot, the Morisita value falls into the [0, 1] range. That means that the point pattern is close to the uniform distribution. Figure 2, therefore, shows that the people with tracking devices are uniformly distributed.
Morisita plot shows that crowd with tracking device uniformly distributed in metro area
Crowd with tracking device gathering in the downtown area
Figure 5 shows a crowd with tracking devices gathering in the downtown area. Figure 3 shows the Morisita plot corresponding to Fig. 5. The highest value of the Moritisa plot is the highest density of the point pattern. The maximum value of Morisita index is close to 8. It shows that people with tracking devices are gathering in some area.
For the downtown case, the maximum of Morisita index values climbs up to 8, which means our raw data was clumped together. For the suburban case, the maximum of Morisita index values falls down to 1.5. The value is significantly smaller than the maximum value 8 in Fig. 3. The result means that the point pattern in this case was not as clumped as in the downtown case.
Figure 7 shows a crowd with tracking devices gathering in the suburban area. Figure 6 shows the Morisita plot corresponding to Fig. 7. The highest value of the Moritisa plot is the highest density of the point pattern. The maximum value of Morisita index is close to 1.5.
Morisita plot about crowd gathering in the suburban area
It also shows that the crowd with tracking devices in Fig. 7 was not as clumped as in Fig. 5.
The Morisita index is superior for comparing the similarities between different samples. If the data is mostly uniformly distributed, the Morisita index value falls between 0 and 1. However, if the data is in a clumped formation, then the Morisita value falls between 1 and N [7]. N is the highest positive number that indicates the highest degree of density. In this case, we use our framework to get the distribution and point pattern of the large-scale tracking data with spatio-temporal marks. Users should check the Morisita value as the real time indicator to determine if it is a gathering event or not. The gathering can be physical or cyber (with IP address mapping). This example shows the function that can help users find social events from point patterns.
The first data set is a sample of the T-Drive trajectory dataset from Microsoft Research [46, 47] that contains a trajectory of 10,357 taxis from 02/02/2008 to 02/08/2008. The total number of points in this dataset is about 15 million and the total distance of the trajectories reaches 9 million km. Taxi drivers are experienced drivers who can usually drive around the metro area. The taxis with tracking devices are mobile sensors probing the behaviour pattern of the subject. So, the taxi tracking record contains the information of both the spatio-temporal pattern and their behaviour patterns.
The second dataset is a twitter data set. It contains data derived from spatio-temporal information of tweets originated from the city of Milan during the months of November and December 2013 [48]. There is no content of the tweets included in this data set.
The simulated data set was used to indicate the behavior of the Morisita index on specific spatial point patterns. The taxi and twitter data set were used to show the Morisita index values generated by the spatio-temporal data.
The taxi and twitter datasets are both typical large scale spatio-temporal datasets. The statistical anomaly event detection and interpretation of new trends and spatio-temporal pattern changes in sequences of social and political events are hidden behind the large scale data. The taxi data is a typical tracking data set with spatio-temporal marks. The twitter data is a typical social media data set with spatio-temporal marks.
Meta-Morisita index architecture
Figure 8 shows the system architecture of the meta-Morisita index based framework. As shown in the figure, we propose a framework able to handle three different tasks: outlier detection for a single subject, anomaly detection for a group of subjects and anomalous social event detection. The outlier detection for a single subject means that the system can detect some anomaly behaviour for a single person, for example, frequent credit card charges in an anomalous location. The anomaly detection for a group of subjects means that the system can identify anomalous subjects by their behaviour, for instance, the person who posts larger amounts of tweets than other people. The anomalous social event detection means that the system can detect social events, such as, network flow burst.
System architecture of meta-Morisita index
We will now describe each of the components of the flow chart of Fig. 8.
During data processing, we extract the highest Morisita value of taxi drivers for each day from the dataset. The Morisita value which is the indicator of density value can be treated as the indicator of behaviour.
First, to identify the outlier, a box plot is employed to generate an outlier list based on the box plot of Morisita values. Second, we employ a clustering method to classify taxi drivers into several groups according to the statistical characteristics of the Morisita value. A K-means algorithm is designed to extract the specific locally convergent and dispersive drivers from the meta-data of point patterns. Finally, we execute a time series analysis using a change point detection algorithm to extract the change point of human convergent and dispersive behaviours from the meta-data of point patterns. We now describe the data processing and anomaly detection parts of our framework in more detail.
Data preprocessing
For a taxi driver, we can construct an individual's behaviour pattern by analysing the point pattern in the time sequence, where the longitude, latitude, and timestamp of a mobile tracking subject and the pattern represent the time when the spatio-temporal points are updated. For two adjacent Morisita values of the records, the time interval is 1 day. We can identify the density of the point pattern during the interval. The Morisita value as the behaviour indicator can be extracted from the spatio-temporal data within every day. For example, the first Morisita index value was collected on 02/02/2008, and the second record was collected on 02/03/2008; we can extract one time series of Morisita value during 02/02/2008–02/08/2008. The time attributes of the two adjacent records are considered to be the time slot. Thus, we can extract the flow matrices (time slots) of Morisita value for 1 week. We use the time series of the Morisita index values to denote the behaviour pattern of taxi drivers in 1 week. For each day, we calculate the Morisita value of each taxi driver, which represents the behaviour pattern of the taxi drivers' moves from day to day during that week. We define the time series of the Morisita value as the largest difference observed during that week. The time series thus provides a behaviour pattern for the taxi driver during the week. The variation of Morisita values during a week can reveal the potential function of the anomaly detection. The clustering analysis of the Morisita values can be used as in the following section to identify convergent and dispersive behaviours.
Anomaly behaviour detection
Box plot method for outlier detection in a single subject In order to detect anomalies in a single subject's behavior, we used a box plot and studied the outliers in that plot.
Clustering method for anomaly group detection For anomaly group detection, we used a clustering method, namely k-means [49]. k-means clustering is a method of vector quantization, originally from signal processing, that is popular for cluster analysis in data mining. k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. This results in a partitioning of the data space into Voronoi cells. In this work, k-means was used with the value k = 4 which provided the best visualization. Since the box plot approach could also be used to detect anomalies in a group situation, we validated the results obtained by k-means with the output of the box-plot.
Time series analysis for social event detection In order to detect social events, we performed time series analysis using the PELT (Pruned Exact Linear Time) [50, 51] algorithm which is a change point detection method. The PELT algorithm is a multiple change point method that is both computationally efficient and flexible in its application [52]. It has been shown that under certain conditions, especially when the number of change points is increasing linearly with n, the computational efficiency of PELT is O(n). As a result, we can detect change points efficiently from the time series of the behaviour pattern of our data sets.
We applied our full framework to the taxi driver data set and illustrated its usefulness on the Twitter data. The first two parts of this section describe our analysis on individual taxi drivers and groups of taxi drivers, respectively. The third part is an initial illustration of our framework on the twitter data set.
The "Experimental methods" section concerns the application of the meta-Morisita index. The "Results and discussion" section concerns the performance analysis of our meta-analysis method based on the Morisita index. The "Experimental methods" section describes the proposed process and demonstrates its effectiveness. The "Results and discussion" section evaluates our meta-Morisita based approach by comparing its performance to that of the most popular clustering algorithms from machine learning.
Behaviour pattern for taxi driver
Table 1 shows spatio-temporal data from taxi driver records. The data was obtained from Microsoft Research [46, 47].
Saptio-temporal tracking record of taxi driver
Taxi_ID
The first column in Table 1 is the ID of the Taxi. The second column is the timestamp of the tracking data. The third and fourth columns are the coordinates of the Taxi. The data was collected in 5 min intervals.
Morista value generated by tracking data per day
Morisita_Value
Table 2 shows the information from Table 1 mapped onto the Morisita representation. The first column in Table 2 is the ID of the Taxi. The second column is the highest Morisita value of tracking data per day for each driver. The third column is the date.
In this first study, we use the box plot method to process the metadata of point patterns before detecting the anomalies. The metadata values are designed to determine the abnormal values according to their statistical characteristics; the method is extensively applied to classify and visualize the abnormal points in the metadata. We chose this method to indicate the outliers of the metadata of the point patterns and marked each outlier using the corresponding metadata value. Figure 9 shows the box plot of the tracking metadata of 30 taxi drivers.
Box plot about metadata of tracking records of 30 taxi drivers
Given that we want to indicate these outliers according to the statistical characteristics of their density values, the method will return Morisita values as density values. The density value indicates that the taxi driver displayed anomalous behaviour on the day the outlier was recorded. From the box plot, we can find the outliers, such as Taxi 549 on 02/07/2008. The Morisita value is extremely high on that day. That means Taxi 549 displayed anomalous behaviour on that day. Taxi 549, Taxi 493, Taxi 1011, and Taxi 430 have box plots that are different from other taxi drivers.
Statistical analysis for meta-data of spatio-temporal data from taxi drivers
As mentioned in "Anomaly behaviour detection" section, anomaly detection for a group of drivers was conducted using clustering and time series analysis based on meta-data of point patterns, where the output from the meta-data of point patterns is converted to a simple time series data using a Morisita value. We used K-means as the unsupervised learning method to cluster the group of drivers. Also, we use change point detection to detect the change point of the behaviour pattern.
In particular, we used PELT (Pruned Extract Linear Time) [50] as the change point detection approach. PELT was used to detect the exact moment of breakout when the algorithms report a change in distribution (if at all), along with precision, recall, and the F-measure.
Anomaly detection for the behaviour pattern of taxi drivers
In this framework we chose K-means to cluster the group of drivers. Figure 10 shows the 2D representation of a clustering plot displaying the behaviour patterns of 30 taxi drivers using the K-means algorithm. Figure 11 shows the four-cluster plot displaying the behaviour patterns of 30 taxi drivers using K-means algorithms.
2D representation of clustering about 30 taxi drivers
4 Cluster k-means clustering about 30 taxi drivers
As a result of K-means clustering, Taxi 549, Taxi 493, Taxi 1011, Taxi 430 were grouped as anomalies. The box plot was used to validate the cluster analysis. The K-means results were shown to match the results of the box plot.
Spatial distribution of two taxi drivers in Beijing city between 02/02/2008 and 02/08/2008 display at the real map. The pink lines are the read road network. The red dots are the location of taxis
Figure 12 shows the spatial distribution of Taxi 549 and Taxi 234 in the city of Beijing between 02/02/2008 and 02/08/2008 . The pink lines represent the road network. The red dots are the location of taxis. Taxi 549 is marked as anomaly taxi driver as the result of K-means analysis. The median values of Morisita index are 8000 and 400 for Taxi 549 and 243 in Fig. 9. Taxi 549's tracking record are more clumped than normal taxis in the upper east area (airport). When looking at the spatial distribution it is clear that Taxi 549 behaves differently from normal driver Taxi 234.
Change point detection for groups of taxi drivers
In statistical analysis, change detection or change point detection tries to identify times when the probability distribution of a stochastic process or time series changes. In general the problem concerns both detecting whether or not a change has occurred, or whether several changes might have occurred, and identifying the times of any such changes. In this framework we choose the PELT (Pruned Exact Linear Time) [50] method to detect the change point of the behaviour patterns of taxi drivers.
Figure 13 shows the change point value of the time series of Morisita values between 02/02/2008–02/08/2008. The vertical red lines mark the change point.
Change point detection between 02/02/2008–02/08/2008
Figure 14 shows the decomposition of the additive time series between 02/02/2008–02/08/2008.
Decomposition of additive time series between 02/02/2008–02/08/2008
As a result of change point detection, Fig. 13 shows that 02/06/2008 is the date at which the maximum change point in the behaviour patterns of taxi drivers was observed. It turns out that 02/06/2008 was the Eve of the Chinese New Year. This may explain the high density value of the point patterns, which may have been caused by the high traffic volume during the holiday. Figure 14 also shows that every morning represents a change point in the behaviour patterns of taxi drivers. These two observations validate the usefulness of our approach since they show that the events detected by our approach correspond to actual events. In the future, this approach could be used to detect spontaneous gatherings resulting from incidents such as accidents, natural disasters or spontaneous demonstrations and could help alert emergency services and security patrols faster, thus providing greater security.
Analysis for twitter data
We now turn to the analysis of the second data set, the twitter data set. Figure 15 shows the spatial distribution of tweets in Milan, Italy at 9 a.m. and 8 p.m. on 1 December 2013 [53]. The pink lines represent the road network. The red dots are the location of twitter users.
Location of every tweets in Milan city at 9 a.m. and 8 p.m. at 12/01/2013 display at the real map. The pink lines are the read road network. The red dots are the location of twitter users
Figure 16 shows the line chart of changes in Morisita values over time for this twitter data on 12/01/2013. The line chart in Fig. 16 shows that the Morisita values is higher at night than during the day time. This may be explained by the fact that at night (see Fig. 15, right plot), twitter users do not move around as much, but instead stay put in a concentrated area of the city whereas during the day (see Fig. 15, left plot), they are more dispersed and, therefore, tweet from various parts of the city. This, once again, is not particularly novel and useful information here, but it illustrates how gatherings of Twitter users in a single location can be detected, which could be useful for security reasons.
Line chart of Morisita value of twitter data in Milan, Italy in 12/01/2013
Figure 17 shows the spatial distribution of tweets in Milan, Italy at 12/02/2013 (Monday) and 12/07/2013 (Saturday) [53]. The pink lines represent the road network. The red dots are the location of twitter users.
Location of every tweets in Milan city at 12/02/2013 and 12/07/2013 display at the real map. The pink lines are the read road network. The red dots are the location of twitter users
Figure 18 shows the line chart of changes in Morisita value over time for this twitter data between 12/01/2013 and 12/12/2013. The line chart in Fig. 18 shows that the Morisita value is higher during the weekend than during the working day. The Morisita value on the Monday (12/02/2013 and 12/09/2013) were local minima. This may be explained by the fact that on weekends (see Fig. 17, right plot), twitter users do not move around as much, but instead stay put in a concentrated area of the city whereas during the working day (see Fig. 17, left plot), they are more dispersed and, therefore, tweet from various parts of the city. This, once again, is not particularly novel and useful information here, but it illustrates how gatherings of Twitter users in a single location can be detected, which could be useful for security reasons.
Line chart of Morisita value of twitter data in Milan, Italy between 12/01/2013 and 12/12/2013
The user information has been masked in the raw data by the original distributor due to privacy reason. We can, therefore, not analyse individual twitter user based on the meta-Morisita index method. So we use Taxi data as substitute. The report based on a single Taxi user can be found in "Statistical analysis for meta-data of spatio-temporal data from taxi drivers" section. Using the same learning method as the one used on Taxi data, the anomaly in twitter users could be detected efficiently provided that data on individual users is made available.
Once again, given the lack of individual twitter data, we could not perform a full quantitative analysis on that data set. Instead, we performed a full quantitative analysis of the Taxi data set. Our experimental environment is based on an eight-core server equipped with Intel(R) Xeon(R) CPU E5-1630 v4 @ 3.70 GHz and 16 GB memory. The version of operating system is Ubuntu 16.04.1.
Time analysis
In this section we recorded the time analysis result of the performance evaluation of different algorithms on our Taxi database.
Performance Comparison among meta-Morisita index, K-means clustering, density-based spatial clustering of applications with noise (DBSCAN) and expectation maximization (EM) Clustering algorithm
Figure 19 shows the elapsed time of different analysis methods. The four different methods in the comparison are K-means, density-based spatial clustering of applications with noise (DBSCAN), the expectation maximization (EM) Clustering algorithm and the meta-Morisita index algorithm.
There is no significant difference when processing small scale data such as the data containing less than \(10^6\) points. The meta-Morisita index , however, obtains better performance when dealing with large scale data such as when the data contains \(10^8\) points. Thus, the experiment shows that the meta-Morisita index obtains better performance than traditional clustering methods for large scale spatio-temporal data. This is because the meta-Morisita index retrieves the characteristic value of different time windows for the whole spatio-temporal data set. Then the machine learning algorithm runs on the meta-data only, which significantly reduces the computational complexity. On the other hand, the other clustering algorithms like K-means compute their results directly based on the entire large scale data set. The remaining question concerns the quality of the results obtained by the Meta-Morisita index. We now explore this question by comparing the results obtained by each of the methods just considered to a reference method.
Reference method
Map algebra [54] is a basic set-based algorithm that manipulates the geospatial data. Several algebraic operations like addition, subtraction, etc. can be performed on two or more raster layers of similar dimensions. The output of the map algebra primitive operations is a new raster layer (map). Map algebra operations work on four different classes: local, focal, global and zonal. The operations on raster cells and pixels are local operations. The operations on the entire layer are focal operations. The operations on the cells which have the same value are zonal operations. In Geographic Information Systems (GIS), map algebra is implemented by script or procedure. All the operations are displayed on the map. Map algebra calculates the exact number of incidents (here, Taxi occurrences, but could also be, number of tweets, etc.) that occur in a particular location. While map algebra can give us exact results, it is not practical to use in large scale analyses, which is why alternative methods, such as clustering methods and meta-Morisita analysis, were sought and map algebra only used as a reference for a small sample of data as a sanity check. In this section we use map algebra as the referential method to detect clusters in the Taxi data. The number of the clusters will be recorded as the reference value.
Figure 20 shows the point pattern of Taxi 39 on 02/07/2008. This figure is generated by map algebra operations. The behaviour of the taxi driver can be visualized by the density value of the spatio-temporal tracking record. In this map the density value has been displayed with color. Red corresponds to a high density value, and blue corresponds to a low density value. In this case we manually count the number of clusters of size greater than or equal to 3. This count is recorded in Table 3 as the reference value.
Taxi 39 tracking record. The number represents the count of the points which generated by map algebra
Accuracy, precision, recall evaluation
In this section we recorded the accuracy, precision and recall rate of the four algorithms previously considered.
Number of cluster identified in the sample, cluster size ≥ 3
Map algebra
K-means
Expectation maximization(EM)
Meta-Morisita
The first column in Table 3 is the ID of the Taxi. The other entries represent the number of clusters which have been detected by different algorithms. The result are also visualized in Fig. 21. The assumption is that the closer the match in number of clusters detected, the closer the actual match is between detected and reference clusters.
The number of the clusters being detected by meta-Morisita index, K-means clustering, DBSCAN and EM Clustering algorithm. The size of the cluster is great or equal to 3
Evaluation result of different clustering algorithms
Evaluation method
Expectation maximization (EM)
Accuracy rate
Precision rate
Recall rate
The first column in Table 4 is the evaluation metric under consideration. The other entries in the table are the results obtained for these metrics by the four methods under consideration. The result was also visualized in Figure 22.
Evaluation comparison among K-means clustering, DBSCAN and EM Clustering algorithm
Figure 21 shows the number of clusters (size ≥ 3) obtained by different analysis methods. The four different methods in the comparison are K-means, DBSCAN, EM Clustering and meta-Morisita index. The reference value for the number of clusters for each taxi is shown by the purple curve. The meta-Morisita value is shown in red. The figure clearly shows that the red curve is the closest match to the purple curve in Fig. 21.
Figure 22 shows the evaluation of different analysis methods. Once again, the four different methods in the comparison are K-Means, DBSCAN, EM Clustering and meta-Morisita Index. The K-means and EM obtain similar results. They both yield a high number of false negatives (FN). DBSCAN is a little different. It detects all the clusters but obtains too many False Positives (FP).
Map-Algebra algorithms [54] have been used as the reference value since they manually compute occurrences in the same location. The clusters in meta-Morisita index are detected by the number of points falling into the same quadrat. As a result we can see that the meta-Morisita index significantly improves the accuracy, precision and recall rate compared with other learning algorithms. The reason is that the meta-Morisita index algorithm comes from spatial statistics which is similar to map algebra. Furthermore, the Euclidean distance used in the machine learning algorithms caused instability in the clustering result for different data samples. The corresponding concept in Morisita index is the smallest diameter of the quadrat, which is a constant value for geospatial data with the same significant numbers. For example, coordinates (in WGS84 geodetic datum) with four significant values like (Longitude: 116.2936, Latitude 39.9227) , the scale of the data is about 10 m. That constant measure ensures that the meta-Morisita value is stable for different data samples.
Figure 23 shows the relationship between the point pattern and the Morisita index. The outlier is the cluster which contains 146 points and is displayed in red on the picture. This cluster has been detected by the Morisita index. However the Morisita index is based on random sampling. As a result, duplicate clusters have also been erroneously detected by the Morisita index which suggests that more than one cluster of this type is present. Please note that the errors made by the meta-Morisita index occurred only in the in the vicinity of the outlier area where the density of the cluster is too high for several clusters to form.
The relationship between the point density and the Morisita index in large area. The point has highest Morisita value in the Morisita plot corresponds to the location which have highest density in the map
The availability of large-scale spatio-temporal data sets (e.g., social media, vehicle or cellphone tracking data, financial network log) provides the opportunity and challenge to study behaviour patterns to better understand the interactions between cyber and physical events. In this paper, we explore tracking data by investigating the spatio-temporal patterns of taxi drivers and twitter users. A brief work flow is proposed to identify and extract spatio-temporal patterns of outliers based on meta-data of the tracking data. Two case studies of Beijing, China and Milan, Italy are employed to test the proposed method; multiple typical spatio-temporal convergent and dispersive patterns are identified in the large area. We discuss the spatio-temporal distribution of these patterns in different functional areas to obtain better knowledge of the behaviour pattern.
In the paper "Statistical Modeling: The Two Cultures" [55], Breiman compared the data and algorithm modeling cultures. The framework we presented is the combination of spatial statistics and machine learning. The Morisita index is a data modeling approach for spatial data in statistics. However the original Morisita index does not provide the ability to learn. In our work, the Morisita index (data model) has been used to generate the characteristic value of raw data, then learning approaches (algorithm model) were applied to meta data generated by the Morisita index. Our framework thus combines both the advantage of machine learning and spatial statistics. At the same time meta-Morisita index prevents the high computational complexity of current clustering algorithms applied to spatial data.
The findings derived from this study provide insights about the location, time, intensity of the taxi drivers in Beijing and twitter users in Milan, which is helpful for mining behaviour pattern and surveillance for cyber-physical subjects. The identified patterns can help government agencies and urban administrations make targeted adjustments to monitoring cyber-physical events with high anomaly activity as a way to improve the efficiency of the methods used to maintain security in society. In addition, the findings can be used as a reference for understanding subject behaviour. For example, if we only know the spatio-temporal distribution of the active areas of a city, it is possible to have a general understanding of daily subject behaviour and dispersion in other cities according to the discussion in "Statistical analysis for meta-data of spatio-temporal data from taxi drivers" section.
In the future, we will use spatio-temporal statistical models to analyse intelligence information about the behaviour of each subject, to determine the spatio-temporal interactions among different areas of the city, and to explore the behaviour patterns among different social roles, which can provide in-depth knowledge regarding the interactions between subjects and their social roles.
Our plan is to collect tracking data with different social roles, such as Teacher, Police Officer, UPS drivers, etc. We will train the system on each social role separately in order to learn behavior patterns from each category and help us detect behaviors that do not fall into expected categories and may be considered suspicious.
In general, the framework provides a universal solution for spatio-temporal analytic tasks beyond the meta-data of spatial point pattern, which is needed for statisticians and researchers. The Morisita index was selected as the indicator of daily behaviour from spatial point pattern. Multiple analytic methods have been used as efficient statistical computing approaches to accommodate multiple spatial-temporal data sources and data schemas. The statistical analyses beyond the meta-data of spatial point pattern, which work to reduce the computational complexity for large scale spatio-temporal data, are flexible enough to be added to existing spatio-temporal data warehouse systems. Using this framework is a more convenient, flexible, and scalable way for data analysts and statisticians to process and analyse large-scale cyber security data with spatio-temporal marks.
A spatial outlier refers to a point whose non-spatial attribute values are significantly different from the values of their spatial neighbors [3]. A hotspot refers to points that show intermittent spatial repetitiveness [4].
The term quadrat in ecology and geography means a plot to isolate a standard size of area to study the distribution of an item over a large area [9].
EM is a much broader statistical estimation method than simple clustering algorithm. It is a general modelling process which can be applied to clustering.
ZY and NJ conceived of and designed this study. ZY analyzed and drafted the manuscript. NJ gave many valuable suggestions about the machine learning algorithms and help editing the language of the manuscript. Both authors read and approved the final manuscript.
The data sets supporting the results of this article are included within the article
All authors have approved the manuscript and agree with its submission to the journal.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Department of Computer Science, American University, Washington D.C., USA
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Kosmas Balanos
Kosmas Balanos (Greek: Κοσμάς Μπαλάνος) (1731–1808) was a Greek mathematician, author and school director. He continued the work of his father Balanos Vasilopoulos,[1] and was among Greece's leading scholars of his time.[2]
Kosmas Balanos
Born1731
Ioannina, Ottoman Empire
Died1808
Ioannina, Ottoman Empire
Occupation(s)Mathematician
head of the Gouma (or Balanaia) School
Parent
• Balanos Vasilopoulos (father)
Life
He was born in Ioannina, a center of the 17th–18th-century modern Greek Enlightenment movement. Balanos was the first son of the scholar Balanos Vasilopoulos and became a priest like his father had done before him.[1] He taught at various Greek-language schools of the Ottoman Empire, initially in Thessaly, and then in Thessaloniki. Around 1760 he succeeded his father as director of the Gouma School in Ioannina.[1] During the 1790s, the Gouma school faced serious financial difficulties, but Balanos managed to find new sponsors among the prosperous Ioannite diaspora and especially the Zosimades brothers. After about 40 years in the Gouma school, Balanos left his post in 1799 to his brother Konstantinos.[1]
Balanos, as a conservative scholar, used archaic Greek in his work and rejected the use of the Demotic, the vernacular form of the Greek language. He also became involved in a personal conflict with the progressive scholar Athanasios Psalidas, schoolmaster of the Kaplaneios School in the same city, whom he denounced to the local ruler, Ali Pasha, as an atheist and voltairianist.[3]
Work
In 1798 Balanos published in Vienna the work Έκθεσις συνοπτικής αριθμητικής, αλγέβρης και χρονολογίας (Concise Exposition of Arithmetics, Algebra and Chronology).[4] He wrote the philosophical work Περί ελλείψεως των παραλειπομένων φωτών παρά τοις αρίστοις των ποιητών και των καταλογάδην συγγραψάντων (On the omission of the unmentioned lights by the excellent among the poets and among those whο wrote prose).[1] Moreover, he published his father Balanos Vasilopoulos' work Έκθεσις ακριβεστάτη της Αριθμητικής (Most Precise Exposition of Arithmetic, Venice, 1803).
After his death, his work Αντιπελάργησις (Antipelargisis, Against the Stork) was published in 1816, in which he rejected his father's claim to have solved the problem of doubling the cube, i.e. finding the cube root of 2 by a graphical method (which was later shown to be impossible).[4]
He composed a historical work about Epirus, in which Ioannina was the urban center, Ιστορικό της Ηπείρου (History of Epirus), which remains unpublished.[1] Balanos also wrote various texts about the social conditions of the region he lived, like describing the forced 18th century Islamizations of local Christians by the Ottoman authorities.[5] Moreover, he composed theological and philosophical works, as well as school textbooks. Many of these works remained unpublished and were burned with the destruction of the Balanos family library during Ali Pasha's defeat by the Sultan's forces in 1822.[1]
References
1. Μπαλάνος Κοσμάς 1731, Ιωάννινα – 1807/8, Ιωάννινα Archived 2012-04-02 at the Wayback Machine Ελληνομνήμων. University of Athens database.
2. Wax effigies of Kosmas Balanos and Athanasios Psalidas Archived 2015-11-28 at the Wayback Machine, Pavlos Vrellis Greek History Museum.
3. Yannaras, Christos (2006). Orthodoxy and the West: Hellenic self-identity in the modern age. Holy Cross Orthodox Press. ISBN 978-1-885652-81-2.
4. Phili, Christine. "Greek mathematical publications in Vienna in the 18th–19th centuries" (PDF). Mathematics in the Austrian-Hungarian Empire. Proceedings of a Symposium Held in Budapest on August 1, 2009 During the XXIII ICHST.: 137–148.
5. Rathberger, Hrsg. von Oliver-Jens Schmitt. Red.: Andreas (2010). Religion und Kultur im albanischsprachigen Südosteuropa (1., Aufl. ed.). Frankfurt am Main: Lang. p. 84. ISBN 978-3-631-60295-9.
External links
• Works of Kosmas Balanos:
• Έκθεσις Συνοπτική Aριθμητικής, Άλγεβρας και Χρονολογίας (Vienna, 1798)
• Αντιπελάργησις (Vienna, 1816)
Authority control: National
• Greece
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\begin{document}
\title{Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities} \shorttitle{Stability in logarithmic Sobolev and related interpolation inequalities} \author{Jean Dolbeault\affil{1} and Giuseppe Toscani\affil{2}} \abbrevauthor{Dolbeault, J., and Toscani, G.} \headabbrevauthor{J.~Dolbeault and G.~Toscani} \address{ \affilnum{1} Ceremade, UMR CNRS nr.~7534, Universit\'e Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex~16, France, \email{[email protected]},\\ \affilnum{2} Department of Mathematics, University of Pavia, via Ferrata 1, 27100 Pavia, Italy,\newline\email{[email protected]}.}
\begin{abstract} \hspace*{-2.8pt}\parbox{17cm}{This paper is devoted to improvements of functional inequalities based on scalings and written in terms of relative entropies. When scales are taken into account and second moments fixed accordingly, deficit functionals provide ex\-pli\-cit stability measurements, \emph{i.e.}, bound with explicit constants distances to the manifold of optimal functions. Various re\-sults are obtained for the Gaussian logarithmic Sobolev inequality and its Euclidean counterpart, for the Gaussian generalized Poincar\'e inequalities and for the Gagliardo-Nirenberg inequalities. As a consequence, faster convergence rates in diffusion equations (fast diffusion, Ornstein-Uhlenbeck and porous medium equations) are obtained.}\end{abstract} \maketitle
\noindent \emph{Keywords:\/} Sobolev inequality; logarithmic Sobolev inequality; Gaussian isoperimetric inequality; generalized Poincar\'e inequalities; Gagliardo-Nirenberg inequalities; interpolation; entropy -- entropy production inequalities; extremal functions; optimal constants; relative entropy; generalized Fisher information; entropy power; stability; improved functional inequalities; fast diffusion equation; Ornstein-Uhlenbeck equation; porous medium equation; rates of convergence \par
\noindent \emph{Mathematics Subject Classification (2010):\/} 26D10; 46E35; 58E35 \par
\section{Introduction}\label{Sec:Intro}
Several papers have recently been devoted to \emph{improvements of the logarithmic Sobolev inequality}. \cite{2014arXiv1403.5855L} use the Stein discrepancy. Closer to our approach is \cite{2014arXiv1408.2115B,2014arXiv1410.6922F}, who exploit the difference between the inequality of \cite[Inequality~(2.3)]{MR0109101} and the logarithmic Sobolev inequality to get a correction term in terms of the \emph{Fisher information} functional. What we do here first is to emphasize the role of scalings and prefer to rely on \cite{MR479373} for a \emph{scale invariant form of the logarithmic Sobolev inequality} on the Euclidean space. We also make the choice to get a remainder term that involves the entropy functional and is very appropriate for stability issues. This allows us to deduce striking results in terms of rates of convergence for the Ornstein-Uhlenbeck equation. Writing the improvement in terms of the entropy has several advantages: contraints on the second moment are made clear, improvements can be extended to all generalized Poincar\'e inequalities for Gaussian measures, which interpolate between the Poincar\'e inequality and the logarithmic Sobolev inequality, and stability results with fully explicit constants can be stated: see for instance Corollary~\ref{Cor:Interpolation}, with an explicit bound of the distance to the manifold of all Gaussian functions given in terms of the so-called \emph{deficit functional}. This is, for the logarithmic Sobolev inequality, the exact analogue of the result of \cite{MR1124290} for Sobolev's inequality.
However, putting the emphasis on scalings has other advantages, as the method easily extends to a nonlinear setting. We are henceforth in a position to get \emph{improved entropy -- entropy production inequalities associated with fast diffusion flows} based on the scale invariant forms of the associated Gagliardo-Nirenberg inequalities, which cover a well-known family of inequalities that contain the logarithmic Sobolev inequality, and Sobolev's inequality as an endpoint. This is not a complete surprise because such improvements were known from \cite{MR3103175} using detailed properties of the fast diffusion equation. By writing the entropy -- entropy production inequality in terms of the relative entropy functional and a generalized Fisher information, we deduce from the scaling properties of Gagliardo-Nirenberg inequalities a correction term involving the square of the relative entropy, and this is much simpler than using the properties of the nonlinear flow. The method also works in the porous medium case, which is new, provides clear evidences on the role of the second moment, and finally explains the fast rates of convergence in relative entropy that can be observed in the initial regime, away from Barenblatt equilibrium or self-similar states.
The reader interested in further considerations on \emph{improvements of the logarithmic Sobolev inequality} is invited to refer to \cite{2014arXiv1403.5855L} and \cite{2014arXiv1408.2115B,2014arXiv1410.6922F} for probabilistic point of view and a measure of the defect in terms of Wasserstein's distance, and to \cite{MR1849347} for earlier results. Much more can also be found in \cite{MR3155209}. Not all Gagliardo-Nirenberg-Sobolev inequalities are covered by our remarks and we shall refer to \cite{MR3177378} and references therein for the spectral point of view and its applications to the Schr\"odinger operator. The logarithmic Sobolev inequality in scale invariant form is equivalent to the \emph{Gaussian isoperimetric inequality}: a study of the corresponding deficit can be found in \cite{Mossel-Neeman}. In the perspective of information theory, we refer to \cite{MR3034582,2014arXiv1410.2722T} for a recent account on a concavity property of \emph{entropy powers} that involves the isoperimetric inequality. It is not possible to quote all earlier related contributions but at least let us point two of them: the correction to the logarithmic Sobolev inequality by an entropy term involving the Wiener transform in \cite[Theorem~6]{MR1132315}, and the \emph{HWI inequality} by \cite{MR1760620}.
\emph{Gagliardo-Nirenberg inequalities} (see \cite{MR0102740,MR0109940}) have been related with fast diffusion or porous media equations in the framework of the so-called entropy methods by \cite{MR1940370}. Also see the papers by \cite{MR1777035,MR1842429,MR1986060} for closely related issues. The message is simple: optimal rates of convergence measured in relative entropy are equivalent to best constant in the inequalities written in entropy -- entropy production form. Later improvements have been obtained on asymptotic rates of convergence by \cite{BBDGV,BDGV,1004,DKM}. A key observation of \cite{1004} is the fact that optimizing a relative entropy with respect to scales determines the second moment. This observation was then exploited by \cite{MR3103175} to get a first \emph{explicit} improvement in the framework of Gagliardo-Nirenberg inequalities. Notice that many papers on improved interpolation inequalities use the estimate of \cite{MR1124290} with the major drawback that the value of the constant is not known. As a consequence the improved inequality, faster convergence rates for the solution to the fast diffusion equation were obtained and a new phenomenon, a delay, was shown by \cite{2014arXiv1408.6781D}. Inspired by \cite{MR1768665}, \cite{MR3200617} studied the $p$-th R\'enyi entropy and observed that the corresponding isoperimetric inequality is a Gagliardo-Nirenberg inequality in scale invariant form. Various consequences for the solutions to the evolution equations have been drawn in \cite{carrillo2014renyi} and \cite{DTS}, which are strongly related with the present paper but can all be summarized in a simple sentence: scales are important and a better adjustment than the one given by the asymptotic regime gives sharper estimates. The counterpart in the present paper is that taking into account the scale invariant form of the inequality automatically improves on the inequality obtained by a simple entropy -- entropy production method. Let us give some explanations.
At a formal level, the strategy of our paper goes as follows. Let us consider a \emph{generalized entropy} functional~$\mathcal E$, which is assumed to be nonnegative, and a \emph{generalized Fisher information} functional $\mathcal I$. We further assume that they are related by a functional inequality of the form \[ \mathcal I-\lambda\,\mathcal E\ge0\,. \] We denote by $\lambda$ the optimal proportionality constant. If the inequality is not in scale invariant form, we will prove in various cases that there exists a convex function $\varphi$, leaving from $\varphi(0)=0$ with $\varphi'(0)=\lambda$ such that $\mathcal I\ge\varphi(\mathcal E)$. Hence we have found an \emph{improved functional inequality} in the sense that \[ \mathcal I-\lambda\,\mathcal E\ge\varphi(\mathcal E)-\lambda\,\mathcal E=\psi(\mathcal E) \] where $\psi(\mathcal E)$ is nonnegative and can be used to measure the distance to the optimal functions. This is a stability result. The left hand side, which is called the deficit functional in the literature, is now controlled from below by a nonlinear function of the entropy functional. A precise distance can be obtained by the Pinsker-Csisz\'ar-Kullback inequality, which is no more than a Taylor expansion at order two, and some generalizations.The key observation is that the optimization under scaling (in the Euclidean space) amounts to adjust the second moment (in the Euclidean space but also in spaces with finite measure, like the Gaussian measure, after some changes of variables).
At this point it is worth to emphasize the difference in our approach compared to the one of \cite{2014arXiv1408.2115B} for the logarithmic Sobolev inequality. What the authors do is that they write the improved inequality as $\varphi^{-1}(\mathcal I)\ge\mathcal E$ and deduce that \[ \mathcal I-\lambda\,\mathcal E\ge\mathcal I-\lambda\,\varphi^{-1}(\mathcal I) \] where the right hand side is again nonnegative because $\varphi^{-1}$ is concave and $\lambda\,(\varphi^{-1})'(0)=1$. This is of course a stronger form of the inequality, as it controls the distance to the manifold of optimal functions in a stronger norm, for instance. However, it is to a large extend useless for the applications that are presented in this paper, as the estimate in terms of the entropy is what matters, for instance, for application in evolution equations.
We shall apply our strategy to the logarithmic Sobolev inequality in Section~\ref{Sec:logSob}, to the generalized Poincar\'e inequalities for Gaussian measures in Section~\ref{Sec:Beckner} and to some Gagliardo-Nirenberg inequalities in Section~\ref{Sec:GN}. Each of these inequalities can be established by the \emph{entropy -- entropy production} method. By considering the Ornstein-Uhlenbeck equation in the first two cases, and the fast diffusion / porous medium equation in the third case, it turns out that $\frac{d\mathcal E}{dt}=-\,\mathcal I$ and \[ -\,\frac d{dt}\(\mathcal I-\lambda\,\mathcal E\)=\mathcal R\ge0\,. \] Hence, if $\lim_{t\to\infty}\(\mathcal I-\lambda\,\mathcal E\)=0$, this shows with no additional assumption that $\int_0^\infty\mathcal R[v(t,\cdot)]\,dt$ is a measure of the distance to the optimal functions. \emph{Improved functional inequalities} follow by ODE techniques if one is able to relate $\mathcal R$ with $\mathcal E$ and $\mathcal I$. This is the method which has been implemented for instance in \cite{MR2152502,pre05312043,MR3103175} and it is well adapted when the diffusion equation can be seen as the gradient flow of $\mathcal E$ with respect to a distance. Typical distances are the Wasserstein distance for the logarithmic Sobolev inequality or the Gagliardo-Nirenberg inequalities, and \emph{ad hoc} distances in case of the generalized Poincar\'e inequalities. See \cite{MR1617171,MR1842429,MR2448650,MR3023408} for more details on gradient flow issues. Improvements can also be obtained when $\frac{d\mathcal E}{dt}$ differs from $-\,\mathcal I$: we refer to \cite{MR2381156,DEKL} for interpolation inequalities on compact manifolds, or to \cite{MR3227280} for improvements of Sobolev's inequality based on the Hardy-Littlewood-Sobolev functional. This makes the link with the famous improvement obtained by \cite{MR1124290}, and also \cite{MR2538501}, but so far no \emph{entropy -- entropy production} method has been able to provide an improvement in such a critical case. For completeness, let us mention that other methods can be used to obtain improved inequalities, which are based on variational methods like in \cite{MR1124290}, on symmetrization techniques like in \cite{MR2538501} or on spectral methods connected with heat flows like in \cite{MR2375056}. Here we shall simply rely on convexity estimates and the interplay of entropy -- entropy production inequalities with their scale invariant counterparts.
A very interesting feature of \emph{improved functional inequalities} in the framework of the \emph{entropy -- entropy production} method is that the entropy decays faster than expected by considering only the asymptotic regime. In that sense, the improved inequality capture an initial rate of convergence which is faster than the asymptotic one. This has already been observed for fast diffusion equations in \cite{MR3103175} with a phenomenon of delay that has been studied in \cite{2014arXiv1408.6781D} and by \cite{carrillo2014renyi}, by resorting to the concept of R\'enyi entropy. A remarkable fact is that the inequality is improved by choosing a scale (in practice by imposing a constraint on the second moment) without requesting anything on the first moment, again something that clearly distinguishes the improvements obtained here from what can be guessed by looking at the asymptotic problem as $t\to\infty$. Details and statements on these consequences for diffusion equations have been collected in Section~\ref{Sec:Diffusion}.
\section{Stability results for the logarithmic Sobolev inequality}\label{Sec:logSob}
Let $d\mu=\mu\,dx$ be the normalized Gaussian measure, with $\mu(x)=(2\,\pi)^{-d/2}\,e^{-|x|^2/2}$, on the Euclidean space ${\mathbb R}^d$ with $d\ge1$. The \emph{Gaussian logarithmic Sobolev inequality} reads \be{Ineq:LogSobGaussian}
\irdmu{|\nabla u|^2}\ge\frac12\,\irdmu{|u|^2\,\log|u|^2}
\end{equation} for any function $u\in\mathrm H^1({\mathbb R}^d,d\mu)$ such that $\irdmu{|u|^2}=1$. This inequality is equivalent to the \emph{Euclidean logarithmic Sobolev inequality in scale invariant form} \be{Ineq:LogSobEuclideanWeissler}
\frac d2\,\log\(\frac2{\pi\,d\,e}\ird{|\nabla w|^2}\)\ge\ird{|w|^2\,\log|w|^2} \end{equation} that can be found in \cite[Theorem~2]{MR479373} in the framework of scalings, but is also the one that can be found in \cite[Inequality~(2.3)]{MR0109101} or in \cite[Inequality~(26)]{MR1132315}. See \cite{2014arXiv1408.2115B,2014arXiv1410.6922F} and \cite{Tos2014} for more comments. The equivalence of~\eqref{Ineq:LogSobGaussian} and~\eqref{Ineq:LogSobEuclideanWeissler} is well known but involves some scalings and we will give a short proof below for completeness. Next, let us consider the function \be{Eqn:varphi} \varphi(t):=\frac d4\,\left[\exp\(\frac{2\,t}d\)-1-\frac{2\,t}d\right]\quad\forall\,t\in{\mathbb R}\,. \end{equation} Our first result is an improvement of \eqref{Ineq:LogSobGaussian}, based on the comparison of~\eqref{Ineq:LogSobGaussian} with~\eqref{Ineq:LogSobEuclideanWeissler}, which combines ideas of \cite{MR2294794} and \cite{2014arXiv1410.6922F}. It goes as follows.
\begin{prop}\label{Thm:LogSob} With $\varphi$ defined by \eqref{Eqn:varphi}, we have \begin{multline}\label{Ineq:LogSobGaussianImproved}
\irdmu{|\nabla u|^2}-\frac12\,\irdmu{|u|^2\,\log|u|^2}\ge\varphi\(\irdmu{|u|^2\,\log|u|^2}\)\\
\forall\,u\in\mathrm H^1({\mathbb R}^d,d\mu)\quad\mbox{such that}\quad\irdmu{|u|^2}=1\quad\mbox{and}\quad\irdmu{|x|^2\,|u|^2}=d\,. \end{multline} \end{prop}
Inequality~\eqref{Ineq:LogSobGaussianImproved} is an improvement of \eqref{Ineq:LogSobGaussian} because $\varphi(t)\ge\frac{t^2}{2\,d}$ for any $t\in{\mathbb R}$ and, by the Pinsker-Csisz\'ar-Kullback inequality, \[
\irdmu{|u|^2\,\log|u|^2}\ge\frac14\,\(\irdmu{\Big|\,|u|^2-1\,\Big|}\)^2\quad\forall\,u\in\mathrm L^2({\mathbb R}^d,d\mu)\quad\mbox{such that}\quad\nrmu u2=1\,. \] See \cite{MR0213190,Csiszar67,Kullback67} for a proof of this inequality. \begin{proof} To emphasize the role of scalings, let us give a proof of Proposition~\ref{Thm:LogSob}, which follows the strategy of~\cite[Proposition~2, p.~694]{MR2294794}.
As a preliminary step, we recover the scale invariant, Euclidean, version of the logarithmic Sobolev inequality from~\eqref{Ineq:LogSobGaussian}. Let $v:=u\,\sqrt\mu$. We observe that $\ird{|v|^2}=1$ and $\ird{|x|^2\,|v|^2}=d$. With one integration by parts, we get that \be{Ineq:LogSobGaussianEuclidean}
\ird{|\nabla v|^2}\ge\frac12\,\ird{|v|^2\,\log|v|^2}+\frac d4\,\log(2\,\pi\,e^2) \end{equation} which is the standard \emph{Euclidean logarithmic Sobolev inequality} established in~\cite{Gross75} (also see \cite{Federbush} for an earlier related result). This inequality is not invariant under scaling. By considering $w$ such that $v(x)=\lambda^{d/2}\,w(\lambda\,x)$, we get that \[
\lambda^2\ird{|\nabla w|^2}-\frac d2\,\log\lambda\ge\frac12\,\ird{|w|^2\,\log|w|^2}+\frac d4\,\log(2\,\pi\,e^2)\,. \]
holds for any $v\in\mathrm H^1({\mathbb R}^d,d\mu)$ such that $\nrm v2=1$. An optimization on the scaling parameter shows that $4\,\lambda^2\,\ird{|\nabla w|^2}=d$ and establishes the scale invariant form of the logarithmic Sobolev inequality, \be{Ineq:LogSobWeissler}
\frac d2\,\log\(\frac2{\pi\,d\,e}\ird{|\nabla w|^2}\)\ge\ird{|w|^2\,\log|w|^2}\quad\forall\,w\in\mathrm H^1({\mathbb R}^d,dx)\quad\mbox{such that}\quad\nrm w2=1\,, \end{equation} which is equivalent to \eqref{Ineq:LogSobEuclideanWeissler}. This inequality can also be written as \[
\ird{|\nabla w|^2}\ge\frac12\,\pi\,d\,e\,\exp\(\frac 2d\ird{|w|^2\,\log|w|^2}\)\,. \]
If we redefine $u$ such that $w=u\,\sqrt\mu$ and assume that $\nrm w2=1$, $\ird{|x|^2\,w}=d$, we have shown that \be{Eqn:LogSobGaussianSimple}
\irdmu{|\nabla u|^2}\ge\frac d4\left[\exp\(\frac 2d\irdmu{|u|^2\,\log|u|^2}\)-1\right]\,.
\end{equation} Inequality~\eqref{Ineq:LogSobGaussianImproved} follows by substracting $\frac12\,\irdmu{|u|^2\,\log|u|^2}$ from both sides of the inequality, which is more or less the idea that has been exploited by \cite{2014arXiv1410.6922F}. \end{proof}
Consider a nonnegative function $f\in\mathrm L^1_2({\mathbb R}^d):=\left\{g\in\mathrm L^1({\mathbb R}^d)\,:\,\ird{|x|^2\,g}<\infty\right\}$ and, assuming that $\ird f>0$, define \be{Def:M-theta}
M_f:=\ird f\,,\quad\theta_{\!f}:=\frac 1d\,\frac{\ird{|x|^2\,f}}{M_f}\,. \end{equation} Let us define the Gaussian function \[
\mu_f(x):=\frac{M_f}{(2\,\pi\,\theta_{\!f})^{d/2}}\,e^{-\frac{|x|^2}{2\,\theta_{\!f}}}\quad\forall\,x\in{\mathbb R}^d\,. \] We shall denote by $\mathrm L^1_2({\mathbb R}^d)$ the space of integrable functions on ${\mathbb R}^d$ with finite second moment.
\begin{lem}\label{Lem:Euclidean} Assume that $f$ is a nontrivial, nonnegative function in $f\in\mathrm L^1_2({\mathbb R}^d)$ such that $\nabla\sqrt f\in\mathrm L^2({\mathbb R}^d)$. With $\theta_{\!f}$, $\mu_f$ and $\varphi$ defined by \eqref{Eqn:varphi} and \eqref{Def:M-theta}, we have \be{Ineq:ImprovevdEuclidean}
\frac{\theta_{\!f}}2\ird{\frac{|\nabla f|^2}f}-\ird{f\,\log f}-\frac d2\,\log\(2\,\pi\,e^2\,\theta_{\!f}\)\,\ird f\ge2\,\varphi\left[\,\ird{f\,\log\(\frac f{\mu_f}\)}\right]\,. \end{equation} \end{lem}
\begin{proof} Let $v$ be such that $\lambda^d\,f(\lambda\,x)=|u(x)|^2\,\mu(x)$ with $\lambda^2=\theta_{\!f}$, $\mu(x)=(2\,\pi)^{-d/2}\,e^{-|x|^2/2}$, and apply Proposition~\ref{Thm:LogSob}.\end{proof}
The Gaussian function $\mu_f$ is the minimizer of the \emph{relative entropy} \[
\mathsf e[f|\mu]:=\ird{\left[f\,\log\(\frac f\mu\)-(f-\mu)\right]} \] w.r.t.~all Gaussian functions in \[
\mathcal M:=\big\{\mu(x)=\frac M{(2\,\pi\,\theta)^{d/2}}\,:\,e^{-\frac{|x|^2}{2\,\theta}}\,,\;M>0\,,\;\theta>0\big\}\,, \] that is, we have the identity \[
\ird{f\,\log\(\frac f{\mu_f}\)}=\mathsf e[f|\mu_f]=\min\left\{\mathsf e[f|\mu]\,:\,\mu\in\mathcal M\right\}\,. \] Also notice that $\mu_f$ is the minimizer of the \emph{relative Fisher information} w.r.t.~all Gaussian functions of mass $M_f$: \[
\ird{\big|\,\nabla\sqrt{f/\mu_f}\,\big|^2}=\min\left\{\ird{\big|\,\nabla\sqrt{f/\mu}\,\big|^2}\,:\,\mu(x)=\frac{M_f}{(2\,\pi\,\theta)^{d/2}}\,e^{-\frac{|x|^2}{2\,\theta}}\,,\theta>0\right\}\,. \]
Recall that by the Pinsker-Csisz\'ar-Kullback inequality, the r.h.s.~in~\eqref{Ineq:ImprovevdEuclidean} provides an explicit stability result in $\mathrm L^1_+({\mathbb R}^d,dx)$ that can be written as \[
\mathsf e[f|\mu_f]\ge\frac1{4\,M_f}\,\nrm{f-\mu_f}1^2\quad\forall\,f\in\mathrm L^1_+({\mathbb R}^d,dx)\,. \] Combined with the observation that $\varphi$ is nondecreasing and $\varphi(t)\ge\frac{t^2}{2\,d}$ for any $t\in{\mathbb R}$, we have shown the following global stability result.
\begin{cor}\label{Cor:Interpolation} Assume that $f$ is a nontrivial, nonnegative function in $f\in\mathrm L^1_2({\mathbb R}^d)$ such that $\nabla\sqrt f\in\mathrm L^2({\mathbb R}^d)$. With $\theta_{\!f}$, $\mu_f$ and $\varphi$ defined by \eqref{Eqn:varphi} and \eqref{Def:M-theta}, we have \begin{multline*}
\frac{\theta_{\!f}}2\ird{\frac{|\nabla f|^2}f}-\ird{f\,\log f}-\frac d2\,\log\(2\,\pi\,e^2\,\theta_{\!f}\)\,\ird f\\
\ge2\,\min_{\mu\in\mathcal M}\varphi\big(\mathsf e[f|\mu]\big)=2\,\varphi\big(\mathsf e[f|\mu_f]\big)\ge\frac{\nrm{f-\mu_f}1^4}{16\,M_f^2}\,. \end{multline*} \end{cor}
\section{An improved version of the generalized Poincar\'e inequalities for Gaussian measures}\label{Sec:Beckner}
We consider the inequalities introduced by W.~Beckner in \cite[theorem~1]{MR954373}. If $\mu(x)=(2\,\pi)^{-d/2}\,e^{-|x|^2/2}$, then for any $p\in[1,2)$ we have \be{Ineq:Beckner} \nrmu u2^2-\nrmu up^2\le(2-p)\,\nrmu{\nabla u}2^2\quad\forall\,u\in\mathrm H^1({\mathbb R}^d,d\mu)\,. \end{equation} These inequalities interpolate between the Poincar\'e inequality ($p=1$ case) and the logarithmic Sobolev inequality, which is achieved by dividing both sides of the inequality by $(2-p)$ and passing to the limit as $p\to2$. Some improvements were obtained already obtained in \cite{MR2152502,MR2375056,MR2201954}. What we gain here is that the improvement takes place also in the limit case as $p\to2$ and is consistent with the results of Proposition~\ref{Thm:LogSob}.
Let us define \[ \varphi_p(x):=\frac d4\left[(1-x)^{-\frac{2\,p}{d\,(2-p)}}-1\right]\quad\forall\,x\in[0,1]\,. \]
\begin{cor}\label{Cor:Beckner} Assume that $u\in\mathrm H^1({\mathbb R}^d,d\mu)$ is such that $\irdmu{|u|^2\,|x|^2}=d\,\nrmu u2^2$. With the above notation, for any $p\in[1,2)$ we have \be{Ineq:BecknerImproved}
\irdmu{|\nabla u|^2}\ge\nrmu u2^2\,\varphi_p\(\frac{\nrmu u2^2-\nrmu up^2}{\nrmu u2^2}\)\,. \end{equation} \end{cor}
By homogeneity we can assume that $\nrmu u2=1$. The reader is invited to check that \[
\lim_{p\to2}\varphi_p\(1-\nrmu up^2\)=\frac d4\(e^{\frac 2d\mathsf E[u]}-1\)\quad\mbox{where}\quad\mathsf E[u]:=\irdmu{\frac{|u|^2}{\nrmu u2^2}\,\log\(\frac{|u|^2}{\nrmu u2^2}\)}\,. \] The proof of Corollary~\ref{Cor:Beckner} is a straightforward consequence of~\eqref{Eqn:LogSobGaussianSimple} and of the following estimate.
\begin{lem}\label{Lem:Equivalence} For any $p\in[1,2)$ and any function $u\in\mathrm L^p\cap\mathrm L^2({\mathbb R}^d,d\mu)$, we have \[ \frac{\nrmu u2^2}{\nrmu up^2}\le \exp\(\tfrac{2-p}p\,\mathsf E[u]\) \] and, as a consequence, for any $u\in\mathrm L^p\cap\mathrm L^2({\mathbb R}^d,d\mu)$ such that $\nrmu u2=1$, we obtain \be{Ineq:equiv}
\nrmu u2^2-\nrmu up^2\le\frac{2-p}p\irdmu{|u|^2\,\log|u|^2}\,. \end{equation} \end{lem}
\begin{proof} The proof relies on an idea that can be found in \cite{MR1796718} and goes as follows. Let us consider the function \[ k(s):=s\,\log\(\irdmu{u^\frac2s}\)\,. \] Derivatives are such that \begin{eqnarray*} &&\frac12\,k'(s)=\log\(\irdmu{u^\frac2s}\)-\frac1s\,\frac{\irdmu{u^\frac2s\,\log u}}{\irdmu{u^\frac2s}}\,,\\
&&\frac{s^3}4\(\irdmu{u^\frac2s}\)^2\,k''(s)=\irdmu{u^\frac2s}\irdmu{u^\frac2s\,|\log u|^2}-\(\irdmu{\log u\,u^\frac2s}\)^2\,,\\ \end{eqnarray*} hence proving that $k$ is convex by the Cauchy-Schwarz inequality. As a consequence we get that \[ k'(1)\le\frac{k(s)-k(1)}{s-1}\quad\forall\,s>1\,. \] Applied with $s=2/p$, this proves that \[
-\irdmu{|u|^2\,\log\(\frac{|u|^2}{\nrmu u2^2}\)}\le\frac p{2-p}\,\nrmu u2^2\,\log\(\frac{\nrmu up^2}{\nrmu u2^2}\), \] from which we deduce the second inequality in \eqref{Ineq:equiv} after observing that $-\log x\ge 1-x$.\end{proof}
The result of Corollary~\ref{Cor:Beckner} deserves a comment. As $x\to0$, $\varphi_p(x)\sim\frac p2\,\frac x{2-p}$, so that we do not recover the optimal constant in~\eqref{Ineq:Beckner} in the asymptotic regime corresponding to $\nrmu up/\nrmu u2\to1$, that is when $u$ approaches a constant, because of the factor $\frac p2$. On the other hand, \eqref{Ineq:BecknerImproved} is a strict improvement compared to~\eqref{Ineq:Beckner} as soon as $\nrmu up^2/\nrmu u2^2<x_\star(p)$ where $x_\star(p)$ is the unique solution to $\varphi_p(x)=\frac x{2-p}$ in $(0,1)$. Let $\Phi_p$ be the function defined by \be{Phi_p} \Phi_p(x)=\varphi_p(x)\quad\mbox{if}\quad x\in(0,x_\star(p))\,,\quad\Phi_p(x)=\frac x{2-p}\quad\mbox{if}\quad x\in[x_\star(p),1]\,. \end{equation} Collecting these estimates of~\eqref{Ineq:Beckner} and~\eqref{Ineq:BecknerImproved}, we can write that \[
\irdmu{|\nabla u|^2}\ge\nrmu u2^2\,\Phi_p\(\frac{\nrmu u2^2-\nrmu up^2}{\nrmu u2^2}\) \]
for any function $u\in\mathrm H^1({\mathbb R}^d,d\mu)$ such that $\irdmu{|u|^2\,|x|^2}=d\,\nrmu u2^2$. This is an improvement with respect to~\eqref{Ineq:Beckner} because $\Phi_p(x)\ge\frac x{2-p}$, with a strict inequality if $x<x_\star(p)$.
The right hand side in~\eqref{Ineq:BecknerImproved} controls the distance to the constants. Indeed, using for instance H\"older's estimates, it is easy to check that \[
\nrmu u2^2-\nrmu up^2\ge\nrmu u2^2-\nrmu u1^2=\irdmu{|u-\overline u|^2} \]
with $\overline u=\irdmu{|u|}$. Sharper estimates based for instance on variants of the Pinsker-Csisz\'ar-Kullback inequality can be found in \cite{MR1951784,BDIK}.
\section{Stability results for some Gagliardo-Nirenberg inequalities}\label{Sec:GN}
\subsection{A first case: \texorpdfstring{$q>1$}{q>1}}\label{Sec:GN1} We study the case of Gagliardo-Nirenberg inequalities \be{GN1} \nrm{\nabla w}2^\vartheta\,\nrm w{q+1}^{1-\vartheta}\ge\mathsf C_{\rm{GN}}\,\nrm w{2q} \end{equation} with $\vartheta=\frac dq\,\frac{q-1}{d+2-q\,(d-2)}$. The value of the optimal constant has been established in \cite{MR1940370} (also see \cite{Gunson91} for an earlier but partial contribution).
Let us start with some elementary observations on convexity. Consider two positive constants $a$ and $b$. Let us define \[ \zeta=\frac b{a+b}\,,\quad\kappa=\(\frac ab\)^\zeta+\(\frac ba\)^{1-\zeta}=\frac{a+b}{a^{1-\zeta}\,b^\zeta}\,. \] Next let us take three positive numbers, $A$, $B$, and $C$ such that $A^\zeta\,B^{1-\zeta}\ge C$ and consider the function \[ h(\lambda)=\lambda^a\,A+\lambda^{-b}\,B-\kappa\,C\,. \] The function $h$ reaches its minimum at $\lambda=\lambda_*:=\(\frac{b\,B}{a\,A}\)^\frac1{a+b}$ and it is straightforward to check that \[ h(1)\ge\inf_{\lambda>0}h(\lambda)=h(\lambda_*)=\kappa\(A^\zeta\,B^{1-\zeta}-C\)\,. \] This computation determines the choice of $\kappa$. Using the assumption $A^\zeta\,B^{1-\zeta}\ge C$, we get the estimate \[ A+B-\kappa\,C\ge C^\frac1\zeta\,B^{1-\frac1\zeta}+B-\kappa\,C=\varphi(B_*-B) \] where \[ B_*:=C\(\frac{1-\zeta}\zeta\)^\zeta \] and \be{Eqn:VarphiFD} \varphi(s):=C^\frac1\zeta\,\left[(B_*-s)^{1-\frac1\zeta}-B_*^{1-\frac1\zeta}\right]-s\,. \end{equation} Indeed, $\zeta=\frac b{a+b}$ leads to the identity
\[
\kappa\,C = B_* + C^{\frac 1\zeta}\,B_*^{1-\frac 1\zeta}\,.
\] Note that $\varphi$ is a nonnegative strictly convex function such that $\varphi(0)=0$ and $\varphi''(s)>\varphi''(0)=\frac{1-\zeta}{\zeta^2}\,C^\frac1\zeta\,B_*^{-1-\frac1\zeta}$ for any $s\in(0,B_*)$.
We apply these preliminary computations with \begin{eqnarray*} &&a=\frac dq-(d-2)\,,\quad b=d\,\frac{q-1}{2\,q}\\
&&A=\frac14\,(q^2-1)\ird{|\nabla w|^2}\,,\quad B=\beta\ird{|w|^{q+1}}\,,\quad\beta=\frac{2\,q}{q-1}-d\\ &&C=(\tfrac14\,(q^2-1))^\zeta\,\beta^{1-\zeta}\,\(\mathsf C_{\rm{GN}}\,\nrm w{2q}\)^\alpha\,,\quad \alpha=q+1-\zeta\,(q-1) \end{eqnarray*} for any $q\in\big(1,\frac d{d-2}\big)$. With $\mathcal K:=(\frac14\,(q^2-1))^\zeta\,\beta^{1-\zeta}\,\kappa$, the functional \[
\mathsf J[w]:=\frac14\,(q^2-1)\ird{|\nabla w|^2}+\beta\ird{|w|^{q+1}}-\mathcal K\,\mathsf C_{\rm{GN}}^\alpha\(\ird{|w|^{2q}}\)^\frac\alpha{2q} \]
is nonnegative and achieves its minimum at $w_*(x)=(1+|x|^2)^\frac1{1-q}$. Hence we have that \[ \mathsf J[w]\ge\mathsf J[w_*]=0 \] and this inequality is equivalent to \eqref{GN1}, after an optimization under scaling. Notice that $\vartheta=2\,\zeta/\alpha$.
\begin{thm}\label{Thm:GN1} With the above notations and $\varphi$ given by~\eqref{Eqn:VarphiFD}, we have \be{Ineq:GNImproved1}
\mathsf J[w]\ge\varphi\left[\beta\(\ird{|w_*|^{q+1}}-\ird{|w|^{q+1}}\)\right]
\end{equation} for any $w\in\mathrm L^{q+1}({\mathbb R}^d)$ such that $\ird{|\nabla w|^2}<\infty$ and $\ird{|w|^{2q}\,|x|^2}=\ird{w_*^{2q}\,|x|^2}$. \end{thm}
\begin{proof} The reader is invited to check that, with the above notations, \[
B_*-B=\beta\(\ird{|w_*|^{q+1}}-\ird{|w|^{q+1}}\)\,. \] \end{proof}
As a last remark in this section, let us observe that the logarithmic Sobolev inequality appears as a limit case of the entropy -- entropy production inequality, and that~\eqref{Ineq:LogSobEuclideanWeissler} is also obtained by taking the limit as $q\to1$ in Gagliardo-Nirenberg inequalities in~\eqref{GN1}: see \cite{MR1940370} for details. Also, when $d\ge2$, the convexity of $\varphi$ is lost as $q\to\frac d{d-2}$, which corresponds to Sobolev's inequality. This shows the consistancy of our method.
\subsection{A second case: \texorpdfstring{$q<1$}{q<1}}\label{Sec:GN2}
Now we study the case of Gagliardo-Nirenberg inequalities \be{GN2} \nrm{\nabla w}2^\vartheta\,\nrm w{2q}^{1-\vartheta}\ge\mathsf C_{\rm{GN}}\,\nrm w{q+1}
\end{equation} with $q=\frac 1{2\,p-1}<1$, $\vartheta=\frac d{1+q}\,\frac{1-q}{d-q\,(d-2)}$ and we denote by $\nrm w{2q}$ the quantity $\(\ird{|w|^{2q}}\)^\frac1{2q}$ for any $q\in(0,1)$, even for $q<1/2$ (in that case, it is only a semi-norm).
Our elementary estimates have to be adapted. Consider two positive constants $a$ and $b$, with $a>b$. Let us define \[ \eta=\frac b{a-b}\,,\quad\kappa=\(\frac ba\)^{\eta}-\(\frac ba\)^{1+\eta}=\frac{a-b}{b^{-\eta}\,a^{1+\eta}}\,. \] Next let us take three positive numbers, $A$, $B$, and $C$ such that $A^{-\eta}\,B^{1+\eta}\le C$ and consider the function \[ h(\lambda)=\lambda^a\,A-\lambda^b\,B+\kappa\,C\,. \] The function $h$ reaches its minimum at $\lambda=\lambda_*:=\(\frac{b\,B}{a\,A}\)^\frac1{a-b}$ and it is straightforward to check that \[ h(\lambda)\ge h(\lambda_*)=\kappa\(C-A^{-\eta}\,B^{1+\eta}\)\,. \] Using the assumption $A^{-\eta}\,B^{1+\eta}\le C$, we get the estimate \[ A-B+\kappa\,C\ge C^{-\frac1\eta}\,B^{1+\frac1\eta}-B+\kappa\,C=\varphi(B-B_*) \] where \[ B_*:=C\(\frac\eta{1+\eta}\)^\eta \] and \be{Eqn:VarphiPM} \varphi(s)=C^{-\frac1\eta}\,\left[(B_*+s)^{1+\frac1\eta}-B_*^{1+\frac1\eta}\right]-s \end{equation} is a nonnegative strictly convex function such that $\varphi(0)=0$ and $\varphi''(s)>\varphi''(0)>0$ for any $s>0$.
We apply these preliminary computations with \begin{eqnarray*} &&a=\frac dq-(d-2)\,,\quad b=d\,\frac{1-q}{2\,q}\\
&&A=\frac14\,(q^2-1)\ird{|\nabla w|^2}\,,\quad B=\beta\ird{|w|^{q+1}}\,,\quad\beta=\frac{2\,q}{1-q}+d\\ &&C=(\tfrac14\,(q^2-1))^{-\eta}\,\beta^{1+\eta}\,\(\mathsf C_{\rm{GN}}\)^{-(q+1)(1+\eta)}\,\nrm w{2q}^\alpha\,,\quad \alpha=q+1+\eta\,(q-1) \end{eqnarray*}
for any $q\in(0,1)$. With $\mathcal K:=(\tfrac14\,(q^2-1))^{-\eta}\,\beta^{1+\eta}\,\kappa$, the functional \[
\mathsf J[w]:=\frac14\,(q^2-1)\ird{|\nabla w|^2}-\beta\ird{|w|^{q+1}}+\mathcal K\,\(\mathsf C_{\rm{GN}}\)^{-(q+1)(1+\eta)}\(\ird{|w|^{2q}}\)^\frac\alpha{2q} \]
is nonnegative and achieves its minimum at $w_*(x)=(1-|x|^2)_+^\frac1{q-1}$. Hence we have that \[ \mathsf J[w]\ge\mathsf J[w_*]=0 \] and this inequality is equivalent to \eqref{GN2}, after an optimization under scaling. Notice that $\vartheta=\frac{2\,\eta}{(q+1)\,(1+\eta)}$.
\begin{thm}\label{Thm:GN2} With the above notations and $\varphi$ given by~\eqref{Eqn:VarphiPM}, we have \be{Ineq:GNImproved2}
\mathsf J[w]\ge\varphi\left[\beta\(\ird{|w|^{q+1}}-\ird{|w_*|^{q+1}}\)\right]\quad\forall\,w\in\mathrm L^{q+1}({\mathbb R}^d)\quad\mbox{such that}\quad\ird{|\nabla w|^2}<\infty\,. \end{equation} \end{thm}
\begin{proof} The proof is similar to the proof of Theorem~\ref{Thm:GN1} except that the roles of $\ird{|w_*|^{q+1}}$ and $\ird{|w_*|^{2q}}$ are exchanged. \end{proof}
\section{Some consequences for diffusion equations}\label{Sec:Diffusion}
\subsection{Linear case: the Ornstein-Uhlenbeck equation}
Let us consider the Ornstein-Uhlenbeck equation (or backward Kolmogorov equation) \be{Eqn:OU} \frac{\partial f}{\partial t}=\Delta f-x\cdot\nabla f
\end{equation} with initial datum $f_0\in\mathrm L^1_+({\mathbb R}^d,(1+|x|^2)\,d\mu$ and define the \emph{entropy} as \[ \mathcal E[f]:=\irdmu{f\,\log f}\,. \] Using~\eqref{Ineq:LogSobGaussian}, a standard computation shows that a solution $f=(t,\cdot)$ to~\eqref{Eqn:OU} satisfies \[
\frac d{dt}\mathcal E[f]=-\,4\irdmu{|\nabla\sqrt f|^2}\le-\,2\,\mathcal E[f]\,, \] thus proving that \be{Estim:StandardOU} \mathcal E[f(t,\cdot)]\le\mathcal E[f_0]\,e^{-2t}\quad\forall\,t\ge0\,. \end{equation} It is well known that $M=\irdmu{f(t,\cdot)}$ does not depend on $t\ge0$. Since the second moment evolves according to \[
\frac d{dt}\irdmu{f\,|x|^2}=2\irdmu{f\,(d-|x|^2)}\,, \]
if we assume that $\irdmu{f_0\,|x|^2}=d\,M$, then we get also that $\irdmu{f(t,\cdot)\,|x|^2}=d\,M$ for any $t\ge0$.
\begin{thm}\label{Thm:RateLinear} Let $d\ge 1$ and consider a nonnegative solution to \eqref{Eqn:OU} with initial datum $f_0$ such that $\mathcal E[f_0]$ is finite and $\irdmu{f_0\,|x|^2}=d\,\irdmu{f_0}$. Then we have \be{Estim:ImprovedOU} \mathcal E[f(t,\cdot)]\le-\,\frac d2\,\log\left[1-\(1-e^{-\frac 2d\,\mathcal E[f_0]}\)\,e^{-2t}\right]\quad\forall\,t\ge0\,. \end{equation} \end{thm}
\begin{proof} The proof relies on the estimate \[
\frac d{dt}\mathcal E[f]=-\,d\,\irdmu{|\nabla\sqrt f|^2}\ge\frac d4\left[\exp\(\frac 2d\,\mathcal E[f]\)-1\right]\,, \] according to~\eqref{Eqn:LogSobGaussianSimple}.\end{proof} Let us conclude this section on the Ornstein-Uhlenbeck equation with some remarks.\begin{description} \item[(i)] The estimate~\eqref{Estim:ImprovedOU} is better than \eqref{Estim:StandardOU}: if we let $x=e^{-2t}$ and $a=e^{-\frac 2d\,\mathcal E[f_0]}$, then \[ -\,\frac d2\,\log\left[1-\(1-e^{-\frac 2d\,\mathcal E[f_0]}\)\,e^{-2t}\right]\le\mathcal E[f_0]\,e^{-2t} \] for any $t\ge0$ is equivalent to prove that $h(x)=1-a^x-(1-a)\,x$ is nonnegative for any $a\ge0$ and any $x\in(0,1)$. This is indeed the case because $h(0)=h(1)=0$ and $h''(x)=-\,a^x\,(\log x)^2<0$. \item The improvement degenerates as $\mathcal E[f_0]\to0_+$. Indeed, we may observe that, for a given $t\ge0$, \[ -\,\frac d2\,\log\left[1-\(1-e^{-\frac 2d\,\mathcal E_0}\)\,e^{-2t}\right]\sim\mathcal E_0\,e^{-2t}\quad\mbox{as}\quad\mathcal E_0\to0_+\,. \] \item[(ii)] Similar results can be obtained using the \emph{generalized Poincar\'e inequalities for Gaussian measures} of Section~\ref{Sec:Beckner}. With $q=2/p$, we get that \[
\frac d{dt}\irdmu{\frac{f^q-M^q}{q-1}}=-\,\frac4q\irdmu{|\nabla f^{q/2}|^2}\le\Psi_p\(\irdmu{\frac{f^q-M^q}{q-1}}\) \] with $\Psi_p(t):=\max\big\{2,\frac{2\,p}{2-p}\,\varphi_p(\frac{2-p}p\,t)\big\}=\frac{2\,p}{2-p}\,\Phi_p(\frac{2-p}p\,t)$ with $\Phi_p$ as in~\eqref{Phi_p}, and thus get an improvement of the standard estimate \[ \irdmu{\frac{f^q-M^q}{q-1}}\le\irdmu{\frac{f_0^q-M^q}{q-1}}\;e^{-2t}\quad\forall\,t\ge0 \] if $\irdmu{\frac{f_0^q-M^q}{q-1}}<\frac p{2-p}\,x_\star(p)$.
\item[(iii)] None of the above improvements requires that $\irdmu{f\,x}=0$. If this condition is added at $t=0$, it is preserved by the flow corresponding to \eqref{Eqn:OU} and spectral methods allows to prove that \[ \irdmu{\frac{f^q-M^q}{q-1}}\le\irdmu{\frac{f_0^q-M^q}{q-1}}\;e^{-2\lambda t}\quad\forall\,t\ge0 \] where $\lambda$ is the best constant in the inequality \[ \nrmu u2^2-\nrmu up^2\le\frac1\lambda\,\nrmu{\nabla u}2^2 \]
for any $u\in\mathrm H^1({\mathbb R}^d,d\mu)$ such that $\irdmu{u\,(1,x,|x|^2-d)}=(0,0,0)$. According to \cite[Theorem~2.4]{MR2375056}, we know that $\lambda\ge\frac{\lambda_3}{1-(p-1)^{\lambda_3/\lambda_1}}$ where $(\lambda_i)_{i\ge0}$ are the eigenvalues of the Ornstein-Uhlenbeck operator $-\Delta+x\cdot\nabla$. Standard results on the harmonic oscillator allow us to prove that $\lambda_i=i$, where the eigenspaces associated with $i=0$, $i=1$ and $i=2$ are generated respectively by the constants, $x_i$ with $i=1$, $2$,... $d$ and $|x|^2-d$. \end{description}
\subsection{Nonlinear case: the fast diffusion equation}
The inequality $\mathsf J[w]\ge0$ in Section~\ref{Sec:GN1} is also known as the entropy -- entropy-production inequality for the fast diffusion equation, as was shown in \cite{MR1940370}. Also see \cite{MR2065020} for a review on these methods. Here are some details. Let us consider the free energy \[ \mathcal F[v]:=\frac1{p-1}\ird{\(v^p-\mathfrak B^p-\,p\,\mathfrak B^{p-1}\,(v-\mathfrak B)\)} \] where the Barenblatt profile $\mathfrak B$ is defined by \[
\mathfrak B(x)=\(1+|x|^2\)^\frac 1{p-1}\quad\forall\,x\in{\mathbb R}^d\,. \] and has mass \[ M_*:=\ird{\mathfrak B}=\pi^\frac d2\,\frac{\Gamma\left(\frac1{1-p}-\frac d2\right)}{\Gamma\left(\frac1{1-p}\right)}\,. \] Next we can define the generalized Fisher information by \[
\mathcal I[v]:=\frac p{1-p}\ird{v\left|\,\nabla v^{p-1}-2\,x\right|^2} \] and consider the deficit functional \[ \mathcal J[v]:=\mathcal I[v]-4\,\mathcal F[v]\,. \] We may also define the temperature as \[
\Theta[v]:=\frac 1d\,\frac{\ird{|x|^2\,v}}{\ird v}\,. \] It turns out that $\mathcal J[v]=\mathsf J[w]$ if $v^{p-\frac12}=w$ and $q=\frac1{2\,p-1}$. Notice that $q\in\big(1,\frac d{d-2}\big)$ is equivalent to $p\in(p_1,1)$ with $p_1:=\frac{d-1}d$. Recall that $\beta=\frac{2\,q}{q-1}-d=\frac1{1-p}-d$. If $\Theta[v]=\Theta[\mathfrak B]$, we observe that $\mathcal F[v]:=\frac1{p-1}\ird{\(v^p-\mathfrak B^p\)}$ because $\ird{\mathfrak B^{p-1}\,(v-\mathfrak B)}=0$. Theorem~\ref{Thm:GN1} can be rephrased in terms of $v$ as follows.
\begin{cor}\label{Cor:GN1} Let $p\in(p_1,1)$. Assume that $v$ is a nonnegative function in $\mathrm L^1({\mathbb R}^d)$ is such that $v^p$ and $v\,|x|^2$ are both in $\mathrm L^1({\mathbb R}^d)$, and $\nabla v$ is in $\mathrm L^2({\mathbb R}^d)$. With the above notations and $\varphi$ defined by \eqref{Eqn:VarphiFD} we have \[ \mathcal J[v]\ge\varphi\big(\beta\,(1-p)\,\mathcal F[v]\big)\quad\mbox{if}\quad\ird v=M_*\quad\mbox{and}\quad\Theta[v]=\Theta[\mathfrak B]\,. \] \end{cor}
Since $\varphi''(s)\ge\varphi''(0)$ for any admissible $s\ge0$, we have in particular that \[ \mathcal J[v]\ge\kappa\,\big(\mathcal F[v]\big)^2 \] under the assumptions of Corollary~\ref{Cor:GN1}, with $\kappa=\frac 12\,\beta^2\,(1-p)^2\,\varphi''(0)$. Hence Corollary~\ref{Cor:GN1} allows to recover \cite[Theorem~8]{MR3103175} with a much simpler proof.
In a second step, we can get rid of the constraints on the mass and on the second moment. Let us define \[ \Theta_*:=\Theta[\mathfrak B]\,. \] If we write that \[ u(x)=\lambda^{1+\alpha\,d}\,\sigma^{-\frac d2}\,v\(\lambda^\alpha\,x/\sqrt\sigma\)\quad\mbox{with}\quad\lambda=\frac M{M_*} \] for some $\alpha\in{\mathbb R}$ to be determined later and choose $\sigma>0$ such that \[
\sigma=\sigma[u]:=\lambda^{-(1+2\alpha)}\,\frac{\ird{u\,|x|^2}}{\ird{\mathfrak B\,|x|^2}}\,, \] then we can define the corresponding \emph{Barenblatt profile} by \[ \mathfrak B_{M,\sigma}(x)=\lambda^{1+\alpha\,d}\,\sigma^{-\frac d2}\,\mathfrak B\(\lambda^\alpha\,x/\sqrt\sigma\)\quad\forall\,x\in{\mathbb R}^d\,, \] and the relative entropy and the relative Fisher information respectively by \[ \mathcal F_{M,\sigma}[u]:=\frac1{p-1}\ird{\(u^p-\mathfrak B_{M,\sigma}^p-\,p\,\mathfrak B_{M,\sigma}^{p-1}\,(u-\mathfrak B_{M,\sigma})\)} \] and, with $p_c:=\frac{d-2}d$, \[
\mathcal I_{M,\sigma}[u]:=\frac p{1-p}\,\sigma^{\frac d2\,(p-p_c)}\ird{u\left|\,\nabla u^{p-1}-\nabla\mathfrak B_{M,\sigma}^{p-1}\right|^2}\,. \]
The Barenblatt profile $\mathfrak B_{M,1}(x)$ takes the form $(C_M+|x|^2)^{1/(p-1)}$ for some positive constant $C_M=\lambda^\frac{1+\alpha\,d}{p-1}$ if and only if \[ \alpha=\frac{1-p}{2-d\,(1-p)}\,. \] With this choice, the functionals $\mathcal F_{M,\sigma}[u]$ and $\mathcal I_{M,\sigma}[u]$ have the same scaling properties, so that we get \[ \mathcal J_{M,\sigma}[u]:=\mathcal I_{M,\sigma}[u]-4\,\mathcal F_{M,\sigma}[u]=\lambda^\frac{2\,p-\,d\,(1-p)}{2-d\,(1-p)}\,\sigma^{\frac d2\,(1-p)}\,\mathcal J[v]\,. \] By expressing the values of $\lambda$ and $\sigma$ in terms of $M$ and $\Theta[u]$, we have that \[ \lambda=\frac M{M_*}\quad\mbox{and}\quad\sigma=\sigma[u]=\(\frac{M_*}M\)^\frac{2\,(2-p)-\,d\,(1-p)}{2-\,d\,(1-p)}\,\frac{\Theta[u]}{\Theta_*}\,, \] \[ \lambda^\frac{2\,p-\,d\,(1-p)}{2-d\,(1-p)}\,\sigma^{\frac d2\,(1-p)}=\mathsf h(M,\Theta[u])\quad\mbox{with}\quad\mathsf h(M,\Theta):=\(\frac M{M_*}\)^{\frac{d^2\,(1-p)^2-\,2\,d\,(p^2-4\,p+3)+4\,p}{2\,[2-\,d\,(1-p)]}}\,\(\frac\Theta{\Theta_*}\)^{\frac d2\,(1-p)} \] We can now rephrase Corollary~\ref{Cor:GN1} for a general function $u$ as follows.
\begin{cor}\label{Cor:GNUnscaled} Let $p\in(p_1,1)$. Assume that $u$ is a nonnegative function in $\mathrm L^1({\mathbb R}^d)$ is such that $u^p$ and $u\,|x|^2$ are both in $\mathrm L^1({\mathbb R}^d)$, and $\nabla u$ is in $\mathrm L^2({\mathbb R}^d)$. With the same notations as in Corollary~\ref{Cor:GN1} and $\varphi$ defined by \eqref{Eqn:VarphiFD} we have \[ \mathcal J[u]_{M,\sigma}\ge\mathsf h(M,\Theta[u])\,\varphi\(\beta\,(1-p)\,\frac{\mathcal F_{M,\sigma}[u]}{\mathsf h(M,\Theta[u])}\)\quad\mbox{with}\quad\sigma=\sigma[u]\,. \] \end{cor}
The choice $\sigma=\sigma[u]$ is remarkable because \[ \mathcal F_{M,\sigma[u]}[u]=\inf_{\sigma>0}\mathcal F_{M,\sigma}[u]\,, \] so that $\mathfrak B_{M,\sigma[u]}$ is the \emph{best matching Barenblatt profile}, among all Barenblatt profiles with mass $M$ and characteristic scale $\sigma>0$, when measured in relative entropy. See \cite{1004,MR3103175} for more details.
One of the interests of the statement of Corollary~\ref{Cor:GNUnscaled} is that the free energy $\mathcal F_{M,\sigma[u]}[u]$ is an explicit distance to the manifold of the optimal functions for the Gagliardo-Nirenberg inequalities. We have indeed the following Csisz\'ar-Kullback inequality.
\begin{thm}\cite[Theorem~4]{MR3103175}\label{Thm:CK} Let $d\ge 1$, $p\in(d/(d+2),1)$ and assume that $u$ is a non-negative function in $\mathrm L^1({\mathbb R}^d)$ such that $u^p$ and $x\mapsto |x|^2\,u$ are both integrable on ${\mathbb R}^d$. If $\nrm u1=M$, then \[
\mathcal F_{M,\sigma[u]}[u]\ge\frac{p\,\sigma[u]^{\frac d2(1-p)}}{8\ird{\mathfrak B_{M,1}^p}}\(C_M\nrm{u-\mathfrak B_{M,\sigma[u]}}1+\frac1{\sigma[u]}\,\ird{|x|^2\,|u-\mathfrak B_{M,\sigma[u]}|}\)^2\,. \] \end{thm}
The fast diffusion equation written in self-similar variables is \be{FDE} \frac{\partial u}{\partial t}+\nabla\left[v\cdot\(\sigma^{\frac d2\,(p-p_c)}\,\nabla v^{p-1}-2\,x\)\right]=0 \end{equation} and it is equivalent to the fast diffusion equation $\frac{\partial v}{\partial t}=\Delta v^p$ up to a rescaling: see for instance \cite{MR1940370,BBDGV} when $\sigma$ is taken constant and \cite{1004,MR3103175,DTS} when $\sigma=\sigma[u(t,\cdot)]$ depends on $t$. By using Corollary~\ref{Cor:GNUnscaled}, we get that \[ \frac d{dt}\mathcal F_{M,\sigma[u(t,\cdot)]}[u(t,\cdot)]=-\,\mathcal I_{M,\sigma[u(t,\cdot)]}[u(t,\cdot)] \] while a direct computation shows that \[ \frac d{dt}\sigma[u(t,\cdot)]=-\,\frac{2\,(1-p)^2}{p\,M\,\Theta[\mathfrak B_{M,1}]}\,\sigma[u(t,\cdot)]^{\frac d2(p-p_c)}\,\mathcal F_{M,\sigma[u(t,\cdot)]}[u(t,\cdot)]\le0\,. \] Altogether this establishes a faster convergence rate of the solutions towards the Barenblatt profiles than one would get using the entropy -- entropy production inequality $\mathcal J_{M,\sigma[u]}[u]\ge0$, and even a better rate that the one found in \cite{MR3103175,DTS} because $\varphi''(s)>\varphi''(0)$. Details are out of the scope of this paper and a simplified method will appear in \cite{DTS}.
\subsection{Nonlinear case: the porous medium equation}
Now we turn our attention to the inequality $\mathsf J[w]\ge0$ in Section~\ref{Sec:GN2} which was also studied in \cite{MR1940370}. With $p>1$, we may consider the free energy \[
\mathcal F[v]:=\frac1{p-1}\ird{\(v^p-\mathfrak B^p+\,p\,|x|^2\,(v-\mathfrak B)\)} \] for any nonnegative function $v$ such that $\ird v=\ird{\mathfrak B}$, where the Barenblatt profile $\mathfrak B$ is now defined by \[
\mathfrak B(x)=\(1-|x|^2\)_+^\frac 1{p-1}\quad\forall\,x\in{\mathbb R}^d\,. \] and has mass \[ M_*:=\pi^\frac d2\,\frac{\Gamma\left(\frac p{p-1}\right)}{\Gamma\left(\frac p{p-1}+\frac d2\right)}\,. \] Next we can define the generalized Fisher information by \[
\mathcal I[v]:=\frac p{p-1}\ird{v\left|\,\nabla v^{p-1}-2\,x\right|^2} \] and consider the deficit functional \[ \mathcal J[v]:=\mathcal I[v]-4\,\mathcal F[v]\,. \] As before, we may also define the temperature as \[
\Theta[v]:=\frac 1d\,\frac{\ird{|x|^2\,v}}{\ird v}\,. \] It turns out that $\mathcal J[v]=\mathsf J[w]$ if $v^{p-\frac12}=w$ and $q=\frac1{2\,p-1}$. Notice that $q\in(0,1)$ is equivalent to $p\in(1,\infty)$. Recall that $\beta=\frac{2\,q}{1-q}+d=\frac1{p-1}+d$. If $\Theta[v]=\Theta[\mathfrak B]$, we observe that $\mathcal F[v]:=\frac1{p-1}\ird{\(v^p-\mathfrak B^p\)}$. Theorem~\ref{Thm:GN2} can be rephrased in terms of $v$ as follows.
\begin{cor}\label{Cor:GN2} Let $p\in(1,+\infty)$. Assume that $v$ is a nonnegative function in $\mathrm L^1({\mathbb R}^d)$ is such that $v^p$ and $v\,|x|^2$ are both in $\mathrm L^1({\mathbb R}^d)$, and $\nabla v$ is in $\mathrm L^2({\mathbb R}^d)$. With the above notations and $\varphi$ defined by \eqref{Eqn:VarphiPM} we have \[ \mathcal J[v]\ge\varphi\big(\beta\,(p-1)\,\mathcal F[v]\big)\quad\mbox{if}\quad\ird v=M_*\quad\mbox{and}\quad\Theta[v]=\Theta[\mathfrak B]\,. \] \end{cor}
Since $\varphi''(s)\ge\varphi''(0)$ for any admissible $s\ge0$, we have in particular that \[ \mathcal J[v]\ge\kappa\,\big(\mathcal F[v]\big)^2 \] under the assumptions of Corollary~\ref{Cor:GN2}, with $\kappa=\frac 12\,\beta^2\,(1-p)^2\,\varphi''(0)$. The result of Corollary~\ref{Cor:GN2} is new.
In a second step, we can get rid of the constraints on the mass and on the second moment. Let us define \[ \Theta_*:=\Theta[\mathfrak B]\,. \] If we write that \be{uv:rescaling} u(x)=\lambda^{1+\alpha\,d}\,\sigma^{-\frac d2}\,v\(\lambda^\alpha\,x/\sqrt\sigma\)\quad\mbox{with}\quad\lambda=\frac M{M_*} \end{equation} for some $\alpha\in{\mathbb R}$ to be determined later and choose $\sigma>0$ such that \[
\sigma=\sigma[u]:=\lambda^{-(1+2\alpha)}\,\frac{\ird{u\,|x|^2}}{\ird{\mathfrak B\,|x|^2}}\,, \] then we can define the corresponding \emph{Barenblatt profile} by \[ \mathfrak B_{M,\sigma}(x)=\lambda^{1+\alpha\,d}\,\sigma^{-\frac d2}\,\mathfrak B\(\lambda^\alpha\,x/\sqrt\sigma\)\quad\forall\,x\in{\mathbb R}^d\,, \] and the relative entropy and the relative Fisher information respectively by \[
\mathcal F_{M,\sigma}[u]:=\frac1{p-1}\ird{\(u^p-\mathfrak B_{M,\sigma}^p+\,p\,\sigma^{-\frac d2\,(p-p_c)}\,|x|^2\,(u-\mathfrak B_{M,\sigma})\)} \] and \[
\mathcal I_{M,\sigma}[u]:=\frac p{p-1}\,\sigma^{\frac d2\,(p-p_c)}\ird{u\left|\,\nabla u^{p-1}-\,2\,x\,\sigma^{-\frac d2\,(p-p_c)}\right|^2}\,. \]
where $p_c=\frac{d-2}d$. The Barenblatt profile $\mathfrak B_{M,1}(x)$ takes the form $(C_M-|x|^2)_+^{1/(p-1)}$ for some positive constant $C_M=\lambda^\frac{1+\alpha\,d}{p-1}$ if and only if \[ \alpha=-\,\frac{p-1}{2+d\,(p-1)}\,. \] With this choice, the functionals $\mathcal F_{M,\sigma}[u]$ and $\mathcal I_{M,\sigma}[u]$ have the same scaling properties, so that we get \[ \mathcal J_{M,\sigma}[u]:=\mathcal I_{M,\sigma}[u]-4\,\mathcal F_{M,\sigma}[u]=\lambda^\frac{2\,p+\,d\,(p-1)}{2+d\,(p-1)}\,\sigma^{\frac d2\,(1-p)}\,\mathcal J[v]\,. \] By expressing the values of $\lambda$ and $\sigma$ in terms of $M$ and $\Theta[u]$, we have that \[ \lambda=\frac M{M_*}\quad\mbox{and}\quad\sigma=\sigma[u]=\(\frac{M_*}M\)^\frac{2\,(2-p)-\,d\,(1-p)}{2-\,d\,(1-p)}\,\frac{\Theta[u]}{\Theta_*}\,, \] \[ \lambda^\frac{2\,p-\,d\,(1-p)}{2-d\,(1-p)}\,\sigma^{\frac d2\,(1-p)}=\mathsf h(M,\Theta[u])\quad\mbox{with}\quad\mathsf h(M,\Theta):=\(\frac M{M_*}\)^{\frac{d^2\,(1-p)^2-\,2\,d\,(p^2-4\,p+3)+4\,p}{2\,[2-\,d\,(1-p)]}}\,\(\frac\Theta{\Theta_*}\)^{\frac d2\,(1-p)} \] We can now rephrase Corollary~\ref{Cor:GN2} for a general function $u$ as follows.
\begin{cor}\label{Cor:GNUnscaled2} Let $p\in(1,+\infty)$. Assume that $u$ is a nonnegative function in $\mathrm L^1({\mathbb R}^d)$ is such that $u^p$ and $u\,|x|^2$ are both in $\mathrm L^1({\mathbb R}^d)$, and $\nabla u$ is in $\mathrm L^2({\mathbb R}^d)$. With the same notations as in Corollary~\ref{Cor:GN2} and $\varphi$ defined by \eqref{Eqn:VarphiPM} we have \[ \mathcal J[u]_{M,\sigma}\ge\mathsf h(M,\Theta[u])\,\varphi\(\beta\,(p-1)\,\frac{\mathcal F_{M,\sigma}[u]}{\mathsf h(M,\Theta[u])}\)\quad\mbox{with}\quad\sigma=\sigma[u]\,. \] \end{cor}
For the same reason as in the fast diffusion case, the choice $\sigma=\sigma[u]$ is remarkable because \[ \mathcal F_{M,\sigma[u]}[u]=\inf_{\sigma>0}\mathcal F_{M,\sigma}[u]\,, \] so that $\mathfrak B_{M,\sigma[u]}$ is the \emph{best matching Barenblatt profile}, among all Barenblatt profiles with mass $M$ and characteristic scale $\sigma>0$, when measured in relative entropy.
One of the interests of the statement of Corollary~\ref{Cor:GNUnscaled2} is that the free energy $\mathcal F_{M,\sigma[u]}[u]$ is an explicit distance to the manifold of the optimal functions for the Gagliardo-Nirenberg inequalities. Variants of the Pinsker-Csisz\'ar-Kullback inequality can be found in \cite{MR1777035,MR1940370} and allow to control $\nrm{u-\mathfrak B_{M,\sigma[u]}}1$ under additional assumptions. Here we shall give a simpler result which goes as follows.
\begin{thm}\label{Thm:CK2} Let $d\ge 1$, $p\in(1,\infty)$ and assume that $u$ is a non-negative function in $\mathrm L^1({\mathbb R}^d)$ such that $u^p$ and $x\mapsto |x|^2\,u$ are both integrable on ${\mathbb R}^d$. With previous notations, we have that \[ \mathcal F_{M,\sigma[u]}[u]\ge\frac p{p-1}\,\min\{1,p-1\}\,\nrm{u-\mathfrak B_{M,\sigma[u]}}p^p\,. \] \end{thm}
\begin{proof} Let us consider the function \[
\chi(t):=t^p-1-p\,(t-1)-c_p\,|t-1|^p \] with $c_p:=\min\{1,p-1\}$. The reader is invited to check that for some $t_p\in(0,1)$, $\chi''(t)<0$ for any $t\in(0,t_p)$, $\chi''(t)>0$ for any $t\in(t_p,1)\cup(1,+\infty)$. Since $\chi(0)=\chi(1)=0$ and $\chi'(0)=0$, we get that $\chi(t)\ge0$ for any $t\ge0$. As above, let us define $v$ by the rescaling~\eqref{uv:rescaling} and consider \[
\mathcal F[v]=\frac 1{p-1}\int_{|x|\le1}\chi\(\frac v{\mathfrak B}\)\mathfrak B^p\;dx+\frac 1{p-1}\int_{|x|>1}\(u^p-p\,(1-|x|^2)\,u\)\;dx\,. \] It is straightforward to check that \[
\mathcal F[v]=\frac{c_p}{p-1}\int_{|x|\le1}\left|v-\mathfrak B\right|^p\;dx+\frac 1{p-1}\int_{|x|>1}u^p\;dx\ge c_p\,\nrm{v-\mathfrak B}p^p \] and we conclude using the identities \[ \mathcal F_{M,\sigma[u]}[u]=\mathsf h(M,\Theta[u])\,\mathcal F[v]\quad\mbox{and}\quad\nrm{u-\mathfrak B_{M,\sigma[u]}}p^p=\mathsf h(M,\Theta[u])\,\nrm{v-\mathfrak B}p^p\,. \] \end{proof}
The porous medium equation written in self-similar variables is \be{PME} \frac{\partial u}{\partial t}=\nabla\left[v\cdot\(\sigma^{\frac d2\,(p-p_c)}\,\nabla v^{p-1}+2\,x\)\right]=0 \end{equation} and it is equivalent to the porous medium equation $\frac{\partial v}{\partial t}=\Delta v^p$ up to a rescaling: as in \cite{1004,MR3103175,DTS} we may choose $\sigma=\sigma[u(t,\cdot)]$ to depend on $t$. By using Corollary~\ref{Cor:GNUnscaled2}, we get that \[ \frac d{dt}\mathcal F_{M,\sigma[u(t,\cdot)]}[u(t,\cdot)]=-\,\mathcal I_{M,\sigma[u(t,\cdot)]}[u(t,\cdot)] \] while a direct computation shows that \[ \frac d{dt}\sigma[u(t,\cdot)]=-\,\frac{2\,(p-1)^2}{p\,M\,\Theta[\mathfrak B_{M,1}]}\,\sigma[u(t,\cdot)]^{\frac d2(p-p_c)}\,\mathcal F_{M,\sigma[u(t,\cdot)]}[u(t,\cdot)]\le0\,. \] Altogether this establishes a faster convergence rate of the solutions towards the Barenblatt profiles than one would get using the entropy -- entropy production inequality $\mathcal J_{M,\sigma[u]}[u]\ge0$. This is new. More details based on a simpler method will appear in \cite{DTS}.
As a concluding remark, which is valid for the porous medium case, for the fast diffusion case, and also for the linear case of the Ornstein-Uhlenbeck equation, we may observe that two moments are involved while the center of mass $\ird{x\,u}$ is not supposed to be equal to $0$. In that sense our measurement of the distance to the manifold of optimal functions by the deficit functional is an explicit but still a rough estimate, which is definitely not optimal at least in the perturbation regime, that is, close to the optimal functions, which is precisely the regime that was considered in \cite{MR1124290}.
\par
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\begin{spacing}{0.9} \noindent{\small{\bf Acknowlegments.} This work has been partially supported by the projects \emph{STAB}, \emph{NoNAP} and \emph{Kibord} of the French National Research Agency (ANR). J.D.~thanks the Department of Mathematics of the University of Pavia for inviting him and G.~Savar\'e for stimulating discussions. The authors thank F.~Bolley for pointing them a missing reference. \par
\noindent{\small\copyright\,2014 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.}}\end{spacing}
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\begin{center} {\Large\bf Frame difference families and resolvable balanced incomplete block designs\footnote{Supported by NSFC under Grant $11471032$, and Fundamental Research Funds for the Central Universities under Grant $2016$JBM$071$, $2016$JBZ$012$ (T. Feng), NSFC under Grant 11771227, and Zhejiang Provincial Natural Science Foundation of China under Grant LY17A010008 (X. Wang).}}
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Simone Costa$^a$, Tao Feng$^b$, Xiaomiao Wang$^c$\\[2ex] {\footnotesize $^a$Dipartimento DICATAM, Universit\`a degli Studi di Brescia, Via Valotti 9, I-25123 Brescia, Italy}\\ {\footnotesize $^b$Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P. R. China}\\ {\footnotesize $^c$Department of Mathematics, Ningbo University, Ningbo 315211, P. R. China}\\ {\footnotesize [email protected], [email protected], [email protected]} \vskip12pt
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\noindent {\bf Abstract:} Frame difference families, which can be obtained via a careful use of cyclotomic conditions attached to strong difference families, play an important role in direct constructions for resolvable balanced incomplete block designs. We establish asymptotic existences for several classes of frame difference families. As corollaries new infinite families of 1-rotational $(pq+1,p+1,1)$-RBIBDs over $\mathbb{F}_{p}^+ \times \mathbb{F}_{q}^+$ are derived, and the existence of $(125q+1,6,1)$-RBIBDs is discussed. We construct $(v,8,1)$-RBIBDs for $v\in\{624,1576,2976,5720,5776,10200,14176,24480\}$, whose existence were previously in doubt. As applications, we establish asymptotic existences for an infinite family of optimal constant composition codes and an infinite family of strictly optimal frequency hopping sequences.
\noindent {\bf Keywords}: frame difference family; resolvable balanced incomplete block design; strong difference family; partitioned difference family; constant composition code; frequency hopping sequence
\noindent {\bf Mathematics Subject Classification:} 05B30; 94B25
\section{Introduction} Throughout this paper, sets and multisets will be denoted by curly braces $\{\ \}$ and square brackets $[\ ]$, respectively. When we emphasize a set or a multiset is fixed with an ordering, we regard it as a sequence, and denote it by $(\ )$. Every union will be understood as {\em multiset union} with multiplicities of elements preserved. $A{\cup}A{\cup}\cdots{\cup}A$ ($h$ times) will be denoted by $\underline{h}A$. If $A$ and $B$ are multisets defined on a multiplicative group, then $A\cdot B$ denotes the multiset $[ab:a\in A,b\in B]$.
For a positive integer $v$, we abbreviate $\{0,1,\dots,v-1\}$ by $\mathbb{Z}_v$ or $I_v$, with the former indicating that a cyclic group of this order is acting.
A $(v,k,\lambda)$-BIBD $(${\em balanced incomplete block design}$)$ is a pair $(V,\cal{A})$ where $V$ is a set of $v$ {\em points} and $\cal A$ is a collection of $k$-subsets of $X$ $($called {\em blocks}$)$ such that every $2$-subset of $X$ is contained in exactly $\lambda$ blocks of $\cal A$. A $(v,k,\lambda)$-BIBD $(V,\cal{A})$ is said to be {\em resolvable}, or briefly a $(v,k,\lambda)$-RBIBD, if there exists a partition $\cal R$ of $\cal A$ $($called a {\em resolution}$)$ into {\em parallel classes}, each of which is a partition of $V$.
A powerful idea to obtain resolvable designs is given by the use of a special class of relative difference families: frame difference families. This concept was put forward by M. Buratti in 1999 \cite{b99}.
Let $(G,+)$ be an abelian group of order $g$ with a subgroup $N$ of order $n$. A $(G,N,k,\lambda)$ {\em relative difference family} $(DF)$, or $(g,n,k,\lambda)$-DF over $G$ relative to $N$, is a family $\mathfrak{B}=[B_1,B_2,\dots,B_r]$ of $k$-subsets of $G$ such that the list $$\Delta \mathfrak{B}:=\bigcup_{i=1}^r[x-y:x,y\in B_i, x\not=y]=\underline{\lambda}(G\setminus N),$$ i.e., every element of $G\setminus N$ appears exactly $\lambda$ times in the multiset $\Delta \mathfrak{B}$ while it has no element of $N$. The members of $\mathfrak{B}$ are called {\em base blocks} and the number $r$ equals to $\lambda(g-n)/(k(k-1))$. A $(G,\{0\},k,\lambda)$-DF is said to be a {\em difference set} if it contains only one base block, written simply as $(G,k,\lambda)$-DS or $(g,k,\lambda)$-DS over $G$. The {\em complement} of a $(g,k,\lambda)$-DS over $G$ with the base block $B$ is a $(g,g-k,g-2k+\lambda)$-DS over $G$ with the base block $G\setminus B$.
Let $\mathfrak{F}$ be a $(g,n,k,\lambda)$-DF over $G$ relative to $N$. $\mathfrak{F}$ is a {\em frame difference family} $(FDF)$ if it can be partitioned into $\lambda n/(k-1)$ subfamilies $\mathfrak{F}_1,\mathfrak{F}_2,\ldots,\mathfrak{F}_{\lambda n/(k-1)}$ such that each $\mathfrak{F}_i$ has size of $(g-n)/(nk)$, and the union of base blocks in each $\mathfrak{F}_i$ is a system of representatives for the nontrivial cosets of $N$ in $G$. When $\lambda n=k-1$, a $(g,n,k,\lambda)$-FDF is said to be {\em elementary}.
The following proposition reveals the relation between frame difference families and resolvable designs, which can be seen as a corollary by combining the results of Theorem 1.1 in \cite{b99} and Theorem 5.11 in \cite{gm}. We outline the proof for completeness.
\begin{proposition}\label{FDFtoRBIBD}
If there exist a $(G,N,k,\lambda)$-FDF and a $(|N|+1,k,\lambda)$-RBIBD, then there exists a $(|G|+1,k,\lambda)$-RBIBD. \end{proposition}
\noindent{\bf Proof \ } Let $\mathfrak{F}$ be a $(G,N,k,\lambda)$-FDF, which can be partitioned into $\lambda |N|/(k-1)$ subfamilies $\mathfrak{F}_1,\mathfrak{F}_2,\ldots,\mathfrak{F}_{\lambda |N|/(k-1)}$ such that $\bigcup_{F\in \mathfrak{F}_i, h\in N}(F+h)=G\setminus N$ for each $1\leq i\leq \lambda|N|/(k-1)$. Set $$\mathfrak{P}_i=\{F+h:F\in \mathfrak{F}_i, h\in N\}$$
for $1\leq i\leq \lambda|N|/(k-1)$. Let $S$ be a complete system of representatives for the cosets of $N$ in $G$. For each $s\in S$, construct a $(|N|+1,k,\lambda)$-RBIBD on $(N+s)\cup \{\infty\}$, where $\infty\not\in G$. It has $\lambda |N|/(k-1)$ parallel classes, written as $\mathfrak{Q}_{s,i}$, $1\leq i\leq \lambda|N|/(k-1)$. It is readily checked that $(\mathfrak{P}_i+s)\cup \mathfrak{Q}_{s,i}$, $1\leq i\leq \lambda|N|/(k-1)$ and $s\in S$, constitute all parallel classes of a $(|G|+1,k,\lambda)$-RBIBD, which is defined on $G\cup \{\infty\}$. \hphantom{.}
$\Box$\medbreak
An {\em automorphism group} of a $(v,k,\lambda)$-RBIBD $(V,{\cal A})$ with $\cal R$ as its resolution is a group of permutations on $V$ leaving $\cal A$ and $\cal R$ invariant, respectively. A $(v,k,\lambda)$-RBIBD is said to be {\em $1$-rotational} over a group $G$ of order $v-1$ if it admits $G$ as an automorphism group fixing one point and acting sharply transitively on the others.
By revisiting the proof of Proposition \ref{FDFtoRBIBD}, one can have the following proposition.
\begin{proposition}\label{FDFtoRBIBD 1-rotational}
Suppose there exists a $(G,N,k,\lambda)$-FDF. If there is a $1$-rotational $(|N|+1,k,\lambda)$-RBIBD over $N$, then there is a $1$-rotational $(|G|+1,k,\lambda)$-RBIBD over $G$. \end{proposition}
The target of this paper is to construct frame difference families and (1-rotational) resolvable designs via strong difference families. Let $\mathfrak{S}=[F_1,F_2,\dots,F_s]$ with $F_i=(f_{i,0},f_{i,1},\dots,f_{i,k-1})$ for $1\leq i\leq s$, be a family of $s$ multisets of size $k$ defined on a group $(G,+)$ of order $g$. We say that $\mathfrak{S}$ is a $(G,k,\mu)$ {\em strong difference family}, or a $(g,k,\mu)$-SDF over $G$, if the list $$\Delta \mathfrak{S}:=\bigcup_{i=1}^s [f_{i,a}-f_{i,b}: 0\leq a,b\leq k-1, a\not=b]=\underline{\mu} G,$$ i.e., every element of $G$ (0 included) appears exactly $\mu$ times in the multiset $\Delta \mathfrak{S}$. The members of $\mathfrak{S}$ are called {\em base blocks} and the number $s$ equals to $\mu g/(k(k-1))$. Note that $\mu$ is necessarily even since the element $0\in G$ is expressed in even ways as differences in any multiset.
\begin{proposition}\label{prop:nece SDF}
A $(G,k,\mu)$-SDF exists only if $\mu$ is even and $\mu |G|\equiv 0 \pmod{k(k-1)}$. \end{proposition}
The concept of strong difference families was introduced in \cite{b99} and revisited in \cite{bg,m}. It is useful in the constructions of relative difference families and BIBDs (cf. \cite{cfw}), and perfect cycle decompositions (cf. \cite{bcw}). Many direct constructions for RBIBDs in the literature are obtained by the use of certain suitable SDFs explicitly or implicitly (cf. \cite{b97,b99,bf,bz}).
Although many authors have worked on the existence of $(v,8,1)$-RBIBDs, there are still $66$ open cases for small values of $v$ (see Table 4 in \cite{ga} or Table 7.41 in \cite{ag}). In Section 2 we shall show that, with a careful application of cyclotomic conditions attached to a strong difference family, we can establish the existence of three new $(v,8,1)$-RBIBDs for $v\in\{624,1576,2976\}$. Then via known recursive constructions for RBIBDs, we obtain another five new $(v,8,1)$-RBIBDs for $v\in\{5720,5776,10200,14176,24480\}$.
M. Buratti, J. Yan and C. Wang \cite{byw} proved that any $(k-1,k,kt)$-SDF can lead to a $((k-1)p,k-1,k,1)$-FDF for any sufficiently large prime $p$ and $p\equiv kt+1 \pmod{2kt}$. We shall generalize their result in Section 3 (see Theorem \ref{thm:FDF-2}).
In Sections 4 and 5, we shall prove that, if the initial SDF has some particular patterns, then the lower bound on $q$ can be reduced greatly. Theorem \ref{thm:DF-4} generalizes Construction A in \cite{bf}, and Theorems \ref{thm:DF-3} and \ref{thm:Singer DS-2} generalize Construction B in \cite{bf}. As corollaries of Theorems \ref{thm:DF-3}-\ref{thm:Singer DS-2} and \ref{thm:DF-4}, Theorems \ref{thm:BIBD-3}-\ref{thm:BIBD-4} give new 1-rotational $(pq+1,p+1,1)$-RBIBDs. Theorem \ref{6,1RBIBD2} presents a new infinite family of $(v,6,1)$-RBIBDs.
As applications, in Section 6, we derive new optimal constant composition codes and new strictly optimal frequency hopping sequences.
\section{Basic lemma and new $(v,8,1)$-RBIBDs}
Let $q$ be a prime power. As usual we denote by $\mathbb{F}_q$ the finite field of order $q$, by $\mathbb{F}_q^+$ its additive group, by $\mathbb{F}^*_q$ its multiplicative group, by $\mathbb{F}_{q}^{\Box}$ the set of nonzero squares, and by $\mathbb{F}_{q}^{\not\Box}$ nonsquares in $\mathbb{F}_{q}$.
If $q\equiv 1 \pmod{e}$, then $C_0^{e,q}$ will denote the group of nonzero $e$th powers of $\mathbb{F}_q$ and once a primitive element $\omega$ of $\mathbb{F}_q$ has been fixed, we set $C_i^{e,q}=\omega^i\cdot C_0^{e,q}$ for $i=0,1,\ldots,e-1$. We refer to the cosets $C_0^{e,q},C_1^{e,q},\ldots,C_{e-1}^{e,q}$ of $C_0^{e,q}$ in $\mathbb{F}^*_q$ as the {\em cyclotomic classes of index $e$}. Let $A$ be a multisubset of $\mathbb{F}_q^*$. If each cyclotomic coset $C_l^{e,q}$ for $l\in I_e$ contains exactly $\lambda$ elements of $A$, then $A$ is said to be a {\em $\lambda$-transversal} for these cosets. If $A$ is a $1$-transversal, $A$ is often referred to as a {\em representative system for the cosets} of $C_0^{e,q}$ in $\mathbb{F}_q^*$. The following lemma allows us to obtain frame different families by using strong difference families.
\begin{lemma}\label{lem:FDF-2}
Let $q\equiv 1 \pmod {e}$ be a prime power and $d|e$. Let $S$ be a representative system for the cosets of $C_0^{e,q}$ in $C_0^{d,q}$. Let $d(q-1)\equiv 0\pmod{ek}$ and $t=d(q-1)/ek$. Suppose that there exists a $(G,k,kt\lambda)$-$SDF$ $\mathfrak{S}=[F_1,F_2,\ldots, F_n]$, where $\lambda|G|\equiv 0\pmod{k-1}$ and $F_i=(f_{i,0},f_{i,1},\dots,f_{i,k-1})$, $1\leq i\leq n$. If there exists a partition $\mathcal{P}$ of base blocks of $\mathfrak{S}$ into $\lambda|G|/(k-1)$ multisets, each of size $t$, such that one can choose appropriate multiset $[\Phi_1,\Phi_2,\dots,\Phi_n]$ of ordered $k$-subsets of $\mathbb{F}_q^*$ with $\Phi_i=(\phi_{i,0},\phi_{i,1},\dots,\phi_{i,k-1})$, $1\leq i \leq n$, satisfying that \begin{itemize} \item[$(1)$] $\bigcup_{i=1}^n [\phi_{i,a}-\phi_{i,b}: f_{i,a}-f_{i,b}=h, (a,b)\in I_k\times I_k, a\not=b]=C_0^{e,q}\cdot D_h$ for each $h\in G$, where $D_h$ is a $\lambda$-transversal for the cosets of $C_0^{d,q}$ in $\mathbb{F}_q^*$, \item[$(2)$] $\bigcup_{i:F_i\in P} [\phi_{i,a}: a\in I_k]=C_0^{e,q}\cdot E_P$ for each $P \in\mathcal{P}$, where $E_P$ is a representative system for the cosets of $C_0^{d,q}$ in $\mathbb{F}_q^*$, \end{itemize} then $$ \mathfrak{F}=[B_i\cdot \{(1,s)\}: 1\leq i\leq n, s\in S]$$ is a $(G\times \mathbb{F}_q^+,G\times \{0\},k,\lambda)$-FDF, where $B_i=\{(f_{i,0},\phi_{i,0}),(f_{i,1},\phi_{i,1}),\dots,(f_{i,k-1},\phi_{i,k-1})\}$. \end{lemma}
\noindent{\bf Proof \ } Since $n=\lambda kt|G|/(k(k-1))$ and $t=d(q-1)/ek$, we have
$$|\mathfrak{F}|= n\cdot \frac{e}{d}=\frac{\lambda|G|(q-1)}{k(k-1)},$$
which coincides with the number of base blocks of a $(G\times \mathbb{F}_q^+,G\times \{0\},k,\lambda)$-FDF. According to $\mathcal{P}$, we can partition $\mathfrak{F}$ into $\lambda|G|/(k-1)$ subfamilies $$\mathfrak{F}_P=[B_i\cdot \{(1,s)\}: F_i\in P, s\in S],$$
where $P\in \mathcal{P}$. Each subfamily contains $|P|\times|S|=te/d=(q-1)/k$ base blocks. Because of Condition (2), $$\bigcup_{i: F_i\in P}\bigcup_{s\in S} B_i\cdot\{(1,s)\}$$ forms a representative system for the nontrivial cosets of $G\times \{0\}$ in $G\times \mathbb{F}_q^+$. Finally it is readily checked that \begin{eqnarray*} \Delta \mathfrak{F}=\hspace{-3mm}&\hspace{-3mm}& \bigcup_{s\in S} \bigcup^n_{i=1}(\Delta B_i\cdot\{(1,s)\})=\bigcup_{s\in S} \bigcup^n_{i=1} [(f_{i,a}-f_{i,b},(\phi_{i,a}-\phi_{i,b})\cdot s): (a,b)\in I_k\times I_k, a\not=b]\\ =\hspace{-3mm}&\hspace{-3mm}&\bigcup_{s\in S}[ \{h\}\times (C_0^{e,q}\cdot D_h\cdot \{s\}): h\in G ]={\underline\lambda} (G\times \mathbb{F}_q^*). \end{eqnarray*} Therefore $\mathfrak{F}$ is a $(G\times \mathbb{F}_q^+,G\times \{0\},k,\lambda)$-FDF. \hphantom{.}
$\Box$\medbreak
\subsection{A 1-rotational $(624,8,1)$-RBIBD} In this subsection we shall apply Lemma \ref{lem:FDF-2} with $e=q-1$ to present a new RBIBD. Note that when $e=q-1$, $C_0^{e,q}=\{1\}$ and $S=C_0^{d,q}$.
\begin{lemma}\label{FDF1} There exists an elementary $(\mathbb{Z}_{7}\times \mathbb{F}_{89}^+,\mathbb{Z}_{7}\times \{0\},8,1)$-FDF. \end{lemma}
\noindent{\bf Proof \ } Take the $(\mathbb{Z}_{7},8,8)$-SDF containing the unique base block $[0,0,1,1,2,2,4,4]$ as the first components of base blocks of the required FDF. Let $$B=\{(0,1),(0,20),(1,14),(1,58),(2,18),(2,61),(4,26),(4,73)\}.$$
Then applying Lemma \ref{lem:FDF-2} with $G=\mathbb{Z}_{7}$, $q=89$, $e=88$, $d=8$, $k=8$ and $\lambda=1$ which yield $|\mathcal{P}|=1$ and $t=1$, we have $$\mathfrak{F}=[B\cdot (1,s):s\in C_0^{8,89}]$$
forms an elementary $(\mathbb{Z}_{7}\times \mathbb{F}_{89}^+,\mathbb{Z}_{7}\times \{0\},8,1)$-FDF. It is readily checked that each $D_h$, $h\in\mathbb{Z}_{7}$, is a representative system for the cosets of $C_0^{8,89}$ in $\mathbb{F}_{89}^*$ (for example $D_0=\{19,42,43,44,45,46,47,70\}$ and $D_1=\{3,4,13,38,47,49,57,83\}$). The unique $E_P=\{1$, $20,14,58,18,61,26,73\}$ is also a representative system for the cosets of $C_0^{8,89}$ in $\mathbb{F}_{89}^*$. \hphantom{.}
$\Box$\medbreak
\begin{theorem}\label{624} There exists a 1-rotational $(624,8,1)$-RBIBD over $\mathbb{Z}_{623}$. \end{theorem}
\noindent{\bf Proof \ } By Lemma \ref{FDF1} there exists a $(\mathbb{Z}_{7}\times \mathbb{F}_{89}^+,\mathbb{Z}_{7}\times \{0\},8,1)$-FDF. Apply Proposition \ref{FDFtoRBIBD 1-rotational} with a trivial 1-rotational $(8,8,1)$-RBIBD to obtain a 1-rotational $(624,8,1)$-RBIBD over $\mathbb{Z}_{7}\times \mathbb{F}_{89}^+$ that is isomorphic to $\mathbb{Z}_{623}$. \hphantom{.}
$\Box$\medbreak
\subsection{A $(v,8,1)$-RBIBD for $v\in\{1576,2976\}$} In this subsection we shall apply Lemma \ref{lem:FDF-2} with $e=\frac{q-1}{4}$ to present two new RBIBDs. \begin{lemma}\label{lem:SDF(63,8,8)} There exists a $(\mathbb{Z}_{p},8,8)$-SDF for $p\in \{63,119\}$. \end{lemma}
\noindent{\bf Proof \ } For $p=63$, take \begin{center} \begin{tabular}{lll} $F_1=[20,20,-20,-20,29,29,-29,-29]$, \\ $F_2=F_3=F_4=F_5=[0,1,3,7,19,34,42,53]$,\\ $F_6=F_7=F_8=F_9=[0,1,4,6,26,36,43,51]$. \end{tabular} \end{center} \noindent Then the multiset $[F_i: 1\leq i\leq 9]$ forms a $(\mathbb{Z}_{63},8,8)$-SDF.
For $p=119$, take \begin{center} \begin{tabular}{lll} $F_1=[20,20,-20,-20,29,29,-29,-29],$\\ $F_2=F_3=F_4=F_5=[0,1,42,28,101,97,94,114],$\\ $F_6=F_7=F_8=F_9=[0,1,12,23,41,85,104,106],$\\ $F_{10}=F_{11}=F_{12}=F_{13}=[0,2,5,17,37,47,68,76],$\\ $F_{14}=F_{15}=F_{16}=F_{17}=[0,4,10,38,54,62,86,93].$ \end{tabular} \end{center} \noindent Then the multiset $[F_i: 1\leq i\leq 17]$ forms a $(\mathbb{Z}_{119},8,8)$-SDF. \hphantom{.}
$\Box$\medbreak
\begin{lemma}\label{lem:63 25} There exists a $(\mathbb{Z}_{p}\times \mathbb{F}_{25}^+,\mathbb{Z}_{p}\times \{0\},8,1)$-FDF for $p\in \{63,119\}$. \end{lemma} \noindent{\bf Proof \ } Take the $(\mathbb{Z}_{p},8,8)$-SDF from Lemma \ref{lem:SDF(63,8,8)} as the first components of base blocks of the required $(\mathbb{Z}_{p}\times\mathbb{F}_{25}^+,\mathbb{Z}_{p}\times\{0\},8,1)$-FDF. Take $x^2-x+2$ to be a primitive polynomial of degree $2$ over $\mathbb{F}_{5}$ and $\omega$ to be a primitive root in $\mathbb{F}_{25}$. Let $\xi=\omega^6$. For $p=63$, let \begin{center} \begin{tabular}{l} $B_1= \{(20,1),(20,-1),(-20,\xi),(-20,-\xi),(29, \omega),(29,-\omega),(-29,\omega\xi),(-29,-\omega\xi)\},$\\ $B_2= \{(0,1),(1,\omega^{17}),(3,\omega^{12}),(7,\omega^{5}),(19,\omega^{23}),(34,\omega^{11}),(42,\omega^{18}),(53,\omega^{6})\},$\\ $B_6= \{(0,1),(1,\omega^{11}),(4,\omega^{5}),(6,\omega^{12}),(26,\omega^{23}),(36,\omega^{18}),(43,\omega^{17}),(51,\omega^{6})\},$\\ $B_3= B_2\cdot\{(1,-1)\},$\ \ \ \ \ \ \ \ \ \ \ $B_4=B_2\cdot\{(1,\xi)\}$,\ \ \ \ \ \ \ \ \ \ \ $B_5= B_2\cdot\{(1,-\xi)\}$,\\ $B_7=B_6\cdot\{(1,-1)\}$,\ \ \ \ \ \ \ \ \ \ \ $B_8= B_6\cdot\{(1,\xi)\},$\ \ \ \ \ \ \ \ \ \ \ $B_9=B_6\cdot\{(1,-\xi)\}$.\\ \end{tabular} \end{center} For $p=119$, let \begin{center} \begin{tabular}{l} $B_1=\{(20,1),(20,-1),(-20,\xi),(-20,-\xi),(29,\omega),(29,-\omega),(-29,\omega\xi),(-29,-\omega\xi)\},$\\ $B_2=\{(0,\omega),(1,\omega^7),(42,1),(28,\omega^6),(101,\omega^{18}),(97,\omega^{13}),(94,\omega^{12}),(114,\omega^{19})\},$\\ $B_6=\{(0,1),(1,\omega^{12}),(12,\omega),(23,\omega^{18}),(41,\omega^{13}),(85,\omega^7),(104,\omega^6),(106,\omega^{19})\},$\\ $B_{10}=\{(0,\omega),(2,\omega^7),(5,\omega^{12}),(17,\omega^6),(37,\omega^{19}),(47,\omega^{18}),(68,1),(76,\omega^{13})\},$\\ $B_{14}=\{(0,\omega),(4,\omega^4),(10,\omega^{10}),(38,\omega^7),(54,\omega^{22}),(62,\omega^{19}),(86,\omega^{16}),(93,\omega^{13})\}.$\\ $B_3= B_2\cdot\{(1,-1)\},$\ \ \ \ \ \ \ \ \ \ \ $B_4=B_2\cdot\{(1,\xi)\}$,\ \ \ \ \ \ \ \ \ \ \ $B_5= B_2\cdot\{(1,-\xi)\}$,\\ $B_7=B_6\cdot\{(1,-1)\}$,\ \ \ \ \ \ \ \ \ \ \ $B_8= B_6\cdot\{(1,\xi)\},$\ \ \ \ \ \ \ \ \ \ \ $B_9=B_6\cdot\{(1,-\xi)\}$,\\ $B_{11}= B_{10}\cdot\{(1,-1)\},$\ \ \ \ \ \ \ \ \ $B_{12}=B_{10}\cdot\{(1,\xi)\}$,\ \ \ \ \ \ \ \ \ $B_{13}= B_{10}\cdot\{(1,-\xi)\}$,\\ $B_{15}=B_{14}\cdot\{(1,-1)\}$,\ \ \ \ \ \ \ \ \ $B_{16}=B_{14}\cdot\{(1,\xi)\},$\ \ \ \ \ \ \ \ \ $B_{17}=B_{14}\cdot\{(1,-\xi)\}$.\\ \end{tabular} \end{center}
Let $S$ be a representative system for the cosets of $C_0^{6,25}=\{1,-1,\xi,-\xi\}$ in $C_0^{2,25}$. Then, applying Lemma \ref{lem:FDF-2} with $G=\mathbb{Z}_{p}$, $q=25$, $e=6$, $d=2$, $k=8$ and $\lambda=1$ which yield $|\mathcal{P}|=p/7$ and $t=1$, we have a $(\mathbb{Z}_{p}\times \mathbb{F}_{25}^+,\mathbb{Z}_{p}\times \{0\},8,1)$-FDF for $p\in \{63,119\}$. Note that for any $i$, $1 \leq i \leq p/7$, $\bigcup_{s\in S} B_i\cdot\{(1,s)\}$ forms a representative system for the nontrivial cosets of $\mathbb{Z}_{p}\times \{0\}$ in $\mathbb{Z}_{p}\times \mathbb{F}_{25}^+$. \hphantom{.}
$\Box$\medbreak
\begin{theorem}\label{thm:RBIBD(1576)} There exists a $(1576,8,1)$-RBIBD and a $(2976,8,1)$-RBIBD. \end{theorem} \noindent{\bf Proof \ } By Lemma \ref{lem:63 25}, there exists a $(\mathbb{Z}_{p}\times \mathbb{F}_{25}^+,Z_{p}\times \{0\},8,1)$-FDF for $p\in \{63,119\}$. Apply Proposition \ref{FDFtoRBIBD} with a $(p+1,8,1)$-RBIBD, which exists by Table 7.41 in \cite{ag}, to obtain a $(1576,8,1)$-RBIBD and a $(2976,8,1)$-RBIBD. \hphantom{.}
$\Box$\medbreak
\subsection{Five new RBIBDs via recursive constructions}
A {\em transversal design} is a triple $(V,\mathcal{G},\mathcal{B})$, where $V$ is a set of $km$ points, $\mathcal{G}$ is a partition of $V$ into $k$ {\em groups}, each of size $m$, and $\mathcal{B}$ is a set of $k$-subsets $($called {\em blocks}$)$ of $V$ satisfying every pair of $V$ is contained either in exactly one group or in exactly one block, but not both. Such a design is denoted by a TD$(k,m)$.
It is well known that the existence of a TD$(k,m)$ is equivalent to the existence of $k-2$ mutually orthogonal Latin squares of order $m$. A TD$(q+1,q)$ exists for any prime power $q$ (cf. Theorem 6.44 in \cite{StinsonBook}), and a TD$(10,48)$ exists by Theorem 2.1 in \cite{ac}.
\begin{lemma}\label{aGre}{\rm (Lemma 4.9 in \cite{ga})} Suppose there exist a TD$(10,m)$ and a $(56m+8,8,1)$-RBIBD. For any given $0\leq n\leq m$, if there exists a $(56n+8,8,1)$-RBIBD, then there exists a $(56(9m+n)+8,8,1)$-RBIBD. \end{lemma}
\begin{lemma}\label{aGreBis}{\rm (Lemma 4.34 in \cite{ga})} Suppose there exist a TD$(9,8n)$, a $(56n+8,8,1)$-RBIBD and a $(56m+8,8,1)$-RBIBD. Then there exists a $(56(8mn+n)+8,8,1)$-RBIBD. \end{lemma}
\begin{theorem}\label{5776} There exists a $(v,8,1)$-RBIBD for $v\in\{5720,5776,10200,24480\}$. \end{theorem}
\noindent{\bf Proof \ } Apply Lemma \ref{aGreBis} with $m=2$ and $n=6$ to obtain a $(5720,8,1)$-RBIBD, where the needed $(v,8,1)$-RBIBDs for $v\in\{120,344\}$ are from Table 7.41 in \cite{ag}.
For $v\in\{5776,10200,24480\}$, apply Lemma \ref{aGre} with $(m,n)\in\{(11,4),(19,11),(48,5)\}$ to obtain a $(v,8,1)$-RBIBD, where the needed $(624,8,1)$-RBIBD is from Theorem \ref{624} and the needed $(u,8,1)$-RBIBDs for $u\in\{232,288,1072,2696\}$ are from Table 7.41 in \cite{ag}. \hphantom{.}
$\Box$\medbreak
Let $(H,+)$ be an abelian group of order $h$. An $(H,k,\lambda)$ {\em difference matrix} $($briefly, $(H,k,\lambda)$-DM$)$ is a $k\times h\lambda$ matrix $D=(d_{ij})$ with entries from $H$ so that for each $1\leq i<j\leq k$ the multiset $$\{ d_{il}-d_{jl}: 1\leq l\leq h\lambda \}$$ contains every element of $H$ exactly $\lambda$ times. An $(H,k,1)$-DM is {\em homogeneous} if its each row is a permutation of elements of $H$.
The property of a difference matrix is preserved even if one add any element of $H$ to all entries in any row or column of the difference matrix. Then, w.l.o.g., all entries in the first row in a difference matrix are zero. Such a difference matrix is said to be {\em normalized}. Any normalized difference matrix can yield a homogeneous difference matrix by deleting its first row. Thus the existence of a homogeneous $(H,k-1,1)$-DM is equivalent to that of an $(H,k,1)$-DM. The multiplication table for the finite field $\mathbb{F}_q$ is an $(\mathbb{F}_q^+,q,1)$-DM \cite{d}.
The following construction is a variation of standard recursive construction for difference families (see for example Theorem 6.1 in \cite{byw}).
\begin{construction}\label{recursive} Suppose there exists a $(G,N,k,1)$-FDF. If there exists a homogeneous $(H,k,1)$-DM, then there exists a $(G\times H,N\times H,k,1)$-FDF. \end{construction}
\begin{theorem}\label{thm:RBIBD(14176)} There exists a $(14176,8,1)$-RBIBD. \end{theorem}
\noindent{\bf Proof \ } Take a $(\mathbb{Z}_{63}\times \mathbb{F}_{25}^+,Z_{63}\times \{0\},8,1)$-FDF from Lemma \ref{lem:63 25}. Then apply Construction \ref{recursive} with a homogeneous $(\mathbb{F}_{9}^+,8,1)$-DM to obtain a $(\mathbb{Z}_{63}\times \mathbb{F}_{25}^+\times \mathbb{F}_{9}^+,Z_{63}\times \{0\}\times \mathbb{F}_{9}^+,8,1)$-FDF. Finally apply Proposition \ref{FDFtoRBIBD} with a $(568,8,1)$-RBIBD, which exists by Table 7.41 in \cite{ag}, to obtain a $(14176,8,1)$-RBIBD. \hphantom{.}
$\Box$\medbreak
\section{Asymptotic existence of FDFs}
Throughout this paper we always write \begin{eqnarray*}\label{Q(e,m)} Q(d,m)=\frac{1}{4}(U+\sqrt{U^2+4d^{m-1}m})^2, \mbox{ where } U=\sum_{h=1}^m {m \choose h}(d-1)^h(h-1) \end{eqnarray*} for given positive integers $d$ and $m$. The following theorem characterizes existences of elements satisfying certain cyclotomic conditions in a finite field.
\begin{theorem}\label{thm:cyclot bound} {\rm \cite{bp,cj}} Let $q\equiv 1 \pmod{d}$ be a prime power, let $B=\{b_0,b_1,\dots, b_{m-1}\}$ be an arbitrary m-subset of $\mathbb{F}_q$ and let $(\beta_0,\beta_1,\dots,\beta_{m-1})$ be an arbitrary element of $\mathbb{Z}_d^m$. Set $X=\{x\in \mathbb{F}_q: x-b_i\in C^{d,q}_{\beta_i} \mbox{ for } i=0,1,\dots,m-1\}$. Then $X$ is not empty for any prime power $q\equiv 1 \pmod{d}$ and $q>Q(d,m)$. \end{theorem}
Abel and Buratti \cite{ab0} announced Theorem \ref{thm:cyclot bound} without proving it. The case of $m=3$ in Theorem \ref{thm:cyclot bound} was first shown by Buratti \cite{b02}. Then a proof similar to that of $m=3$ allows Chang and Ji \cite{cj}, and Buratti and Pasotti \cite{bp} to generalize this result to any $m$. Theorem \ref{thm:cyclot bound} is derived from Weil's Theorem (see \cite{ln}, Theorem 5.41) on multiplicative character sums and plays an essential role in the asymptotic existence problem for difference families (cf. \cite{cwz}).
The idea of the following lemma is from Theorem 4.1 in \cite{byw}.
\begin{lemma}\label{thm:FDF-1}
If there exists a $(G,k,kt\lambda)$-SDF with $\lambda|G|\equiv 0\pmod{k-1}$, then there exists a $(G\times \mathbb{F}_q^+,G\times \{0\},k,\lambda)$-FDF \begin{itemize} \item for any even $\lambda$ and any prime power $q\equiv 1 \pmod{kt}$ with $q>Q(kt,k)$; \item for any odd $\lambda$ and any prime power $q\equiv kt+1 \pmod{2kt}$ with $q>Q(kt,k)$. \end{itemize} \end{lemma}
\noindent{\bf Proof \ } By assumption one can take a $(G,k,kt\lambda)$-SDF $\mathfrak{S}=[F_{1},F_{2},\dots,F_{n}]$, where $n=t\lambda|G|/(k-1)$ and $F_i=[f_{i,0},f_{i,1},\ldots,f_{i,k-1}]$, $1\leq i\leq n$. To apply Lemma \ref{lem:FDF-2} with $e=q-1$ and $d=kt$, we need to give a partition $\mathcal{P}$ of base blocks of $\mathfrak{S}$ into $\lambda|G|/(k-1)$ multisets, each of size $t$, such that one can choose appropriate multiset $[\Phi_1,\Phi_2,\dots,\Phi_n]$ of ordered $k$-subsets of $\mathbb{F}_q^*$ with $\Phi_i=(\phi_{i,0},\phi_{i,1},\dots,\phi_{i,k-1})$, $1\leq i \leq n$, satisfying that \begin{itemize} \item[$(1)$] $\bigcup_{i=1}^n [\phi_{i,a}-\phi_{i,b}: f_{i,a}-f_{i,b}=h, (a,b)\in I_k\times I_k, a\not=b]=D_h$ for each $h\in G$, \item[$(2)$] $\bigcup_{i: F_i\in P} [\phi_{i,a}: a\in I_k]= E_P$ for each $P \in\mathcal{P}$, \end{itemize} where $D_h$ is a $\lambda$-transversal for the cosets of $C_0^{d,q}$ in $\mathbb{F}_q^*$ and $E_P$ is a representative system for the cosets of $C_0^{d,q}$ in $\mathbb{F}_q^*$.
Thus if one can choose an appropriate mapping $\pi$ acting on symbolic expressions satisfying that \begin{itemize} \item[$(1')$] $\bigcup_{i=1}^n [\pi(\phi_{i,a}-\phi_{i,b}): f_{i,a}-f_{i,b}=h, (a,b)\in I_k\times I_k, a\not=b]=\underline{\lambda}I_d$ for each $h\in G$, \item[$(2')$] $\bigcup_{i:F_i\in P} [\pi(\phi_{i,a}): a\in I_k]=I_d$ for each $P \in\mathcal{P}$. \end{itemize} and can choose appropriate elements of $\Phi_i$, $1\leq i\leq n$, such that these elements are consistent with the mapping $\pi$, i.e., $\pi$ can be seen as a function from $\mathbb{F}_q^*$ to $\mathbb{Z}_d$ satisfying $\pi(x)=\theta$ if $x\in C^{d,q}_\theta$, then one can apply Lemma \ref{lem:FDF-2} to obtain a $(G\times \mathbb{F}_q^+,G\times \{0\},k,\lambda)$-FDF.
This procedure can always be done. First we need to choose an appropriate mapping $\pi$ satisfying Conditions $(1')$ and $(2')$. Condition $(2')$ can be satisfied easily. The key is how to meet Condition $(1')$. Let $G_2$ denote the subgroup of $\{h\in G:2h=0\}$. When $\lambda$ is even, we can specify $\pi$ to satisfy $$\left\{ \begin{array}{ll} \bigcup_{i=1}^n [\pi(\phi_{i,a}-\phi_{i,b}): f_{i,a}-f_{i,b}=h, (a,b)\in I_k\times I_k, a\not=b]=\underline{\lambda}I_d, & h\in G\setminus G_2,\\\\ \bigcup_{i=1}^n [\pi(\phi_{i,a}-\phi_{i,b}): f_{i,a}-f_{i,b}=h, (a,b)\in I_k\times I_k, a<b]=\underline{\frac{\lambda}{2}} I_d, & h\in G_2.\\ \end{array} \right. $$ When $\lambda$ is odd, we can specify $\pi$ to satisfy $$\left\{ \begin{array}{ll} \bigcup_{i=1}^n [\pi(\phi_{i,a}-\phi_{i,b}): f_{i,a}-f_{i,b}=h, (a,b)\in I_k\times I_k, a\not=b]=\underline{\lambda}I_d, & h\in G\setminus G_2,\\\\ \bigcup_{i=1}^n [\pi(\phi_{i,a}-\phi_{i,b}): f_{i,a}-f_{i,b}=h, (a,b)\in I_k\times I_k, a<b]={\underline \lambda} I_{\frac{d}{2}}, & h\in G_2.\\ \end{array} \right. $$ Note that when $\lambda$ is odd, by Proposition \ref{prop:nece SDF}, the existence of the given $(G,k,kt\lambda)$-SDF implies $d=kt$ is even. When $q\equiv d+1 \pmod{2d}$, $-1\in C_{\frac{d}{2}}^{d,q}$.
Once $\pi$ is fixed, one can apply Theorem \ref{thm:cyclot bound} and Lemma \ref{lem:FDF-2} to obtain the required $(G\times \mathbb{F}_q^+,G\times \{0\},k,\lambda)$-FDF. \hphantom{.}
$\Box$\medbreak
\begin{theorem}\label{thm:FDF-2}
If there exists a $(G,k,kt\lambda)$-SDF with $\lambda|G|\equiv 0\pmod{k-1}$, then there exists a $(G\times \mathbb{F}_q^+,G\times \{0\},k,\lambda)$-FDF \begin{itemize} \item for any even $\lambda$ and any prime power $q\equiv 1 \pmod{kt}$ with $q>Q(kt,k)$; \item for any odd $\lambda$ and any prime power $q\equiv krt+1 \pmod{2krt}$ with $q>Q(krt,k)$, where $r$ is any positive integer. \end{itemize} \end{theorem}
\noindent{\bf Proof \ } For even $\lambda$, the conclusion is straightforward by Lemma \ref{thm:FDF-1}. For odd $\lambda$, if a $(G,k,kt\lambda)$-SDF exists with $n$ base blocks, then there exists a $(G,k,krt\lambda)$-SDF with $rn$ base blocks for any positive integer $r$, which means by Lemma \ref{thm:FDF-1} that there exists a $(G\times \mathbb{F}_q^+,G\times \{0\},k,\lambda)$-FDF for any prime power $q\equiv krt+1 \pmod{2krt}$ with $q>Q(krt,k)$. \hphantom{.}
$\Box$\medbreak
\section{FDFs from SDFs with particular patterns}
The application of Theorem \ref{thm:FDF-2} results in a huge lower bound on $q$. To reduce the lower bound, we shall request the initial SDF has some special patterns. If an SDF only contains one base block, then it is referred to as a {\em difference multiset} (cf. \cite{b99}) or a {\em regular difference cover} (cf. \cite{abms}).
\begin{lemma}\label{lem:SDF-Paley}{\rm \cite{b99}} \begin{itemize} \item[$(1)$] Let $p\equiv 3\pmod{4}$ be a prime power. Then ${\underline 2}(\{0\}\cup\mathbb{F}^{\Box}_p)$ is an $(\mathbb{F}_p^+,p+1,p+1)$-SDF $($called Paley difference multiset of the second type$)$.
\item[$(2)$] Let $p$ be an odd prime power. Set $X_1={\underline 2}(\{0\}\cup\mathbb{F}^{\Box}_p)$ and $X_2={\underline 2}(\{0\}\cup\mathbb{F}^{\not \Box}_p)$. Then $[X_1,X_2]$ is an $(\mathbb{F}_p^+,p+1,2p+2)$-SDF $($called Paley strong difference family of the third type$)$.
\item[$(3)$] Given twin prime powers $p>2$ and $p+2$, the set $(\mathbb{F}^{\Box}_p\times \mathbb{F}^{\Box}_{p+2})\cup (\mathbb{F}^{\not\Box}_p\times \mathbb{F}^{\not\Box}_{p+2})\cup(\mathbb{F}_p\times\{0\})$ is a $(p(p+2),\frac{p(p+2)-1}{2},\frac{p(p+2)-3}{4})$-DS over $\mathbb{F}_p^+\times \mathbb{F}_{p+2}^+$. Let $D$ be its complement. Then ${\underline 2}D$ is a $(p(p+2),p(p+2)+1,p(p+2)+1)$ difference multiset $($called twin prime power difference multiset$)$.
\item[$(4)$] Given any prime power $p$ and any integer $m\geq3$, there is a $(\frac{p^m-1}{p-1},\frac{p^{m-1}-1}{p-1},\frac{p^{m-2}-1}{p-1})$ difference set over $\mathbb{Z}_{\frac{p^m-1}{p-1}}$. Let $D$ be its complement. Then ${\underline p}D$ is a $(\frac{p^m-1}{p-1},p^m,p^m(p-1))$ difference multiset $($called Singer difference multiset$)$. \end{itemize} \end{lemma}
By Theorem \ref{thm:FDF-2} we can easily obtain an infinite family of FDFs from each of the SDFs in Lemma \ref{lem:SDF-Paley}. For example, by the second type Paley $(\mathbb{F}_p^+,p+1,p+1)$-SDF, we get
\begin{corollary}\label{cor:Paley DS-1} Let $p\equiv 3\pmod{4}$ be a prime power. Then there exists an $(\mathbb{F}_{p}^+ \times \mathbb{F}_{q}^+,\mathbb{F}_{p}^+\times \{0\},p+1,1)$-FDF for any prime power $q\equiv p+2 \pmod{2(p+1)}$ and $q>Q(p+1,p+1)$. \end{corollary}
The lower bound on $q$ in Corollary \ref{cor:Paley DS-1} is huge even if $p$ is small. For example, if $p=11$, then $Q(12,12)=7.94968\times 10^{27}$. Thus it would be meaningful to develop a new technique to reduce the bound.
\begin{lemma}\label{lem:DF-3} Let $G$ be an additive group of odd order $l$. Suppose that there exists a $(G,l+1,l+1)$-SDF whose unique base block $(f_{0},f_1,\ldots,f_{l})=$ \begin{eqnarray*}\label{2st Paley} (x_0,x_0,x_1,x_1,\ldots,x_{\frac{l-1}{2}},x_{\frac{l-1}{2}}), \end{eqnarray*} where $x_0,x_1,\ldots,x_{(l-1)/2}$ are distinct elements of $G$. Let $q$ be a prime power satisfying $q\equiv 1\pmod{l+1}$ and let $d=(l+1)/2$ . Suppose that one can choose an appropriate multiset $(\phi_{0},\phi_1,\ldots,\phi_{l})=$ \begin{eqnarray*}\label{2st Paley-y} (y_0,-y_0,y_1,-y_1,\ldots,y_{\frac{l-1}{2}},-y_{\frac{l-1}{2}}) \end{eqnarray*} such that $\{y_0,y_1,\ldots,y_{(l-1)/2}\}\subseteq {\mathbb F}_{q}^*$ and for each $h\in G$, \begin{eqnarray*}\label{2st Paley-dh} [\phi_{a}-\phi_{b}:f_{a}-f_{b}=h,(a,b)\in I_{l+1}\times I_{l+1},a\neq b]=\{1,-1\}\cdot D_h, \end{eqnarray*} where $D_h$ is a representative system for the cosets of $C_0^{d,q}$ in $\mathbb{F}_{q}^*$. Let $S$ be a representative system for the cosets of $\{1,-1\}$ in $C_0^{d,q}$. Let $B=\{(f_0,\phi_0),(f_1,\phi_1),\ldots,(f_{l},\phi_{l})\}.$ Then $$\mathfrak{F}=[B\cdot\{(1,s)\}:s\in S]$$ forms an elementary $(G\times \mathbb{F}_{q}^+,G\times \{0\},l+1,1)$-FDF. \end{lemma}
\noindent{\bf Proof \ } Since $d$ is a divisor of $l+1$ and $q\equiv 1\pmod{l+1}$, $d$ is also a divisor of $q-1$. This makes $C_0^{d,q}$ meaningful. The assumption $q\equiv 1\pmod{l+1}$ ensures $-1\in C_0^{d,q}$. Since $q$ is odd, $y_i\neq -y_i$ for any $0\leq i\leq (l-1)/2$, $B\cdot\{(1,s)\}$ is a set of size $l+1$ for any $s\in S$. Then applying Lemma \ref{lem:FDF-2} with $e=(q-1)/2$, $d=(l+1)/2$, $k=l+1$ and $\lambda=1$ which yield $|\mathcal{P}|=1$ and $t=1$, we have a $(G\times \mathbb{F}_{q}^+,G\times \{0\},l+1,1)$-FDF. Note that $\{1,-1\}=C_0^{(q-1)/2,q}$ and $\{y_0,-y_0,y_1,-y_1,\ldots,y_{\frac{l-1}{2}},-y_{\frac{l-1}{2}}\}=\{1,-1\}\cdot \frac{1}{2}\cdot D_0$. \hphantom{.}
$\Box$\medbreak
\begin{lemma}\label{lem:DF-3-Dh} Follow the notation in Lemma $\ref{lem:DF-3}$. \begin{itemize} \item[$(1)$] W.l.o.g., $D_0=2\cdot\{y_0,y_1,\ldots,y_{(l-1)/2}\}$. \item[$(2)$] For each $h\in G\setminus\{0\}$, let $$T_h=[\phi_{a}-\phi_{b}:f_{a}-f_{b}=h,(a,b)\in I_{l+1}\times I_{l+1},a\neq b].$$ Then $T_h=\{1,-1\}\cdot D_h$ for some $D_h\subset \mathbb{F}_{q}$ and the size of $D_h$ is $(l+1)/2$. Furthermore, $D_h=D_{-h}$ and w.l.o.g., $D_h$ consists of elements of type $y_i\pm y_j$. \item[$(3)$] Let $T$ be a representative system for the cosets of $\{1,-1\}$ in $G\setminus\{0\}$. Any element of type $y_i\pm y_j$ must be contained in a unique $D_h$ for some $h\in T$ $($note that the term ``element'' here is a symbolic expression; for example $y_1-y_2$ and $y_3+y_4$ are different element but they may have the same value$)$. \end{itemize}\end{lemma}
\noindent{\bf Proof \ } The verification is straightforward. \hphantom{.}
$\Box$\medbreak
Begin with SDFs from Lemma \ref{lem:SDF-Paley}(1), (3) and (4), and then apply Lemma \ref{lem:DF-3}, \ref{lem:DF-3-Dh}, and Theorem \ref{thm:cyclot bound}. We have the following theorems. Note that we take $p=2$ when use Lemma \ref{lem:SDF-Paley}(4).
\begin{theorem}\label{thm:DF-3} Let $p\equiv 3\pmod{4}$ be a prime power. Then there exists an elementary $(\mathbb{F}_{p}^+ \times \mathbb{F}_{q}^+,\mathbb{F}_{p}^+\times \{0\},p+1,1)$-FDF for any prime power $q\equiv 1 \pmod{p+1}$ and $q>Q((p+1)/2,p)$. \end{theorem}
\begin{theorem}\label{thm:Twin DS-2} Let $p$ and $p+2$ be twin prime powers satisfying $p>2$. Then there exists an elementary $(\mathbb{F}_{p}^+ \times \mathbb{F}_{p+2}^+\times\mathbb{F}_{q}^+,\mathbb{F}_{p}^+ \times \mathbb{F}_{p+2}^+\times \{0\},p(p+2)+1,1)$-FDF for any prime power $q\equiv 1 \pmod{p(p+2)+1}$ and $q>Q((p(p+2)+1)/2,p(p+2))$. \end{theorem}
\begin{theorem}\label{thm:Singer DS-2} Let $m\geq3$ be an integer. Then there exists an elementary $(\mathbb{Z}_{2^m-1} \times \mathbb{F}_{q}^+,\mathbb{Z}_{2^m-1} \times \{0\},2^m,1)$-FDF for any prime power $q\equiv 1 \pmod{2^{m}}$ and $q>Q(2^{m-1},2^{m}-1)$. \end{theorem}
Compared with Corollary \ref{cor:Paley DS-1}, Theorem \ref{thm:DF-3} not only reduce the lower bound on $q$ but also relax the congruence condition on $q$. Take for example $p=11$. By Theorem \ref{thm:DF-3} the bound is $Q(6,11)=8.77844\times 10^{18}$, which is much smaller than $Q(12,12)$.
Theorems \ref{thm:DF-3} and \ref{thm:Singer DS-2} can be seen as a generalization of M. Buratti and N. Finizio's construction in \cite{bf} for $(\mathbb{Z}_{7} \times \mathbb{Z}_{q},\mathbb{Z}_{7}\times \{0\},8,1)$-FDFs, where $q$ is a prime.
\begin{lemma}\label{lem:DF-4} Let $p\equiv 1\pmod{4}$ be a prime power and $d=p+1$. Let $q$ be a prime power satisfying $q\equiv 1\pmod{2p+2}$. Let $\omega$ be a generator of $\mathbb{F}_{p}$. Take the third type Paley $(\mathbb{F}_{p}^+,p+1,2p+2)$-SDF from Lemma $\ref{lem:SDF-Paley}(2)$ whose base blocks are: \begin{center} \begin{tabular}{l} $(f_{10},f_{11},\ldots,f_{1p})=(0,0,\omega^2,\omega^2,\omega^4,\omega^4,\ldots,\omega^{p-1},\omega^{p-1}),$\\ $(f_{20},f_{21},\ldots,f_{2p})=(0,0,\omega,\omega,\omega^3,\omega^3,\ldots,\omega^{p-2},\omega^{p-2}).$ \end{tabular} \end{center} Suppose that one can choose appropriate multisets \begin{center} \begin{tabular}{l} $(\phi_{10},\phi_{11},\ldots,\phi_{1p})=(y_0,-y_0,y_1,-y_1,y_2,-y_2,\ldots,y_{\frac{p-1}{2}},-y_{\frac{p-1}{2}}),$\\ $(\phi_{20},\phi_{21},\ldots,\phi_{2p})=(y_{\frac{p+1}{2}},-y_{\frac{p+1}{2}},y_{\frac{p+3}{2}},-y_{\frac{p+3}{2}},\ldots,y_{p},-y_{p}),$ \end{tabular} \end{center} such that $\{y_0,y_1,\ldots,y_{p}\}\subseteq {\mathbb F}_{q}^*$ and for each $h\in \mathbb{F}_{p}$, \begin{eqnarray*}\label{2st Paley-dh} \bigcup_{i=1}^2[\phi_{ia}-\phi_{ib}:f_{ia}-f_{ib}=h,(a,b)\in I_{p+1}\times I_{p+1},a\neq b]=\{1,-1\}\cdot D_h, \end{eqnarray*} where $D_h$ is a representative system for the cosets of $C_0^{d,q}$ in $\mathbb{F}_{q}^*$. Let $S$ be a representative system for the cosets of $\{1,-1\}$ in $C_0^{d,q}$. Let $B_i=\{(f_{i0},\phi_{i0}),(f_{i1},\phi_{i1}),\ldots,(f_{ip},\phi_{ip})\},$ where $i=1,2$. Then $$\mathfrak{F}=\bigcup_{i=1}^2[B_i\cdot\{(1,s)\}:s\in S]$$ forms an elementary $(\mathbb{F}_{p}^+\times \mathbb{F}_{q}^+,\mathbb{F}_{p}^+\times \{0\},p+1,1)$-FDF. \end{lemma}
\noindent{\bf Proof \ } Applying Lemma \ref{lem:FDF-2} with $G=\mathbb{F}_{p}^+$, $e=(q-1)/2$, $d=p+1$, $k=p+1$ and $\lambda=1$ which yield $|\mathcal{P}|=1$ and $t=2$, we obtain a $(\mathbb{F}_{p}^+\times \mathbb{F}_{q}^+,\mathbb{F}_{p}^+\times \{0\},p+1,1)$-FDF. Note that $\{1,-1\}=C_0^{(q-1)/2,q}$ and $\{y_0,-y_0,y_1,-y_1,\ldots,y_p,-y_p\}=\{1,-1\}\cdot \frac{1}{2}\cdot D_0$. \hphantom{.}
$\Box$\medbreak
\begin{lemma}\label{lem:DF-4-Dh} Follow the notation in Lemma $\ref{lem:DF-4}$. \begin{itemize} \item[$(1)$] W.l.o.g., $D_0=2\cdot\{y_0,y_1,y_2,\ldots,y_{p}\}$. \item[$(2)$] For each $h\in \mathbb{F}_{p}^*$, let $$T_h=\bigcup_{i=1}^2[\phi_{ia}-\phi_{ib}:f_{ia}-f_{ib}=h,(a,b)\in I_{p+1}\times I_{p+1},a\neq b].$$ Then $T_h=\{1,-1\}\cdot D_h$ for some $D_h\subset \mathbb{F}_{q}$ and the size of $D_h$ is $p+1$. Furthermore, $D_h=D_{-h}$ and w.l.o.g., $D_h$ consists of elements of type $y_i\pm{y_j}$. \item[$(3)$] Let $T$ be a representative system for the cosets of $\{1,-1\}$ in $\mathbb{F}_{p}^*$. Any element of type $y_i\pm{y_j}$ must be contained in a unique $D_h$ for some $h\in T$. \end{itemize}\end{lemma}
\noindent{\bf Proof \ } The verification is straightforward. \hphantom{.}
$\Box$\medbreak
Combining the results of Lemmas \ref{lem:DF-4} and \ref{lem:DF-4-Dh}, and then applying Theorem \ref{thm:cyclot bound}, we have
\begin{theorem}\label{thm:DF-4} Let $p\equiv 1\pmod{4}$ be a prime power. Then there exists an elementary $(\mathbb{F}_{p}^+ \times \mathbb{F}_{q}^+,\mathbb{F}_{p}^+\times \{0\},p+1,1)$-FDF for any prime power $q\equiv 1 \pmod{2p+2}$ and $q>Q(p+1,p)$. \end{theorem}
Theorem \ref{thm:DF-4} can be seen as a generalization of M. Buratti and N. Finizio's construction in \cite{bf} for $(\mathbb{Z}_{5} \times \mathbb{Z}_{q},\mathbb{Z}_{5}\times \{0\},6,1)$-FDFs, where $q$ is a prime.
Start from the FDFs in Theorems \ref{thm:DF-3}, \ref{thm:Twin DS-2}, \ref{thm:Singer DS-2} and \ref{thm:DF-4}. Then apply Proposition \ref{FDFtoRBIBD 1-rotational} with a trivial 1-rotational $(k,k,1)$-RBIBD. We obtain the following theorems.
\begin{theorem}\label{thm:BIBD-3} There exists a 1-rotational $(pq+1,p+1,1)$-RBIBD over $\mathbb{F}_{p}^+ \times \mathbb{F}_{q}^+$ for any prime powers $p$ and $q$ with $p\equiv 3 \pmod{4}$, $q\equiv 1\pmod{p+1}$ and $q>Q((p+1)/2,p)$. \end{theorem}
\begin{theorem}\label{thm:Twin BIBD-2} Let $p$ and $p+2$ be twin prime powers satisfying $p>2$. There exists a 1-rotational $(p(p+2)q+1,p(p+2)+1,1)$-RBIBD over $\mathbb{F}_{p}^+ \times\mathbb{F}_{p+2}^+ \times \mathbb{F}_{q}^+$ for any prime power $q\equiv 1 \pmod{p(p+2)+1}$ and $q>Q((p(p+2)+1)/2,p(p+2))$. \end{theorem}
\begin{theorem}\label{thm:Singer BIBD-2} There exists a 1-rotational $((2^m-1)q+1,2^m,1)$-RBIBD over $\mathbb{Z}_{2^m-1} \times \mathbb{F}_{q}^+$ for any integer $m\geq3$ and any prime power $q\equiv 1\pmod{2^m}$ and $q>Q(2^{m-1},2^m-1)$. \end{theorem}
\begin{theorem}\label{thm:BIBD-4} There exists a 1-rotational $(pq+1,p+1,1)$-RBIBD over $\mathbb{F}_{p}^+ \times \mathbb{F}_{q}^+$ for any prime powers $p$ and $q$ with $p\equiv 1 \pmod{4}$, $q\equiv 1\pmod{2p+2}$ and $q>Q(p+1,p)$. \end{theorem}
\section{A family of RBIBDs with block size $6$}
\begin{lemma}\label{125SDF} There exists a $(\mathbb{Z}_{125},6,6)$-SDF. \end{lemma}
\noindent{\bf Proof \ } Take \begin{center} \begin{tabular}{ll} $A_1=[0,0,19,19,71,71]$,& $A_2=A_3=[0,10,28,51,78,97]$,\\ $A_4=A_5=[0,3,62,75,86,110]$, & $A_6=A_7=[0,5,12,58,70,112],$\\ $A_8=A_{9}=[0,7,27,44,70,96]$, & $A_{10}=A_{11}=[0,1,42,93,85,45],$\\ $A_{12}=A_{13}=[0,1,100,104,109,88]$, & $A_{14}=A_{15}=[0,1,90,81,21,32]$,\\ $A_{16}=A_{17}=[0,3,16,40,46,50]$, & $A_{18}=A_{19}=[0,2,7,29,35,68],$\\ $A_{20}=A_{21}=[0,2,8,57,102,116]$, & $A_{22}=A_{23}=[0,2,22,32,36,96],$\\ $A_{24}=A_{25}=[0,8,23,38,72,86].$ & \end{tabular} \end{center} Then the multiset $[A_i: 1\leq i \leq 25]$ forms a $(\mathbb{Z}_{125}, 6,6)$-SDF. \hphantom{.}
$\Box$\medbreak
Applying Theorem \ref{thm:FDF-2} with a $(\mathbb{Z}_{125},6,6)$-SDF, we can obtain a $(\mathbb{Z}_{125}\times \mathbb{F}_q^+,\mathbb{Z}_{125}\times \{0\},6,1)$-FDF for any prime power $q\equiv 7 \pmod{12}$ and $q>Q(6,6)$. But the bound $Q(6,6)=3.4829\times 10^{10}$ is a little big. We shall reduce the bound by supplying a refined construction in this section.
\begin{lemma}\label{125FDF} Let $q\equiv 7 \pmod{12}$ be a prime power. Take the $(\mathbb{Z}_{125},6,6)$-SDF given in Lemma $\ref{125SDF}$, whose base blocks are $A_1,A_2,\ldots,A_{25}$. Suppose one can choose appropriate multisets $$C_1=(y_{11},-y_{11},y_{12},-y_{12},y_{13},-y_{13}),$$ $$C_{2i}=(y_{2i,1},-y_{2i,1},y_{2i,2},-y_{2i,2},y_{2i,3},y_{2i,4}),\hspace{0.5 cm}C_{2i+1}=-C_{2i},$$ where $1\leq i\leq 12$, such that each $C_j$, $1\leq j\leq 25$, is a representative system for the cosets of $C_0^{6,q}$ in $\mathbb{F}_q^*$. Furthermore, write $A_j=(a_{j1},a_{j2},a_{j3},a_{j4},a_{j5},a_{j6})$, $C_j=(c_{j1},c_{j2},c_{j3},c_{j4},c_{j5},c_{j6})$, and $$B_j=\{(a_{j1},c_{j1}),(a_{j2},c_{j2}),(a_{j3},c_{j3}),(a_{j4},c_{j4}),(a_{j5},c_{j5}),(a_{j6},c_{j6})\},$$ where $1\leq j\leq 25$. Set $$\bigcup_{j=1}^{25} \Delta B_j=\bigcup_{l\in \mathbb{Z}_{125}} \{l\}\times \Delta_l.$$ If for any $l\in \mathbb{Z}_{125}$, $\Delta_l$ is a representative system for the cosets of $C_0^{6,q}$ in $\mathbb{F}_q^*$, then $$\mathfrak{F}=[B_j\cdot \{(1,\alpha)\}: \alpha\in C_0^{6,q}, 1\leq j\leq 25]$$ forms a $(\mathbb{Z}_{125}\times \mathbb{F}_{q}^+,\mathbb{Z}_{125}\times \{0\},6,1)$-FDF. \end{lemma}
\noindent{\bf Proof \ } Apply Lemma \ref{lem:FDF-2} with $G=\mathbb{Z}_{125}$, $e=q-1$, $d=6$, $k=6$ and $\lambda=1$ which yield $|\mathcal{P}|=25$ and $t=1$ to obtain the required a $(\mathbb{Z}_{125}\times \mathbb{F}_{q}^+,\mathbb{Z}_{125}\times \{0\},6,1)$-FDF. \hphantom{.}
$\Box$\medbreak
\begin{lemma}\label{lem:125p} Follow the notation in Lemma $\ref{125FDF}$. \begin{itemize} \item[$(1)$] $C_1=\{1,-1\}\cdot \{y_{11},y_{12},y_{13}\}.$ \item[$(2)$] For $1\leq i\leq 12$, $C_{2i}=(\{1,-1\}\cdot\{y_{2i,1},y_{2i,2}\})\cup\{y_{2i,3},y_{2i,4}\}$ and $C_{2i+1}=(\{1,-1\}\cdot \{y_{2i,1},y_{2i,2}\})\cup\{-y_{2i,3},-y_{2i,4}\}$. \item[$(3)$] For each $l\in \mathbb{Z}_{125}$, $\Delta_l=\Delta_{-l}=\{1,-1\}\cdot D_l$ for some $D_l\subset \mathbb{F}_q$ and the size of $D_l$ is $3$. \item[$(4)$] W.l.o.g., $D_0=2\cdot\{y_{11},y_{12},y_{13}\}$. \item[$(5)$] For each $l\in \mathbb{Z}_{125}^*$, w.l.o.g., $D_l$ consists of elements having the following types
\begin{center}
\begin{tabular}{l}
$(I)$ $y_{1,t_1}\pm y_{1,t_2}$ for some $1\leq t_1\neq t_2\leq 3$; \\
$(II)$ $y_{2i,r}\pm y_{2i,s}$ for some $1\leq i\leq 12$, $r\in\{3,4\}$ and $s\in\{1,2\}$;\\
$(III)$ $y_{2i,4}\pm y_{2i,3}$ for some $1\leq i\leq 12$;\\
$(IV)$ $y_{2i,2}\pm y_{2i,1}$ for some $1\leq i\leq 12$; \\
$(V)$ $2y_{2i,s}$ for some $1\leq i\leq 12$ and $s\in\{1,2\}$.
\end{tabular}
\end{center} \item[$(6)$] Any element of Types $(I)$, $(II)$, $(III)$ and $(V)$ is contained in a unique $D_l$ for some $1\leq l\leq 62$. \item[$(7)$] Any element of Type $(IV)$ is contained in exactly two different $D_l$'s, say $D_{l_1}$ and $D_{l_2}$, for some $1\leq l_1\neq l_2\leq 62$. \end{itemize} \end{lemma}
\noindent{\bf Proof \ } It is readily checked that $(1)$-$(6)$ hold. The verification for $(7)$ is a little more complicated, which relies heavily on the given $(\mathbb{Z}_{125},6,6)$-SDF. For example, since $B_2=\{(0,y_{21}),(10,-y_{21}),(28,y_{22}),(51,-y_{22}),(78,y_{23}),(97,y_{24})\}$, we have $y_{22}-y_{21}\in D_{28}$ and $D_{41}$; $y_{22}+y_{21}\in D_{18}$ and $D_{51}$. By tedious calculation, one can check $(7)$. For convenience, we list each $D_l$ explicitly for $0\leq l\leq 62$ in Table \ref{tab3}. \hphantom{.}
$\Box$\medbreak
\begin{table}[t]{\tabcolsep 0.05in {\small \begin{tabular}{ll} $D_0=[2y_{1,1},2y_{1,2},2y_{1,3}],$& $D_1=[2y_{10,1},2y_{12,1},2y_{14,1}],$\\ $D_2=[2y_{18,1},2y_{20,1},2y_{22,1}],$& $D_3=[2y_{4,1},y_{10,4}-y_{10,2},2y_{16,1}],$\\ $D_4=[2y_{12,2},y_{16,4}-y_{16,3},y_{22,3}+y_{22,2}],$ & $D_5=[2y_{6,1},y_{12,3}+y_{12,2},y_{18,2}+y_{18,1}],$\\ $D_6=[y_{16,3}+y_{16,2},y_{18,3}+y_{18,2},y_{20,2}+y_{20,1}],$ & $D_7=[y_{6,2}+y_{6,1},2y_{8,1},y_{18,2}-y_{18,1}],$\\ $D_8=[y_{10,3}+y_{10,2},y_{20,2}-y_{20,1},2y_{24,1}],$ & $D_9=[y_{12,3}-y_{12,2},2y_{14,2},y_{20,4}-y_{20,1}],$\\ $D_{10}=[2y_{2,1},y_{16,4}+y_{16,2},2y_{22,2}],$ & $D_{11}=[y_{4,3}+y_{4,2},y_{14,4}-y_{14,3},y_{20,4}+y_{20,1}],$\\ $D_{12}=[y_{6,2}-y_{6,1},y_{6,3}+y_{6,2},y_{12,4}-y_{12,2}],$ & $D_{13}=[2y_{4,2},y_{6,4}-y_{6,1},y_{16,2}+y_{16,1}],$\\ $D_{14}=[y_{20,4}-y_{20,3},y_{22,3}-y_{22,2},y_{24,4}-y_{24,3}],$ & $D_{15}=[y_{4,4}-y_{4,1},2y_{24,2},y_{24,2}+y_{24,1}],$\\ $D_{16}=[y_{12,3}-y_{12,1},y_{12,4}+y_{12,2},y_{16,2}-y_{16,1}],$ & $D_{17}=[2y_{8,2},y_{12,3}+y_{12,1},y_{20,4}-y_{20,2}],$\\ $D_{18}=[y_{2,2}+y_{2,1},y_{4,4}+y_{4,1},y_{6,4}+y_{6,1}],$ & $D_{19}=[y_{1,2}+y_{1,1},y_{1,2}-y_{1,1},y_{2,4}-y_{2,3}],$\\ $D_{20}=[y_{8,2}+y_{8,1},y_{14,3}+y_{14,1},y_{22,2}+y_{22,1}],$ & $D_{21}=[y_{12,2}+y_{12,1},y_{12,4}-y_{12,3},y_{14,3}-y_{14,1}],$\\ $D_{22}=[y_{12,2}-y_{12,1},2y_{18,2},y_{22,2}-y_{22,1}],$ & $D_{23}=[2y_{2,2},y_{20,3}-y_{20,1},y_{24,2}-y_{24,1}],$\\ $D_{24}=[y_{4,3}-y_{4,2},y_{4,4}-y_{4,3},2y_{16,2}],$ & $D_{25}=[y_{6,4}-y_{6,2},y_{12,2}-y_{12,1},y_{20,3}+y_{20,1}],$\\ $D_{26}=[y_{8,3}+y_{8,2},y_{8,4}-y_{8,3},y_{12,2}+y_{12,1}],$ & $D_{27}=[y_{2,3}+y_{2,2},y_{8,2}-y_{8,1},y_{18,2}-y_{18,1}],$\\ $D_{28}=[y_{2,2}-y_{2,1},y_{2,4}-y_{2,1},y_{18,3}-y_{18,2}],$ & $D_{29}=[y_{8,4}-y_{8,1},y_{18,2}+y_{18,1},y_{22,4}-y_{22,1}],$\\ $D_{30}=[y_{16,3}-y_{16,2},y_{22,2}-y_{22,1},y_{24,2}-y_{24,1}],$ & $D_{31}=[y_{14,4}+y_{14,1},y_{20,3}-y_{20,2},y_{22,4}+y_{22,1}],$\\ $D_{32}=[y_{10,2}+y_{10,1},y_{14,4}-y_{14,1},y_{22,2}+y_{22,1}],$ & $D_{33}=[y_{10,2}-y_{10,1},y_{18,3}+y_{18,1},y_{18,4}-y_{18,3}],$\\ $D_{34}=[y_{16,4}-y_{16,2},y_{22,3}+y_{22,1},y_{24,3}+y_{24,2}],$ & $D_{35}=[y_{4,4}+y_{4,2},y_{14,2}-y_{14,1},y_{18,3}-y_{18,1}],$\\ $D_{36}=[y_{8,4}+y_{8,1},y_{14,2}+y_{14,1},y_{22,3}-y_{22,1}],$ & $D_{37}=[y_{8,2}-y_{8,1},y_{12,4}-y_{12,1},y_{16,2}-y_{16,1}],$\\ $D_{38}=[y_{2,4}+y_{2,1},y_{12,4}+y_{12,1},y_{24,2}+y_{24,1}],$ & $D_{39}=[y_{4,3}-y_{4,1},y_{18,4}+y_{18,2},y_{24,4}-y_{24,1}],$\\ $D_{40}=[y_{10,3}-y_{10,1},y_{10,4}-y_{10,3},y_{16,2}+y_{16,1}],$ & $D_{41}=[y_{2,2}-y_{2,1},y_{10,2}+y_{10,1},y_{10,3}+y_{10,1}],$\\ $D_{42}=[y_{4,3}+y_{4,1},y_{6,4}-y_{6,3},y_{10,2}-y_{10,1}],$ & $D_{43}=[y_{8,3}-y_{8,2},y_{10,3}-y_{10,2},y_{16,3}+y_{16,1}],$\\ $D_{44}=[y_{8,2}+y_{8,1},y_{10,4}+y_{10,1},y_{14,2}+y_{14,1}],$ & $D_{45}=[y_{10,4}-y_{10,1},y_{14,2}-y_{14,1},y_{20,3}+y_{20,2}],$\\ $D_{46}=[y_{2,4}+y_{2,2},2y_{6,2},y_{16,3}-y_{16,1}],$ & $D_{47}=[y_{2,3}-y_{2,1},y_{16,4}+y_{16,1},y_{24,4}+y_{24,1}],$\\ $D_{48}=[y_{4,4}-y_{4,2},y_{10,4}+y_{10,2},y_{24,4}+y_{24,2}],$ & $D_{49}=[y_{14,4}+y_{14,2},2y_{20,2},y_{24,3}-y_{24,2}],$\\ $D_{50}=[y_{2,3}-y_{2,2},y_{4,2}+y_{4,1},y_{16,4}-y_{16,1}],$ & $D_{51}=[y_{2,2}+y_{2,1},2y_{10,2},y_{22,4}-y_{22,2}],$\\ $D_{52}=[y_{1,3}-y_{1,2},y_{1,3}+y_{1,2},y_{8,4}+y_{8,2}],$ & $D_{53}=[y_{4,2}-y_{4,1},y_{6,2}-y_{6,1},y_{24,3}-y_{24,1}],$\\ $D_{54}=[y_{1,3}-y_{1,1},y_{1,3}+y_{1,1},y_{6,4}+y_{6,2}],$ & $D_{55}=[y_{6,3}-y_{6,1},y_{8,3}-y_{8,1},y_{20,2}-y_{20,1}],$\\ $D_{56}=[y_{2,4}-y_{2,2},y_{8,4}-y_{8,2},y_{14,3}-y_{14,2}],$ & $D_{57}=[y_{2,3}+y_{2,1},y_{18,4}-y_{18,1},y_{20,2}+y_{20,1}],$\\ $D_{58}=[y_{6,2}+y_{6,1},y_{6,3}-y_{6,2},y_{14,4}-y_{14,2}],$ & $D_{59}=[y_{4,2}+y_{4,1},y_{18,4}+y_{18,1},y_{20,4}+y_{20,2}],$\\ $D_{60}=[y_{6,3}+y_{6,1},y_{14,3}+y_{14,2},y_{22,4}-y_{22,3}],$ & $D_{61}=[y_{18,4}-y_{18,2},y_{22,4}+y_{22,2},y_{24,3}+y_{24,1}],$\\ $D_{62}=[y_{4,2}-y_{4,1},y_{8,3}+y_{8,1},y_{24,4}-y_{24,2}].$\\ \end{tabular}}}\caption{$D_l$, $0\leq l\leq 62$}\label{tab3} \end{table}
\begin{theorem}\label{thm:125p} There exists a $(\mathbb{Z}_{125}\times \mathbb{F}_{q}^+,\mathbb{Z}_{125}\times \{0\},6,1)$-FDF for any prime $q\equiv 7 \pmod{12}$ and $q>43$. \end{theorem}
\noindent{\bf Proof \ } Since $q\equiv 7 \pmod{12}$, $-1\in C^{6,q}_3$. By Lemma \ref{lem:125p} (1)-(4), if one can choose an appropriate mapping $g$ acting on symbolic expressions satisfying that \begin{itemize} \item $\{g(y_{11}),g(y_{12}),g(y_{13})\}=\{0,1,2\}$, \item $g(y_{2i,1})\neq g(y_{2i,2})$ for $1\leq i\leq 12$, \item $\{g(d):\ d\in D_l\}$ is $\{0,1,2\}$ for each $1\leq l\leq 62$, \end{itemize} and can choose appropriate elements of $C_j$, $1\leq j\leq 25$, such that these elements are consistent with the mapping $g$, i.e., $g$ can be seen as a function from $\mathbb{F}_q^*$ to $\mathbb{Z}_3$ satisfying $g(x)=\theta$ if $x\in C^{3,q}_\theta$, then one can apply Lemma \ref{125FDF} to obtain a $(\mathbb{Z}_{125}\times \mathbb{F}_{q}^+,\mathbb{Z}_{125}\times \{0\},6,1)$-FDF. Note that once the above second condition is satisfied, let $\alpha_i=\mathbb{Z}_3\setminus\{f(y_{2i,1}),f(y_{2i,2})\}$, $1\leq i\leq 12$, and we require $y_{2i,3}$ and $y_{2i,4}$ belong to different cosets $C^{6,q}_{\alpha_i}$ and $C^{6,q}_{\alpha_i+3}$.
By Lemma \ref{lem:125p} (6) and (7), the key to pick up an appropriate mapping $g$ is to assign values of $g$ for elements of Types (IV) and (V) (note that elements of Type (V) is contained in a unique $D_l$ for some $1\leq l\leq 62$ and is related with some $C_{2i}$, $1\leq i\leq 12$). We here give explicit values of $g$ for elements of Types (IV) and (V) in Table \ref{tab2}. Then combining Table \ref{tab3}, one can give values of $g$ for elements of Types (I), (II) and (III). For example, for $D_{33}=[y_{10,2}-y_{10,1},y_{18,3}+y_{18,1},y_{18,4}-y_{18,3}]$, by Table \ref{tab2}, $g(y_{10,2}-y_{10,1})=1$, so it suffices to require $\{g(y_{18,3}+y_{18,1}),g(y_{18,4}-y_{18,3})\}=\{0,2\}$.
\begin{table}[h]\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline $i$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ & $12$\\ \hline $g(2y_{2i,1})$ & $1$ & $1$ & $1$ & $2$ & $1$ & $2$ & $0$ & $0$ & $1$ & $2$ & $0$ & $2$\\ \hline $g(2y_{2i,2})$ & $2$ & $2$ & $2$ & $1$ & $2$ & $1$ & $2$ & $1$ & $0$ & $0$ & $2$ & $0$\\ \hline $g(y_{2i,2}-y_{2i,1})$ & $2$ & $1$ & $2$ & $1$ & $1$ & $2$ & $2$ & $2$ & $0$ & $1$ & $1$ & $0$\\ \hline $g(y_{2i,2}+y_{2i,1})$ & $1$ & $1$ & $1$ & $1$ & $1$ & $0$ & $2$ & $1$ & $2$ & $0$ & $0$ & $1$\\ \hline \end{tabular}\caption{the values of $g$ for elements of Types (IV) and (V) }\label{tab2} \end{table}
Once $g$ is fixed, one can apply Theorem \ref{thm:cyclot bound} and Lemma \ref{125FDF} to obtain a $(\mathbb{Z}_{125}\times \mathbb{F}_{q}^+,\mathbb{Z}_{125}\times \{0\},6,1)$-FDF for any prime $q\equiv 7 \pmod{12}$ and $q>Q(3,7)=6.43306\times 10^7$. Note that the number $7$ in $Q(3,7)$ is from the fact that the number of cyclotomic conditions on $y_{2i,s}$ is $7$ for $1\leq i\leq 12$ and $s\in\{1,2\}$; the number of cyclotomic conditions on $y_{2i,r}$ is $6$, $1\leq i\leq 12$, $r\in\{3,4\}$; the number of cyclotomic conditions on $y_{1,t}$ is $5$, $t\in\{1,2,3\}$.
For primes $q\equiv 7 \pmod{12}$ and $50023 \leq q\leq Q(3,7)$, by computer search, we can pick up appropriate elements of $C_j$, $1\leq j\leq 25$, such that they are consistent with the mapping $g$ given in Table \ref{tab2}.
On the other hand, elements of $C_j$ do not have to be what they look like in Lemma \ref{125FDF}. It's enough to require each $C_j$, $1\leq j\leq 25$, is a representative system for the cosets of $C_0^{6,q}$ in $\mathbb{F}_q^*$, such that each $\Delta_l$, $l\in \mathbb{Z}_{125}$, is also a representative system for the cosets of $C_0^{6,q}$ in $\mathbb{F}_q^*$. If we allow $C_j$ to vary among the multisets that satisfy the required conditions, by the use of computer, we can also find appropriate elements of $C_j$, $1\leq j\leq 25$, for primes $q\equiv 7 \pmod{12}$ and $43<q<50023$. For example for $p=67$, we can take \begin{center} \begin{tabular}{ll} $C_1=\{ 1, -1, 6, -6, 7, -7 \}$,& $C_2=\{ 1, -1, 2, -2, 4, 20 \}$,\\ $C_4=\{ 1, -1, 2, -2, 4, 11 \}$,& $C_6=\{ 1, -1, 17, -17, 12, 29 \}$,\\ $C_8=\{ 2, -2, 1, -1, 4, 32 \}$,& $C_{10}=\{ 1, -1, 30, -30, 12, 35 \}$,\\ $C_{12}=\{ 2, -2, 5, -5, 20, 4 \}$,& $C_{14}=\{ 1, 43, 13, 19, 4, 46 \}$,\\ \end{tabular} \end{center}
\begin{center} \begin{tabular}{ll} $C_{16}=\{ 1, 5, 13, 16, 31, 36 \}$,& $C_{18}=\{ 1, 3, 10, 44, 46, 33 \}$,\\ $C_{20}=\{ 1, 53, 17, 50, 63, 21 \}$,& $C_{22}=\{ 2, 37, 63, 4, 42, 9 \}$,\\ $C_{24}=\{ 2, 6, 53, 1, 35, 12 \}$. \end{tabular} \end{center} and $C_{2i+1}=-C_{2i}$ for $1\leq i\leq 12$. The interested reader may get a copy of these data from the authors. \hphantom{.}
$\Box$\medbreak
\begin{lemma}\label{lem:5p}{\rm (Theorem $4.4$ in \cite{bf})} There exists a $(\mathbb{Z}_{5}\times \mathbb{F}_{q}^+,\mathbb{Z}_{5}\times \{0\},6,1)$-FDF for any prime $q\equiv 1 \pmod{12}$ and $q>37$. \end{lemma}
\begin{theorem}\label{125FDF by recur} There exists a $(\mathbb{F}_{125}^+\times \mathbb{F}_{q}^+,\mathbb{F}_{125}^+\times \{0\},6,1)$-FDF for any prime $q\equiv 1 \pmod{12}$ and $q>37$. \end{theorem}
\noindent{\bf Proof \ } By Lemma \ref{lem:5p}, there exists a $(\mathbb{Z}_{5}\times \mathbb{F}_{q}^+,\mathbb{Z}_{5}\times \{0\},6,1)$-FDF for any prime $q\equiv 1 \pmod{12}$ and $q>37$. Start from this FDF and then apply Construction \ref{recursive} with a homogeneous $(\mathbb{F}_{25}^+,6,1)$-DM to obtain a $(\mathbb{Z}_{5}\times \mathbb{F}_{q}^+\times \mathbb{F}_{25}^+,\mathbb{Z}_{5}\times \{0\}\times \mathbb{F}_{25}^+,6,1)$-FDF. Note that $\mathbb{Z}_{5}\times \mathbb{F}_{25}^+$ is isomorphic to $\mathbb{F}_{125}^+$. \hphantom{.}
$\Box$\medbreak
\begin{theorem}\label{6,1RBIBD2} \begin{itemize} \item[$(1)$] There exists a $(125q+1,6,1)$-RBIBD for any prime $q\equiv 7 \pmod{12}$ and $q>43$. \item[$(2)$] There exists a 1-rotational $(125q+1,6,1)$-RBIBD over $\mathbb{F}_{125}^+\times \mathbb{F}_{q}^+$ for any prime $q\equiv 1 \pmod{12}$ and $q>37$. \end{itemize} \end{theorem}
\noindent{\bf Proof \ } (1) For any prime $q\equiv 7 \pmod{12}$ and $q>43$, by Theorem \ref{thm:125p}, there exists a $(\mathbb{Z}_{125}\times \mathbb{F}_{q}^+,\mathbb{Z}_{125}\times \{0\},6,1)$-FDF. Then apply Proposition \ref{FDFtoRBIBD} with a $(126,6,1)$-RBIBD (a unital design (cf. \cite{b})) to get a $(125q+1,6,1)$-RBIBD.
(2) By Theorem \ref{125FDF by recur}, there exists a $(\mathbb{F}_{125}^+\times \mathbb{F}_{q}^+,\mathbb{F}_{125}^+\times \{0\},6,1)$-FDF for any prime $q\equiv 1 \pmod{12}$ and $q>37$. Then apply Proposition \ref{FDFtoRBIBD 1-rotational} with a 1-rotational $(126,6,1)$-RBIBD over $\mathbb{F}_{125}^+$, which exists by Example 16.92 in \cite{ab}, to get a 1-rotational $(125q+1,6,1)$-RBIBD over $\mathbb{F}_{125}^+\times \mathbb{F}_{q}^+$. \hphantom{.}
$\Box$\medbreak
\section{Applications}
In this section we establish asymptotic existences for optimal constant composition codes and strictly optimal frequency hopping sequences by the use of frame difference families obtained in this paper.
\subsection{Partitioned difference families and constant composition codes}
Let $(G,+)$ be an abelian group of order $g$ with a subgroup $N$ of order $n$. A $(G,N,K,\lambda)$ {\em partitioned relative difference family} $(PRDF)$ is a family $\mathfrak{B}=[B_1,B_2,\dots,B_r]$ of $G$ such that the elements of $\mathfrak{B}$ form a partition of $G\setminus N$, and the list $$\Delta \mathfrak{B}:=\bigcup_{i=1}^r[x-y:x,y\in B_i, x\not=y]=\underline{\lambda}(G\setminus N),$$
where $K$ is the multiset $\{|B_i|:1\leq i\leq r\}$. When $N=\{0\}$, a $(G,\{0\},K,\lambda)$-PRDF is called a {\em partitioned difference family} and simply written as a $(G,K,\lambda)$-PDF. The members of $\mathfrak{B}$ are called {\em base blocks}.
We often use an exponential notation to describe the multiset $K$: a $(G,N,[k_1^{u_1} k_2^{u_2}$ $\cdots k_l^{u_l}],\lambda)$-PRDF is a PRDF in which there are $u_j$ base blocks of size $k_j$, $1\leq j\leq l$.
\begin{proposition}\label{FDFtoPDF}
If there exists an elementary $(G,N,k,1)$-FDF with $|N|=k-1$, then there exists a $(G,[(k-1)^1 k^{s}],k-1)$-PDF, where $s=(|G|-k+1)/k$. \end{proposition}
\noindent{\bf Proof \ } Let $\mathfrak{F}$ be an elementary $(G,N,k,1)$-FDF with $|N|=k-1$. Then $\mathfrak{F}$ satisfies $\bigcup_{F\in \mathfrak{F}, h\in N}(F+h)=G\setminus N$. Set $$\mathfrak{B}=\{F+h:F\in \mathfrak{F}, h\in N\}\cup \{N\}.$$
Then $\mathfrak{B}$ forms a $(G,[(k-1)^1 k^{s}],k-1)$-PDF, where $s=(|G|-k+1)/k$. \hphantom{.}
$\Box$\medbreak
Combining the results of Lemma \ref{FDF1}, Theorems \ref{thm:DF-3}, \ref{thm:Twin DS-2}, \ref{thm:Singer DS-2}, \ref{thm:DF-4} and Proposition \ref{FDFtoPDF}, we have
\begin{theorem}\label{cor:PDF} \begin{itemize} \item[(1)] There exists a $(\mathbb{Z}_{7}\times \mathbb{F}_{89}^+,[7^1 8^{77}],7)$-PDF.
\item[(2)] There exists an $(\mathbb{F}_{p}^+ \times \mathbb{F}_{q}^+,[p^1 (p+1)^{s}],p)$-PDF \begin{itemize} \item for any prime power $p\equiv 3\pmod{4}$ and any prime power $q\equiv 1 \pmod{p+1}$ with $q>Q((p+1)/2,p)$; \item for any prime power $p\equiv 1\pmod{4}$ and any prime power $q\equiv 1 \pmod{2p+2}$ with $q>Q(p+1,p)$, \end{itemize} where $s=p(q-1)/(p+1)$.
\item[(3)] Let $p$ and $p+2$ be twin prime powers satisfying $p>2$. There exists an $(\mathbb{F}_{p}^+ \times \mathbb{F}_{p+2}^+\times\mathbb{F}_{q}^+,[(p(p+2))^1 (p(p+2)+1)^{s}],p(p+2))$-PDF for any prime power $q\equiv 1 \pmod{p(p+2)+1}$ and $q>Q((p(p+2)+1)/2,p(p+2))$, where $s=p(p+2)(q-1)/(p(p+2)+1)$.
\item[(4)] There exists a $(\mathbb{Z}_{2^m-1} \times \mathbb{F}_{q}^+,[(2^m-1)^1 (2^m)^{s}],2^m-1)$-PDF for any integer $m\geq3$ and any prime power $q\equiv 1 \pmod{2^{m}}$ with $q>Q(2^{m-1},2^{m}-1)$, where $s=(2^m-1)(q-1)/2^m$. \end{itemize} \end{theorem}
Partitioned difference families were explicitly introduced in \cite{dy} to construct optimal constant composition codes, which can be used in the MFSK modulation of power line communications (cf. \cite{pvyh}). For more information on partitioned difference families and its relationship with other topics such as zero-difference balanced functions (cf. \cite{zty}), the interested reader may refer to \cite{lwg}. We remark that recently M. Buratti \cite{b17} established a construction for partitioned difference families by the use of Hadamard strong difference families.
Let $Q=\{0,1,\ldots,q-1\}$ be an alphabet with $q$ symbols. An $(n,M,d,[\omega_0$, $\omega_1,\ldots,$ $\omega_{q-1}])_q$ constant composition code $($CCC$)$ is a subset ${\cal C}\subseteq Q^n$ with size $M$ and minimum Hamming distance $d$, such that the symbol $i$, $i\in Q$, appears exactly $\omega_i$ times in each codeword of $\cal C$.
Since there is no essential difference among the symbols of $Q$, one often regards $[\omega_0,\omega_1,\ldots,\omega_{q-1}]$ as a multiset and denotes it by $[k_1^{u_1}k_2^{u_2}\cdots k_l^{u_l}]$, where $k_j$ appears $u_j$ times in $[\omega_0,\omega_1,\ldots,\omega_{q-1}]$, $1\leq j\leq l$. Let $A_q(n,d,[\omega_0,\omega_1,\ldots,\omega_{q-1}])$ be the maximal size of an $(n,M, d,[\omega_0,\omega_1,\ldots,\omega_{q-1}])_q$-CCC.
\begin{proposition}\label{prop:CCC}{\rm \cite{lvc}} If $nd-n^2+(\omega_0^2+\omega_1^2+\cdots+\omega_{q-1}^2)>0$, then \begin{eqnarray}\label{ccc-bound} A_q(n,d,[\omega_0,\omega_1,\ldots,\omega_{q-1}])\leq \frac{nd}{nd-n^2+(\omega_0^2+\omega_1^2+\cdots+\omega_{q-1}^2)}. \end{eqnarray} \end{proposition}
A constant composition code attaining the bound (\ref{ccc-bound}) is called {\em optimal}. The following proposition indicates that optimal CCCs can be derived from PDFs.
\begin{proposition}\label{prop:CCC-PDF-relation}{\rm (Construction $6$ in \cite{dy})}
If a $(G,[k_1^{u_1}k_2^{u_2}\cdots k_l^{u_l}],\lambda)$-PDF exists, then there is an optimal $(|G|,|G|,|G|-\lambda,[k_1^{u_1}k_2^{u_2}\cdots k_l^{u_l}])_q$-CCC meeting the bound (\ref{ccc-bound}), where $q=\sum_{j=1}^l u_j$. \end{proposition}
Applying Proposition \ref{prop:CCC-PDF-relation} with PDFs from Theorem \ref{cor:PDF}, we have
\begin{theorem}\label{thm:ccc} \begin{itemize} \item[(1)] There exists an optimal $(623,623,616,[7^1 8^{77}])_{78}$-CCC. \item[(2)] There exists an optimal $(pq,pq,p(q-1),[p^1 (p+1)^{s}])_{s+1}$-CCC \begin{itemize} \item for any prime power $p\equiv 3\pmod{4}$ and any prime power $q\equiv 1 \pmod{p+1}$ with $q>Q((p+1)/2,p)$; \item for any prime power $p\equiv 1\pmod{4}$ and any prime power $q\equiv 1 \pmod{2p+2}$ with $q>Q(p+1,p)$, \end{itemize} where $s=p(q-1)/(p+1)$.
\item[(3)] Let $p$ and $p+2$ be twin prime powers satisfying $p>2$. There exists an optimal $(p(p+2)q,p(p+2)q,p(p+2)(q-1),[(p(p+2))^1 (p(p+2)+1)^{s}])_{s+1}$-PDF for any prime power $q\equiv 1 \pmod{p(p+2)+1}$ and $q>Q((p(p+2)+1)/2,p(p+2))$, where $s=p(p+2)(q-1)/(p(p+2)+1)$.
\item[(4)] There exists an optimal $((2^m-1)q,(2^m-1)q,(2^m-1)(q-1),[(2^m-1)^1 (2^m)^{s}])_{s+1}$-PDF for any integer $m\geq3$ and any prime power $q\equiv 1 \pmod{2^{m}}$ with $q>Q(2^{m-1},2^{m}-1)$, where $s=(2^m-1)(q-1)/2^m$. \end{itemize} \end{theorem}
\subsection{Frequency hopping sequences}
Frequency hopping multiple-access has been widely used in the modern communication systems such as ultrawideband, military communications and so on (cf. \cite{pd,yb}).
Let $F=\{f_0,f_1,\ldots,f_{l-1}\}$ be a set $($called an {\em alphabet}$)$ of $l\geq 2$ available frequencies. A sequence $X=\{x(t)\}_{t=0}^{n-1}$ is called a {\em frequency hopping sequence} $($FHS$)$ of length $n$ over $F$ if $x(t)\in F$ for any $0\leq t\leq n-1$.
For any FHS $X=\{x(t)\}_{t=0}^{n-1}$, {\em the partial Hamming autocorrelation} function of X for a correlation window length $L$ starting at $j$ is defined by \begin{eqnarray}\label{autocorrelation}
H_{X,X}(\tau;j|L)=\sum_{t=j}^{j+L-1}h[x(t),x(t+\tau)],0\leq \tau\leq n-1, \end{eqnarray} where $1\leq L\leq n$, $0\leq j\leq n-1$, $h[a,b]=1$ if $a=b$ and 0 otherwise, and the addition is performed modulo $n$. If $L=n$, the partial Hamming correlation function defined in (\ref{autocorrelation}) becomes the {\em conventional periodic Hamming correlation} (cf. \cite{lg}).
For any FHS $X=\{x(t)\}_{t=0}^{n-1}$ and any given $1\leq L\leq n$, define
$$H(X;L)=\max_{0\leq j<n} \max_{1\leq \tau <n}\{H_{X,X}(\tau;j|L)\}.$$
\begin{proposition}\label{prop:FHS}{\rm \cite{czyt}} Let $X$ be an FHS of length $n$ over an alphabet of size $l$. Then, for each window length $L$ with $1\leq L\leq n$, \begin{eqnarray}\label{FHS-bound} H(X;L)\geq \left\lceil\frac{L}{n}\left\lceil\frac{(n-\epsilon)(n+\epsilon-l)}{l(n-1)}\right\rceil\right\rceil, \end{eqnarray} where $\epsilon$ is the least nonnegative residue of $n$ modulo $l$. \end{proposition}
Let $X$ be an FHS of length $n$ over an alphabet $F$. It is said to be {\em strictly optimal} if the bound $(\ref{FHS-bound})$ in Proposition \ref{prop:FHS} is met for any $1\leq L\leq n$.
\begin{proposition}{\rm (Theorem $3.7$ in \cite{bj})}
Let $k$ and $v$ be positive integers satisfying $k+1|v-1$. Then there exists a strictly optimal FHS of length $kv$ over an alphabet of size $(kv+1)/(k+1)$ if and only if there exists an elementary $(kv,k,k+1,1)$-FDF over $\mathbb{Z}_{kv}$. \end{proposition}
Combining the results of Lemma \ref{FDF1}, Theorems \ref{thm:DF-3}, \ref{thm:Twin DS-2}, \ref{thm:Singer DS-2}, \ref{thm:DF-4} and Proposition \ref{prop:FHS}, we have the following theorem. Note that to apply Proposition \ref{prop:FHS}, the needed FDFs must be defined on a cyclic group.
\begin{theorem}\label{thm:FHS} \begin{itemize} \item[(1)] There exists a strictly optimal FHS of length $623$ over an alphabet of size $78$. \item[(2)] There exists a strictly optimal FHS of length $pq$ over an alphabet of size $(pq+1)/(p+1)$ \begin{itemize} \item for any prime $p\equiv 3\pmod{4}$ and any prime $q\equiv 1 \pmod{p+1}$ with $q>Q((p+1)/2,p)$; \item for any prime $p\equiv 1\pmod{4}$ and any prime $q\equiv 1 \pmod{2p+2}$ with $q>Q(p+1,p)$. \end{itemize}
\item[(3)] Let $p$ and $p+2$ be twin primes. There exists a strictly optimal FHS of length $p(p+2)q$ over an alphabet of size $(p(p+2)q+1)/(p(p+2)+1)$ for any prime $q\equiv 1 \pmod{p(p+2)+1}$ and $q>Q((p(p+2)+1)/2,p(p+2))$.
\item[(4)] There exists a strictly optimal FHS of length $(2^m-1)q$ over an alphabet of size $((2^m-1)q+1)/2^m$ for any integer $m\geq3$ and any prime $q\equiv 1 \pmod{2^{m}}$ with $q>Q(2^{m-1},2^{m}-1)$. \end{itemize} \end{theorem}
\section{Concluding remarks}
By a careful application of cyclotomic conditions attached to strong difference families, this paper establishes (asymptotic) existences of several classes of frame difference families, which are used to derive new resolvable balanced incomplete block designs, new optimal constant composition codes and new strictly optimal frequency hopping sequences.
We believe that starting from those CCCs or FHSs obtained in Section 6, and applying appropriate known recursive constructions in the literature, one can obtain more new existence results on them. For example, by Construction 3 in \cite{lwg} and via similar technique in the proof of Theorems 18 and 19 in \cite{lwg}, one can obtain more new PDFs, which can yield new CCCs; apply Theorem 6.8 in \cite{bj} to obtain more new FHSs, and so on.
Frame difference families can be seen as special resolvable difference families (cf. \cite{b97,byw}). By using a $(44,4,5,2)$ resolvable difference family, which is not a frame difference family, M. Buratti, J. Yan and C. Wang \cite{byw} presented the first example of a $(45,5,2)$-RBIBD. Thus an interesting future direction is to establish constructions, especially direct constructions, for resolvable difference families.
Most of our results in this paper rely heavily on existences of elements satisfying certain cyclotomic conditions in a finite field, and Theorem \ref{thm:cyclot bound} just supplies us a way to ensure existences of such elements. We remark that recently X. Lu improved the lower bound on $q$ in Theorem \ref{thm:cyclot bound} in some circumstances (see Theorem 3 in \cite{l}). He introduced an existence bound $L(d,t)$ for elements $x$ that satisfies the following cyclotomic conditions: \begin{itemize} \item[i)] $x\in \bigcup_{i\in U(\mathbb{Z}_d)}\ C_i^{d, q}$, where $U(\mathbb{Z}_d)$ is the set of all units in $\mathbb{Z}_d$; \item[ii)] $x^{d-c_j}(a_jx+b_j)\in C_{0}^{d, q}$ for $1\leq j\leq t-1$. \end{itemize} One possible development of this paper could be to find suitable constructions that use his result.
\end{document} | arXiv |
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Emergent gravity from Eguchi-Kawai reduction
Journal of High Energy Physics, Mar 2017
Edgar Shaghoulian
Abstract Holographic theories with a local gravitational dual have a number of striking features. Here I argue that many of these features are controlled by the Eguchi-Kawai mechanism, which is proposed to be a hallmark of such holographic theories. Higher-spin holographic duality is presented as a failure of the Eguchi-Kawai mechanism, and its restoration illustrates the deformation of higher-spin theory into a proper string theory with a local gravitational limit. AdS/CFT is used to provide a calculable extension of the Eguchi-Kawai mechanism to field theories on curved manifolds and thereby introduce "topological volume independence." Finally, I discuss implications for a general understanding of the extensivity of the Bekenstein-Hawking-Wald entropy.
https://link.springer.com/content/pdf/10.1007%2FJHEP03%282017%29011.pdf
Received: December Emergent gravity from Eguchi-Kawai reduction Santa Barbara 0 CA 0 U.S.A. 0 Open Access 0 c The Authors. 0 0 Department of Physics, University of California Holographic theories with a local gravitational dual have a number of striking features. Here I argue that many of these features are controlled by the Eguchi-Kawai mechanism, which is proposed to be a hallmark of such holographic theories. Higherspin holographic duality is presented as a failure of the Eguchi-Kawai mechanism, and its restoration illustrates the deformation of higher-spin theory into a proper string theory with a local gravitational limit. AdS/CFT is used to provide a calculable extension of the Eguchi-Kawai mechanism to eld theories on curved manifolds and thereby introduce topological volume independence." Finally, I discuss implications for a general understanding of the extensivity of the Bekenstein-Hawking-Wald entropy. AdS-CFT Correspondence; Gauge-gravity correspondence; Gauge Symmetry 1 Introduction 1.1 Summary of results 2 Center symmetry and Wilson loops 3 Reproducing gravitational phase structure/sparse spectra/extended 5 Higher-spin theory as a failure of the Eguchi-Kawai mechanism 6 Learning about the Eguchi-Kawai mechanism from gravity Center symmetry stabilization and translation symmetry breaking 6.2 Extending the Eguchi-Kawai mechanism to curved backgrounds 7 Extensivity of the Bekenstein-Hawking-Wald entropy Reproducing additional features of AdS gravity Reducing or blowing up models The necessity of the Eguchi-Kawai mechanism for holographic gauge theories 28 range of validity of the Cardy formula Extended range of validity of Cardy formula Sparse spectra in holographic CFTs SL(2; Z) family of black holes 3.4 SL(d; Z) family of black holes 4 Correlation functions and entanglement entropy Correlation functions Two-point functions M -point functions 4.4 Entanglement/Renyi entropies 3.1 3.2 3.3 8 Discussion Outlook A SL(d; Z) B Four-point function sample calculation C Validity of gravitational description Holographic theories with a local gravitational dual have several remarkable features that can be read o by analyzing (semi-)classical gravity in Anti-de Sitter space (AdS). To understand the emergence of gravity, it is important to understand precisely in the language of eld theory what mechanism is responsible for these features. Much of the work in this direction has focused on constraints from conformal eld theory (CFT). Conformality is not an essential feature of holography. On the other hand, every holographic theory to date can be understood as a large-N gauge theory. It is therefore natural to leverage whatever power such a structure brings us. This brings us to the idea of Eguchi-Kawai reduction. The proposal of Eguchi and Kawai was that large-N SU(N ) lattice gauge theory could be reduced to a matrix model living on a single site of the lattice [1]. This equivalence was postulated through an analysis of the Migdal-Makeenko loop equations (the SchwingerDyson equations for Wilson loop correlation functions) [2, 3] and assumed the preservation of center symmetry in the gauge theory. However, it was immediately noticed [4] that the center symmetry is spontaneously broken at weak coupling, disallowing the consistency of the reduction with a continuum limit. The authors of [4] further proposed the rst in a long list of modi cations to the gauge theory in an attempt to prevent center symmetry from spontaneously breaking. Their proposal is known as the quenched Eguchi-Kawai model, further studied in [5], where the eigenvalues of the link matrices were frozen to a centersymmetric distribution. Another proposed variant is known as the twisted Eguchi-Kawai model, wherein each plaquette in Wilson's action is \twisted" (multiplied by) an element of the center of the gauge group [6]. Numerical studies have shown these early modi cations fail at preserving center symmetry as well [7{10]. Let us turn to the continuum. Whether or not center symmetry is preserved is often checked analytically by pushing the theory into a weakly coupled regime and calculating the one-loop Coleman-Weinberg potential for the Wilson loop around the compact direction. This is an order parameter for the center symmetry, and a nonvanishing value indicates a breaking of center symmetry. An early analytic calculation of the Coleman-Weinberg potential indicates the center-symmetry-breaking nature of Yang-Mills theories [11]. Nevertheless, there are a few tricks that seem to work at suppressing any center-breaking phase transitions: a variant of the original twisted Eguchi-Kawai model [12], deforming the action by particular double-trace terms [13], or considering adjoint fermions with periodic boundary conditions [14]. For a modern review see [15]. In this work, we will not be concerned with suppressing center-breaking phase transitions. Instead, we will focus on implications of the Eguchi-Kawai mechanism within centersymmetric phases. This will not be a restriction to the con ned phase since we will be considering center symmetry with respect to spatial and thermal cycles. As we will be working in the continuum, let us formulate the continuum version of the Eguchi-Kawai mechanism. Consider a d-dimensional large-N gauge theory compacti ed on M symmetry at the Lagrangian level. If translation symmetry and center symmetry are not spontaneously broken along a given S1, then correlation functions of appropriate singletrace, gauge-invariant operators are independent of the size of that S1 at leading order in (S1)k with center N . We will review these notions in the rest of the introduction and spend section 4 elaborating on which sorts of observables are \appropriate." This is often called large-N volume independence, where \volume" in particular refers to the size of the center-symmetric S1s. The Eguchi-Kawai mechanism is a robust, nonperturbative property of large-N gauge theories that preserve certain symmetries. Famously, large-N gauge theories also play a starring role in holographic duality. Curiously both contexts involve emergent spacetime in radically di erent ways. In this work we will be interested in what predictions the Eguchi-Kawai mechanism makes about gravity in AdS. Since the proposal concerns only leading-in-N observables, we will be dealing exclusively with the (semi-)classical gravity limit in AdS. A simple example illustrating the mechanism at work is the temperatureindependence of the free energy density on M S1 at leading order in N in the con ned S1). In AdS/CFT, this occurs because the thermal partition function is given by the contribution of thermal AdS below the HawkingPage phase transition, whose on-shell action has an overall factor of inverse temperature . When the theory decon nes, the free energy density becomes a nontrivial function volume independence, see equation (3.1). We will spend the next section reviewing introductory material, ending with the central tool of this work, which is that a smooth, translation-invariant gravitational description implies center symmetry preservation along all but one cycle. Center symmetry can spontaneously break along a given cycle as its size is varied, but there must only ever be one cycle which breaks the symmetry. We will refer to these transitions as center-symmetryswapping transitions (CSSTs). The rest of the paper will leverage this structure to learn primarily about universal features of gravity, but also to learn about the Eguchi-Kawai mechanism in large-N gauge theories. For some previous work exploring the Eguchi-Kawai mechanism in holography, see [16{19]. Summary of results Our primary tool will be that a smooth, translation-invariant gravitational description of a state or density matrix in a toroidally compacti ed CFT preserves center symmetry along all but one cycle. We will use this to produce the following universal features of gravity in AdS: (a) an extended range of validity of the general-dimensional Cardy formula, (b) the exact phase structure (including thermal and quantum phase transitions) with a toroidally compacti ed boundary, (c) a sparse spectrum of light states on the torus, (d) leading-in-N connected correlators will be given by the method of images under smooth quotients of the spacetime, which reproduces the behavior of tree-level Witten diagrams, and (e) extensivity of the entropy for spherical/hyperbolic/planar black holes which dominate the canonical ensemble; for planar black holes this implies the Bekenstein-Hawking-Wald area law. (a)(c) are closely related and can be found in section 3, (d) can be found in section 4, and (e) can be found in section 7. Using gravity to learn about the Eguchi-Kawai mechanism, we will nd new center-stabilizing structures for strongly coupled holographic theories and propose an extension of the mechanism to curved backgrounds in section 6. Center symmetry and Wilson loops Consider pure Yang-Mills theory on manifold Md 1 SU(N )) with nontrivial center C (for example ZN ): This theory is invariant under the gauge symmetry S = F a = @ A S1 with gauge group G (for example S1 ! G a map from our spacetime into the gauge group. The eld strength Let us consider the function g to be periodic along the S1 only up to an element of the gauge group: g(x; + ) = g(x; )h for h 2 G. For A to remain periodic we need h 1A (x; )h. But this requires h 2 C so we can commute it past A h 1. So we see that we can consistently maintain twisted gauge transformations as long as and cancel it against we twist by an element of the center. The action above is invariant under these extended gauge transformations. The space of physical states are constrained to be singlets under the usual gauge group G but not under the twisted gauge transformations. In particular, Wilson loops which wrap an S1, which will henceforth be referred to as Polyakov loops, transform under the generalized gauge transformation. To see this, consider the pathordered exponential, i.e. the holonomy of the connection, around the S1: x( + ; ) = P exp The P stands for path. We will refer to the trace of this object as the Polyakov loop, which for ordinary gauge transformations causes g and g 1 to annihilate by cylicity. For twisted gauge transformations, however, we are left with The W stands for Polyakov. Thus the expectation value of a Polyakov loop can serve as an order parameter for the spontaneous breaking of center symmetry. We will always take our trace in the fundamental representation, since the vanishing of the expectation value of such a loop is necessary and su cient for the preservation of center symmetry, independent of the matter content. Contrast this with the case of rectangular Wilson loops (traces of path-ordered exponentials where the path traces out a large rectangle instead of wrapping an S1) where the trace needs to be evaluated in the same representation as that of the matter content to access the energy required to decon ne the matter. Even for matter in vectorlike representations that break center symmetry, there is Let us now specify to gauge group SU(N ). The center of the gauge group is ZN , by which of the N representations of ZN it falls under. This is called the N -ality of the representation, and it is determined by counting the number of boxes mod N of the Young tableau of the representation. The addition of matter to our gauge theory explicitly breaks the center symmetry of the Lagrangian unless the matter is in a representation of vanishing N -ality [20]. Fundamental representations have N -ality 1 and therefore explicitly break center symmetry. Adjoint representations, on the other hand, have vanishing N -ality and therefore preserve center symmetry. an e ective emergence of the symmetry as N ! 1 as long as the number of vectorlike avors is kept nite. This is simply because quarks decouple at leading order and one is left with the pure Yang-Mills theory. Interestingly, by orientifold dualities, even matrix representations (which break center symmetry and for which the matter does not decouple) have an emergent center symmetry at in nite N [21, 22]. Calculating Wilson loops in AdS. There is a simple prescription for calculating the expectation value of a Wilson loop in the fundamental representation of the gauge theory using classical string theory. One calculates e SNG for the Nambu-Goto action SNG for a Euclidean string worldsheet which ends on the contour of the Wilson loop C [23]. Let us specify to Polyakov loops wrapping an S1 on the boundary. Notice that if this circle example where this criterion distinguishes con ned and decon ned phases. The thermally stable (i.e. large) AdS-Schwarzschild black hole, which has a thermal circle which caps o in the interior, admits a string worldsheet and therefore gives a nonvanishing Polyakov loop expectation value. This indicates a decon ned phase, which is appropriate as the AdS-Schwarzschild black hole is the correct background for the gauge theory at high temperature. Thermal global AdS, however, has a thermal circle which does not cap o in the interior and therefore gives a vanishing Polyakov loop expectation value. This indicates a con ned phase, which is appropriate for the theory at low temperature. Indeed, the bulk canonical phase structure for pure gravity indicates a transition between these two backgrounds when the inverse temperature is of order the size of the sphere. Similarly, the entropy transitions from O(1) in the con ned phase (no black hole horizon) to O(N 2) in the decon ned phase (yes black hole horizon). There is one more basic geometric fact we will need. Consider an asymptotically Euclidean AdSd+1 spacetime with toroidal boundary conditions. Preserving translation invariance along the non-radial directions | a necessary condition for the Eguchi-Kawai mechanism to work | gives a metric of the form ds2 = g (r ! 1) = r2 To avoid conical singularities (e.g. metrics which look locally like r2(d 21 + d 22)), no more than one of the boundary circles can cap o in the interior of the spacetime. While it may be possible that none of the boundary circles cap o in the interior (say through the internal manifold capping o instead), I do not know of any smooth, geodesically complete examples. We will therefore not consider this possibility, so in our context exactly one cycle caps o and the other d 1 circles remain nite-sized. This motivates the following simple yet extremely powerful statement. In any smooth, translation-invariant geometric description, the expectation value of Polyakov loops in the fundamental representation vanish in 1 of the directions. For theories with an explicit center symmetry, this means that we will have volume independence along d 1 directions as discussed in the introduction. Appropriate observables will therefore be independent of the sizes of the circles. For the gravitational description to be valid, the circles in the interior need to remain above string scale. For a translation of this criterion into eld theory language, and in particular a discussion of Eguchi-Kawai reduction to zero size, see appendix C. Just like the original Eguchi-Kawai example of pure Yang-Mills, our theory will of course decon ne, as signaled by the Hawking-Page phase transition in the bulk. This is sometimes called partial Eguchi-Kawai reduction, since the reduction only holds in the center-symmetric phase. We will refer to the \Eguchi-Kawai mechanism" and \large-N volume independence" to describe this state of a airs. (Large-N volume independence refers in particular to independence of the size of center-symmetric S1s, not necessarily the overall volume.) From our point of view, the decon nement transition is just a centersymmetry-swapping transition (CSST) from the thermal cycle to a spatial cycle. It remains true that d 1 of the cycles preserve center symmetry. CSSTs can also occur between spatial cycles as they are varied. In this case, the transition is unrelated to con nement of degrees of freedom, since the entropy is O(1) before and after the transition. It instead signals a quantum phase transition, which can take place at zero temperature. Interestingly, this quantum phase transition persists up to a critical temperature. Reproducing gravitational phase structure/sparse spectra/extended range of validity of the Cardy formula We will now show that the semiclassical phase structure of gravity in AdS is implied by our center symmetry structure. Consider an asymptotically AdSd+1 spacetime with toroidal boundary conditions. The cycle lengths will be denoted L1; : : : ; Ld with = L1. We will pick thermal periodicity conditions for any bulk matter along all cycles and will comment at the end about di erent periodicity conditions. Assuming a smooth and translationinvariant description, the phase structure implied by gravity can succinctly be written in terms of the free energy density as log Z(L1; : : : ; Ld) where "vac is a pure positive number (independent of any length scales) characterizing the vacuum energy on S1 d 2 as Evac=V "vac=Ld for spatial volume V [24], and Lmin is the length of the smallest cycle. This is the phase structure independent of the precise bulk theory of di eomorphism-invariant gravity, as long as we maintain translation invariance and consider the thermal ensemble. Like in AdS3, all the data about higher curvature terms is packaged into "vac. Notice that the triviality of this phase structure implies highly unorthodox eld theory behavior. The phase structure (3.1) implies thermal phase transitions as the thermal cycle becomes the smallest cycle. There are also quantum phase transitions when two spatial cycles are smaller than the rest of the cycles (including ), and the larger of the two is changed to become smaller. These are quantum phase transitions because they can (and do) occur when ! 1, so they are not driven by thermal uctuations. These quantum phase transitions, however, persist at nite temperature. Finally, in any given phase the functional form of the free energy density is independent of all cycle lengths except for one! Much of [25] was focused on reproducing this structure in eld theory, and we refer the reader to that work to see the many nuances involved. We now turn to the gauge theory. We will see that our framework gives (3.1) immediately, thereby locating the points where phase transitions occur and the precise functional form of the free energy in all phases. Consider a eld theory with our assumed center symmetry structure, which is that all but one cycle preserve center symmetry. We also have thermal periodicity conditions for the matter elds along all cycles, since this will give thermal periodicity conditions for the bulk matter elds and preserve modular S invariance between any pair of cycles. Notice that by extensivity of the free energy and modular invariance [24, 26], we have f (L1 ! 0; L2; : : : ; Ld) = Since the free energy density is supposed to be independent of the center-symmetry preserving directions, we deduce that the L1 cycle breaks center symmetry. This is consistent with the expected decon nement of the theory. Now let us consider varying any of the cycle sizes. As long as there is no center-symmetry-swapping transition (CSST), f (L1; : : : ; Ld) continues to depend only on L1. Since the theory is scale invariant, this xes the L1 dependence and we continue to have the behavior (3.2). Finally, any CSST that occurs between has to occur when L = L by the modular symmetry between all cycles. So, when cycle lengths are equal, they must be symmetric: either they both preserve the center or they are undergoing a CSST. They cannot both break the center since only one cycle can ever break the center in our framework. Using the above facts that f (L1; : : : ; Ld) can only change its functional form at CSSTs and that two cycles which have equal length must have the same center-symmetry structhe symmetry between cycles there must be a CSST between L1 and the next-smallest cycle when they become equal. As L1 is increased further, it is a center-preserving cycle passing other center-preserving cycles, so no more CSSTs can occur and the free energy density remains unchanged. Starting from an arbitrary torus, with an arbitrary cycle taken asymptotically small, this argument produces for us the entire phase structure (3.1). What about the case where we do not preserve the symmetry between cycles? An interesting example of this is if we pick bulk fermions to be periodic along some cycles. In the gravitational picture these cycles are not allowed to cap o in the interior since this would not lead to a consistent spin structure. Thus, the phase structure is just as in (3.1), where now Lmin minimizes only over the cycles with antiperiodic bulk fermions. We will comment more on the eld theory implications of this in section 6.1. To predict this bulk phase structure, we need to supplement our assumption of d 1 cycles preserving center symmetry with an assumption about which cycles preserve center symmetry for all cycle sizes. These cycles can then never undergo CSSTs with other cycles. By repeating the arguments above, we can reproduce this modi ed bulk phase structure. Extended range of validity of Cardy formula Holographic gauge theories, in addition to having the remarkable phase structure exhibited above, have an extended range of validity of the general-dimensional Cardy formula. The Cardy formula in higher dimensions was derived in [24, 26] and reproduces the entropy of toroidally compacti ed black branes at asymptotically high energy: This precisely mimics how the two-dimensional Cardy formula [27] reproduces the entropy of BTZ black holes at asymptotically high energy [28]. Large N operates as a thermodynamic limit that can transform our statements about the canonical partition function into the microcanonical density of states (this is discussed for example in the appendices of [25, 29]). We nd that the Cardy formula is not valid only asymptotically, but instead is valid down to E = 1)Evac, which in canonical variables is at a symmetric point = Li;min where Li;min is the smallest spatial cycle. This is precisely the energy at which the HawkingPage phase transition between the toroidally compacti ed black brane and the toroidally compacti ed AdS soliton occurs in the bulk! Similar arguments in the case of non-conformal branes should give an extended range of validity for the Cardy formula of [30]. Sparse spectra in holographic CFTs A sparse spectrum is often invoked as a fundamental requirement of holographic CFTs, and we have several avenues of thought that lead to this conclusion. Here we will be concerned with the sparseness necessary to reproduce the phase structure of gravity [25, 29], not with the sparseness necessary to decouple higher-spin elds [31]. We have already reproduced the complete phase structure (3.1). By the arguments in [25, 29] this implies a sparse low-lying spectrum 1)Evac) . exp (Li;min(E where Li;min is the smallest spatial cycle. To roughly recap the argument of [25], modular constraints on the vacuum energy coupled with the phase structure imply vacuum domination along all cycles except the smallest one. But to be vacuum dominated means that excited states do not contribute to the partition function. This leads to the constraint above, which is really a constraint on the entire spectrum, but is written as above since for 1)Evac we have a precise functional form for the density of states: it takes the higher-dimensional Cardy form, which trivially satis es the Hagedorn bound above. One can also access additional sparseness data by investigating di erent boundary conditions. To point out the simplest case, consider super Yang-Mills theory in a given number of dimensions with fermions having periodic boundary conditions along one cycle and antiperiodic boundary conditions along another cycle. Then modular covariance will equate a thermal partition function ZNS;R with a twisted partition function ZR;NS (twisted by ( 1)F ), which will access the twisted density of states F (E). By similar steps as performed above, one will conclude a sparseness bound for this twisted density of states. The fact that preserving center symmetry can imply a supersymmetry-like bound is carefully discussed in a non-supersymmetric context in [32, 33]. SL(2; Z) family of black holes In this section and the next we will consider the case of twists between the cycles of the torus. We will begin with three bulk dimensions, where there is an extended family of solutions known as the SL(2; Z) family of black holes, rst discussed in [34] and elaborated upon in [35]. They give an in nite number of phases, instead of the two we usually consider in Lorentzian signature, and we can check volume independence in each of the phases individually. Twists do not seem to be considered in the literature on large-N volume independence, but we will show that volume independence continues to hold. A general SL(2; Z) black hole has a unique contractible cycle, sometimes called an A-cycle. The non-contractible cycle (sometimes called a B-cycle) is only additively de ned, since for any B-cycle one can construct another B-cycle by winding around the A-cycle n times (n 2 Z) while going over the original B cycle. The usual convention is to set this winding number to zero. Due to this in nity is contractible in the interior [35]. Here Z acts as + n for modular parameter . This data is given by two relatively prime integers (c; d) with c 0. We also need to include the famous examples (0; 1) (thermal AdS3) and (1; 0) (BTZ). In the rest of this section we will ignore numerical prefactors in the free energy density and will only track the dependence on cycle lengths. Let us consider the simplest cases rst, thermal AdS3 and BTZ, both with zero angular potential. This means 1= are pure imaginary. We have Thermal AdS : f ( ; L) BTZ : f ( ; L) These exhibit volume independence for the center-symmetry preserving (i.e. noncontractible) cycles. Let us now add an angular potential , which makes Thermal rotating AdS : L 2=L2 + 2=L2 = We again get consistent results, since the lengths of the contractible cycles of thermal rotating AdS and rotating BTZ are L and p 2 + 2, respectively. The general SL(2; Z) black hole can be given in a frame where the modular parameter is (a +b)=(c +d), the contractible cycle z = z + L(a + b). Their lengths are given as z +L(c +d) and the non-contractible jSA1j = pd2L2 + 2cdL + c2( 2 + 2); jSB1j = pb2L2 + 2abL + a2( 2 + 2) : (3.9) The free energy density is found, for general = i =L + =L, to be d2L2 + 2cdL + c2( 2 + 2) Notice that a and b enter into the size of the non-contractible cycle, but the condition expected since the physically distinct states should only care about c; d by the arguments above. We therefore nd for the general SL(2; Z) geometry that the free energy density exhibits volume independence. SL(d; Z) family of black holes There exists an unexplored analog to the SL(2; Z) family of black holes in higher dimensions, which I will call the SL(d; Z) family of black holes. For a review of some salient points about conformal eld theory on Td and SL(d; Z), see appendix A. The bulk topology is that of a solid d-torus, with a unique contractible cycle. Winding a B-cycle by an A-cycle is topologically trivial. A \small" bulk di eomorphism, i.e. one continuously connected to the identity, can undo this winding. However, winding a Bcycle by another B-cycle leads to a true winding number and is topologically distinct. This corresponds to a large di eomorphism in the bulk. Thus, as in the two-dimensional case, we only need to sum over a subgroup of the full SL(d; Z), because B-cycles are only 1, n 2 Z and V~d the xed contractible cycle vector. As reviewed in appendix A, the V~i represent lattice vectors that de ne the quotient of the plane that gives us the torus Td. Our \seed" solution in three bulk dimensions was global AdS3 at nite temperature and nite angular velocity. In higher dimensions our seed solution will be the AdS soliton, with all spatial directions compacti ed, arbitrary twists turned on (including both twists between spatial directions and time-space twists, interpreted as angular velocities), and the geometry described above should give an SL(d; Z)-invariant partition function. Ignoring the important issue of convergence of this sum, we can see that the invariance is naively guaranteed since the seed solution and its images are independently invariant under the Z we mod out by. In other words, the analog of Z0;1( ) from the previous section, call it Z0(V~1; : : : V~d), and its images are invariant under shifts V~i ! V~i + nV~d. Anyway, this restricted sum is not important for our purposes. It is su cient to show that an arbitrary element of the SL(d; Z) family has a free energy density that depends only on the contractible cycle. The simplest case is the AdS soliton at nite temperature with spatial directions compacti ed, which has free energy density where Ld is the length of the contractible cycle. This is volume-independent as required. Twisting any of the non-contractible directions by any of the other directions by any amount does not change this answer. Thus, the general AdS soliton with arbitrary angular potentials and spatial twists exhibits volume independence with respect to the non-contractible cycles. We can now consider SL(d; Z) images of this geometry. The general SL(d; Z) image geometry has global Killing vector elds for all the nonVol(Td) R drF (r; r~h) where r~h is a parameter xed by the size of the dth cycle and F (r; r~h) is some function. Thus, twists can only enter into Vol(Td), but torus volumes are invariant under twists. Higher-order corrections in the Newton constant GN will bring in a dependence on the twists, as the momentum quantization of perturbative torus depends on the twists. In this way we see that volume-independence will break down density a little more carefully. Consider a general twisted seed geometry, with the contractible direction chosen to lie along the dth direction, speci ed by lattice vectors de ning the twists: L2k = j=1 i=1 = 666 ... det(A) = +1 to give Pid=1 a1i id Pid=1 adi id We can compose d(d 1)=2 rotations in the d(d 1)=2 two-planes to make this matrix upper triangular. This will allow us to identify the new modular parameter matrix This will not change the lengths of the cycles, which are given as where Ld gives the length of the contractible direction. The volume of the resulting torus Vol(ATd) = det(A ) = det(A) det( ) = Y In particular, it is unchanged by the SL(d; Z) transformation. The free energy density is exhibiting volume independence in the center-symmetric directions. with the case of no twist in that direction. This is because there exists a bulk di eomorphism, continuously connected to the identity, which induces this twist on the boundary. Twists in non-contractible directions, however, correspond to large gauge transformations cycle is not su cient. We still have a reduction in moduli, with d2 1) = d(d numbers specifying distinct geometries. Interestingly, the distinct geometries obtained by twisting non-contractible directions by other non-contractible directions do not di er in their classical on-shell action. Correlation functions and entanglement entropy In this section we will discuss the implications of the Eguchi-Kawai mechanism for correlation functions and Renyi entropies. As usual, the statements are restricted to leading order in N , meaning tree-level Witten diagrams in the bulk. We will only consider volume independence with respect to a single direction for conceptual clarity; generalization to multiple directions is straightforward. For correlation functions we will see that position space correlators must be given by the method-of-images under smooth quotients, as in (4.10). The connection between large-N reduced correlation functions and the role of the method of images in AdS has previously been explored in the stringy (zero 't Hooft coupling) limit in [16, 17], although there are several points of deviation from the present work. Correlation functions Let us assume that we are volume-independent with respect to a single direction. Then connected correlation functions of local, single-trace, gauge-invariant, neutral-sector observables will be volume independent at leading order in N . Nonlocal operators like Wilson loops can also be treated as long as they have trivial winding around the cycle. One term that may need explanation is \neutral-sector." We will explain brie y below; for details see [14]. Consider the theory on R S1 as we vary the circle size from some length L to some other length L0. A given operator in the theory of size L can be decomposed as n= 1 On=Le2 inx=L : sector" operators, and it is their correlation functions which are volume-independent. For independent. While this may seem like a severe restriction, we will only be concerned with nite-size results from in nite-size results, and all momenta in commensurate with some momentum in in nite size. write this precisely as O(1=N 2(M purely adjoint theory shows that the connected correlator of M single-trace operators is N M 1 in front to isolate the leading contribution to the connected correlator. But the basic point is clear: the statement is about the rst order in N that is expected to have a nonvanishing answer by large-N counting. If it vanishes, no statements are made about the leading nonvanishing order. This is what the limit above makes precise in a pure adjoint theory. We will not worry about the various cases of large-N counting, because within AdS/CFT the leading-in-N diagrams are given by tree-level diagrams in the bulk. It is only these diagrams we wish to make a statement about. We will therefore use as our primary tool the equality hOn1=LOn2=L : : : OnM =LiL = J M 1hOn1=LOn2=L : : : OnM =LiJL with the caveat that this is the leading-in-N piece of a connected correlator left implicit. To see the e ect on a general correlation function of local operators, it will su ce to consider the two-point function. We consider the Fourier representation of the nite-size hO(x)O(y)iL = e 2 i(nx+my)=LhOn=LOm=LiL e 2 i(nx+my)=LJ hOn=LOm=LiJL ; where in the second line we used (4.3). We could immediately use translation invariance to write the correlator as a function of only the separation x y, but to make generalization to higher-point correlators clear we will keep the dependence until the end. We can now simplify this expression by transforming the momentum-space correlator in size J L to position space and evaluating the various sums and integrals: hO(x)O(y)iL = dy0e 2 i(nx+my)=Le2 i(nx0+my0)=LJ hO(x0)O(y0)iJL This generalizes to hO(x1) : : : O(xM )iL = hO(x1 + n1L) : : : O(xM + nM L)iJL : The converse is also true. That is, starting from the method-of-images form of a position space correlator above, one can show (4.3). Altogether, volume-independence of neutralsector correlators is true if and only if nite-size correlators are obtained by the method of images from correlators in a larger size. Two-point functions To focus on the simplest case, consider the equal-time two-point function in a translationinvariant two-dimensional theory. Say we want to construct the nite-size correlator from the in nite-size correlator. We begin from (4.10) and use translation invariance, which says that our correlator is only a function of the distance between the two insertion points: hO(x)O(y)iL = hO(x y) O(0)iL = m))L) O(0)iJL where we used the J L-periodicity of the size-J L correlator. To compare to the in nite-size correlator we can take J ! 1 in a particular way: y) O(0)iL = lim y + nL) O(0)iJL + hO(x hO(x + nL)O(y + mL)iJL : ni=0 = lim n= (J 1) e 2 i(nx+my)=Le2 i(nx0+my0)=LhO(x0)O(y0)iJL mL)hO(x0)O(y0)iJL Notice that taking this limit will give us the correlator on the semi-in nite line with semiin nite periodicity. Doubling it (and picking up a factor of 2 just as in the factor of J that comes from relating two-point functions in size L to size J L) gives us the real-line correlator. We thus have our nal result n= 1 y) O(0)iL = Now we compare to gravity in AdS. Conformal eld theory correlators, at leading order in N , are obtained by extrapolating the bulk-to-bulk propagator to the boundary. Since the bulk-to-bulk propagator for free elds satis es a Green function equation, we can the propagator after performing an arbitrary smooth quotient by the method of images. This gives precisely the form of correlator above, which for example in the famous case of the BTZ black hole takes the form [36] hO(t; )O(0; 0)i = n= 1 cosh 2 t for operators of dimension . Notice that this sums over spatial images but not thermal images. For thermal AdS3, which is obtained instead as a quotient in the Euclidean time direction, we would sum over thermal images but not over spatial images. In each case, the correlator is given by a sum over images with respect to the center-preserving direction. This is exactly what is predicted by our arguments above. Furthermore, we see that the \free-ness" of large-N theories is not su cient by itself to imply that the correlator should be a sum over images, since there is no sum over images in the center-breaking direction. M -point functions For higher-point functions, recall that we focus only on diagrams in the bulk that do not have any loops. Any given contribution to the tree-level M -point function is constructed out of M bulk-to-boundary propagators K and n < M bulk-to-bulk propagators G. This means there are n + 1 interaction vertices in the bulk. An illustrative case of tree-level (leading in N ) and loop level (subleading in N ) diagrams is depicted in The position space correlation function can be written schematically as hO(x1) : : : O(xM )iAdS = AdS i=1 where boundary points are denoted by small x and bulk points by big X. From here we hOn1=L OnM =LiAdS= = j j M 1hOn1=L OnM =LiAdS : Before we outline the proof of this we need the following facts. The bulk-to-bulk propagator satis es a Green function equation since the bulk theory is free at this order (leading in N ). The bulk-to-boundary propagator is obtained by a certain limit of the bulk-to-bulk propagator where one of its points is pulled to the boundary. Thus, both propagators There are many more diagrams contributing at this order. Right: a loop-level Witten diagram, which contributes at rst subleading order in N to the nine-point function. It is constructed out interaction vertices. There are again many more diagrams contributing at this order. can be obtained on a smooth quotient of our AdS background by the method of images. Finally, in momentum space, the integrals over spacetime give n + 1 momentum-conserving delta functions since there are no loops in the bulk. The general proof of (4.19) is notationally clumsy and would ruin the already regretful aesthetics of this paper, so we will provide an outline of the general proof here and give a sample calculation in appendix B. The left-hand-side is evaluated by an inverse Fourier transform of the position space expression. The position space expression is written in by those in AdS by the method of images. These propagators are then transformed into and integrals are re-ordered at will and this expression is simpli ed down to an integral over the bulk radial interaction vertices zi. The right-hand-side is evaluated in the same way, except its propagators are never replaced with other propagators. This leads to (4.19). Explicit details for a four-point function can be found in appendix B. So we see that the behavior of tree-level perturbation theory in AdSd+1 under generic, smooth quotients of spacetime is reproduced. Notice that bulk loops are made of bulk-tobulk propagators as well, but their momenta are not xed and instead are integrated over. This leads to a non-universal answer, since there are bulk-to-bulk propagators in the AdS by the usual method of images trick, the sums are over di erent momenta and cannot be carried out in general. Entanglement/Renyi entropies Another place where volume independence crops up is in the calculation of entanglement entropy of theories dual to gravity in AdSd+1. For simplicity I will restrict to AdS3. Recall that the Ryu-Takayanagi prescription dictates that the entanglement entropy is given by the regularized area of a minimal surface that is anchored on the entangling surface on the AdS boundary [37]. Consider a spatial interval of size ` on a spatial circle of size L at temperature T . For entangling surfaces at xed time for static states or density matrices, the minimal surface will lie on a constant bulk time slice. This makes it clear that in the con ned phase, which is thermal AdS3, the Ryu-Takayanagi answer will be independent of the center-preserving thermal circle of size T : SEE = . Note that it is not given as a sum over thermal images like in the case of correlation functions. It is instead completely independent of the thermal cycle size. In the decon ned phase, i.e. above the Hawking-Page CSST, we get an answer independent of the center-preserving spatial circle of size L: SEE = The minimization inherent in the Ryu-Takayanagi prescription is the reason why we do not sum over images and so get exact volume independence. (There is a proposal that the image minimal surfaces instead contribute to entanglement between internal degrees of freedom, coined \entwinement" [38].) Apparently, single-interval entanglement entropy is an appropriate neutral-sector \observable" that obeys large-N volume independence. As shown by a bulk calculation in [39], volume-dependence appears at rst subleading order in the central charge c (the proxy for N in two-dimensional theories). Volume-dependence also appears at leading order in the central charge in the Renyi entropies, but not in any trivial way as in the local correlators of the previous section. The Renyi entropies must not be neutral-sector observables. The Renyi entropy in this context is related to the free energy on higher-genus handlebodies; the analytic continuation connecting to the original torus to de ne the entanglement entropy is therefore special. It is interesting that in the cases where we have a volume-independent object, it is the entanglement entropy and not any of the higher Renyi entropies. This may be related to the fact that it is the entanglement entropy that naturally geometrizes in the bulk, or to the fact that it is a good ensemble observable (or these two could be the same thing). Higher-spin theory as a failure of the Eguchi-Kawai mechanism We have presented large-N volume independence along all but one cycle of toroidal compacti cations as a necessary condition for a eld theory to have a local gravitational dual. This is discussed further in section 8.3. Higher-spin theories are a good example of how things go wrong if this does not occur, and provide additional evidence for this conjecture. Higher-spin theories in AdS are nonlocal on the scale of the AdS curvature. There are a zoo of higher-spin theories, so let us analyze one of the simplest cases. Consider the parity-invariant Type-A non-minimal Vasiliev theory with Neumann boundary conditions for the bulk scalar eld [40{42]. This is a theory that can be expanded around an AdS4 background and has elds of all non-negative integer spin. It is proposed to be dual to the three-dimensional, free U(N ) vector model of a scalar eld restricted to the singlet sector [43]. The singlet projection is performed by weakly gauging the U(N ) symmetry with a Chern-Simons gauge eld. The Chern-Simons-matter theory does not enjoy large-N volume independence. In fact, given that the matter is in the fundamental representation, it does not even have center symmetry at the Lagrangian level. However, there is a simple procedure for deforming such theories into close cousins with explicit center symmetry at the Lagrangian level. This is discussed for example in [14]. First we add a global U(Nf ) avor symmetry to the boundary theory, and then we weakly gauge it and change the representation of the matter to be in the bifundamental. Such a theory has explicit center symmetry at the Lagrangian level now exist single-trace, gauge-invariant operators made up of arbitrarily long strings of the bifundamental elds, which did not exist in the previous theory. These are the objects associated to the string states in the bulk. This procedure, with some more bells and whistles (the bells and whistles being an appropriate amount of supersymmetry), is precisely what takes these vector models into the more mature ABJ theory [44, 45]. The bulk interpretation of this procedure is also straightforward and deforms the higher-spin theory into its more mature cousin, string theory. The addition of the global avor symmetry is the addition of Chan-Paton factors to the higher-spin theory, which implies upgrading the spin-1 bulk gauge eld to a nonabelian U(Nf ) gauge eld, with all other elds transforming in the adjoint of U(Nf ). The gauging is then a familiar procedure in AdS/CFT whereby the boundary conditions of this bulk gauge eld are changed. In fact, this entire story is just that of the ABJ triality beautifully painted in [46], whereby the higher-spin \bits" are conjectured to bind together into the strings of ABJ theory. All I would like to highlight is that the deformations that were necessary to connect to a theory with a local gravitational limit included deforming to a theory with an explicitly center-symmetric Lagrangian and center-symmetric phases leads to a lifting [47] of the light states present in vector models [48]. It may be interesting to explore what other deformations of the vector models can introduce center symmetry and the particular center symmetry structure that is a hallmark of classical gravity. This may shed light on how to deform the set of proposed higher-spin dualities for de Sitter space [49{51] to an Einstein-like dual. In the context of de Sitter, the deformation discussed above leads to a \tachyonic catastrophe" in the bulk, as discussed in [52], and does not seem to give a viable option. Learning about the Eguchi-Kawai mechanism from gravity In this section, we will shift our focus and analyze what gravity teaches us about the Eguchi-Kawai mechanism. Center symmetry stabilization and translation symmetry breaking Although this was discussed in previous sections, we would like to emphasize that the bulk gravitational description gives us a way to predict whether volume independence is upheld in particular holographic gauge theories. rst nontrivial statement is that center symmetry can be broken along at most one cycle for any given con guration of cycle sizes. The second nontrivial statement is that there are simple ways to preserve center symmetry along a given cycle for any cycle size which remains larger than string scale in the bulk. In particular, periodic bulk fermions and antiperiodic bulk scalars prevent cycles from capping o in the bulk, as this is an inconsistent spin structure. These cases therefore preserve center symmetry beyond the CSST points which correspond to gravitational Hawking-Page transitions. This argument does not explicitly rely on the representation theory, where the fermions are in the adjoint and the bifundamental, respectively). The bulk matter is made of gauge-invariant combinations of the boundary periodicity conditions of the bulk matter will be correlated with the periodicity conditions of the boundary elds. For example, bulk fermions are constructed by taking single-trace gauge-invariant operators consisting of an odd number of boundary fermionic elds (e.g. ]). Therefore, bulk fermions with periodic spin structure imply boundary fermions with periodic spin structure. A similar statement is true for antiperiodic bulk scalars. Higher-spin dualities, however, o er an interesting case where the bulk theory is purely bosonic while the boundary theory can be purely fermionic. The quantum-mechanically generated potentials for the gauge eld holonomies can be straightforwardly calculated at weak coupling, see for example [11, 53]. From the weakly coupled point of view, for (3 + 1)-dimensional SU(N ) Yang-Mills theories, preserving center symmetry with non-adjoint periodic fermions or antiperiodic scalars of any representation is not possible. The only choice that works is periodic adjoint fermions. Interestingly, for periodic adjoint fermions (which we will have for super Yang-Mills theories) we seem to preserve center symmetry at strong coupling as well. But there is a small catch. At weak coupling, one would need to make the fermions periodic along all k cycles of Tk At strong coupling, however, this will not give us a background well-described by gravity alone, since it will be the toroidally compacti ed Poincare patch with circles shrinking to substringy scales near the horizon. To have a proper gravitational description, we would need to make the fermions antiperiodic along one of the cycles (or the scalars periodic). In this case, we will still preserve center symmetry along all the cycles that have periodic fermions, but this does not match what happens at weak coupling. 1A calculation of the Casimir energy in N = 4 super Yang-Mills on T2 R2 [54], for example, shows that we lose volume independence along both cycles if the fermion is periodic along only one cycle. We have not made any comments about operator expectation values and correlation functions within a grand canonical ensemble, say for turning on a chemical potential for some global symmetry. In this case, one can spontaneously break translation invariance, in which case the Eguchi-Kawai mechanism fails [55]. There exist holographic examples of such spatially modulated phases [56{58]. Extending the Eguchi-Kawai mechanism to curved backgrounds An important question about the Eguchi-Kawai mechanism is whether it extends to curved backgrounds. The original Eguchi-Kawai mechanism, and most modern proofs of largeN volume independence, rely on a lattice regularization which we do not have on curved backgrounds (although see [59] for some progress in the case of spherical backgrounds). We will set this aside for the moment as a technical issue. We will see that the natural uplift of volume independence to curved backgrounds is what I will call \topological volume independence." We will make this notion precise by de ning an order parameter (which will again be the expectation value of a Polyakov loop) and checking in gravitational examples that \topological volume independence" is indeed realized. eld theory. We already have some hints from eld theory about what the Eguchi-Kawai mechanism on curved manifolds should look like. The rst hint comes from the perturbative intuition for volume independence on torus compacti cations. In particular, mesons and glueballs form the con ned phase degrees of freedom (baryons have masses that scale with N and can be ignored for our purposes), and interactions between theory behaves as if it is free. The con ned phase degrees of freedom are therefore incapable of communicating with their images to discover they are in a toroidal box. This intuition, however, is valid even in a curved box. This seems to suggest the size of the manifold should again not be relevant even if it is curved. But curved backgrounds have local curvature which can vary as you change the overall size of the background, e.g. increasing the radius of a sphere. There is no reason the mesons and glueballs cannot feel this local curvature at leading order in N and thereby (for maximally symmetric manifolds like a sphere of hyperboloid) would know the overall size of the compact manifold on which they live. So it seems we should not expect a totally general uplift to curved backgrounds. The second hint comes from thinking about volume independence in toroidal compacti cations as a generalized orbifold projection, where one orbifolds by a discrete translation group [14]. (The language of orbifolds here is conventional but everything is really a smooth quotient.) Generic changes in the overall size of curved backgrounds cannot be thought of this way, so we again see that we cannot expect a totally general uplift to Combining the two hints above provides a compelling case for what kind of setup has a chance of maintaining a useful notion of volume independence. One begins with a curved background and considers smooth quotients that change the volume of the manifold. Such operations do not change the local curvature and maintain the picture of volume-changing as an orbifold procedure. This therefore utilizes the two hints above. We can now check that gravity provides a calculable setup where this proposal for the Eguchi-Kawai mechanism on curved backgrounds can be checked to be valid. The simplest case to analyze is the conformal eld theory on any simply connected manifold, like the sphere or the hyperboloid. As an illustrative example, we will investigate the family of lens spaces formed by smooth quotients of S3, although our results are general. For any smooth quotients of simply connected manifolds, we will see that the Polyakov loop expectation value continues to serve as an order parameter for center symmetry. Holographic realization of the Eguchi-Kawai mechanism on curved manifolds. Holographic gauge theories in the gravitational limit realize all of the intuition of the above. They explicitly show that naive volume-independence on curved backgrounds does not hold. Furthermore, they show that topological volume independence does hold when interpreted in the above sense! To see that naive volume independence on curved backgrounds does not hold, we can consider an observable as basic as the zero-point function, or the free energy density. We saw that for torus-compacti ed holographic theories, the free energy density was volumeindependent due to the thermodynamics of black branes. For holographic theories on a sphere or the hyperboloid, this is no longer the case. The relevant bulk geometries are the spherical and hyperbolic black holes. The key di erence between these geometries and the black brane is that the horizon radius is not proportional to the Hawking temperature. Instead, we have This means that the Bekenstein-Hawking area law, which scales as rd 1 and gives the thermal entropy of the CFT, is not extensive in eld theory variables (i.e. does not scale as T d 1). Here it is important to keep in mind that the theories we are considering are xing the temperature dependence xes the volume dependence. Moreover changing the radius of the sphere or hyperboloid can equally well be regarded as changing we do not have extensivity of the thermal entropy or the free energy, unless rh ! 1 which pushes us into the black brane limit. Furthermore, correlation functions in these backgrounds have nontrivial volume-dependence. While the ideas of large-N volume independence do not apply, there may still be a lower-dimensional matrix model description of the higher-dimensional theory, see e.g. [60{62] Both of these problems are solved by considering the smooth orbifolds suggested in the previous section. The entropy density (or free energy density) becomes appropriately volume-independent because smooth orbifolds of the spatial manifold cannot be interpreted as changes in the temperature. Thus, the nonlinear relation between horizon radius and temperature is not a problem. Said another way, we consider a setup where our eld theory is on a manifold M d 1 and its thermal ensemble at high temperature (i.e. the decon ned theory) is dominated by a black object with horizon topology M for the eld theory on a sphere, plane, or hyperboloid. The quotient of the manifold Md 1 d 1. This is what happens by some freely acting group changes the Bekenstein-Hawking entropy as follows: SBH = We see from this formula that the eld theory's entropy density and free energy density is appropriately independent of such changes in volume, as long as no CSST occurs (more on this possibility below). How about correlation functions? As we saw before, these are constructed by bulk Witten diagrams, whose atoms are bulk-to-bulk and bulk-to-boundary propagators. These objects again obey a Green function equation in the bulk, meaning any orbifold of the background geometry can be dealt with by summing over orbifold images. As long as we remain at leading order in N , meaning we do not consider bulk loops, the correlator will pick up a trivial volume dependence fully determined by the volume-dependence before quotienting. We have analyzed volume independence in the decon ned phase of the theory, where the relevant bulk geometries which dominate the thermal ensemble are given by black holes with some horizon topology. Uplifting the intuition from our torus-compacti ed theories, we should expect to nd nontrivial volume-dependence and temperature-independence in the con ned phase of the theory. We will address this in the next section. It is interesting that the gravitational description and the eld theory description give the same hints as to what sort of generalization to curved backgrounds should work. In particular, we discussed how from the eld theory point of view we should expect volume-changing orbifolds to be the natural uplift of the Eguchi-Kawai mechanism to curved backgrounds. Gravity gives the exact same intuition, and furthermore it explicitly demonstrates that it works, at least for the types of observables considered above. Order parameter on curved manifolds and testing topological volume independence. For any simply connected manifold M d 1, the quotient by some freely acting gives a manifold with nontrivial fundamental group isomorphic to . This means that we can wrap a Polyakov loop on the existing nontrivial cycle and could reasonably expect that its expectation value continues to serve as a good order parameter. We will see in a concrete example that this is the case. To illustrate the point, consider the family of lens spaces L(p; 1) which have < 4 =p : (6.3) d 23=p = Volume independence for lens spaces can now be stated in terms very close to that of the generalized orbifold projections used to discuss volume independence for torus compacti cations. Just as we vary the size of a circle in a torus compacti cation by shifting its periodicity, in this case we move between lens spaces by changing the periodicity of the coordinate. To maintain a smooth quotient we need p 2 Z+ so these are discrete changes. by the change in the circle. We can wrap a Polyakov loop around the circle due to the nontrivial homotopy, and it is again the expectation value of this loop which we propose serves as our order parameter. Let us turn to the gravity picture. In the decon ned phase, the orbifolded circle is non-contractible in the bulk, which implies a vanishing Polyakov loop expectation value and therefore volume independence: ds2 = + r2d 23=p : As we showed in the previous section, topological volume independence is indeed realized in the free energy density and correlation functions. How about the con ned phase? The naive geometry for the con ned phase is obtained by taking a quotient of global AdS. This geometry has a conical singularity at the origin which is not well-described within gravity. For antiperiodic fermions along the orbifolded circle (with even p > 2), it has been proposed that closed string tachyon condensation regularizes the geometry into what is called the Eguchi-Hanson-AdS soliton [63, 64]. This geometry has the orbifolded circle smoothly capping o in the interior, giving a nonvanishing expectation value to the Polyakov loop. There is a decon ning CSST at inverse temperature c = 2 8p2 + 20)3=2 (This corrects the expression given in (4.14) of [65].) In the con ned phase, an analysis of the Eguchi-Hanson soliton shows that we have topological volume-dependence with respect to the spatial manifold and volume-independence with respect to the thermal circle! This picture of topological volume independence is also found in ABJM theory through a nontrivial calculation utilizing supersymmetric localization on lens spaces [66]. An intuitive way to understand the absence of nite-size e ects is to transmute the connections along the orbifolded circle is discussed in e.g. [67]. The topological volume independence that we discuss seems to be controlling the relaon S2, as discussed on the gravity side in [68] and the eld theory side in [69]. An important distinction we draw here from previous work is that the precise pattern of center symmetry breaking/preservation in the gravitational picture is not realized at weak coupling. It would be fascinating to carry out weakly coupled tests of our proposal for topological volume independence of gauge theory on quotients of simply connected manifolds. A simple case to analyze is that of (3 + 1)-dimensional gauge theory on a lens space. In particular, our arguments (and weakly coupled intuition from an ordinary circle compacti cation of at space) suggest that periodic adjoint fermions along the Hopf ber of the lens space should lead to topological volume independence at weak coupling. Extensivity of the Bekenstein-Hawking-Wald entropy The Bekenstein-Hawking area law is a universal formula in Einstein gravity that applies to black hole horizons, cosmological horizons, and in a certain sense to spacetime itself. Let us restrict the discussion to black hole horizons and focus on the scaling with area, ignoring the AdS, since this corresponds to the asymptotically high-temperature limit of the eld theory where the entropy should become extensive [70]. As discussed in the previous section, in this limit the scaling of the eld theory entropy with the spatial volume maps directly to the scaling with the area of the horizon in the bulk. The Eguchi-Kawai mechanism, when manifested as the volume-independence of entropy density, seems to be exactly the sort of tool necessary to provide a general mechanism for the area law. But there are several puzzling and ultimately insurmountable features in trying to pinpoint an exact scaling with area purely from the Eguchi-Kawai mechanism (except for large toroidally compacti ed black branes in AdS). We will instead see that the mechanism explains a more general \area" law: the extensivity of the Bekenstein-Hawking-Wald entropy.2 Before considering higher curvature corrections, however, let us investigate how the Bekenstein-Hawking area law is at least consistent with the Eguchi-Kawai mechanism, even if not predicted by it. In AdSd+1/CFTd, we may ask why toroidally compacti ed black branes above the Hawking-Page phase transition have no subextensive piece in their classical entropy. Fixing to a spatial torus, as ! 0 we expect to get an entropy scaling of the conformal eld theory . Since the bulk Hawking temperature scales as T rh, this gives S d 1Vd 1 in bulk variables, which is precisely the Bekenstein-Hawking area law. However, h as the temperature is lowered we should generically expect subextensive corrections to the thermal entropy, which would spoil the universal area law in the bulk since T tained for black branes at any temperature. However, the Eguchi-Kawai mechanism saves the day, and implies that no such corrections can appear until one undergoes a CSST, whose location can be determined as discussed in section 3. This uses the Eguchi-Kawai mechanism to generalize Witten's explanation of the Bekenstein-Hawking area law to all toroidally compacti ed black branes above the Hawking-Page transition. Of course, if a periodic spin structure is chosen for the fermions along all spatial cycles, then no such transition appears in the gravitational regime and we can explain the area law for arbitrary toroidally compacti ed black branes. This is just a recap of what was shown more carefully in section 3. What about the Bekenstein-Hawking area law for black hole horizons with curvature, like the spherical or hyperbolic black holes in AdS? Again adopting center-symmetry preservation along the orbifolding cycle (up to any CSST) as our working assumption, we deduce that the entropy density in the eld theory is volume-independent in the orbifold2The language here and in the literature is very confusing. We refer to the Bekenstein-Hawking entropy as extensive even though it is very famously subextensive. By this we mean extensive in horizon area not volume. Also, the Wald entropy is sometimes referred to as providing subextensive corrections to the Bekenstein-Hawking area law, by which it is meant terms that do not scale with the area of the event horizon. When we refer to the extensivity of the Wald or Bekenstein-Hawking-Wald entropy, we mean the fact that it can be written as an integral of a local quantity over the horizon of the black hole. We will discuss this further below. ing direction. The orbifolding direction is a discrete direction, indexed by an integer p in the previous section. Any potential analytic continuation to complex p is on very shaky ground, but the Bekenstein-Hawking area law for the original spherical or hyperbolic black hole may be understood by analytic continuation from the discrete family of quotiented geometries. This is akin to understanding entanglement entropy through the discrete Renyi family, although there the analytic continuation is on much rmer footing. If these ideas are correct, then they provide a mechanism for the area law for large black holes with horizon topology which dominate the canonical ensemble for some dual eld theory on background . What about small black holes? Here the interpretation in terms of plasma balls in the dual large-N gauge theory may be useful [71]. It may then be true that the Eguchi-Kawai mechanism applies to this decon ned plasma ball in a way which maps to the area law in the bulk, as we saw for large black holes above. Stringy corrections and extensivity of the Wald entropy. We can ask about subleading order in the 't Hooft coupling , which should correspond to bulk stringy corrections. One way these stringy corrections manifest themselves is as higher-curvature corrections to the bulk Einstein gravity. The Polyakov loop analysis remains the same and continues to indicate center symmetry preservation along d 1 cycles. Thus a center-symmetry analysis in the eld theory predicts that for any planar/spherical/hyperbolic black holes, the entropy density should be volume-independent in any smooth orbifolding direction. To check this, we can look at zero-point functions like the entropy density. Since we have higher-curvature corrections we need to use the Wald formula for black hole entropy. For toroidally compacti ed black branes, the area law is maintained although the coe cient can change. For spherical or hyperbolic black holes, we have corrections to the BekensteinHawking area law which do not scale with the area of the horizon. This seems to be in contradiction with the Eguchi-Kawai mechanism. To address this, let us step back for a There is a spiritually correct but technically incorrect holographic explanation of the Bekenstein-Hawking area law that is often given. It says that the scaling with area is because there is a holographic dual theory in one lower dimension with the same entropy, and its entropy is scaling with volume as it should be. This captures the holographic spirit, but in general it is technically incorrect as can be seen in many ways. If the area maps to a eld theory volume, does the 1/GN map to temperature? This is of course wrong. Even in the cases where the area does map rigorously to volume, like toroidally compacti ed black branes, why does the eld theory not exhibit any subextensive corrections to its entropy? This we explained within our framework of large-N volume independence. Finally, what about higher curvature corrections? In the bulk the entropy picks up what are sometimes confusingly called \subextensive corrections to the Bekenstein-Hawking area law" from the Wald entropy formula. This ruins the Bekenstein-Hawking area law. Interpreted as bulk stringy corrections and therefore as corrections in the gauge coupling of a dual eld theory, why should going to weaker coupling ruin extensivity? These issues are clari ed by recalling that the Wald entropy is an integral over the event horizon and is therefore extensive. Consider a black hole with metric ansatz ds2 = is independent of r and t. This does not capture the most general case but will su ce for the argument. The Wald entropy for a general di eomorphism-invariant higher-curvature theory of gravity with Lagrangian density L is given as an integral along is the binormal to the horizon. The corrections implied by the Wald entropy are terms that do not scale as rhd 1, which is the scaling of the Bekenstein-Hawking entropy. But notice that the general theory will still scale with the volume of : SW Vol( ). This is what we mean by extensivity, which as before can be thought of in terms of quotients of SW = ! Md 1= =) SW ! SW =j j : In this sense the general Wald entropy | therefore the entropy in an arbitrary di eomorphism-invariant theory of classical gravity | is just as extensive as the Bekenstein-Hawking entropy. For black branes this means that the Wald entropy To bring this extensivity of curved horizons into clearer focus, consider quantum (subleading in GN , i.e. subleading in N ) corrections to the Bekenstein-Hawking-Wald entropy. At rst order, these are logarithmic in the area of the event horizon: SW + log(SW ) + : : : : The correction neither scales with the area of the horizon nor with Vol( ). It is truly This discussion should make clear that the gravity that emerges from our center symmetry analysis is not necessarily Einstein gravity. Nevertheless, it would be fascinating if somehow the stringency of this center symmetry structure necessitated a CFT with an Einstein gravity dual. One way this could occur is by requiring a sparse higher spin spectrum [31] | recently shown to give c a for the anomaly coe cients c and a in four-dimensional CFTs [72] | just as it required a sparse spectrum of low-lying states to reproduce the extended range of validity of the general-dimensional Cardy formula. In this spirit, it is encouraging that restoration of a center symmetry plays an important role in deforming higher-spin theory (within which the higher spin elds cannot be made sparse) into ABJ theory (within which they can). Reproducing additional features of AdS gravity We have shown that several universal features of AdS gravity can be reproduced with the starting assumption of center symmetry preservation along all but one cycle in a large-N theory (and the suitable generalization of this statement to curved backgrounds as discussed before). However, there are still several features that we would like to explain. A powerful technical assumption in the context of reproducing universal features of gravity in AdS3/CFT2 is that of Virasoro vacuum block domination of the four-point function on the sphere. This is expected to be a valid assumption in large-c theories with a sparse light spectrum and sparse low-lying operator-product-expansion (OPE) coe cients. This suggests that it might be implied by our framework. More precisely, consider a fourpoint function hO1(1); O2(z)O3(1)O4(0)i, which can be decomposed into representations of the Virasoro algebra (i.e. into Virasoro blocks) by inserting a complete set of states. It is believed that taking c ! 1 with external and internal operator dimensions scaling with c leads to an exponentiation of the Virasoro block [73, 74]: i = 1; 2; 3; 4 ; where hp is the internal operator dimension. Now taking z ! 0 leads to vacuum block leading OPE singularity from bringing together O2 and O4: F (c; hp; hi; z) = zhp h2 h4 (1 + O(z)) : In holographic theories, vacuum block dominance | like the Cardy formula we discussed in (3) | seems to have an extended range of validity, which in this case means for a range of z beyond the asymptotic limit z ! 0. This requires a sparseness bound both on the spectrum of states and on the operator product expansion coe cients. Our framework requires large c to begin with and reproduces a sparse light spectrum as discussed in section 3. Data about the OPE coe cients is also accessible in this framework since treelevel Witten diagrams have bulk interactions. Concretely, one may hope to analyze more carefully volume independence for the blocks between the sphere and the torus, possibly using the tools of [75{79]. An orthogonal clue that vacuum block dominance may be implied by this framework is a calculation of the entanglement entropy in a heavy microstate on a circle [80{82], which gives an answer independent of the size of the circle! Accessing some quantity or feature which directly exhibits the smooth, geometric nature of the bulk is another natural goal for this framework. The singularities of [83] are one such feature that indicate a sharp geometric structure. Reducing or blowing up models The strong coupling description of holographic theories makes manifest that one can achieve full volume-independence (i.e. preserve center symmetry for all cycle sizes) along directions with periodic (antiperiodic) boundary conditions for fermions (bosons), as long as one direction has the opposite boundary conditions and caps o in the interior. then perform a large-N reduction of these theories down to matrix quantum mechanical theories, i.e. (0 + 1)-dimensional theories. For a discussion of the validity of the reduction down to zero size, see appendix C. This captures physics in both con ned and decon ned phases. When describing thermal physics in the gravitational limit, there will always be one direction that does not reduce, prohibiting the reduction to a matrix model description, i.e. a (0 + 0)-dimensional theory. (See [84] for a discussion of subtleties in dimensionally reducing volume-independent theories.) Blowing up low-dimensional models is another interesting direction to pursue, especially in light of recent developments in low-dimensional models like the Sachdev-Ye-Kitaev (SYK) model, which captures some features of AdS2 gravity. The addition of avor to the SYK model [85] gives it the necessary ingredient to be blown up into a higher-dimensional model by the methods of [86, 87]. (See also [88, 89] for a di erent kind of blow-up.) The necessity of the Eguchi-Kawai mechanism for holographic gauge theI have intermittently referred to the Eguchi-Kawai mechanism as a necessary feature of holographic gauge theories. In a certain sense, this is obviously ridiculous. Center symmetal matter eld, although we still have a controlled gravitational description of the infrared physics. In this case, what I really mean is that there exists a theory which at large N is equivalent to the one with a single fundamental eld, but which has center symmetry at the Lagrangian level. More simply, the fundamental matter decouples at leading order in N , so the center symmetry is emergent at in nite N . As explored heavily in the literature on large-N volume independence and mentioned in the introduction, orbifold/orientifold dualities in many cases imply an emergent center symmetry at in nite N , even when centerbreaking matter does not naively decouple [21, 22]. It is this generalized emergent sense in which the Eguchi-Kawai mechanism is necessary. In other words, there is a possibility that center symmetry (whether existing explicitly or emergent) is playing an indispensable role in realizing the precise form of volume independence necessary to admit a gravitational description. Absent conclusive evidence to the contrary, I conjecture this to be the case. It would be nice to have a formalism centered around center symmetry that does not use the crutch of gauge theory, which may be an unnecessary redundancy of description.3 Interesting cases to study, which may teach us about large-N equivalences, are that of the D1-D5 system and of attempts at describing unquenched avor in AdS/CFT. At the orbifold point, the D1-D5 theory can be thought of as a free symmetric orbifold CFT. It is a gauge theory, but the gauge group is SN which has a trivial center. Nevertheless, this theory seems to have at least some aspects of large-N volume independence. It realizes the phase structure of gravity, and certain correlators can be written as a sum over images [91]. Indeed, the physics of long strings/short strings and sharp transitions (see for example [92, 93]). The case of unquenched avor requires keeping Nf =N N ! 1, which means the avor does not decouple at leading order in N . If there is a 3It was pointed out to me by Brian Willett that center symmetry can be discussed in the language of one-form global symmetries, without the need for a Lagrangian, as developed in [90]. smooth gravitational description in AdS (or some similarly warped spacetime), then the nature of nite-size e ects should be analyzed. There are many directions to pursue with these ideas in the context of AdS/CFT, only some of which were addressed above. Taking a broader view of the subject, it is clear that holographic dualities which have rules like those of AdS/CFT will have similar volumeindependent structure in correlation functions and phase structures. It is remarkable that rst introduced by Eguchi and Kawai is relevant only in the context of largeN gauge theories, and even then only at leading order in N . It is as if it was tailor-made to explain classical gravity, whether within AdS or with some other asymptotia. Indeed, one universal feature of classical gravity we can hang our hats on, robust against changes in asymptotia, is the extensivity of the Bekenstein-Gibbons-Hawking-Wald entropy. The universality of this simple formula only exists at leading order in GN , and we saw that in the context of AdS/CFT it maps to universal volume-independence at leading order in N for certain black holes. It is natural to conjecture that the same mechanism is controlling the entropy for all black holes, although as discussed in the main text this statement should be interpreted with care. The capability of these ideas in addressing classical gravity more this is a useful and technically accurate perspective beyond AdS/CFT remains to be seen. I am greatly indebted to Aleksey Cherman for his many patient explanations of modern developments regarding the Eguchi-Kawai mechanism. I would like to thank Tarek Anous, Aleksey Cherman, and Raghu Mahajan for useful conversations and comments on a draft. I would also like to thank Dionysios Anninos, David Berenstein, William Donnelly, David Gross, Gary Horowitz, Nabil Iqbal, Zohar Komargodski, Don Marolf, Mark Srednicki, Tomonori Ugajin, Mithat Unsal and Brian Willett for useful conversations. In this section we will review some basic points about SL(d; Z), the mapping class group of Td. When d is even, we will want to consider PSL(d; Z) instead, obtained by quotienting by the center f1; 1g. For simplicity we will just refer to the group as SL(d; Z) with this Naively, the torus is parameterized by d arbitrary real vectors V1; : : : ; Vd in ddimensional space. However, we can use global rotational invariance to eliminate Pd 1 overall size modulus. The torus now has d2 1 = (d 1)(d + 2)=2 real moduli. Calling the coordinates x1; : : : ; xd, we have a twist modulus ij between xi and all xj with i=1 i = i < j, and a size modulus ii for d 1 of the cycles xi. Keeping the overall size modulus explicit, we can arrange the moduli in terms of the following lattice vectors: 2 V~1 3 6 V~2 77 u1 = 66 U1 = 660 1 U2 = 660 0 1 Generators. In this section we will list four sets of generators of SL(d; Z) and show them to be equivalent. Our rst two sets of generators of SL(d; Z) can be written as u2 = 660 0 1 d matrices. The small u's can be shown to generate the big U 's and vice versa. The relations for e.g. d = 4 are U1 = u1 1; U2 = u1 1u2u1 2u2u1u2u1 1u2 1u1u2 1u1u2 1u1 1u2u1 1u2u1u2 1u 1 Generating the small u's by the big U 's is obtained by swapping u $ U . We will henceforth stick with the big U 's. U1 cyclically permutes all the entries of a vector while U2 twists the rst vector by an integer amount in the direction of the second vector. The power d + 1 on Another set of generators can be given by a simple generalization of the usual S and T generators familiar from SL(2; Z). In this case, we simply have Sij and Tij along any pair of directions i < j. Beware the notation: Sij is a d d matrix for any given i; j, not the fi; jgth element of a matrix S. Confusingly, S Transposes and T Shears! Better to think of it as S Swaps and T Twists. So we have the elementary row switching (with a minus sign, conventionally placed in the upper triangular part) and upper-triangular row addition (with integer entry) transformations. To see their action more explicitly as matrix multiplication, imagine arranging the lattice vectors row by row into a d-dimensional matrix. Then, for example, T25 twists direction two by an integer in direction transposes lattice vectors as V~1 ! are more diagrams contributing at this order, including the one with the four bulk-to-boundary propagators meeting at a single interaction vertex in the interior. twists in any direction. These include the upper-triangular Tij from the previous section and upper-twists can also generate U1 and U2 as U1 = (S12)(S23) (Sd 1;d) and U2 = T12. Four-point function sample calculation Here we calculate the tree-level contribution to the four-point function illustrated in gure 2. We will calculate it in an AdS background where one direction has size L and another AdS background where the same direction has size J L for J 2 Z+. We will suppress all We rst calculate the correlator in size J L. We have O(x4)iJL = Fourier transforming gives K(s1=J L) K(s4=J L)G(s5=J L); where i = 1; : : : ; 5. Evaluating the x5 and x6 integrals gives s1+s2;s5 s3+s4; s5 e 2JLi (s1x1+ +s4x4)K(s1=J L) K(s4=J L)G(s5=J L) : X J 2L2 s1+s2+s3+s4;0e 2JLi (s1x1+ +s4x4)K(s1=JL) K(s4=JL)G((s1+s2)=JL) ; transform with respect to the variables xi. Recall that the discrete transforms in nite size hO(n1=L) : : : O(n4=L)iJL = X e 2JLi (s1x1+:::s4x4)+ 2Li (n1x1+ +n4x4)K(s1=JL) K(s4=JL)G((s1 + s2)=JL) : Evaluating the integrals and then the sum gives dz5 dz6K(n1=L) : : : K(n4=L)G((n1 + n2)=L) n1+n2+n3+n4;0 : f (x) = X e2 inx=Lf (n=L) =) f (n=L) = dx e 2 inx=Lf (x) : 1 Z L e 2JLi (n01(x1 x5)+ +n04(x4 x6)+n05(x5 x6))K(n01=JL) K(n04=JL)G(n05=JL) (B.14) Now we consider the correlator in size L, where we replace the bulk-to-bulk propagator and the bulk-to-boundary propagators with those of size JL by the method of images: O(x4)iL = ni=0 x5)K(x2 + n2L x6)K(x4 + n4L x6)G(x5 + n5L ni=0 n0i= 1 e 2JLi (n01(x1+n1L x5)+ +n04(x4+n4L x6)+n05(x5+n5L x6))K(n01=JL) K(n04=JL)G(n05=JL) : Switching the two sums and evaluating the sums over ni gives n0i= 1 for arbitrary integer si. Evaluating the sums over n0i gives si= 1 e 2Li (s1(x1 x5)+ +s4(x4 x6)+s5(x5 x6))K(s1=L) K(s4=L)G(s5=L) : s1+s2;s5 s3+s4; s5 e 2Li (s1x1+ +s4x4+s5x5)K(s1=L) K(s4=L)G(s5=L) : s1+s2+s3+s4;0 e 2Li (s1x1+ +s4x4)K(s1=L) K(s4=L)G((s1 + s2)=L) : Performing the x5 and x6 integrals gives si= 1 Performing the sum over s5 gives si= 1 = J L 1 Z L hO(n1=L) O(n4=L)iL = dz5 dz6K(n1=L) K(n4=L)G((n1 + n2)=L) n1+n2+n3+n4;0 : (B.20) This is our nal answer for the correlator in size L. Comparing this answer to (B.8) gives us hO(n1=L) O(n4=L)iL = J 3hO(n1=L) O(n4=L)iJL as predicted by (4.19). This calculation should make clear that (4.19) is correct diagram-by-diagram in the bulk. Moreover, any bulk-to-bulk propagator with momenta that need to be integrated over, as would be the case for loop diagrams, would ruin this structure. This is expected since the presence of such propagators signals a subleading-in-N Witten diagram, for which volume-independence does not apply. Validity of gravitational description For our gravitational description to be valid, we need to deal with smooth geometries and keep cycle sizes larger than string scale. The rst criterion is simply because singularities are not well-described within gravity. The second criterion is because stringy excitations (e.g. strings that wrap the cycles) will become important for cycles that are string scale. In this case, one needs to T-dualize along the small cycle to blow it up. The language here is a bit confusing, as T-dualizing takes us from a valid IIB gravity description to a valid IIA gravity description, but we are concerned with maintaining a valid gravity description in the same frame throughout. Maintaining validity of the gravitational description depends on the periodicity conditions chosen for the matter elds. To be very concrete, let us consider the duality between Type IIB string theory in AdS5 pacti ed on a spatial three-torus of cycle lengths Li. First consider the case where the matter elds are given supersymmetry-preserving boundary conditions along the spatial cycles. In this case the ground state geometry is given by the Poincare patch with periodic identi cations in the spatial directions. But this means that the cycles become arbitrarily small as the horizon is approached, necessitating a breakdown of the IIB gravity description. This was the case analyzed in [18]. However, nite temperature is di erent and necessitates a discussion of the order of limits taken. The Euclidean geometry is that of the black brane: ds2 = f (r) = r2(1 (rh=r))4; rh, the S5 is suppressed, and tE gives the inverse temperature. The minimum proper size of a given cycle i occurs at rh. This size must be bigger than the string scale `s, which gives us the condition `s =) Here we have brought in the 't Hooft coupling . We see that we can make Li arbitrarily small and maintain validity of the gravitational description as long as we take rst. In other words, we do not scale any cycle sizes with the 't Hooft coupling as we take the strong coupling limit The case we were more preoccupied with in the text, especially in section 3, is that of modular U1-invariant boundary conditions. This means supersymmetry-breaking boundary conditions along all cycles. As we saw, this implies that when a cycle size is the smallest, it caps o in the interior. The geometry that dominates is either the black brane or the AdS soliton, whose Euclidean continuations are identical. The condition above therefore where L ;min is the minimum cycle size. By de nition we have L ;min < 1, so this condition is satis ed trivially. Any time a cycle tries to become substringy, it instead caps o . Mixed boundary conditions which preserve some subgroup of the full modular U1 invariance are analyzed similarly. 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Edgar Shaghoulian. Emergent gravity from Eguchi-Kawai reduction, Journal of High Energy Physics, 2017, 11, DOI: 10.1007/JHEP03(2017)011 | CommonCrawl |
Find an integer $x$ such that $\frac{2}{3} < \frac{x}{5} < \frac{6}{7}$.
Multiplying all expressions in our chain of inequalities by $5$, we have $$\frac{10}{3} < x < \frac{30}{7}.$$ Writing this in terms of mixed numbers, we have $$3\frac13 < x < 4\frac27.$$ The only integer $x$ satisfying this chain of inequalities is $\boxed{4}$. | Math Dataset |
What does a subatomic charge actually mean?
I was recently reading a popular science book called The Canon - The Beautiful Basics of Science by Natalie Angier, and it talks about subatomic particles like protons, neutrons and electrons in chapter 3. I came across this section on subatomic charges that made me wonder about the nature of the positive and negative charges that we associate with protons and electrons respectively.
When you talk about a fully charged battery, you probably have in mind a battery loaded with a stored source of energy that you can slip into the compartment of your digital camera to take many exciting closeups of flowers. In saying that the proton and electron are charged particles while the neutron is not, however, doesn't mean that the proton and electron are little batteries of energy compared to the neutron. A particles's charge is not a measure of the particles's energy content. Instead, the definition is almost circular. A particle is deemed charged by its capacity to attract or repel other charged particles.
I found this definition/description a bit lacking, and I still don't grasp the nature of a "subatomic charge", or what do physicists mean when they say that a proton is positively charged and electron is negatively charged?
electrons atoms charge protons neutrons
AnupAnup
$\begingroup$ The same thing they mean when they talk about macroscopic objects being (electrically) charged. Indeed, macroscopic charges are explained in terms of microscopic charges, but at some level you must get down to "because these behaviors are observed and are correctly modeled with this math". $\endgroup$
– dmckee --- ex-moderator kitten
When physicists say that a particle has electric charge, they mean that it is either a source or sink for electric fields, and that such a particle experiences a force when an electric field is applied to them.
In a sense, a single pair of charged particles are a battery, if you arrange them correctly and can figure out how to get them to do useful work for you. It is the tendency for charged particles to move in an electric field that lets us extract work from them.
A typical electronic device uses moving electrons to generate magnetic fields (moving electrons cause currents, and currents generate magnetic fields) and these magnetic fields can move magnets, causing a motor to turn. What is happening at a fundamental level is that an electric field is being applied (via the potential across the battery) that is causing those electrons to move.
If I wanted a magnetic field to be generated, I could get one from a single pair of charges, say, two protons placed next to one another. The protons will repel (like charges repel) and fly away from each other. These moving protons create a current (moving charge) which creates a magnetic field.
Your author is right when he says that charges attract or repel other charges. To help connect it to more familiar concepts, consider this: The negative end of your battery terminal attracts electrons and the positive end repels them. (The signs of battery terminals are actually opposite the conventional usage of positive and negative when referring to elementary charges. As a physicist, I blame electrical engineers.) The repelled and attracted electrons start moving, and these moving electrons can be used to do work.
KDNKDN
To expand on my comments, electrical phenomena includes a force between electrically charged objecs of $\vec{F}_1 = -k\frac{q_1 q_2}{r^2_{1,2}}\hat{r}_{1,2}$ where $q_1$ and $q_2$ are the charges on two objects, $r_{1,2}$ is the postion vector from object 1 to object 2 and $k$ is a constant of proportionality that depends on the system of units you choose.
We can measure those forces very simply in a macroscopic setting and with a little more work in a microscopic setting.
And this interaction (and the rest of electrodynamics) is essentially how we define electrical charges at all scales.
The "little more work" I refer to above means things like scattering experiments where you throw one charged particle past another and look at how the angle of deflection varies with the impact parameter.
dmckee --- ex-moderator kittendmckee --- ex-moderator kitten
$1$. The nature of a "subatomic charge" "The reason for the electric charge is that the quarks couple to a U(1) gauge vector-potential also known as the photon, which endows the quark with an electric charge. .. The theory encompassing the quarks and their properties is called Yang-Mills theory with associated Lie algebras U(1), SU(2), and S(3) which are called gauge groups."
quoted from Krchov2000 See Yang-Mills theory (Wikipedia).
However, I can't just say that charge is a fundamental property which we must just accept to be true without an explanation.Rather, the explanation is too complicated to just 'answer', you need to study entire subjects to get the 'real' explanation. You need to learn stuff like Yang-Mills theory to actually understand the nature of subatomic charge.
$2$. What do physicists mean when they say that a proton is positively charged and electron is negatively charged?
Charge refers to a property. When an electron is charged with a certain magnitude or charge, then we have a certain magnitude of electrical force etc. according to equations like $F=\frac {kq_1 q_2}{r^2} $, where $q_1$ and $q_2$ are the charges of two particles.
The positive and negative charges are just by definitions: if we wanted, we could define protons to be negatively charged and electrons to be positively charged. What really matters is to understand that protons and electrons have opposite charges.
raindropraindrop
$\begingroup$ While this is correct I'm not sure that it is the best approach to take with questioner is unsure of what is meant by "charge". $\endgroup$
Not the answer you're looking for? Browse other questions tagged electrons atoms charge protons neutrons or ask your own question.
What does "charge" actually mean?
A basic confusion about what is an atom
Fundamental difference between neutron and proton
Why is an electron negatively charged, and what is the difference between negative and positive charges?
What is meant by protons having positive charge?
How accurately is it known that protons have the same charge as electrons?
Why does this quiz question say that protons and electrons do not combine to form neutrons?
Does the electron capture process ever occur in hydrogen? | CommonCrawl |
\begin{definition}[Definition:P-Product Metric/Real Number Plane]
Let $\R^2$ be the real number plane.
Let $p \in \R_{\ge 1}$.
The '''$p$-product metric''' on $\R^2$ is defined as:
:$\ds \map {d_p} {x, y} := \sqrt [p] {\size {x_1 - y_1}^p + \size {x_2 - y_2}^p}$
where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.
\end{definition} | ProofWiki |
by Rudolf Kohulák. Published on 28 April 2016.
Due to the immense popularity of tea, many heated arguments have been started in kitchens, at dinner tables and indeed on the internet concerning the proper way of making it. There is a lot of confusion surrounding tea–making, including—but not limited to—the question of the right brewing temperature, the amount of milk one should add and the correct order of adding milk and hot water. Clearly there's way too many variables and one can easily get lost in all that madness. On top of this, looking for advice on the internet results in an inundation of contradictory advice and very hysterical arguments. So let's put all the emotions aside and seek refuge in the realms of maths and science.
Recently, students from the University of Leicester came up with a formula for the perfect brew. According to them, one should add 200ml of boiled water, let it brew for 2 minutes, then add 10ml of milk and wait 6 minutes for it to reach its optimum temperature of 60°C. Which is all very nice but not everyone can be bothered to measure the exact amount of milk and water and, ultimately, we don't live in an ideal world where one can start making a cup of tea safe in the knowledge that there will be no interruption halfway through. Since we at Chalkdust are mainly concerned with practical consumer advice, let us consider the scenario described below (in the discussion that follows we shall assume that milk is added after the water. Partly because if you don't, it would make the whole article meaningless, but mostly because any other practice is simply wrong and should not be encouraged in any way).
We've all been there, you boil the water, pour it into the cup and suddenly the doorbell rings. It could be the postman, your neighbour asking for some sugar, or a religious enthusiast trying to save you from eternal damnation. Either way, it's going to be a few minutes before you can enjoy your tea. Obviously, you would like to maximise your chances of your tea still being nice and hot upon your return. What should you do? Add milk now and answer the door or pour the milk later once the nuisance business has been taken care of?
To start answering the question, we need to have an idea of how an object cools down. According to Newton's law of cooling, the rate of cooling of an object is negatively proportional to the difference between its temperature and the temperature of the environment.
This model assumes that all the thermodynamic properties of the system, (such as the heat capacity, thermal conductivity, etc) can be 'hidden' in that one constant k. But what is its value? The best way to find out is to perform some experiments. So we bought some thermometers, put the kettle on and made some measurements. We measured the temperature of the room (which was quite cold, only 16°C!) and made a note of the temperature of the boiled water in 1-minute intervals. Then the next step is relatively simple. We need to fit a value for k that minimises the error between the measured temperatures and the values predicted by the model. For our purposes, we chose the sum squared error as our measure and asked Microsoft Excel do the rest of the work for us. This gave us a value of k equal to roughly 0.03.
where $\alpha$ is the volume of liquid one relative to the whole mixture.
But does this actually work? To find out we once again grabbed the thermometers and boiled some water. We tested three scenarios: $t_m$ = 0, 5 and 10. Due to lack of time and tiredness, we decided to perform all three of them at once. However, due to a lack of manpower and equipment we could not measure all three of them at exactly the same time, so we measured with a 20s time delay between the three cases. After having done that, we randomised the positions of the cups and their associated time delay and repeated the experiments.
The results are plotted in the figure above. It is clear that the model gives a slightly wrong temperature for the smaller times. This is probably due to the fact that the cooling effects of the containers are not accounted for. However, putting that aside, the predictions are NOT TOO BAD. According to the measurements (and indeed the model), putting in the milk earlier resulted in higher final temperatures.
So while the maths needed lots of constants and parameter values, our advice to you—unlike that of the students at the University of Leicester—is simple: the next time the doorbell rings while you are making your daily brew, add the milk immediately!
Rudolf Kohulák is a PhD student at UCL working on the modelling of freeze-drying processes.
We created hot ice from scratch, a solution that remains liquid even below its freezing point! | CommonCrawl |
Infinite translation surfaces with infinitely generated Veech groups
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Structure of attractors for $(a,b)$-continued fraction transformations
October 2010, 4(4): 693-714. doi: 10.3934/jmd.2010.4.693
Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory
Maxime Zavidovique 1,
Unité de Mathématiques Pures et Appliquées, École Normale Supérieure de Lyon, siteMonod, UMR CNRS 5669, 46, allée d'Italie, 69364 LYON Cedex 07, France
Received April 2010 Revised October 2010 Published January 2011
In this article, following [29], we study critical subsolutions in discrete weak KAM theory. In particular, we establish that if the cost function $c: M \times M\to \R$ defined on a smooth connected manifold is locally semiconcave and satisfies twist conditions, then there exists a $C^{1,1}$ critical subsolution strict on a maximal set (namely, outside of the Aubry set). We also explain how this applies to costs coming from Tonelli Lagrangians. Finally, following ideas introduced in [18] and [26], we study invariant cost functions and apply this study to certain covering spaces, introducing a discrete analog of Mather's $\alpha$ function on the cohomology.
Keywords: Aubry Mather theory, Hamiltonian and Lagrangian dynamics., Ilmanen's lemma, twist maps, critical subsolutions, Discrete weak KAM theory.
Mathematics Subject Classification: Primary: 37J50; Secondary: 37E40, 37B2.
Citation: Maxime Zavidovique. Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory. Journal of Modern Dynamics, 2010, 4 (4) : 693-714. doi: 10.3934/jmd.2010.4.693
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Maxime Zavidovique | CommonCrawl |
List of triangle topics
This list of triangle topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or in triangular arrays such as Pascal's triangle or triangular matrices, or concretely in physical space. It does not include metaphors like love triangle in which the word has no reference to the geometric shape.
Geometry
• Triangle
• Acute and obtuse triangles
• Altern base
• Altitude (triangle)
• Area bisector of a triangle
• Angle bisector of a triangle
• Angle bisector theorem
• Apollonius point
• Apollonius' theorem
• Automedian triangle
• Barrow's inequality
• Barycentric coordinates (mathematics)
• Bernoulli's quadrisection problem
• Brocard circle
• Brocard points
• Brocard triangle
• Carnot's theorem (conics)
• Carnot's theorem (inradius, circumradius)
• Carnot's theorem (perpendiculars)
• Catalogue of Triangle Cubics
• Centroid
• Ceva's theorem
• Cevian
• Circumconic and inconic
• Circumscribed circle
• Clawson point
• Cleaver (geometry)
• Congruence (geometry)
• Congruent isoscelizers point
• Contact triangle
• Conway triangle notation
• CPCTC
• Delaunay triangulation
• de Longchamps point
• Desargues' theorem
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The (incomplete) Bell polynomials from a triangular array of polynomials (see also Polynomial sequence).
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| Wikipedia |
Social Science Research in the Arab World and Beyond pp 51–86Cite as
Bivariate Analysis: Associations, Hypotheses, and Causal Stories
Mark Tessler2
First Online: 04 October 2022
Part of the SpringerBriefs in Sociology book series (BRIEFSSOCY)
Every day, we encounter various phenomena that make us question how, why, and with what implications they vary. In responding to these questions, we often begin by considering bivariate relationships, meaning the way that two variables relate to one another. Such relationships are the focus of this chapter.
3.1 Description, Explanation, and Causal Stories
There are many reasons why we might be interested in the relationship between two variables. Suppose we observe that some of the respondents interviewed in Arab Barometer surveys and other surveys report that they have thought about emigrating, and we are interested in this variable. We may want to know how individuals' consideration of emigration varies in relation to certain attributes or attitudes. In this case, our goal would be descriptive, sometimes described as the mapping of variance. Our goal may also or instead be explanation, such as when we want to know why individuals have thought about emigrating.
Description means that we seek to increase our knowledge and refine our understanding of a single variable by looking at whether and how it varies in relation to one or more other variables. Descriptive information makes a valuable contribution when the structure and variance of an important phenomenon are not well known, or not well known in relation to other important variables.
Returning to the example about emigration, suppose you notice that among Jordanians interviewed in 2018, 39.5 percent of the 2400 men and women interviewed reported that they have considered the possibility of emigrating.
Our objective may be to discover what these might-be migrants look like and what they are thinking. We do this by mapping the variance of emigration across attributes and orientations that provide some of this descriptive information, with the descriptions themselves each expressed as bivariate relationships. These relationships are also sometimes labeled "associations" or "correlations" since they are not considered causal relationships and are not concerned with explanation.
Of the 39.5 percent of Jordanians who told interviewers that they have considered emigrating, 57.3 percent are men and 42.7 percent are women. With respect to age, 34 percent are age 29 or younger and 19.2 percent are age 50 or older. It might have been expected that a higher percentage of respondents age 29 or younger would have considered emigrating. In fact, however, 56 percent of the 575 men and women in this age category have considered emigrating. And with respect to destination, the Arab country most frequently mentioned by those who have considered emigration is the UAE, named by 17 percent, followed by Qatar at 10 percent and Saudi Arabia at 9.8 percent. Non-Arab destinations were mentioned more frequently, with Turkey named by 18.1 percent, Canada by 21.1 percent, and the U.S. by 24.2 percent.
With the variables sex, age, and prospective destination added to the original variable, which is consideration of emigration, there are clearly more than two variables under consideration. But the variables are described two at a time and so each relationship is bivariate.
These bivariate relationships, between having considered emigration on the one hand and sex, age, and prospective destination on the other, provide descriptive information that is likely to be useful to analysts, policymakers, and others concerned with emigration. They tell, or begin to tell, as noted above, what might-be migrants look like and what they are thinking. Still additional insight may be gained by adding descriptive bivariate relationships for Jordanians interviewed in a different year to those interviewed in 2018. In addition, of course, still more information and possibly a more refined understanding, may be gained by examining the attributes and orientations of prospective emigrants who are citizens of other Arab (and perhaps also non-Arab) countries.
With a focus on description, these bivariate relationships are not constructed to shed light on explanation, that is to contribute to causal stories that seek to account for variance and tell why some individuals but not others have considered the possibility of emigrating. In fact, however, as useful as bivariate relationships that provide descriptive information may be, researchers usually are interested as much if not more in bivariate relationships that express causal stories and purport to provide explanations.
Explanation and Causal Stories
There is a difference in the origins of bivariate relationships that seek to provide descriptive information and those that seek to provide explanatory information. The former can be thought to be responding to what questions: What characterizes potential emigrants? What do they look like? What are their thoughts about this or that subject? If the objective is description, a researcher collects and uses her data to investigate the relationship between two variables without a specific and firm prediction about the relationship between them. Rather, she simply wonders about the "what" questions listed above and believes that finding out the answers will be instructive. In this case, therefore, she selects the bivariate relationships to be considered based on what she thinks it will be useful to know, and not based on assessing the accuracy of a previously articulated causal story that specifies the strength and structure of the effect that one variable has on the other.
A researcher is often interested in causal stories and explanation, however, and this does usually begin with thinking about the relationship between two variables, one of which is the presumed cause and the other the presumed effect. The presumed cause is the independent variable, and the presumed effect is the dependent variable. Offering evidence that there is a strong relationship between two variables is not sufficient to demonstrate that the variables are likely to be causally related, but it is a necessary first step. In this respect it is a point of departure for the fuller, probably multivariate analysis, required to persuasively argue that a relationship is likely to be causal. In addition, as discussed in Chap. 4, multivariate analysis often not only strengthens the case for inferring that a relationship is causal, but also provides a more elaborate and more instructive causal story. The foundation, however, on which a multivariate analysis aimed at causal inference is built, is a bivariate relationship composed of a presumed independent variable and a presumed dependent variable.
A hypothesis that posits a causal relationship between two variables is not the same as a causal story, although the two are of course closely connected. The former specifies a presumed cause, a presumed determinant of variance on the dependent variable. It probably also specifies the structure of the relationship, such as linear as opposed to non-linear, or positive (also called direct) as opposed to negative (also called inverse).
On the other hand, a causal story describes in more detail what the researcher believes is actually taking place in the relationship between the variables in her hypothesis; and accordingly, why she thinks this involves causality. A causal story provides a fuller account of operative processes, processes that the hypothesis references but does not spell out. These processes may, for example, involve a pathway or a mechanism that tells how it is that the independent variable causes and thus accounts for some of the variance on the dependent variable. Expressed yet another way, the causal story describes the researcher's understandings, or best guesses, about the real world, understandings that have led her to believe, and then propose for testing, that there is a causal connection between her variables that deserves investigation. The hypothesis itself does not tell this story; it is rather a short formulation that references and calls attention to the existence, or hypothesized existence, of a causal story. Research reports present the causal story as well as the hypothesis, as the hypothesis is often of limited interpretability without the causal story.
A causal story is necessary for causal inference. It enables the researcher to formulate propositions that purport to explain rather than merely describe or predict. There may be a strong relationship between two variables, and if this is the case, it will be possible to predict with relative accuracy the value, or score, of one variable from knowledge of the value, or score, of the other variable. Prediction is not explanation, however. To explain, or attribute causality, there must be a causal story to which a hypothesized causal relationship is calling attention.
An instructive illustration is provided by a recent study of Palestinian participation in protest activities that express opposition to Israeli occupation.Footnote 1 There is plenty of variance on the dependent variable: There are many young Palestinians who take part in these activities, and there are many others who do not take part. Education is one of the independent variables that the researcher thought would be an important determinant of participation, and so she hypothesized that individuals with more education would be more likely to participate in protest activities than individuals with less education.
But why would the researcher think this? The answer is provided by the causal story. To the extent that this as yet untested story is plausible, or preferably, persuasive, at least in the eyes of the investigator, it gives the researcher a reason to believe that education is indeed a determinant of participation in protest activities in Palestine. By spelling out in some detail how and why the hypothesized independent variable, education in this case, very likely impacts a person's decision about whether or not to protest, the causal story provides a rationale for the researcher's hypothesis.
In the case of Palestinian participation in protest activities, another investigator offered an insightful causal story about the ways that education pushes toward greater participation, with emphasis on its role in communication and coordination.Footnote 2 Schooling, as the researcher theorizes and subsequently tests, integrates young Palestinians into a broader institutional environment that facilitates mass mobilizations and lowers informational and organizational barriers to collective action. More specifically, she proposes that those individuals who have had at least a middle school education, compared to those who have not finished middle school, have access to better and more reliable sources of information, which, among other things, enables would-be protesters to assess risks. More schooling also makes would-be protesters better able to forge inter-personal relationships and establish networks that share information about needs, opportunities, and risks, and that in this way facilitate engaging in protest activities in groups, rather than on an individual basis. This study offers some additional insights to be discussed later.
The variance motivating the investigation of a causal story may be thought of as the "variable of interest," and it may be either an independent variable or a dependent variable. It is a variable of interest because the way that it varies poses a question, or puzzle, that a researcher seeks to investigate. It is the dependent variable in a bivariate relationship if the researcher seeks to know why this variable behaves, or varies, as it does, and in pursuit of this objective, she will seek to identify the determinants and drivers that account for this variance. The variable of interest is an independent variable in a particular research project if the researcher seeks to know what difference it makes—on what does its variance have an impact, of what other variable or variables is it a driver or determinant.
The variable in which a researcher is initially interested, that is to say the variable of interest, can also be both a dependent variable and an independent variable. Returning to the variable pertaining to consideration of emigration, but this time with country as the unit of analysis, the variance depicted in Table 3.1 provides an instructive example. The data are based on Arab Barometer surveys conducted in 2018–2019, and the table shows that there is substantial variation across twelve countries. Taking the countries together, the mean percentage of citizens that have thought about relocating to another country is 30.25 percent. But in fact, there is very substantial variation around this mean. Kuwait is an outlier, with only 8 percent having considered emigration. There are also countries in which only 21 percent or 22 percent of the adult population have thought about this, figures that may be high in absolute terms but are low relative to other Arab countries. At the other end of the spectrum are countries in which 45 percent or even 50 percent of the citizens report having considered leaving their country and relocating elsewhere.
Table 3.1 Percentage considering emigration
The very substantial variance shown in Table 3.1 invites reflection on both the causes and the consequences of this country-level variable, aggregate thinking about emigration. As a dependent variable, the cross-country variance brings the question of why the proportion of citizens that have thought about emigrating is higher in some countries than in others; and the search for an answer begins with the specification of one or more bivariate relationships, each of which links this dependent variable to a possible cause or determinant. As an independent variable, the cross-country variance brings the question of what difference does it make—of what is it a determinant or driver and what are the consequences for a country if more of its citizens, rather than fewer, have thought about moving to another country.
3.2 Hypotheses and Formulating Hypotheses
Hypotheses emerge from the research questions to which a study is devoted. Accordingly, a researcher interested in explanation will have something specific in mind when she decides to hypothesize and then evaluate a bivariate relationship in order to determine whether, and if so how, her variable of interest is related to another variable. For example, if the researcher's variable of interest is attitude toward gender equality and one of her research questions asks why some people support gender equality and others do not, she might formulate the hypothesis below to see if education provides part of the answer.
Hypothesis 1. Individuals who are better educated are more likely to support gender equality than are individuals who are less well-educated.
The usual case, and the preferred case, is for an investigator to be specific about the research questions she seeks to answer, and then to formulate hypotheses that propose for testing part of the answer to one or more of these questions. Sometimes, however, a researcher will proceed without formulating specific hypotheses based on her research questions. Sometimes she will simply look at whatever relationships between her variable of interest and a second variable her data permit her to identify and examine, and she will then follow up and incorporate into her study any findings that turn out to be significant and potentially instructive. This is sometimes described as allowing the data to "speak." When this hit or miss strategy of trial and error is used in bivariate and multivariate analysis, findings that are significant and potentially instructive are sometimes described as "grounded theory." Some researchers also describe the latter process as "inductive" and the former as "deductive."
Although the inductive, atheoretical approach to data analysis might yield some worthwhile findings that would otherwise have been missed, it can sometimes prove misleading, as you may discover relationships between variables that happened by pure chance and are not instructive about the variable of interest or research question. Data analysis in research aimed at explanation should be, in most cases, preceded by the formulation of one or more hypotheses. In this context, when the focus is on bivariate relationships and the objective is explanation rather than description, each hypothesis will include a dependent variable and an independent variable and make explicit the way the researcher thinks the two are, or probably are, related. As discussed, the dependent variable is the presumed effect; its variance is what a hypothesis seeks to explain. The independent variable is the presumed cause; its impact on the variance of another variable is what the hypothesis seeks to determine.
Hypotheses are usually in the form of if-then, or cause-and-effect, propositions. They posit that if there is variance on the independent variable, the presumed cause, there will then be variance on the dependent variable, the presumed effect. This is because the former impacts the latter and causes it to vary.
An illustration of formulating hypotheses is provided by a study of voting behavior in seven Arab countries: Algeria, Bahrain, Jordan, Lebanon, Morocco, Palestine, and Yemen.Footnote 3 The variable of interest in this individual-level study is electoral turnout, and prominent among the research questions is why some citizens vote and others do not. The dependent variable in the hypotheses proposed in response to this question is whether a person did or did not vote in the country's most recent parliamentary election. The study initially proposed a number of hypotheses, which include the two listed here and which would later be tested with data from Arab Barometer surveys in the seven countries in 2006–2007. We will return to this illustration later in this chapter.
Hypothesis 1: Individuals who have used clientelist networks in the past are more likely to turn out to vote than are individuals who have not used clientelist networks in the past.
Hypothesis 2: Individuals with a positive evaluation of the economy are more likely to vote than are individuals with a negative evaluation of the economy.
Another example pertaining to voting, which this time is hypothetical but might be instructively tested with Arab Barometer data, considers the relationship between perceived corruption and turning out to vote at the individual level of analysis.
The normal expectation in this case would be that perceptions of corruption influence the likelihood of voting. Even here, however, competing causal relationships are plausible. More perceived corruption might increase the likelihood of voting, presumably to register discontent with those in power. But greater perceived corruption might also actually reduce the likelihood of voting, presumably in this case because the would-be voter sees no chance that her vote will make a difference. But in this hypothetical case, even the direction of the causal connection might be ambiguous. If voting is complicated, cumbersome, and overly bureaucratic, it might be that the experience of voting plays a role in shaping perceptions of corruption. In cases like this, certain variables might be both independent and dependent variables, with causal influence pushing in both directions (often called "endogeneity"), and the researcher will need to carefully think through and be particularly clear about the causal story to which her hypothesis is designed to call attention.
The need to assess the accuracy of these hypotheses, or any others proposed to account for variance on a dependent variable, will guide and shape the researcher's subsequent decisions about data collection and data analysis. Moreover, in most cases, the finding produced by data analysis is not a statement that the hypothesis is true or that the hypothesis is false. It is rather a statement that the hypothesis is probably true or it is probably false. And more specifically still, when testing a hypothesis with quantitative data, it is often a statement about the odds, or probability, that the researcher will be wrong if she concludes that the hypothesis is correct—if she concludes that the independent variable in the hypothesis is indeed a significant determinant of the variance on the dependent variable. The lower the probability of being wrong, of course, the more confident a researcher can be in concluding, and reporting, that her data and analysis confirm her hypothesis.
Exercise 3.1
Hypotheses emerge from the research questions to which a study is devoted. Thinking about one or more countries with which you are familiar: (a) Identify the independent and dependent variables in each of the example research questions below. (b) Formulate at least one hypothesis for each question. Make sure to include your expectations about the directionality of the relationship between the two variables; is it positive/direct or negative/inverse? (c) In two or three sentences, describe a plausible causal story to which each of your hypotheses might call attention.
Does religiosity affect people's preference for democracy?
Does preference for democracy affect the likelihood that a person will vote?Footnote 4
Since its establishment in 2006, the Arab Barometer has, as of spring 2022, conducted 68 social and political attitude surveys in the Middle East and North Africa. It has conducted one or more surveys in 16 different Arab countries, and it has recorded the attitudes, values, and preferences of more than 100,000 ordinary citizens.
The Arab Barometer website (arabbarometer.org) provides detailed information about the Barometer itself and about the scope, methodology, and conduct of its surveys. Data from the Barometer's surveys can be downloaded in either SPSS, Stata, or csv format. The website also contains numerous reports, articles, and summaries of findings.
In addition, the Arab Barometer website contains an Online Data Analysis Tool that makes it possible, without downloading any data, to find the distribution of responses to any question asked in any country in any wave. The tool is found in the "Survey Data" menu. After selecting the country and wave of interest, click the "See Results" tab to select the question(s) for which you want to see the response distributions. Click the "Cross by" tab to see the distributions of respondents who differ on one of the available demographic attributes.
The charts below present, in percentages, the response distributions of Jordanians interviewed in 2018 to two questions about gender equality. Below the charts are questions that you are asked to answer. These questions pertain to formulating hypotheses and to the relationship between hypotheses and causal stories.
For each of the two distributions, do you think (hypothesize) that the attitudes of Jordanian women are:
About the same as those of Jordanian men
More favorable toward gender equality than those of Jordanian men
Less favorable toward gender equality than those of Jordanian men
For each of the two distributions, do you think (hypothesize) that the attitudes of younger Jordanians are:
About the same as those of older Jordanians
More favorable toward gender equality than those of older Jordanians
Less favorable toward gender equality than those of older Jordanians
Restate your answers to Questions 1 and 2 as hypotheses.
Give the reasons for your answers to Questions 1 and 2. In two or three sentences, make explicit the presumed causal story on which your hypotheses are based.
Using the Arab Barometer's Online Analysis Tool, check to see whether your answers to Questions 1 and 2 are correct. For those instances in which an answer is incorrect, suggest in a sentence or two a causal story on which the correct relationship might be based.
In which other country surveyed by the Arab Barometer in 2018 do you think the distributions of responses to the questions about gender equality are very similar to the distributions in Jordan? What attributes of Jordan and the other country informed your selection of the other country?
In which other country surveyed by the Arab Barometer in 2018 do you think the distributions of responses to the questions about gender equality are very different from the distributions in Jordan? What attributes of Jordan and the other country informed your selection of the other country?
We will shortly return to and expand the discussion of probabilities and of hypothesis testing more generally. First, however, some additional discussion of hypothesis formulation is in order. Three important topics will be briefly considered. The first concerns the origins of hypotheses; the second concerns the criteria by which the value of a particular hypothesis or set of hypotheses should be evaluated; and the third, requiring a bit more discussion, concerns the structure of the hypothesized relationship between an independent variable and a dependent variable, or between any two variables that are hypothesized to be related.
Origins of Hypotheses
Where do hypotheses come from? How should an investigator identify independent variables that may account for much, or at least some, of the variance on a dependent variable that she has observed and in which she is interested? Or, how should an investigator identify dependent variables whose variance has been determined, presumably only in part, by an independent variable whose impact she deems it important to assess.
Previous research is one place the investigator may look for ideas that will shape her hypotheses and the associated causal stories. This may include previous hypothesis-testing research, and this can be particularly instructive, but it may also include less systematic and structured observations, reports, and testimonies. The point, very simply, is that the investigator almost certainly is not the first person to think about, and offer information and insight about, the topic and questions in which the researcher herself is interested. Accordingly, attention to what is already known will very likely give the researcher some guidance and ideas as she strives for originality and significance in delineating the relationship between the variables in which she is interested.
Consulting previous research will also enable the researcher to determine what her study will add to what is already known—what it will contribute to the collective and cumulative work of researchers and others who seek to reduce uncertainty about a topic in which they share an interest. Perhaps the researcher's study will fill an important gap in the scientific literature. Perhaps it will challenge and refine, or perhaps even place in doubt, distributions and explanations of variance that have thus far been accepted. Or perhaps her study will produce findings that shed light on the generalizability or scope conditions of previously accepted variable relationships. It need not do any of these things, but that will be for the researcher to decide, and her decision will be informed by knowledge of what is already known and reflection on whether and in what ways her study should seek to add to that body of knowledge.
Personal experience will also inform the researcher's search for meaningful and informative hypotheses. It is almost certainly the case that a researcher's interest in a topic in general, and in questions pertaining to this topic in particular, have been shaped by her own experience. The experience itself may involve many different kinds of connections or interactions, some more professional and work-related and some flowing simply and perhaps unintentionally from lived experience. The hypotheses about voting mentioned earlier, for example, might be informed by elections the researcher has witnessed and/or discussions with friends and colleagues about elections, their turnout, and their fairness. Or perhaps the researcher's experience in her home country has planted questions about the generalizability of what she has witnessed at home.
All of this is to some extent obvious. But the take-away is that an investigator should not endeavor to set aside what she has learned about a topic in the name of objectivity, but rather, she should embrace whatever personal experience has taught her as she selects and refines the puzzles and propositions she will investigate. Should it happen that her experience leads her to incorrect or perhaps distorted understandings, this will be brought to light when her hypotheses are tested. It is in the testing that objectivity is paramount. In hypothesis formation, by contrast, subjectivity is permissible, and, in fact, it may often be unavoidable.
A final arena in which an investigator may look for ideas that will shape her hypotheses overlaps with personal experience and is also to some extent obvious. This is referenced by terms like creativity and originality and is perhaps best captured by the term "sociological imagination." The take-away here is that hypotheses that deserve attention and, if confirmed, will provide important insights, may not all be somewhere out in the environment waiting to be found, either in the relevant scholarly literature or in recollections about relevant personal experience. They can and sometimes will be the product of imagination and wondering, of discernments that a researcher may come upon during moments of reflection and deliberation.
As in the case of personal experience, the point to be retained is that hypothesis formation may not only be a process of discovery, of finding the previous research that contains the right information. Hypothesis formation may also be a creative process, a process whereby new insights and proposed original understandings are the product of an investigator's intellect and sociological imagination.
Crafting Valuable Hypotheses
What are the criteria by which the value of a hypothesis or set of hypotheses should be evaluated? What elements define a good hypothesis? Some of the answers to these questions that come immediately to mind pertain to hypothesis testing rather than hypothesis formation. A good hypothesis, it might be argued, is one that is subsequently confirmed. But whether or not a confirmed hypothesis makes a positive contribution depends on the nature of the hypothesis and goals of the research. It is possible that a researcher will learn as much, and possibly even more, from findings that lead to rejection of a hypothesis. In any event, findings, whatever they may be, are valuable only to the extent that the hypothesis being tested is itself worthy of study.
Two important considerations, albeit somewhat obvious ones, are that a hypothesis should be non-trivial and non-obvious. If a proposition is trivial, suggesting a variable relationship with little or no significance, discovering whether and how the variables it brings together are related will not make a meaningful contribution to knowledge about the determinants and/or impact of the variance at the heart of the researcher's concern. Few will be interested in findings, however rigorously derived, about a trivial proposition. The same is true of an obvious hypothesis, obvious being an attribute that makes a proposition trivial. As stated, these considerations are themselves somewhat obvious, barely deserving mention. Nevertheless, an investigator should self-consciously reflect on these criteria when formulating hypotheses. She should be sure that she is proposing variable relationships that are neither trivial nor obvious.
A third criterion, also somewhat obvious but nonetheless essential, has to do with the significance and salience of the variables being considered. Will findings from research about these variables be important and valuable, and perhaps also useful? If the primary variable of interest is a dependent variable, meaning that the primary goal of the research is to account for variance, then the significance and salience of the dependent variable will determine the value of the research. Similarly, if the primary variable of interest is an independent variable, meaning that the primary goal of the research is to determine and assess impact, then the significance and salience of the independent variable will determine the value of the research.
These three criteria—non-trivial, non-obvious, and variable importance and salience—are not very different from one another. They collectively mean that the researcher must be able to specify why and how the testing of her hypothesis, or hypotheses, will make a contribution of value. Perhaps her propositions are original or innovative; perhaps knowing whether they are true or false makes a difference or will be of practical benefit; perhaps her findings add something specific and identifiable to the body of existing scholarly literature on the subject. While calling attention to these three connected and overlapping criteria might seem unnecessary since they are indeed somewhat obvious, it remains the case that the value of a hypothesis, regardless of whether or not it is eventually confirmed, is itself important to consider, and an investigator should, therefore, know and be able to articulate the reasons and ways that consideration of her hypothesis, or hypotheses, will indeed be of value.
Hypothesizing the Structure of a Relationship
Relevant in the process of hypothesis formation are, as discussed, questions about the origins of hypotheses and the criteria by which the value of any particular hypothesis or set of hypotheses will be evaluated. Relevant, too, is consideration of the structure of a hypothesized variable relationship and the causal story to which that relationship is believed to call attention.
The point of departure in considering the structure of a hypothesized variable relationship is an understanding that such a relationship may or may not be linear. In a direct, or positive, linear relationship, each increase in the independent variable brings a constant increase in the dependent variable. In an inverse, or negative, linear relationship, each increase in the independent variable brings a constant decrease in the dependent variable. But these are only two of the many ways that an independent variable and a dependent variable may be related, or hypothesized to be related. This is easily illustrated by hypotheses in which level of education or age is the independent variable, and this is relevant in hypothesis formation because the investigator must be alert to and consider the possibility that the variables in which she is interested are in fact related in a non-linear way.
Consider, for example, the relationship between age and support for gender equality, the latter measured by an index based on several questions about the rights and behavior of women that are asked in Arab Barometer surveys. A researcher might expect, and might therefore want to hypothesize, that an increase in age brings increased support for, or alternatively increased opposition to, gender equality. But these are not the only possibilities. Likely, perhaps, is the possibility of a curvilinear relationship, in which case increases in age bring increases in support for gender equality until a person reaches a certain age, maybe 40, 45, or 50, after which additional increases in age bring decreases in support for gender equality. Or the researcher might hypothesize that the curve is in the opposite direction, that support for gender equality initially decreases as a function of age until a particular age is reached, after which additional increases in age bring an increase in support.
Of course, there are also other possibilities. In the case of education and gender equality, for example, increased education may initially have no impact on attitudes toward gender equality. Individuals who have not finished primary school, those who have finished primary school, and those who have gone somewhat beyond primary school and completed a middle school program may all have roughly the same attitudes toward gender equality. Thus, increases in education, within a certain range of educational levels, are not expected to bring an increase or a decrease in support for gender equality. But the level of support for gender equality among high school graduates may be higher and among university graduates may be higher still. Accordingly, in this hypothetical illustration, an increase in education does bring increased support for gender equality but only beginning after middle school.
A middle school level of education is a "floor" in this example. Education does not begin to make a difference until this floor is reached, and thereafter it does make a difference, with increases in education beyond middle school bringing increases in support for gender equality. Another possibility might be for middle school to be a "ceiling." This would mean that increases in education through middle school would bring increases in support for gender equality, but the trend would not continue beyond middle school. In other words, level of education makes a difference and appears to have explanatory power only until, and so not after, this ceiling is reached. This latter pattern was found in the study of education and Palestinian protest activity discussed earlier. Increases in education through middle school brought increases in the likelihood that an individual would participate in demonstrations and protests of Israeli occupation. However, additional education beyond middle school was not associated with greater likelihood of taking part in protest activities.
This discussion of variation in the structure of a hypothesized relationship between two variables is certainly not exhaustive, and the examples themselves are straightforward and not very complicated. The purpose of the discussion is, therefore, to emphasize that an investigator must be open to and think through the possibility and plausibility of different kinds of relationships between her two variables, that is to say, relationships with different structures. Bivariate relationships with several different kinds of structures are depicted visually by the scatter plots in Fig. 3.4.
These possibilities with respect to structure do not determine the value of a proposed hypothesis. As discussed earlier, the value of a proposed relationship depends first and foremost on the importance and salience of the variable of interest. Accordingly, a researcher should not assume that the value of a hypothesis varies as a function of the degree to which it posits a complicated variable relationship. More complicated hypotheses are not necessarily better or more correct. But while she should not strive for or give preference to variable relationships that are more complicated simply because they are more complicated, she should, again, be alert to the possibility that a more complicated pattern does a better job of describing the causal connection between the two variables in the place and time in which she is interested.
This brings the discussion of formulating hypotheses back to our earlier account of causal stories. In research concerned with explanation and causality, a hypothesis for the most part is a simplified stand-in for a causal story. It represents the causal story, as it were. Expressing this differently, the hypothesis states the causal story's "bottom line;" it posits that the independent variable is a determinant of variance on the dependent variable, and it identifies the structure of the presumed relationship between the independent variable and the dependent variable. But it does not describe the interaction between the two variables in a way that tells consumers of the study why the researcher believes that the relationship involves causality rather than an association with no causal implications. This is left to the causal story, which will offer a fuller account of the way the presumed cause impacts the presumed effect.
3.3 Describing and Visually Representing Bivariate Relationships
Once a researcher has collected or otherwise obtained data on the variables in a bivariate relationship she wishes to examine, her first step will be to describe the variance on each of the variables using the univariate statistics described in Chap. 2. She will need to understand the distribution on each variable before she can understand how these variables vary in relation to one another. This is important whether she is interested in description or wishes to explore a bivariate causal story.
Once she has described each one of the variables, she can turn to the relationship between them. She can prepare and present a visual representation of this relationship, which is the subject of the present section. She can also use bivariate statistical tests to assess the strength and significance of the relationship, which is the subject of the next section of this chapter.
Contingency Tables
Contingency tables are used to display the relationship between two categorical variables. They are similar to the univariate frequency distributions described in Chap. 2, the difference being that they juxtapose the two univariate distributions and display the interaction between them. Also called cross-tabulation tables, the cells of the table may present frequencies, row percentages, column percentages, and/or total percentages. Total frequencies and/or percentages are displayed in a total row and a total column, each one of which is the same as the univariate distribution of one of the variables taken alone.
Table 3.2, based on Palestinian data from Wave V of the Arab Barometer, crosses gender and the average number of hours watching television each day. Frequencies are presented in the cells of the table. In the cell showing the number of Palestinian men who do not watch television at all, row percentage, column percentage, and total percentage are also presented. Note that total percentage is based on the 10 cells showing the two variables taken together, which are summed in the lower right-hand cell. Thus, total percent for this cell is 342/2488 = 13.7. Only frequencies are given in the other cells of the table; but in a full table, these four figures – frequency, row percent, column percent and total percent – would be given in every cell.
Table 3.2 Hours watching television by gender (Palestine, Wave V)
Compute the row percentage, the column percentage, and the total percentage in the cell showing the number of Palestinian women who do not watch television at all.
Describe the relationship between gender and watching television among Palestinians that is shown in the table. Do the television watching habits of Palestinian men and women appear to be generally similar or fairly different? You might find it helpful to convert the frequencies in other cells to row or column percentages.
Stacked Column Charts and Grouped Bar Charts
Stacked column charts and grouped bar charts are used to visually describe how two categorical variables, or one categorical and one continuous variable, relate to one another. Much like contingency tables, they show the percentage or count of each category of one variable within each category of the second variable. This information is presented in columns stacked on each other or next to each other. The charts below show the number of male Palestinians and the number of female Palestinians who watch television for a given number of hours each day. Each chart presents the same information as the other chart and as the contingency table shown above (Fig. 3.1).
Fig. 3.1
Stacked column charts and grouped bar charts comparing Palestinian men and Palestinian women on hours watching television
Box Plots and Box and Whisker Plots
Box plots, box and whisker plots, and other types of plots can also be used to show the relationship between one categorical variable and one continuous variable. They are particularly useful for showing how spread out the data are. Box plots show five important numbers in a variable's distribution: the minimum value; the median; the maximum value; and the first and third quartiles (Q1 and Q2), which represent, respectively, the number below which are 25 percent of the distribution's values and the number below which are 75 percent of the distribution's values. The minimum value is sometimes called the lower extreme, the lower bound, or the lower hinge. The maximum value is sometimes called the upper extreme, the upper bound, or the upper hinge. The middle 50 percent of the distribution, the range between Q1 and Q3 that represents the "box," constitutes the interquartile range (IQR). In box and whisker plots, the "whiskers" are the short perpendicular lines extending outside the upper and lower quartiles. They are included to indicate variability below Q1 and above Q3. Values are usually categorized as outliers if they are less than Q1 − IQR*1.5 or greater than Q3 + IQR*1.5. A visual explanation of a box and whisker plot is shown in Fig. 3.2a and an example of a box plot that uses actual data is shown in Fig. 3.2b.
The box plot in Fig. 3.2b uses Wave V Arab Barometer data from Tunisia and shows the relationship between age, a continuous variable, and interpersonal trust, a dichotomous categorical variable. The line representing the median value is shown in bold. Interpersonal trust, sometimes known as generalized trust, is an important personal value. Previous research has shown that social harmony and prospects for democracy are greater in societies in which most citizens believe that their fellow citizens for the most part are trustworthy. Although the interpersonal trust variable is dichotomous in Fig. 3.2b, the variance in interpersonal trust can also be measured by a set of ordered categories or a scale that yields a continuous measure, the latter not being suitable for presentation by a box plot. Figure 3.2b shows that the median age of Tunisians who are trusting is slightly higher than the median age of Tunisians who are mistrustful of other people. Notice also that the box plot for the mistrustful group has an outlier.
(a) A box and whisker plot. (b) Box plot comparing the ages of trusting and mistrustful Tunisians in 2018
Line plots may be used to visualize the relationship between two continuous variables or a continuous variable and a categorical variable. They are often used when time, or a variable related to time, is one of the two variables. If a researcher wants to show whether and how a variable changes over time for more than one subgroup of the units about which she has data (looking at men and women separately, for example), she can include multiple lines on the same plot, with each line showing the pattern over time for a different subgroup. These lines will generally be distinguished from each other by color or pattern, with a legend provided for readers.
Line plots are a particularly good way to visualize a relationship if an investigator thinks that important events over time may have had a significant impact. The line plot in Fig. 3.3 shows the average support for gender equality among men and among women in Tunisia from 2013 to 2018. Support for gender equality is a scale based on four questions related to gender equality in the three waves of the Arab Barometer. An answer supportive of gender equality on a question adds +.5 to the scale and an answer unfavorable to gender equality adds −.5 to the scale. Accordingly, a scale score of 2 indicates maximum support for gender equality and a scale score of −2 indicates maximum opposition to gender equality.
Line plot showing level of support for gender equality among Tunisian women and men in 2013, 2016, and 2018
Scatter plots are used to visualize a bivariate relationship when both variables are numerical. The independent variable is put on the x-axis, the horizontal axis, and the dependent variable is put on the y-axis, the vertical axis. Each data point becomes a dot in the scatter plot's two-dimensional field, with its precise location being the point at which its value on the x-axis intersects with its value on the y-axis. The scatter plot shows how the variables are related to one another, including with respect to linearity, direction, and other aspects of structure. The scatter plots in Fig. 3.4 illustrate a strong positive linear relationship, a moderately strong negative linear relationship, a strong non-linear relationship, and a pattern showing no relationship.Footnote 5 If the scatter plot displays no visible and clear pattern, as in the lower left hand plot shown in Fig. 3.4, the scatter plot would indicate that the independent variable, by itself, has no meaningful impact on the dependent variable.
Scatter plots showing bivariate relationships with different structures
Scatter plots are also a good way to identify outliers—data points that do not follow a pattern that characterizes most of the data. These are also called non-scalar types. Figure 3.5 shows a scatter plot with outliers.
Outliers can be informative, making it possible, for example, to identify the attributes of cases for which the measures of one or both variables are unreliable and/or invalid. Nevertheless, the inclusion of outliers may not only distort the assessment of measures, raising unwarranted doubts about measures that are actually reliable and valid for the vast majority of cases, they may also bias bivariate statistics and make relationships seem weaker than they really are for most cases. For this reason, researchers sometimes remove outliers prior to testing a hypothesis. If one does this, it is important to have a clear definition of what is an outlier and to justify the removal of the outlier, both using the definition and perhaps through substantive analysis. There are several mathematical formulas for identifying outliers, and researchers should be aware of these formulas and their pros and cons if they plan to remove outliers.
If there are relatively few outliers, perhaps no more than 5–10 percent of the cases, it may be justifiable to remove them in order to better discern the relationship between the independent variable and the dependent variable. If outliers are much more numerous, however, it is probably because there is not a significant relationship between the two variables being considered. The researcher might in this case find it instructive to introduce a third variable and disaggregate the data. Disaggregation will be discussed in Chap. 4.
A scatter plot with outliers marked in red
Exercise 3.4 Exploring Hypotheses through Visualizing Data: Exercise with the Arab Barometer Online Analysis Tool
Go to the Arab Barometer Online Analysis Tool (https://www.arabbarometer.org/survey-data/data-analysis-tool/)
Select Wave V and a country that interests you
Select "See Results"
Select "Social, Cultural and Religious topics"
Select "Religion: frequency: pray"
Questions: What does the distribution of this variable look like? How would you describe the variance?
Click on "Cross by," then
Select "Show all variables"
Select "Kind of government preferable" and click
Select "Options," then "Show % over Row total," then "Apply"
Questions: Does there seem to be a relationship between religiosity and preference for democracy? If so, what might explain the relationship you observe—what is a plausible causal story? Is it consistent with the hypothesis you wrote for Exercise 3.1?
What other variables could be used to measure religiosity and preference for democracy? Explore your hypothesis using different items from the list of Arab Barometer variables
Do these distributions support the previous results you found? Do you learn anything additional about the relationship between religiosity and preference for democracy?
Now it is your turn to explore variables and variable relationships that interest you!
Pick two variables that interest you from the list of Arab Barometer variables. Are they continuous or categorical? Ordinal or nominal? (Hint: Most Arab Barometer variables are categorical, even if you might be tempted to think of them as continuous. For example, age is divided into the ordinal categories 18–29, 30–49, and 50 and more.)
Do you expect there to be a relationship between the two variables? If so, what do you think will be the structure of that relationship, and why?
Select the wave (year) and the country that interest you
Select one of your two variables of interest
Click on "Cross by," and then select your second variable of interest.
On the left side of the page, you'll see a contingency table. On the right side at the top, you'll see several options to graphically display the relationship between your two variables. Which type of graph best represents the relationship between your two variables of interest?
Do the two variables seem to be independent of each other, or do you think there might be a relationship between them? Is the relationship you see similar to what you had expected
3.4 Probabilities and Type I and Type II Errors
As in visual presentations of bivariate relationships, selecting the appropriate measure of association or bivariate statistical test depends on the types of the two variables. The data on both variables may be categorical; the data on both may be continuous; or the data may be categorical on one variable and continuous on the other variable. These characteristics of the data will guide the way in which our presentation of these measures and tests is organized. Before briefly describing some specific measures of association and bivariate statistical tests, however, it is necessary to lay a foundation by introducing a number of terms and concepts. Relevant here are the distinction between population and sample and the notions of the null hypothesis, of Type I and Type II errors, and of probabilities and confidence intervals. As concepts, or abstractions, these notions may influence the way a researcher thinks about drawing conclusions about a hypothesis from qualitative data, as was discussed in Chap. 2. In their precise meaning and application, however, these terms and concepts come into play when hypothesis testing involves the statistical analysis of quantitative data.
To begin, it is important to distinguish between, on the one hand, the population of units—individuals, countries, ethnic groups, political movements, or any other unit of analysis—in which the researcher is interested and about which she aspires to advance conclusions and, on the other hand, the units on which she has actually acquired the data to be analyzed. The latter, the units on which she actually has data, is her sample. In cases where the researcher has collected or obtained data on all of the units in which she is interested, there is no difference between the sample and the population, and drawing conclusions about the population based on the sample is straightforward. Most often, however, a researcher does not possess data on all of the units that make up the population in which she is interested, and so the possibility of error when making inferences about the population based on the analysis of data in the sample requires careful and deliberate consideration.
This concern for error is present regardless of the size of the sample and the way it was constructed. The likelihood of error declines as the size of the sample increases and thus comes closer to representing the full population. It also declines if the sample was constructed in accordance with random or other sampling procedures designed to maximize representation. It is useful to keep these criteria in mind when looking at, and perhaps downloading and using, Arab Barometer data. The Barometer's website gives information about the construction of each sample. But while it is possible to reduce the likelihood of error when characterizing the population from findings based on the sample, it is not possible to eliminate entirely the possibility of erroneous inference. Accordingly, a researcher must endeavor to make the likelihood of this kind of error as small as possible and then decide if it is small enough to advance conclusions that apply to the population as well as the sample.
The null hypothesis, frequently designated as H0, is a statement to the effect that there is no meaningful and significant relationship between the independent variable and the dependent variable in a hypothesis, or indeed between two variables even if the relationship between them has not been formally specified in a hypothesis and does not purport to be causal or explanatory. The null hypothesis may or may not be stated explicitly by an investigator, but it is nonetheless present in her thinking; it stands in opposition to the hypothesized variable relationship. In a point and counterpoint fashion, the hypothesis, H1, posits that the variables are significantly related, and the null hypothesis, H0, replies and says no, they are not significantly related. It further says that they are not related in any meaningful way, neither in the way proposed in H1 nor in any other way that could be proposed.
Based on her analysis, the researcher needs to determine whether her findings permit rejecting the null hypothesis and concluding that there is indeed a significant relationship between the variables in her hypothesis, concluding in effect that the research hypothesis, H1, has been confirmed. This is most relevant and important when the investigator is basing her analysis on some but not all of the units to which her hypothesis purports to apply—when she is analyzing the data in her sample but seeks to advance conclusions that apply to the population in which she is interested. The logic here is that the findings produced by an analysis of some of the data, the data she actually possesses, may be different than the findings her analysis would hypothetically produce were she able to use data from very many more, or ideally even all, of the units that make up her population of interest.
This means, of course, that there will be uncertainty as the researcher adjudicates between H0 and H1 on the basis of her data. An analysis of these data may suggest that there is a strong and significant relationship between the variables in H1. And the stronger the relationship, the more unlikely it is that the researcher's sample is a subset of a population characterized by H0 and that, therefore, the researcher may consider H1 to have been confirmed. Yet, it remains at least possible that the researcher's sample, although it provides strong support for H1, is actually a subset of a population characterized by the null hypothesis. This may be unlikely, but it is not impossible, and so, therefore, to consider H1 to have been confirmed is to run the risk, at least a small risk, of what is known as a Type I error. A Type I error is made when a researcher accepts a research hypothesis that is actually false, when she judges to be true a hypothesis that does not characterize the population of which her sample is a subset. Because of the possibility of a Type I error, even if quite unlikely, researchers will often write something like "We can reject the null hypothesis," rather than "We can confirm our hypothesis."
Another analysis related to voter turnout provides a ready illustration. In the Arab Barometer Wave V surveys in 12 Arab countries,Footnote 6 13,899 respondents answered a question about voting in the most recent parliamentary election. Of these, 46.6 percent said they had voted, and the remainder, 53.4 percent, said they had not voted in the last parliamentary election.Footnote 7 Seeking to identify some of the determinants of voting—the attitudes and experiences of an individual that increase the likelihood that she will vote, the researcher might hypothesize that a judgment that the country is going in the right direction will push toward voting. More formally:
H1. An individual who believes that her country is going in the right direction is more likely to vote in a national election than is an individual who believes her country is going in the wrong direction.
Arab Barometer surveys provide data with which to test this proposition, and in fact there is a difference associated with views about the direction in which the country is going. Among those who judged that their country is going in the right direction, 52.4 percent voted in the last parliamentary election. By contrast, among those who judged that their country is going in the wrong direction, only 43.8 percent voted in the last parliamentary election.
This illustrates the choice a researcher faces when deciding what to conclude from a study. Does the analysis of her data from a subset of her population of interest confirm or not confirm her hypothesis? In this example, based on Arab Barometer data, the findings are in the direction of her hypothesis, and differences in voting associated with views about the direction the country is going do not appear to be trivial. But are these differences big enough to justify the conclusion that judgements about the country's path going forward are a determinant of voting, one among others of course, in the population from which her sample was drawn? In other words, although this relationship clearly characterizes the sample, it is unclear whether it characterizes the researcher's population of interest, the population from which the sample was drawn.
Unless the researcher can gather data on the entire population of eligible voters, or at least almost all of this population, it is not possible to entirely eliminate uncertainty when the researcher makes inferences about the population of voters based on findings from the subset, or sample, of voters on which she has data. She can either conclude that her findings are sufficiently strong and clear to propose that the pattern she has observed characterizes the population as well, and that H1 is therefore confirmed; or she can conclude that her findings are not strong enough to make such an inference about the population, and that H1, therefore, is not confirmed. Either conclusion could be wrong, and so there is a chance of error no matter which conclusion the researcher advances.
The terms Type I error and Type II error are often used to designate the possible error associated with each of these inferences about the population based on the sample. Type I error refers to the rejection of a true null hypothesis. This means, in other words, that the investigator could be wrong if she concludes that her finding of a strong, or at least fairly strong, relationship between her variables characterizes Arab voters in the 12 countries in general, and if she thus judges H1 to have been confirmed when the population from which her sample was drawn is in fact characterized by H0. Type II error refers to acceptance of a false null hypothesis. This means, in other words, that the investigator could be wrong if she concludes that her finding of a somewhat weak relationship, or no relationship at all, between her variables characterizes Arab voters in the 12 countries in general, and that she thus judges H0 to be true when the population from which her sample was drawn is in fact characterized by H1.
In statistical analyses of quantitative data, decisions about whether to risk a Type I error or a Type II error are usually based on probabilities. More specifically, they are based on the probability of a researcher being wrong if she concludes that the variable relationship—or hypothesis in most cases—that characterizes her data, meaning her sample, also characterizes the population on which the researcher hopes her sample and data will shed light. To say this in yet another way, she computes the odds that her sample does not represent the population of which it is a subset; or more specifically still, she computes the odds that from a population that is characterized by the null hypothesis she could have obtained, by chance alone, a subset of the population, her sample, that is not characterized by the null hypothesis. The lower the odds, or probability, the more willing the researcher will be to risk a Type I error.
There are numerous statistical tests that are used to compute such probabilities. The nature of the data and the goals of the analysis will determine the specific test to be used in a particular situation. Most of these tests, frequently called tests of significance or tests of statistical significance, provide output in the form of probabilities, which always range from 0 to 1. The lower the value, meaning the closer to 0, the less likely it is that a researcher has collected and is working with data that produce findings that differ from what she would find were she to somehow have data on the entire population. Another way to think about this is the following:
If the researcher provisionally assumes that the population is characterized by the null hypothesis with respect to the variable relationship under study, what is the probability of obtaining from that population, by chance alone, a subset or sample that is not characterized by the null hypothesis but instead shows a strong relationship between the two variables;
The lower the probability value, meaning the closer to 0, the less likely it is that the researcher's data, which support H1, have come from a population that is characterized by H0;
The lower the probability that her sample could have come from a population characterized by H0, the lower the possibility that the researcher will be wrong, that she will make a Type I error, if she rejects the null hypothesis and accepts that the population, as well as her sample, is characterized by H1;
When the probability value is low, the chance of actually making a Type I error is small. But while small, the possibility of an error cannot be entirely eliminated.
If it helps you to think about probability and Type I and Type II error, imagine that you will be flipping a coin 100 times and your goal is to determine whether the coin is unbiased, H0, or biased in favor of either heads or tails, H1. How many times more than 50 would heads have to come up before you would be comfortable concluding that the coin is in fact biased in favor of heads? Would 60 be enough? What about 65? To begin to answer these questions, you would want to know the odds of getting 60 or 65 heads from a coin that is actually unbiased, a coin that would come up heads and come up tails roughly the same number of times if it were flipped many more than 100 times, maybe 1000 times, maybe 10,000. With this many flips, would the ratio of heads to tails even out. The lower the odds, the less likely it is that the coin is unbiased. In this analogy, you can think of the mathematical calculations about an unbiased coin's odds of getting heads as the population, and your actual flips of the coin as the sample.
But exactly how low does the probability of a Type I error have to be for a researcher to run the risk of rejecting H0 and accepting that her variables are indeed related? This depends, of course, on the implications of being wrong. If there are serious and harmful consequences of being wrong, of accepting a research hypothesis that is actually false, the researcher will reject H0 and accept H1 only if the odds of being wrong, of making a Type I error, are very low.
There are some widely used probability values, which define what are known as "confidence intervals," that help researchers and those who read their reports to think about the likelihood that a Type I error is being made. In the social sciences, rejecting H0 and running the risk of a Type I error is usually thought to require a probability value of less than .05, written as p < .05. The less stringent value of p < .10 is sometimes accepted as sufficient for rejecting H0, although such a conclusion would be advanced with caution and when the consequences of a Type I error are not very harmful. Frequently considered safer, meaning that the likelihood of accepting a false hypothesis is lower, are p < .01 and p < .001. The next section introduces and briefly describes some of the bivariate statistics that may be used to calculate these probabilities.
3.5 Measures of Association and Bivariate Statistical Tests
The following section introduces some of the bivariate statistical tests that can be used to compute probabilities and test hypotheses. The accounts are not very detailed. They will provide only a general overview and refresher for readers who are already fairly familiar with bivariate statistics. Readers without this familiarity are encouraged to consult a statistics textbook, for which the accounts presented here will provide a useful guide. While the account below will emphasize calculating these test statistics by hand, it is also important to remember that they can be calculated with the assistance of statistical software as well. A discussion of statistical software is available in Appendix 4.
Parametric and Nonparametric Statistics
Parametric and nonparametric are two broad classifications of statistical procedures. A parameter in statistics refers to an attribute of a population. For example, the mean of a population is a parameter. Parametric statistical tests make certain assumptions about the shape of the distribution of values in a population from which a sample is drawn, generally that it is normally distributed, and about its parameters, that is to say the means and standard deviations of the assumed distributions. Nonparametric statistical procedures rely on no or very few assumptions about the shape or parameters of the distribution of the population from which the sample was drawn. Chi-squared is the only nonparametric statistical test among the tests described below.
Degrees of freedom (df) is the number of values in the calculation of a statistic that are free to vary. Statistical software programs usually give degrees of freedom in the output, so it is generally unnecessary to know the number of the degrees of freedom in advance. It is nonetheless useful to understand what degrees of freedom represent. Consistent with the definition above, it is the number of values that are not predetermined, and thus are free to vary, within the variables used in a statistical test.
This is illustrated by the contingency tables below, which are constructed to examine the relationship between two categorical variables. The marginal row and column totals are known since these are just the univariate distributions of each variable. df = 1 for Table 3.3a, which is a 4-cell table. You can enter any one value in any one cell, but thereafter the values of all the other three cells are determined. Only one number is not free to vary and thus not predetermined. df = 2 for Table 3.3b, which is a 6-cell table. You can enter any two values in any two cells, but thereafter the values of all the other cells are determined. Only two numbers are free to vary and thus not predetermined. For contingency tables, the formula for calculating df is:
$$ \mathrm{Number}\ \mathrm{of}\ \mathrm{columns}-1\ {\left(\mathrm{minus}\ 1\right)}^{\ast }\ \mathrm{Number}\ \mathrm{of}\ \mathrm{rows}-1\ \left(\mathrm{minus}\ 1\right) $$
Table 3.3 Computing Degrees of Freedom for a Contingency Table
Chi-Squared
Chi-squared, frequently written X2, is a statistical test used to determine whether two categorical variables are significantly related. As noted, it is a nonparametric test. The most common version of the chi-squared test is the Pearson chi-squared test, which gives a value for the chi-squared statistic and permits determining as well a probability value, or p-value. The magnitude of the statistic and of the probability value are inversely correlated; the higher the value of the chi-squared statistic, the lower the probability value, and thus the lower the risk of making a Type I error—of rejecting a true null hypothesis—when asserting that the two variables are strongly and significantly related.
The simplicity of the chi-squared statistic permits giving a little more detail in order to illustrate several points that apply to bivariate statistical tests in general. The formula for computing chi-squared is given below, with O being the observed (actual) frequency in each cell of a contingency table for two categorical variables and E being the frequency that would be expected in each cell if the two variables are not related. Put differently, the distribution of E values across the cells of the two-variable table constitutes the null hypothesis, and chi-squared provides a number that expresses the magnitude of the difference between an investigator's actual observed values and the values of E.
The computation of chi-squared involves the following procedures, which are illustrated using the data in Table 3.4.
The values of O in the cells of the table are based on the data collected by the investigator. For example, Table 3.4 shows that of the 200 women on whom she collected information, 85 are majoring in social science.
The value of E for each cell is computed by multiplying the marginal total of the column in which the cell is located by the marginal total of the row in which the cell is located divided by N, N being the total number of cases. For the female students majoring in social science in Table 3.4, this is: 200 * 150/400 = 30,000/400 = 75. For the female students majoring in math and natural science in Table 3.4, this is: 200 * 100/400 = 20,000/400 = 50.
The difference between the value of O and the value of E is computed for each cell using the formula for chi-squared. For the female students majoring in social science in Table 3.4, this is: (85–75)2/75 = 102/75 = 100/75 = 1.33. For the female students majoring in math and natural science, the value resulting from the application of the chi-squared is: (45–50)2/50 = 52/75 = 25/75 = .33.
The values in each cell of the table resulting from the application of the chi-squared formula are summed (Σ). This chi-squared value expresses the magnitude of the difference between a distribution of values indicative of the null hypothesis and what the investigator actually found about the relationship between gender and field of study. In Table 3.4, the cell for female students majoring in social science adds 1.33 to the sum of the values in the eight cells, the cell for female students majoring in math and natural science adds .33 to the sum, and so forth for the remaining six cells.
Table 3.4 Fields of study of 400 hypothetical male and female university students: testing the hypothesis that female university students are less likely to major in math and natural science than male university students
A final point to be noted, which applies to many other statistical tests as well, is that the application of chi-squared and other bivariate (and multivariate) statistical tests yields a value with which can be computed the probability that an observed pattern does not differ from the null hypothesis and that a Type I error will be made if the null hypothesis is rejected and the research hypothesis is judged to be true. The lower the probability, of course, the lower the likelihood of an error if the null hypothesis is rejected.
Prior to the advent of computer assisted statistical analysis, the value of the statistic and the number of degrees of freedom were used to find the probability value in a table of probability values in an appendix in most statistics books. At present, however, the probability value, or p-value, and also the degrees of freedom, are routinely given as part of the output when analysis is done by one of the available statistical software packages.
Table 3.5 shows the relationship between economic circumstance and trust in the government among 400 ordinary citizens in a hypothetical country. The observed data were collected to test the hypothesis that greater wealth pushes people toward greater trust and less wealth pushes people toward lesser trust. In the case of all three patterns, the probability that the null hypothesis is true is very low. All three patterns have the same high chi-squared value and low probability value. Thus, the chi-squared and p-values show only that the patterns all differ significantly from what would be expected were the null hypothesis true. They do not show whether the data support the hypothesized variable relationship or any other particular relationship.
As the three patterns in Table 3.5 show, variable relationships with very different structures can yield similar or even identical statistical test and probability values, and thus these tests provide only some of the information a researcher needs to draw conclusions about her hypothesis. To draw the right conclusion, it may also be necessary for the investigator to "look at" her data. For example, as Table 3.5 suggests, looking at a tabular or visual presentation of the data may also be needed to draw the proper conclusion about how two variables are related.
How would you describe the three patterns shown in the table, each of which differs significantly from the null hypothesis? Which pattern is consistent with the research hypothesis? How would you describe the other two patterns? Try to visualize a plot of each pattern.
Table 3.5 The relationship between economic circumstance and trust in government: testing the hypothesis that greater wealth pushes toward greater trust
Pearson Correlation Coefficient
The Pearson correlation coefficient, more formally known as the Pearson product-moment correlation, is a parametric measure of linear association. It gives a numerical representation of the strength and direction of the relationship between two continuous numerical variables. The coefficient, which is commonly represented as r, will have a value between −1 and 1. A value of 1 means that there is a perfect positive, or direct, linear relationship between the two variables; as one variable increases, the other variable consistently increases by some amount. A value of −1 means that there is a perfect negative, or inverse, linear relationship; as one variable increases, the other variable consistently decreases by some amount. A value of 0 means that there is no linear relationship; as one variable increases, the other variable neither consistently increases nor consistently decreases.
It is easy to think of relationships that might be assessed by a Pearson correlation coefficient. Consider, for example, the relationship between age and income and the proposition that as age increases, income consistently increases or consistently decreases as well. The closer a coefficient is to 1 or −1, the greater the likelihood that the data on which it is based are not the subset of a population in which age and income are unrelated, meaning that the population of interest is not characterized by the null hypothesis. Coefficients very close to 1 or −1 are rare; although it depends on the number of units on which the researcher has data and also on the nature of the variables. Coefficients higher than .3 or lower than −.03 are frequently high enough, in absolute terms, to yield a low probability value and justify rejecting the null hypothesis. The relationship in this case would be described as "statistically significant."
Estimating Correlation Coefficients from scatter plots
Look at the scatter plots in Fig. 3.4 and estimate the correlation coefficient that the bivariate relationship shown in each scatter plot would yield.
Explain the basis for each of your estimates of the correlation coefficient.
Spearman's Rank-Order Correlation Coefficient
The Spearman's rank-order correlation coefficient is a nonparametric version of the Pearson product-moment correlation. Spearman's correlation coefficient, (ρ, also signified by rs) measures the strength and direction of the association between two ranked variables.
Bivariate Regression
Bivariate regression is a parametric measure of association that, like correlation analysis, assesses the strength and direction of the relationship between two variables. Also, like correlation analysis, regression assumes linearity. It may give misleading results if used with variable relationships that are not linear.
Regression is a powerful statistic that is widely used in multivariate analyses. This includes ordinary least squares (OLS) regression, which requires that the dependent variable be continuous and assumes linearity; binary logistic regression, which may be used when the dependent variable is dichotomous; and ordinal logistic regression, which is used with ordinal dependent variables. The use of regression in multivariate analysis will be discussed in the next chapter. In bivariate analysis, regression analysis yields coefficients that indicate the strength and direction of the relationship between two variables. Researchers may opt to "standardize" these coefficients. Standardized coefficients from a bivariate regression are the same as the coefficients produced by Pearson product-moment correlation analysis.
T-Test
The t-test, also sometimes called a "difference of means" test, is a parametric statistical test that compares the means of two variables and determines whether they are different enough from each other to reject the null hypothesis and risk a Type I error. The dependent variable in a t-test must be continuous or ordinal—otherwise the investigator cannot calculate a mean. The independent variable must be categorical since t-tests are used to compare two groups.
An example, drawing again on Arab Barometer data, tests the relationship between voting and support for democracy. The hypothesis might be that men and women who voted in the last parliamentary election are more likely than men and women who did not vote to believe that democracy is suitable for their country. Whether a person did or did not vote would be the categorical independent variable, and the dependent variable would be the response to a question like, "To what extent do you think democracy is suitable for your country?" The question about democracy asked respondents to situate their views on a 11-point scale, with 0 indicating completely unsuitable and 10 indicating completely suitable.
Focusing on Tunisia in 2018, Arab Barometer Wave V data show that the mean response on the 11-point suitability question is 5.11 for those who voted and 4.77 for those who did not vote. Is this difference of .34 large enough to be statistically significant? A t-test will determine the probability of getting a difference of this magnitude from a population of interest, most likely all Tunisians of voting age, in which there is no difference between voters and non-voters in views about the suitability of democracy for Tunisia. In this example, the t-test showed p < .086. With this p-value, which is higher than the generally accepted standard of .05, a researcher cannot with confidence reject the null hypotheses, and she is unable, therefore, to assert that the proposed relationship has been confirmed.
This question can also be explored at the country level of analysis with, for example, regime type as the independent variable. In this illustration, the hypothesis is that citizens of monarchies are more likely than citizens of republics to believe that democracy is suitable for their country. Of course, a researcher proposing this hypothesis would also advance an associated causal story that provides the rationale for the hypothesis and specifies what is really being tested. To test this proposition, an investigator might merge data from surveys in, say, three monarchies, perhaps Morocco, Jordan, and Kuwait, and then also merge data from surveys in three republics, perhaps Algeria, Egypt, and Iraq. A t-test would then be used to compare the means of people in republics and people in monarchies and give the p-value.
A similar test, the Wilcoxon-Mann-Whitney test, is a nonparametric test that does not require that the dependent variable be normally distributed.
Analysis of variance, or ANOVA, is closely related to the t-test. It may be used when the dependent variable is continuous and the independent variable is categorical. A one-way ANOVA compares the mean and variance values of a continuous dependent variable in two or more categories of a categorical independent variable in order to determine if the latter affects the former.
ANOVA calculates the F-ratio based on the variance between the groups and the variance within each group. The F-ratio can then be used to calculate a p-value. However, if there are more than two categories of the independent variable, the ANOVA test will not indicate which pairs of categories differ enough to be statistically significant, making it necessary, again, to look at the data in order to draw correct conclusions about the structure of the bivariate relationships. Two-way ANOVA is used when an investigator has more than two variables.
Table 3.6 presents a summary list of the visual representations and bivariate statistical tests that have been discussed. It reminds readers of the procedures that can be used when both variables are categorical, when both variables are numerical/continuous, and when one variable is categorical and one variable is numerical/continuous.
Bivariate Statistics and Causal Inference
It is important to remember that bivariate statistical tests only assess the association or correlation between two variables. The tests described above can help a researcher estimate how much confidence her hypothesis deserves and, more specifically, the probability that any significant variable relationships she has found characterize the larger population from which her data were drawn and about which she seeks to offer information and insight.
The finding that two variables in a hypothesized relationship are related to a statistically significant degree is not evidence that the relationship is causal, only that the independent variable is related to the dependent variable. The finding is consistent with the causal story that the hypothesis represents, and to that extent, it offers support for this story. Nevertheless, there are many reasons why an observed statistically significant relationship might be spurious. The correlation might, for example, reflect the influence of one or more other and uncontrolled variables. This will be discussed more fully in the next chapter. The point here is simply that bivariate statistics do not, by themselves, address the question of whether a statistically significant relationship between two variables is or is not a causal relationship.
Table 3.6 Bivariate visual representations and bivariate statistical tests for pairs of variables possessing particular characteristics
Only an Introductory Overview
As has been emphasized throughout, this chapter seeks only to offer an introductory overview of the bivariate statistical tests that may be employed when an investigator seeks to assess the relationship between two variables. Additional information will be presented in Chap. 4. The focus in Chap. 4 will be on multivariate analysis, on analyses involving three or more variables. In this case again, however, the chapter will provide only an introductory overview. The overviews in the present chapter and the next provide a foundation for understanding social statistics, for understanding what statistical analyses involve and what they seek to accomplish. This is important and valuable in and of itself. Nevertheless, researchers and would-be researchers who intend to incorporate statistical analyses into their investigations, perhaps to test hypotheses and decide whether to risk a Type I error or a Type II error, will need to build on this foundation and become familiar with the contents of texts on social statistics. If this guide offers a bird's eye view, researchers who implement these techniques will also need to expose themselves to the view of the worm at least once.
Chapter 2 makes clear that the concept of variance is central and foundational for much and probably most data-based and quantitative social science research. Bivariate relationships, which are the focus of the present chapter, are building blocks that rest on this foundation. The goal of this kind of research is very often the discovery of causal relationships, relationships that explain rather than merely describe or predict. Such relationships are also frequently described as accounting for variance. This is the focus of Chap. 4, and it means that there will be, first, a dependent variable, a variable that expresses and captures the variance to be explained, and then, second, an independent variable, and possibly more than one independent variable, that impacts the dependent variable and causes it to vary.
Bivariate relationships are at the center of this enterprise, establishing the empirical pathway leading from the variance discussed in Chap. 2 to the causality discussed in Chap. 4. Finding that there is a significant relationship between two variables, a statistically significant relationship, is not sufficient to establish causality, to conclude with confidence that one of the variables impacts the other and causes it to vary. But such a finding is necessary.
The goal of social science inquiry that investigates the relationship between two variables is not always explanation. It might be simply to describe and map the way two variables interact with one another. And there is no reason to question the value of such research. But the goal of data-based social science research is very often explanation; and while the inter-relationships between more than two variables will almost always be needed to establish that a relationship is very likely to be causal, these inter-relationships can only be examined by empirics that begin with consideration of a bivariate relationship, a relationship with one variable that is a presumed cause and one variable that is a presumed effect.
Against this background, with the importance of two-variable relationships in mind, the present chapter offers a comprehensive overview of bivariate relationships, including but not only those that are hypothesized to be causally related. The chapter considers the origin and nature of hypotheses that posit a particular relationship between two variables, a causal relationship if the larger goal of the research is explanation and the delineation of a causal story to which the hypothesis calls attention. This chapter then considers how a bivariate relationship might be described and visually represented, and thereafter it discusses how to think about and determine whether the two variables actually are related.
Presenting tables and graphs to show how two variables are related and using bivariate statistics to assess the likelihood that an observed relationship differs significantly from the null hypothesis, the hypothesis of no relationship, will be sufficient if the goal of the research is to learn as much as possible about whether and how two variables are related. And there is plenty of excellent research that has this kind of description as its primary objective, that makes use for purposes of description of the concepts and procedures introduced in this chapter. But there is also plenty of research that seeks to explain, to account for variance, and for this research, use of these concepts and procedures is necessary but not sufficient. For this research, consideration of a two-variable relationship, the focus of the present chapter, is a necessary intermediate step on a pathway that leads from the observation of variance to explaining how and why that variance looks and behaves as it does.
Dana El Kurd. 2019. "Who Protests in Palestine? Mobilization Across Class Under the Palestinian Authority." In Alaa Tartir and Timothy Seidel, eds. Palestine and Rule of Power: Local Dissent vs. International Governance. New York: Palgrave Macmillan.
Yael Zeira. 2019. The Revolution Within: State Institutions and Unarmed Resistance in Palestine. New York: Cambridge University Press.
Carolina de Miguel, Amaney A. Jamal, and Mark Tessler. 2015. "Elections in the Arab World: Why do citizens turn out?" Comparative Political Studies 48, (11): 1355–1388.
Question 1: Independent variable is religiosity; dependent variable is preference for democracy. Example of hypothesis for Question 1: H1. More religious individuals are more likely than less religious individuals to prefer democracy to other political systems. Question 2: Independent variable is preference for democracy; dependent variable is turning out to vote. Example of hypothesis for Question 2: H2. Individuals who prefer democracy to other political systems are more likely than individuals who do not prefer democracy to other political systems to turn out to vote.
Mike Yi. "A complete Guide to Scatter Plots," posted October 16, 2019 and seen at https://chartio.com/learn/charts/what-is-a-scatter-plot/
The countries are Algeria, Egypt, Iraq, Jordan, Kuwait, Lebanon, Libya, Morocco, Palestine, Sudan, Tunisia, and Yemen. The Wave V surveys were conducted in 2018–2019.
Not considered in this illustration are the substantial cross-country differences in voter turnout. For example, 63.6 of the Lebanese respondents reported voting, whereas in Algeria the proportion who reported voting was only 20.3 percent. In addition to testing hypotheses about voting in which the individual is the unit of analysis, country could also be the unit of analysis, and hypotheses seeking to account for country-level variance in voting could be formulated and tested.
Department of Political Science, University of Michigan, Ann Arbor, MI, USA
Mark Tessler
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Cite this chapter
Tessler, M. (2023). Bivariate Analysis: Associations, Hypotheses, and Causal Stories. In: Social Science Research in the Arab World and Beyond. SpringerBriefs in Sociology. Springer, Cham. https://doi.org/10.1007/978-3-031-13838-6_3
DOI: https://doi.org/10.1007/978-3-031-13838-6_3
Publisher Name: Springer, Cham
Online ISBN: 978-3-031-13838-6
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Brain Structure and Function
June 2018 , Volume 223, Issue 5, pp 2157–2179 | Cite as
Post-mortem inference of the human hippocampal connectivity and microstructure using ultra-high field diffusion MRI at 11.7 T
Justine Beaujoin
Nicola Palomero-Gallagher
Fawzi Boumezbeur
Markus Axer
Jeremy Bernard
Fabrice Poupon
Daniel Schmitz
Jean-François Mangin
Cyril Poupon
The human hippocampus plays a key role in memory management and is one of the first structures affected by Alzheimer's disease. Ultra-high magnetic resonance imaging provides access to its inner structure in vivo. However, gradient limitations on clinical systems hinder access to its inner connectivity and microstructure. A major target of this paper is the demonstration of diffusion MRI potential, using ultra-high field (11.7 T) and strong gradients (750 mT/m), to reveal the extra- and intra-hippocampal connectivity in addition to its microstructure. To this purpose, a multiple-shell diffusion-weighted acquisition protocol was developed to reach an ultra-high spatio-angular resolution with a good signal-to-noise ratio. The MRI data set was analyzed using analytical Q-Ball Imaging, Diffusion Tensor Imaging (DTI), and Neurite Orientation Dispersion and Density Imaging models. High Angular Resolution Diffusion Imaging estimates allowed us to obtain an accurate tractography resolving more complex fiber architecture than DTI models, and subsequently provided a map of the cross-regional connectivity. The neurite density was akin to that found in the histological literature, revealing the three hippocampal layers. Moreover, a gradient of connectivity and neurite density was observed between the anterior and the posterior part of the hippocampus. These results demonstrate that ex vivo ultra-high field/ultra-high gradients diffusion-weighted MRI allows the mapping of the inner connectivity of the human hippocampus, its microstructure, and to accurately reconstruct elements of the polysynaptic intra-hippocampal pathway using fiber tractography techniques at very high spatial/angular resolutions.
Diffusion MRI Human hippocampus Structural connectivity Neurite density Microstructure imaging
The online version of this article ( https://doi.org/10.1007/s00429-018-1617-1) contains supplementary material, which is available to authorized users.
The human hippocampal formation plays a critical role in learning and memory. Its regions appear to be specialized for preferential functions, such as the specific involvement of the dentate gyrus (DG) and the Cornu Ammonis (CA) subfield 3 in pattern separation and completion, respectively, and that of CA2 in social memory (Leutgeb et al. 2007; Hitti and Siegelbaum 2014). Furthermore, the structure and connectivity of hippocampal regions and layers are known to be selectively affected by multiple neurological disorders such as Alzheimer's disease or temporal lobe epilepsy, as well as by the normal process of aging (Zhou et al. 2008; Wang 2006; Dinkelacker et al. 2015; Coras 2014; Prull et al. 200; Wilson et al. 2006). Although anterior–posterior (ventral–dorsal in rodents) functional differences within the hippocampal formation have been reported in humans and experimental animals (Fanselow and Dong 2010; Poppenk et al. 2013), detailed anatomical studies are still not available in humans. The complexity of the hippocampus makes it one of the most mysterious regions in the central nervous system and also one of the most extensively studied. The boundaries between different hippocampal subfields have been described in the neuroanatomy literature using cytoarchitectonic features that require histological staining and microscopic resolution to visualize (Gloor 1997; Amaral and Insausti 1990; Duvernoy 2005), but there are still discrepancies concerning whether it makes sense to separate some regions or not.
Few in vivo imaging techniques are available to investigate the human hippocampus. Magnetic resonance imaging (MRI), and more precisely anatomical MRI (aMRI) and diffusion MRI (dMRI), remain the key modalities used nowadays. Studies using aMRI were first employed to segment the hippocampus and enable volumetric analysis to define early markers of Alzheimer's disease or depression (Jack et al. 1992; Videbech and Ravnkilde 2004; Boutet 2014). Several studies have also been conducted using diffusion tensor imaging (DTI) on clinical MRI systems to investigate the connectivity of the human hippocampus to better understand the anatomo-functional mapping of the limbic system (Adnan et al. 2015; Dinkelacker et al. 2015; Zeineh et al. 2012). However, clinical MRI systems have inherent limitations that prevent one from reaching ultra-high spatial and angular resolutions, mainly due to the characteristics of the available gradient coils. The recent development of the Connectome gradient, able to provide 300 mT/m, could offer an alternative, but it is still limited in spatial resolution (1.25 mm) due to the static field being kept moderate (3T) to prevent the strong mechanical constraints that would occur at ultra-high field (McNab 2013; Setsompop 2013).
Alternatively, ex vivo MR imaging can be performed using ultra-high field preclinical scanners. In addition to the ultra-high static field, these systems can be equipped with very strong gradients that can reach 1000 mT/m. The b-values can thus exceed 10,000 \(\hbox {s}/\hbox {mm}^{2}\), diffusion times can be short enough to reach short diffusion time regimes, and diffusion gradient pulses become closer to Dirac shapes. Finally, contrary to in vivo scans that cannot exceed a couple of hours, ex vivo imaging does not suffer from such a limitation. Specifically, ex vivo studies have thus been carried out with medial temporal lobe (MTL) samples containing the hippocampus (Shepherd et al. 2007; Augustinack 2010; Coras 2014; Colon-Perez 2015; Modo et al. 2016), where most of the authors carried out DTI-based tractography to map some of the larger connections of the hippocampus, such as the perforant pathway (Augustinack 2010; Coras 2014). Although they remain landmark studies, these studies rely on a model known to present strong limitations. First, it cannot model multiple fiber populations within a voxel, like bundle crossings or kissings, which is a weakness especially in the case of the hippocampus, because it contains multiple fiber crossings reflecting its complex circuitry. Second, diffusion tensor features like fractional anisotropy (FA) and mean diffusivity (MD) are inherently non-specific, and a reduction in their value can be associated with different types of microstructural changes. For example, a reduction in FA can be due to demyelination, edema, increased neurite dispersion, or other microstructural changes (Takahashi 2002; Beaulieu 2009; De Santis et al. 2014).
To model multiple fiber populations within a voxel, numerous reconstruction techniques have been developed during the last decades. Regarding modeling of regions with a complex fiber architecture, High Angular Resolution Diffusion Imaging (HARDI) is probably the most widely adopted (see Tournier et al. (2011) for a review of such models). HARDI models produce maps of orientation distribution functions (ODF), the peaks of which characterize the diffusion displacement profile. Since the hippocampus has a very complex fiber architecture with crossings, kissings, and splittings of fibers, it becomes mandatory to use HARDI models to robustly infer its structural connectivity using tractography. To our knowledge, HARDI models have only been used once on a human hippocampus, by Colon-Perez (2015), and not to study the human hippocampal connectivity in its entirety.
Several reconstruction techniques have recently emerged in diffusion MRI to characterize the tissue microstructure, yielding new applications aiming at doing virtual biopsy, and known as diffusion MR microscopy. They rely on the development of multi-compartmental models that estimate, for each voxel, the fraction of each compartments and its key characteristics (e.g., density or dimension). The first established technique relying on the Composite Hindered And Restricted ModEl of Diffusion (CHARMED) introduced by Assaf and Basser (2005) assumed a diffusion attenuation resulting from three compartments. This model laid the foundations of the two next techniques aiming at providing estimates of the axon diameter and density, i.e., AxCaliber (Assaf et al. 2008) and ActiveAx (Alexander et al. 2010), as well as their improvements (Zhang et al. 2011; De Santis et al. 2016). More recently, the Neurite Orientation Dispersion and Density Imaging (NODDI) reconstruction technique was introduced (Zhang et al. 2012) to quantify axon and dendrite densities (collectively known as neurites). These parameters have been shown to provide more specific characteristics of brain tissue microstructure than the quantitative parameters derived from the DTI model.
For the first time, we demonstrate on a human medial temporal lobe sample that ex vivo ultra-high field with strong gradients MRI at 11.7 T and 780 mT/m gives the opportunity to map not only the complex anatomy of the hippocampus, but also its inner connectivity and its organization at the mesoscopic scale. First, we investigate the use of combined anatomical and diffusion MRI to segment the inner structures of the hippocampus and the adjacent entorhinal cortex. Second, we demonstrate that HARDI allows one to accurately reconstruct elements of the polysynaptic pathway of the hippocampal formation. Third, we show that diffusion MR microscopy is a powerful technique, that gives access to insights about cell populations of the hippocampal tissues by revealing the laminar structure of the cornu ammonis (CA).
Tissue sample and container
The study was carried out on a post-mortem right temporal lobe from an 87-year-old male with no abnormal neuropathology findings obtained from the donor program of the Institute of Anatomy, Rostock, Germany. The specimen measured approximately \(38\times 50\times 55\ \hbox {mm}^{3}\), and contained the whole hippocampal formation and some of its surrounding structures. It was fixed with \(4\%\) formalin buffer 36 h after death and stored 38 months until further processing.
Prior to MRI acquisition, the sample was transferred for 1 week to a 0.1 M phosphate-buffered saline solution at \(4\,^{\circ }\hbox {C}\) to be rehydrated. Since acquisitions take place at room temperature (approximately \(20\,^{\circ }\hbox {C}\)), the specimen was placed into the imaging container 4–5 h prior to scanning and transferred to the magnet room to stabilize its temperature to \(20\,^{\circ }\hbox {C}\). This process is required to avoid effects related to temperature variations that would induce modifications of the local \(T_{2}\) relaxation time and the local apparent diffusion coefficient (ADC) up to a factor of 1.5–2 between 4 and \(20\,^{\circ }\hbox {C}\) (Thelwall et al. 2006). Note that the ADC of this post-mortem sample being scanned at \(20\,^{\circ }\hbox {C}\), instead of \(37\,^{\circ }\hbox {C}\) as for in vivo imaging, was already reduced by a factor of about 2. Furthermore, to avoid drying of the tissue during the experiment and to reduce imaging artifacts, the sample was immersed in a proton-free fluid, Fluorinert (FC-40, 3M Company, USA). This fluid does not provide any NMR signal and shares a similar susceptibility coefficient to the one of brain tissues, enabling one to avoid the induction of static magnetic field variations close to the interfaces between air and tissue that would induce local geometrical and intensity distortions. A dedicated container was manufactured to exactly fit the inner diameter of the MR coil antenna with a specific design aimed to prevent the formation of air bubbles that would be responsible for severe susceptibility-induced imaging artifacts. The suspension of the sample within its container is guaranteed by a plastic funnel that does not induce any MR signal and avoids motion artifacts during the commutation of the strong diffusion gradients.
MRI hardware
All the acquisitions were performed on a preclinical 11.7 T Bruker MRI system (BioSpec 117/16 USR Bruker MRI, Ettlingen, Germany) equipped with strong gradients (maximum gradient \(\hbox {magnitude} = 780\,\hbox {mT}/\hbox {m}\), slew-\(\hbox {rate} = 9660\ \hbox {T}/\hbox {m}/\hbox {s}\)) using a 60 mm transmit/receive volume coil. Although surface coils are known to provide better SNRs than volume coils, the 60 mm volume coil was preferred, because it corresponded to the best trade-off between the field-of-view (FOV) coverage and the dimensions of the sample. Furthermore, it enabled preservation of a relative homogeneity of the signal through its entire FOV.
Determination of the magnetic and diffusion properties of the sample
It was mandatory to calibrate the distributions of the magnetic and diffusion properties of the hippocampus sample to adequately tune the target diffusion-weighted multiple-shell imaging protocol required to apply the NODDI model. To this aim, a calibration MRI protocol was established including a series of experiments to infer the histograms of the \(T_{2}\) transverse relaxation time and the mean diffusivity D. Their analysis helps to define the maximum value of the diffusion sensitization b to be used. For a conventional Pulsed-Gradient-Spin-Echo (PGSE) diffusion-weighted imaging sequence, \(\hbox {b} =(\mathbf |G| \gamma \delta )^{2}(\varDelta -\delta /3)=\mathbf |q| ^{2}\tau\) with the approximation of rectangular gradients, with G the applied diffusion gradient vector, \(\gamma\) the nuclear gyromagnetic ratio for water protons, \(\delta\) the duration time of gradient pulses, \(\varDelta\) the time between two pulses, and \(\tau\) the diffusion time.
To reach high b-values, one can increase the gradient strength G, the gradient width \(\delta\), or the separation time between the two gradient pulses \(\varDelta\). However, to keep the gradient pulses close enough to Dirac pulses, thus preserving a Fourier relationship between the diffusion propagator and the diffusion NMR signal with respect to the q wavevector, \(\delta\) has to be kept at its minimum possible value, 4.3 ms in our case. Consequently, either \(\varDelta\) or G have to be raised to increase the diffusion sensitization. The side effect of increasing \(\varDelta\) is a net increase of the echo time \(T_\text {E}\). However, the PGSE sequence yields an NRM signal that integrates not only an exponential diffusion decay \(e^{-bD}\), but also a further exponential decay \(e^{-T_\text {E}/T_{2}}\) linked to the \(T_{2}\) transverse relaxation inherited from the spin-echo scheme present in a PGSE sequence.
An estimation of D and \(T_{2}\) was carried out to determine the range of usable echo time \(T_\text {E}\) and diffusion sensitization b, to avoid excessive signal loss.
Histogram of the transverse relaxation time \(T_{2}\)
Fixed tissues generally suffer from a significant reduction of their transverse relaxation time drastically reducing the time window available to acquire the signal with a good signal-to-noise ratio (SNR) (Pfefferbaum et al. 2004). To map the transverse relaxation time at each voxel of our sample, a standard multi-spin-multi-echo (MSME) pulse sequence (Meiboom and Gill 1958) was used. In total, 12 echoes were collected corresponding to 12 echo times linearly spaced between 6.4 and 76.8 ms. Imaging parameters for this sequence were: isotropic spatial resolution of \(300\,\upmu \hbox {m}\), 12 averages, \(\hbox {TR} = 16,000\,\hbox {ms}\), and a total scan duration of 10 h 14 min. Collected MSME data were then used to fit the log-linear model corresponding to the \(T_{2}\)-decay with a Levenberg–Marquardt algorithm carefully initialized to get robust estimates. The histogram of the \(T_{2}\) quantitative values computed from the voxels included in a precomputed mask of the sample and depicts two main modes of \(T_{2}\). The mode corresponding to the lower value is identified as the white matter \(T_{2}\) (\(T_{2w}\) \(\approx\) 36.3 ms), and that corresponding to the higher value is identified as grey matter \(T_{2}\) (\(T_{2g}\) \(\approx\) 46.4 ms): transverse relaxation time is lower where fibers are more concentrated, i.e., in the white matter. As the hippocampus is mainly composed of grey matter, we assumed its \(T_{2}\) value closer to 46 ms.
Histogram of the mean diffusivity D
Post-mortem tissues depict diffusion coefficients significantly lower (two to five times lower) than in vivo values. Consequently, probing the anisotropy of the diffusion process requires the use of inversely proportional higher b values, to obtain a comparable diffusion contrast to in vivo images (D'Arceuil et al. 2007). A histogram of the mean diffusivity D was inferred using the DTI model from a diffusion-weighted data set acquired with a single-shell sampling of the q-space at \(b=4500\, \hbox {s}/\hbox {mm}^2\) along 60 uniformly distributed diffusion directions, \(\hbox {TR}/\hbox {TE} = 9000/24.2\,\hbox {ms}\), \(\varDelta /\delta = 14.4/4.3\,\hbox {ms}\), matrix size: \(192 \times 192 \times 176\), and an isotropic resolution of \(300\,\upmu\)m. The distribution indicates a mean diffusivity of \(0.16 \times 10^{-3}\ \hbox {mm}^{2}/\hbox {s}\).
Determination of maximum \(T_\text {E}\) and b values
To prevent an excessive loss of signal, a lower acceptable limit of the product \(e^{\frac{-\text{TE}}{T_{2}}}\ e^{-bD}\) was set to 0.05, equally distributed over the two signal decays:
$$\begin{aligned} \left\{ \begin{array}{ccc} e^{\frac{-\text {TE}_{\text {max}}}{T_{2}}} &{} \ge &{} \sqrt{0.05}\\ e^{-b_{\text {max}}D} &{} \ge &{} \sqrt{0.05},\\ \end{array}\right. \end{aligned}$$
and hence
$$\begin{aligned} \left\{ \begin{array}{cccc} \text {TE}_{\text {max}} &{} \le &{} 69 &{}\\ e^{-b_{\text {max}}D} &{} \le &{} 9361 &{} \hbox {s}/\hbox {mm}^{2}.\\ \end{array}\right. \end{aligned}$$
In practice, \(TE_{\text {max}}\) was set to 59.1ms, which enabled a \(b_{\text {max}}\) of 10,000 \(\hbox {s}/\hbox {mm}^{2}\).
Imaging protocol
The imaging protocol included anatomical and diffusion scans. The anatomical scan was tuned to reach a very high spatial resolution to perform accurate manual segmentation of the hippocampal subfields and of the entorhinal cortex.
Anatomical scan
The anatomical image was acquired using a standard \(T_{2}\)-weighted spin-echo sequence with isotropic spatial resolution of \(200\,\upmu \hbox {m}\), matrix \(268 \times 268\); 256 slices, thickness \(200\,\upmu \hbox {m}\), \(\hbox {TR}/\hbox {TE}\) = 12,000\(/26.5\,\hbox {ms}\), 15 averages, and a total scan time of 10 h 46 min.
Diffusion scan
Diffusion data were acquired using a conventional PGSE sequence. The protocol included a first single-shell high angular resolution diffusion imaging (HARDI) data set used to infer the structural connectivity of the human hippocampal formation, to which a multiple-shell hybrid diffusion imaging (HYDI) data set was added to infer quantitative microstructural features using diffusion MR microscopy.
The structural connectivity was established from the HARDI data set collected at \(\hbox {b} = 4500\ \hbox {s}/\hbox {mm}^{2}\) along 500 directions uniformly distributed over a sphere. It was split into 15 blocks of 32 directions and one of 16 because of the memory limitation of the system. For each block, 6 \({b} = 0\) images were acquired. Scanning parameters were: isotropic spatial resolution of \(300\,\upmu \hbox {m}\); \(\hbox {TR}/\hbox {TE} = 9000/24.2\,\hbox {ms}\); \(\varDelta /\delta = 14.4/4.3\,\hbox {ms}\); matrix size: \(192 \times 192\); 176 slices; and a total scan time of 8 days and 18 h. The b value was calibrated taking into account the reduction of the average diffusivity from \(D = 0.7 \times 10^{-3}\,\hbox {mm}^2/\hbox {s}\) in vivo to \(D = 0.16 \times 10^{-3}\,\hbox {mm}^2/\hbox {s}\) in our case, to compensate for the loss of contrast due to the observed reduction factor. As mentioned earlier, the data set was used first to perform tractography, but also to map microstructural features when merged with the next HYDI data set.
In addition to the HARDI data set, a further multiple-shell HYDI data set was acquired for 3 different shells corresponding to 3 different diffusion sensitization at \(\hbox {b} = 4500\, \hbox {s}/\hbox {mm}^{2}\), \(\hbox {b} = 7500\, \hbox {s}/\hbox {mm}^{2}\) and \(\hbox {b}\) = 10,000 \(\hbox {s}/\hbox {mm}^{2}\). The choice of the b-values was carried out based on the usual sensitizations taken for NODDI or ActiveAx models with a scaling factor of around 3 applied to account for the attenuation of the diffusivities observed ex vivo. For each shell, 60 diffusion-weighted volumes were acquired along 60 uniformly distributed diffusion directions. The acquisition was divided into two blocks of 30 directions each. Each block lasted 12 h 57 min, giving a total of 3 days and 7 h. The parameters were tuned as listed in Table 1. It is important at this step to note that the three shells were acquired with linearly increasing separation times \(\varDelta\) of 14.4, 30.0, and 45.0 ms while keeping the gradient pulse width to \(\delta = 4.3\,\hbox {ms}\), to vary the diffusion time. This choice was motivated by the willingness to be able to use alternative models to NODDI in the future, such as ActiveAx, which also requires sampling of the diffusion time. A further specificity of the HYDI protocol was to use the minimum echo time for each shell and not to impose the largest of the three minimum echo times. As a consequence, the data stemming from each shell have to be preprocessed to remove the \(T_{2}\)-decay dependency, which is possible thanks to the quantitative \(T_{2}\) calibration performed previously to characterize the magnetic and diffusion properties of the hippocampus sample.
Parameters for the NODDI data set
b (\(\hbox {s}/\hbox {mm}^{2}\))
\(N_{\text {dir}}\)
TR (ms)
TE (ms)
\(\varDelta\) (ms)
G (mT/m)
Delineation of the inner and surrounding structure of the hippocampus
In-house developed software (by FP), PtkVoi, was used to trace the hippocampal subfields and surrounding structures. It was performed manually by two independent experts from the \(T_{2}\)-weighted anatomical scan, using anatomical landmarks and following the different strategies prescribed and detailed in the literature (Insausti 1998; Duvernoy 2005; Wisse et al. 2012; Insausti and Amaral 2012; Boutet 2014).
Segmentations were traced in the coronal plane and the three-dimensional consistency was ensured by checking the axial and sagittal planes, as well as the 3D shapes of the delineated structures with respect to their known morphology. Considering the main intra-hippocampal circuits, we endeavored to identify: the entorhinal cortex, the dentate gyrus (including the hilus), CA2/CA3, CA1, the alveus, the fimbria, and the subicular complex.
Inner structures of the hippocampus
The number of inner segmented structures of the hippocampus resulted from a trade-off between the feasibility to identify their boundaries on the high-resolution anatomical MRI data (thus depending on their contrast to noise ratio), and the target regions involved in the circuits of interest. The segmentation process used a coarse to fine strategy involving a first step in which the hippocampal head, body, and tail were identified, followed by a second step to delineate regions and/or layers.
Coarse scale: segmentation of head, body, and tail
The segregation of the hippocampus in three parts was based on the protocol presented in Boutet (2014). The very anterior limit is not present in our sample, because the head was partly cut. The most anterior part of the body corresponds to the first coronal slice where the median part of the uncus is no longer visible. The most posterior coronal slice of the body was identified at the level of the enlargement of the fimbria and the loss of the specific C-shape of the body.
Fine scale: segmentation of subregions
In a second step, we segmented the following structures (3):
Alveus and fimbria. These two structures belong to the white matter and are visible in the anatomical \(T_{2}\)-weighted MRI data by the hypointense contrast at the level of the outer boundary of the hippocampus. This specific contrast was exploited to guide their delineation. In addition, the change in orientation of the underlying fibers between the alveus (oriented mainly parallel to the coronal plane) and the fimbria (oriented mainly parallel to the sagittal plane) can be easily identified in diffusion ODF fields, thus allowing the identification of the boundary between the alveus and the fimbria. The lateral boundary of the fimbria was set at the end of the fibers orientation shift zone, where the fibers are entirely contained in the sagittal plane (white line in Fig. 1). The inferior lateral boundary of the alveus was set at the junction with the collateral eminence of the lateral ventricle. The separation between the alveus and the fimbria was only possible in the body and the tail of the hippocampus. At the level of the hippocampal head, all white matter was associated with the alveus.
The lacunosum-molecular layer of the CA1–CA3 regions. This zone can easily be identified in the anatomical \(T_{2}\)-weighted MRI scan as a dark line in the CA region of the hippocampus, corresponding to an area of very low concentration of neural bodies (which are mainly located in the pyramidal layer).
The pyramidal and radiatum layers of the CA1 region. This region is the thickest Ammon's field, while CA2 is the thinnest, thus easily enabling definition of the boundary between both portions of the hippocampus. The border with the subicular complex could often be identified based on differences in grey values, since the CA1 region appeared brighter when compared to the adjacent subicular component. In the tail region of the hippocampus, where differences in grey values were only very subtle, we applied the method suggested by Boutet (2014), which consists of tracing the largest diameter of the hilum and a perpendicular line passing by the medium of the diameter that corresponds to the target boundary.
The pyramidal layer and radiatum of the CA2 and CA3 regions. These two regions were merged into a single ROI (CA2/CA3), because the boundary between them is almost impossible to identify in MR images. Furthermore, not only CA3 neurons project to the CA1 region, but also CA2 pyramids make strong excitatory synaptic contacts with CA1 neurons (Chevaleyre and Siegelbaum 2010). The limit with the dentate gyrus was defined by the end of the ribbon-like aspect of the Ammon's horn (Boutet 2014).
The dentate gyrus. Its boundaries were defined by the other structures already segmented and the cisterna ambiens enlarging the cerebro-spinal fluid-filled space lateral to the cerebral crus.
The subicular complex. The lateral limit of this region was identified based on differences in cortical thickness, as described by Wisse et al. (2012). Thus, the border was defined by drawing a line between the most medial part of the grey matter and the most medial part of the white matter of the temporal stem.
The entorhinal cortex. Its lateral limit is slightly upstream of the collateral sulcus, itself located under the collateral eminence. In Wisse et al. (2012), the posterior border of the entorhinal cortex was set 0.7 mm beyond the hippocampal head which corresponds to four slices for our spatial resolution.
Close view of the shift in fiber orientation between alveus and fimbria. Yellow arrowheads highlight typical inferior–superior orientation of ODFs in the alveus (fibers in the coronal plane, as shown on upper left). Orange arrowheads highlight typical anterior–posterior orientation of ODFs in the fimbria (fibers in the sagittal plane, as shown on upper left). White line shows the boundary between alveus and fimbria, defined within the transition zone corresponding to the shift in the fiber orientation
Connectivity and microstructure mapping
As mentioned earlier, the diffusion-weighted imaging protocol included the acquisition of a multiple-shell HYDI data set at three different b values (4500, 7500, 10,000 \(\hbox {s}/\hbox {mm}^{2}\)) with minimum echo times set for each individual shell. This choice was motivated by the optimization of the SNR per shell, but it requires to correct for the \(T_{2}\)-weighting equal to \(e^{\frac{-\rm {TE}}{T_{2}}}\) that differs from one shell to another. To this aim, a correction was applied to the two larger shells at 7500 and 10,000 \(\hbox {s}/\hbox {mm}^{2}\) that consisted of multiplying voxelwise all the diffusion-weighted images by the compensating factors \(e{^{\frac{\varDelta {\text{TE }}}{ T_{2} (v) }}}\) for each voxel v. This resulted in an intensity comparable to the one obtained with an acquisition done with the TE used at \({b} = 4500\,\hbox {s}/\rm {mm}^{2}\) for all the shells.
The conventional PGSE sequence is not sensitive to field inhomogeneities in comparison with its corresponding echoplanar version, such that the collected DW data are free from distortions induced by eddy currents or susceptibility effects. Consequently, there is no need to apply any correction. In addition, the measured SNR established to 9.9, 7.6, and 4.2 for the three shells, respectively, corresponding to an increasing b-value, are large enough to approximate the Rician noise distribution by a Gaussian distribution, thus allowing the use of conventional mean-square estimators without loss of information. Finally, all the acquisitions were performed during 13 consecutive days avoiding the need to remove the sample from its container, and thus avoiding the presence of any motion between two diffusion-sensitized volumes. Consequently, the remaining transformation exiting between the two anatomical and diffusion data sets turns to be the simple rigid transformation between their two fields of view. This transformation was inferred using the registration tool of the Connectomist toolbox (Duclap et al. 2012; Assaf 2013) based on a mutual information matching criterion optimized using a standard Nelder–Mead simplex algorithm. After registration, the two data sets perfectly matched, allowing to navigate between the structures inferred from the anatomical scans and the connectivity or microstructural quantitative maps inferred from diffusion MRI scans.
Choice of the local reconstruction model
Inference of the local orientation distribution function of the diffusion process or of the fiber orientation distribution can be done using two classes of HARDI techniques: modeldependent reconstructions and model-free reconstructions. Among model-dependent reconstructions, that impose a specific impulse response of a fiber bundle to the diffusion process are the Ball and Sticks model of Behrens (2003), the Constrained Spherical Deconvolution of Tournier and Calamante (2007) or the Sharpening Deconvolution Transform of Descoteaux et al. (2009). Among model-free reconstructions, that do not make any assumption about the response of an heterogeneous population to the diffusion process, are the Diffusion Spectrum Imaging model of Wedeen et al. (2000), the analytical Q-ball model of Descoteaux et al. (2007), the Diffusion Orientation Transform of Özarslan et al. (2006), or the latest Simple Harmonic Oscillator-Based Reconstruction and Estimation (SHORE) propagator model of Özarslan et al. (2013).
The current trend is to use spherical deconvolution approaches to obtain sharp fiber orientation distributions resulting from the deconvolution of ODFs using an auto-estimated convolution kernel. While this approach is suitable to reconstruct the connectivity of the entire brain, it cannot be considered for this study of the inner hippocampal connectivity where the complex configuration of fibers does not enable the definition of an adequate convolution kernel: there is no equivalent of the corpus callosum within the hippocampal complex. Taking this into consideration, the analytical Q-ball model (Descoteaux et al. 2007; Descoteaux 2010) was adopted in our study. This model relies on the decomposition of the DW signal onto a modified spherical harmonics basis, linked to the decomposition of the ODF onto the same basis by the Funk–Hecke matrix. Its use is relevant in our case, because preclinical diffusion MRI allows one to use high diffusion sensitizations (\(\ge \,4500\ \hbox {s}/\hbox {mm}^2\)) with short echo times (thus preventing severe \(T_{2}\)-decay and consequently SNR loss) yielding sharp ODFs. The DTI model was also computed to compare with the aQBI model.
Inference of the structural connectivity using diffusion MRI
The diffusion analyses were performed using the Connectomist toolbox. The HARDI data set was used to compute fields of ODFs stemming from the analytical Q-ball model and from the DTI model, as well as conventional quantitative maps stemming from the tensor model, including the FA map, the MD maps, and the color-encoded direction (CED) map. The analytical Q-ball model reconstruction was applied using a spherical harmonics order 8 and a regularization factor \(\lambda = 0.006\), as defined in Descoteaux et al. (2007).
A streamline regularized deterministic (SRD) and a streamline regularized probabilistic (SRP) tractography algorithms (Perrin et al. 2005) were applied to the whole sample using the maps of aQBI and DTI ODFs previously computed, with the following parameters: 8 seeds per voxel yielding dense tractograms containing 11.6 millions of fibers, forward step \(70\,\upmu \hbox {m}\), maximum solid angle \(30^{\circ }\), minimum/maximum fiber length 0.5/100 mm to avoid loops, and discard spurious fibers. This results in four tractograms (DTI/SRD, QBI/SRD, DTI/SRP, and QBI/SRP) that will be used to analyze the differences in the inference of the structural connectivity with respect to the model and to the fiber tracking algorithm.
Following the connectomics approach introduced in Hagmann (2010) to macroscopically describe the level of connectivity between two sets of regions of interest, we computed, for the deterministic and the probabilistic tractogram obtained with the aQBI model, a \(22 \times 22\) connectivity matrix that, for each pair of hippocampal subfields, counts the number of connections present in the former tractogram and linking them. The matrix is symmetric, because efferent and afferent projections cannot be distinguished with diffusion MRI, and the exclusion of self-connections implies a zero diagonal line. This connectivity matrix gives a concise overview of the main hubs of connections present in the hippocampus. In practice, whole brain studies performed on clinical systems rely on data suffering from low SNRs at high b value, causing overweighting of short reconstructed fibers with respect to long ones. A simple way to counterbalance such effects is to normalize each point of the matrix by the logarithm of the average length of fibers connecting to the concerned subregions. In our case, the hippocampus remains a small structure internally connected with relatively short fibers (average length 34.15 mm; standard deviation of length 21.85 mm) and tractography was relying on a high SNR diffusion MRI data set, thus less prone to fiber tracking degenerescence. Consequently, there is no need to apply any normalization of the connectivity matrix.
Finally, elements of the trisynaptic pathway, one of the most extensively studied pathways of the brain, were reconstructed from the four tractograms. The trisynaptic pathway, as presented in Duvernoy (2005), is composed of three elements:
Perforant path: axons from neurons in the entorhinal cortex that make synaptic contacts with dendrites of the granule cells in the dentate gyrus, located in the molecular layer of the dentate gyrus.
Mossy fibers: the granule cells of the dentate gyrus project to the pyramids of the CA3 (and partly also CA2) region. Synaptic contacts are located in the lucidum layer, which is between the pyramidal and radiatum layers of CA3.
Schaffer collaterals: pyramids of the CA3 region send their axons via the alveus–fimbria–fornix to the mammillary bodies. In addition, collaterals of these axons terminate on the dendrites of CA1 pyramids. These synaptic contacts are located in the lacunosum-molecular layer of CA1.
The four tractograms (QBI/SRP, QBI/SRD, DTI/SRP, and DTI/SRD) were used to extract mossy fibers and the perforant pathway with the aim of performing a comparison of the accuracy of aQBI versus DTI as well as SRD versus SRP tractography algorithms in reconstructing known pathways. To extract trisynaptic elements from the four tractograms, analysis pipelines were developed to intersect the connectograms with the starting and ending regions of interest corresponding to the termination of each element, plus a set of intermediate regions crossed by fibers to avoid the selection of false positives. The four tractograms were analyzed using a single filtering pipeline for each pathway.
Inference of tissue microstructure using diffusion MRI
Because of its practical implementation, from the acquisition point of view with respect to alternative models such as ActiveAx and AxCaliber, the NODDI model has become very popular to map the tissue microstructure in vivo in the frame of clinical applications (Kunz 2014; Chang 2015; Jelescu et al. 2015; Kodiweera et al. 2016). This model was specifically designed to map the biodistribution of dendrites and axons in the brain. It has been mostly applied in vivo in human using low-field conventional MRI scanners, limited to the millimeter spatial resolution. However, some ex vivo preclinical studies have also been performed using NODDI in animal models [mice in Sepehrband et al. (2015) and monkeys in Alexander et al. (2010)]. However, to our knowledge, this is the first time that this model is used to explore the human hippocampus ex vivo. The NODDI model consists of four diffusive compartments (Alexander et al. 2010; Zhang et al. 2012) with no exchange between them. Each compartment contributes to the global diffusion attenuation A resulting from a linear combination of the individual signal attenuations associated with each compartment:
\(A_{\text {ic}}\), the signal attenuation stemming from the compartment of highly restricted water molecules trapped within axons and dendrites (i.e., neurites) modeled as cylinders of zero diameter (i.e., sticks) and characterized by a volume fraction \(f_{\text {ic}}\),
\(A_{\text {ec}}\), the signal attenuation stemming from the extra-cellular compartment of water molecules surrounding the neurites characterized by a volume fraction \(f_{\text {ec}}\). This compartment is modeled by a cylindrically symmetric tensor, assuming a Gaussian anisotropic diffusion independent from the diffusion time,
\(A_{\text {iso}}\), the signal attenuation stemming from the CSF compartment containing free molecules with an isotropic displacement probability and characterized by a volume fraction \(f_{\text {iso}}\),
\(A_{\text {stat}}\), the signal attenuation stemming from the compartment of stationary water molecules trapped within glial cells modeled as spheres of zero diameter (i.e., points) and characterized by a volume fraction \(f_{\text {stat}}\). This additional compartment results from the process of fixation, in particular with the \(4\%\) formaldehyde, that reduces the membrane permeability of glial cells due to an interaction with aquaporin channels (Thelwall et al. 2006).
The net diffusion signal attenuation A corresponds to the following linear combination:
$$\begin{aligned} A = f_{\text {ic}}\cdot a._{\text {ic}} + f_{\text {ec}}\cdot a._{\text {ec}} + f_{\text {iso}}\cdot a._{\text {iso}} + f_{\text {stat}}\cdot a._{\text {stat}} \end{aligned}$$
with \(f_{\text {ic}} + f_{\text {ec}} + f_{\text {stat}} + f_{\text {iso}} = 1\).
The signal from the stationary population remains unattenuated by diffusion weighting, yielding \(A_{\text {stat}} = 1\). Equation (3) can, therefore, be written as follows:
$$\begin{aligned} \begin{aligned} A&=(1-f_{\text {iso}}) \left[ f_{\text {ic}}^{'}\cdot a._{\text {ic}}+f_{\text {ec}}^{'}\cdot a._{\text {ec}} + f_{\text {stat}}^{'} \right] + f_{\text {iso}}\cdot a._{\text {iso}}\\&=(1-f_{\text {iso}})\left[ (1-f_{\text {stat}}^{'})(f_{\text {ic}}^{*}\cdot a._{\text {ic}}+f_{\text {ec}}^{*}\cdot a._{\text {ec}}) + f_{\text {stat}}^{'} \right] \\&\quad +\,f_{\text {iso}}\cdot a._{\text {iso}}\\ \end{aligned} \end{aligned}$$
with \(f_{\text {ic}}^{'} + f_{\text {ec}}^{'} + f_{\text {stat}}^{'} = 1\) and \(f_{\text {ic}}^{*} + f_{\text {ec}}^{*} = 1\), hence:
$$\begin{aligned} \begin{aligned} A&= (1-f_{\text {iso}})\left[ (1-f_{\text {stat}}^{'})(f_{\text {ic}}^{*}\cdot a._{\text {ic}}+(1-f_{\text {ic}}^{*})A_{\text {ec}}) + f_{\text {stat}}^{'} \right] \\&\quad +f_{\text {iso}}\cdot a._{\text {iso}}\\ \end{aligned} \end{aligned}$$
with \(f_{\text {stat}} = (1-f_{\text {iso}}) f_{\text {stat}}^{'}\) and \(f_{\text {ic}} = (1-f_{\text {iso}})(1-f_{\text {stat}}^{'}) f_{\text {ic}}^{*}\).
To speed up the fitting procedure, some parameters were fixed as suggested by Zhang et al. (2012). Watson's distribution was preferred to Bingham's distribution. Whereas, for in vivo studies, intrinsic and isotropic diffusivities are usually set to 1.7 \(\times 10^{-3}\) and \(3.0 \times 10^{-3}\ \hbox {mm}^{2}/\hbox {s}\), respectively (Zhang et al. 2012), they were set to \(0.16 \times 10^{-3}\ \hbox {mm}^{2}/\hbox {s}\) corresponding to the mean diffusivity found in the grey matter of our sample and to the diffusion coefficient of water at \(20\,^{\circ }\hbox {C}\), i.e., \(2.0 \times 10^{-3} \hbox {mm}^{2}/\hbox {s}\).
Statistics of the intra-cellular volume fraction \(f_{\text {ic}}\) (assumed to represent the neurite density) were analyzed for each segmented structure of the hippocampus from the histogram of its values within the structure, to study their variation according to the structure of interest.
Anatomical MRI/3D rendering of anatomical hippocampal structures
The anatomical T2-weighted MRI data set (\(200\ \upmu \hbox {m}\), Fig. 2a, b) presented a very good contrast and SNR (33.7), thus enabling accurate delineation of the entorhinal cortex and of several components of the hippocampal complex: dentate gyrus, pyramidal and radiatum layers of the CA1 and CA2/CA3 regions, lacunosum-molecular layer, alveus, fimbria, and subicular complex. Figure 3 shows series of coronal sections with all the delineated areas. A three-dimensional rendering of the manual segmentation of the hippocampal regions and layers as well as of the entorhinal cortex is available in Supplementary Material. The accuracy of the segmentations is a key factor to successfully discriminate the fiber tracts connecting them.
Raw images obtained with the anatomical acquisition at \(200\ \upmu \hbox {m}\) (a, b), and with the diffusion acquisition at \({b} = 4500\ \hbox {s}/\hbox {mm}^{2}\) (c)
The upper figure shows a sagittal view with references to all the coronal images. Coronal images of the hippocampal formation are shown in an anterior-to-posterior direction from a to h. The head is displayed in a–d, the body in e and f, and the tail in g and h. The segmentation is shown in a\(^{'}\)–h\(^{'}\)
DTI and Q-ball imaging
Figures 4, 5 depict the obtained color-encoded direction map, as well as the Q-ball ODF field and the tractogram obtained with a probabilistic algorithm superimposed on the \(T_{2}\)(\({b}=0\)) reference map. The two figures were obtained using the HARDI data set. The color-encoded maps shown in Fig. 4 reveal a plethora of fine anatomical details and the ODF peaks shown in Fig. 5b, c seem in good agreement with the underlying structural connectivity.
The high anisotropy in the fimbria, oriented in the sagittal plane (orange arrowheads in Figs. 1, 4), can be related to efferent axons from CA3, CA1, and the subicular complex, along with afferent axons from structures in the diencephalon and basal forebrain. These fibers run parallel to the septal–temporal axis of the hippocampus. This main orientation is also visible in Fig. 4a, where the fimbria appears in blue near the head, then pink, and almost red when it goes towards the fornix.
Regarding the alveus, oriented in the coronal plane (yellow arrowheads in Figs. 1 and 4), it contains the axons from the CA1 region and the subicular complex, reaching the fimbria through the alveus in an oblique septal direction. The warping of the fibers around the surface of the hippocampus is also visible in Fig. 4c, where the alveus appears green and then pink when it gets close to the fimbria.
The fiber orientation observed in the pyramidal layer (Figs. 4, 5) can be attributed to the projection of the large apical dendrites of CA1 and CA3 through the lucidum (only in CA3) and radiatum layers towards their termination in the lacunosum-molecular layer. It is also affected by the perforant pathway. Orthogonally to the projection of CA1, CA2, and CA3 dendrites towards the lacunosum-molecular layer, the Schaffer collaterals runs from CA3 to CA1 (probably corresponding to the red part in the pyramidal layer from CA3 to CA1 in Fig. 4c). Voxels between CA3 and CA1 contain multiple fiber crossings, which cannot be resolved by the DTI model. This demonstrates the relevance of using the HARDI/Q-ball model. Figure 5b shows ODFs recovering multiple fiber crossings in the pyramidal layer with a shape revealing two main peaks (one for each principal direction). Figure 5c clearly depicts ODFs with two main peaks that can also be attributed to the Schaffer collateral crossing the projection of CA1, CA2, and CA3 dendrites towards the lacunosum-molecular layer.
In the lacunosum-molecular layer, and to a lesser extent in the radiatum layer, the apical dendrites of CA-pyramids diverge orthogonally into the terminal arborizations that tend to run parallel to the hippocampal sulcus. This corresponds to the area in the radiatum and lacunosum-molecular layers with ODF orthogonal to the coronal plane (pink arrowheads in Figs. 4c, 5b).
Hence, although color-encoded maps obtained with the DTI model show the main directions in each voxel and lead to a partial understanding of the fibers pathways, HARDI/QBall model is mandatory to resolve fiber crossings involved in the Schaffer collaterals and the perforant pathway.
Color-encoded diffusion directions at \(300\,\upmu \hbox {m}\) and 500 directions. The white dotted lines indicate the correspondence between axial (a), sagittal (b), and coronal (c) slices. Orange arrowheads point at the fimbria, yellow arrowheads at the alveus, and pink arrowhead at the lacunosum-molecular layer. The color-encoding cross at the bottom left of the image depicts the direction of largest displacement probability orientation. \({A} \, \hbox {anterior}\), \({P}\, \hbox {posterior}\), \({M}\, \hbox {medial}\), \({L} \,\hbox {lateral}\), \({S}\, \hbox {superior}\), \({I} \,\hbox {inferior}\)
On the left, fusion of the \(T_{2}\)(\({b}=0\)) image and the color-encoded diffusion directions map obtained with the DTI model (a). The color-encoding cross at the bottom left depicts the direction of largest displacement probability orientation. On the right, the orientation diffusion functions field computed with the Q-ball model (b) with a zoom of ODFs showing a crossing at the level of the Schaffer collateral (c), and the SRP tractogram calculated with this field (d), superimposed on (a). Pink arrowhead point at the radiatum and lacunosum-molecular layers with ODF orthogonal to the coronal plane
Analysis of the connectivity matrix and reconstruction of elements of the polysynaptic pathway
Figures 6, 7 show the connectivity matrices of the hippocampal substructures, obtained with the streamline regularized deterministic (Fig. 6) and the streamline regularized probabilistic algorithms (Fig. 7) applied with Q-ball model. As a general trend, the two matrices reveal a higher level of connectivity in the head compared to the body and the tail. The high connectivity of the lacunosum-molecular layer with CA1 can be explained by the location of the apical dendrites of pyramidal neurons of CA1 in this layer. As regards the connectivity between CA1 and the alveus in the head, this is likely due to pyramids of the CA1 region that send their axons via the alveus–fimbria–fornix to the mammillary bodies and representing one source of output from the hippocampus. The same applies to connectivity between CA2/CA3 and the alveus. Moreover, the high connectivity between CA1 and the alveus can be due not only to the presence in this structure of efferent axons from CA1 pyramids, but also to the basal dendrites of the pyramidal neurons bending into the alveus, as described in both polysynaptic and direct pathways (Duvernoy 2005). It also reveals connections that extend through the length of the hippocampus. Connections within each structure are also found. The subicular complex of the body is connected with the subicular complex of the head and the subicular complex of the tail. This is also the case for the dentate gyrus or CA2/CA3. This is in agreement with a primate study (Kondo et al. 2008), showing that the projections of the dentate gyrus extend bidirectionally along much of the length of the hippocampus. Finally, longitudinal connectivity also occurs between related regions like the entorhinal cortex in the hippocampal head and subicular complex in its body. Thus, while the subfields are usually studied in the coronal plane, it appears that connections between subfields extend both in cross section and longitudinally. All these results are in agreement with a recent literature review (Strange et al. 2014) showing a gradient of connectivity that varies along the length of the hippocampus. Finally, the two matrices also depict different connectivity levels. For instance, the level of connectivity between the dentate gyrus and CA2/CA3 appears lower with probabilistic tractography than with deterministic tractography. Conversely, the level of connectivity between CA1 and the alveus is lower with the deterministic approach. These observations would benefit from further analysis and comparison with a gold standard, which is beyond the scope of this study.
Connectivity matrix of the hippocampal substructures obtained with the streamline regularized deterministic model. Each matrix element represents the number of fibers connecting the ROIs indicated by the column and by the row. Self-connections are excluded, which implies a zero diagonal line in the matrix. The map is symmetric, because efferent and afferent projections cannot be distinguished with diffusion MRI. The heat scale represents the number of fibers connecting the ROIs
Connectivity matrix of the hippocampal substructures obtained with the probabilistic model. Each matrix element represents the number of fibers connecting the ROIs indicated by the column and by the row. Self-connections are excluded, which implies a zero diagonal line in the matrix. The map is symmetric, because efferent and afferent projections cannot be distinguished with diffusion MRI. The heat scale represents the number of fibers connecting the ROIs
Figure 8 shows how elements of the polysynaptic pathway could be extracted using high field/strong gradients diffusion MR-based tractography with the QBI/SRP tractogram, and also illustrates the differences in the inference of the structural connectivity with respect to the model and to the fiber tracking algorithm. Mossy fibers could be extracted using the four approaches. Although probabilistic approaches display fibers with a more realistic distribution of their origins along the granular layer of the dentate gyrus. By contrast, when it comes to the extraction of the perforant pathway, probabilistic approaches were the only one to give satisfying results, thus playing in favour of the use of a probabilistic fiber tracking technique. The QBI/SRD method leads to a bundle with very few fibers and the DTI/SRD tractogram does not allow the extraction of any fiber of the perforant pathway. This can be attributed to the fact that probabilistic tractography with multiple fiber orientations shows more robustness to noise and more sensitivity than the standard deterministic tractography as it explores all possible options, allowing to temporarily select suboptimal directions during the streamlining process (Behrens et al. 2007). The comparison of the QBI/SRP and the DTI/SRP results shows the benefit arising from the use of Q-ball model, since the resulting bundle displays more fibers and with a more regular distribution.
3D-rendering of the perforant pathway and mossy fibers in the body of the hippocampus extracted from four tractograms (DTI/SRD, QBI/SRD, DTI/SRP, QBI/SRP) a–d mossy fibers in the body, from the granular layer of the dentate gyrus to the lucidum layer of CA3. 3D-rendering only shows the body segmentation; e, f perforant pathway from the entorhinal cortex to the molecular layer of the dentate gyrus in the body of the hippocampus. 3D-rendering shows the entire segmentation (head, body, and tail) with the same color code as the one presented in Fig. 3 and a transparency of 0.2, except for the entorhinal cortex (opaque grey) and the dentate gyrus in the body (opaque dark blue)
Analysis of the hippocampal tissue microstructure
Investigating neurite density in the hippocampus
Neurite density gradient in hippocampal grey matter. The intra-cellular volume fraction obtained with NODDI model was set with a limit at 0.2 to highlight gradients in grey matter regions. White circle indicates a region of higher intra-cellular volume fraction in grey matter. The heat scale represents the intra-cellular volume fraction. \({A} \, \hbox {anterior}\), \({P}\, \hbox {posterior}\), \({M}\, \hbox {medial}\), \({L} \,\hbox {lateral}\)
Neurite density gradient in hippocampal white matter. The intra-cellular volume fraction map obtained with NODDI model was determined with maximum limit set at 0.7 to highlight gradients in white matter regions where the axonal density is high. White circle indicates a zone of higher density in the posterior hippocampus. The heat scale represents the intra-cellular volume fraction. \({A} \,\hbox {anterior}\), \({P}\, \hbox {posterior}\), \({S} \,\hbox {superior}\), \({I} \, \hbox {inferior}\)
As already mentioned, the restricted intra-cellular volume fraction inferred from the NODDI model corresponds to the neurite density. Within the hippocampus, Figs. 9 and 10 clearly depict a positive gradient of this neurite density in the grey matter from the posterior to the anterior part of the hippocampus and, conversely, a negative gradient in the white matter.
Mean intra-cellular volume fractions were computed for each segmented structure and reported in Table 2. Table 2 also assesses the positive gradient of intra-cellular fraction in the anterior part of the hippocampus, with a higher neurite density in grey matter regions, especially in the subicular complex and CA1. This result is consistent with the results obtained with the tractography, which also revealed a higher connectivity in the head than in the body or tail portions of the hippocampus (Figs. 6, 7). Only CA2/CA3 does not follow the gradient described for the grey matter regions. That might be due to the fact that cell packing density within CA2/CA3 varies along the rostro-caudal axis of the hippocampus. It has already been described to be higher in the body than in the head part in monkeys (Willard et al. 2013). Partial volume effects occurring with the segmentation can also have an impact, e.g., voxels labeled as CA2/CA3, but actually containing white matter.
Table 2 also shows the increased intra-cellular fraction in the tail, with a higher neurite density in white matter, i.e., the alveus and fimbria. This result can be interpreted on the basis of the polysynaptic pathway, since the principal outputs to the cortex merge along the different rostro-caudal levels of the hippocampus in the fimbria, which could explain the gradual increase in axonal density in the fimbria along the anterior–posterior direction.
Mean intra-cellular fraction for each label
Mean \(f_{\text {ic}}\)
Entorhinal cortex
Subicular complex
CA2/CA3
Dentate gyrus
Lacunosum-molecular layer
Fimbria
Investigating the laminar structure of the Ammon's horn
The main hippocampal layers, referred to as layer I, II, and III in Fig. 11c taken from Duvernoy (2005), can be segmented from \(T_{2}\)-weighted images, but their contrast does not provide an indisputable boundary for each layer. As depicted in Fig. 11a, the boundary remains partially defined. Green dotted lines are the boundaries inferred only with the \(T_{2}\)-weighted contrast and, in orange, limits obtained with the combination of the T2-weighted image and the neurite density map (Fig. 11b). Figure 11b clearly demonstrates that the intra-cellular volume fraction of the neurite population significantly enhances the contrast between layers and facilitates their segregation. This is particularly true when considering the delineation of the boundary separating layers II and III. Furthermore, the underlying physical principle driving this novel contrast mechanism is coherent with the anatomical knowledge. Layer I, with a high neurite density, corresponds to the alveus and to the oriens layers. Layer II, in contrast, shows a very poor neurite density, and corresponds to the pyramidal layer, mostly composed of the somas of neurons. Layer III, adjoining the vestigial hippocampal sulcus, appears with a higher neurite density than the pyramidal layer, which is consistent with the fact that it contains arborizations of the apical dendrites of pyramidal neurons, and corresponding to the molecular zone of the CA region, i.e., to the radiatum and lacunosum-molecular layers.
The intra-cellular volume fraction map thus shows new contrasts, consistent with histology compartments, that could be applied in the future to other cortical brain to improve the quality of the segmentation at the level of cortical layers.
Comparison of the contrast between the three hippocampal layers using a a standard anatomical \(T_{2}\)-weighted spin-echo sequence and b the intra-cellular volume fraction map obtained with NODDI model. The heat scale represents the intra-cellular volume fraction. The theoretical layers in cornu Ammonis are drawn in c, adapted from Duvernoy (2005). Their boundaries have been recognized, based on the T2-weighted image, with the green dotted lines in a and also with the help of the intra-cellular volume fraction map as shown with the orange dotted lines in a and the white dotted lines in b
The main contributions of this work are:
The presentation of a novel ultra-high field \(T_{2}\)-weighted and diffusion MRI protocol providing high spatial and angular resolution data. It appears to be useful to propose a novel segmentation approach of the human hippocampus subfields that combines the available sources of contrast to enhance their segregation.
A proof-of-concept that ultra-high field high angular resolution diffusion imaging (UHF-HARDI) can robustly map the inner connectivity of the hippocampus and give evidence of a higher level of connectivity in the uncal region than in the body or tail portions. This result holds regardless of the tractography approach. The inference of known pathways was provided to illustrate the potential of UHF-HARDI. The comparison between the results obtained from the four tractograms highlights the need for using a probabilistic tractography algorithm and, when it comes to the reconstruction of more complex pathways like the perforant pathway, the benefit arising from the use of Q-ball model.
A proof-of-concept that ultra-high field diffusion MR microscopy could play an increasing role in the future to decipher the cytoarchitecture of the hippocampus. In our case, it provides evidence of a rostro-caudal heterogeneity that could be associated with differences in gene expression patterns and could support the long-axis functional specialization of the human hippocampus (Strange et al. 2014). Indeed, the anterior portion of the hippocampus, but not its posterior one, is activated by tasks probing emotional and motivational aspects of cognitive processes (Kim and Fanselow 1992; Viard et al. 2011). Furthermore, whereas the caudal portion of the hippocampus is involved in the encoding of memories or in the local component of spatial representation, its rostral part plays a crucial role in retrieval processes and in the global component of spatial representation (Poppenk et al. 2013; Zeidman et al. 2015; Zeidman and Maguire 2016).
Comparison with existing studies
Several studies have been published to design a robust segmentation pipeline of the hippocampal substructures. Most of those relying on MRI were designed to exploit a single contrast, either \(T_{2}\) or \(T_{2}^{*}\) weighting (Wisse et al. 2012; Boutet 2014; Adler et al. 2014; Yushkevich 2009). The segmentation pipeline proposed in this study combines several contrasts stemming from \(T_{2}\)-weighted anatomical MRI and diffusion-weighted MRI to take benefit from all the available information. Not only the scalar information of diffusion MRI data were exploited, but also the angular profiles of ODFs. Their singularities could be used to better identify the boundaries between substructures of the hippocampal formation when the \(T_{2}\) contrast did not provide enough valuable information, e.g., the transition from alveus to fimbria.
Most existing studies aiming at investigating hippocampal microstructure rely on a diffusion imaging protocol acquired using a single-shell sampling of the q-space, therefore, not compatible with the use of the novel models emerging from the field of diffusion MR microscopy, requiring multiple-shell HYDI acquisition schemes. Most of them used the diffusion tensor model (DTI) and investigated the variations of FA, ADC, or just the contrast of the \(T_{2}\)-weighted image between subregions and/or the layers of the hippocampus (Shepherd et al. 2007; Coras 2014). Unfortunately, those invariant DTI-based scalar features are not specific to a particular cellular organization. A decrease in FA may correspond either to the degenerescence of axons, or to their spreading. A decrease of the ADC may correspond either to a reduction of the axon diameter or to disorders in the extra-cellular space. For instance, Coras (2014) showed the seven hippocampal layers (i.e., alveus, sr pyramidale, sr radiatum, sr lacunosum, sr moleculare, as well as the sr moleculare and granule cell layer of the DG) identified with the contrast of an \(T_{2}\)-weighted image for a healthy sample. Sclerotic hippocampal samples depicted only four layers, and the non-specificity of the method prevented from establishing the causes of this alteration. The HYDI acquisition scheme implemented in our study allowed us to use more advanced models of diffusion MRI giving more specific features to characterize the microstructure, like the neurite density inferred from the NODDI model. Such quantitative features lead to a better understanding of the variations occurring in the rostro-caudal direction at the cellular level and make correlations with the long-axis functional specialization of the human hippocampus.
Established pathways have already been reconstructed or identified in other studies. Zeineh et al. (2012) reconstructed, from in vivo data, the best-known pathways of the medial temporal lobe including the perforant pathway and the Schaffer collaterals. Coras (2014) also identified the perforant pathway on a DTI-based tractogram and Augustinack (2010) reconstructed it from ex vivo DTI-based data. However, our study was one of the first to go beyond diffusion tensor imaging to probe the circuits and the inner connectivity of the human hippocampus. Assuming a Gaussian distribution of water molecule displacements, DTI can support only one fiber population per voxel, thus being unable to render complex fiber configurations like crossings, kissings, or splittings. HARDI models, to which the analytical Q-ball belongs, were designed to go beyond this limitation, and are particularly suitable for the hippocampus where such configurations are likely to happen. Finally, it is the first time that connectivity matrices were used to assess the gradient of connectivity existing along the rostro-caudal axis of the hippocampus. Such observations were hypothesized from functional studies, but to our knowledge, have never been investigated from a structural point of view using diffusion MRI and tools coming from the field of connectomics.
In this study, we have established methods to delineate subfields and substructures of the hippocampal formation to infer their structural connectivity and their microstructure. We have given evidence that using diffusion-based microstructural maps enables the segmentation of smaller hippocampal structures, like its lamination. Investigating this potential should be generalized in the future to develop a novel MRI-based post-mortem atlas of the hippocampal complex. We have also demonstrated that UHF diffusion MRI using a preclinical system allows the reconstruction of hippocampal known pathways, like the perforant path and the mossy fibers. To go a step further, clustering techniques applied to the connectogram would provide clusters of co-localized fibers sharing similar geometries and belonging to the same white matter tract. Combined with the integration of more samples to better capture the intersubject variability, fiber clustering should accelerate the construction of a probabilistic atlas of the hippocampal inner connectivity. However, this construction is out of the scope of this paper.
In the present study, the sample is fixed by immersion, and there is a risk for the fixation to be inhomogeneous, as the fixative has to diffuse from the surface to the deepest structures. The time it takes for fixative to permeate through the brain can lead to higher fixation time for the deepest structures, inducing higher degradation of the tissue, for instance from autolysis. This inhomogeneity could be a confound when reporting gradients in microstructure maps. To minimise this risk, the sample was immersed 2 years, which is enough to entirely fix the tissue. In case of high inhomogeneities resulting from the fixation, clear non-anatomical borders would appear in the structural MRI images. Since no kinds of severe contrasts (independent from anatomical structures) were observed in the high-resolution MRI measurements, the homogeneity of the fixation can be assumed. In addition, the hippocampus is located in the periphery of the brain. There is then little risk of fixation inhomogeneities. However, it is impossible to completely eliminate this confound with one specimen and a further study would benefit from having more samples.
Another aspect that could impact the diffusion contrast and the quality of our results is the choice of the diffusion sensitization. Given the reduction factor of the mean diffusivity from \({D} = 0.7\times 10^{-3}\ \hbox {mm}^2/\hbox {s}\) (standard value reported in vivo) to \({D} = 0.16\times 10^{-3}\ \hbox {mm}^{2}/\hbox {s}\), the diffusion attenuation at \({b}=4500\,\hbox {s}/\hbox {mm}^{2}\) should be equivalent to that of an in vivo scan performed at \({b}=1000\ \hbox {s}/\hbox {mm}^{2}\). A higher diffusion sensitization is generally recommended to obtain sharp ODFs (Hess et al. 2006). The use of a single shell at \(4500\,\hbox {s}/\hbox {mm}^2\) was motivated by the literature, that typically mention the use of b values of at least \(4000\,\hbox {s}/\hbox {mm}^2\) to scan post-mortem pieces. In particular, Dyrby et al. (2011) demonstrated that any HARDI acquisition with a b value between 2000 and \(8000\,\hbox {s}/\hbox {mm}^2\) allows for the inference of multiple fiber populations from ex vivo fixed specimens scanned at room temperature, with an optimal value around \(4000\,\hbox {s}/\hbox {mm}^2\). Furthermore, the application of a probabilistic fiber tracking method contributed to more robustly manage fiber crossings than deterministic approaches. It would also be of great value to investigate alternative reconstructions using the acquired HYDI data set, to go beyond HARDI models. Advanced models could be considered, like the mean average propagator (MAP-MRI) reconstruction (Özarslan et al. 2013) or the fiber orientation distribution (FOD) reconstruction (Jeurissen et al. 2014), relying both on a multiple-shell sampling of the q-space. On the one hand, MAP-MRI would provide the estimation of further information like the return-to-origin probability, sensitive to compartment sizes, or non-Gaussianity, providing insights about the tissue complexity. On the other hand, FODs inferred from multiple-shell acquisitions are based on a multi-tissue constrained spherical deconvolution that would provide more precise fiber orientation estimates at the interface between tissues, thus yielding improved tractograms. In further works, such HYDI-based models may improve the quality of the obtained tractograms. Nevertheless, this investigation is beyond the scope of this study.
Diffusion MR microscopy has become a growing topic of interest in the diffusion MRI community, and models are constantly improving. Nowadays, the NODDI model has become very popular due to its easy implementation from the acquisition protocol to the analysis pipeline. Alternative models should be investigated in the future, in particular those probing further features like the mean axon diameter or the myelin water fraction. Because we took a special care to establish an acquisition protocol densely sampling the q and diffusion time spaces, the ActiveAx model could be investigated in the future using our HYDI diffusion data set. Its investigation is ongoing, but beyond the scope of this preliminary study. The latest improvements of the model now integrate a time dependence for the extra-axonal space that should allow to finely probe maps of the mean axon diameter within the hippocampus with fewer bias.
Validation and comparison with alternative modalities
Ex vivo MRI is able to bridge the gap from the in vivo world to meso-scale configurations with a spatial resolution of 100–\(200\ \upmu \hbox {m}\). In this study, we chose to limit our spatial resolution to 200–\(300\ \upmu \hbox {m}\), respectively, for the anatomical and DW images to reach high b values (10,000\(\ \hbox {s}/\hbox {mm}^{2}\)) with a reasonable SNR to explore the properties of the tissue.
Novel optical methods are able to go down even further, to the microscopic scale. These methods include, in particular, optical coherence tomography (OCT), serial two-photon (STP) tomography, and 3D-polarized light imaging (3D-PLI). STP tomography (Ragan 2012) combines fluorescence imaging with two-photon microscopy, but requires the use of histochemical dyes to label the cells with type-specific fluorescent proteins, which is not the case of the two following methods, therefore, being sensitive only to the targeted cell populations. OCT (Magnain 2015) is a high-resolution (up to \(1\,\upmu \hbox {m}\) in plane) optical technique analogous to ultrasound imaging as it measures the backscattered light of the sample, and is sensitive to differences in the refraction index in tissue. 3D-PLI (Axer 2011) gives the opportunity to observe the 3D orientation of the myelinated fibers without any staining procedure, thanks to the birefringence of the myelin sheath with an in-plane resolution of \(1.3\,\upmu \hbox {m}\) and slices of \(70\,\upmu \hbox {m}\). In contrast with other optical methods, whole human brain imaging is feasible, even if axons with a diameter at the range of the spatial resolution cannot be distinguished. This optical technique has already been applied to ex vivo human hippocampi by Zeineh et al. (2016), but the results have only been compared with in vivo DTI-based color-encoded maps. Despite their remarkable spatial resolution, optical methods also present inherent limitations compared with MRI. First, the sample has to be cut into slices for PLI and STP, and at least into blocks with a flat surface for OCT (Magnain 2014). Furthermore, contrary to dMRI, no real 3D acquisition is possible. Given the extremely large image size, supercomputing facilities are then required to precisely align the serial sections and produce three-dimensional reconstructions. Finally, diffusion MRI gives access to quantitative microstructural characteristics, like axonal density and diameter, which is not easily feasible using the novel optical methods described above.
Despite its own limitations, 3D-PLI can probably be considered as mature enough for the validations of diffusion MRI (Zeineh et al. 2016; Mollink et al. 2017). The human hippocampus sample scanned in the frame of this study is actually being analyzed using the 3D-PLI setup of our research partner. Tractography will be also performed from the 3D-PLI data and compared to the results of this work.
Clinical prospects
This work has been done at an intermediate mesoscopic scale, between the micrometer obtained with optical methods and the millimeter obtained with in vivo MRI. It prefigures what could be achieved with ultra-high field clinical MRI. This is just the beginning of a new era of brain exploration. Advances in knowledge thanks to the ex vivo study, as well as microstructural models and hardware improvement, will allow, in fine, to consider a translational approach to reach the in vivo clinical routine.
Nowadays, there is no atlas of the human hippocampus connectivity and its microstructure at the mesoscopic scale. However, it would be of greatest value to clinical and cognitive neurosciences. Several tools are available to segment the hippocampus from MRI data [the object-based ROI module of the Anatomist software (http://www.brainvisa.info/index.html) in Boutet (2014), FIRST (FSL) or Freesurfer in Morey (2009)], and numerical atlases of the human hippocampus subfields have been established (Chupin 2009; Yushkevich 2010; Iglesias 2015). However, to our knowledge, no numerical atlas of their connectivity or their microstructure has been established. However, in most pathologies affecting the hippocampus, there is a need to better understand which subfields are affected, at what rate, and if the modifications induced by the pathology affect the neuronal cell bodies and/or their connections. For instance, Coras (2014) showed, in the case of hippocampal sclerosis (HS), that type 1 and 2 depicted different rates of cell loss with a more pronounced cell loss in CA3 and CA4 in type 1. Both kinds of HS samples depicted a contrast that did not allow the discrimination of the seven hippocampal layers contrary to normal samples, likely because of pathological shrinkage and fiber alteration.
Regarding structural modifications occurring with the normal process of aging, it is known that the hippocampus undergoes a particular volume decrease with age. MRI studies have suggested that, in typical aging, volume loss is more specific to CA1 and DG/CA3 subregions (Mueller and Weiner 2009). This volume loss is probably not due to neuronal cell loss (Riddle 2007), but rather to synapse loss (Burke and Barnes 2006), and occurs especially in the cortical inputs into the hippocampus such as the perforant pathway (Yassa et al. 2011). Wilson et al. (2006) also suggested that changes strengthen the auto-associative network of the CA3, amplifying the completion pattern (retrieval of previously stored information from a partial cue). The subject studied in this paper was an 87-year-old male. Regarding our results, that would mean that the outputs of the entorhinal cortex may be less significant than in a young hippocampus. As self-connections are excluded, there is no impact of the strengthening of the auto-associative network of CA3. Comparing these results to others obtained with a young hippocampus could highlight the reduction of the perforant path induced by the process of normal aging.
From a fundamental point of view, having access to a fine description of the hippocampal anatomy, including its subfields, its connectivity, and its microstructure, and being able to perform functional imaging using various memory tasks, opens the way to an improved functional neuroanatomy of the sensory, short-term, working, and long-term memories. Better understanding the neural networks driving these various cognitive processes might be useful to design, for instance, novel educational tools to improve the efficacy of young children to learn.
Finally, the protocol established for the human hippocampus could be generalized for the entire brain, and ensuing findings may help to push forward tractography algorithms. One of the limitations in tractography is that when a technique shows high sensitivity, i.e., a high rate of true positives, it most likely will show low specificity, i.e., a high rate of false positives (Thomas et al. 2014). Therefore, adding constraints arising from anatomical priors, like fine connectivity or microstructural characteristics, is intended to drastically reduce false positives creation.
This study forms a proof-of-concept of how ultra-high field MRI with strong gradients can be applied to analyze hippocampal connectivity and microstructure ex vivo. It introduces a unique acquisition and segmentation protocol, and demonstrates that diffusion-weighted MRI offers a new opportunity to map the inner structural connectivity and microstructure of the human hippocampus, in good agreement with histology and current functional studies. The tractography and microstructure models highlight a higher connectivity and neurite density in the anterior hippocampus, whereas the intra-cellular volume fraction map reveals the laminar structure of the Ammon's horn and could be used to improve segmentation protocols. In the future, these results could be of potential benefit to better correlate hippocampal atrophy, observed at low field in Alzheimer's patients, with modifications of its inner connectivity and neurite density.
This project has received funding from the European Union's Horizon 2020 Framework Programme for Research and Innovation under Grant Agreement no. 720270 (Human Brain Project SGA1).
Supplementary material 1 (mp4 139142 KB)
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
1.CEA NeuroSpin/UNIRSGif-sur-YvetteFrance
2.Université Paris-SaclayOrsayFrance
3.France Life ImagingOrsayFrance
4.Forschungszentrum JülichJülichGermany
5.Department of Psychiatry, Psychotherapy and Psychosomatics, Medical FacultyRWTH AachenAachenGermany
6.CEA NeuroSpin/UNATIGif-sur-YvetteFrance
7.CATI Neuroimaging Platformhttp://catineuroimaging.com
Beaujoin, J., Palomero-Gallagher, N., Boumezbeur, F. et al. Brain Struct Funct (2018) 223: 2157. https://doi.org/10.1007/s00429-018-1617-1
Received 17 February 2017 | CommonCrawl |
Orthogonal matrix
From formulasearchengine
In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e.
QTQ=QQT=I,{\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,}
where I is the identity matrix.
This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse:
QT=Q−1,{\displaystyle Q^{\mathrm {T} }=Q^{-1},\,}
An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q*) and therefore normal (Q*Q = QQ*) in the reals. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the dot product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation or reflection. In other words, it is a unitary transformation.
The set of n × n orthogonal matrices forms a group O(n), known as the orthogonal group. The subgroup SO(n) consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.
The complex analogue of an orthogonal matrix is a unitary matrix.
3 Elementary constructions
3.1 Lower dimensions
3.2 Higher dimensions
3.3 Primitives
4.1 Matrix properties
4.2 Group properties
4.3 Canonical form
4.4 Lie algebra
5 Numerical linear algebra
5.1 Benefits
5.2 Decompositions
5.3 Randomization
5.4 Nearest orthogonal matrix
6 Spin and pin
7 Rectangular matrices
An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. Although we consider only real matrices here, the definition can be used for matrices with entries from any field. However, orthogonal matrices arise naturally from dot products, and for matrices of complex numbers that leads instead to the unitary requirement. Orthogonal matrices preserve the dot product,[1] so, for vectors u, v in an n-dimensional real Euclidean space
u⋅v=(Qu)⋅(Qv){\displaystyle {\mathbf {u}}\cdot {\mathbf {v}}=\left(Q{\mathbf {u}}\right)\cdot \left(Q{\mathbf {v}}\right)\,}
where Q is an orthogonal matrix. To see the inner product connection, consider a vector v in an n-dimensional real Euclidean space. Written with respect to an orthonormal basis, the squared length of v is vTv. If a linear transformation, in matrix form Qv, preserves vector lengths, then
vTv=(Qv)T(Qv)=vTQTQv.{\displaystyle {\mathbf {v}}^{\mathrm {T} }{\mathbf {v}}=(Q{\mathbf {v}})^{\mathrm {T} }(Q{\mathbf {v}})={\mathbf {v}}^{\mathrm {T} }Q^{\mathrm {T} }Q{\mathbf {v}}.}
Thus finite-dimensional linear isometries—rotations, reflections, and their combinations—produce orthogonal matrices. The converse is also true: orthogonal matrices imply orthogonal transformations. However, linear algebra includes orthogonal transformations between spaces which may be neither finite-dimensional nor of the same dimension, and these have no orthogonal matrix equivalent.
Orthogonal matrices are important for a number of reasons, both theoretical and practical. The n×n orthogonal matrices form a group under matrix multiplication, the orthogonal group denoted by O(n), which—with its subgroups—is widely used in mathematics and the physical sciences. For example, the point group of a molecule is a subgroup of O(3). Because floating point versions of orthogonal matrices have advantageous properties, they are key to many algorithms in numerical linear algebra, such as QR decomposition. As another example, with appropriate normalization the discrete cosine transform (used in MP3 compression) is represented by an orthogonal matrix.
Below are a few examples of small orthogonal matrices and possible interpretations.
[1001](identity transformation){\displaystyle {\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}\qquad ({\text{identity transformation}})}
An instance of a 2×2 rotation matrix:
R(16.26∘)=[cosθ−sinθsinθcosθ]=[0.96−0.280.280.96](rotation by 16.26∘){\displaystyle R(16.26^{\circ })={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}={\begin{bmatrix}0.96&-0.28\\0.28&\;\;\,0.96\\\end{bmatrix}}\qquad ({\text{rotation by }}16.26^{\circ })}
[100−1](reflection across x-axis){\displaystyle {\begin{bmatrix}1&0\\0&-1\\\end{bmatrix}}\qquad ({\text{reflection across }}x{\text{-axis}})}
[0−0.80−0.600.80−0.360.480.600.48−0.64](rotoinversion:axis (0,−3/5,4/5), angle 90∘){\displaystyle {\begin{bmatrix}0&-0.80&-0.60\\0.80&-0.36&\;\;\,0.48\\0.60&\;\;\,0.48&-0.64\end{bmatrix}}\qquad \left({\begin{aligned}&{\text{rotoinversion:}}\\&{\text{axis }}(0,-3/5,4/5),{\text{ angle }}90^{\circ }\end{aligned}}\right)}
[0001001010000100](permutation of coordinate axes){\displaystyle {\begin{bmatrix}0&0&0&1\\0&0&1&0\\1&0&0&0\\0&1&0&0\end{bmatrix}}\qquad ({\text{permutation of coordinate axes}})}
Elementary constructions
Lower dimensions
The simplest orthogonal matrices are the 1×1 matrices [1] and [−1] which we can interpret as the identity and a reflection of the real line across the origin.
The 2×2 matrices have the form
[ptqu],{\displaystyle {\begin{bmatrix}p&t\\q&u\end{bmatrix}},}
which orthogonality demands satisfy the three equations
1=p2+t2,1=q2+u2,0=pq+tu.{\displaystyle {\begin{aligned}1&=p^{2}+t^{2},\\1&=q^{2}+u^{2},\\0&=pq+tu.\end{aligned}}}
In consideration of the first equation, without loss of generality let p = cos θ, q = sin θ; then either t = −q, u = p or t = q, u = −p. We can interpret the first case as a rotation by θ (where θ = 0 is the identity), and the second as a reflection across a line at an angle of θ/2.
[cosθ−sinθsinθcosθ] (rotation), [cosθsinθsinθ−cosθ] (reflection){\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}{\text{ (rotation), }}\qquad {\begin{bmatrix}\cos \theta &\sin \theta \\\sin \theta &-\cos \theta \\\end{bmatrix}}{\text{ (reflection)}}}
The special case of the reflection matrix with θ=90° generates a reflection about the line 45° line y=x and therefore exchanges x and y; it is a permutation matrix, with a single 1 in each column and row (and otherwise 0):
[0110].{\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}.}
The identity is also a permutation matrix.
A reflection is its own inverse, which implies that a reflection matrix is symmetric (equal to its transpose) as well as orthogonal. The product of two rotation matrices is a rotation matrix, and the product of two reflection matrices is also a rotation matrix.
Regardless of the dimension, it is always possible to classify orthogonal matrices as purely rotational or not, but for 3×3 matrices and larger the non-rotational matrices can be more complicated than reflections. For example,
[−1000−1000−1] and [0−1010000−1]{\displaystyle {\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}}{\text{ and }}{\begin{bmatrix}0&-1&0\\1&0&0\\0&0&-1\end{bmatrix}}}
represent an inversion through the origin and a rotoinversion about the z axis.
[cos(α)cos(γ)−sin(α)sin(β)sin(γ)−sin(α)cos(β)−cos(α)sin(γ)−sin(α)sin(β)cos(γ)cos(α)sin(β)sin(γ)+sin(α)cos(γ)cos(α)cos(β)cos(α)sin(β)cos(γ)−sin(α)sin(γ)cos(β)sin(γ)−sin(β)cos(β)cos(γ)]{\displaystyle {\begin{bmatrix}\cos(\alpha )\cos(\gamma )-\sin(\alpha )\sin(\beta )\sin(\gamma )&-\sin(\alpha )\cos(\beta )&-\cos(\alpha )\sin(\gamma )-\sin(\alpha )\sin(\beta )\cos(\gamma )\\\cos(\alpha )\sin(\beta )\sin(\gamma )+\sin(\alpha )\cos(\gamma )&\cos(\alpha )\cos(\beta )&\cos(\alpha )\sin(\beta )\cos(\gamma )-\sin(\alpha )\sin(\gamma )\\\cos(\beta )\sin(\gamma )&-\sin(\beta )&\cos(\beta )\cos(\gamma )\end{bmatrix}}}
Rotations become more complicated in higher dimensions; they can no longer be completely characterized by one angle, and may affect more than one planar subspace. It is common to describe a 3×3 rotation matrix in terms of an axis and angle, but this only works in three dimensions. Above three dimensions two or more angles are needed, each associated with a plane of rotation.
However, we have elementary building blocks for permutations, reflections, and rotations that apply in general.
The most elementary permutation is a transposition, obtained from the identity matrix by exchanging two rows. Any n×n permutation matrix can be constructed as a product of no more than n − 1 transpositions.
A Householder reflection is constructed from a non-null vector v as
Q=I−2vvTvTv.{\displaystyle Q=I-2{{\mathbf {v}}{\mathbf {v}}^{\mathrm {T} } \over {\mathbf {v}}^{\mathrm {T} }{\mathbf {v}}}.}
Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of v. This is a reflection in the hyperplane perpendicular to v (negating any vector component parallel to v). If v is a unit vector, then Q = I − 2vvT suffices. A Householder reflection is typically used to simultaneously zero the lower part of a column. Any orthogonal matrix of size n×n can be constructed as a product of at most n such reflections.
A Givens rotation acts on a two-dimensional (planar) subspace spanned by two coordinate axes, rotating by a chosen angle. It is typically used to zero a single subdiagonal entry. Any rotation matrix of size n×n can be constructed as a product of at most n(n − 1)/2 such rotations. In the case of 3×3 matrices, three such rotations suffice; and by fixing the sequence we can thus describe all 3×3 rotation matrices (though not uniquely) in terms of the three angles used, often called Euler angles.
A Jacobi rotation has the same form as a Givens rotation, but is used to zero both off-diagonal entries of a 2×2 symmetric submatrix.
Matrix properties
A real square matrix is orthogonal if and only if its columns form an orthonormal basis of the Euclidean space Rn with the ordinary Euclidean dot product, which is the case if and only if its rows form an orthonormal basis of Rn. It might be tempting to suppose a matrix with orthogonal (not orthonormal) columns would be called an orthogonal matrix, but such matrices have no special interest and no special name; they only satisfy MTM = D, with D a diagonal matrix.
The determinant of any orthogonal matrix is +1 or −1. This follows from basic facts about determinants, as follows:
1=det(I)=det(QTQ)=det(QT)det(Q)=(det(Q))2.{\displaystyle 1=\det(I)=\det(Q^{\mathrm {T} }Q)=\det(Q^{\mathrm {T} })\det(Q)=(\det(Q))^{2}\,\!.}
The converse is not true; having a determinant of +1 is no guarantee of orthogonality, even with orthogonal columns, as shown by the following counterexample.
[20012]{\displaystyle {\begin{bmatrix}2&0\\0&{\frac {1}{2}}\end{bmatrix}}}
With permutation matrices the determinant matches the signature, being +1 or −1 as the parity of the permutation is even or odd, for the determinant is an alternating function of the rows.
Stronger than the determinant restriction is the fact that an orthogonal matrix can always be diagonalized over the complex numbers to exhibit a full set of eigenvalues, all of which must have (complex) modulus 1.
Group properties
The inverse of every orthogonal matrix is again orthogonal, as is the matrix product of two orthogonal matrices. In fact, the set of all n×n orthogonal matrices satisfies all the axioms of a group. It is a compact Lie group of dimension n(n − 1)/2, called the orthogonal group and denoted by O(n).
The orthogonal matrices whose determinant is +1 form a path-connected normal subgroup of O(n) of index 2, the special orthogonal group SO(n) of rotations. The quotient group O(n)/SO(n) is isomorphic to O(1), with the projection map choosing [+1] or [−1] according to the determinant. Orthogonal matrices with determinant −1 do not include the identity, and so do not form a subgroup but only a coset; it is also (separately) connected. Thus each orthogonal group falls into two pieces; and because the projection map splits, O(n) is a semidirect product of SO(n) by O(1). In practical terms, a comparable statement is that any orthogonal matrix can be produced by taking a rotation matrix and possibly negating one of its columns, as we saw with 2×2 matrices. If n is odd, then the semidirect product is in fact a direct product, and any orthogonal matrix can be produced by taking a rotation matrix and possibly negating all of its columns. This follows from the property of determinants that negating a column negates the determinant, and thus negating an odd (but not even) number of columns negates the determinant.
Now consider (n+1)×(n+1) orthogonal matrices with bottom right entry equal to 1. The remainder of the last column (and last row) must be zeros, and the product of any two such matrices has the same form. The rest of the matrix is an n×n orthogonal matrix; thus O(n) is a subgroup of O(n + 1) (and of all higher groups).
[0O(n)⋮00⋯01]{\displaystyle {\begin{bmatrix}&&&0\\&O(n)&&\vdots \\&&&0\\0&\cdots &0&1\end{bmatrix}}}
Since an elementary reflection in the form of a Householder matrix can reduce any orthogonal matrix to this constrained form, a series of such reflections can bring any orthogonal matrix to the identity; thus an orthogonal group is a reflection group. The last column can be fixed to any unit vector, and each choice gives a different copy of O(n) in O(n+1); in this way O(n+1) is a bundle over the unit sphere Sn with fiber O(n).
Similarly, SO(n) is a subgroup of SO(n+1); and any special orthogonal matrix can be generated by Givens plane rotations using an analogous procedure. The bundle structure persists: SO(n) ↪ SO(n+1) → Sn. A single rotation can produce a zero in the first row of the last column, and series of n−1 rotations will zero all but the last row of the last column of an n×n rotation matrix. Since the planes are fixed, each rotation has only one degree of freedom, its angle. By induction, SO(n) therefore has
(n−1)+(n−2)+⋯+1=n(n−1)/2{\displaystyle (n-1)+(n-2)+\cdots +1=n(n-1)/2}
degrees of freedom, and so does O(n).
Permutation matrices are simpler still; they form, not a Lie group, but only a finite group, the order n! symmetric group Sn. By the same kind of argument, Sn is a subgroup of Sn+1. The even permutations produce the subgroup of permutation matrices of determinant +1, the order n!/2 alternating group.
Canonical form
More broadly, the effect of any orthogonal matrix separates into independent actions on orthogonal two-dimensional subspaces. That is, if Q is special orthogonal then one can always find an orthogonal matrix P, a (rotational) change of basis, that brings Q into block diagonal form:
PTQP=[R1⋱Rk](n even),PTQP=[R1⋱Rk1](n odd).{\displaystyle P^{T}QP={\begin{bmatrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{bmatrix}}\ (n{\text{ even}}),\ P^{T}QP={\begin{bmatrix}R_{1}&&&\\&\ddots &&\\&&R_{k}&\\&&&1\end{bmatrix}}\ (n{\text{ odd}}).}
where the matrices R1,...,Rk are 2×2 rotation matrices, and with the remaining entries zero. Exceptionally, a rotation block may be diagonal, ±I. Thus, negating one column if necessary, and noting that a 2×2 reflection diagonalizes to a +1 and −1, any orthogonal matrix can be brought to the form
PTQP=[R1⋱Rk00±1⋱±1],{\displaystyle P^{T}QP={\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}},}
The matrices R1,...,Rk give conjugate pairs of eigenvalues lying on the unit circle in the complex plane; so this decomposition confirms that all eigenvalues have absolute value 1. If n is odd, there is at least one real eigenvalue, +1 or −1; for a 3×3 rotation, the eigenvector associated with +1 is the rotation axis.
Lie algebra
Suppose the entries of Q are differentiable functions of t, and that t = 0 gives Q = I. Differentiating the orthogonality condition
QTQ=I{\displaystyle Q^{\mathrm {T} }Q=I\,\!}
Q˙TQ+QTQ˙=0{\displaystyle {\dot {Q}}^{\mathrm {T} }Q+Q^{\mathrm {T} }{\dot {Q}}=0}
Evaluation at t = 0 (Q = I) then implies
Q˙T=−Q˙.{\displaystyle {\dot {Q}}^{\mathrm {T} }=-{\dot {Q}}.}
In Lie group terms, this means that the Lie algebra of an orthogonal matrix group consists of skew-symmetric matrices. Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal matrix (in fact, special orthogonal).
For example, the three-dimensional object physics calls angular velocity is a differential rotation, thus a vector in the Lie algebra so(3){\displaystyle {\mathfrak {so}}(3)} tangent to SO(3). Given ω = (xθ,yθ,zθ), with v = (x,y,z) a unit vector, the correct skew-symmetric matrix form of ω is
Ω=[0−zθyθzθ0−xθ−yθxθ0].{\displaystyle \Omega ={\begin{bmatrix}0&-z\theta &y\theta \\z\theta &0&-x\theta \\-y\theta &x\theta &0\end{bmatrix}}.}
The exponential of this is the orthogonal matrix for rotation around axis v by angle θ; setting c = cos θ/2, s = sin θ/2,
exp(Ω)=[1−2s2+2x2s22xys2−2zsc2xzs2+2ysc2xys2+2zsc1−2s2+2y2s22yzs2−2xsc2xzs2−2ysc2yzs2+2xsc1−2s2+2z2s2].{\displaystyle \exp(\Omega )={\begin{bmatrix}1-2s^{2}+2x^{2}s^{2}&2xys^{2}-2zsc&2xzs^{2}+2ysc\\2xys^{2}+2zsc&1-2s^{2}+2y^{2}s^{2}&2yzs^{2}-2xsc\\2xzs^{2}-2ysc&2yzs^{2}+2xsc&1-2s^{2}+2z^{2}s^{2}\end{bmatrix}}.}
Numerical analysis takes advantage of many of the properties of orthogonal matrices for numerical linear algebra, and they arise naturally. For example, it is often desirable to compute an orthonormal basis for a space, or an orthogonal change of bases; both take the form of orthogonal matrices. Having determinant ±1 and all eigenvalues of magnitude 1 is of great benefit for numeric stability. One implication is that the condition number is 1 (which is the minimum), so errors are not magnified when multiplying with an orthogonal matrix. Many algorithms use orthogonal matrices like Householder reflections and Givens rotations for this reason. It is also helpful that, not only is an orthogonal matrix invertible, but its inverse is available essentially free, by exchanging indices.
Permutations are essential to the success of many algorithms, including the workhorse Gaussian elimination with partial pivoting (where permutations do the pivoting). However, they rarely appear explicitly as matrices; their special form allows more efficient representation, such as a list of n indices.
Likewise, algorithms using Householder and Givens matrices typically use specialized methods of multiplication and storage. For example, a Givens rotation affects only two rows of a matrix it multiplies, changing a full multiplication of order n3 to a much more efficient order n. When uses of these reflections and rotations introduce zeros in a matrix, the space vacated is enough to store sufficient data to reproduce the transform, and to do so robustly. (Following Template:Harvtxt, we do not store a rotation angle, which is both expensive and badly behaved.)
A number of important matrix decompositions Template:Harv involve orthogonal matrices, including especially:
QR decomposition
M = QR, Q orthogonal, R upper triangular.
Singular value decomposition
M = UΣVT, U and V orthogonal, Σ non-negative diagonal.
Eigendecomposition of a symmetric matrix (Decomposition according to Spectral theorem)
S = QΛQT, S symmetric, Q orthogonal, Λ diagonal.
Polar decomposition
M = QS, Q orthogonal, S symmetric non-negative definite.
Consider an overdetermined system of linear equations, as might occur with repeated measurements of a physical phenomenon to compensate for experimental errors. Write Ax = b, where A is m×n, m > n. A QR decomposition reduces A to upper triangular R. For example, if A is 5×3 then R has the form
R=[⋆⋆⋆0⋆⋆00⋆000000].{\displaystyle R={\begin{bmatrix}\star &\star &\star \\0&\star &\star \\0&0&\star \\0&0&0\\0&0&0\end{bmatrix}}.}
The linear least squares problem is to find the x that minimizes ‖Ax − b‖, which is equivalent to projecting b to the subspace spanned by the columns of A. Assuming the columns of A (and hence R) are independent, the projection solution is found from ATAx = ATb. Now ATA is square (n×n) and invertible, and also equal to RTR. But the lower rows of zeros in R are superfluous in the product, which is thus already in lower-triangular upper-triangular factored form, as in Gaussian elimination (Cholesky decomposition). Here orthogonality is important not only for reducing ATA = (RTQT)QR to RTR, but also for allowing solution without magnifying numerical problems.
In the case of a linear system which is underdetermined, or an otherwise non-invertible matrix, singular value decomposition (SVD) is equally useful. With A factored as UΣVT, a satisfactory solution uses the Moore-Penrose pseudoinverse, VΣ+UT, where Σ+ merely replaces each non-zero diagonal entry with its reciprocal. Set x to VΣ+UTb.
The case of a square invertible matrix also holds interest. Suppose, for example, that A is a 3×3 rotation matrix which has been computed as the composition of numerous twists and turns. Floating point does not match the mathematical ideal of real numbers, so A has gradually lost its true orthogonality. A Gram-Schmidt process could orthogonalize the columns, but it is not the most reliable, nor the most efficient, nor the most invariant method. The polar decomposition factors a matrix into a pair, one of which is the unique closest orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular. (Closeness can be measured by any matrix norm invariant under an orthogonal change of basis, such as the spectral norm or the Frobenius norm.) For a near-orthogonal matrix, rapid convergence to the orthogonal factor can be achieved by a "Newton's method" approach due to Template:Harvtxt (1990), repeatedly averaging the matrix with its inverse transpose. Template:Harvtxt has published an accelerated method with a convenient convergence test.
For example, consider a non-orthogonal matrix for which the simple averaging algorithm takes seven steps
[3175]→[1.81250.06253.43752.6875]→⋯→[0.8−0.60.60.8]{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}1.8125&0.0625\\3.4375&2.6875\end{bmatrix}}\rightarrow \cdots \rightarrow {\begin{bmatrix}0.8&-0.6\\0.6&0.8\end{bmatrix}}}
and which acceleration trims to two steps (with γ = 0.353553, 0.565685).
[3175]→[1.41421−1.060661.060661.41421]→[0.8−0.60.60.8]{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}1.41421&-1.06066\\1.06066&1.41421\end{bmatrix}}\rightarrow {\begin{bmatrix}0.8&-0.6\\0.6&0.8\end{bmatrix}}}
Gram-Schmidt yields an inferior solution, shown by a Frobenius distance of 8.28659 instead of the minimum 8.12404.
[3175]→[0.393919−0.9191450.9191450.393919]{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}0.393919&-0.919145\\0.919145&0.393919\end{bmatrix}}}
Some numerical applications, such as Monte Carlo methods and exploration of high-dimensional data spaces, require generation of uniformly distributed random orthogonal matrices. In this context, "uniform" is defined in terms of Haar measure, which essentially requires that the distribution not change if multiplied by any freely chosen orthogonal matrix. Orthogonalizing matrices with independent uniformly distributed random entries does not result in uniformly distributed orthogonal matrices{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}, but the QR decomposition of independent normally distributed random entries does, as long as the diagonal of R contains only positive entries. Template:Harvtxt replaced this with a more efficient idea that Template:Harvtxt later generalized as the "subgroup algorithm" (in which form it works just as well for permutations and rotations). To generate an (n + 1)×(n + 1) orthogonal matrix, take an n×n one and a uniformly distributed unit vector of dimension n + 1. Construct a Householder reflection from the vector, then apply it to the smaller matrix (embedded in the larger size with a 1 at the bottom right corner).
Nearest orthogonal matrix
The problem of finding the orthogonal matrix Q{\displaystyle Q} nearest a given matrix M{\displaystyle M} is related to the Orthogonal Procrustes problem. There are several different ways to get the unique solution, the simplest of which is taking the singular value decomposition of M{\displaystyle M} and replacing the singular values with ones. Another method expresses the R{\displaystyle R} explicitly but requires the use of a matrix square root:[2]
Q=M(MTM)−12{\displaystyle Q=M(M^{\mathrm {T} }M)^{-{\frac {1}{2}}}}
This may be combined with the Babylonian method for extracting the square root of a matrix to give a recurrence which converges to an orthogonal matrix quadratically:
Qn+1=2M(Qn−1M+MTQn)−1{\displaystyle Q_{n+1}=2M(Q_{n}^{-1}M+M^{\mathrm {T} }Q_{n})^{-1}}
where Q0=M{\displaystyle Q_{0}=M} . These iterations are stable provided the condition number of M{\displaystyle M} is less than three.[3]
Spin and pin
A subtle technical problem afflicts some uses of orthogonal matrices. Not only are the group components with determinant +1 and −1 not connected to each other, even the +1 component, SO(n), is not simply connected (except for SO(1), which is trivial). Thus it is sometimes advantageous, or even necessary, to work with a covering group of SO(n), the spin group, Spin(n). Likewise, O(n) has covering groups, the pin groups, Pin(n). For n > 2, Spin(n) is simply connected, and thus the universal covering group for SO(n). By far the most famous example of a spin group is Spin(3), which is nothing but SU(2), or the group of unit quaternions.
The Pin and Spin groups are found within Clifford algebras, which themselves can be built from orthogonal matrices.
Rectangular matrices
If Q is not a square matrix, then the conditions QTQ = I and QQT = I are not equivalent. The condition QTQ = I says that the columns of Q are orthonormal. This can only happen if Q is an m×n matrix with n ≤ m. Similarly, QQT = I says that the rows of Q are orthonormal, which requires n ≥ m.
There is no standard terminology for these matrices. They are sometimes called "orthonormal matrices", sometimes "orthogonal matrices", and sometimes simply "matrices with orthonormal rows/columns".
Orthogonal group
Rotation (mathematics)
Skew-symmetric matrix, a matrix whose transpose is its negative
Symplectic matrix
Unitary matrix
↑ "Paul's online math notes"Template:Citation broken, Paul Dawkins, Lamar University, 2008. Theorem 3(c)
↑ "Finding the Nearest Orthonormal Matrix", Berthold K. P. Horn, MIT.
↑ "Newton's Method for the Matrix Square Root", Nicholas J. Higham, Mathematics of Computation, Volume 46, Number 174, 1986.
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Tutorial and Interactive Program on Orthogonal Matrix
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Strong neutron pairing in core+4n nuclei (1803.04777)
A. Revel, F.M. Marques, O. Sorlin, T. Aumann, C. Caesar, M. Holl, V. Panin, M. Vandebrouck, F. Wamers, H. Alvarez-Pol, L. Atar, V. Avdeichikov, S. Beceiro-Novo, D. Bemmerer, J. Benlliure, C. A. Bertulani, J. M. Boillos, K. Boretzky, M. J. G. Borge, M. Caamano, E. Casarejos, W.N. Catford, J. Cederkäll, M. Chartier, L. Chulkov, D. Cortina-Gil, E. Cravo, R. Crespo, U. Datta Pramanik, P. Diaz Fernandez, I. Dillmann, Z. Elekes, J. Enders, O. Ershova, A. Estrade, F. Farinon, L. M. Fraile, M. Freer, D. Galaviz, H. Geissel, R. Gernhauser, P. Golubev, K. Göbel, J. Hagdahl, T. Heftrich, M. Heil, M. Heine, A. Heinz, A. Henriques, A. Hufnagel, A. Ignatov, H.T. Johansson, B. Jonson, J. Kahlbow, N. Kalantar-Nayestanaki, R. Kanungo, A. Kelic-Heil, A. Knyazev, T. Kroll, N. Kurz, M. Labiche, C. Langer, T. Le Bleis, R. Lemmon, S. Lindberg, J. Machado, J. Marganiec, A. Movsesyan, E. Nacher, M. Najafi, E. Nikolskii, T. Nilsson, C. Nociforo, S. Paschalis, A. Perea, M. Petri, S. Pietri, R. Plag, R. Reifarth, G. Ribeiro, C. Rigollet, M. Roder, D. Rossi, D. Savran, H. Scheit, H. Simon, I. Syndikus, J. T. Taylor, O. Tengblad, R. Thies, Y. Togano, P. Velho, V. Volkov, A. Wagner, H. Weick, C. Wheldon, G. Wilson, J. S. Winfield, P. Woods, D. Yakorev, M. Zhukov, A. Zilges, K. Zuber
March 13, 2018 nucl-ex
The emission of neutron pairs from the neutron-rich $N\!=\!12$ isotones $^{18}$C and $^{20}$O has been studied by high-energy nucleon knockout from $^{19}$N and $^{21}$O secondary beams, populating unbound states of the two isotones up to 15~MeV above their two-neutron emission thresholds. The analysis of triple fragment-$n$-$n$ correlations shows that the decay $^{19}$N$(-1p)^{18}$C$^*\!\rightarrow^{16}$C+$n$+$n$ is clearly dominated by direct pair emission. The two-neutron correlation strength, the largest ever observed, suggests the predominance of a $^{14}$C core surrounded by four valence neutrons arranged in strongly correlated pairs. On the other hand, a significant competition of a sequential branch is found in the decay $^{21}$O$(-1n)^{20}$O$^*\!\rightarrow^{18}$O+$n$+$n$, attributed to its formation through the knockout of a deeply-bound neutron that breaks the $^{16}$O core and reduces the number of pairs.
Effective proton-neutron interaction near the drip line from unbound states in $^{25,26}$F (1707.07995)
M. Vandebrouck, A. Lepailleur, O. Sorlin, T. Aumann, C. Caesar, M. Holl, V. Panin, F. Wamers, S.R. Stroberg, J.D. Holt, F. De Oliveira Santos, H. Alvarez-Pol, L. Atar, V. Avdeichikov, S. Beceiro-Novo, D. Bemmerer, J. Benlliure, C. A. Bertulani, S.K. Bogner, J.M. Boillos, K. Boretzky, M.J.G. Borge, M. Caamano, E. Casarejos, W. Catford, J. Cederkäll, M. Chartier, L. Chulkov, D. Cortina-Gil, E. Cravo, R. Crespo, U. Datta Pramanik, P. Diaz Fernandez, I. Dillmann, Z. Elekes, J. Enders, O. Ershova, A. Estrade, F. Farinon, L.M. Fraile, M. Freer, D. Galaviz, H. Geissel, R. Gernhauser, J. Gibelin, P. Golubev, K. Göbel, J. Hagdahl, T. Heftrich, M. Heil, M. Heine, A. Heinz, A. Henriques, H. Hergert, A. Hufnagel, A. Ignatov, H.T. Johansson, B. Jonson, J. Kahlbow, N. Kalantar-Nayestanaki, R. Kanungo, A. Kelic-Heil, A. Knyazev, T. Kröll, N. Kurz, M. Labiche, C. Langer, T. Le Bleis, R. Lemmon, S. Lindberg, J. Machado, J. Marganiec, F.M. Marques, A. Movsesyan, E. Nacher, M. Najafi, E. Nikolskii, T. Nilsson, C. Nociforo, S. Paschalis, A. Perea, M. Petri, S. Pietri, R. Plag, R. Reifarth, G. Ribeiro, C. Rigollet, M. Röder, D. Rossi, D. Savran, H. Scheit, A. Schwenk, H. Simon, I. Syndikus, J. Taylor, O. Tengblad, R. Thies, Y. Togano, P. Velho, V. Volkov, A. Wagner, H. Weick, C. Wheldon, G. Wilson, J.S. Winfield, P. Woods, D. Yakorev, M. Zhukov, A. Zilges, K. Zuber
July 25, 2017 nucl-ex
Background: Odd-odd nuclei, around doubly closed shells, have been extensively used to study proton-neutron interactions. However, the evolution of these interactions as a function of the binding energy, ultimately when nuclei become unbound, is poorly known. The $^{26}$F nucleus, composed of a deeply bound $\pi0d\_{5/2}$ proton and an unbound $\nu0d\_{3/2}$ neutron on top of an $^{24}$O core, is particularly adapted for this purpose. The coupling of this proton and neutron results in a $J^{\pi} = 1^{+}\_1 - 4^{+}\_1$ multiplet, whose energies must be determined to study the influence of the proximity of the continuum on the corresponding proton-neutron interaction. The $J^{\pi} = 1^{+}\_1, 2^{+}\_1,4^{+}\_1$ bound states have been determined, and only a clear identification of the $J^{\pi} =3^{+}\_1$ is missing.Purpose: We wish to complete the study of the $J^{\pi} = 1^{+}\_1 - 4^{+}\_1$ multiplet in $^{26}$F, by studying the energy and width of the $J^{\pi} =3^{+}\_1$ unbound state. The method was firstly validated by the study of unbound states in $^{25}$F, for which resonances were already observed in a previous experiment.Method: Radioactive beams of $^{26}$Ne and $^{27}$Ne, produced at about $440A$\,MeV by the FRagment Separator at the GSI facility, were used to populate unbound states in $^{25}$F and $^{26}$F via one-proton knockout reactions on a CH$\_2$ target, located at the object focal point of the R$^3$B/LAND setup. The detection of emitted $\gamma$-rays and neutrons, added to the reconstruction of the momentum vector of the $A-1$ nuclei, allowed the determination of the energy of three unbound states in $^{25}$F and two in $^{26}$F. Results: Based on its width and decay properties, the first unbound state in $^{25}$F is proposed to be a $J^{\pi} = 1/2^-$ arising from a $p\_{1/2}$ proton-hole state. In $^{26}$F, the first resonance at 323(33)~keV is proposed to be the $J^{\pi} =3^{+}\_1$ member of the $J^{\pi} = 1^{+}\_1 - 4^{+}\_1$ multiplet. Energies of observed states in $^{25,26}$F have been compared to calculations using the independent-particle shell model, a phenomenological shell-model, and the ab initio valence-space in-medium similarity renormalization group method.Conclusions: The deduced effective proton-neutron interaction is weakened by about 30-40\% in comparison to the models, pointing to the need of implementing the role of the continuum in theoretical descriptions, or to a wrong determination of the atomic mass of $^{26}$F.
Determination of the Neutron-Capture Rate of 17C for the R-process Nucleosynthesis (1604.05832)
M. Heine, S. Typel, M.-R. Wu, T. Adachi, Y. Aksyutina, J. Alcantara, S. Altstadt, H. Alvarez-Pol, N. Ashwood, T. Aumann, V. Avdeichikov, M. Barr, S. Beceiro-Novo, D. Bemmerer, J. Benlliure, C. A. Bertulani, K. Boretzky, M. J. G. Borge, G. Burgunder, M. Caamano, C. Caesar, E. Casarejos, W. Catford, J. Cederkäll, S. Chakraborty, M. Chartier, L. V. Chulkov, D. Cortina-Gil, R. Crespo, U. Datta Pramanik, P. Diaz Fernandez, I. Dillmann, Z. Elekes, J. Enders, O. Ershova, A. Estrade, F. Farinon, L. M. Fraile, M. Freer, M. Freudenberger, H. O. U. Fynbo, D. Galaviz, H. Geissel, R. Gernhäuser, K. Göbel, P. Golubev, D. Gonzalez Diaz, J. Hagdahl, T. Heftrich, M. Heil, A. Heinz, A. Henriques, M. Holl, G. Ickert, A. Ignatov, B. Jakobsson, H. T. Johansson, B. Jonson, N. Kalantar-Nayestanaki, R. Kanungo, A. Kelic-Heil, R. Knöbel, T. Kröll, R. Krücken, J. Kurcewicz, N. Kurz, M. Labiche, C. Langer, T. Le Bleis, R. Lemmon, O. Lepyoshkina, S. Lindberg, J. Machado, J. Marganiec, G. Martínez-Pinedo, V. Maroussov, M. Mostazo, A. Movsesyan, A. Najafi, T. Neff, T. Nilsson, C. Nociforo, V. Panin, S. Paschalis, A. Perea, M. Petri, S. Pietri, R. Plag, A. Prochazka, A. Rahaman, G. Rastrepina, R. Reifarth, G. Ribeiro, M. V. Ricciardi, C. Rigollet, K. Riisager, M. Röder, D. Rossi, J. Sanchez del Rio, D. Savran, H. Scheit, H. Simon, O. Sorlin, V. Stoica, B. Streicher, J. T. Taylor, O. Tengblad, S. Terashima, R. Thies, Y. Togano, E. Uberseder, J. Van de Walle, P. Velho, V. Volkov, A. Wagner, F. Wamers, H. Weick, M. Weigand, C. Wheldon, G. Wilson, C. Wimmer, J. S. Winfield, P. Woods, D. Yakorev, M. V. Zhukov, A. Zilges, K. Zuber
April 20, 2016 nucl-ex, astro-ph.SR
With the R$^{3}$B-LAND setup at GSI we have measured exclusive relative-energy spectra of the Coulomb dissociation of $^{18}$C at a projectile energy around 425~AMeV on a lead target, which are needed to determine the radiative neutron-capture cross sections of $^{17}$C into the ground state of $^{18}$C. Those data have been used to constrain theoretical calculations for transitions populating excited states in $^{18}$C. This allowed to derive the astrophysical cross section $\sigma^{*}_{\mathrm{n}\gamma}$ accounting for the thermal population of $^{17}$C target states in astrophysical scenarios. The experimentally verified capture rate is significantly lower than those of previously obtained Hauser-Feshbach estimations at temperatures $T_{9}\leq{}1$~GK. Network simulations with updated neutron-capture rates and hydrodynamics according to the neutrino-driven wind model as well as the neutron-star merger scenario reveal no pronounced influence of neutron capture of $^{17}$C on the production of second- and third-peak elements in contrast to earlier sensitivity studies.
Systematic investigation of projectile fragmentation using beams of unstable B and C isotopes (1603.00323)
R. Thies, A. Heinz, T. Adachi, Y. Aksyutina, J. Alcantara-Núñes, S. Altstadt, H. Alvarez-Pol, N. Ashwood, T. Aumann, V. Avdeichikov, M. Barr, S. Beceiro-Novo, D. Bemmerer, J. Benlliure, C. A. Bertulani, K. Boretzky, M. J. G. Borge, G. Burgunder, M. Caamano, C. Caesar, E. Casarejos, W. Catford, J. Cederkäll, S. Chakraborty, M. Chartier, L. V. Chulkov, D. Cortina-Gil, R. Crespo, U. Datta, P. Díaz Fernández, I. Dillmann, Z. Elekes, J. Enders, O. Ershova, A. Estradé, F. Farinon, L. M. Fraile, M. Freer, M. Freudenberger, H. O. U. Fynbo, D. Galaviz, H. Geissel, R. Gernhäuser, K. Göbel, P. Golubev, D. Gonzalez Diaz, J. Hagdahl, T. Heftrich, M. Heil, M. Heine, A. Henriques, M. Holl, G. Ickert, A. Ignatov, B. Jakobsson, H. T. Johansson, B. Jonson, N. Kalantar-Nayestanaki, R. Kanungo, R. Knöbel, T. Kröll, R. Krücken, J. Kurcewicz, N. Kurz, M. Labiche, C. Langer, T. Le Bleis, R. Lemmon, O. Lepyoshkina, S. Lindberg, J. Machado, J. Marganiec, V. Maroussov, M. Mostazo, A. Movsesyan, A. Najafi, T. Nilsson, C. Nociforo, V. Panin, S. Paschalis, A. Perea, M. Petri, S. Pietri, R. Plag, A. Prochazka, A. Rahaman, G. Rastrepina, R. Reifarth, G. Ribeiro, M. V. Ricciardi, C. Rigollet, K. Riisager, M. Röder, D. Rossi, J. Sanchez del Rio, D. Savran, H. Scheit, H. Simon, O. Sorlin, V. Stoica, B. Streicher, J. T. Taylor, O. Tengblad, S. Terashima, Y. Togano, E. Uberseder, J. Van de Walle, P. Velho, V. Volkov, A. Wagner, F. Wamers, H. Weick, M. Weigand, C. Wheldon, G. Wilson, C. Wimmer, J. S. Winfield, P. Woods, D. Yakorev, M. V. Zhukov, A. Zilges, K. Zuber
March 2, 2016 nucl-ex
Background: Models describing nuclear fragmentation and fragmentation-fission deliver important input for planning nuclear physics experiments and future radioactive ion beam facilities. These models are usually benchmarked against data from stable beam experiments. In the future, two-step fragmentation reactions with exotic nuclei as stepping stones are a promising tool to reach the most neutron-rich nuclei, creating a need for models to describe also these reactions. Purpose: We want to extend the presently available data on fragmentation reactions towards the light exotic region on the nuclear chart. Furthermore, we want to improve the understanding of projectile fragmentation especially for unstable isotopes. Method: We have measured projectile fragments from 10,12-18C and 10-15B isotopes colliding with a carbon target. These measurements were all performed within one experiment, which gives rise to a very consistent dataset. We compare our data to model calculations. Results: One-proton removal cross sections with different final neutron numbers (1pxn) for relativistic 10,12-18C and 10-15B isotopes impinging on a carbon target. Comparing model calculations to the data, we find that EPAX is not able to describe the data satisfactorily. Using ABRABLA07 on the other hand, we find that the average excitation energy per abraded nucleon needs to be decreased from 27 MeV to 8.1 MeV. With that decrease ABRABLA07 describes the data surprisingly well. Conclusions: Extending the available data towards light unstable nuclei with a consistent set of new data have allowed for a systematic investigation of the role of the excitation energy induced in projectile fragmentation. Most striking is the apparent mass dependence of the average excitation energy per abraded nucleon. Nevertheless, this parameter, which has been related to final-state interactions, requires further study.
First measurement of several $\beta$-delayed neutron emitting isotopes beyond N=126 (1511.01296)
R. Caballero-Folch, C. Domingo-Pardo, J. Agramunt, A. Algora, F. Ameil, A. Arcones, Y. Ayyad, J. Benlliure, I.N. Borzov, M. Bowry, F. Calvino, D. Cano-Ott, G. Cortés, T. Davinson, I. Dillmann, A. Estrade, A. Evdokimov, T. Faestermann, F. Farinon, D. Galaviz, A.R. García, H. Geissel, W. Gelletly, R. Gernhäuser, M.B. Gómez-Hornillos, C. Guerrero, M. Heil, C. Hinke, R. Knöbel, I. Kojouharov, J. Kurcewicz, N. Kurz, Y. Litvinov, L. Maier, J. Marganiec, T. Marketin, M. Marta, T. Martínez, G. Martínez-Pinedo, F. Montes, I. Mukha, D.R. Napoli, C. Nociforo, C. Paradela, S. Pietri, Zs. Podolyák, A. Prochazka, S. Rice, A. Riego, B. Rubio, H. Schaffner, Ch. Scheidenberger, K. Smith, E. Sokol, K. Steiger, B. Sun, J.L. Taín, M. Takechi, D. Testov, H. Weick, E. Wilson, J.S. Winfield, R. Wood, P. Woods, A. Yeremin
Nov. 4, 2015 nucl-ex, nucl-th
The $\beta$-delayed neutron emission probabilities of neutron rich Hg and Tl nuclei have been measured together with $\beta$-decay half-lives for 20 isotopes of Au, Hg, Tl, Pb and Bi in the mass region N$\gtrsim$126. These are the heaviest species where neutron emission has been observed so far. These measurements provide key information to evaluate the performance of nuclear microscopic and phenomenological models in reproducing the high-energy part of the $\beta$-decay strength distribution. In doing so, it provides important constraints to global theoretical models currently used in $r$-process nucleosynthesis.
Nuclear astrophysics with radioactive ions at FAIR (1310.1632)
R. Reifarth, S. Altstadt, K. Göbel, T. Heftrich, M. Heil, A. Koloczek, C. Langer, R. Plag, M. Pohl, K. Sonnabend, M. Weigand, T. Adachi, F. Aksouh, J. Al-Khalili, M. AlGarawi, S. AlGhamdi, G. Alkhazov, N. Alkhomashi, H. Alvarez-Pol, R. Alvarez-Rodriguez, V. Andreev, B. Andrei, L. Atar, T. Aumann, V. Avdeichikov, C. Bacri, S. Bagchi, C. Barbieri, S. Beceiro, C. Beck, C. Beinrucker, G. Belier, D. Bemmerer, M. Bendel, J. Benlliure, G. Benzoni, R. Berjillos, D. Bertini, C. Bertulani, S. Bishop, N. Blasi, T. Bloch, Y. Blumenfeld, A. Bonaccorso, K. Boretzky, A. Botvina, A. Boudard, P. Boutachkov, I. Boztosun, A. Bracco, S. Brambilla, J. Briz Monago, M. Caamano, C. Caesar, F. Camera, E. Casarejos, W. Catford, J. Cederkall, B. Cederwall, M. Chartier, A. Chatillon, M. Cherciu, L. Chulkov, P. Coleman-Smith, D. Cortina-Gil, F. Crespi, R. Crespo, J. Cresswell, M. Csatlós, F. Déchery, B. Davids, T. Davinson, V. Derya, P. Detistov, P. Diaz Fernandez, D. DiJulio, S. Dmitry, D. Doré, J. Dueṅas, E. Dupont, P. Egelhof, I. Egorova, Z. Elekes, J. Enders, J. Endres, S. Ershov, O. Ershova, B. Fernandez-Dominguez, A. Fetisov, E. Fiori, A. Fomichev, M. Fonseca, L. Fraile, M. Freer, J. Friese, M. G. Borge, D. Galaviz Redondo, S. Gannon, U. Garg, I. Gasparic, L. Gasques, B. Gastineau, H. Geissel, R. Gernhäuser, T. Ghosh, M. Gilbert, J. Glorius, P. Golubev, A. Gorshkov, A. Gourishetty, L. Grigorenko, J. Gulyas, M. Haiduc, F. Hammache, M. Harakeh, M. Hass, M. Heine, A. Hennig, A. Henriques, R. Herzberg, M. Holl, A. Ignatov, A. Ignatyuk, S. Ilieva, M. Ivanov, N. Iwasa, B. Jakobsson, H. Johansson, B. Jonson, P. Joshi, A. Junghans, B. Jurado, G. Körner, N. Kalantar, R. Kanungo, A. Kelic-Heil, K. Kezzar, E. Khan, A. Khanzadeev, O. Kiselev, M. Kogimtzis, D. Körper, S. Kräckmann, T. Kröll, R. Krücken, A. Krasznahorkay, J. Kratz, D. Kresan, T. Krings, A. Krumbholz, S. Krupko, R. Kulessa, S. Kumar, N. Kurz, E. Kuzmin, M. Labiche, K. Langanke, I. Lazarus, T. Le Bleis, C. Lederer, A. Lemasson, R. Lemmon, V. Liberati, Y. Litvinov, B. Löher, J. Lopez Herraiz, G. Münzenberg, J. Machado, E. Maev, K. Mahata, D. Mancusi, J. Marganiec, M. Martinez Perez, V. Marusov, D. Mengoni, B. Million, V. Morcelle, O. Moreno, A. Movsesyan, E. Nacher, M. Najafi, T. Nakamura, F. Naqvi, E. Nikolski, T. Nilsson, C. Nociforo, P. Nolan, B. Novatsky, G. Nyman, A. Ornelas, R. Palit, S. Pandit, V. Panin, C. Paradela, V. Parkar, S. Paschalis, P. Pawłowski, A. Perea, J. Pereira, C. Petrache, M. Petri, S. Pickstone, N. Pietralla, S. Pietri, Y. Pivovarov, P. Potlog, A. Prokofiev, G. Rastrepina, T. Rauscher, G. Ribeiro, M. Ricciardi, A. Richter, C. Rigollet, K. Riisager, A. Rios, C. Ritter, T. Rodríguez Frutos, J. Rodriguez Vignote, M. Röder, C. Romig, D. Rossi, P. Roussel-Chomaz, P. Rout, S. Roy, P. Söderström, M. Saha Sarkar, S. Sakuta, M. Salsac, J. Sampson, J. Sanchez del Rio Saez, J. Sanchez Rosado, S. Sanjari, P. Sarriguren, A. Sauerwein, D. Savran, C. Scheidenberger, H. Scheit, S. Schmidt, C. Schmitt, L. Schnorrenberger, P. Schrock, R. Schwengner, D. Seddon, B. Sherrill, A. Shrivastava, S. Sidorchuk, J. Silva, H. Simon, E. Simpson, P. Singh, D. Slobodan, D. Sohler, M. Spieker, D. Stach, E. Stan, M. Stanoiu, S. Stepantsov, P. Stevenson, F. Strieder, L. Stuhl, T. Suda, K. Sümmerer, B. Streicher, J. Taieb, M. Takechi, I. Tanihata, J. Taylor, O. Tengblad, G. Ter-Akopian, S. Terashima, P. Teubig, R. Thies, M. Thoennessen, T. Thomas, J. Thornhill, G. Thungstrom, J. Timar, Y. Togano, U. Tomohiro, T. Tornyi, J. Tostevin, C. Townsley, W. Trautmann, T. Trivedi, S. Typel, E. Uberseder, J. Udias, T. Uesaka, L. Uvarov, Z. Vajta, P. Velho, V. Vikhrov, M. Volknandt, V. Volkov, P. von Neumann-Cosel, M. von Schmid, A. Wagner, F. Wamers, H. Weick, D. Wells, L. Westerberg, O. Wieland, M. Wiescher, C. Wimmer, K. Wimmer, J. S. Winfield, M. Winkel, P. Woods, R. Wyss, D. Yakorev, M. Yavor, J. Zamora Cardona, I. Zartova, T. Zerguerras, I. Zgura, A. Zhdanov, M. Zhukov, M. Zieblinski, A. Zilges, K. Zuber
Oct. 6, 2013 nucl-ex, astro-ph.IM
The nucleosynthesis of elements beyond iron is dominated by neutron captures in the s and r processes. However, 32 stable, proton-rich isotopes cannot be formed during those processes, because they are shielded from the s-process flow and r-process beta-decay chains. These nuclei are attributed to the p and rp process. For all those processes, current research in nuclear astrophysics addresses the need for more precise reaction data involving radioactive isotopes. Depending on the particular reaction, direct or inverse kinematics, forward or time-reversed direction are investigated to determine or at least to constrain the desired reaction cross sections. The Facility for Antiproton and Ion Research (FAIR) will offer unique, unprecedented opportunities to investigate many of the important reactions. The high yield of radioactive isotopes, even far away from the valley of stability, allows the investigation of isotopes involved in processes as exotic as the r or rp processes.
Approaching the precursor nuclei of the third r-process peak with RIBs (1309.3047)
C. Domingo-Pardo, R. Caballero-Folch, J. Agramunt, A. Algora, A. Arcones, F. Ameil, Y. Ayyad, J. Benlliure, M. Bowry, F. Calviño, D. Cano-Ott, G. Cortés, T. Davinson, I. Dillmann, A. Estrade, A. Evdokimov, T. Faestermann, F. Farinon, D. Galaviz, A. García-Rios, H. Geissel, W. Gelletly, R. Gernhäuser, M.B. Gómez-Hornillos, C. Guerrero, M. Heil, C. Hinke, R. Knöbel, I. Kojouharov, J. Kurcewicz, N. Kurz, Y. Litvinov, L. Maier, J. Marganiec, M. Marta, T. Martínez, G. Martínez-Pinedo, B.S. Meyer, F. Montes, I. Mukha, D.R. Napoli, Ch. Nociforo, C. Paradela, S. Pietri, Z. Podolyák, A. Prochazka, S. Rice, A. Riego, B. Rubio, H. Schaffner, Ch. Scheidenberger, K. Smith, E. Sokol, K. Steiger, B. Sun, J.L. Taín, M. Takechi, D. Testov, H. Weick, E. Wilson, J.S. Winfield, R. Wood, P. Woods, A. Yeremin
Sept. 13, 2013 nucl-ex
The rapid neutron nucleosynthesis process involves an enormous amount of very exotic neutron-rich nuclei, which represent a theoretical and experimental challenge. Two of the main decay properties that affect the final abundance distribution the most are half-lives and neutron branching ratios. Using fragmentation of a primary $^{238}$U beam at GSI we were able to measure such properties for several neutron-rich nuclei from $^{208}$Hg to $^{218}$Pb. This contribution provides a short update on the status of the data analysis of this experiment, together with a compilation of the latest results published in this mass region, both experimental and theoretical. The impact of the uncertainties connected with the beta-decay rates and with beta-delayed neutron emission is illustrated on the basis of $r$-process network calculations. In order to obtain a reasonable reproduction of the third $r$-process peak, it is expected that both half-lives and neutron branching ratios are substantially smaller, than those based on FRDM+QRPA, commonly used in $r$-process model calculations. Further measurements around $N\sim126$ are required for a reliable modelling of the underlying nuclear structure, and for performing more realistic $r$-process abundance calculations.
Deuterium chemistry in protoplanetary disks II The inner 30 AU (0908.1114)
K. Willacy, P. Woods
Aug. 7, 2009 astro-ph.SR
We present the results of models of the chemistry, including deuterium, in the inner regions of protostellar disks. We find good agreement with recent gas phase observations of several (non--deuterated) species. We also compare our results with observations of comets and find that in the absence of other processing e.g. in the accretion shock at the surface of the disk, or by mixing in the disk, the calculated D/H ratios in ices are higher than measured and reflect the D/H ratio set in the molecular cloud phase. Our models give quite different abundances and molecular distributions to other inner disk models because of the differences in physical conditions in the model disk. This emphasizes how changes in the assumptions about the density and temperature distribution can radically affect the results of chemical models.
A Possible Magnetar Nature for IGR J16358-4726 (astro-ph/0610768)
S. K. Patel, J. Zurita, M. Del Santo, M. Finger, C. Kouveliotou, D. Eichler, E. Gogus, P. Ubertini, R. Walter, P. Woods, C. A. Wilson, S. Wachter, A. Bazzano
Oct. 26, 2006 astro-ph
We present detailed spectral and timing analysis of the hard x-ray transient IGR J16358-4726 using multi-satellite archival observations. A study of the source flux time history over 6 years, suggests that lower luminosity transient outbursts can be occuring in intervals of at most 1 year. Joint spectral fits of the higher luminosity outburst using simultaneous Chandra/ACIS and INTEGRAL/ISGRI data reveal a spectrum well described by an absorbed power law model with a high energy cut-off plus an Fe line. We detected the 1.6 hour pulsations initially reported using Chandra/ACIS also in the INTEGRAL/ISGRI light curve and in subsequent XMM-Newton observations. Using the INTEGRAL data we identified a spin up of 94 s (dP/dt = 1.6E-4), which strongly points to a neutron star nature for IGR J16358-4726. Assuming that the spin up is due to disc accretion, we estimate that the source magnetic field ranges between 10^13 - 10^15 G, depending on its distance, possibly supporting a magnetar nature for IGR J16358-4726.
Long term spectral variability in the Soft Gamma-ray Repeater SGR 1900+14 (astro-ph/0609319)
A. Tiengo, P. Esposito, S. Mereghetti, L. Sidoli, D. Goetz, M. Feroci, R. Turolla, S. Zane, G.L. Israel, L. Stella, P. Woods
Sept. 12, 2006 astro-ph
We present a systematic analysis of all the BeppoSAX data of SGR1900+14. The observations spanning five years show that the source was brighter than usual on two occasions: ~20 days after the August 1998 giant flare and during the 10^5 s long X-ray afterglow following the April 2001 intermediate flare. In the latter case, we explore the possibility of describing the observed short term spectral evolution only with a change of the temperature of the blackbody component. In the only BeppoSAX observation performed before the giant flare, the spectrum of the SGR1900+14 persistent emission was significantly harder and detected also above 10 keV with the PDS instrument. In the last BeppoSAX observation (April 2002) the flux was at least a factor 1.2 below the historical level, suggesting that the source was entering a quiescent period.
Five years of SGR 1900+14 observations with BeppoSAX (astro-ph/0609078)
P. Esposito, S. Mereghetti, A. Tiengo, L. Sidoli, M. Feroci, P. Woods
Sept. 4, 2006 astro-ph
We present a systematic analysis of all the BeppoSAX data of the soft gamma-ray repeater SGR 1900+14: these observations allowed us to study the long term properties of the source quiescent emission. In the observation carried out before the 1998 giant flare the spectrum in the 0.8-10 keV energy range was harder and there was evidence for a 20-150 keV emission, possibly associated with SGR 1900+14. This possible hard tail, if compared with the recent INTEGRAL detection of SGR 1900+14, has a harder spectrum (power-law photon index ~1.6 versus ~3) and a 20-100 keV flux ~4 times larger. In the last BeppoSAX observation (April 2002), while the source was entering the long quiescent period that lasted until 2006, the 2-10 keV flux was ~25% below the historical level. We also studied in detail the spectral evolution during the 2001 flare afterglow. This was characterized by a softening that can be interpreted in terms of a cooling blackbody-like component.
Precise Localization of the Soft Gamma Repeater SGR 1627-41 and the Anomalous X-ray Pulsar AXP 1E1841-045 with Chandra (astro-ph/0408266)
S. Wachter, S. Patel, C. Kouveliotou, P. Bouchet, F. Ozel, A. Tennant, P. Woods, K. Hurley, W. Becker, P. Slane
Aug. 13, 2004 astro-ph
We present precise localizations of AXP 1E1841-045 and SGR 1627-41 with Chandra. We obtained new infrared observations of SGR 1627-41 and reanalyzed archival observations of AXP 1E1841-045 in order to refine their positions and search for infrared counterparts. A faint source is detected inside the error circle of AXP 1E1841-045. In the case of SGR 1627-41, several sources are located within the error radius of the X-ray position and we discuss the likelihood of one of them being the counterpart. We compare the properties of our candidates to those of other known AXP and SGR counterparts. We find that the counterpart candidates for SGR 1627-41 and SGR 1806-20 would have to be intrinsically much brighter than AXPs to have detectable counterparts with the observational limits currently available for these sources. To confirm the reported counterpart of SGR 1806-20, we obtained new IR observations during the July 2003 burst activation of the source. No brightening of the suggested counterpart is detected, implying that the counterpart of SGR 1806-20 remains yet to be identified.
Discovery of a New Transient Magnetar Candidate: XTE J1810-197 (astro-ph/0310665)
A. I. Ibrahim, C. Markwardt, J. Swank, S. Ransom, M. Roberts, V. Kaspi, P. Woods, S. Safi-Harb, S. Balman, W. Parke, C. Kouveliotou, K. Hurley, T. Cline
We report the discovery of a new X-ray pulsar, XTE J1810-197. The source was serendipitously discovered on 2003 July 15 by the Rossi X-ray Timing Explorer (RXTE) while observing the soft gamma repeater SGR 1806-20. The pulsar has a 5.54 s spin-period and a soft spectrum (photon index ~ 4). We detect the source in earlier RXTE observations back to 2003 January. These show that a transient outburst began between 2002 November 17 and 2003 January 23 and that the pulsar has been spinning down since then, with a high rate Pdot ~ 10^-11 s/s showing significant timing noise, but no evidence for Doppler shifts due to a binary companion. The rapid spin-down rate and slow spin-period imply a super-critical magnetic field B=3x10^14 G and a young characteristic age < 7600 yr. These properties are strikingly similar to those of anomalous X-ray pulsars and soft gamma repeaters, making the source a likely new magnetar. A follow-up Chandra observation provided a 2".5 radius error circle within which the 1.5 m Russian-Turkish Optical Telescope RTT150 found a limiting magnitude of R_c=21.5, in accord with other recently reported limits. The source is present in archival ASCA and ROSAT data as well, at a level 100 times fainter than the \~ 3 mCrab seen in 2003. This suggests that other X-ray sources that are currently in a state similar to the inactive phase of XTE J1810-197 may also be unidentified magnetars awaiting detection via a similar activity.
The 2001 April Burst Activation of SGR 1900+14: X-ray afterglow emission (astro-ph/0306091)
M. Feroci, S. Mereghetti, P. Woods, C. Kouveliotou, E. Costa, D.D. Frederiks, S.V. Golenetskii, K. Hurley, E. Mazets, P. Soffitta, M. Tavani
June 4, 2003 astro-ph
After nearly two years of quiescence, the soft gamma-ray repeater SGR 1900+14 again became burst-active on April 18 2001, when it emitted a large flare, preceded by few weak and soft short bursts. After having detected the X and gamma prompt emission of the flare, BeppoSAX pointed its narrow field X-ray telescopes to the source in less than 8 hours. In this paper we present an analysis of the data from this and from a subsequent BeppoSAX observation, as well as from a set of RossiXTE observations. Our data show the detection of an X-ray afterglow from the source, most likely related to the large hard X-ray flare. In fact, the persistent flux from the source, in 2-10 keV, was initially found at a level $\sim$5 times higher than the usual value. Assuming an underlying persistent (constant) emission, the decay of the excess flux can be reasonably well described by a t$^{-0.9}$ law. A temporal feature - a $\sim$half a day long bump - is observed in the decay light curve approximately one day after the burst onset. This feature is unprecedented in SGR afterglows. We discuss our results in the context of previous observations of this source and derive implications for the physics of these objects.
The BeppoSAX View on the 2001 Reactivation of SGR 1900+14 (astro-ph/0112239)
M. Feroci, S. Mereghetti, E. Costa, J.J.M. in 't Zand, P. Soffitta, T. Cline, R. Duncan, M. Finger, S.V. Golenetskii, K. Hurley, C. Kouveliotou, P. Li, E. Mazets, M. Tavani, C. Thompson, P. Woods
Dec. 11, 2001 astro-ph
After a couple of years of quiescence, the soft gamma repeater SGR 1900+14 suddenly reactivated on 18 April 2001, with the emission of a very intense, long and modulated flare, only second in intensity and duration to the 27 August 1998 giant flare. BeppoSAX caught the large flare with its Gamma Ray Burst Monitor and with one of the Wide Field Cameras. The Wide Field Cameras also detected X-ray bursting activity shortly before the giant flare. A target of opportunity observation was started only 8 hours after the large flare with the Narrow Field Instruments, composed of two 60-ks long pointings. These two observations show an X-ray afterglow of the persistent SGR 1900+14 source, decaying with time according to a power law of index -0.6.
The Timing Evolution of 4U 1630-47 During its 1998 Outburst (astro-ph/9912028)
S. W. Dieters, T. Belloni, E. Kuulkers, P. Woods, W. Cui, S. N. Zhang, M. van der Klis, J. van Paradijs, J. Swank, W.H.G. Lewin, C. Kouveliotou
Dec. 1, 1999 astro-ph
We report on the evolution of the timing of 4U1630-47 during its 1998 outburst using data obtained with the Rossi X-ray Timing Explorer (RXTE). The count rate and position in hardness-intensity, color-color diagrams and simple spectral fits are used to track the concurrent spectral changes. The source showed seven distinct types of timing behavior, most of which show differences with the canonical black hole spectral/timing states. In marked contrast to previous outbursts, we find quasi-periodic oscillation (QPO) signals during nearly all stages of the outburst with frequencies between 0.06Hz and 14Hz and a remarkable variety of other characteristics. In particular we find large (up to 23% rms) amplitude QPO on the early rise. Later, slow 0.1Hz semi- regular short (~ 5 sec), 9 to 16% deep dips dominate the light curve. At this time there are two QPOs, one stable near 13.5Hz and the other whose frequency drops from 6--8Hz to ~4.5 Hz during the dips. BeppoSAX observations during the very late declining phase show 4U1630-47 in a low state.
ASCA Observation of the quiescent X-ray counterpart to SGR1627-41 (astro-ph/9909355)
K. Hurley, T. Strohmayer, P. Li, C. Kouveliotou, P. Woods, J. van Paradijs, T. Murakami, D. Hartmann, I. Smith, M. Ando, A. Yoshida, M. Sugizaki
We present a 2 - 10 keV ASCA observation of the field around the soft gamma repeater SGR1627-41. A quiescent X-ray source was detected in this observation whose position was consistent both with that of a recently discovered BeppoSAX X-ray source and with the Interplanetary Network localization for this SGR. In 2 - 10 keV X-rays, the spectrum of the X-ray source may be fit equally well by a power law, blackbody, or bremsstrahlung function, with unabsorbed flux ~5 x 10^-12 erg cm^-2 s^-1. We do not confirm a continuation of a fading trend in the flux, and we find no evidence for periodicity, both noted in the earlier BeppoSAXobservations.
Precise Interplanetary Network Localization of a New Soft Gamma Repeater, SGR1627-41 (astro-ph/9903268)
K. Hurley, C. Kouveliotou, P. Woods, E. Mazets, S. Golenetskii, D. D. Fredericks, T. Cline, J. van Paradijs
March 17, 1999 astro-ph
We present Ulysses, KONUS-WIND, and BATSE observations of bursts from a new soft gamma repeater which was active in 1998 June and July. Triangulation of the bursts results in a ~ 1.8 degree by 16 '' error box whose area is ~ 7.6 arcminutes^2, which contains the Galactic supernova remnant G337.0-0.1. This error box intersects the position of a BeppoSAX X-ray source which is also consistent with the position of G337.0-0.1 (Woods et al. 1999), and is thought to be the quiescent counterpart to the repeater. If so, the resulting error box is ~ 2 ' by 16 '' and has an area of ~ 0.6 arcminutes^2. The error box location within the supernova remnant suggests that the neutron star has a transverse velocity of ~ 200 - 2000 km/s.
Reactivation and Precise IPN Localization of the Soft Gamma Repeater SGR1900+14 (astro-ph/9811411)
K. Hurley, C. Kouveliotou, P. Woods, T. Cline, P. Butterworth, E. Mazets, S. Golenetskii, D. Frederics
Nov. 25, 1998 astro-ph
In 1998 May, the soft gamma repeater SGR1900+14 emerged from several years of quiescence and emitted a series of intense bursts, one with a time history unlike any previously observed from this source. Triangulation using Ulysses, BATSE, and KONUS data give a 1.6 square arcminute error box near the galactic supernova remnant G42.8+0.6. This error box contains a quiescent soft X-ray source which is probably a neutron star associated with the soft repeater. | CommonCrawl |
A horse 24 feet from the center of a merry-go-round makes 32 revolutions. In order to travel the same distance, how many revolutions would a horse 8 feet from the center have to make?
The radius of the circular path of the horse closer to the center is $\frac{1}{3}$ of the radius of the path of the horse farther from the center. Since circumference is directly proportional to radius, the length of shorter path is $\frac{1}{3}$ of the length of the longer path. Therefore, 3 times as many revolutions must be made to go the same distance, which is $32\times3=\boxed{96}$ revolutions. | Math Dataset |
Repunit
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for "repeated unit" and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers.[note 1]
Repunit prime
No. of known terms11
Conjectured no. of termsInfinite
First terms11, 1111111111111111111, 11111111111111111111111
Largest known term(1086453−1)/9
OEIS index
• A004022
• Primes of the form (10^n − 1)/9
A repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes. As of May 2023, the largest known prime number 282,589,933 − 1, the largest probable prime R8177207 and the largest elliptic curve primality-proven prime R86453 are all repunits in various bases.
Definition
The base-b repunits are defined as (this b can be either positive or negative)
$R_{n}^{(b)}\equiv 1+b+b^{2}+\cdots +b^{n-1}={b^{n}-1 \over {b-1}}\qquad {\mbox{for }}|b|\geq 2,n\geq 1.$
Thus, the number Rn(b) consists of n copies of the digit 1 in base-b representation. The first two repunits base-b for n = 1 and n = 2 are
$R_{1}^{(b)}={b-1 \over {b-1}}=1\qquad {\text{and}}\qquad R_{2}^{(b)}={b^{2}-1 \over {b-1}}=b+1\qquad {\text{for}}\ |b|\geq 2.$
In particular, the decimal (base-10) repunits that are often referred to as simply repunits are defined as
$R_{n}\equiv R_{n}^{(10)}={10^{n}-1 \over {10-1}}={10^{n}-1 \over 9}\qquad {\mbox{for }}n\geq 1.$
Thus, the number Rn = Rn(10) consists of n copies of the digit 1 in base 10 representation. The sequence of repunits base-10 starts with
1, 11, 111, 1111, 11111, 111111, ... (sequence A002275 in the OEIS).
Similarly, the repunits base-2 are defined as
$R_{n}^{(2)}={2^{n}-1 \over {2-1}}={2^{n}-1}\qquad {\mbox{for }}n\geq 1.$
Thus, the number Rn(2) consists of n copies of the digit 1 in base-2 representation. In fact, the base-2 repunits are the well-known Mersenne numbers Mn = 2n − 1, they start with
1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, ... (sequence A000225 in the OEIS).
Properties
• Any repunit in any base having a composite number of digits is necessarily composite. Only repunits (in any base) having a prime number of digits might be prime. This is a necessary but not sufficient condition. For example,
R35(b) = 11111111111111111111111111111111111 = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001,
since 35 = 7 × 5 = 5 × 7. This repunit factorization does not depend on the base-b in which the repunit is expressed.
• If p is an odd prime, then every prime q that divides Rp(b) must be either 1 plus a multiple of 2p, or a factor of b − 1. For example, a prime factor of R29 is 62003 = 1 + 2·29·1069. The reason is that the prime p is the smallest exponent greater than 1 such that q divides bp − 1, because p is prime. Therefore, unless q divides b − 1, p divides the Carmichael function of q, which is even and equal to q − 1.
• Any positive multiple of the repunit Rn(b) contains at least n nonzero digits in base-b.
• Any number x is a two-digit repunit in base x − 1.
• The only known numbers that are repunits with at least 3 digits in more than one base simultaneously are 31 (111 in base-5, 11111 in base-2) and 8191 (111 in base-90, 1111111111111 in base-2). The Goormaghtigh conjecture says there are only these two cases.
• Using the pigeon-hole principle it can be easily shown that for relatively prime natural numbers n and b, there exists a repunit in base-b that is a multiple of n. To see this consider repunits R1(b),...,Rn(b). Because there are n repunits but only n−1 non-zero residues modulo n there exist two repunits Ri(b) and Rj(b) with 1 ≤ i < j ≤ n such that Ri(b) and Rj(b) have the same residue modulo n. It follows that Rj(b) − Ri(b) has residue 0 modulo n, i.e. is divisible by n. Since Rj(b) − Ri(b) consists of j − i ones followed by i zeroes, Rj(b) − Ri(b) = Rj−i(b) × bi. Now n divides the left-hand side of this equation, so it also divides the right-hand side, but since n and b are relatively prime, n must divide Rj−i(b).
• The Feit–Thompson conjecture is that Rq(p) never divides Rp(q) for two distinct primes p and q.
• Using the Euclidean Algorithm for repunits definition: R1(b) = 1; Rn(b) = Rn−1(b) × b + 1, any consecutive repunits Rn−1(b) and Rn(b) are relatively prime in any base-b for any n.
• If m and n have a common divisor d, Rm(b) and Rn(b) have the common divisor Rd(b) in any base-b for any m and n. That is, the repunits of a fixed base form a strong divisibility sequence. As a consequence, If m and n are relatively prime, Rm(b) and Rn(b) are relatively prime. The Euclidean Algorithm is based on gcd(m, n) = gcd(m − n, n) for m > n. Similarly, using Rm(b) − Rn(b) × bm−n = Rm−n(b), it can be easily shown that gcd(Rm(b), Rn(b)) = gcd(Rm−n(b), Rn(b)) for m > n. Therefore, if gcd(m, n) = d, then gcd(Rm(b), Rn(b)) = Rd(b).
Factorization of decimal repunits
(Prime factors colored red means "new factors", i. e. the prime factor divides Rn but does not divide Rk for all k < n) (sequence A102380 in the OEIS)[2]
R1 =1
R2 =11
R3 =3 · 37
R4 =11 · 101
R5 =41 · 271
R6 =3 · 7 · 11 · 13 · 37
R7 =239 · 4649
R8 =11 · 73 · 101 · 137
R9 =32 · 37 · 333667
R10 =11 · 41 · 271 · 9091
R11 =21649 · 513239
R12 =3 · 7 · 11 · 13 · 37 · 101 · 9901
R13 =53 · 79 · 265371653
R14 =11 · 239 · 4649 · 909091
R15 =3 · 31 · 37 · 41 · 271 · 2906161
R16 =11 · 17 · 73 · 101 · 137 · 5882353
R17 =2071723 · 5363222357
R18 =32 · 7 · 11 · 13 · 19 · 37 · 52579 · 333667
R19 =1111111111111111111
R20 =11 · 41 · 101 · 271 · 3541 · 9091 · 27961
R21 =3 · 37 · 43 · 239 · 1933 · 4649 · 10838689
R22 =112 · 23 · 4093 · 8779 · 21649 · 513239
R23 =11111111111111111111111
R24 =3 · 7 · 11 · 13 · 37 · 73 · 101 · 137 · 9901 · 99990001
R25 =41 · 271 · 21401 · 25601 · 182521213001
R26 =11 · 53 · 79 · 859 · 265371653 · 1058313049
R27 =33 · 37 · 757 · 333667 · 440334654777631
R28 =11 · 29 · 101 · 239 · 281 · 4649 · 909091 · 121499449
R29 =3191 · 16763 · 43037 · 62003 · 77843839397
R30 =3 · 7 · 11 · 13 · 31 · 37 · 41 · 211 · 241 · 271 · 2161 · 9091 · 2906161
Smallest prime factor of Rn for n > 1 are
11, 3, 11, 41, 3, 239, 11, 3, 11, 21649, 3, 53, 11, 3, 11, 2071723, 3, 1111111111111111111, 11, 3, 11, 11111111111111111111111, 3, 41, 11, 3, 11, 3191, 3, 2791, 11, 3, 11, 41, 3, 2028119, 11, 3, 11, 83, 3, 173, 11, 3, 11, 35121409, 3, 239, 11, ... (sequence A067063 in the OEIS)
Repunit primes
The definition of repunits was motivated by recreational mathematicians looking for prime factors of such numbers.
It is easy to show that if n is divisible by a, then Rn(b) is divisible by Ra(b):
$R_{n}^{(b)}={\frac {1}{b-1}}\prod _{d|n}\Phi _{d}(b),$
where $\Phi _{d}(x)$ is the $d^{\mathrm {th} }$ cyclotomic polynomial and d ranges over the divisors of n. For p prime,
$\Phi _{p}(x)=\sum _{i=0}^{p-1}x^{i},$
which has the expected form of a repunit when x is substituted with b.
For example, 9 is divisible by 3, and thus R9 is divisible by R3—in fact, 111111111 = 111 · 1001001. The corresponding cyclotomic polynomials $\Phi _{3}(x)$ and $\Phi _{9}(x)$ are $x^{2}+x+1$ and $x^{6}+x^{3}+1$, respectively. Thus, for Rn to be prime, n must necessarily be prime, but it is not sufficient for n to be prime. For example, R3 = 111 = 3 · 37 is not prime. Except for this case of R3, p can only divide Rn for prime n if p = 2kn + 1 for some k.
Decimal repunit primes
Rn is prime for n = 2, 19, 23, 317, 1031, 49081, 86453 ... (sequence A004023 in OEIS). On April 3, 2007 Harvey Dubner (who also found R49081) announced that R109297 is a probable prime.[3] On July 15, 2007, Maksym Voznyy announced R270343 to be probably prime.[4] Serge Batalov and Ryan Propper found R5794777 and R8177207 to be probable primes on April 20 and May 8, 2021, respectively.[5] As of their discovery each was the largest known probable prime. On March 22, 2022 probable prime R49081 was eventually proven to be a prime.[6] On May 15, 2023 probable prime R86453 was eventually proven to be a prime.[7]
It has been conjectured that there are infinitely many repunit primes[8] and they seem to occur roughly as often as the prime number theorem would predict: the exponent of the Nth repunit prime is generally around a fixed multiple of the exponent of the (N−1)th.
The prime repunits are a trivial subset of the permutable primes, i.e., primes that remain prime after any permutation of their digits.
Particular properties are
• The remainder of Rn modulo 3 is equal to the remainder of n modulo 3. Using 10a ≡ 1 (mod 3) for any a ≥ 0,
n ≡ 0 (mod 3) ⇔ Rn ≡ 0 (mod 3) ⇔ Rn ≡ 0 (mod R3),
n ≡ 1 (mod 3) ⇔ Rn ≡ 1 (mod 3) ⇔ Rn ≡ R1 ≡ 1 (mod R3),
n ≡ 2 (mod 3) ⇔ Rn ≡ 2 (mod 3) ⇔ Rn ≡ R2 ≡ 11 (mod R3).
Therefore, 3 | n ⇔ 3 | Rn ⇔ R3 | Rn.
• The remainder of Rn modulo 9 is equal to the remainder of n modulo 9. Using 10a ≡ 1 (mod 9) for any a ≥ 0,
n ≡ r (mod 9) ⇔ Rn ≡ r (mod 9) ⇔ Rn ≡ Rr (mod R9),
for 0 ≤ r < 9.
Therefore, 9 | n ⇔ 9 | Rn ⇔ R9 | Rn.
Algebra factorization of generalized repunit numbers
If b is a perfect power (can be written as mn, with m, n integers, n > 1) differs from 1, then there is at most one repunit in base-b. If n is a prime power (can be written as pr, with p prime, r integer, p, r >0), then all repunit in base-b are not prime aside from Rp and R2. Rp can be either prime or composite, the former examples, b = −216, −128, 4, 8, 16, 27, 36, 100, 128, 256, etc., the latter examples, b = −243, −125, −64, −32, −27, −8, 9, 25, 32, 49, 81, 121, 125, 144, 169, 196, 216, 225, 243, 289, etc., and R2 can be prime (when p differs from 2) only if b is negative, a power of −2, for example, b = −8, −32, −128, −8192, etc., in fact, the R2 can also be composite, for example, b = −512, −2048, −32768, etc. If n is not a prime power, then no base-b repunit prime exists, for example, b = 64, 729 (with n = 6), b = 1024 (with n = 10), and b = −1 or 0 (with n any natural number). Another special situation is b = −4k4, with k positive integer, which has the aurifeuillean factorization, for example, b = −4 (with k = 1, then R2 and R3 are primes), and b = −64, −324, −1024, −2500, −5184, ... (with k = 2, 3, 4, 5, 6, ...), then no base-b repunit prime exists. It is also conjectured that when b is neither a perfect power nor −4k4 with k positive integer, then there are infinity many base-b repunit primes.
The generalized repunit conjecture
A conjecture related to the generalized repunit primes:[9][10] (the conjecture predicts where is the next generalized Mersenne prime, if the conjecture is true, then there are infinitely many repunit primes for all bases $b$)
For any integer $b$, which satisfies the conditions:
1. $|b|>1$.
2. $b$ is not a perfect power. (since when $b$ is a perfect $r$th power, it can be shown that there is at most one $n$ value such that ${\frac {b^{n}-1}{b-1}}$ is prime, and this $n$ value is $r$ itself or a root of $r$)
3. $b$ is not in the form $-4k^{4}$. (if so, then the number has aurifeuillean factorization)
has generalized repunit primes of the form
$R_{p}(b)={\frac {b^{p}-1}{b-1}}$
for prime $p$, the prime numbers will be distributed near the best fit line
$Y=G\cdot \log _{|b|}\left(\log _{|b|}\left(R_{(b)}(n)\right)\right)+C,$
where limit $n\rightarrow \infty $, $G={\frac {1}{e^{\gamma }}}=0.561459483566...$
and there are about
$\left(\log _{e}(N)+m\cdot \log _{e}(2)\cdot \log _{e}{\big (}\log _{e}(N){\big )}+{\frac {1}{\sqrt {N}}}-\delta \right)\cdot {\frac {e^{\gamma }}{\log _{e}(|b|)}}$
base-b repunit primes less than N.
• $e$ is the base of natural logarithm.
• $\gamma $ is Euler–Mascheroni constant.
• $\log _{|b|}$ is the logarithm in base $|b|$
• $R_{(b)}(n)$ is the $n$th generalized repunit prime in baseb (with prime p)
• $C$ is a data fit constant which varies with $b$.
• $\delta =1$ if $b>0$, $\delta =1.6$ if $b<0$.
• $m$ is the largest natural number such that $-b$ is a $2^{m-1}$th power.
We also have the following 3 properties:
1. The number of prime numbers of the form ${\frac {b^{n}-1}{b-1}}$ (with prime $p$) less than or equal to $n$ is about $e^{\gamma }\cdot \log _{|b|}{\big (}\log _{|b|}(n){\big )}$.
2. The expected number of prime numbers of the form ${\frac {b^{n}-1}{b-1}}$ with prime $p$ between $n$ and $|b|\cdot n$ is about $e^{\gamma }$.
3. The probability that number of the form ${\frac {b^{n}-1}{b-1}}$ is prime (for prime $p$) is about ${\frac {e^{\gamma }}{p\cdot \log _{e}(|b|)}}$.
History
Although they were not then known by that name, repunits in base-10 were studied by many mathematicians during the nineteenth century in an effort to work out and predict the cyclic patterns of repeating decimals.[11]
It was found very early on that for any prime p greater than 5, the period of the decimal expansion of 1/p is equal to the length of the smallest repunit number that is divisible by p. Tables of the period of reciprocal of primes up to 60,000 had been published by 1860 and permitted the factorization by such mathematicians as Reuschle of all repunits up to R16 and many larger ones. By 1880, even R17 to R36 had been factored[11] and it is curious that, though Édouard Lucas showed no prime below three million had period nineteen, there was no attempt to test any repunit for primality until early in the twentieth century. The American mathematician Oscar Hoppe proved R19 to be prime in 1916[12] and Lehmer and Kraitchik independently found R23 to be prime in 1929.
Further advances in the study of repunits did not occur until the 1960s, when computers allowed many new factors of repunits to be found and the gaps in earlier tables of prime periods corrected. R317 was found to be a probable prime circa 1966 and was proved prime eleven years later, when R1031 was shown to be the only further possible prime repunit with fewer than ten thousand digits. It was proven prime in 1986, but searches for further prime repunits in the following decade consistently failed. However, there was a major side-development in the field of generalized repunits, which produced a large number of new primes and probable primes.
Since 1999, four further probably prime repunits have been found, but it is unlikely that any of them will be proven prime in the foreseeable future because of their huge size.
The Cunningham project endeavours to document the integer factorizations of (among other numbers) the repunits to base 2, 3, 5, 6, 7, 10, 11, and 12.
Demlo numbers
D. R. Kaprekar has defined Demlo numbers as concatenation of a left, middle and right part, where the left and right part must be of the same length (up to a possible leading zero to the left) and must add up to a repdigit number, and the middle part may contain any additional number of this repeated digit.[13] They are named after Demlo railway station (now called Dombivili) 30 miles from Bombay on the then G.I.P. Railway, where Kaprekar started investigating them. He calls Wonderful Demlo numbers those of the form 1, 121, 12321, 1234321, ..., 12345678987654321. The fact that these are the squares of the repunits has led some authors to call Demlo numbers the infinite sequence of these,[14] 1, 121, 12321, ..., 12345678987654321, 1234567900987654321, 123456790120987654321, ..., (sequence A002477 in the OEIS), although one can check these are not Demlo numbers for p = 10, 19, 28, ...
See also
• All one polynomial — Another generalization
• Goormaghtigh conjecture
• Repeating decimal
• Repdigit
• Wagstaff prime — can be thought of as repunit primes with negative base $b=-2$
Footnotes
Notes
1. Albert H. Beiler coined the term "repunit number" as follows:
A number which consists of a repeated of a single digit is sometimes called a monodigit number, and for convenience the author has used the term "repunit number" (repeated unit) to represent monodigit numbers consisting solely of the digit 1.[1]
References
1. Beiler 2013, pp. 83
2. For more information, see Factorization of repunit numbers.
3. Harvey Dubner, New Repunit R(109297)
4. Maksym Voznyy, New PRP Repunit R(270343)
5. OEIS: A004023
6. "PrimePage Primes: R(49081)". PrimePage Primes. 2022-03-21. Retrieved 2022-03-31.
7. "PrimePage Primes: R(86453)". PrimePage Primes. 2023-05-16. Retrieved 2023-05-16.
8. Chris Caldwell. "repunit". The Prime Glossary. Prime Pages.
9. Deriving the Wagstaff Mersenne Conjecture
10. Generalized Repunit Conjecture
11. Dickson & Cresse 1999, pp. 164–167
12. Francis 1988, pp. 240–246
13. Kaprekar 1938a, 1938b, Gunjikar & Kaprekar 1939
14. Weisstein, Eric W. "Demlo Number". MathWorld.
References
• Beiler, Albert H. (2013) [1964], Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover Recreational Math (2nd Revised ed.), New York: Dover Publications, ISBN 978-0-486-21096-4
• Dickson, Leonard Eugene; Cresse, G.H. (1999), History of the Theory of Numbers, Volume I: Divisibility and primality (2nd Reprinted ed.), Providence, RI: AMS Chelsea Publishing, ISBN 978-0-8218-1934-0
• Francis, Richard L. (1988), "Mathematical Haystacks: Another Look at Repunit Numbers", The College Mathematics Journal, 19 (3): 240–246, doi:10.1080/07468342.1988.11973120
• Gunjikar, K. R.; Kaprekar, D. R. (1939), "Theory of Demlo numbers" (PDF), Journal of the University of Bombay, VIII (3): 3–9
• Kaprekar, D. R. (1938a), "On Wonderful Demlo numbers", The Mathematics Student, 6: 68
• Kaprekar, D. R. (1938b), "Demlo numbers", J. Phys. Sci. Univ. Bombay, VII (3)
• Kaprekar, D. R. (1948), Demlo numbers, Devlali, India: Khareswada
• Ribenboim, Paulo (1996-02-02), The New Book of Prime Number Records, Computers and Medicine (3rd ed.), New York: Springer, ISBN 978-0-387-94457-9
• Yates, Samuel (1982), Repunits and repetends, FL: Delray Beach, ISBN 978-0-9608652-0-8
External links
• Weisstein, Eric W. "Repunit". MathWorld.
• The main tables of the Cunningham project.
• Repunit at The Prime Pages by Chris Caldwell.
• Repunits and their prime factors at World!Of Numbers.
• Prime generalized repunits of at least 1000 decimal digits by Andy Steward
• Repunit Primes Project Giovanni Di Maria's repunit primes page.
• Smallest odd prime p such that (b^p-1)/(b-1) and (b^p+1)/(b+1) is prime for bases 2<=b<=1024
• Factorization of repunit numbers
• Generalized repunit primes in base -50 to 50
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List of prime numbers
| Wikipedia |
\begin{definition}[Definition:Cyclic Group/Notation]
A '''cyclic group''' with $n$ elements is often denoted $C_n$.
Some sources use the notation $\sqbrk g$ or $\gen g$ to denote the '''cyclic group''' generated by $g$.
From Integers Modulo m under Addition form Cyclic Group, $\struct {\Z_m, +_m}$ is a '''cyclic group'''.
Thus $\struct {\Z_m, +_m}$ often taken as the archetypal example of a '''cyclic group''', and the notation $\Z_m$ is used.
This is justified as, from Cyclic Groups of Same Order are Isomorphic, $\Z_m$ is isomorphic to $C_m$.
In certain contexts $\Z_m$ is particularly useful, as it allows results about '''cyclic groups''' to be demonstrated using number theoretical techniques.
\end{definition} | ProofWiki |
\begin{document}
\lstset{language=C,
basicstyle=\ttfamily\scriptsize,
keywordstyle=\bfseries,
commentstyle=\slshape,
frameround = fttt,
emptylines={1},
showlines= true,
showspaces=false,
showstringspaces=false}
\title{Unconstrained Recursive Importance Sampling}
\begin{abstract} We propose an \emph{unconstrained} stochastic approximation method of finding the optimal measure change (in an \emph{a priori} parametric family) for Monte Carlo simulations. We consider different parametric families based on the Girsanov theorem and the Esscher transform (or exponential-tilting). In a multidimensional Gaussian framework, Arouna uses a projected Robbins-Monro procedure to select the parameter minimizing the variance (see \cite{ARO}). In our approach, the parameter (scalar or process) is selected by a classical Robbins-Monro procedure without projection or truncation. To obtain this unconstrained algorithm we intensively use the regularity of the density of the law without assume smoothness of the payoff. We prove the convergence for a large class of multidimensional distributions and diffusion processes.
We illustrate the effectiveness of our algorithm via pricing a Basket payoff under a multidimensional NIG distribution, and pricing a barrier options in different markets. \end{abstract}
\noindent {\em Key words: Stochastic algorithm, Robbins-Monro, Importance sampling, Esscher transform, Girsanov, NIG distribution, Barrier options.}
\ni {\em 2000 Mathematics Subject Classification: 65C05, 65B99, 60H35.}
\noindent \section{Introduction} The basic problem in Numerical Probability is to {\em optimize} some way or another the computation by a Monte Carlo simulation of a real quantity $m$ known by a probabilistic representation \begin{equation*}
m = \esp{F(X)} \end{equation*}
where $X:(\Omega,{\cal A}, {\math P})\to (E,|\,.\,|_E)$ is a random vector having values in a Banach space $E$ and $F:E\to {\math R}$ is a Borel function (and $F(X)$ is square integrable). The space $E$ is ${\math R}^d$ but can also be a functional space of paths of a process $X=(X_t)_{t\in [0,T]}$. However, in this introduction section, we will first focus on the finite dimensional case $E={\math R}^d$.
Assume that $X$ has an absolutely continuous distribution ${\math P}_{_X}(dx)=p(x)\lambda_d(dx)$ ($\lambda_d$ denotes the Lebesgue measure on $({\math R}^d,{\cal B}or({\math R}^d))$) and that $F\!\in L^2({\math P}_{_X})$ with ${\math P}(F(X)\neq 0)>0$ (otherwise the expectation is clearly $0$ and the problem is meaningless). Furthermore we assume that the probability density $p$ is {\em everywhere positive} on ${\math R}^d$.
The paradigm of importance sampling applied to a parametrized family of distributions is the following: consider the family of absolutely continuous probability distributions $\pi_\theta(dx):=p_\theta(x) dx$, $\theta \!\in \Theta$, such that $p_\theta(x) > 0$, $\lambda_d(dx)$-$a.e.$ . One may assume without loss of generality that $\Theta$ is an open non empty connected subset of ${\math R}^q$ containing $0$ so that $p_0=p$. In fact we will assume throughout the paper that $\Theta={\math R}^q$. Then for any ${\math R}^d$-valued random variable $X^{(\theta)}$ with distribution $\pi_\theta$, we have \begin{equation}\label{CamMar}
\esp{F(X)} = \esp{F(X^{(\theta)})\frac{p(X^{(\theta)})}{p_\theta(X^{(\theta)})}}. \end{equation}
Among all these random variables having the same expectation $m=\esp{F(X)}$, the one with the lowest variance is the one with the lowest quadratic norm: minimizing the variance amounts to finding the parameter $\theta^* $ solution (if any) to the following minimization problem \begin{equation*}
\min_{\theta\!\in {\math R}^q} V(\theta) \end{equation*} where, for every $\theta\!\in {\math R}^q$, \begin{equation}\label{V=E}
V(\theta) := \esp{F^2(X^{(\theta)}) \frac{p^2(X^{(\theta)})}{p_\theta^2(X^{(\theta)})}}
= \esp{F^2(X) \frac{p(X)}{p_\theta(X)}} \le +\infty. \end{equation} A typical situation is importance sampling by mean translation in a finite dimensional Gaussian framework \emph{i.e.} \begin{equation*}
X^{(\theta)} = X+\theta,\quad p(x) = \frac{e^{-\frac{\abs{x}^2}{2}}}{(2\pi)^{\frac d2}}, \quad p_\theta(x)=p(x-\theta) \quad \mbox{ and }\quad V(\theta)= e^{-\abs{\theta}^2}\esp{F^2(X) e^{-2\psca{\theta,X}}}. \end{equation*} Then the second equality in~(\ref{V=E}) is simply the Cameron-Martin formula. This specific framework is very important for applications, especially in Finance, and was the starting point of the new interest for recursive importance sampling procedures, mainly initiated by Arouna in~\cite{ARO} (see further on). In fact, as long as variance reduction is concerned, one can consider a more general framework without extra effort. As a matter of fact, if the distributions $p_{\theta}$ satisfy \begin{align}\label{H1} \tag{${\cal H}_1$}
\begin{cases}
(i) & \forall \,x\! \in {\math R}^d,\quad \theta \mapsto p_{\theta} (x) \text{ is $\log$-concave} \\
(ii) & \forall\, x\! \in {\math R}^d,\quad \lim_{\abs{\theta} \rightarrow +\infty} p_{\theta}(x) = 0,
\qquad \text{or}\qquad
\forall x \in {\math R}^d,\quad \lim_{\abs{\theta} \rightarrow +\infty} \frac{p_{\theta}(x)}{p^2_{\theta/2}(x)} = 0,
\end{cases} \end{align} and $F$ satisfies $\esp{F^2(X)\frac{p(X)}{p_\theta(X)}} < +\infty$ for every $\theta\!\in {\math R}^q$, then (see Proposition~\ref{prop_intro} below), the function $V$ is finite, convex, goes to infinity at infinity. As a consequence $\argmin V=\{\nabla V=0\}$ {\em is non empty}. Assumption $(ii)$ can be localized by by considering that one the two conditions holds only on a Borel set $C$ of ${\math R}^d$ such that ${\math P}_{_X}(C\cap\{F\neq 0\})>0$. If $\theta\mapsto p_\theta(x)$ is {\em strictly} $\log$-concave for every $x$ in a Borel set $B$ such that ${\math P}{_X}\!\br{B\cap\{F\neq 0\}}>0$, then $V$ is strictly convex and $\argmin V=\{\nabla V=0\}$ is reduced to a single $\theta^*\!\in {\math R}^q$. These results follow from the second representation of $V$ as an expectation in~(\ref{V=E}) which is obtained by a second change of probability (the reverse one). For notational convenience we will temporarily assume that $\argmin V=\{\theta^*\}$ in this introduction section, although our main result needs no such restriction.
A classical procedure to approximate $\theta^*$ is the so-called Robbins-Monro algorithm. This is a recursive stochastic algorithm (see~(\ref{RM}) below) which can be seen as a stochastic counterpart of deterministic recursive zero search procedures like the Newton-Raphson one. It can be formally implemented provided the gradient of the (convex) target function $V$ admits a representation as an expectation. Since we have no {\em a priori} knowledge about the regularity of $F$ (\footnote{When $F$ is smooth enough alternative approaches have been developed based on some large deviation estimates which provide a good approximation of $\theta^*$ by deterministic optimization methods (see~\cite{GLHESH}).}) and do not wish to have any, we are naturally lead to {\em formally} differentiate the second representation of $V$ in~(\ref{V=E}) to obtain a representation of $\nabla V$ as
\begin{equation} \label{grad1}
\nabla V(\theta) = \esp{F^2(X)\frac{p(X)}{p_\theta(X)} \frac{\nabla_{\!\theta} \,p_{\theta}(X)}{p_{\theta}(X)}}. \end{equation}
Then, if we consider the function $\bar{H}_V(\theta, x)$ such that $\nabla V(\theta) = {\math E}\left( \bar{H}_V(\theta, X) \right)$ naturally defined by~(\ref{grad1}), the derived Robbins-Monro procedure writes \begin{equation}\label{RM} \tag{AlgoRM}
\theta_{n+1} = \theta_n - \gamma_{n+1} \bar{H}_V(\theta_n, X_{n+1}), \end{equation} with $(\gamma_n)_{n \ge 0}$ a {\em step} sequence decreasing to 0 (at an appropriate rate), $(X_n)_{n \ge 0}$ a sequence of i.i.d. random variables with distribution $p(x) \lambda_d(d x)$. To establish the convergence of a Robbins-Monro procedure to $\theta^* = \argmin V$ requires seemingly not so stringent assumptions. We mean by that: not so different from those needed in a deterministic framework. However, one of them turns out to be quite restrictive for our purpose: the sub-linear growth assumption in quadratic mean \begin{equation} \label{NEC} \tag{NEC}
\forall \theta\!\in {\math R}^d, \qquad \normLp{2}{\bar{H}_{V}(\theta, X)} \le C (1 + \abs{\theta}). \end{equation} which is the stochastic counterpart of the classical non-explosion condition needed in a deterministic framework. In practice, this condition is almost never satisfied in our framework due to the behaviour of the term $\frac{p(x)}{p_{\theta}(x)}$ as $\theta$ goes to infinity.
\ms The origin of recursive importance sampling as briefly described above goes back to Kushner and has recently been brought back to light in a Gaussian framework by Arouna in~\cite{ARO}. However, as confirmed by the numerical experiments carried out by several authors (\cite{ARO, KAW, LEL}), the regular Robbins-Monro procedure~(\ref{RM}) does suffer from a structural instability coming from the violation of~(\ref{NEC}). This phenomenon is quite similar to the behaviour of the explicit discretization schemes of an $ODE$ $\equiv \; \dot x=h(x)$ when $h$ has a super-linear growth at infinity. Furthermore, in a probabilistic framework no ``implicit scheme" can be devised in general. Then the only way out {\em mutatis mutandis} is to kill the procedure when it comes close to explosion and to restart it with a smaller step sequence. Formally, this can be described as some repeated projections or truncations when the algorithm leaves a slowly growing compact set waiting for stabilization which is shown to occur $a.s.$. Then, the algorithm behaves like a regular Robbins-Monro procedure. This is the so-called ``Projection \`a la Chen" avatar of the Robbins-Monro algorithm, introduced by Chen in~\cite{CHE, CHEetal} and then investigated by several authors (see $e.g.$ \cite{ANMOPR,LEL}) Formally, repeated projections ``\`a la Chen" can be written as follows: \begin{equation*} \tag{AlgoP}
\theta_{n+1} = \Pi_{K_{\sigma(n)}} \ac{\theta_n - \bar{H}_V(\theta_n, X_{n+1})} \end{equation*} where $\Pi_{K_{\sigma(n)}}$ denotes the projection on the convex compact $K_{\sigma(n)}$ ($K_p$ is increasing to ${\math R}^d$ as $p\to \infty$). In~\cite{LEL} is established a a Central Limit Theorem for this version of the recursive variance reduction procedure. Some extensions to non Gaussian framework have been carried out by Arouna in his PhD thesis (with some applications to reliability) and more recently to the marginal distributions of a L\'evy processes by Kawai in~\cite{KAW}.
However, convergence occurs for this procedure after a long ``stabilization phase" \dots provided that the sequence of compact sets have been specified in an appropriate way. This specification turns out to be a rather sensitive phase of the ``tuning" of the algorithm to be combined with that of the step sequence.
\ms In this paper, we show that as soon as the growth of $F$ at infinity can be explicitly controlled, it is always possible to design a regular Robbins-Monro algorithm which $a.s.$ converges to a variance minimizer $\theta^*$ with no risk of explosion (and subsequently no need of repeated projections).
To this end the key is to introduce a {\em third} change of probability in order to control the term $\frac{p(x)}{p_{\theta}(x)}$. In a Gaussian framework this amounts to switching the parameter $\theta$ from the density $p$ to the function $F$ by a third mean translation. This of course corresponds to a new function $\bar{H}_V$ but can also be interpreted {\em a posteriori} as a way to introduce an {\em adaptive} step sequence (in the spirit of~\cite{LEM}).
In terms of formal importance sampling, we introduce a new positive density $q_\theta$ (everywhere positive on $\{p>0\}$) so that the gradient writes \begin{equation} \label{grad2}
\nabla V(\theta) = \esp{F^2(\widetilde X^{(\theta)}) \frac{p^2(\widetilde X^{(\theta)})}{p_\theta(\widetilde X^{(\theta)}) q_\theta(\widetilde X^{(\theta)})} \frac{\nabla p_{\theta}(\widetilde X^{(\theta)})}{p_{\theta}(\widetilde X^{(\theta)})}} = \esp{\widetilde H_V(\theta,\widetilde X^{(\theta)})}, \end{equation} where $\widetilde X^{(\theta)} \sim q_{\theta}(x) d x$. The ``weight" $\displaystyle \frac{p^2(\widetilde X^{(\theta)})}{p_\theta(\widetilde X^{(\theta)}) q_\theta(\widetilde X^{(\theta)})} \frac{\nabla p_{\theta}(\widetilde X^{(\theta)})}{p_{\theta}(\widetilde X^{(\theta)})}$ may seem complicated but the r\^ole of the density $q_\theta$ is to control the critical term $\frac{p^2(x)}{p_\theta(x) q_\theta(x)}$ by a (deterministic) quantity only depending on $\theta$. Then we can replace $\widetilde H_V$ by a function $H(\theta,x)= \delta(\theta)\, \widetilde H_V(\theta,x)$ in the above Robbins-Monro procedure~(\ref{RM}) where $\delta $ is a positive function used to control the behaviour of $\widetilde H_V(\theta,x)$ for large values of $x$ (note that $\ac[2]{\esp[2]{H(.,\widetilde X^{(\theta)})} = 0} = \ac{\nabla V = 0}$).
We will first illustrate this paradigm in a finite dimensional setting with parametrized importance sampling procedures: the mean translation and the Esscher transform which coincide for Gaussian vectors on which a special emphasis will be put. Both cases correspond to a specific choice of $q_\theta$ which significantly simplifies the expression of the weight.
As a second step, we will deal with an infinite dimensional setting (path-dependent diffusion like processes) where we will rely on the Girsanov transform to play the role of mean translator. To be more precise, we want now to compute $\esp{F(X)}$ where $X$ is a path-dependent diffusion process and $F$ is a functional defined on the space ${\cal C}([0,T],{\math R}^d)$ of continuous functions defined on $[0,T]$. We consider a $d$-dimensional It\^o process $X=(X_t)_{t\in [0,T]}$ solution of the path-dependent SDE \begin{equation*} \tag{$E_{b,\sigma,W}$}
\dd X_t= b(t,X^t) \dd t + \sigma(t,X^t) \dd W_t,\quad X_0=x\!\in {\math R}^d, \end{equation*} where $W=(W_t)_{t\in [0,T]}$ is a $q$-dimensional standard Brownian motion, $X^t:=(X_{t\wedge s})_{s\in [0,T]}$ is the stopped process at time $t$, $b:[0,T]\times {\cal C}([0,T],{\math R}^d)\to {\math R}^d$ and $\sigma:[0,T]\times {\cal C}([0,T],{\math R}^d)\to {\cal M}(d,q)$ are Lipschitz with respect to the $\normsup{\;.\;}$ on the space $ {\cal C}([0,T],{\math R}^d)$ and continuous in $(t,x)\!\in [0,T]\times {\cal C}([0,T],{\math R}^d)$ (see~\cite{ROWI} for more details about these path-dependent SDE's).
Let $\varphi$ be a fixed borel bounded functional on ${\cal C}([0,T],{\math R}^d)$ with values in ${\cal M}(q, p)$ (where $p \ge 1$ is a free integral parameter). Then a Girsanov transform yields that for every $\theta\!\in L^2_{T,p}:=L^2([0,T],{\math R}^p)$, \begin{equation*}
\esp{F(X)} = \esp{F(X^{(\theta)}) e^{-\int_0^T\!\psca{\varphi(X^{(\theta),s}) \theta(s), \dd W_s} - \frac{1}{2} \normLTp{q}{\varphi(X^{(\theta)}{}^{,.})\theta}^2}} \end{equation*} where $X^{(\theta)}$ is the solution to $(E_{b+\sigma\varphi\theta,\sigma})$. The functional to be minimized is now $$ V(\theta) = \esp{F(X^{(\theta)})^2 e^{-2\int_0^T\!\psca{\varphi(X^{(\theta),s}) \theta(s), \dd W_s} - \normLTp{q}{\varphi(X^{(\theta)}{}^{,.})\theta}^2}},\qquad \theta\!\in L^2_{T, p}.
$$
In practice we will only minimize $V$ over a {\em finite dimensional subspace} of $E={\rm span}\{e_1,\ldots,e_m\}\subset L^2_{T, p}$.
The paper is organized as follows. Section \ref{dimfinie} is devoted to the finite dimensional setting where we recall the main tool including a slight extension of the Robbins-Monro theorem in the Subsection \ref{argmin-target} and the gaussian case investigated in \cite{ARO} is revisited to emphasize the new aspects of our algorithm in the Subsection \ref{gaussian_case}.
In Section 2 we successively investigate the translation for log-concave distributions probability and the Esscher transform. In Section \ref{diffusion} we introduce a functional version of our algorithm based on the Girsanov theorem to deal the SDE. In Section 4 we provide some comments on the practical implementation and in Section \ref{numerical} some numerical experiments are carried out on some option pricing problems.
\ss \ni {\sc Notations:} $\bullet$ We will denote by $S>0$ the fact that a symmetric matrix $S$ is positive definite.
\ni $|\,.\,|$ will denote the canonical Euclidean norm on ${\math R}^m$ and $\langle\,.,\,.\,\rangle$ will denote the canonical inner product.
\ni $\bullet$ The real constant $C>0$ denotes a positive real constant that may vary from line to line.
\ni $\bullet$ $\normLTp{p}{f}:= \left(\int_0^T f_1^2(t)+\cdots+f_p^2(t)dt \right)^{\frac 12}$ if $f=(f_1,\ldots,f_p)$ is an ${\math R}^p$-valued (class of) Borel function(s).
\section{The finite-dimensional setting} \label{dimfinie} \subsection{$\boldsymbol \argmin \,\boldsymbol V$ as a target} \label{argmin-target}
\begin{Pro} \label{prop_intro} Suppose \eqref{H1} holds. \\ Then the function $V$ defined by~(\ref{V=E}) is convex and $\lim_{\abs{\theta} \rightarrow +\infty} V(\theta) = +\infty$. As a consequence \[
\argmin V= \{\nabla V=0\}\neq \emptyset. \] \end{Pro} \noindent {\bf Proof.} By the change of probability $\frac{d \pi_{\theta}}{d \lambda_d}$ we have $V(\theta) = \esp{F^2(X)\frac{p(X)}{p_\theta(X)}}$. Let $x$ fixed in ${\math R}^d$. The function $(\theta \mapsto \log p_\theta(x))$ is concave, hence $\log (1/p_\theta(x))=-\log p_\theta(x)$ is convex so that, owing to the Young Inequality, the function $\frac{1}{p_\theta(x)}$ is convex since it is non-negative.
To prove that $V$ tends to infinity as $\abs{\theta}$ goes to infinity, we consider two cases: \begin{itemize}
\item[--] If $\ds \lim_{\abs{\theta} \rightarrow +\infty} p_\theta(x) = 0$ for every $x \in {\math R}^d$, the result is trivial by Fatou's Lemma.
\item[--] If $\ds \lim_{\abs{\theta} \rightarrow +\infty} \frac{p_\theta(x)}{p^2_{\theta/2}(x)} = 0$ for every $x \in {\math R}^d$, we apply the reverse H\"older inequality with conjugate exponents $(\frac{1}{3},-\frac{1}{2})$ to obtain \begin{align*}
V(\theta) & \ge \esp[5]{F^{2/3}(X) \pa[4]{\frac{p^2_{\theta/2}(X)}{p(X)p_\theta(X)}}^{\frac{1}{3}}}^3 \esp{\pa{\frac{p(X)}{p_{\theta/2}(X)}}^{-1}}^{-2}, \\
& \ge \esp[5]{F^{2/3}(X) \pa[4]{\frac{p^2_{\theta/2}(X)}{p(X)p_\theta(X)}}^{\frac{1}{3}}}^3, \end{align*} ($p$ and $p_\theta$ are probability density functions). One concludes again by Fatou's Lemma. $\cqfd$ \end{itemize}
\ms The set $\argmin V$, or to be precise, the random vectors taking values in $\argmin V$ will the target(s) of our new algorithm. If $V$ is strictly convex, $e.g.$ if $$\prob{X\!\in\{p_.(x) \mbox{ strictly $\log$-concave and } F(x)\neq 0\}} > 0,$$ then $\argmin V=\{\theta^*\}$. Nevertheless this will not be necessary owing to the combination of the two results that follow.
\begin{Lem}\label{Un}Let $U:{\math R}^d\to {\math R}_+$ be a convex differentiable function, then \begin{equation*}
\forall\, \theta,\, \theta'\!\in {\math R}^d,\qquad \psca{\nabla U(\theta)-\nabla U(\theta'), \theta-\theta'} \ge 0. \end{equation*} Furthermore, if $\argmin U$ is nonempty, it is a convex closed set (which coincide with $\{\nabla U=0\}$) and \begin{equation*}
\forall\, \theta\!\in {\math R}^d\setminus \argmin U, \;\forall\, \theta^*\!\in \argmin U, \qquad \psca{\nabla U(\theta), \theta-\theta^*}> 0. \end{equation*} \end{Lem}
A sufficient (but in no case necessary) condition for a nonnegative convex function $U$ to attain a minimum is that $\lim_{|x|\to \infty} U(x)=+\infty$.
Now we pass to the statement of the convergence theorem on which we will rely throughout the paper. It is a slight variant of the regular Robbins-Monro procedure whose proof is rejected in an annex.
\begin{Thm}\label{ThmRZ} (Extended Robbins-Monro Theorem) Let $H:{\math R}^q\times{\math R}^d\to {\math R}^d$ a Borel function and $X$ an ${\math R}^d$-valued random vector such that $\esp{\abs{H(\theta,X)}}<+\infty$ for every $\theta\!\in {\math R}^d$. Then set \begin{equation*}
\forall\, \theta\!\in {\math R}^d,\qquad h(\theta)= \esp{H(\theta,X)}. \end{equation*} Suppose that the function $h$ is continuous and that ${\cal T}^*:=\{h=0\}$ satisfies \begin{equation}\label{RMmeanreverting}
\forall\, \theta\!\in {\math R}^d\setminus {\cal T}^* ,\; \forall\, \theta^*\!\in{\cal T}^*, \qquad \psca{\theta-\theta^*,h(\theta)} > 0. \end{equation} Let $\g=(\g_n)_{n\ge 1}$ be a sequence of gain parameters satisfying \begin{equation}\label{StepCond}
\sum_{n\ge 1} \g_n=+\infty \qquad \mbox{ and }\qquad \sum_{n\ge 1} \g^2_n<+\infty. \end{equation} Suppose that \begin{equation}\label{LinGrowth} \tag{NEC}
\forall\, \theta\!\in {\math R}^d,\qquad \esp{\abs{H(\theta,X)}^2} \le C(1+\abs{\theta}^2) \end{equation} (which implies $\abs{h(\theta)}^2\le C(1+\abs{\theta}^2)$).
Let $(X_n)_{n\ge 1}$ be an i.i.d. sequence of random vectors having the distribution of $X$, a random vector $\theta_0$, independent of $(X_n)_{n\ge 1}$ satisfying $\esp{|\theta_0|^2} <+\infty$, all defined on the same probability space $(\Omega,{\cal A}, {\math P})$. Then, the recursive procedure defined by \begin{equation}\label{Algo}
\theta_n = \theta_{n-1}-\g_{n+1} H(\theta_n,X_{n+1}),\qquad n \ge 1 \end{equation} satisfies: \begin{equation}
\exists \, \theta_{_\infty}:(\Omega,{\cal A})\to {\cal T}^*,\; \theta_{_\infty}\!\in L^2({\math P}),\quad \mbox{such that} \quad \theta_n \stackrel{a.s.}{\longrightarrow} \theta_{_\infty}.\hskip 1,5 cm \end{equation} The convergence also holds in $L^p({\math P})$, $p\!\in (0,2)$. \end{Thm}
The proof is postponed to the Appendix at the end of the paper. The natural way to apply this theorem for our purpose is the following: \begin{itemize}
\item[--] {\sc Step~1}: we will show that the convex function $V$ in~(\ref{V=E}) is differentiable with a gradient $\nabla V$ having a representation as an expectation formally given $\nabla V (\theta)= \esp{\nabla_{\theta} v(\theta,X)}$.
\item[--] {\sc Step~2}: then set $H(\theta,x):= \rho(\theta)\nabla_{\theta} v(\theta,x)$ where $\rho$ is a (strictly) positive function on ${\math R}^q$. As a matter of fact, with the notations of the above theorem \begin{equation*}
\psca{\theta-\theta^*, h(\theta)} = \delta(\theta) \psca{\theta-\theta^*, \nabla V(\theta)}, \end{equation*} so that ${\cal T}^*= \argmin V$ and \eqref{RMmeanreverting} is satisfied (set $U:=V$ in Lemma~\ref{Un}).
\item[--] {\sc Step~3}: Specify in an appropriate way the function $\delta$ so that the linear quadratic growth assumption is satisfied. This is the sensitive point that will lead us to modify the structure more deeply by finding a new representation of $\nabla V$ as an expectation not directly based on the local gradient $\nabla_{\theta} v(\theta,x)$. \end{itemize}
\subsection{A first illustration: the Gaussian case revisited} \label{gaussian_case} The Gaussian is the framework of~\cite{ARO}. It is also a kind of introduction to the infinite dimensional diffusion setting investigated in Section~\ref{diffusion}. In the Gaussian case, the natural importance sampling density is the translation of the gaussian density: $p_{\theta}(x) = p(x-\theta)$ for $\theta \in {\math R}^d$ ($i.e.$ $q=d$). We have \begin{equation*}
p_\theta(x) = e^{-\frac{\abs{\theta}^2}{2} + \psca{\theta, X}} p(x). \end{equation*}
The assumption \eqref{H1} is clearly satisfied by the Gaussian density, and we assume that $F$ satisfies $\esp{ F^2(X) e^{- \psca{\theta, X}}} < +\infty$ so that $V$ is well defined.
In~\cite{ARO}, Arouna considers the function $\bar{H}_V(\theta, x)$ defined by \begin{equation*}
\bar{H}_V(\theta, x) = F^2(x) e^{\frac{\abs{\theta}^2}{2} - \psca{\theta, x}}(\theta-x). \end{equation*} It is clear that the condition \eqref{NEC} is not satisfied even if we simplify this function by $e^{\abs{\theta}^2/2}$ (which does not modify the problem).
\ni {\em A first approach:} When $F(X)$ have finite moments of any order, a naive way to control directly $\normLp{2}{\bar{H}_V(\theta_n, X_{n+1})}$ by an explicit deterministic function of $\theta$ (in order to rescale it) is to proceed as follows: one derives from H\"older Inequality that for every couple $(r,s)$, $r,\,s>1$ of conjugate exponents \begin{equation*}
\normLp{2}{\bar{H}_V(\theta, X)} \le
\normLp{2r}{(\theta-X)F^2(X)} \normLp{2s}{e^{-\psca{\theta, X}}} e^{\frac{\abs{\theta}^2}{2}}. \end{equation*} Setting $r=1+\frac{1}{\varepsilon}$ and $s=1+\varepsilon$, yields \begin{align*}
\normLp{2}{\bar{H}_{V}(\theta, X)}
& \le\pa{\normLp{2(1+\frac{1}{\varepsilon})}{X F^2(X)} +
\normLp{2(1+\frac{1}{\varepsilon})}{F^2(X)}\abs{\theta}}
e^{\pa{\frac{3}{2}+\varepsilon}\abs{\theta}^2}. \end{align*}
Then, $\bar{H}_{\varepsilon}(\theta, x):=e^{-(\frac{3}{2}+\varepsilon)\abs{\theta}^2}\bar{H}_{V}(\theta, x)$ satisfies the condition \eqref{NEC} and theoretically the standard Robbins-Monro algorithm implemented with $\bar{H}_{\varepsilon}$ $a.s.$ converges and no projection nor truncation is needed. Numerically, the solution is not satisfactory because the correcting factor $e^{-\pa{\frac{3}{2}+\varepsilon}\abs{\theta}^2}$ goes to zero much too fast as $\theta$ goes to infinity: if at any iteration at the beginning of the procedure $\theta_n$ is sent ``too far", then it is frozen instantly. If $\varepsilon$ is too small it will simply not prevent explosion. The tuning of $\varepsilon$ becomes quite demanding and payoff dependent. This is in complete contradiction with our aim of {\em a self-controlled variance reducer}. A more robust approach needs to be developed. On the other hand this kind of behaviour suggests that we are not in the right asymptotics to control $\normLp{2}{\bar{H}_V(\theta_n, X_{n+1})}$.
Note however that when $F$ is bounded with a compact support, then one can set $\varepsilon=0$ and the above approach provides an efficient answer to our problem.
\ni {\em A general approach:} We consider the density \begin{equation*}
q_\theta(x) = \frac{p^2(x)}{p(x-\theta)} = p(x+\theta). \end{equation*} By \eqref{grad2}, we have \begin{equation*}
\nabla V(\theta) = \esp{F^2(\widetilde X^{(\theta)}) \frac{\nabla p(\widetilde X^{(\theta)}-\theta)}{p(\widetilde X^{(\theta)}-\theta)}}, \end{equation*} with $\widetilde X^{(\theta)} \sim p(x+\theta) d x$, \emph{i.e.} $\widetilde X^{(\theta)} = X-\theta$. Since $p$ is the Gaussian density, we have $\frac{\nabla p(x)}{p(x)} = - x$. As a consequence, the function $H_V$ defined by \begin{equation*}
H_V(\theta, x) = F^2(x-\theta) (2\theta - x), \end{equation*}
provides a representation $\nabla V(\theta) = \esp{H_V(\theta, X)}$ of the gradient $\nabla V$. As soon as $F$ is bounded, this function satisfies the condition \eqref{NEC}. Otherwise, we note that thanks to this new change of variable the parameter $\theta$ lies now {\em inside} the payoff function $F$ and that the {\em exponential term} has disappeared from the expectation. If we have an {\em a priori} control on the function $F(x)$ as $|x|$ goes to infinity, say \begin{equation*}
\exists \lambda \in {\math R}_+, \quad \forall x \in {\math R}^d, \quad \abs{F(x)} \le c_F e^{\lambda \abs{x}}, \end{equation*} then we can consider the function $H_{\lambda}(\theta, x) = e^{-\lambda \abs{\theta}} H_V(\theta, x)$ which satisfies \begin{align*} \begin{split}
\normLp{2}{H_\lambda(\theta, X)} & \le c_F^2 \normLp{2}{e^{2\lambda \abs{X}} (2 \theta - X)}, \\
& \le C (1 + \abs{\theta}).
\end{split} \end{align*}
The resulting Robbins-Monro algorithm reads \begin{equation*}
\theta_{n+1} = \theta_n - \gamma_{n+1} e^{-\lambda \abs{\theta_n}} F^2(X_{n+1} - \theta_n) (2 \theta_n - X_{n+1}). \end{equation*}
We no longer to tune the correcting factor and one verifies on simulations that it does not suffer from freezing in general. In case of a too dissymmetric function $F$ this may still happen but a {\em self-controlled} variant is proposed in Section~\ref{translation} below to completely get rid of this effect (which cannot be compared to an explosion).
\subsection{Translation of the mean: the general strongly unimodal case}\label{translation} We consider importance sampling by mean translation, namely we set \begin{equation*}
\forall x \in {\math R}^d, \quad p_\theta(x) = p(x-\theta), \end{equation*} for $\theta \in {\math R}^d$.
In this section we assume that \begin{equation*}
p \mbox{ is $\log$-concave and } \lim_{|x|\to \infty} p(x)=0, \end{equation*} so that \eqref{H1} holds. Moreover, we make the following additional assumption on the probability density $p$ \begin{equation} \tag{${\cal H}^{tr}_a$} \label{H2} \exists\,a\!\in[1,2] \; \mbox{ such that } \;\left\{
\begin{array}{ll}(i)& \frac{\left|\nabla p\right|}{p}(x)=O(|x|^{a-1})
\; \mbox{ as }\; |x|\to \infty \\ (ii)&\exists \delta>0,\; \log
p(x)+\delta |x|^a\;\mbox{ is convex.} \end{array}\right. \end{equation}
First we will use~(\ref{V=E}) to differentiate $V$ since \begin{Pro} Suppose \eqref{H1} and \eqref{H2} are satisfied and the function $F$ satisfies \begin{equation}\label{CondDiff}
\forall\, \theta \in {\math R}^d, \quad \esp{F^2(X) \frac{p(X)}{p(X-\theta)}} < +\infty \quad\text{ and }\quad \forall\, C>0,\quad \esp{F^2(X)e^{C|X|^{a-1}}} < +\infty. \end{equation} Then $V$ is finite and differentiable on ${\math R}^d$ with a gradient given by \begin{align}\label{gradV1}
\nabla V(\theta) & = \esp{F^2(X)\frac{p(X)}{p^2(X-\theta)}\nabla p(X-\theta)}\\
& = \esp{F^2(X-\theta)\frac{p^2(X-\theta)}{p(X)p(X-2\theta)}\frac{\nabla p(X-2\theta)}{p(X-2\theta)}}.\label{gradV2} \end{align} \end{Pro}
\noindent {\bf Proof.} The formal differentiation to get~(\ref{gradV1}) from~(\ref{V=E}) is obvious. So it remains to check the domination property for $\theta$ lying inside a compact set. Let $x\!\in {\math R}^d$ and $\theta\!\in {\math R}^d$. The $\log$-concavity of $p$ implies that \[ \log p(x)\le \log p(x-\theta) +\frac{\psca{\nabla p(x-\theta),\theta}}{p(x-\theta)} \] so that \[
0\le \frac{p(x)}{p(x-\theta)}\le \exp{\frac{|\nabla
p(x-\theta)|}{p(x-\theta)}|\theta|}. \] Using the assumption~\eqref{H2} yields, for every $\theta\!\in B(0,R)$, \begin{align*} F^2(X)\frac{p(X)}{p^2(X-\theta)}\nabla p(X-\theta)
&\le F^2(X)(A|X|^{a-1}+B)e^{(A|X-\theta|^{a-1}+B)|\theta|}\\
& \le C_{R}F^2(X) e^{C'|X|^{a-1}}=:g(X)\!\in L^1({\math P}) \end{align*} To derive the second expression \eqref{gradV2} for the gradient, we proceed as follows: an elementary change of variable shows that \begin{align*} \nabla V(\theta)&= \int_{{\math R}^d}F^2(x)\frac{p(x)}{p^2(x-\theta)}p(x)\nabla p(x-\theta)dx\\ &= \int_{{\math R}^d}F^2(x-\theta)\frac{p^2(x-\theta)}{p^2(x-2\theta)}\nabla p(x-2\theta)dx\\ &= \esp{F^2(X-\theta)\frac{p^2(X-\theta)}{p(X)p(X-2\theta)}\frac{\nabla p(X-2\theta)}{p(X-2\theta)}}.\cqfd \end{align*}
\ni {\bf Remark.} The second change of variable (in~\eqref{V=E}) has been processed to withdraw the parameter $\theta$ from the possible non smooth function $F$ to make possible the differentiation of $V$ (since $p$ is smooth). The second expression~\eqref{gradV2} results form a {\em third} translation of the variable in order to plug back the parameter $\theta$ into the function $F$ which in common applications has a known controlled growth rate at infinity.
This last statement may look strange at a first glance since $\theta$ appears in the ``weight" term of the expectation that involves the probability density $p$. However, when $X\stackrel{d}{=}{\cal N}(0;1)$, this term can be controlled easily since it reduces to \[ \frac{p^2(x-\theta)}{p(x)p(x-2\theta)}\frac{\nabla p(x-2\theta)}{p(x-2\theta)}= e^{\theta^2}(2\theta-x). \] The following lemma shows that, more generally in our strongly unimodal setting, if \eqref{H1} and \eqref{H2} are satisfied, this ``weight" can always be controlled by a deterministic function of $\theta$.
\begin{Lem}\label{lemmetec} If \eqref{H2} holds, then there exists two real constants $A$, $B$ such that \begin{equation}\label{IneqTech}
\frac{p^2(x-\theta)}{p(x)p(x-2\theta)}\frac{|\nabla p(x-2\theta)|}{p(x-2\theta)}\le e^{2\delta |\theta|^a}\left(A |x|^{a-1} + A |\theta|^{a-1}+B\right). \end{equation} \end{Lem}
\noindent {\bf Proof.} Let $f$ be the convex function defined on ${\math R}^d$ by $f(x)= \log p(x)+\delta|x|^a$. Then, for every $x,\, \theta\!\in {\math R}^d$, \[ \log\left(\frac{p^2(x-\theta)}{p(x)p(x-2\theta)}\right)= \log\left(\frac{f^2(x-\theta)}{f(x)f(x-2\theta)}\right)+
\delta\left(|x|^a+|x-2\theta|^a -2|x-\theta|^a\right) \] Note that $x-\theta= \frac 12 (x+(x-2\theta))$. Then, using the $\log$-convexity of $f$ and the elementary inequality \[
\forall\, u,\, v\!\in {\math R}^d,\quad |u|^a+|v|^a \le 2
\left(\left|\frac{u+v}{2}\right|^a+\left|\frac{u-v}{2}\right|^a\right) \] (valid if $a\!\in (0,2]$) yields \begin{equation} \label{control_p_trans}
\frac{p^2(x-\theta)}{p(x)p(x-2\theta)}\le e^{2\delta |\theta|^a}. \end{equation} One concludes by the point $(i)$ of \eqref{H2}. $\cqfd$
\bs \ni {\bf Remark.} Thus the normal distribution satisfies \eqref{H2} with $a=2$ and $\delta=1/2$. Moreover, note that the last inequality in the above proof holds as an equality.
\bs Now we are in position to derive an unconstraint (extended) Robbins-Monro algorithm to minimize the function $V$, provided the function $F$ satisfies a sub-multiplicative control property, in which $c>0$ is a real parameter and $\widetilde F$ a function from ${\math R}^d $ to ${\math R}_+$, such that, namely \begin{equation} \tag{${\cal H}^{tr}_{c}$} \label{H3}
\left\{\begin{array}{ll}
\forall\, x,\, y\!\in {\math R}^d, & |F(x)|\le \widetilde F(x) \; \mbox{ and }\;
\widetilde F(x+y)\le C(1+\widetilde F(x))^c(1+\widetilde F(y))^c\\ \\
& \quad \esp{|X|^{2(a-1)}\widetilde F(X)^{4c}}<+\infty .
\end{array}\right.
\end{equation}
\noindent{\bf Remark.}
Assumption \eqref{H3} seems almost non-parametric. However, its field of application is somewhat limited by~(\ref{H2}) for the following reason: if there exists a positive real number $\eta>0$ such that $x\mapsto \log p(x) +\eta |x|^a$ is concave, then $p(x) \le Ce^{-\eta |x|^a}(|x|+1)$ for some real constant $C>0$; which in turn implies that the function $\widetilde F$ in \eqref{H3} needs to satisfy $\widetilde F(x)\le C'e^{\lambda |x|^b}$ for some $b\!\in (0,a)$ and some $\lambda>0$. (Then $c=c_b$ with $c_b=1$ if $b\!\in[0, 1]$ and $c_b = 2^{\frac b2}$ if $b\!\in (1,a)$, when $a>1$).
\begin{Thm}\label{ThmMT} Suppose $X$ and $F$ satisfy \eqref{H1}, \eqref{H2},~(\ref{CondDiff}) and \eqref{H3} for some parameters $a\!\in (0,2]$, $b\!\in(0,a)$ and $\lambda>0$, and that the step sequence $(\gamma_n)_{n\ge 1}$ satisfies the usual decreasing step assumption \[ \sum_{n\ge 1}\g_n=+\infty \qquad \mbox{ and } \qquad \sum_{n\ge 1}\g^2_{n+1} <+\infty. \] Then the recursive procedure defined by \begin{equation}\label{AlgoRM}
\theta_{n+1}=\theta_n-\g_{n+1}H(\theta_n, X_{n+1}),\qquad \theta_0\!\in {\math R}^d \end{equation} where $(X_n)_{n\ge 1}$ is an i.i.d. sequence with the same distribution as $X$ and \begin{equation}\label{H-a} H(\theta,x):=
\frac{F^2(x-\theta)}{1+\widetilde F(-\theta)^{2c}}e^{-2\delta
|\theta|^a}\frac{p^2(x-\theta)}{p(x)p(x-2\theta)}\frac{\nabla p(x-2\theta)}{p(x-2\theta)}, \end{equation}
$a.s.$ converges toward an $\argmin V$-valued (square integrable) random variable $\theta^*$. \end{Thm}
\ni {\bf Proof.} In order to apply Theorem~\ref{ThmRZ}, we have to check the following fact:
-- {\em Mean reversion}: The mean function of the procedure defined by~(\ref{AlgoRM}) reads \begin{equation*}
h(\theta) = \esp{H(\theta,X)} = \frac{e^{-2\delta |\theta|^a}}{1+\widetilde F(-\theta)^{2c}}\nabla V(\theta) \end{equation*} so that ${\cal T}^*:=\{h=0\}=\{\nabla V =0\}$ and if $\theta^*\!\in {\cal T}^*$ and $\theta\!\in {\math R}^d\setminus {\cal T}^*$, \begin{equation*}
\psca{\theta-\theta^*, h(\theta) } = \frac{e^{-2\delta |\theta|^a}}{1+\widetilde F(-\theta)^{2c}} \psca{\nabla V(\theta), \theta-\theta^*} >0 \end{equation*} for every $\theta\neq \theta^*$.
-- {\em Linear growth of $\theta \mapsto \|H(\theta,X)\|_{_2}$}: All our efforts in the design of the procedure are motivated by this Assumption~(\ref{LinGrowth}) which prevents explosion. This condition is clearly fulfilled by $H$ since \begin{align*}
\esp{\abs{H(\theta,X)}}^2 &= \frac{e^{-4\delta |\theta|^a}}{(1+\widetilde F(-\theta)^{2c})^2}\esp{F^4(X-\theta)\left(\frac{p^2(X-\theta)}{p(X)p(X-2\theta)}\frac{|\nabla p(X-2\theta)|}{p(X-2\theta)}\right)^2}, \\
&\le C e^{-4\delta |\theta|^a} \esp{(1+\widetilde F(X)^{2c})^2 (A(|X|^{a-1}+|\theta|^{a-1})+B)^2}, \end{align*} where we used Assumption~\eqref{H3} in the first line and Inequality~(\ref{IneqTech}) from Lemma~\ref{lemmetec} in the second line. One derives that there exists a real constant $C>0$ such that \begin{equation*}
\esp{\abs{H(\theta,X)}}^2 \le C \esp{ \widetilde F(X)^{4c}(1+|X|)^{2(a-1)}} (1+|\theta|^{2(a-1)}). \end{equation*} This provides the expected condition since~\eqref{H3} holds. $\cqfd$
\subsubsection*{Examples of distributions} \begin{itemize} \item{The normal distribution.} Its density is given on ${\math R}^d$ by \[
p(x)= (2\pi)^{-\frac d2}e^{-|x|^2/2},\qquad x\!\in {\math R}^d. \] so that \eqref{H1} is satisfied as well as \eqref{H2} for $a=2$, $\delta=\frac12$. Assumption \eqref{H3} is satisfied iff $(b,\lambda)\!\in(0,2)\times (0,\infty)\cup\{2\}\times (0,\frac 12)$. Then the function $H$ has a particularly simple form \[
H(\theta,x) =e^{-\frac{\lambda}{2}|\theta|^b }F^2(x-\theta) (2\theta-x) \] \item {\em The hyper-exponential distributions} \[
p(x) = C_{d,a,\sigma}e^{-\frac{|x|^a}{\sigma^a}}P(x), \quad a\!\in [1,2] \] where $P$ is polynomial function. This wide family includes the normal distributions, the Laplace distribution, the symmetric gamma distributions, etc.
\item {\em The logistic distribution} Its density on the real line is given by \[ p(x) =\frac{e^x}{(e^x+1)^2} \] \eqref{H1} is satisfied as well as \eqref{H2} for $a=1+\eta$ ($\eta\!\in(0,1)$), $\delta>0$. Assumption \eqref{H3} is satisfied iff $(b,\lambda)\!\in(0,1)\times (0,\infty)\cup\{1\}\times (0,1)$. \end{itemize}
\subsection{Exponential change of measure: the Esscher transform}\label{Esscher} A second classical approach is to consider an exponential change of measure (or Esscher transform). This transformation has already been consider for that purpose in~\cite{KAW} to extend the procedure with repeated projections introduced in~\cite{ARO}. We denote by $\psi$ the cumulant generating function (or log--Laplace) of $X$ \emph{i.e.} $\psi(\theta) = \log \esp{e^{\psca{\theta, X}}}$. We assume that $\psi(\theta)<+\infty$ for every $\theta\!\in {\math R}^d$ (which implies that $\psi$ is an infinitely differentiable convex function) and define \begin{equation*}
p_\theta(x) = e^{\psca{\theta,x} - \psi(\theta)} p(x), \quad x \in {\math R}^d. \end{equation*} Let $X^{(\theta)}$ denote any random variable with distribution $p_\theta$.
We assume that $\psi$ satisfies \begin{equation} \label{Hes} \tag{${\cal H}^{es}_\delta$}
\lim_{\abs{\theta}\rightarrow +\infty} \psi(\theta)-2\psi\pa[3]{\frac{\theta}{2}} = +\infty \qquad \text{and} \qquad \exists\,\delta>0,\quad \theta \mapsto \psi(\theta) - \delta \abs{\theta}^2 \text{ is concave}. \end{equation}
One must be aware that what follows makes sense as a variance reduction procedure only if the distribution of $X^{(\theta)}$ can be simulated at the same cost as $X$ or at least at a reasonable cost $i.e.$ \begin{equation}\label{ReXEsscher} X^{(\theta)}= g(\theta,\xi),\quad \xi :(\Omega,{\cal A}, {\math P})\to {\cal X} \end{equation} where $ {\cal X}$ is a Borel subset of a metric space and $g:{\math R}^d\times {\cal X}$ is an explicit Borel function. By \eqref{V=E}, the potential $V$ to be minimized is $V(\theta) = \esp{F^2(X) e^{-\psca{\theta,X} + \psi(\theta)}}.$
\begin{Pro} Suppose $\psi$ satisfies \eqref{Hes} and $F$ satisfies \begin{equation} \label{hypo-F-es}
\forall \theta \in {\math R}^d,\quad \esp{\abs{X} F^2(X) e^{\psca{\theta,X}}} < +\infty. \end{equation} Then \eqref{H1} is fulfilled and the function $V$ is differentiable on ${\math R}^d$ with a gradient given by \begin{align} \label{formal_diff_esscher}
\nabla V(\theta) &= \esp{\pa{\nabla \psi(\theta) - X} F^2(X) e^{-\psca{\theta, X} + \psi(\theta)}}, \\
&= \esp{\pa[2]{\nabla \psi(\theta) - X^{(-\theta)}} F^2(X^{(-\theta)})} e^{\psi(\theta)-\psi(-\theta)}, \label{diff_esscher} \end{align} where $\ds \nabla \psi(\theta) = \frac{\esp{X e^{\psca{\theta,X}}}}{\esp{e^{\psca{\theta,X}}}}$. \end{Pro}
\ni {\bf Proof.} The function $\psi$ is clearly log-convex so that $\theta \mapsto p_\theta(x)$ is $\log$-concave for every $x\!\in {\math R}^d$. On the other hand, by \eqref{Hes} we have $\lim \frac{p^2_{\theta/2}(x)}{p_\theta(x)} = +\infty$ for every $x\!\in {\math R}^d$, and \eqref{H1} is fulfilled.
The formal differentiation to get \eqref{formal_diff_esscher} is obvious and is made rigorous by applying the assumption on $F$. The second expression \eqref{diff_esscher} of the gradient uses a third change of variable \begin{align*}
\nabla V(\theta) & = \int_{{\math R}^d}\! \pa{\nabla \psi(\theta) - x} F^2(x) e^{-\psca{\theta,x} + \psi(\theta)} p(x) dx, \\
& = \int_{{\math R}^d}\! \pa{\nabla \psi(\theta) - x} F^2(x) e^{\psi(\theta) - \psi(-\theta)} p_{-\theta}(x) dx, \\
&= \esp{\pa[2]{\nabla \psi(\theta) - X^{(-\theta)}} F^2(X^{(-\theta)})} e^{\psi(\theta)-\psi(-\theta)}.\cqfd \end{align*}
\begin{Thm} We assume that $\psi$ satisfies~\eqref{Hes} and $F$ satisfies~\eqref{hypo-F-es} and \begin{equation*}
\forall x \in {\math R}^d, \quad \abs{F(x)} \le C e^{\frac{\lambda}{4} \abs{x}} \qquad \text{and}\qquad \esp{\abs{X}^2 e^{\lambda \abs{X}}} < +\infty.
\end{equation*} Then the recursive procedure \begin{equation*}
\left\{\begin{array}{lcl}X_{n+1}^{(\theta_n)}&=& g(\theta_n,\xi_{n+1})\\ \theta_{n+1}&=&\theta_n -\g_{n+1}H(\theta_n, X_{n+1}^{(-\theta_n)}),\quad n\ge 0, \quad \theta_0\!\in {\math R}^d
\end{array}\right. \end{equation*} where $(\xi_n)_{n\ge 1}$ is an i.i.d. sequence with the same distribution as $\xi$ in~(\ref{ReXEsscher}) and \begin{equation*}
H(\theta, x) := e^{-\frac{\lambda}{2}\sqrt{d} \abs{\nabla \psi(-\theta)}} F^2(x) \pa[2]{\nabla \psi(\theta) - x} \end{equation*} $a.s.$ converges toward an $\argmin V$-valued (square integrable) random vector $\theta^*$ . \end{Thm}
\ni {\bf Proof.} We have to check the linear growth of the function $\theta \mapsto \normLp{2}{H(\theta, X^{(-\theta)})}$ (condition \eqref{NEC}). We have \begin{align}
\esp{\abs{H(\theta, X^{(-\theta)})}^2} & = e^{-\lambda\sqrt{d} \abs{\nabla \psi(-\theta)}} \esp{F^4(X^{(-\theta)}) \abs[2]{\nabla \psi(\theta) - X^{(-\theta)}}^2}, \notag \\
& \le C e^{-\lambda\sqrt{d} \abs{\nabla \psi(-\theta)}} \esp{e^{\lambda \abs{X^{(-\theta)}}}\abs[2]{\nabla \psi(\theta) - X^{(-\theta)}}^2}, \notag \\
& \le C e^{-\lambda \sqrt{d}\abs{\nabla \psi(-\theta)}} \pa{\abs{\nabla \psi(\theta)}^2 \esp{e^{\lambda \abs{X^{(-\theta)}}}} + \esp{\abs{X^{(-\theta)}}^2 e^{\lambda \abs{X^{(-\theta)}}}}}. \label{esscher-check-NEC} \end{align}
First, by the following inequality \begin{equation*}
\forall x \in {\math R}^d, \quad e^{\lambda \abs{x}} \le \prod_{j=1}^d \pa{e^{\lambda x_j} + e^{-\lambda x_j}}
= \sum_{J\subset\ac{1,\dots,d}} e^{\lambda \pa{\sum_{j \in J} \psca{e_j, x} + \sum_{j \in J^c} \psca{e_j, x}}} \end{equation*} we have $\ds e^{\lambda \abs{x}} \le \sum_{J \subset\ac{1,\dots,n}} e^{\lambda \psca{e_J, x}}$ where $(e_J)_j = 1$ if $j \in J$ or $-1$ if $j \in J^c$. With this notation, we have \begin{align*}
\esp{e^{\lambda \abs{X^{(-\theta)}}}} &\le \sum_{J\subset\ac{1,\dots,d}} \esp{e^{\lambda \psca{e_J, X^{(-\theta)}}}} = \sum_{J\subset\ac{1,\dots,d}} e^{\psi(\lambda e_J-\theta) - \psi(-\theta)}. \end{align*}
By the concavity of $\psi-\delta\,|\,.\,|^2$, we have \[
\forall\, u,\, v\!\in {\math R}^d,\qquad \psi(u+v)-\psi(u)\le \langle \nabla \psi(u),v\rangle +\delta\,|v|^2 \] so that \begin{equation}
\esp{e^{\lambda \abs{X^{(-\theta)}}}} \le \sum_{J\subset\ac{1,\dots,d}} e^{\lambda \psca{\nabla \psi(-\theta), e_J}+\delta\lambda^2|e_J|^2} \le C_{d,\lambda,\delta} e^{\lambda \sqrt{d} \abs{\nabla \psi(-\theta)}}. \label{esscher-check-NEC-1} \end{equation}
In the same way, we have \begin{align}
\esp{\abs{X^{(-\theta)}}^2 e^{\lambda \abs{X^{(-\theta)}}}} &\le \sum_{J\subset\ac{1,\dots,d}} \esp{\abs{X^{(\lambda e_J-\theta)}}^2} e^{\psi(\lambda e_J-\theta) - \psi(-\theta)}, \notag \\
&\le C_{d,\lambda,\delta} e^{\lambda \sqrt{d} \abs{\nabla \psi(-\theta)}}\sum_{J\subset\ac{1,\dots,d}} \esp{\abs{X^{(\lambda e_J-\theta)}}^2}. \label{esscher-check-NEC-2} \end{align} Now, by differentiation of $\psi$ it is easy to check that \begin{equation*}
\forall \theta \in {\math R}^d, \quad \Dp \psi(\theta) = \frac{\int x^{\otimes 2} e^{\psca{\theta, x}} p(x) \dd x}{e^{\psi(\theta)}} - \nabla \psi(\theta)^{\otimes 2}, \end{equation*} which implies \begin{equation*}
\esp{\abs{X^{(\lambda e_J - \theta)}}^2} = \Tr\pa{\Dp \psi(\lambda e_J - \theta)} + \Tr\pa{\nabla \psi(\lambda e_J - \theta)^{\otimes 2}}. \end{equation*} The assumption \eqref{Hes} implies that $0\le D^2\psi(\theta)\le 2\delta\,I_d$ (for the partial order on symmetric matrices induced by nonnegative symmetric matrices) then $D^2\psi(\theta)$ is a bounded function of $\theta\!\in {\math R}^d$ and in turn $\nabla(\theta)$ has a linear growth by the fundamental formula of calculus. Consequently, for every $J \subset \pa{1,\dots,d}$, \begin{equation*}
\esp{\abs{X^{(\lambda e_J - \theta)}}^2} \le C \pa{1 + \abs{\theta}^2}. \end{equation*} Plugging this into \eqref{esscher-check-NEC-2} and using \eqref{esscher-check-NEC-1} and \eqref{esscher-check-NEC} we obtain $\esp{\abs{H(\theta, X^{(-\theta)})}^2} \le C\pa{1 + \abs{\theta}^2}$.
$\cqfd$
\section{Adaptive variance reduction for diffusions} \label{diffusion} \subsection{Framework and preliminaries} We consider a $d$-dimensional It\^o process $X=(X_t)_{t\in [0,T]}$ solution to the stochastic differential equation (SDE) \begin{equation}\label{SDExt} \tag{$E_{b,\sigma,W}$}
\quad \dd X_t = b(t,X^t) \dd t + \sigma(t,X^t) \dd W_t,\quad X_0=x\!\in {\math R}^d, \end{equation} where $W=(W_t)_{t\in [0,T]}$ is a $q$-dimensional standard Brownian motion, $X^t:=(X_{t\wedge s})_{s\in [0,T]}$ is the stopped process at time $t$, $b:[0,T]\times {\cal C}([0,T],{\math R}^d)\to {\math R}^d$ and $\sigma:[0,T]\times {\cal C}([0,T],{\math R}^d)\to {\cal M}(d,q)$ are measurable with respect to the canonical predictable $\sigma$-field on $[0,T]\times {\cal C}([0,T],{\math R}^d) $. For further details we refer to~\cite{ROWI}, p. 124-130.
Thus, if $b(t,x^t)=\beta(t,x(t))$ and $\sigma(t,x^t)=\vartheta(t,x(t))$ for every $x\!\in {\cal C}([0,T],{\math R}^d)$, $X$ is a usual diffusion process with drift $\beta$ and diffusion coefficient $\vartheta$. \ss
If $b(t,x^t)=\beta(t, x(\underline t))$ and $\sigma(t,x^t)=\vartheta(t,x(\underline t))$ for every $x\!\in {\cal C}([0,T],{\math R}^d)$ where $\underline t:= \lfloor \frac {tn}{T}\rfloor \frac Tn $, then $X$ is {\em the continuous Euler scheme with step} $T/n$ of the above diffusion with drift $\beta$ and diffusion coefficient $\vartheta$.
An easy adaptation of standard proofs for regular SDE's show (see~\cite{ROWI}) that strong existence and uniqueness of solutions for~(\ref{SDExt}) follows from the following assumption \begin{align} \label{Hbs} \tag{${\cal H}_{b, \sigma}$}
\begin{cases}
(i) & b(.,0) \mbox{ and }\sigma(.,0) \mbox { are continuous}\\
(ii) & \forall\, t\!\in [0,T],\; \forall\, x,\, y\!\in {\cal C}([0,T],{\math R}^d),\, \abs{b(t,y)-b(t,x)}+ \norm{\sigma(t,y)-\sigma(t,x)} \le C_{b,\sigma} \normsup{x-y}.
\end{cases} \end{align}
Our aim is to devise an adaptive variance reduction method inspired from Section~\ref{dimfinie} for the computation of \begin{equation*}
\esp{F(X)} \end{equation*} where $F$ is an Borel functional defined on ${\cal C}([0,T],{\math R}^d)$ such that \begin{equation}\label{nonvide}
\prob{F^2(X)>0}>0 \quad \text{and}\quad F(X)\!\in L^2({\math P}). \end{equation} In this functional setting, Girsanov Theorem will play the role of the invariance of Lebesgue measure by translation. The translation process that we consider in this section is of the form $\Theta(t, X^t)$ where $\Theta$ is defined for every $\xi\!\in {\cal C}([0,T],{\math R}^d)$ and $\theta\!\in L^2_{T,p}$ by \begin{equation*}
\Theta(t, \xi) := \varphi(t, \xi^t)\, \theta_t, \quad \text{ with } \quad \varphi : [0,T] \times {\cal C}([0,T],{\math R}^d) \rightarrow {\cal M}(q, p), \end{equation*} a bounded Borel function and $\theta\!\in L^2_{T,p}$ (represented by a Borel function) for $p \ge 1$. In the sequel, we use the following notations $\varphi_t(\xi) := \varphi(t, \xi^t)$, \begin{equation*}
\Theta_t := \Theta(t, X^t), \quad \Theta^{(\theta)}_t := \Theta(t, X^{(\theta),t}), \quad \text{and} \quad \Theta^{(-\theta)}_t := \Theta(t, X^{(-\theta),t}), \end{equation*} where $X^{(\pm \theta)}$ denotes {\color{black} the solution} to $(E_{b \pm \sigma\Theta, \sigma,W})$.
First we need the following standard abstract lemma. \begin{Lem} \label{Girs} Suppose $(H_{b,\sigma})$ holds. \\ The SDE $(E_{b+\sigma \Theta, \sigma, W})$ satisfies the {\color{black}weak existence and uniqueness} assumptions and for every non negative Borel functional $G:{\cal C}([0,T],{\math R}^{d+1}) \to {\math R}_+$ and $f \in {\cal C}([0,T],{\math R}^{q})$ we have, with the above notations, \begin{multline*}
\esp[4]{G \pa[3]{X, \int_0^.\!\! \psca{f(s, X^s), \dd W_s}}}
= \esp[4]{G \pa[3]{X^{(\theta)}, \int_0^.\!\! \psca{f(s, X^{(\theta),s}), \dd W_s}
+ \int_0^. \!\!\psca[1]{f, \Theta}(s, X^{(\theta),s}) \dd s} \\
\times e^{- \int_0^T\!\! \psca{\Theta^{(\theta)}_s, \dd W_s} -\frac{1}{2} \normLTp{q}{\Theta^{(\theta)}}^2}}, \end{multline*} and \begin{multline*}
\esp[4]{G \pa[3]{X^{(\theta)}, \int_0^. \!\!\psca{f(s, X^{(\theta),s}), \dd W_s}}}
= \esp[4]{G \pa[3]{X, \int_0^. \!\!\psca{f(s, X^s), \dd W_s}
- \int_0^. \!\!\psca[1]{f, \Theta}(s, X^s) \dd s}\\
\times e^{\int_0^T \!\!\psca{\Theta_s, \dd W_s} -\frac{1}{2} \normLTp{q}{\Theta}^2}}, \end{multline*}
\end{Lem}
\ni {\bf Proof.} This is a straightforward application of Theorem 1.11, p.372 (and the remark that immediately follows) in~\cite{REYO} once noticed that $(t,\omega)\mapsto b(t,X^t(\omega))$, $(t,\omega)\mapsto \sigma(t,X^t(\omega))$ and $(t,\omega)\mapsto \Theta(t,X^t(\omega))$ are predictable processes with respect to the completed filtration of $W$. $\cqfd$
\ni {\bf Remarks.} $\bullet$ The Dol\'eans exponential $\left(e^{\int_0^t\! \psca{\Theta_s, \dd W_s} -\frac{1}{2} \normLTp{q}{\Theta}^2} \right)_{t\in[0,T]}$ is a true martingale for any $\theta\!\in L^2_{T,p}$.
\ni $\bullet$ In fact, still following the above cited remark form~\cite{REYO}, the above lemma holds true if we replace $\Theta$ by any progressively measurable process $\tilde \Theta$ such that $\esp{e^{\frac 12 \int_0^T \!\abs{\tilde \Theta(s,\omega)}^2 \dd s}} < +\infty$.
It follows from the first identity in Lemma~\ref{Girs} that for every bounded Borel function $\varphi:[0,T] \times {\cal C}([0,T], {\math R}^d) \to {\cal M}(q, p)$ and for every $\theta\!\in L^2_{_{T,p}}$ \begin{equation*}
\esp{F(X)} = \esp{F(X^{(\theta)})
e^{- \int_0^T \!\psca{\Theta^{(\theta)}_s, \dd W_s} -\frac{1}{2} \normLTp{q}{\Theta^{(\theta)}}^2}}, \end{equation*} (set $G(x,y)=F(x)$). So, finding the estimator with the lowest variance amounts to solving the minimization problem \begin{equation*}
\min_{\theta \in L^2_{T,p}} V(\theta) \quad
\text{where} \quad V(\theta) := \esp[4]{F^2(X^{(\theta)}) e^{-2\int_0^T \!\psca{\Theta^{(\theta)}_s, \dd W_s} - \normLTp{q}{\Theta^{(\theta)}}^2}}, \end{equation*} Using Lemma~\ref{Girs} with $G(x,y)=F^2(x) e^{-2 y(T) - \normLTp{q}{\varphi(., x^.) \theta}^2}$ and $f = \Theta$ yields \begin{equation}\label{V3}
V(\theta) = \esp[4]{F^2(X) e^{-\int_0^T \!\psca{\Theta_s, \dd W_s} + \frac{1}{2}\normLTp{q}{\Theta}^2}}. \end{equation}
\begin{Pro} Assume ${\math E}\, F(X)^{2+\eta}<+\infty$ for some $\eta>0$ as well as Assumptions~(\ref{nonvide}) and $(H_{b,\sigma})$. Then function $V$ is finite on $L^2_{T,p}$ and $\log$-convex.
\begin{itemize}
\item[$(a)$] Assume that the bounded matrix-valued Borel function $\varphi$ satisfies that $\varphi(s, X^s)$ has a non-atomic kernel on the event $\{F(X)>0\}$ $i.e.$ \begin{equation}\label{nonatomic}
\prob{\{\exists \, \theta\!\in L^2_{T,p}\setminus\{0\}\;\mbox{ s.t. } \theta(s) \!\in \mbox{Ker} \, \varphi(s, X^s)\; \dd s\mbox{-a.e. and } F^2(X)>0 \}}=0
\end{equation}
then for every finite dimensional subspace $E\subset L^2_{_T}$, $\ds \lim_{\normLTp{p}{\theta} \to +\infty, \theta \in E} V(\theta)=+\infty$. If furthermore \begin{equation}\label{General}
\inf_{\normLTp{p}{\theta} = 1} \int_0^T \theta(s)^*\esp{ \varphi(s, X^s)^*\varphi(s, X^s) \indic[1]{F^2(X) > 0}} \theta(s) \dd s > 0, \end{equation} then $\ds \lim_{\normLTp{p}{\theta} \to +\infty} V(\theta)=+\infty$.
\item[$(b)$] The function $V$ is differentiable at every $\theta \!\in L^2_{T,p}$ and the differential $\DD V(\theta) \!\in L^2_{T,p}$ is characterized on every $\psi\!\in L^2_{T,p}$ by \begin{align}
& \pscaLTp{p}{\DD V(\theta), \psi}
= \esp[4]{F^2(X) e^{-\int_0^T\!\! \psca{\Theta_s, \dd W_s} + \frac{1}{2}\normLTp{q}{\Theta}^2}
\pa[3]{ \pscaLTp[1]{p}{\Theta, \varphi(., X^.) \psi} - \int_0^T\!\! \psca{\varphi(s, X^s) \psi_s, \dd W_s}}}, \notag \\
& \; = \esp[4]{F^2(X^{(-\theta)}) e^{\normLp{L^2_{T,p}}{\Theta^{(-\theta)}}^2}
\pa[3]{2 \pscaLTp[1]{p}{\Theta^{(-\theta)}, \varphi(X^{(-\theta)}{}^{,.}) \psi} - \int_0^T\!\! \psca{\varphi(X^{(-\theta),s}) \psi_s, \dd W_s}}}.\label{Diff2} \end{align}
\end{itemize} \end{Pro}
\noindent{\bf Remarks.} $\bullet$ For practical implementation, the ``finite dimensional" statement is the only result of interest since it ensures that $\argmin_{|E}\neq \emptyset$.
\noindent $\bullet$ If $p=q$ and $\varphi=I_q$, the ``infinite-dimensional" assumption is always satisfied.
\ni {\bf Proof.} $(a)$ As concerns the function $V$, we rely on Equality~(\ref{V3}). Set $r= 1+2/\eta$. Owing to the H\"older Inequality, showing that this function is finite on the whole space $L^2_{T,q}$ amounts to proving that \begin{equation*}
\esp[4]{e^{\frac{r}{2} \normLTp{q}{\Theta}^2 - r\int_0^T \langle\Theta_s ,\dd W_s\rangle}}
\le e^{\normsup{\varphi}^2 \normLTp{p}{\theta} r(r+1)/2} < +\infty. \end{equation*}
To show that $V$ goes to infinity at infinity, one proceeds as follows. Using the trivial equality \begin{equation*}
e^{-\int_0^T\!\psca{\Theta_s, \dd W_s} + \frac{1}{2}\normLTp{q}{\Theta}^2} = \\
\pa[4]{e^{-\frac{1}{2} \int_0^T\!\psca{\Theta_s, \dd W_s} + \frac{1}{8}\normLTp{q}{\Theta}^2}}^2
e^{\frac{1}{4} \normLTp{q}{\Theta}^2} \end{equation*} and the reverse H\"older inequality with conjugate exponents $(\frac{1}{3}, -\frac{1}{2})$ we obtain \begin{align*}
V(\theta) & \ge \esp[4]{F^{2/3}(X) e^{\frac{1}{12} \normLTp{q}{\Theta}}}^3 \esp[4]{e^{\frac{1}{2} \int_0^T\!\psca{\Theta_s,\dd W_s} - \frac{1}{8}\normLTp{q}{\Theta}^2}}^{-2}, \\
& \ge \esp[4]{F^{2/3}(X) e^{\frac{1}{12} \normLTp{q}{\Theta}^2}}^3 \end{align*} by the martingale property of the Dol\'eans exponential. Let $\varepsilon > 0$ such that $\prob{F^2(X)\ge \varepsilon} > 0$. We have then $V(\theta) \ge \varepsilon^{1/3} \esp[3]{\indic[1]{F^2(X) \ge \varepsilon}e^{\frac{1}{12} \normLTp{q}{\Theta}}}^3$, and by the conditional Jensen inequality \begin{align*}
V(\theta) &\ge \varepsilon^{1/3} \esp[4]{\indic[1]{F^2(X) \ge \varepsilon}e^{\frac{1}{12} \espc[2]{\normLTp{q}{\Theta}^2}{F^2(X) \ge \varepsilon}}}^3\\
&= \esp[4]{\indic[1]{F^2(X) \ge \varepsilon}e^{\frac{1}{12} \prob{F^2(X) \ge \varepsilon} \esp[2]{\normLTp{q}{\Theta}^2 \indic[1]{F^2(X) \ge \varepsilon}}}}^3 . \end{align*}
Now \begin{equation*}
\esp{\normLTp{q}{\Theta}^2 \indic[1]{F^2(X) \ge \varepsilon}} = \int_0^T \theta(s)^* \esp{\varphi_s(X^s)^* \varphi_s(X^s) \indic[1]{F^2(X) \ge \varepsilon}} \theta(s) \dd s \ge 0. \end{equation*} The assumption~(\ref{nonatomic}) implies that, for every $\theta\!\in L^2_{T,p}$, \begin{equation*}
\int_0^T \theta(s)^*\,\esp{\varphi_s(X^s)^* \,\varphi_s(X^s) \indic[1]{F^2(X) \ge \varepsilon}} \theta(s) \dd s \ge \int_0^T \theta(s)^*\,\esp{\varphi_s(X^s)^* \,\varphi_s(X^s) \indic[1]{F^2(X) \ge 0}} \theta(s) \dd s > 0, \end{equation*} so that if $\theta$ runs over the compact sphere of a finite dimensional subspace $E$ of $L^2_{T,p}$ \begin{equation*}
\inf_{\normLTp{p}{\theta} = 1, \theta\in E}\int_0^T \theta(s)^*\, \esp{\varphi_s(X^s)^*\, \varphi_s(X^s) \indic[1]{F^2(X) \ge \varepsilon}} \theta(s) \dd s > 0, \end{equation*} so that \begin{equation*}
\lim_{\normLTp{p}{\theta} \to \infty, \theta\in E} \esp[3]{\normLTp{q}{\Theta}^2 \indic[1]{F^2(X) \ge \varepsilon}} = +\infty, \end{equation*} and one concludes by Fatou's Lemma using that $\prob{F^2(X) \ge \varepsilon} > 0$. The second claim easily follows from Assumption~(\ref{General}).
\ms \ni $(b)$ As a first step, we show that the random functional $\Phi(\theta):=\frac{1}{2} \normLTp{q}{\Theta} - \int_0^T \psca{\Theta(s), \dd W_s}$ from $L^2_{T,p}$ into $L^r({\math P})$ ($r\!\in [1,\infty)$), is differentiable. Indeed, it from the below inequality, \begin{equation}\label{diffL2Lp}
\forall\, \theta,\, \psi\!\in L^2_{T, p}, \qquad \abs{\Phi(\theta+\psi) - \Phi(\theta) - \pscaLTp{p}{\DD \Phi(\theta), \psi}} \le \normsup{\varphi}^2 \normLTp{p}{\theta} \normLTp{p}{\psi} \end{equation} where $\psi\mapsto \pscaLTp{p}{\DD \Phi(\theta), \psi} = \int_0^T\!\psca{\varphi_s(X^s) \theta(s), \varphi_s(X^s) \psi(s)} \dd s - \int_0^T\!\psca{\varphi_s(X^s) \psi(s), \dd W_s}$ is clearly a bounded random functional from $L^2_{T,p}$ into $L^r({\math P})$, with an operator norm $\trnorm{\DD \Phi(\theta)}_{L^2_{T,p}, L^r({\math P})} \le \normsup{\varphi}^2 \normLTp{p}{\theta} + c_p \normsup{\varphi}$ ($c_p\!\in (0,+\infty)$ (this follows from H\"older and \emph{B.D.G.} inequalities).
Then, we derive that $\theta \mapsto e^{\Phi(\theta)}$ is differentiable form $L^2_{T,p}$ into every $L^r({\math P})$ with differential $e^{\Phi(\theta)} \DD \Phi(\theta)$. This follows from standard computation based on~(\ref{diffL2Lp}), the elementary inequality $|e^u-1-u|\le \frac 12 u^2(e^{u}+e^{-u})$ and the fact that \begin{align*}
\normLp[4]{r}{\abs[3]{\int_0^T \! \psca{\varphi_s(X^s) \psi(s), \dd W_s}} e^{\int_0^T\!\psca{\phi(X^s) \theta(s),\dd W_s}}}
& \le \normLp[4]{2r}{\int_0^T\!\psca{\varphi_s(X^s)\psi(s), \dd W_s}} \normLp[4]{2p}{e^{\int_0^T\!\psca{\varphi_s(X^s) \theta(s),\dd W_s}}}, \\
& \le c_p \normsup{\varphi} \normLTp{p}{\theta} \normLTp{p}{\psi}, \end{align*} where we used both H\"older and \emph{B.D.G.} inequality.
One concludes that $\theta\mapsto V(\theta)= \esp{F(X)^2e^{\Phi(\theta)}}$ is differentiable by using the $(L^2_{_{T,p}},L^r({\math P}))$--differentiability of $e^{\Phi(\theta)}$ with $r=1+\frac{\eta}{2}$.
The second form of the gradient is obtained by a Girsanov transform using Lemma~\ref{Girs}. $\cqfd$
\subsection{Design of the algorithm}
In view of a practical implementation of the procedure we are lead to consider some non trivial finite dimensional subspaces $E$ of $L^2_{T,p}$. The function $V$ being strictly $\log$-convex on $E$ and going to infinity as $\normLTp{p}{\theta}$ goes to infinity, $\theta \!\in E$, the restriction of $V$ on $E$ attains a minimum $\theta^*_E$ which {\em de facto} becomes the target of the procedure. Furthermore, for every $\theta \!\in E$, $\DD V_{|E}(\theta)= \DD V(\theta)_{|E}$ and the quadratic function $L(\theta):= \normLTp{p}{\theta-\theta^*_{_E}}$ is a Lyapunov function for the problem.
Like for the static framework investigated in Section~\ref{translation}, our algorithm will be based on the representation~(\ref{Diff2}) for the differential $\DD V$ of $V$: in this representation the variance reducer $\theta$ appears inside the functional $F$ which makes easier a control at infinity in order to prevent from any early explosion of the procedure. However, to this end we need to control the discrepancy between $X$ and $X^{(-\theta)}$. This is the purpose of the following Lemma.
\begin{Lem} \label{XmoinsXtheta} Assume $(H_{b,\sigma})$ holds. Let $\varphi$ be a bounded Borel ${\cal M}(q, p)$-valued function defined on $[0,T] \times {\cal C}([0,T], {\math R}^d)$, let $\theta \!\in L^2_{T,p}$ and let $X$ and $X^{(\theta)}$ denote a strong solutions of $E_{b,\sigma,W}$ and $E_{b+\sigma\Theta,\sigma, W}$ driven by the same Brownian motion. Then, for every $r \ge 1$, there exists a real constant $C_{b,\sigma}>0$ such that \begin{equation} \label{thetasigma}
\normLp[4]{r}{\sup_{t \in [0,T]} \abs[2]{X_t - X^{(\theta)}_t}} \le C_{b,\sigma} e^{C_{b, \sigma} T} \normLp[4]{r}{\int_0^T \!\! \abs[2]{\sigma(s, X^{(\theta),s}) \Theta^{(\theta)}_s} \dd s}.
\end{equation} \end{Lem}
\ni{\bf Proof.} The proof follows the lines of the proof of the strong rate of convergence of the Euler scheme (see~$e.g.$~\cite{BOLE}). $\cqfd$
The main result of this section is the following theorem. \begin{Thm} Suppose that Assumption~(\ref{nonvide}) and $(H_{b,\sigma})$ hold.
Let $\varphi$ be a bounded Borel ${\cal M}(q, p)$-valued function (with $p \ge 1$) defined on $[0,T] \times {\cal C}([0,T], {\math R}^d)$, and let $F$ be a functional $F$ satisfying \begin{equation} \tag{$G_{F,\lambda}$}
\forall\, x\!\in {\cal C}([0,T],{\math R}^d),\qquad \abs{F(x)} \le C_{_F}(1+ \normsup{x}^\lambda) \end{equation} for some positive exponent $\lambda>0$ (then $F(X)\!\in L^r({\math P})$ for every $r>0$). Let $E$ be a finite dimensional subspace of $L^2_{T,p}$ spanned by an orthonormal basis $(e_1,\ldots,e_m)$.
Let $\eta > 0$. We define the algorithm by \begin{equation*}
\theta_{n+1} = \theta_n -\g_{n+1} H_{\lambda,\eta}(\theta_n,X^{(-\theta_n)}, W^{(n+1)}) \end{equation*} where $\gamma=(\gamma_n)_{n\ge 1}$ satisfies~(\ref{StepCond}), $(W^{(n)})_{n\ge 1}$ is a sequence of {\color{black}independent Brownian motions for which $X^{(-\theta_n)}={\cal G}(-\theta_n, W^{(n+1)})$ is a strong solution to $(E_{b - \sigma \Theta}, W^{(n+1)})$} and for every standard Brownian motion $W$, every ${\cal F}^W_t$-adapted ${\math R}^p$-valued process $\xi=(\xi_t)_{t\in [0,T]}$, \begin{equation*}
\pscaLTp{p}{H_{\lambda,\eta}(\theta,\xi,W), e_i} =
\Psi_{\lambda, \eta}(\theta, \xi) F^2(\xi) e^{\normLTp{q}{\Theta(.,\xi^.)}} \pa[3]{2 \pscaLTp[1]{q}{\Theta(.,\xi^.), \varphi(., \xi^.) e_i} - \int_0^T\!\!\psca{\varphi(s, \xi^s) e_i(s),\dd W_s}}
\end{equation*} where for $\eta > 0$ \begin{equation*}
\Psi_{\lambda, \eta}(\theta, \xi) = \begin{cases}
\frac{e^{- \normsup{\varphi}\normLTp{p}{\theta}}}{1+\normLTp{q}{\varphi(., \xi^.) \theta}^{2\lambda+\eta}}
& \text{if $\sigma$ is bounded}, \\
e^{- (\normsup{\varphi} + \eta)\normLTp{p}{\theta}} & \text{if $\sigma$ is unbounded}. \\ \end{cases} \end{equation*} Then the recursive sequence $(\theta_n)_{n \ge 1}$ a.s. converges toward an $\argmin V$-valued (squared integrable) random variable $\theta^*$. \end{Thm} {\color{black} \ni {\bf Remark.} For a practical implementation of this algorithm, we must have \emph{for all} Brownian motions $W^{(n+1)}$ a strong solution $X^{(-\theta_n)}$ of $(E_{b - \sigma \Theta}, W^{(n+1)})$. In particular, this is the case if the driver $\varphi$ is locally Lipshitz (in space) or if $X$ is the continuous Euler scheme of a diffusion with step $T/n$ (using the driver $\varphi(t, x^t) = f(t, x(\underline t))$).
Note that if $\varphi$ is continuous (in space) but not necessarily locally Lipshitz, the Euler scheme converges in law to the solution of the SDE. }
\ms \ni {\bf Proof.} When the diffusion coefficient $\sigma$ is bounded, it follows from Lemma~\ref{XmoinsXtheta} that, for every $r\ge 1$, \begin{equation*}
\normLp[4]{r}{\sup_{t \in [0,T]} \abs[2]{X_t - X^{(\theta)}_t}} \le C_{b,\sigma,T} \normsup{\varphi} \normLTp{p}{\theta} \normsup{\sigma}, \end{equation*} where $\normsup{\sigma} = \sup_{(t,x)\in [0,T]\times{\cal C}([0,T], {\math R}^d)} \norm{\sigma(t,x)}$.
First note that for every $\theta,\,\psi\!\in E$, the mean function $h$ of the algorithm reads \begin{equation*}
\pscaLTp{p}{h(\theta), \psi} = \esp{\pscaLTp{p}{H_{\lambda,\eta}(\theta, X^{(-\theta)}, W), \psi}} =
\esp{\frac{e^{- \normsup{\varphi}\normLTp{p}{\theta}}}{1+\normLTp{q}{\Theta^{(-\theta)}}^{2\lambda+\eta}} \pscaLTp{p}{\DD V_{|E}(\theta), \psi}}, \end{equation*} so that, for every $\theta\neq \theta^*_{_E}$, \begin{equation*}
\psca{h(\theta), \theta - \theta^*_{E}} = \esp{\frac{e^{- \normsup{\varphi}\normLTp{p}{\theta}}}{1+\normLTp{q}{\Theta^{(-\theta)}}^{2\lambda+\eta}} \pscaLTp{p}{\DD V_{|E}(\theta), \theta-\theta^*_E}} > 0. \end{equation*}
It remains to check that for every $i\!\in\{1,\ldots,m\}$, $\normLp{2}{H_{\lambda,\eta}(\theta, X^{(-\theta)}, W)} \le C\pa[2]{1 + \normLTp{p}{\theta}}$ to apply the Robbins-Zygmund Lemma which ensures the $a.s.$ convergence of the procedure (see Section \ref{argmin-target}). We first deal with the term $F(X^{(-\theta)})^2\int_0^T\!\psca{\varphi_s(X^{(-\theta),s}) e_i(s), \dd W_s}$. Let $\eta'=\frac{\eta}{2\lambda}>0$. \begin{align*}
\normLp{2}{F(X^{(-\theta)})^2\int_0^T\!\psca{\varphi_s(X^{(-\theta),s}) e_i(s), \dd W_s}}
& \le \normLp{2+\eta'}{F(X^{(-\theta)})^2} \normLp{2(1+1/\eta')}{\int_0^T\!\psca{\varphi_s(X^{(-\theta),s}) e_i(s), \dd W_s}}, \\
& \le \normLp{2+\eta'}{F(X^{(-\theta)})^2} \normLp{1+1/\eta'}{\int_0^T\!\abs{\varphi_s(X^{(-\theta),s}) e_i(s)}^2 \dd s}, \\
& \le \normLp{2+\eta'}{F(X^{(-\theta)})^2} \normsup{\varphi}. \end{align*} Now \begin{align*}
\normLp{2(1+\eta')}{F(X^{(-\theta)})^2}
& \le C \pa{1 + \normLp[2]{4\lambda(1+\eta')}{\normsup[2]{X^{(-\theta)}}}^{2\lambda(1+\eta')}}, \\
& \le C_{\lambda,b,\sigma,T} \pa{1 + \normLp[2]{4\lambda(1+\eta')}{\normsup[2]{X}}^{2\lambda(1+\eta')}+ \normLTp{p}{\theta}^{2\lambda(1+\eta')} \normsup{\varphi}^{2\lambda(1+\eta')} \normsup{\sigma}^{2\lambda(1+\eta')}}, \\
& \le C_{\lambda,b,\sigma,\varphi,T} \pa{1+\normLTp{p}{\theta}^{2\lambda+\eta}}. \end{align*} One shows likewise that \begin{equation*}
\normLp{2}{F(X^{(-\theta)})^2} \le C_{\lambda,b,\sigma,\varphi, T} \pa{1+\normLTp{p}{\theta}^{2\lambda}}. \end{equation*} Combining theses estimates shows that $H_{\lambda,\eta}(\theta, X^{(-\theta)}, W)$ satisfies the linear growth assumption in $L^2({\math P})$.
\ms \ni If $\sigma$ is unbounded it follows from Assumption~\eqref{Hbs} that, for every $(t,x)\!\in [0,T]\times {\cal C}([0,T],{\math R}^d)$, \begin{equation*}
\norm{\sigma(t,x)} \le C_{\sigma} \pa{1+\normsup{x}}. \end{equation*} Elementary computation based on~\eqref{thetasigma} and Lemma~\ref{Girs} yield \begin{align*}
\normLp{r}{\int_0^T \abs{\sigma(s, X^{(\theta),s}) \Theta^{(\theta)}_s}}
& \le C_\sigma \normLp{L^1_{T,p}}{\theta} \normsup{\varphi} \pa{1 + \normLp{r}{\normsup{X} e^{-\frac{r}{2} \normLTp{q}{\frac{\Theta}{r}}^2 + \int_0^T\!\psca{\frac{\Theta_s}{r}, \dd W_s}}}}, \\
& \le C_\sigma \normLp{L^1_{T,p}}{\theta} \normsup{\varphi} \pa{1 + e^{\frac{\normsup{\varphi}}{2 r r'} \normLTp{p}{\theta}^2} \normLp{r(1+r')}{\normsup{X}}}, \\
& \le C_{r,b,\sigma,\varphi} \normLTp{p}{\theta}, \end{align*}
for every $r>0$ (Assumption~\eqref{Hbs} implies that $\normLp{r}{\normsup{X}}<+\infty$ for every $r>0$). Following the same proof to the bounded case, we obtain easily the results with $\Psi_{\lambda, \eta}(\theta, \xi) = \frac{e^{- (\normsup{\varphi}+\eta) \normLTp{p}{\theta}}}{1+\normLTp{q}{\varphi(., \xi^.) \theta}^{2\lambda+\eta}}$. We conclude by noting that $\eta$ is an arbitrary parameter to cancel the denominator. $\cqfd$
\ms \ni {\bf Remark.} If the functional $F$ is bounded ($\lambda = 0$), we prove in the same way that the algorithm without correction, \emph{i.e.} build with $\Psi_{\lambda, \eta} = 1$, a.s. converges.
\section{Additional remarks} For the sake of simplicity we focus in this section on importance sampling by mean translation in a finite dimensional setting (Section~\ref{translation}) although most of the comments below can also be applied at least in the path-dependent diffusions setting.
\subsection{Purely adaptive approach} As proved by Arouna (see \cite{ARO}), we can consider a purely adaptive approach to reduce the variance. It consists to perform the Robbins-Monro algorithm simultaneously with the Monte Carlo approximation. More precisely, estimate $\esp{F(X)}$ by \begin{equation*}
S_N = \frac{1}{N} \sum_{k=1}^N F(X_k + \theta_{k-1}) \frac{p(X_k + \theta_{k-1})}{p(X_k)} \end{equation*} where $X_k$ is the \emph{same innovation} as that used in the Robbins-Monro procedure $\theta_k = \theta_{k-1} - \gamma_k H(\theta_{k-1}, X_k)$. This adaptive Monte Carlo procedure satisfies a Central Limit Theorem with the optimal asymptotic variance \begin{equation*}
\sqrt{N} \pa{S_N - \esp{F(X)}} \xrightarrow{\cal L} {\cal N}(0, \sigma^2_*), \quad \text{whith} \quad \sigma^2_* = V(\theta^*) - \esp{F(X)}^2. \end{equation*}
This approach can be extended to the Esscher transform when we use the same innovation $\xi_k$ (see \eqref{ReXEsscher}) for the Monte Carlo procedure (computing $X^{(\theta_{k-1})}_k = g(\theta_{k-1}, \xi_k)$) and the Robbins-Monro algorithm (computing $X^{(-\theta_{k-1})}_k = g(-\theta_{k-1}, \xi_k)$). Likewise in the functional setting we can combine the variance reduction procedure and the Monte Carlo simulations using the same Brownian motion.
In practice, it is not clear that this adaptive Monte Carlo is better than the naive two stage procedure: performing first Robbins-Monro with a small number of iterations (to get a rough estimate $\theta^*$), then performing the Monte Carlo simulations with this optimized parameter.
\subsection{Weak rate of convergence: Central Limit Theorem (CLT)}
As concerns the rate of convergence, once again this a regular stochastic algorithm behaves as described in usual Stochastic Approximation Theory textbooks like~\cite{KUYI},~\cite{BEMEPR},~\cite{DUF}. So, as soon as the optimal variance reducer set is reduced to a single point $\theta^*$, the procedure satisfies under quite standard assumptions a $CLT$. We will not enter into technicalities at this stage but only try to emphasize the impact of a renormalization factor $g(\theta)$ like $g(\theta):=e^{-\frac{\lambda}{2}|\theta|^b}$ or $g(\theta):=\frac{1}{1+\widetilde F(-\theta)^2}$ induced by the function $F$ on the ``final" rate of convergence of the algorithm toward $\theta^*$. We will assume that $d=1$ and that $X\stackrel{d}{=} {\cal N}(0;1)$ for the sake of simplicity. One can write \begin{equation*}
H(\theta,x)= g(\theta)H_0(\theta,x) \quad\mbox{ where }\quad H_0(\theta,x)=F^2(x-\theta) (2\theta-x) \end{equation*} The function $H_0$ corresponds to the case of a bounded function $F$ (then $\lambda=0$). Under simple integration assumptions, one shows that $V$ is twice differentiable and that \begin{equation*}
V''(\theta) = e^{\frac{|\theta|^2}{2}}\esp{F^2(X)e^{-\theta X}\pa{1+(\theta-X)^2}} \end{equation*}
Consequently the mean functions $h$ and $h_0$ related to $H$ and $H_0$ which read respectively \[
h(\theta)= g(\theta) e^{-|\theta|^2} V'(\theta) \quad\mbox{ and
}\quad h_0(x) = e^{-|\theta|^2} V'(\theta) \] are differentiable at $\theta^*$ and \[
h'(\theta^*)= g(\theta^*) e^{-|\theta^*|^2} V'(\theta^*)\quad\mbox{
and }\quad h_0'(\theta^*)= e^{-|\theta^*|^2} V'(\theta^*) \]
Now, general results about CLT say that if $\g_n =\frac{\alpha}{\beta+n}$, $\alpha,\, \beta>0$ with \begin{equation}\label{CondTCL} \alpha > \frac{1}{2h'(\theta^*)}= \frac{1}{2g(\theta^*) h_0'(\theta^*)} \end{equation} then \[ \sqrt{n}(\theta_n-\theta^*)\stackrel{{\cal L}_{stably}}{\longrightarrow} {\cal N}(0;\Sigma_{\alpha}^*) \] where \begin{equation}\label{asympVar} \Sigma_{\alpha}^*={\rm Var}(H(y^*,Z))\frac{\a^{2}}{2\a h'(y^*)-1}. \end{equation} The mapping $\alpha\mapsto \Sigma_{\alpha}$ reaches its minimum at $\alpha^* = \frac{1}{h'(\theta^*)}=\frac{1}{g(\theta^*) h_0'(\theta^*)}$ leading to the minimal asymptotic variance \[ \Sigma^*=\Sigma^*_{\alpha^*}= \frac{{\rm Var}(H(y^*,Z))}{h'(y^*)^2}= \frac{\esp{H_0(y^*,Z)^2}}{h_0'(y^*)^2}= \frac{\esp{F^4(X)(\theta^*-X)^2e^{-\theta^*X}}}{\esp{F^2(X)(X^2-\theta^*X+1)}^2} \] by homogeneity.
So the optimal rate of convergence of the procedure is not impacted by the use of the normalizing function $g(\theta)$. However, coming back to condition~(\ref{CondTCL}), we see that this assumption on the coefficient $\alpha$ is more stringent since $\frac{1}{g(\theta^*)}>1$ (in practice this factor can be rather large). Consequently, given the fact that $g(\theta^*)$ is unknown to the user, this will induce a blind choice of $\alpha$ biased to higher values. With the well-known consequence in practice that if $\alpha$ is too large the ``\emph{CLT} regime'' will take place later than it would with smaller values. One solution to overcome this contradiction can be to make $\alpha$ depend on $n$ and slowly decrease.
As a conclusion, the algorithm never explodes (and converges) even for strongly unbounded functions $F$ which is a major asset compared to the version of the algorithm based on repeated projections. Nevertheless, the normalizing factor which ensures the non-explosion of the procedure may impact the rate of convergence since it has an influence on the tuning of the step sequence (which is always more or less ``blind" since it depends on the target $\theta^*$. In fact, we did not meet such difficulty in our numerical experiments reported below.
One classical way to overcome this problem can be to introduce the empirical mean of the algorithm implemented with a slowly decreasing step ``\`a la Rupert \& Poliak" (see $e.g.$~\cite{PEL}): Set $\g_n=\frac{c}{n^{r}}$, $\frac 12<r<1$ and \[ \bar \theta_{n+1} :=\frac{\theta_0+\cdots+\theta_{n}}{n+1}= \bar \theta_n -\frac{1}{n+1}(\bar \theta_n-\theta_{n}),\quad n\ge 0 \] where $(\theta_n)_{n\ge 0}$ denotes the regular Robbins-Monro algorithm defined by~(\ref{AlgoRM}) starting at $\theta_0$. Then $(\bar \theta_n)_{n\ge 0}$ converges toward $\theta^*$ and satisfies a CLT with the optimal asymptotic variance~(\ref{asympVar}). See also a variant based on a gliding window developed in~\cite{LEL}.
\subsection{Extension to more general sets of parameters} \label{extension} In many applications (see below with the Spark spread options with the NIG distribution) the natural set of parameters $\Theta$ is not ${\math R}^q$ but an open connected subset of ${\math R}^q$. Nevertheless, as illustrated below, our unconstrained approach still works provided one can proceed a diffeomorphic change of parameter by setting \[ \theta= T(\tilde \theta), \quad \theta \!\in \Theta \]
where $T: {\math R}^q \to \Theta$ is a ${\cal C}^1$-diffeomorphism with a bounded differential ($i.e.$ $\sup_{\tilde \theta} |\!\| DT(\tilde \theta)|\!\|<+\infty$). As an illustration, let us consider the case where the state function $H(\theta,X)$ of the procedure is designed so that $h(\theta):= {\math E}(H(\theta,X))=\rho(\theta) \nabla V(\theta)$ where $V$ is the objective function to be minimized over $\Theta$ and $\rho $ is a bounded {\em positive} Borel function. Then, one replaces $H(\theta,X)$ by $\widetilde H(\tilde\theta,X):=DT(\tilde\theta).H(T(\tilde \theta),X)$ and defines recursively a procedure on ${\math R}^q$ by \[ \tilde \theta_{n+1} = \tilde \theta_{n} -\g_{n+1}\widetilde H(\tilde \theta_n,X_{n+1}). \] In order to establish the $a.s.$ convergence of $\theta_n:=T(\tilde \theta_n)$ to $\argmin V$, one relies on a variant of Robbins-Monro algorithm, namely a stochastic gradient approach (see~\cite{DUF,KUYI} for further details): one defines $U(\tilde\theta)= V(T(\tilde \theta))$ which turns out to be a Lyapunov function for the new algorithm since \[
\langle \nabla U(\tilde \theta), {\math E}(DT(\tilde\theta)H(T(\tilde \theta),X))\rangle = \rho(T(\tilde \theta)) |\nabla U(\tilde \theta)|^2>0\quad \mbox{ on }\quad T^{-1}(\{\nabla V\neq0\}). \]
If $U$ satisfies $\|\widetilde H(\tilde \theta,X)\|_{_2}+|\nabla U(\tilde \theta)
|\le C(1+U(\tilde \theta))^{\frac 12}$ (which is a hidden constraint on the choice of $T$), one shows under the standard ``decreasing" assumption on the step sequence that $U(\tilde \theta_n)\to U_{\infty}\!\in L^1({\math P})$ and $\sum_n\g_{n+1} \rho(\tilde \theta_n) |\nabla U(\tilde \theta_n)|^2<+\infty$.
If $\ds \lim_{\theta\to \partial \Theta}V(\theta)=+\infty$ or $\ds \liminf_{\theta\to \partial \Theta}\rho(T(\theta))|\nabla V( \theta)|^2>0$, one easily derives that ${\rm dist}(\theta_n, \{\nabla V=0\}) \to 0$ $a.s.$ as $n\to\infty$.
\section{Numerical illustrations} \label{numerical} \subsection{Multidimensional setting: the NIG distribution} First we consider a simple case to compare the two algorithms of Section~\ref{dimfinie}. The quantity to compute is \begin{equation*}
\esp{F(X)} = \int_{{\math R}} F(x) \, p_{\NIG}(x;\alpha,\beta,\delta,\mu) \dd x, \end{equation*} where $p_{\NIG}(x; \alpha,\beta,\delta,\mu)$ is the density of $X$ a normal inverse gaussian (NIG) random variable of parameters $(\alpha,\beta,\delta,\mu)$ i.e. $\alpha > 0$, $\abs{\beta} \le \alpha$, $\delta > 0$, $\mu \in {\math R}$, \begin{equation*}
p_{\NIG}(x;\alpha,\beta,\delta,\mu) = \frac{\alpha \delta K_1\pa{\alpha \sqrt{\delta^2 +(x-\mu)^2}}}{\pi\sqrt{\delta^2 +(x-\mu)^2}} e^{\delta \gamma + \beta(x-\mu)}, \\ \end{equation*} where $K_1$ is a modified Bessel function of the second kind and $\gamma = \sqrt{\alpha^2 - \beta^2}$.
We can summarize the two algorithms presented in section \ref{dimfinie}, more precisely the variance reduction based on translation of the density (see Subsection \ref{translation}) and the one based on the Esscher transform (see Subsection \ref{Esscher}), by the following simplified (no computation of the variance) pseudo-code: \begin{center} \begin{minipage}{0.48\textwidth} \centering{Translation (see \ref{translation})} \begin{lstlisting}[frame= single] for n = 0 to M do
X ~ NIG(alpha, beta, mu, delta)
theta = theta - 1/(n+1000)*H1(theta, X) for n = 0 to N do
X ~ NIG(alpha, beta, mu, delta)
mean = mean + F(X) * p(X+theta)/p(X)
\end{lstlisting} \end{minipage}\hspace{1em} \begin{minipage}{0.48\textwidth} \centering{Esscher transform (see \ref{Esscher})} \begin{lstlisting}[frame=single] for n = 0 to M do
X ~ NIG(alpha, beta-theta, mu, delta)
theta = theta - 1/(n+1000)*H2(theta, X) for n = 0 to N do
X ~ NIG(alpha, beta+theta, mu, delta)
mean = mean + F(X) * exp(-theta*X) mean = mean * exp(psi(theta)) \end{lstlisting} \end{minipage}\end{center}
\begin{itemize} \item[--]\emph{Translation case.} We consider the function $H_1$ of the Robbins-Monro procedure of the first algorithm defined by \begin{equation*}
H_1(\theta, X) = e^{-2\abs{\theta}} F^2(X) \frac{p'(X-2\theta)}{p(X)} \pa{\frac{p(X-\theta)}{p(X-2\theta)}}^2, \end{equation*} where an analytic formulation of the derivative $p'$ is easily obtained using the relation on the modified Bessel function $K_1'(x) = \frac{1}{x} K_1(x) - K_2(x)$.
The assumption \eqref{H2} is satisfied with $a = 1$, and our results of Subsection \ref{translation} apply.
\item[--] \emph{Esscher transform.} In the Esscher approach we consider the function $H_2$ defined by \begin{equation*}
H_2(\theta, X) = e^{-\abs{\theta}} F^2(X) \pa{\nabla \psi(\theta) - X}. \end{equation*} Note that $\psi$ is not well defined for every $\theta \in {\math R}^d$. Indeed, the cumulant generating function of the NIG distribution is defined by \begin{equation*}
\psi(\theta) = \mu \theta + \delta\pa{\gamma - \sqrt{\alpha^2 - (\beta + \theta)^2}}, \end{equation*} for every $\theta \in (-\alpha-\beta, \alpha-\beta)$. Moreover, we need $\psi(-\theta)$ to be well defined \emph{i.e.} $\theta \in (-\alpha+\beta, \alpha+\beta)$. To take account of these restrictions, we slightly modify the algorithm parametrization (see Subsection \ref{extension}) $\theta = T(\tilde\theta) := (\beta-\alpha) \frac{\tilde \theta}{\sqrt{1+\tilde\theta^2}}$, and update $\tilde{\theta}\in{\math R}$ in the Robbins-Monro procedure (multiply the function $H_2(T(\tilde\theta), X)$ by the derivative $T'(\tilde\theta) = \frac{\beta-\alpha}{(1+\tilde\theta^2)^{3/2}}$). \end{itemize}
The payoff $F$ is a Call option of strike $K$, $F(X) = 50(e^X-K)_+$. The parameters of the NIG random variable $X$ are $\alpha=2$, $\beta=0.2$, $\delta=0.8$ and $\mu=0.04$. The variance reduction obtained for different value of $K$ are summarized in the tabular \ref{tab-trans-esscher}. The number of iterations in the Robbins-Monro variance reduction procedure is $M=100\,000$ and the number of Monte Carlo iterations is $N=1\,000\,000$. Note that for each strike, the prices are computed using the same pseudo-random number generator initialized with the same \emph{seed}. \begin{table}[ht!]
\begin{center}
\begin{tabular}[t]{l|cc|cr|cr}
\multirow{2}*{K} & \multirow{2}*{mean} & \multirow{2}*{crude var} & var. ratio. & & var. ratio & \\
& & & translation & ($\theta$) & Esscher & ($\theta$) \\
\hline 0.6 & 42.19 & 8538 & 5.885 & (0.791) & 56.484 & (1.322) \\ 0.8 & 34.19 & 8388 & 7.525 & (0.903) & 39.797 & (1.309) \\ 1.0 & 27.66 & 8176 & 9.218 & (0.982) & 32.183 & (1.294) \\ 1.2 & 22.60 & 7930 & 10.068 & (1.017) & 29.232 & (1.280) \\ 1.4 & 18.76 & 7677 & 9.956 & (1.026) & 28.496 & (1.268) \\
\end{tabular}
\caption{Variance reduction for different strikes (one dimensional NIG example).}
\label{tab-trans-esscher} \end{center} \end{table}
To complete this numerical example, Figure \ref{fig-trans-esscher} illustrates the densities obtained after the Rob\-bins-Monro procedure. The deformation provided by the Esscher transform is very impressive in this example. We remark that the Esscher transform modifies the parameter $\beta$ which controls the asymmetric shape of the NIG distribution. \begin{figure}
\caption{Densities of $X$ (crude), $X+\theta$ (translation) and $X^{(\theta)}$ (Esscher) in the case $K=1$.}
\label{fig-trans-esscher}
\end{figure} \subsubsection*{Spark spread} We consider now a exchange option between gas and electricity (called spark spread). We choose to model the price of the energy by the exponential of a $NIG$ distribution. A simplified form of the payoff is then \begin{equation*}
F(X) = 50 (e^{X^{elec}} - c e^{X^{gas}} - K)_+, \end{equation*} where $X^{elec} \sim NIG(2,0.2,0.8,0.04)$ and $X^{gas} \sim NIG(1.4,0.2,0.2,0.04)$ are independent.
The results obtained for different strikes after $300\,000$ iterations of the Robbins-Monro procedure and $3\,000\,000$ iterations of Monte Carlo, are summarized in the Table \ref{tab-spark-spread}. \begin{table}[ht!]
\begin{center}
\begin{tabular}[t]{ll|cccc}
\multirow{2}*{K} & \multirow{2}*{c} & \multirow{2}*{mean} & \multirow{2}*{crude var} & var. ratio. & var. ratio \\
& & & & translation & Esscher \\
\hline
0.4 & 0.2 & 41.021 & 8540.6 & 5.0118 & 25.171 \\
& 0.4 & 32.719 & 8356.9 & 5.1338 & 27.006 \\
& 0.6 & 26.337 & 8112.2 & 4.9752 & 28.062 \\
& 0.8 & 21.556 & 7845.3 & 4.7569 & 29.964 \\
& 1 & 17.978 & 7582 & 4.5575 & 32.849 \\
0.6 & 0.2 & 33.235 & 8378.4 & 5.2609 & 27.455 \\
& 0.4 & 26.534 & 8133.3 & 5.0604 & 28.669 \\
& 0.6 & 21.587 & 7862.7 & 4.8046 & 30.649 \\
& 0.8 & 17.931 & 7595.2 & 4.5839 & 33.656 \\
& 1 & 15.184 & 7344.2 & 4.4064 & 37.489 \\
0.8 & 0.2 & 26.908 & 8160.1 & 5.1366 & 28.876 \\
& 0.4 & 21.725 & 7884.9 & 4.844 & 31.018 \\
& 0.6 & 17.955 & 7612.5 & 4.6031 & 34.166 \\
& 0.8 & 15.156 & 7357.3 & 4.416 & 38.167 \\
& 1 & 13.027 & 7123.9 & 4.2685 & 42.781 \\
\end{tabular}
\caption{Variance reduction for different strikes (spark spread example).}
\label{tab-spark-spread} \end{center} \end{table}
\subsection{Functional setting: Down \& In Call option} We consider a process $(X_t)_{t \ge 0}$ solution of the following diffusion \begin{equation*}
\dd X_t = b(X_t) \dd t + \sigma(X_t) \dd W_t, \quad X_0 = x_0 \in {\math R}. \end{equation*} A Down \& In Call option of strike $K$ and barrier $L$ is a Call of strike $K$ which is activated when the underlying $X$ moves down and hits the barrier $L$. The payoff of such a European option is defined by \begin{equation*}
F(X) = (X_T - K)_+ \indic[3]{\ds \min_{0 \le t \le T} X_t \le L}. \end{equation*} A naive Monte Carlo approach to price this option is to consider an Euler-Maruyama scheme $\bar{X}=(\bar{X}_{t_k})_{k\in\{0,\dots,n\}}$ to discretize $X$ and to approximate $\ds \min_{0 \le t \le T} X_t$ by $\ds \min_{k \in \{0,\dots,n\}} \bar{X}_{t_k}$. It is well known that this approximation of the functional payoff is poor. More precisely, the weak order of convergence cannot be greater than $\frac{1}{2}$ (see \cite{GOB}).
A standard approach is to consider the continuous Euler scheme $\bar{X}^c$ obtained by extrapolation of the Brownian between two instants of discretization. More precisely, for every $t \in [{t_k}, {t_{k+1}}]$, \begin{equation*}
\bar{X}^c_t = \bar{X}^c_{t_k} + b(\bar{X}^c_{t_k}) (t-{t_k}) + \sigma(\bar{X}^c_{t_k})(W_t - W_{t_k}), \quad \bar{X}^c_0 = x_0 \in {\math R}. \end{equation*} By preconditioning, \begin{equation} \label{MC_brownian_interpolation}
\esp{F(X)} = \esp{(\bar{X}_T - K)_+ \pa{1 - \prod_{k = 0}^{N-1} p(\bar{X}_{t_k}, \bar{X}_{t_{k+1}})}}, \end{equation} with $\ds p(x_k, x_{k+1}) = \probc{\min_{t \in [{t_k}, {t_{k+1}}]} \bar{X}^c_t \ge L}{(\bar{X}_{t_k}, \bar{X}_{t_{k+1}}) = (x_k, x_{k+1})}$. Now using the Girsanov Theorem and the law of the Brownian bridge (see for example \cite{GLAS}), we have \begin{align}
\begin{split} \label{def-p}
p(x_k,x_{k+1}) &= 1 - \probc{\min_{t \in [0, t_1]} W_t \le \frac{L-x_k}{\sigma(x_k)}}{W_{t_1} = \frac{x_{k+1}-x_k}{\sigma(x_k)}}, \\
&= \begin{cases} 0 & \text{if $L\ge\min(x_k, x_{k+1})$}, \\
1-e^{-\frac{2(L-x_k)(L-x_{k+1})}{\sigma^2(x_k)({t_{k+1}}-{t_k})}}, & \text{otherwise}.
\end{cases}
\end{split} \end{align}
In the following simulations we consider an Euler scheme of step ${t_k} = k \frac{T}{n}$ with $n = 100$. \subsubsection*{Deterministic case (trivial driver $\varphi \equiv 1$)} We consider three different basis of $L^2([0,1], {\math R})$ \begin{itemize}
\item[--] a polynomial basis composed of the shifted Legendre polynomials $\tilde{P}_n(t)$ defined by
\begin{equation} \tag{ShLeg}\label{shift_legendre}
\forall n \ge 0, \forall t \in [0,1], \quad \tilde{P}_n(t) = P_n(2t - 1) \quad \text{where} \quad P_n(t) = \frac{1}{2^n n!} \frac{\dd^{\,n}}{\dd t^n}\pa{(t^2-1)^n}. \end{equation}
\item[--] the Karhunen-Lo\`eve basis defined by
\begin{equation} \tag{KL} \label{KL}
\forall n \ge 0, \forall t \in [0,1], \quad e_n(t) = \sqrt{2} \sin \pa{\pa[3]{n+\frac{1}{2}} \pi t} \end{equation}
\item[--] the Haar basis defined by
\begin{equation} \tag{Haar} \label{Haar}
\forall n \ge 0, \forall k = 0,\dots,2^n-1, \forall t \in [0,1], \quad \psi_{n,k}(t) = 2^{\frac{k}{2}} \psi(2^k t - n) \end{equation} where $\psi(t) = \begin{cases} 1 & \text{if $t \in [0, \frac{1}{2})$} \\
-1 & \text{if $t \in [\frac{1}{2}, 1)$} \\
0 & \text{otherwise}
\end{cases}$ \end{itemize}
\ms \ni \emph{Black\&Scholes Model} \\ First, we consider the classical Black\&Scholes model. We set the interest rate $r$ to $4\%$ and the volatility $\sigma$ to $70\%$ (which is a high volatility). The strike of the payoff $F$ is set at $K = 115$ and the barrier level at $L = 65$. A crude Monte Carlo (with Brownian bridge interpolation, see \eqref{MC_brownian_interpolation}) give a price of $2.596$ with a variance of $230$ after $500\,000$ trials. Note that the true price of this product is $2.554$.
For different basis, the results of our algorithm are summarized in the table \ref{tab-result-bs}. In the Robbins-Monro procedure, we define the step sequence by $\gamma_n = \frac{1}{n+10x_0^2}$ and set the number of iterations at $50\,000$. \begin{table}[ht!] \label{tab-result-bs} \centering
\begin{tabular}{lc|ccc} Basis & Dim. & Mean & ~ CI $95\%$ ~ & Variance ratio \\ \hline Constant & 1 & 2.5737 & $\pm$0.0230 & 3.4710 \\ \hline ShiftLegendre & 2 & 2.5741 & $\pm$0.0197 & 4.7225 \\ \eqref{shift_legendre} & 4 & 2.5717 & $\pm$0.0193 & 4.9478 \\ & 8 & 2.5717 & $\pm$0.0193 & 4.9494 \\ \hline Karhunen-Lo\`eve & 2 & 2.5678 & $\pm$0.0164 & 6.8644 \\ \eqref{KL} & 4 & 2.5729 & $\pm$0.0160 & 7.1851 \\ & 8 & 2.5705 & $\pm$0.0156 & 7.5218 \\ \hline Haar & 2 & 2.5657 & $\pm$0.0192 & 4.9710 \\ \eqref{Haar} & 4 & 2.5671 & $\pm$0.0163 & 6.9459 \\ & 8 & 2.5663 & $\pm$0.0155 & 7.6574 \\ \end{tabular} \caption{Variance ratio obtained for different basis in the Black\&Scholes model ($K = 115$, $L = 65$, variance of the crude Monte Carlo: 230).} \end{table}
In figure \ref{fig-theta} are depicted the optimal variance reducer when the optimization of $V$ is carried out on $E_m$ for several values of $m$ (2, 4 and 8) in the different basis mentioned above.
\begin{figure}
\caption{Optimal $\theta$ process obtained with different basis by our algorithm using $50\,000$ trials.}
\label{fig-theta}
\end{figure}
\ms \ni \emph{A local volatility Model} \\ To emphasize the generic feature of our algorithm we consider the same product in a local volatility model (inspired by the CEV model) defined by \begin{equation} \label{voloc-model}
\dd x_t = r x_t \dd t + \sigma x_t^\beta \frac{x_t}{\sqrt{1 + x_t^2}} \dd W_t, \end{equation} with $r = 0.04$, $\sigma = 7$ and $\beta = 0.5$.
The price of the Down \& In Call (strike 115, barrier 65) given by a crude Monte Carlo with Brownian interpolation after $500\,000$ trials is $3.194$ and the variance is $206.52$. \begin{table}[ht!] \label{tab-result-voloc} \centering
\begin{tabular}{lc|ccc} Basis & Dim. & Mean & ~ CI $95\%$ ~& Variance ratio \\ \hline Constant & 1 & 3.1836 & $\pm$0.0251 & 2.6297 \\ \hline ShiftLegendre & 2 & 3.1830 & $\pm$0.0223 & 3.3258 \\ \eqref{shift_legendre}& 4 & 3.1815 & $\pm$0.0215 & 3.5670 \\ & 8 & 3.1813 & $\pm$0.0215 & 3.5659 \\ \hline Karhunen-Lo\`eve & 2 & 3.1852 & $\pm$0.0187 & 4.7254 \\ \eqref{KL} & 4 & 3.1862 & $\pm$0.0183 & 4.9385 \\ & 8 & 3.1918 & $\pm$0.0178 & 5.2183 \\ \hline Haar & 2 & 3.1834 & $\pm$0.0215 & 3.5699 \\ \eqref{Haar} & 4 & 3.1871 & $\pm$0.0186 & 4.7896 \\ & 8 & 3.1864 & $\pm$0.0177 & 5.2675 \\ \end{tabular} \caption{Variance ratio obtained for different basis in the local volatility model \eqref{voloc-model} ($K = 115$, $L = 65$, variance of the crude Monte Carlo: 206.52).} \end{table}
\subsubsection*{Adaptive case (non-trivial driver)} We experiment now our algorithm with a non-trivial driver $\varphi$ defined for $t = {t_k}$ by \begin{equation*}
\varphi(t, \xi^t) = \pa[3]{\bar p_k \quad 1-\bar p_k}, \quad \text{with} \quad \bar p_k = \prod_{j=0}^{k-1} p(\xi_{t_j},\xi_{t_{j+1}}), \end{equation*} where $p$ is defined by \eqref{def-p}. Note that $\bar p_k = \probc{\min_{t \in [0, {t_k}]} \xi_t \ge L}{\xi_0,\dots,\xi_{t_k}}$ so that there is no extra-computation compared to the Brownian bridge interpolation.
We set $p = 2$ and $E = ({\math R} \ind_{[0,T]})^2$ so that the optimal parameter $\theta_{t_k} = \alpha \bar p_k + \beta (1-\bar p_k)$ with $(\alpha, \beta) \in {\math R}^2$. The results for different strikes and barrier levels are reported in Table \ref{result-adapt-bs} for the Black\&Scholes model and in Table \ref{result-adapt-voloc} for the local volatility model. The simulation parameters are unchanged.
\begin{table}[ht!] \label{tab-result-adapt-bs} \centering
\begin{tabular}{cc|ccrr|cc} Strike & Barrier & Mean & ~ CI 95$\%$ ~ & \multicolumn{2}{c}{Variance ratio (Crude)} & $\alpha$ & $\beta$ \\ \hline 85 & 65 & 2.5738 & $\pm$0.0115 &\hspace{1.5em} 13.49 & (16.56) & -0.1752 & 1.6685 \\ & 75 & 6.0489 & $\pm$0.0186 & 14.26 & (43.39) & 0.0493 & 1.9191 \\ \hline 95 & 65 & 2.5704 & $\pm$0.0110 & 14.64 & (15.26) & 0.0524 & 1.9987 \\ & 75 & 6.0492 & $\pm$0.0190 & 13.67 & (45.25) & 0.1557 & 2.0560 \\ & 85 & 11.5970 & $\pm$0.0301 & 12.23 & (112.92) & 0.4108 & 2.1226 \\ \hline 105 & 65 & 2.5687 & $\pm$0.0122 & 12.03 & (18.56) & 0.3888 & 2.1423 \\ & 75 & 6.0548 & $\pm$0.0206 & 11.66 & (53.08) & 0.3895 & 2.1720 \\ & 85 & 11.5953 & $\pm$0.0308 & 11.67 & (118.32) & 0.4524 & 2.1608 \\ & 95 & 19.2882 & $\pm$0.0348 & 17.17 & (151.04) & 0.6619 & 1.7910 \\ \hline 115 & 65 & 2.5706 & $\pm$0.0135 & 9.75 & (22.90) & 0.5473 & 1.8903 \\ & 75 & 6.0530 & $\pm$0.0211 & 11.16 & (55.42) & 0.4591 & 1.9371 \\ & 85 & 11.5976 & $\pm$0.0297 & 12.55 & (109.98) & 0.4807 & 2.0008 \\ & 95 & 19.2958 & $\pm$0.0347 & 17.21 & (150.67) & 0.7217 & 1.6380 \\ \end{tabular} \caption{Variance reduction for different strikes and barrier levels in the Black\&Scholes model.} \label{result-adapt-bs} \end{table}
\begin{table}[ht!] \label{tab-result-adapt-voloc} \centering
\begin{tabular}{cc|ccrr|cc} Strike & Barrier & Mean & ~ CI 95$\%$ ~ & \multicolumn{2}{c}{Variance ratio (Crude)} & $\alpha$ & $\beta$ \\ \hline 85 & 65 & 3.1827 & $\pm$0.0127 &\hspace{1.5em} 10.02 & (20.28) & -0.3057 & 1.5522 \\ & 75 & 6.4115 & $\pm$0.0190 & 9.96 & (45.03) & -0.1428 & 1.7985 \\ \hline 95 & 65 & 3.1846 & $\pm$0.0124 & 10.65 & (19.08) & -0.1141 & 1.9139 \\ & 75 & 6.4117 & $\pm$0.0199 & 9.07 & (49.42) & -0.0029 & 1.9814 \\ & 85 & 11.4478 & $\pm$0.0293 & 8.03 & (106.99) & 0.1898 & 1.8937 \\ \hline 105 & 65 & 3.1835 & $\pm$0.0135 & 8.98 & (22.65) & 0.1487 & 1.9628 \\ & 75 & 6.4120 & $\pm$0.0209 & 8.21 & (54.59) & 0.1493 & 2.0060 \\ & 85 & 11.4458 & $\pm$0.0295 & 7.88 & (108.94) & 0.2503 & 1.8737 \\ & 95 & 18.6060 & $\pm$0.0345 & 9.83 & (149.07) & 0.5594 & 1.4343 \\ \hline 115 & 65 & 3.1817 & $\pm$0.0148 & 7.38 & (27.54) & 0.3062 & 1.6884 \\ & 75 & 6.4112 & $\pm$0.0209 & 8.18 & (54.79) & 0.1928 & 1.8119 \\ & 85 & 11.4470 & $\pm$0.0289 & 8.24 & (104.16) & 0.2599 & 1.7430 \\ & 95 & 18.6061 & $\pm$0.0346 & 9.79 & (149.76) & 0.5755 & 1.4313 \\ \end{tabular} \caption{Variance reduction for different strikes and barrier levels in the local volatility model.} \label{result-adapt-voloc} \end{table}
\section{Appendix: proof of Theorem~\ref{ThmRZ}} We propose below the proof of the slight extension of the regular Robbins-Monro algorithm when $\{h=0\}$ is not reduced to a single equilibrium point. The key is still the convergence theorem for non negative super-martingales.
\bs \ni{\bf Proof.} Set ${\cal F}_n := \sigma(\theta_0,Z_1,\ldots,Z_n)$, $n\ge 1$. Let $\theta^*\!\in {\cal T^*}$. Then
\begin{align}
\notag \abs{\theta_{n+1}-\theta^*}^2 &= \abs{\theta_{n}-\theta^*}^2 - 2\g_{n+1} \psca{\theta_{n}-\theta^*, H(\theta_n,Z_{n+1})} + \g_{n+1}^2 \abs{H(\theta_n,Z_{n+1})}^2, \\ \label{IneqL2} & \le \abs{\theta_{n}-\theta^*}^2 - 2\g_{n+1} \psca{\theta_{n}-\theta^*,h(\theta_n)} -2\g_{n+1}\psca{\theta_{n}-\theta^*, \Delta M_{n+1}} + \g_{n+1}^2\abs{H(\theta_n,Z_{n+1})}^2, \end{align} where \begin{equation*}
\Delta M_{n+1} = H(\theta_n,Z_{n+1}) - \espc{H(\theta_n,Z_{n+1})}{{\cal F}_n} = H(\theta_n,Z_{n+1}) - h(\theta_n), \end{equation*}
is an increment of (local) martingale satisfying $\esp{|\Delta M_{n+1}|^2} \le C(1+\esp{|\theta_n-\theta^*|^2})$ owing to the assumptions on $H$ and Schwarz Inequality which also implies that \begin{equation*}
\esp{\abs{\psca{\theta_{n}-\theta^*, H(\theta_n,Z_{n+1}}}} \le \frac 12 \pa{\esp{\abs{\theta_n-\theta^*}^2} + \esp{\abs{H(\theta_n,Z_{n+1})}^2}}
\le C(1+\esp{\abs{\theta_n-\theta^*}^2}, \end{equation*}
for an appropriate real constant $C$. Then, one shows by induction on $n$ from~(\ref{IneqL2}) that $|\theta_n|$ is square integrable for every $n\ge 0$ and that $\Delta M_{n+1}$ is integrable, hence a true martingale increment. Now, one derives from the assumptions~(\ref{StepCond}) and~(\ref{IneqL2}) that \begin{equation*}
S_n = \frac{|\theta_n-\theta^*|^2 + 2\sum_{k=0}^{n-1}\g_{k+1} \psca{\theta_{k}-\theta^*, h(\theta_{k})} + C\sum_{k\ge n+1}\g_k^2}{\prod_{k=1}^n (1+C\g^2_k)}, \end{equation*}
is a (non negative) super-martingale with $S_0=|\theta_0-\theta^*|^2\!\in L^1({\math P})$. This uses the mean-reverting assumption~\eqref{RMmeanreverting}. Hence $S_n$ is ${\math P}$-$a.s.$ converging toward an integrable r.v. $S_{_\infty}$. Consequently, using that $\sum_{k\ge n+1}\g_k^2\to 0$, one gets \begin{equation}\label{CvRZ}
|\theta_n-\theta^*|^2 + 2\sum_{k=0}^{n-1}\g_{k+1} \psca{\theta_k-\theta^*,h(\theta_{k})} \stackrel{a.s.}{\longrightarrow} \widetilde S_{_\infty}=S_{_\infty}\prod_{n\ge1} (1+C_{_L}\g^2_n)\!\in L^1({\math P}). \end{equation}
The super-martingale $(S_n)$ being $L^1$-bounded, one derives likewise that $(|\theta_n-\theta^*|^2)_{n\ge 0}$ is $L^1$-bounded since \begin{equation*}
|\theta_n-\theta^*|^2\le \prod_{k=1}^n (1+C_{_L}\g^2_k) S_n, \quad n\ge 0. \end{equation*} Now, a series with nonnegative terms which is upper bounded by an ($a.s.$) converging sequence, $a.s.$ converges in ${\math R}_+$ so that \begin{equation*}
\sum_{n\ge 0} \g_{n+1} \psca{\theta_n-\theta^*, h(\theta_{n})} < +\infty \qquad {\math P}\mbox{-}a.s. \end{equation*}
It follows from~(\ref{CvRZ}) that, ${\math P}$-$a.s.$, $|\theta_n-\theta^*|^2\stackrel{n\to \infty}{\longrightarrow} L_{_\infty}$ which is integrable since $(|\theta_n-\theta^*|^2)_{n\ge0}$ is $L^1$-bounded and consequently $a.s.$ finite.
Let $L>0$. Set \begin{equation*}
\Omega_{_L}:= \left\{\omega\!\in \Omega,\,\forall\, n\ge 0, |\theta_n(\omega)-\theta^*|\le L\right\}. \end{equation*}
It follows from the $a.s.$ finiteness of $L_{_\infty}$ that $\bigcup_{L>0}\Omega_{_L} = \Omega$ $a.s.$. Now we consider the compact set $K_{_L}= {\cal T}^*\cap \bar B(0,L)$. It is separable so there exists an everywhere dense sequence in $K_{_L}$, denoted for convenience $(\theta^{*,k})_{k\ge 1}$. The above proof shows that ${\math P}$-$a.s.$, for every $k\ge 1$, $|\theta_n-\theta^{*,k}|^2\to \ L^k_{_\infty}<+\infty$ as $n\to \infty$. Then set \begin{equation*}
\Omega'_{_L}:=\left\{\omega\!\in \Omega_{_L},\; \abs{\theta_n(\omega)-\theta^{*,k}}^2 \stackrel{n\to \infty}{\to} L^k_{_\infty}(\omega), k\ge 1, \; \sum_{n\ge 1}\gamma_n \psca{\theta_{n-1}(\omega)-\theta^*, h(\theta_{n-1}(\omega)} <+ \infty\right\} \end{equation*} which satisfies ${\math P}(\Omega'_{_L})= {\math P}(\Omega_{_L})$. Assume $\omega\!\in \Omega'_{_L}$. Up to two successive extractions, there exists a subsequence $\theta_{\phi(n,\omega)}$ such that \begin{equation*}
\psca{\theta_{\phi(n,\omega)}-\theta^{*}, h(\theta_{\phi(n,\omega)}(\omega))} \stackrel{n\to \infty}{\longrightarrow} 0\qquad\mbox{ and } \qquad \theta_{\phi(n,\omega)}(\omega) \stackrel{n\to \infty}{\longrightarrow}\theta_{_\infty}(\omega). \end{equation*} The function $h$ being continuous $\psca{\theta_{_\infty}(\omega)-\theta^{*}, h(\theta_{_\infty}(\omega))}=0$ which implies that $\theta_{_\infty}(\omega)\!\in\{h=0\}$. Hence $\theta_{_\infty}(\omega)\!\in K_{_L}$. Then any limiting value $\theta'_{_\infty}(\omega)$ of the sequence $(\theta_n(\omega))_{n\ge 1}$ will satisfy \begin{equation*}
\forall\, k\ge 1,\quad |\theta'_{_\infty}(\omega)-\theta^{*,k}|= |\theta_{_\infty}(\omega)-\theta^{*,k}|= \sqrt{L^k_{_\infty}(\omega)} \end{equation*} which in turn implies that $\theta'_{_\infty}(\omega)=\theta_{_\infty}(\omega)$ by considering a subsequence $\theta^{*,k'}\to \theta_{_\infty}(\omega)$. So, $\theta_{_\infty}(\omega)$ is the unique limiting value of the sequence $(\theta_n(\omega))_{n\ge 0}$ $i.e.$ $\theta_n(\omega)\to \theta_{_\infty}(\omega)$ as $n\to \infty$. The fact that the resulting random vector $\theta_{_\infty}$ is square integrable follows from Fatou's Lemma and the $L^2$-boundedness of the sequence $(\theta_n-\theta^*)_{n\ge 1}$.$\cqfd$
\end{document} | arXiv |
\begin{document}
\date{} \title[Bilinear control]{Controllability of periodic bilinear quantum systems on infinite graphs }
\author{Ka\"{\i}s Ammari} \address{UR Analysis and Control of PDEs, UR 13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Tunisia} \email{[email protected]}
\author{Alessandro Duca} \address{Institut Fourier, Université Grenoble Alpes, 100 Rue des Mathématiques, 38610 Gières, France} \email{[email protected]}
\begin{abstract} In this work, we study the controllability of the bilinear Schr\"odinger equation on infinite graphs for periodic quantum states. We consider the bilinear Schr\"odinger equation \eqref{mainx1} $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2_p$ composed by functions defined on an infinite graph $\mathscr{G}$ verifying periodic boundary conditions on the infinite edges. The Laplacian $-\Delta$ is equipped with specific boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We present the well-posedness of the \eqref{mainx1}
in suitable subspaces of $D(|\Delta|^{3/2})$ . In such spaces, we study the global exact controllability and we provide examples involving tadpole graphs and star graphs with infinite spokes. \end{abstract}
\subjclass[2010]{35Q40, 93B05, 93C05} \keywords{Bilinear control, infinite graph}
\maketitle
\section{Introduction}\label{intro}
Graph type structures (Figure \ref{fig:1}) have been widely studied for the modeling of phenomena arising in science, social sciences and engineering. Among the many applications to quantum mechanics, they were used to study the dynamics of free electrons in organic molecules starting from the seminal work \cite{198}, the superconductivity in granular and artificial materials \cite{121212}, acoustic and electromagnetic wave-guides networks in \cite{112, 181}, etc.
\begin{figure}
\caption{An infinite graph is an one-dimensional domain composed by vertices (points) connected by edges (segments and half-lines).}
\label{fig:1}
\end{figure}
We consider a particle trapped on a network of wave-guides or wires where some branches are way longer than the others. We model the long branches with half-lines and the remaining ones with segments in order to represent the network by an infinite graph. The nodes of the network are ideal so that the crossing particle is subjected to zero resistance during the motion and we assume that the system is subjected to an external field which plays the role of control.
A natural choice for such setting is to represent the network by an infinite graph $\mathscr{G}$ and the state of the particle by a function $\psi$ with domain $\mathscr{G}$. The state $\psi$ belongs to a suitable Hilbert space $\mathscr{H}$ and the dynamics of the particle is modeled by the bilinear Schr\"odinger equation in $\mathscr{H}$ \begin{align}\label{eq1}i\partial_t\psi(t,x)= A\psi(t,x) +u(t) B\psi(t,x),\ \ \ \ \ \ t\in [0,T],\ x\in \mathscr{G},\end{align} where $A$ is a positive self-adjoint operator. The term $u(t)B$ represents the time dependent external field acting on the system which action is given by the bounded symmetric operator $B$ and its intensity by the control function $u\in L^2((0,T),\mathbb{R})$.
In this work, we consider $\mathscr{H}$ as the Hilbert space composed by $L^2_{loc}$ functions over the graph satisfying periodic boundary conditions on the infinite edges and $A$ is a Laplacian equipped with suitable boundary conditions. We study the controllability of the bilinear Schr\"odinger equation \eqref{eq1} according to the choice of the graph. Our purpose is to analyze when it is possible to control exactly the motion by time-varying the intensity of the external field.
{\noindent \bf \underline{Some biblio}g\underline{ra}p\underline{h}y}
The mathematical analysis of operators defined on networks was preliminarily addressed in \cite{209} by Ruedenber and Scherr. In this work, they studied the dynamics of particular electrons in the conjugated double-bounds organic molecules. These particles move as if they were trapped on a network of wave-guides and the graphs are obtained as the idealization of such structures in the limit where the diameter of the section is much smaller than the length. A similar approach was developed by Saito in \cite{211, 212} where the graphs are obtained as ``shrinking'' domains. For analogous ideas, we refer to the papers \cite{202, 208}.
The controllability of finite-dimensional quantum systems modeled by equations as \eqref{eq1}, when $A$ and $B$ are $N\times N$ Hermitian matrices, is well-known for being linked to the rank of the Lie algebra spanned by $A$ and $B$ (see \cite{basso,corona}). Nevertheless, the Lie algebra rank condition can not be used for infinite-dimensional quantum systems (see \cite{corona}).
The {global approximate controllability} of the bilinear Schr\"odinger equations \eqref{eq1} was proved with different techniques in literature. We refer to \cite{milo,nerse2} for Lyapunov techniques, to \cite{ugo2,ugo3} for adiabatic arguments and to \cite{ugo,nabile} for Lie-Galerking methods.
The {exact controllability} of infinite-dimensional quantum systems is in general more delicate. For instance, the controllability and observability of the linear Schr\"odinger equation are reciprocally dual. Various results were developed by addressing directly or by duality the control problem with multiplier methods \cite{lion,Machty}, microlocal analysis \cite{rauch,burqa,lollo} and Carleman estimates \cite{44,lasiecka,330}. However, a complete theory on networks is far from being formulated. Indeed, the interaction between the different components of the structure may generate unexpected phenomena. For further details on the subject, we refer to \cite{wave}.
An important property of the bilinear Schr\"odinger equation is that its controllability can not be approached with the techniques valid for the linear Schr\"odinger equation. Indeed, the dynamics of \eqref{eq1} is well-known for not being exactly controllable in the Hilbert space $\mathscr{H}$ where it is defined when $B$ is a bounded operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$ (even though it is well-posed in such space). This result was proved by Turinici in \cite{torino} by exploiting the theory developed by Ball, Mardsen and Slemrod in \cite{ball} (see \cite{bal, bal2} for other results on bilinear systems). As a consequence, the classical techniques can not be exploited for the exact controllability of bilinear quantum systems.
The turning point for this kind of studies has been the idea of controlling the equation in specific subspaces of $D(A)$. Preliminarily introduced by Beauchard in \cite{be1}, this approach was mostly popularized by the work \cite{laurent} of Beauchard and Laurent. There, they considered the bilinear Schr\"odinger equation $\eqref{eq1}$ on the interval $\mathscr{G}=(0,1)$ when $\mathscr{H}=L^2((0,1),\mathbb{C})$, $B$ is a suitable multiplication operator and $A=-\Delta_D$ is the Dirichlet Laplacian $$D(-\Delta_D)=H^2((0,1),\mathbb{C})\cap H^1_0((0,1),\mathbb{C})),\ \ \ \ \ -\Delta_D\psi:=-\Delta\psi,\ \ \ \ \forall\psi\in D(-\Delta_D).$$ They proved the {well-posedness} and the {local exact
controllability} of the equation in the space $D(|\Delta_D|^{3/2})$. Afterwards, different works on the subject were developed. We refer to \cite{beauchard2017,mio2} for {global exact controllability} results and \cite{mio2,morgane1,morganerse2} for simultaneous exact controllability results.
The controllability of bilinear quantum systems on graphs was preliminarily addressed by the second author in \cite{mio3,mio4}. There, the bilinear Schr\"odinger equation \eqref{eq1} is considered in the Hilbert space $L^2(\mathscr{G},\mathbb{C})$ with $\mathscr{G}$ a compact graph and $A$ a suitable self-adjoint Laplacian. One of the main difficulties of this framework is due to the nature of the spectrum of $A$. In particular, when we consider its ordered sequence of eigenvalues $(\lambda_k)_{k\in\mathbb{N}^*}$, it is possible to show that there exists $\mathcal{M}\in\mathbb{N}^*$ such that \begin{equation}\label{g13}\begin{split}
\inf_{{k\in\mathbb{N}^*}}|\lambda_{k+\mathcal{M}}-\lambda_k|>0\\ \end{split}\end{equation} (as ensured in \cite[Lemma\ 2.4]{mio3}). Nevertheless, the uniform {spectral gap}
$\inf_{{k\in\mathbb{N}^*}}|\lambda_{k+1}-\lambda_k|> 0$ is only valid when $\mathscr{G}=(0,1)$. This hypothesis was crucial for the techniques adopted in the previous works on bounded intervals, which could not be applied in this framework. To this purpose, new spectral techniques were developed in the works \cite{mio3,mio4} in order to ensure the global exact controllability of the bilinear Schr\"odinger equation \eqref{eq1} on compact graphs.
When we consider the bilinear Schr\"odinger equation \eqref{eq1} on infinite graphs instead, a natural obstacle to the controllability is the loss of localization of the wave packets during the evolution: the dispersion. This effect can be measured by $L^\infty$-time decay, which implies a spreading out of the solutions, due to the time invariance of the $L^2$-norm. Dispersive estimations on infinite graphs can be found in \cite{AAN17,AAN15}. The other side of the same coin is that a self-adjoint Laplacian $A$ on $L^2(\mathscr{G},\mathbb{C})$ where $\mathscr{G}$ is an infinite graph, does not admit compact resolvent and then, the spectral techniques from \cite{mio3,mio4} can not be directly applied to this framework.
Despite the dispersive behavior of the bilinear Schr\"odinger equation \eqref{eq1} on infinite graphs, the authors addressed the problem in \cite{mio5} by exploiting a simple but still effective idea. When $\mathscr{G}$ contains suitable substructures, the Laplacian $A$ admits discrete spectrum corresponding to some specific eigenmodes. Such states are preserved by the dynamics of \eqref{eq1} for suitable choices of $B$ and they are not affected by the dispersive behavior of the equation. By working on the space spanned by such eigenmodes, global exact controllability results for the equation \eqref{eq1} can be ensured in suitable subspaces of $L^2(\mathscr{G},\mathbb{C})$ with $\mathscr{G}$ an infinite graph, as presented in \cite{mio5}. We underline that the considered eigenmodes are supported in compact sub-graphs of $\mathscr{G}$ and then, the result is only valid for suitable states vanishing on the infinite edges of the graph.
From this perspective, our purpose is natural. We aim to carry on the existing theory by proving the controllability of \eqref{eq1} for quantum states that do not vanish on the infinite edges of the graph. In this regard, we consider the bilinear Schr\"odinger equation \eqref{eq1} for periodic functions. This choice allows us to have non-compactly supported eigenmodes and then, to ensure the exact controllability for states also defined on the infinite edges of the graph.
{\noindent \bf \underline{Scheme of the work}}
The paper is organized as follows. In Section \ref{preli}, we introduce the main notations of the work. In Sections \ref{sec2} and Section \ref{sec3}, we respectively prove the global exact controllability when $\mathscr{G}$ is an infinite tadpole graph and an infinite star graph. In the last section, we generalize the previous results to some general infinite graphs.
\section{Preliminaries}\label{preli}
Let $\mathscr{G}$ be a general graph composed by $N$ finite edges $\{e_j\}_{1\leq j\leq N}$ of lengths $\{L_j\}_{1\leq j\leq N}$ and $\widetilde N$ half-lines $\{e_j\}_{N+1\leq j\leq N+\widetilde N}$. Each edge $e_j$ with $j\leq N$ is associated to a coordinate starting from $0$ and going to $L_j$, while $e_j$ with $N+1\leq j\leq N+\widetilde N$ is parametrized with a coordinate starting from $0$ and going to $+\infty$. We consider $\mathscr{G}$ as domain of functions $$f:=(f^1,...,f^{N+\widetilde N}):\mathscr{G}\rightarrow \mathbb{C}, \ \ \ \ \ \ \ \ \ f^j:e_j\rightarrow \mathbb{C},\ \ \ \ \ 1\leq j\leq N+\widetilde N.$$ Let $\{L_j\}_{ N+1\leq j\leq N+\widetilde N}\subset \mathbb{R}^+$. We consider the Hilbert space \begin{align}\label{spazio}L^2_p:=L^2_p(\mathscr{G},\mathbb{C})= \Big(\prod_{j=1}^{N}L^2(e_j,\mathbb{C})\Big)\times \Big(\prod_{j=N+1}^{N+\widetilde N}L^2_p(e_j,\mathbb{C})\Big),\ \ \ \ \ \text{with}\end{align} $$L^2_p(e_j,\mathbb{C})=\Big\{f\in L^2_{loc}(e_j,\mathbb{C})\ :\ f(\cdot)=f\big(\ \cdot\ +2\pi kL_j\big),\ \ \ \ \forall k\in\mathbb{N}^*\Big\},\ \ \ N+1\leq j\leq N+\widetilde N.$$
The Hilbert spaces $L^2_p$ is equipped with the norm $\|\cdot\|_{L^2_p}$ induced by the scalar product $$\langle\psi,\varphi\rangle_{L^2_p}=\sum_{j=1}^{N+\widetilde N}\int_{0}^{L_j}\overline{\psi^j}(x)\varphi^j(x)dx,\ \ \ \ \ \ \forall \psi,\varphi\in L^2_p.$$ We introduce the spaces $$H^s_p:= L^2_p\cap\Bigg(\Big(\prod_{j=1}^{N}H^s(e_j,\mathbb{C})\Big)\times \Big(\prod_{j=N+1}^{N+\widetilde N}H^s_{loc}(e_j,\mathbb{C})\Big)\Bigg)$$ with $s>0$. For $T>0$, we consider the bilinear Schr\"odinger equation in $L^2_p$ \begin{equation}\label{mainx1}\tag{BSE}\begin{split} \begin{cases} i\partial_t\psi(t)=-A\psi(t)+u(t)B\psi(t),\ \ \ \ \ \ \ \ &t\in(0,T),\\ \psi(0)=\psi_0.\\ \end{cases} \end{split} \end{equation} The operator $A$ is a Laplacian equipped with suitable boundary conditions such that $D(A)\subseteq H^2_p$. The operator $B$ is a bounded symmetric operator in $L^2_p$ and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We respectively denote $$\upvarphi:=\{\varphi_k\}_{k\in\mathbb{N}^*},\ \ \ \ \ \ \ \ \ \ \ \ \upmu:=\{\mu_k\}_{k\in\mathbb{N}^*}$$ an orthonormal system of $L^2_p$ made by some eigenfunctions of $A$ and the corresponding eigenvalues. For $s>0$, we define the spaces
$$\mathscr{H}(\upvarphi):=\overline{span\{\varphi_k\ |\ k\in\mathbb{N}^*\}}^{\ L^2_p},$$ \begin{equation}\label{spaces}\begin{split}
&H^s_{\mathscr{G}}(\upvarphi):=\{\psi\in \mathscr{H}(\upvarphi)\ | \ \sum_{k\in\mathbb{N}^*}|k^s\langle\varphi_k,\psi\rangle_{L^2_p}|^2<\infty\},\\
&h^s:=\Big\{\{a_k\}_{k\in\mathbb{N}^*}\in\ell^2(\mathbb{C})\ \big|\ \sum_{k\in\mathbb{N}^*}|k^{s}a_k|^2<\infty\Big\}.\\ \end{split}\end{equation}
We respectively equip $H^s_{\mathscr{G}}(\upvarphi)$ and $h^s$ with the norms $\|\cdot\|_{(s)}=\big({\sum_{k \in
\mathbb{N}^*}}|k^s\langle\varphi_k,\cdot\rangle_{L^2_p}|^2\big)^\frac{1}{2}$ and $$\|{\bf
x}\|_{(s)}=\big(\sum_{k\in\mathbb{N}^*}|k^sx_k|^2\big)^\frac{1}{2},\ \ \ \ \ \ \ \ \ \forall {\bf x}:=(x_k)_{k\in\mathbb{N}^*}\in h^s.$$ \begin{oss} The space $\mathscr{H}(\upvarphi)$ is usually strictly smaller than $L^2_p$. If for instance we consider $\mathscr{G}$ as a ring parametrized from $0$ to $1$ and $\upvarphi=\big\{\sqrt{2}\sin(2k\pi x)\big\}_{k\in\mathbb{N}^*}$, then $\mathscr{H}(\upvarphi)$ is composed by those $L^2_p$ states which are odd with respect to the point $x=1/2$ and clearly $\mathscr{H}(\upvarphi)\subset L_p^2.$ \end{oss} \begin{oss}
Let $\mu_k\sim k^2$ and $c\in \mathbb{R}^+$ be such that $0\not\in\sigma(A+c,\mathscr{H}(\upvarphi))$ (the spectrum of $A+c$ in
the Hilbert space $\mathscr{H}(\upvarphi)$). For every $s>0,$ there exist $C_1,C_2>0$ such that $$C_1\|\psi\|_{(s)}\leq
\||A+c|^{s/2}\psi\|_{L^2_p}\leq C_2\|\psi\|_{(s)},\ \ \ \ \ \ \ \ \forall \psi \in H^s_\mathscr{G}(\upvarphi).$$ \end{oss} Let $\Gamma_T^u$ be the unitary propagator (when it is defined) corresponding to the dynamics of \eqref{mainx1} in the time interval $[0,T]$. \begin{defi}\label{exact} Let $\upvarphi$ be an orthonormal system of $L^2_p$ made by some eigenfunctions of $A$ and $s>0$. The bilinear Schr\"odinger equation \eqref{mainx1} is said to be globally exactly controllable in $H^s_{\mathscr{G}}(\upvarphi)$ when, for every $\psi_1,\psi_2\in H^s_\mathscr{G}(\upvarphi)$
such that $\|\psi_1\|_{L^2_p}=\|\psi_2\|_{L^2_p}$, there exist $T>0$ and $u\in L^2((0,T),\mathbb{R})$ such that
$$\Gamma_T^u\psi_1=\psi_2.$$ \end{defi}
The aim of the work is to study the global exact controllability of the \eqref{mainx1} on infinite graphs in suitable spaces $H^s_{\mathscr{G}}(\upvarphi)$ with $s>0$.
\section{Infinite tadpole graph} \label{sec2}
Let $\mathcal{T}$ be an {\it infinite tadpole graph} composed by two edges $e_1$ and $e_2$. The self-closing edge $e_1$, the \virgolette{head}, is connected to $e_2$ in the vertex $v$ and it is parametrized in the clockwise direction with a coordinate going from $0$ to $1$ (the length of $e_1$). The \virgolette{tail} $e_2$ is an half-line equipped with a coordinate starting from $0$ in $v$ and going to $+\infty$. The tadpole graph presents a natural symmetry axis that we denote by $r$. \begin{figure}
\caption{The parametrization of the infinite tadpole graph and its natural symmetry axis $r$.}
\label{parametrizzazionetadpole1}
\end{figure} Let $L^2_p$ be composed by functions which are periodic on the tail with period $1$, {\it i.e.} $L_2=1$. We consider the bilinear Schr\"odinger equation \eqref{mainx1} in $L^2_p$ with $A=-\Delta$ the Laplacian equipped with {\it Neumann-Kirchhoff\,} boundary conditions in the vertex $v$, {\it i.e.} \begin{equation*}\begin{split} D(A)=\Big\{\psi=(\psi^1,\psi^{2})\in H^2_p\ : \ \psi^1(0)=\psi^1(1)=\psi^2(0),\ \ \ \ \frac{\partial \psi^1}{\partial x}(0)-\frac{\partial \psi^1}{\partial x}(1)+\frac{\partial \psi^2}{\partial x}(0)=0\Big\}.\\ \end{split} \end{equation*} \begin{osss}\label{peculiarity} The chosen operator $A$ is not self-adjoint in the Hilbert space $L^2_p$. This fact is an important peculiarity of this work with respect to the existing ones on bilinear quantum systems. However, we show how to construct subspaces of $L^2_p$ composed by eigenspaces of $A$ where the well-posedness and the controllability can be ensured. \end{osss}
We assume the control field $B:\psi=(\psi^1,\psi^2)\longmapsto (V^1\psi^1,V^2\psi^2)$ being such that $$V^1(x)=x^2(x-1)^2,\ \ \ \ \ \ \ \ \ \ V^2(x)=\sum_{n\in\mathbb{N}}(x-n)^2(x-n-1)^2\upchi_{[n,n+1]}(x).$$ The choice of the potentials $V_1$ and $V_2$ is calibrated so that $B$ preserves
the space $L^2_p$ and $V^1\psi^1\equiv V^2\psi^2|_{[n,n+1]}$ for every $n\in\mathbb{N}$ when
$\psi=(\psi^1,\psi^2)\in L^2_p$ is such that $\psi^1\equiv\psi^2|_{[n,n+1]}$ for every $n\in\mathbb{N}$. In this framework, the \eqref{mainx1} corresponds to the two following Cauchy systems respectively in $L^2(e_1,\mathbb{C})$ and $L^2_p(e_2,\mathbb{C})$ \begin{equation}\tag{BSEt}\label{mainT}\begin{split} \begin{cases} i\partial_t\psi^1=-\Delta\psi^1+uV^1\psi^1,\\ \psi^1(0)=\psi^1_0,\\ \end{cases}\ \ \ \ \ \begin{cases} i\partial_t\psi^2=-\Delta\psi^2+uV^2\psi^2,\\ \psi^2(0)=\psi^2_0.\\ \end{cases} \end{split}
\end{equation} Let $\upvarphi:=\{\varphi_k\}_{k\in\mathbb{N}^*}$ be an orthonormal system of $L^2_p$ made by eigenfunctions of $-\Delta$ and corresponding to the eigenvalues $\upmu:=\{\mu_k\}_{k\in\mathbb{N}^*}$ such that, for every $k\in \mathbb{N}^*\setminus\{1\}$, \begin{equation*}\begin{split}\begin{cases} \varphi_k=\big(\cos({2(k-1)}\pi x),\cos({2(k-1)}\pi x)\big),\ \ \ \ \ \ \ \ \ \ &\mu_k={4(k-1)^2\pi^2},\\ \varphi_1=\big(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\big),\ \ \ \ \ \ \ \ \ \ &\mu_1=0.
\end{cases}\end{split}\end{equation*}
\begin{osss}\label{equivalentdefi} We notice that each $f=(f^1,f^2)\in L^2_p$ belongs to $\mathscr{H}(\upvarphi)$ when \begin{itemize}
\item $f^1$ is symmetric with respect to the symmetry axis $r$ of $\mathcal{T}$;
\item $f^2$ has period $2\pi$ and $f^2|_{[2n\pi,2(n+1)\pi]}\equiv f^1$ for every $n\in\mathbb{N}$.
\end{itemize}
\end{osss}
\begin{prop}\label{lauraT}
Let $\psi_0\in H^{4}_{\mathcal{T}}(\upvarphi)$ and $u\in L^2((0,T),\mathbb{R})$. There
exists a unique mild solution of the (\ref{mainT}) in
$H^{4}_{\mathcal{T}}(\upvarphi)$, i.e. a function $\psi\in C^0\big([0,T],H^{4}_{\mathcal{T}}(\upvarphi)\big)$ such that
\begin{equation}\label{mild}
\psi(t)=e^{i\Delta t}\psi_0-
i\int_0^t e^{i\Delta(t-s)}u(s)B\psi(s)ds .\\
\end{equation}
Moreover, the flow of \eqref{mainT} on $\mathscr{H}(\upvarphi)$ can be extended to a unitary flow $\Gamma_t^{u}$ with respect to the $L^2_p-$norm such that $\Gamma_t^{u}\psi_0=\psi(t)$ for any solution $\psi$ of \eqref{mainT} with initial data $\psi_0\in \mathscr{H}(\upvarphi)$. \end{prop} \begin{proof} {\bf 1) Unitary flow.} We consider Remark \ref{equivalentdefi}. For every $f=(f^1,f^2)\in \mathscr{H}(\upvarphi)$, we notice that $(B f)^1$ inherits from $f^1$ the property of being symmetric with respect to the symmetry axis $r$,
while $(Bf)^2|_{[2n\pi,2(n+1)\pi]}\equiv(B f)^1$ for every $n\in\mathbb{N}$ as $f^2|_{[2n\pi,2(n+1)\pi]}\equiv f^1$ for every $n\in\mathbb{N}$. Now, $(Bf)^2$ has period $2\pi$ and $(Bf)^2(x)=(Bf)(2(n+1)\pi-x)$ for every $n\in\mathbb{N}$ and $x\in[2n\pi,(2n+1)\pi].$ Thus, $Bf=(V^1f^1,V^2f^2)\in \mathscr{H}(\upvarphi)$ for every $f=(f^1,f^2)\in \mathscr{H}(\upvarphi)$ and the control field $B$ preserves $\mathscr{H}(\upvarphi)$. The space $\mathscr{H}(\upvarphi)$ is a Hilbert space where the operator $A$ is self-adjoint and $B$ is bounded symmetric. Thanks to \cite[Theorem\ 2.5]{ball}, the \eqref{mainT} admits a unique solution $\psi\in C^0([0,T],\mathscr{H}(\upvarphi))$ for every $T>0$ and $\psi_0\in\mathscr{H}(\upvarphi)$. The flow of \eqref{mainT} is unitary in $\mathscr{H}(\upvarphi)$ thanks to the following arguments. If $u\in
C^0((0,T),\mathbb{R})$, then $\psi\in C^1((0,T),\mathscr{H}(\upvarphi))$ and $\partial_t\|\psi(t)\|_{L^2_p}^2=0$ from (\ref{mainT}).
Thus $\|\psi(t)\|_{L^2_p} =\|\psi_0\|_{L^2_p}$. The generalization for $u\in L^2((0,T),\mathbb{R})$ follows from a classical density argument, which ensures that the flow of the dynamics of the \eqref{mainT} is unitary in $\mathscr{H}(\upvarphi)$.
\needspace{3\baselineskip}
\noindent
{\bf 2) Regularity of the integral term in the mild solution.} The remaining part of the proof refers to the techniques leading to \cite[Lemma\ 1;\ Proposition\ 2]{laurent} (also adopted in the proof of \cite[Proposition\ 2.1]{mio5}). Let $\psi\in C^0([0,T],H_{\mathcal{T}}^4(\upvarphi))$ with $T>0$. We notice $B\psi(s) \in H_p^{4}\cap H_{\mathcal{T}}^2(\upvarphi)$ for almost every $s\in (0,t)$ and $t\in(0,T)$. Let $G(t,x)=\int_0^t e^{i\Delta(t-s)}u(s)B\psi(s,x)ds$ so that
\begin{equation*}\begin{split}
\|G(t)\|_{(4)}&=\Big(\sum_{k\in\mathbb{N}^*}\Big|k^4\int_0^t e^{i\mu_ks}\langle\varphi_k,u(s)B\psi(s,\cdot)\rangle_{L^2_p}
ds\Big|^2\Big)^\frac{1}{2}.\\
\end{split}\end{equation*}
For $f(s,\cdot):=u(s)B\psi(s,\cdot)$ such that $f=(f^1,f^2)$ and $k\in\mathbb{N}^*\setminus\{1\},$ we have
\begin{equation*}\begin{split}
&\langle\varphi_k,f(s)\rangle_{L^2_p}=-\frac{1}{\mu_k }\Big(\int_{0}^1\varphi_k^1(y)\partial_{x}^2f^1(s,y)dy+\int_{0}^{1}\varphi_k^2(y)\partial_{x}^2f^2(s,y)dy\Big)\\
&=-\frac{2}{\mu_k }\int_{0}^1\varphi_k^1(y)\partial_{x}^2f^1(s,y)dy=\frac{1}{4 (k-1)^3\pi^3 }\int_{0}^1\sin(2 (k-1)\pi x)\partial_{x}^3f^1(s,y)dy\\
&=\frac{1}{8(k-1)^4\pi^4 }\Bigg(\partial_{x}^3f^1(s,1)-\partial_{x}^3f^1(s,0)-\int_{0}^1\cos(2 (k-1)\pi x) \partial_{x}^4f^1(s,y)dy\Bigg).\\
\end{split}\end{equation*}
In the last relations, we considered
$\varphi_k^1(\cdot)\partial_{x}^2f^1(s,\cdot)|_{[0,1]}=\varphi_k^2(\cdot)\partial_{x}^2f^2(s,\cdot)|_{[0,1]}$ as
$\partial_{x}^2f^1(s,\cdot)|_{[0,1]}=\partial_{x}^2f^2(s,\cdot)|_{[0,1]}$. Equivalently to the first point of the proof of \cite[Proposition\ 2.1]{mio5}, there exists $C_1>0$ such that
\begin{equation*}\begin{split}
\|G(t)\|_{(4)}\leq& C_1\Big(\Big\|\int_0^t \big(\partial_{x}^3f^1(s,1)-\partial_{x}^3f^1(s,0)\big)e^{i \mu_{(\cdot)}s}ds
\Big\|_{\ell^2} +\sqrt{t}\|f\|_{L^2((0,t),H^4_p)}\Big).
\end{split}\end{equation*} Thanks \cite[Proposition\ B.6]{mio3}, there exists $C_2(t)>0$ uniformly bounded for $t$ in bounded intervals such
that $\|G(t)\|_{(4)} \leq C_2(t)\|f(\cdot,\cdot)\|_{L^2((0,t),H^4_p)}.$ For every $t\in [0,T]$, the last inequality shows that $G(t)\in H^4_\mathcal{T}(\upvarphi)$ and the provided upper bound is uniform. The Dominated Convergence Theorem leads to $G\in C^0([0,T], H^4_{\mathcal{T}}(\upvarphi))$.
\needspace{3\baselineskip}
\noindent
{\bf 3) Conclusion. } As $Ran(B|_{H_{\mathcal{T}}^{4}}(\upvarphi))\subseteq H_p^{4}\cap H^2_{\mathcal{T}}(\upvarphi)\subseteq H^{4}_p$, we have $B\in \mathcal{L}(H^{4}_{\mathcal{T}}(\upvarphi),H^{4}_p)$ thanks to the arguments of \cite[Remark\ 2.1]{mio1}. Let $\psi_0\in H^{4}_{\mathcal{T}}(\upvarphi)$. We consider the map $$F:\psi \in C^0([0,T],H^{4}_{\mathcal{T}}(\upvarphi))\mapsto\phi\in C^0([0,T],H^{4}_{\mathcal{T}}(\upvarphi)) ,$$
$$\phi(t)=F(\psi)(t)=e^{i\Delta t}\psi_0-\int_0^te^{i\Delta(t-s)}u(s) B\psi(s)ds,\ \ \ \ \ \forall t\in [0,T].$$
For every $\psi_1,\psi_2\in C^0([0,T], H^4_{\mathcal{T}}(\upvarphi))$, from the first point of the proof, there exists $C(t)>0$ uniformly bounded for $t$ lying on bounded intervals such that
\begin{equation*}\begin{split}
&\|F(\psi_1)-F(\psi_2)\|_{L^\infty((0,T),H^{4}_\mathcal{T}(\upvarphi))}\leq\left\|\int_0^{(\cdot)} e^{i\Delta((\cdot)-s)}u(s)
B(\psi_1(s)-\psi_2(s))ds\right\|_{L^\infty((0,T),H^{4}_\mathcal{T}(\upvarphi))}\\
&\leq C(T)\|u\|_{L^2((0,T),\mathbb{R})}{\, \vert\kern-0.25ex\vert\kern-0.25ex\vert\, } B{\, \vert\kern-0.25ex\vert\kern-0.25ex\vert\, }_{\mathcal{L}(H^{4}_{\mathcal{T}}(\upvarphi),H^{4}_p)}
\|\psi_1-\psi_2\|_{L^\infty((0,T),H^{4}_\mathcal{T}(\upvarphi))}.\\
\end{split}\end{equation*}
If $\|u\|_{L^2((0,T),\mathbb{R})}$ is small enough, then $F$ is a contraction and Banach Fixed Point Theorem yields the
existence of $\psi \in C^0([0,T],H^{4}_{\mathcal{T}}(\upvarphi) )$ such that $F(\psi)=\psi.$ When $\|u\|_{L^2((0,T),\mathbb{R})}$ is not sufficiently small, we decompose $(0,T)$ with a sufficiently thin partition $\{t_j\}_{0\leq j\leq n}$ with
$n\in\mathbb{N}^*$ such that each $\|u\|_{L^2((t_{j-1},t_j),\mathbb{R})}$ is so small such that $F$ defined on the interval $[t_{j-1},t_j]$ is a contraction. The well-posedness on $[0,T]$ is defined by gluing each flow defined in every interval of the partition. \qedhere \end{proof}
We are finally ready to present the following global exact controllability result (Definition \ref{exact}). \begin{teorema}\label{globalegirino}
The (\ref{mainT}) is globally exactly controllable in $H^4_\mathcal{T}(\upvarphi)$.
\end{teorema} \begin{proof} The statement is proved by using the arguments adopted in the proof of \cite[Theorem\ 2.2]{mio5}.
\noindent
{\bf 1) Local exact controllability.} We notice that $\varphi_1(T)=e^{-i\mu_1T}\varphi_1=\varphi_1 $ with $T>0$ as the first eigenvalue $\mu_1$ is equal to $0$. For $\epsilon,T>0$, we define
$$O_{\epsilon}^{4}:=\big\{\psi\in H_{\mathcal{T}}^{4}(\upvarphi)\big|\ : \ \|\psi\|_{L^2_p}=1,\ \|\psi
-\varphi_1\|_{(4)}<\epsilon\big\}.$$ We ensure there exist $T,\epsilon>0$ so that, for every $\psi\in O_{\epsilon}^{4}$, there exists $u\in L^2((0,T),\mathbb{R})$ such that $\psi= \Gamma^u_T\varphi_1.$
The result can be proved by showing the surjectivity of the map $\Gamma_T^{(\cdot)}\varphi_1:u\in L^2((0,T),\mathbb{R})\longmapsto \psi \in O_{\epsilon}^{4}$ with $T>0$. Let $$\Gamma_{t}^{(\cdot)}\varphi_1=\sum_{k\in\mathbb{N}^*}{\varphi_k(t)}\langle \varphi_k(t),\Gamma_{t}^{(\cdot)}\varphi_1\rangle_{L^2_p}.$$ We recall the definition of $h^4$ provided in \eqref{spaces}. Let $\alpha$ be the map defined as the sequence with elements $\alpha_{k}(u)=\langle \varphi_k(T), \Gamma_{T}^u\varphi_1\rangle_{L^2_p}$ for $k\in\mathbb{N}^*$ such that
$$\alpha:L^2((0,T),\mathbb{R})\longrightarrow Q:=\{{\bf x}:=\{x_k\}_{k\in\mathbb{N}^*}\in h^4(\mathbb{C})\ |\ \|{\bf
x}\|_{\ell^2}=1\}.$$ The local exact controllability follows from the local surjectivity of $\alpha$ in a neighborhood of $\alpha(0)=\updelta=\{\delta_{k,1}\}_{k\in\mathbb{N}^*}$ with respect to the $h^4-$norm. To this end, we consider the Generalized Inverse Function Theorem and we study the surjectivity of $\gamma(v):=(d_u\alpha(0))\cdot\ v$ the Fréchet derivative of $\alpha$. Let $B_{k,1}:=\langle\varphi_k,B\varphi_1\rangle_{L^2_p}$ with $k\in\mathbb{N}^*$. The map $\gamma$ is the sequence of elements $\gamma_{k}(v):=
-i\int_{0}^Tv(\tau)e^{i(\mu_k-\mu_1)s}d\tau B_{k,1}$ with $k\in\mathbb{N}^*$ so that
$$\gamma:L^2((0,T),\mathbb{R})\longrightarrow T_{\updelta}Q=\{{\bf x}:=\{x_k\}_{k\in\mathbb{N}^*}\in h^4(\mathbb{C})\ |\ ix_1\in\mathbb{R}\}.$$
As $\mu_1=0$, the surjectivity of $\gamma$ corresponds to the solvability of the moments problem
\begin{equation}\begin{split}\label{mome1}
{x_{k}}{B_{k,1}^{-1}}=-i\int_{0}^Tu(\tau)e^{i\mu_k\tau}d\tau,\ \ \ \ \ \ \ \ \ \ \forall \{x_{k}\}_{k\in\mathbb{N}^*}\in T_{\updelta}Q\subset h^4.
\\
\end{split}\end{equation}
By direct computation, there exists $C>0$ such that
$|B_{k,1}|=|\langle\varphi_k,B\varphi_1\rangle_{L^2_p}|\geq\frac{C}{k^4}$ for every $k\in\mathbb{N}^*$ and
$$\big\{x_k B_{k,1}^{-1}\big\}_{k\in\mathbb{N}^*}\in \ell^2,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i{x_{1}}B_{k,1}^{-1}\in\mathbb{R}.$$
In conclusion, the solvability of $(\ref{mome1})$ is guaranteed by \cite[Proposition\ B.5]{mio3} since $$\{ix_k
B_{k,1}^{-1}\}_{k\in\mathbb{N}^*}\in\{\{c_k\}_{k\in\mathbb{N}^*}\in \ell^2\ |\ c_1\in\mathbb{R}\},\ \ \ \ \ \
\inf_{k\in\mathbb{N}^*}|\mu_{k+1}-\mu_k|={4\pi^2}.$$
\noindent
{\bf 2) Global exact controllability.} Let $T,\epsilon>0$ be so that {\bf 1)} is valid. Thanks to Remark \ref{approxT}, for any $\psi_1,\psi_2\in H^{4}_{\mathcal{T}}(\upvarphi)$ such that
$\|\psi_1\|_{L^2_p}=\|\psi_2\|_{L^2_p}=p$, there exist $T_1,T_2>0$, $u_1\in L^2((0,T_1),\mathbb{R})$ and $u_2\in
L^2((0,T_2),\mathbb{R})$ such that $$\|\Gamma^{u_1}_{T_1}p^{-1}\psi_1-\varphi_1\|_{(4)}<{\epsilon},\ \ \ \ \
\|\Gamma^{u_2}_{T_2}p^{-1}\psi_2-\varphi_1\|_{(4)}<{\epsilon}$$
and $ p^{-1}\Gamma^{u_1}_{T_1}\psi_1,p^{-1}\Gamma^{u_2}_{T_2}\psi_2\in O_{\epsilon}^{4}.$ From {\bf 1)}, there exist $u_3,u_4\in L^2((0,T),\mathbb{R})$ such that $$\Gamma_T^{u_3}\Gamma^{u_1}_{T_1}\psi_1=\Gamma_T^{u_4}\Gamma^{u_2}_{T_2}\psi_2=p\varphi_1\ \ \ \
\Longrightarrow \ \ \ \ \ \exists T>0,\ \widetilde u\in L^2((0,\widetilde T),\mathbb{R})\ \ :\ \ \Gamma_{\widetilde T}^{\widetilde u}\psi_1=\psi_2. $$ \end{proof}
\noindent Let $\Phi:=\{\phi_k\}_{k\in\mathbb{N}^*}$ be an orthonormal system of $L^2_p$ made by eigenfunctions of $-\Delta$ and corresponding to the eigenvalues $\Lambda:=\{\lambda_k\}_{k\in\mathbb{N}^*}$ such that $$\phi_k=\big(\sqrt{{2}}\sin({2k}\pi x),0\big),\ \ \ \ \ \ \ \ \ \ \lambda_k={4k^2\pi^2},\ \ \ \ \ \ \ \ \ \ \forall k\in \mathbb{N}^*.$$ We notice that the results \cite[Theorem\ 2.1;\ Theorem\ 2.2]{mio5} are still valid in the current framework and they lead to the following proposition.
\begin{prop}\label{lauraT1}
Let (\ref{mainT}) be considered with $V_1(x)=x(1-x)$ and $V_2=0$ The (\ref{mainT}) is well-posed and globally exactly controllable in $H^3_\mathcal{T}(\Phi)$. \end{prop}
The techniques leading to Proposition \ref{lauraT}, Theorem \ref{globalegirino} and Proposition \ref{lauraT1} also imply the following corollary.
\begin{coro}\label{corogirino}
Let (\ref{mainT}) be considered with $$V^1(x)=x(1-x)+x^2(x-1)^2,\ \ \ V^2(x)=\sum_{n\in\mathbb{N}}(x-n)^2(x-n-1)^2\upchi_{[n,n+1]}(x).$$ The (\ref{mainT}) is well-posed and globally exactly controllable in $H^4_\mathcal{T}(\upvarphi)$ and $H^3_\mathcal{T}(\Phi)$. \end{coro}
\begin{osss}\label{extension} The choice of the lengths $L_1=1$ and $L_2=1$ has been done in order to simplify the theory of the current section. Nevertheless, it is possible to obtain similar results by considering different parameters $L_1$ and $L_2$ such that $L_1/L_2\in\mathbb{Q}$. A very similar situation is considered in the next section for a star graph with infinite spokes. \end{osss}
\section{Star graph with infinite spokes} \label{sec3} Let $\mathscr{S}$ be a {\it star graph} composed by $N$ segments $\{e_j\}_{1\leq j\leq N}$ of lengths $\{L_j\}_{1\leq j \leq N}$ and $\widetilde N$ half-lines $\{e_j\}_{N+1\leq j\leq N+\widetilde N}$. The edges are connected in the internal vertex $v$, while $\{v_j\}_{1\leq j\leq N}$ are the external vertices of $\mathscr{S}$ (those vertices of $\mathscr{S}$ connected with only one edge). Each $e_j$ with $1\leq j\leq N$ is associated to a coordinate starting from $0$ in $v_j$ and going to $L_j$, while $e_j$ with $N+1\leq j\leq N+\widetilde N$ is parametrized with a coordinate starting from $0$ in $v$ and going to infinite. \begin{figure}
\caption{The parametrization of a star graph composed by $N=2$ segments and $\widetilde N=1$ half-lines.}
\end{figure} Let $L^2_p$ be defined in \eqref{spazio}. This space is composed by functions which are periodic on the infinite edges with periods $\{L_j\}_{N+1\leq j\leq N+\widetilde N}$. We consider the bilinear Schr\"odinger equation \eqref{mainx1} in $L^2_p$ and the Laplacian $A=-\Delta$ being equipped with {\it Neumann-Kirchhoff\,} boundary conditions in $v$ and {\it Neumann\,} boundary conditions in $\{v_{j}\}_{1\leq j\leq N}$, {\it i.e.} \begin{equation*}\begin{split} D(A)=&\Big\{\psi=(\psi^1,...,\psi^{N+\widetilde N})\in H^2_p\ \ :\ \ \sum_{j=1}^{N}\frac{\partial \psi^j}{\partial x}(L_j)=\sum_{j=N+1}^{N+\widetilde N}\frac{\partial \psi^j}{\partial x}(0),\\ &\ \ \ \psi\in C^0(\mathscr{S},\mathbb{C}),\ \ \ \ \ \ \frac{\partial \psi^j}{\partial x}(v_j)=0\ \ \ \ \ \forall 1\leq j\leq N\Big\}.\\ \end{split} \end{equation*} Let $B:\psi\in L^2_p\mapsto B\psi=\big((B\psi)^1,...,(B\psi)^{N+\widetilde N}\big)$ be a bounded symmetric operator. The \eqref{mainx1} corresponds to the following Cauchy systems in $L^2(e_j,\mathbb{C})$ when $1\leq j\leq N$ and in $L^2_p(e_j,\mathbb{C})$ when $ N+1\leq j\leq N+\widetilde N$ \begin{equation}\tag{BSEs}\label{mainS}\begin{split} \begin{cases} i\partial_t\psi^j(t)=-\Delta\psi^j(t)+u(t)(B\psi)^j(t),& \ \ \ \ \ \ t\in(0,T),\\ \psi^j(0)=\psi^j_0.\\ \end{cases} \end{split} \end{equation} \begin{osss}\label{peculiarity1} As in Section \ref{sec2}, the chosen operator $A$ is not self-adjoint in the Hilbert space $L^2_p$. The central point here is to seek for the correct framework where the existence of eigenfunctions for $A$ is guaranteed. It is clear that the periodicity conditions on each infinite edge $e_j$ with $N+1\leq j\leq N+\widetilde N$ force any eigenvalue $\lambda$ of $A$ to be of the form $\frac{4k^2\pi^2}{L_j}$ with $k\in\mathbb{N}$. Thus, the eigenvalues $\lambda$ has to be contained in $\bigcap_{ N+1\leq j\leq N+\widetilde N}\big\{\frac{4k^2\pi^2}{L_j}\big\}_{k\in\mathbb{N}^*}$ which has to be non-empty. This is possible for suitable resonant lengths for the edges of the graphs. In the following part of this section we introduce a set of assumptions ensuring this fact. \end{osss} Let $L_{N+1}/L_j\in\mathbb{Q}$ for every $N+2\leq j\leq N+\widetilde N$. We denote by $l_j\in\mathbb{N}^*$ the smallest natural number such that \begin{equation}\label{multipliers}l_j\frac{L_{N+1}}{L_{j}}\in\mathbb{N}^*,\ \ \ \ \ \ \ \text{with}\ \ \ \ 1\leq j\leq N+\widetilde N.\end{equation} Let $n_k:=(k-1)\prod_{j=N+1}^{N+\widetilde N}l_j\frac{L_{N+1}}{L_j}\in\mathbb{N}^*$ for every $k\in\mathbb{N}^*$. We notice $$\bigcap_{j=N+1 }^{N+\widetilde N}\Big\{\frac{2 m\pi}{L_j}\Big\}_{m\in\mathbb{N}^*}= \Big\{\frac{2n_k\pi}{L_{N+1}}\Big\}_{k\in \mathbb{N}^*}.$$
\begin{assumptionA} The numbers $\{L_j\}_{1\leq j\leq N+\widetilde N}$ are such that every ratio $\frac{L_{N+1}}{L_j}\in\mathbb{Q}$ for any
$N+2\leq j\leq N+\widetilde N$. In addition, there exist $J\subseteq\mathbb{N}^*$ with $|J|=+\infty$ and $\{c_j\}_{N+1\leq j\leq N+\widetilde N}$ with $c_j\in[0,L_j]$ for any ${N+1\leq j\leq N+\widetilde N}$ such that $$\sum_{j=1}^{N}\tan\Big(\frac{2 n_k \pi }{L_{N+1}}L_j\Big)=\sum_{j=N+1}^{N+\widetilde N}\tan\Big(\frac{2 n_k \pi }{L_{N+1}} c_j\Big),\ \ \ \ \ \ \ \ \ \forall k\in J.$$ In conclusion, the sequence $(\mu_k)_{k\in\mathbb{N}^*}=\Big(\frac{4 n_k^2 \pi^2 }{L_{N+1}^2}\Big)_{k\in J}$ is such that $\mu_k\sim k^2$, {\it i.e.} there exist $C_1,C_2>0$ such that $$C_1k^2\leq \mu_k\leq C_2 k^2,\ \ \ \ \ \ \forall k\in\mathbb{N}^*.$$ \end{assumptionA}
When Assumptions A are satisfied, we define $\{\varphi_k\}_{k\in \mathbb{N}^*}$ such that \begin{equation}\label{eigen}\begin{split}\begin{cases} \varphi_1^j=\frac{1}{\sqrt{{(N+\widetilde N)L_j}}},\ \ \ \ \ &\forall\ 1\leq j\leq N+\widetilde N,\\ \varphi_k^1=\alpha_k\cos(\sqrt\mu_k x),\\ \varphi_k^j=\alpha_k\frac{\cos(\sqrt\mu_k L_1)}{\cos(\sqrt\mu_k L_j)}\cos(\sqrt\mu_k x),\ \ \ \ \ \ \ \ \ \ \ \ &\forall\ 2\leq j\leq N,\\ \varphi_k^j=\alpha_k\frac{\cos(\sqrt\mu_k L_1)}{\cos(\sqrt\mu_k c_j)}\cos(\sqrt\mu_k (x+c_j)\Big),\ \ \ \ \ \ \ \ \ \ \ \ \ \ &\forall\ N+1\leq j\leq N+\widetilde N\\ \end{cases}\end{split}\end{equation}
with $\alpha_k\in\mathbb{C}$ such that $\|\varphi_k\|_{L^2_p}=1$ and for every $k\in \mathbb{N}^*\setminus\{1\}$.
\begin{lemma} Let $\mathscr{S}$ be a star graph satisfying Assumptions A. The sequence $\{\varphi_k\}_{k\in \mathbb{N}^*}$ is an orthonormal system of $L_p^2$ made by eigenfunctions of the Laplacian $A$ corresponding to the eigenvalues $(\mu_k)_{k\in\mathbb{N}^*}$. \end{lemma} \begin{proof} We notice that any eigenfunction $f=(f^1,...,f^{N+\widetilde N})$ of $A$ corresponding to an eigenvalue $\mu$ has to be such that $f^j$ has period $\frac{2\pi}{L_j}$ for every $N+1\leq j\leq N+\widetilde N$. Thus, $$\sqrt\mu\in\bigcap_{j=N+1 }^{N+\widetilde N}\Big\{\frac{2 m\pi}{L_j}\Big\}_{m\in\mathbb{N}^*}\supseteq \Big\{\frac{2n_k\pi}{L_{N+1}}\Big\}_{k\in J}.$$ Thanks to the Neumann boundary conditions in $\{v_j\}_{j\leq N}$ and to the periodicity conditions in $\{e_j\}_{N+1\leq j\leq N+\widetilde N}$, there exist $c_j\in[0,L_j]$ for any ${N+1\leq j\leq N+\widetilde N}$ such that \begin{equation*}\begin{split}\begin{cases} f^j=\alpha_j\cos(\sqrt\mu x),\ \ \ \ \ \ \ \ &1\leq j\leq N,\\ f^j=\alpha_j\cos(\sqrt\mu (x+c_j))+\beta_j\sin(\sqrt\mu (x+c_j)),\ \ \ \ \ \ \ \ \ &N+1\leq j\leq N+\widetilde N,\\ \end{cases}\end{split}\end{equation*} with suitable $\{\alpha_j\}_{j\neq N+\widetilde N},\{\beta_j\}_{N+1\leq j\neq N+\widetilde N}\subset\mathbb{C}$. The Neumann-Kirchhoff boundary conditions in $v$ yield \begin{equation*}\begin{split}\begin{cases}\alpha_1\cos(\sqrt\mu L_1)=\alpha_j\cos(\sqrt\mu L_j),\ \ \ \ \ \ \ \ \ &\forall 2\leq j\leq N,\\ \alpha_1\cos(\sqrt\mu L_1)=\alpha_j\cos(c_j)+\beta_j\sin(c_j),\ \ \ \ \ \ \ \ \ &\forall N+1\leq j\leq N+\widetilde N.\\\end{cases}\end{split}\end{equation*} When $\beta_j=0$ for every $N+1\leq j\leq N+\widetilde N$, the last identities implies \begin{equation*}\begin{split}\begin{cases} f^1=\alpha_j\cos(\sqrt\mu x),\\ f^j=\alpha_j\frac{\cos(\sqrt\mu L_1)}{\cos(\sqrt\mu L_j)}\cos(\sqrt\mu x),\ \ \ \ \ \ \ \ \ \ \ \ &\forall 2\leq j\leq N,\\ f^j=\alpha_j\frac{\cos(\sqrt\mu L_1)}{\cos(\sqrt\mu c_j)}\cos(\sqrt\mu (x+c_j)),\ \ \ \ \ \ \ \ \ \ \ \ &\forall 2\leq j\leq N.\\ \end{cases}\end{split}\end{equation*} We recall that the numbers $c_j\in[0,L_j]$ for every ${N+1\leq j\leq N+\widetilde N}$ are such that $$\sum_{j=1}^{N}\tan(\sqrt\mu L_j)=\sum_{j=N+1}^{N+\widetilde N}\tan(\sqrt\mu c_j),\ \ \ \ \ \ \ \forall \sqrt\mu\in \Big\{\frac{2n_k\pi}{L_{N+1}}\Big\}_{k\in J}.$$ Thus, the second condition characterizing the Neumann-Kirchhoff boundary conditions is verified when $\beta_j=0$. As a consequence, $\{\varphi_k\}_{k\in \mathbb{N}^*}$ is composed by eigenfunctions of $A$. The orthonormality follows from the fact that $\big\{\cos\big(\frac{2\pi k}{L}x\big)\big\}_{k\in\mathbb{N}^*}$ is an orthogonal family in $L^2([0,L],\mathbb{C})$ with $L>0$. \qedhere \end{proof}
Equivalently to Proposition \ref{lauraT}, we have the following well-posedness result.
\begin{prop}\label{lauraS} Let the star graph $\mathscr{S}$ satisfy Assumptions A. Let $B$ be a bounded symmetric operator in $L^2_p$ such that $$B:\mathscr{H}(\upvarphi)\longrightarrow \mathscr{H}(\upvarphi),\ \ \ \ \ \ \ \ B:H^2_{\mathscr{S}}(\upvarphi)\longrightarrow H^2_{\mathscr{S}}(\upvarphi),\ \ \ \ \ \ \ \ B:H_{\mathscr{S}}^{3}(\upvarphi)\longrightarrow H_p^{3}\cap H^2_{\mathscr{S}}(\upvarphi).$$
Let $\psi_0\in H^{3}_{\mathscr{S}}(\upvarphi)$ and $u\in L^2((0,T),\mathbb{R})$. There
exists a unique mild solution $\psi\in C^0([0,T],H^3_{\mathscr{S}}(\upvarphi))$ of (\ref{mainS}) with initial data $\psi_0$. The flow of \eqref{mainS} on $\mathscr{H}(\upvarphi)$ can be extended to a unitary flow $\Gamma_t^{u}$ with respect to the $L^2_p-$norm such that $\Gamma_t^{u}\psi_0=\psi(t)$ for any solution $\psi$ of \eqref{mainS} with initial data $\psi_0\in \mathscr{H}(\upvarphi)$. \end{prop} \begin{proof} The proof follows from the same arguments adopted in Proposition \ref{lauraT}. First, we notice that $A$ is self-adjoint in $\mathscr{H}(\upvarphi)$ and $B$ is bounded symmetric since $B:\mathscr{H}(\upvarphi)\rightarrow\mathscr{H}(\upvarphi)$. Second, we can define an unitary flow for the dynamics of the equation in $\mathscr{H}(\upvarphi)$ as in the proof of the mentioned proposition.
\needspace{3\baselineskip}
\noindent
{\bf 1) Regularity of the integral term in the mild solution.} Let $\psi\in C^0([0,T],H_{\mathscr{S}}^3(\upvarphi))$ with $T>0$. We notice $B\psi(s) \in H^{3}_p\cap H_{\mathscr{S}}^2(\upvarphi)$ for almost every $s\in (0,t)$ and $t\in(0,T)$. Let $G(t)=\int_0^t e^{i\Delta(t-s)}u(s)B\psi(s,x)ds$ so that
\begin{equation*}\begin{split}
\|G(t)\|_{(3)}&=\Big(\sum_{k\in\mathbb{N}^*}\Big|k^3\int_0^t e^{i\mu_ks}\langle\varphi_k,u(s)B\psi(s,\cdot)\rangle_{L^2_p}
ds\Big|^2\Big)^\frac{1}{2}.\\
\end{split}\end{equation*}
Let $f(s,\cdot):=u(s)B\psi(s,\cdot)$. We define $\partial_x f(s)=(\partial_x f^1(s),...,\partial_x f^N(s))$ the derivative of $f(s)$. Thanks to the validity of Assumptions A, we have $\sqrt\mu_k\sim k$ and there exists $C_1>0$ such that, for every $k\in\mathbb{N}^*\setminus\{1\},$
\begin{equation*}\begin{split}
&\left|k^3\int_0^t e^{i\mu_ks}\langle\varphi_k,f(s)\rangle_{L^2_p} ds\right|
\leq\frac{C_1}{k}\sum_{j=1}^{N+\widetilde N}\left(
\left|\partial_{x}\varphi_k^j(L_j)\int_0^t e^{i\mu_ks}\partial_{x}^2f^j(s,L_j)ds\right|\right.\\
&+\left.\left|\partial_{x}\varphi_k^j(0)\int_0^t e^{i\mu_ks}\partial_{x}^2f^j(s,0)ds\right|+ \left|\int_0^t
e^{i\mu_ks}\int_{0}^{L_j}\partial_{x}\varphi_k^j(y)\partial_{x}^3f^j(s,y)dyds\right|\right).\\
\end{split}\end{equation*}
The argument of \cite[Remark\ 3.4]{mio5} yields that $\partial_{x}^3f(s,\cdot)\in\overline{span\big\{\mu_k^{-1/2}\partial_{x}\varphi_k:\ k\in\mathbb{N}^*\big\}}^{ L^2}$ for almost every $s\in (0,t)$ and $t\in(0,T)$, and there exists $C_2>0$ such that
\begin{equation*}\begin{split}
\|G(t)\|_{(3)}\leq &C_2\sum_{j=1}^{N+\widetilde N}\Big(\Big\|\int_0^t\partial_{x}^2 f^j(s,0)e^{i \mu_{(\cdot)}s}ds
\Big\|_{\ell^2}+\Big\|\int_0^t\partial_{x}^2 f^j(s,L_j)e^{i \mu_{(\cdot)}s}ds \Big\|_{\ell^2}\Big)\\
&+C_2\Big\|\int_0^t\big\langle {\mu_{(\cdot)}^{-1/2}}\partial_x\varphi_{(\cdot)}(s),\partial_{x}^3 f(s)\big\rangle_{L^2_p} e^{i
\mu_{(\cdot)}s}ds\Big\|_{\ell^2}.\\
\end{split}\end{equation*}
From \cite[Proposition\ B.6]{mio3}, there exists $C_3(t)>0$ uniformly bounded for $t$ in bounded intervals such
that $\|G\|_{(3)}\leq C_3(t)\|f(\cdot,\cdot)\|_{L^2((0,t),H^3_p)}.$ The provided upper bounds are uniform and the Dominated Convergence Theorem leads to $G\in C^0([0,T], H^3_{\mathscr{S}}(\upvarphi))$.
\noindent
{\bf 2) Conclusion.} We proceed as in the second point of the proof of Proposition \ref{lauraT}. Let $\psi_0\in H^{3}_{\mathscr{S}}(\upvarphi)$. We consider the map $F:\psi \in C^0([0,T],H^{3}_{\mathscr{S}}(\upvarphi))\mapsto\phi\in C^0([0,T],H^{3}_{\mathscr{S}}(\upvarphi)) $ with
$$\phi(t)=F(\psi)(t)=e^{i\Delta t}\psi_0-\int_0^te^{i\Delta(t-s)}u(s) B\psi(s)ds,\ \ \ \ \ \forall t\in [0,T].$$
Let $L^\infty(H^{3}_\mathscr{S}(\upvarphi)):=L^\infty((0,T),H^{3}_\mathscr{S}(\upvarphi))$. For every $\psi_1,\psi_2\in C^0([0,T],H^{3}_{\mathscr{S}}(\upvarphi))$, thanks to {\bf 1)}, there exists $C(t)>0$ uniformly bounded for $t$ lying on bounded intervals such that
\begin{equation*}\begin{split}
&\|F(\psi_1)-F(\psi_2)\|_{L^\infty(H^{3}_\mathscr{S}(\upvarphi))}\leq C(T)\|u\|_{L^2((0,T),\mathbb{R})}{\, \vert\kern-0.25ex\vert\kern-0.25ex\vert\, }
B{\, \vert\kern-0.25ex\vert\kern-0.25ex\vert\, }_{\mathcal{L}(H^{3}_{\mathscr{S}}(\upvarphi),H_p^{3})} \|\psi_1-\psi_2\|_{L^\infty(H^{3}_\mathscr{S}(\upvarphi))}.\\
\end{split}\end{equation*}
The Banach Fixed Point Theorem leads to the claim as in the mentioned proof. \qedhere \end{proof}
By keeping in mind the definition of global exact controllability provided in Definition \ref{exact}, we present the following result.
\begin{teorema}\label{globalestella} Let the hypotheses of Proposition $\ref{lauraS}$ be satisfied. We also assume that \begin{enumerate}
\item there exists $C>0$ such that $|\langle\varphi_k,B\varphi_1\rangle_{L^2_p}|\geq\frac{C}{k^{3}}$ for every $k\in\mathbb{N}^*$;
\item for every $(j,k),(l,m)\in \mathbb{N}^2$ such that $(j,k)\neq (l,m)$, $j<k$, $l<m$ and $\mu_j-\mu_k=\mu_j-\mu_m,$ it holds $$\langle\varphi_j,B\varphi_j\rangle_{L^2_p}-\langle\varphi_k,B\varphi_k\rangle_{L^2_p}-\langle\varphi_l,B\varphi_l\rangle_{L^2_p}+\langle\varphi_m,B\varphi_m\rangle_{ L^2_p}\neq 0.$$
\end{enumerate}
The (\ref{mainS}) is globally exactly controllable in $H^3_\mathscr{S}(\upvarphi)$.
\end{teorema} \begin{proof}{\bf 1) Local exact controllability.} The statement follows as Theorem \ref{globalegirino}. First, for $\epsilon,T>0$, the local exact controllability in
$O_{\epsilon,T}^{3}:=\big\{\psi\in H_{\mathscr{S}}^{3}(\upvarphi)\big|\ \|\psi\|_{L^2_p}=1,\ \|\psi
-\varphi_1(T)\|_{(3)}<\epsilon\big\}$ with $\varphi_1(T)=e^{-i\mu_1T}\varphi_1$ is ensured by proving the surjectivity of the map
$$\gamma:L^2((0,T),\mathbb{R})\longrightarrow T_{\updelta}Q=\{{\bf x}:=\{x_k\}_{k\in\mathbb{N}^*}\in h^3(\mathbb{C})\ |\ ix_1\in\mathbb{R}\},$$
the sequence of elements $\gamma_{k}(v):=
-i\int_{0}^Tv(\tau)e^{i(\mu_k-\mu_1)s}d\tau B_{k,1}$ with $B_{k,1}:=\langle\varphi_k,B\varphi_1\rangle_{L^2_p}$ for $k\in\mathbb{N}^*$. The surjectivity of $\gamma$ corresponds to the solvability of the moments problem
\begin{equation}\begin{split}\label{mome1bis}
{x_{k}}B_{k,1}^{-1}=-i\int_{0}^Tu(\tau)e^{i(\mu_k-\mu_1)\tau}d\tau,\ \ \ \ \ \ \ \ \ \ \forall \{x_{k}\}_{k\in\mathbb{N}^*}\in T_{\updelta}Q\subset h^3.
\\
\end{split}\end{equation}
As there exists $C>0$ such that $|\langle\varphi_k,B\varphi_1\rangle_{L^2_p}|\geq\frac{C}{k^3}$ for every $k\in\mathbb{N}^*$, we have $\big\{x_k B_{k,1}^{-1}\big\}_{k\in\mathbb{N}^*}\in \ell^2$ and $i{x_{1}}B_{k,1}^{-1}\in\mathbb{R}.$ The solvability of $(\ref{mome1bis})$ is guaranteed by \cite[Proposition\ B.5]{mio3} since
$$\inf_{k\in\mathbb{N}^*}|\mu_{k+1}-\mu_k|\geq {\pi^2}\min\{L_j^{-2}:\ {N+1\leq j\leq N+\widetilde N}\}>0.$$
\noindent
{\bf 2) Global exact controllability.} The global exact controllability in $H^3_\mathscr{S}(\upvarphi)$ is ensured as in the second point of the proof of Theorem \ref{globalegirino} by considering Remark \ref{approxS} instead of Remark \ref{approxT}. \qedhere \end{proof} \begin{oss} Let $\{L_j\}_{1\leq j\leq N+\widetilde N}$ be such that $\frac{L_{N+1}}{L_j}\in\mathbb{Q}\ $ for any $1\leq j\leq N+\widetilde N$. We notice that Assumptions A are satisfied with $c_j=0$ for every $N+1\leq j\leq N+\widetilde N$. Indeed, let $l_j$ be the numbers defined in \eqref{multipliers}. The sequence $$(\mu_k)_{k\in\mathbb{N}^*}:=\Big\{\frac{4\widetilde n_k^2\pi^2}{L_{N+1}^2}\Big\}_{k\in\mathbb{N}^*}\ \ \ \ \ \ \text{with }\ \ \ \ \widetilde n_k:=(k-1)\prod_{j=1}^{N+\widetilde N}l_j\frac{L_{N+1}}{L_j}$$ is composed by eigenvalues. The corresponding eigenfunctions $(\varphi_k)_{k\in\mathbb{N}^*}$ are provided in \eqref{eigen}. In this framework, $$\mu_k\sim k^2,\ \ \ \ \ \ \ \tan(\mu_kL_j)=0\ \ \ \ \ \ \ \ \forall k\in\mathbb{N}^*,\ \ \ 1\leq j\leq N.$$ Thus, the validity of Assumptions A is ensured with $c_j=0$ for every $N+1\leq j\leq N+\widetilde N$. \end{oss}
\begin{oss} Let $\mathscr{S}$ satisfy Assumptions A. We consider $\{L_j\}_{1\leq j\leq N+\widetilde N}$ being such that $\frac{L_{N+1}}{L_j}\in\mathbb{Q}\ $ for any $1\leq j\leq N+\widetilde N$ so that the previous remark is verified. Let $\widetilde B:\psi\longmapsto (V^1\psi^1,...,V^{N+\widetilde N}\psi^{N+\widetilde N})$ be such that \begin{equation*}\begin{split}\begin{cases}V^j(x)=x^2(x-L_j)^2,\ \ \ \ \ &\forall 1\leq j\leq N,\\ V^j(x)=\sum_{n\in\mathbb{N}}(x-nL_j)^2(x-(n+1)L_j)^2\upchi_{[nL_j,(n+1)L_j]}(x), &\forall N+1\leq j\leq N+\widetilde N.\end{cases}\end{split}\end{equation*} If we consider the operator $B$ on $L^2_p$ such that $B\psi=\sum_{j=1}^{+\infty}\varphi_j\langle\varphi_j,\widetilde B\psi\rangle_{L_p^2},$ then the corresponding \eqref{mainS} is well-posed and globally exactly controllable in the space $H^4_\mathscr{S}(\upvarphi)$. The result is proved by using the techniques leading to Proposition \ref{lauraT}, Proposition \ref{lauraS}, Theorem \ref{globalegirino} and Theorem \ref{globalestella}. In the next section, we ensure in the same way the well-posedness and the global exact controllability in $H^s_\mathscr{G}$ for suitable $s\geq 3$ with abstract $\mathscr{G}$ and $B$. \end{oss} \section{Generic graphs} \label{sec4} In this section, we study the controllability of the \eqref{mainx1} for a general graph $\mathscr{G}$ made by $N$ finite edges $\{e_j\}_{1\leq j\leq N}$ of lengths $\{L_j\}_{1\leq j\leq N}$, $\widetilde N$ half-lines $\{e_j\}_{N+1\leq j\leq N+\widetilde N}$ and $M$ vertices $\{v_j\}_{1\leq j\leq M}$. For every vertex $v$, we denote
$N(v):=\big\{l \in\{1,...,N\}\ |\ v\in e_l\big\}$. We respectively call $V_e$ and $V_i$ the external and the internal vertices of $\mathscr{G}$, {\it i.e.}
$$V_e:=\big\{v\in\{v_j\}_{1\leq j\leq M}\ |\ \exists ! e\in\{e_j\}_{1\leq j\leq N}: v\in e\big\},\ \ \ \ \ V_i:=\{v_j\}_{1\leq j\leq M}\setminus V_e.$$
We consider the bilinear Schr\"odinger equation \eqref{mainx1} in $L^2_p$ for a general graph $\mathscr{G}$. The Laplacian $A=-\Delta$ is equipped with {\it Dirichlet} or {\it Neumann} boundary conditions in the external vertices, and the internal vertices are equipped with {\it Neumann-Kirchhoff} boundary conditions. More precisely, a vertex $v\in V_i$ is said to be equipped with Neumann-Kirchhoff boundary conditions when every function $f=(f^1,...,f^N)\in D(A)$ is continuous in $v$ and \begin{equation*}\begin{split}
\sum_{l\in N(v)}\frac{\partial f^l}{\partial x}(v)=0,
\end{split}
\end{equation*} when the derivatives are assumed to be taken in the directions away from the vertex. We respectively call ($\mathcal{D}$), ($\mathcal{N}$) and ($\mathcal{N}\mathcal{K}$) the {\it Dirichlet}, {\it Neumann} and {\it Neumann-Kirchhoff} boundary conditions characterizing $D(A)$.
We say that a vertex $v$ of $\mathscr{G}$ is equipped with one of the previous boundaries, when each $f\in D(A)$ satisfies it in $v$. We say that $\mathscr{G}$ is equipped with ($\mathcal{D}$) (or ($\mathcal{N}$)) when, for every $f\in D(A)$, the function $f$ satisfies ($\mathcal{D}$) (or ($\mathcal{N}$)) in every $v\in V_e$ and verifies ($\mathcal{N}\mathcal{K}$) in every $v\in V_i$. In addition, the graph $\mathscr{G}$ is said to be equipped with ($\mathcal{D}$/$\mathcal{N}$) when, for every $f\in D(A)$ and $v\in V_e$, the function $f$ satisfies ($\mathcal{D}$) or ($\mathcal{N}$) in $v$, and $f$ verifies ($\mathcal{N}\mathcal{K}$) in every $v\in V_i$.
Let $\upvarphi:=\{\varphi_k\}_{k\in\mathbb{N}^*}$ be an orthonormal system of $L^2_p$ made by some eigenfunctions of $A$ and let $\{\mu_{k}\}_{k\in\mathbb{N}^*}$ be the corresponding eigenvalues. Let $[r]$ be the entire part of $r\in\mathbb{R}$. We define $\mathscr{G}(\upvarphi)=\bigcup_{k\in\mathbb{N}^*}supp(\varphi_k)$ and we respectively denote by $V_e(\upvarphi)$ and $V_i(\upvarphi)$ the external and internal vertices of $\mathscr{G}(\upvarphi)$. For $s>0$, we introduce the space \begin{equation*}
\begin{split}
H^s_{\mathcal{N}\mathcal{K}}(\upvarphi):=\Big\{&\psi\in \mathscr{H}(\upvarphi)\cap H^s_p\ |\ \partial_x^{2n}\psi\text{ is continuous in }v,\
\forall n\in\mathbb{N},\ n<\big[({s+1})/{2}\big],\ \forall v\in V_i;\\ & \sum_{j\in N(v)}\partial_{x}^{2n+1}\psi^j(v)=0,\ \forall n\in\mathbb{N},\ n<\big[{s}/{2}\big],\ \forall v\in V_i\Big\}.\\
\end{split} \end{equation*}
\begin{osss}\label{subgraph} We notice the following facts. \begin{itemize}
\item $\mathscr{G}(\upvarphi)$ is a finite or infinite sub-graph of $\mathscr{G}$ whose structure depends on the orthonormal family $\upvarphi$.
\item The functions belonging to $\mathscr{H}(\upvarphi)$, $H^s_\mathscr{G}(\upvarphi)$ and $H^s_{\mathcal{N}\mathcal{K}}(\upvarphi)$ can be considered as functions with domain $\mathscr{G}(\upvarphi)$.
\item $\mathscr{G}(\upvarphi)$ shares some external and internal vertices with $\mathscr{G}$. Its new external vertices are $V_e(\upvarphi)\setminus V_e$.
\item Let $L^2_p(\mathscr{G}(\upvarphi),\mathbb{C})$ be the space defined from the identities \eqref{spazio} by considering the graph
$\mathscr{G}(\upvarphi)$. Each $\varphi_k|_{\mathscr{G}(\upvarphi)}$ is an eigenfunction of a Laplacian $\widetilde A$ defined on $L^2_p(\mathscr{G}(\upvarphi),\mathbb{C})$ as follows. The domain $D(\widetilde A)$ is composed by the restriction in $\mathscr{G}(\upvarphi)$ of those $H^2_p$ functions satisfying $(\mathcal{D})$ in the vertices $V_e(\upvarphi)\setminus V_e$ and verifying the same boundary conditions defining $D(A)$ in the vertices $V_i(\upvarphi)\cup \big(V_e(\upvarphi)\cap V_e\big)$.
\end{itemize}
\end{osss} From now on, when we claim that the vertices of $\mathscr{G}(\upvarphi)$ are equipped with any type of boundary conditions, this is done in the meaning of Remark \ref{subgraph}. Let $\eta>0,$ $a\geq 0$ and $$I:=\{(j,k)\in(\mathbb{N}^*)^2:j< k\}.$$
\needspace{3\baselineskip} \begin{assumptionI}[$\upvarphi,\eta$] Let $B$ be a bounded and symmetric operator in $L^2_p$ satisfying the following conditions.
\begin{enumerate}
\item There exists $C>0$ such that $|\langle\varphi_k,B\varphi_1\rangle_{L^2_p}|\geq\frac{C}{k^{2+\eta}}$ for every $k\in\mathbb{N}^*$.
\item For every $(j,k),(l,m)\in I$ such that $(j,k)\neq(l,m)$ and $\mu_j-\mu_k=\mu_j-\mu_m,$ it holds $\langle\varphi_j,B\varphi_j\rangle_{L^2_p}-\langle\varphi_k,B\varphi_k\rangle_{L^2_p}-\langle\varphi_j,B\varphi_j\rangle_{L^2_p}+\langle\varphi_m,B\varphi_m\rangle_{L^2_p }\neq 0.$
\end{enumerate}
\end{assumptionI}
\begin{assumptionII}[$\upvarphi,\eta,a$] We have $B:\mathscr{H}(\upvarphi)\rightarrow\mathscr{H}(\upvarphi)$ and
$Ran(B|_{H^2_{\mathscr{G}}(\upvarphi)})\subseteq H^2_{\mathscr{G}}(\upvarphi).$ In addition, one of the following points is satisfied.
\begin{enumerate}
\item When $\mathscr{G}(\upvarphi)$ is equipped with ($\mathcal{D}$/$\mathcal{N}$) and $a+\eta\in(0, 3/2)$, there exists
$d\in[\max\{a+\eta,1\},3/2)$ such that $$Ran(B|_{H_{\mathscr{G}}^{2+d}(\upvarphi)})\subseteq H^{2+d}_p\cap H^2_{\mathscr{G}}(\upvarphi).$$
\item When $\mathscr{G}(\upvarphi)$ is equipped with ($\mathcal{N}$) and $a+\eta\in(0, 7/2)$, there exist
$d\in[\max\{a+\eta,2\},7/2)$ and $d_1\in(d,7/2)$ such that $$Ran(B|_{ H^{d_1}_{\mathcal{N}\mathcal{K}}(\upvarphi)})\subseteq
H^{d_1}_{\mathcal{N}\mathcal{K}}(\upvarphi),\ \ \ \ \ Ran(B|_{H_{\mathscr{G}}^{2+d}(\upvarphi)})\subseteq H^{2+d}_p\cap H^{1+d}_{\mathcal{N}\mathcal{K}}(\upvarphi)\cap H^2_{\mathscr{G}}(\upvarphi).$$
\item When $\mathscr{G}$ is equipped with ($\mathcal{D}$) and $a+\eta\in(0, 5/2)$, there exists
$d\in[\max\{a+\eta,1\},5/2)$ such that $$Ran(B|_{H_{\mathscr{G}}^{2+d}(\upvarphi)})\subseteq H^{2+d}_p\cap H^{1+d}_{\mathcal{N}\mathcal{K}}(\upvarphi)\cap H^2_{\mathscr{G}}(\upvarphi).$$ If $d\geq 2$, then there exists $d_1\in(d,5/2)$ such
that $Ran(B|_{H^{d_1}_p\cap \mathscr{H}(\upvarphi)})\subseteq H^{d_1}_p\cap \mathscr{H}(\upvarphi).$
\end{enumerate} \end{assumptionII}
From now on, we omit the terms $\upvarphi,$ $\eta$ and $a$ from the notations of Assumptions I and Assumptions II when their are not relevant.
We are finally ready to present some {\it interpolation properties} for the spaces $H^s_{\mathscr{G}}(\upvarphi)$ with $s>0$. \begin{prop}\label{bor}Let $\upvarphi:=\{\varphi_k\}_{k\in\mathbb{N}^*}$ be an orthonormal system of $L^2_p$ made by eigenfunctions of $A$.
\noindent
{\bf 1)} If the graph $\mathscr{G}(\upvarphi)$ is equipped with ($\mathcal{D}$/$\mathcal{N}$), then
$$H^{s_1+s_2}_{\mathscr{G}}(\upvarphi)=H_{\mathscr{G}}^{s_1}(\upvarphi)\cap H^{s_1+s_2}_p \ \ \ \text{for}\ \ \ s_1\in\mathbb{N},\ s_2\in[0,1/2).$$
\noindent
{\bf 2)} If the graph $\mathscr{G}(\upvarphi)$ is equipped with ($\mathcal{N}$), then
$$H^{s_1+s_2}_{\mathscr{G}}(\upvarphi)=H_{\mathscr{G}}^{s_1}(\upvarphi)\cap H^{s_1+s_2}_{\mathcal{N}\mathcal{K}}(\upvarphi) \ \ \ \text{for}\ \ \ s_1\in 2\mathbb{N}\,\ s_2\in[0,3/2).$$
\noindent
{\bf 3)} If the graph $\mathscr{G}(\upvarphi)$ is equipped with ($\mathcal{D}$), then
$$H^{s_1+s_2+1}_{\mathscr{G}}(\upvarphi)=H_{\mathscr{G}}^{s_1+1}(\upvarphi)\cap H^{s_1+s_2+1}_{\mathcal{N}\mathcal{K}}(\upvarphi) \ \ \ \text{for}\ \ \ s_1\in 2\mathbb{N},\ s_2\in[0,3/2).$$ \end{prop} \begin{proof} Let us start by considering the first point of the statement. We denote by $\{e_j\}_{j\leq N_1}$ the finite edges composing $\mathscr{G}(\upvarphi)$, while $\{e_j\}_{N_1+1\leq j\leq N_1+\widetilde N_1}$ are its infinite edges corresponding to the periods $\{L_j\}_{N_1+1\leq j\leq N_1+\widetilde N_1}$. We define a compact graph $\widetilde \mathscr{G}(\upvarphi)$ from $\mathscr{G}(\upvarphi)$ as follows (see Figure \ref{parametrizzazionetadpole2} for further details). For every $N_1+1\leq j\leq N_1+\widetilde N_1$, we cut the edge $e_j$ at distance $L_j$ from the internal vertex of $\mathscr{G}(\upvarphi)$ where $e_j$ is connected. As $\widetilde \mathscr{G}(\upvarphi)$ is a compact graph, the space $L^2_p(\widetilde \mathscr{G}(\upvarphi),\mathbb{C})$ corresponds to $L^2(\widetilde \mathscr{G}(\upvarphi),\mathbb{C})$. There, we consider a self-adjoint Laplacian $\widetilde A$ being defined as follows. Every internal vertex of $\widetilde \mathscr{G}(\upvarphi)$ is equipped with Neumann-Kirchhoff boundary conditions. Every external vertex of $\widetilde \mathscr{G}(\upvarphi)$ belonging to $V_e(\upvarphi)$ is equipped with the same boundary conditions of $ \mathscr{G}(\upvarphi)$, while every other external vertex is equipped with ($\mathcal{D}$).
Finally, we denote by $H^s_{\widetilde \mathscr{G}(\upvarphi)}:=D(|\widetilde A|^\frac{s}{2})$ for every $s>0$. \begin{figure}
\caption{The figure represents an example of definition of the compact graph $\widetilde \mathscr{G}(\upvarphi)$ (on the right) from a specific infinite graph $\mathscr{G}(\upvarphi)$ (on the left) composed by $ N_1=11$ finite edges and $\widetilde N_1=2$ infinite edges. We also underline the boundary conditions characterizing $D(\widetilde A)$ in $\widetilde \mathscr{G}(\upvarphi)$.}
\label{parametrizzazionetadpole2}
\end{figure}
\noindent Afterwards, for every edge $e_j$ with $N_1+1\leq j\leq N_1+\widetilde N_1$, we define a ring $\widetilde e_j$ having length $L_j$. We consider on $L^2(\widetilde e_j,\mathbb{C})$ a self-adjoint Laplacian $A_j$ with domain $D(A_j)=H^2(\widetilde e_j,\mathbb{C})$ and we
denote by $H^s_{\widetilde e_j}:=D(|A_j|^\frac{s}{2})$ for every $s>0.$ On $L^2((0,L_j),\mathbb{C})$ with $N_1+1\leq j\leq N_1+\widetilde N_1$, we consider a Dirichlet Laplacian $A^\mathcal{D}_j$ and Neumann Laplacian $A^\mathcal{N}_j$, while we call, for every $s>0$,
$$H^s_{e_j^\mathcal{D}}:=D(|A^\mathcal{D}_j|^\frac{s}{2}),\ \ \ \ \ \ \ \ \ \ \ \ H^s_{e_j^\mathcal{N}}:=D(|A^\mathcal{N}_j|^\frac{s}{2}).$$ Now, for every $\psi=(\psi^1,...,\psi^{N_1+\widetilde N_1})\in H^{s_1+s_2}_{\mathscr{G}}(\upvarphi)$ with $s_1\in\mathbb{N}$ and $s_2\in[0,1/2)$, there exist \begin{equation*}\begin{split} \begin{cases}\psi_1=(\psi_1^1,...,\psi_1^{N_1+\widetilde N_1})\in H^{s_1+s_2}_{\widetilde \mathscr{G}(\upvarphi)}, \ \ \ \ \ \ \ \\ f^j\in H^{s_1+s_2}_{\widetilde e_j},\ \ \ &\forall N_1+1\leq j\leq N_1+\widetilde N_1,\\ g^j\in H^{s_1+s_2}_{e_j^\mathcal{D}},\ \ \ &\forall N_1+1\leq j\leq N_1+\widetilde N_1,\\ h^j\in H^{s_1+s_2}_{ e_j^\mathcal{N}},\ \ \ &\forall N_1+1\leq j\leq N_1+\widetilde N_1,\end{cases} \end{split} \end{equation*} \begin{equation*}\begin{split} \text{such that} \ \ \begin{cases} \psi^j\equiv\psi^j_1,\ \ \ \ \ &\forall 1\leq j\leq N_1,\\ \psi^j(x)=\psi^j_1(x)+g^j(x)+h^j(x),\ \ \ \ &\forall x\in (0,L_j),\ \ \ N_1+1\leq j\leq N_1+\widetilde N_1,\\ \psi^j(x)=f^j\big(x-\big[\frac{x}{L_j }\big]\big),\ \ \ \ &\forall x\in (L_j,+\infty),\ \ \ N_1+1\leq j\leq N_1+\widetilde N_1.\\ \end{cases} \end{split} \end{equation*} The last decomposition yields that $H^{s_1+s_2}_{\mathscr{G}}(\upvarphi)$ can be identified with a suitable subspace of $$H^{s_1+s_2}_{\widetilde \mathscr{G}(\upvarphi)}\times\prod_{j=N_1+1}^{N_1+\widetilde N_1} \big(H^{s_1+s_2}_{\widetilde e_j}\times H^{s_1+s_2}_{e_j^\mathcal{D}}\times H^{s_1+s_2}_{e_j^\mathcal{N}}\big) .$$ Thanks to the first point of \cite[Proposition\ 4.2]{mio3}, we have \begin{equation*}\begin{split} \begin{cases} H^{s_1+s_2}_{\widetilde \mathscr{G}(\upvarphi)}=H^{s_1}_{\widetilde \mathscr{G}(\upvarphi)}\cap H^{s_1+s_2}(\widetilde \mathscr{G}(\upvarphi),\mathbb{C}),\\ H^{s_1+s_2}_{\widetilde e_j}=H^{s_1}_{\widetilde e_j}\cap H^{s_1+s_2}(\widetilde e_j,\mathbb{C}),\ \ \ \ \ \ \ &\forall N_1+1\leq j\leq N_1+\widetilde N_1,\\ H^{s_1+s_2}_{e_j^\mathcal{D}}=H^{s_1}_{e_j^{\mathcal{D}}}\cap H^{s_1+s_2}((0,L_j),\mathbb{C}),\ \ \ \ \ \ \ &\forall N_1+1\leq j\leq N_1+\widetilde N_1,\\ H^{s_1+s_2}_{e_j^\mathcal{N}}=H^{s_1}_{e_j^{\mathcal{N}}}\cap H^{s_1+s_2}((0,L_j),\mathbb{C}),\ \ \ \ \ \ \ &\forall N_1+1\leq j\leq N_1+\widetilde N_1.\\ \end{cases} \end{split} \end{equation*} The last relations imply that, for every $\psi\in H^{s_1+s_2}_{\mathscr{G}}(\upvarphi)$ with $s_1\in\mathbb{N}$ and $s_2\in[0,1/2)$, there holds $\psi\in H^{s_1}_{\mathscr{G}}(\upvarphi)\cap H^{s_1+s_2}_p$ achieving the proof of the first point of the proposition. The second and the third statement follow from the same techniques by respectively using the second and third point of \cite[Proposition\ 4.2]{mio3}.\qedhere \end{proof}
In the following theorem, we collect the well-posedness and the controllability result for the bilinear Schr\"odinger equation in this general framework. The well-posedness is proved exactly as \cite[Proposition\ 3.3]{mio5} by using Proposition \ref{bor} instead of \cite[Proposition\ 3.2]{mio5}. The controllability result subsequently follows from the same arguments of \cite[Theorem\ 3.6]{mio5} by considering Proposition \ref{approx} instead of \cite[Proposition\ B.2]{mio5}.
\begin{teorema}\label{generale} Let $\upvarphi:=\{\varphi_k\}_{k\in\mathbb{N}^*}$ be an orthonormal system of $L^2_p$ made by some eigenfunctions of $A$ and let $\{\mu_{k}\}_{k\in\mathbb{N}^*}$ be the corresponding eigenvalues.
\noindent {\bf 1)} Let the couple $(A,B)$ satisfy Assumptions II$(\upvarphi,\eta,\tilde d)$ with $\eta>0$ and $\tilde d\geq 0$. Let $d$ be introduced in Assumptions II and $\mu_k\sim k^2$. For every $\psi_0\in H^{2+d}_{\mathscr{G}}(\upvarphi)$ and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. There
exists a unique mild solution $\psi\in C_0([0,T],H^{s}_{\mathscr{G}}(\upvarphi))$ of the \eqref{mainx1}. In addition, the flow of \eqref{mainx1} on $\mathscr{H}(\upvarphi)$ can be extended to a unitary flow $\Gamma_t^{u}$ with respect to the $L^2_p-$norm such that $\Gamma_t^{u}\psi_0=\psi(t)$ for any solution $\psi$ of \eqref{mainx1} with initial data $\psi_0\in \mathscr{H}(\upvarphi)$.
\noindent
{\bf 2)} If there exist $C>0$ and $\tilde d\geq 0$ such that
$$ |\mu_{k+1}-\mu_k|\geq C k^{-{\tilde d}},\ \ \ \ \ \ \ \ \ \ \ \ \forall k\in\mathbb{N}^*$$
and if $(A,B)$ satisfies Assumptions I$(\upvarphi,\eta)$ and Assumptions II$(\upvarphi,\eta,\tilde d)$ for $\eta>0$, then the \eqref{mainx1} is globally exactly controllable in $H^{s}_{\mathscr{G}}(\upvarphi)$ for $s=2+d$ with $d$ from Assumptions II. \end{teorema}
\noindent {\bf Acknowledgments.} The second author was financially supported by the ISDEEC project by ANR-16-CE40-0013.
\appendix\section{Global approximate controllability } Let us denote by $U(\mathscr{H})$ the space of the unitary operators on a Hilbert space $\mathscr{H}.$ \begin{defi} Let $\upvarphi:=\{\varphi_k\}_{k\in\mathbb{N}^*}$ be an orthonormal system of $L^2_p$ made by some eigenfunctions of $A$. The \eqref{mainx1} is said to be globally approximately controllable in $H_{\mathscr{G}}^{s}(\upvarphi)$ with $s>0$ if the following assertion is verified. For every $\epsilon>0$, $\psi\in H^{s}_{\mathscr{G}}(\upvarphi)$ and $\widehat\Gamma\in U(\mathscr{H}(\upvarphi))$ such that $\widehat\Gamma\psi\in H^{s}_{\mathscr{G}}(\upvarphi)$, there exist $T>0$ and $u\in L^2((0,T),\mathbb{R})$ such that
$$\|\widehat\Gamma\psi-\Gamma^u_T\psi\|_{(s)}<\epsilon.$$ \end{defi} \begin{prop}\label{approx} Let $\upvarphi:=\{\varphi_k\}_{k\in\mathbb{N}^*}$ be an orthonormal system of $L^2_p$ made by some eigenfunctions of $A$. If the hypotheses of Theorem \ref{generale} are satisfied, then the \eqref{mainx1} is globally approximately controllable in $H^{s}_{\mathscr{G}}(\upvarphi)$ for $s=2+d$ with $d$ from Assumptions II. \end{prop} \begin{proof} The proof is the same of \cite[Proposition\ B.2]{mio5}.\qedhere\end{proof} \begin{osss}\label{approxT} Let us consider the framework introduced in Section \ref{sec2} with $\mathcal{T}$ an infinite tadpole graph.
As Proposition $\ref{approx}$, the problem (\ref{mainT}) is globally approximately controllable in $H_{\mathcal{T}}^{4}(\upvarphi)$ when the hypotheses of Theorem $\ref{globalegirino}$ are verified. Indeed, for every $(j,k),(l,m)\in \{(j,k)\in(\mathbb{N}^*)^2:j< k\}$ so that $(j,k)\neq(l,m)$ and such that $\mu_j-\mu_k-\mu_j+\mu_m={\pi^2}(j^2-k^2-l^2+m^2)=0,$ there exists $C>0$ such that
\begin{equation*}\begin{split}
&\langle \varphi_j,B\varphi_j\rangle_{L^2_p}-\langle\varphi_k,B\varphi_k\rangle_{L^2_p}-\langle\varphi_l,B\varphi_l\rangle_{L^2_p}+\langle\varphi_m,B\varphi_m\rangle_{L^2_p} =C(j^{-4}-k^{-4}-l^{-4}+m^{4}) \neq 0.
\end{split}\end{equation*}
Finally, the arguments leading to Proposition $\ref{approx}$ also ensure the claim. \end{osss} \begin{osss}\label{approxS} Let us consider the framework introduced in Section \ref{sec3} with $\mathscr{S}$ a star graph composed by a finite number of edges of finite or infinite length.
Equivalently to Remark \ref{approxT}, the (\ref{mainS}) is globally approximately controllable in $H_{\mathscr{S}}^{3}(\upvarphi)$ when the hypotheses of Theorem \ref{globalestella} are verified. Indeed, for every $(j,k),(l,m)\in \{(j,k)\in(\mathbb{N}^*)^2:j< k\}$ so that $(j,k)\neq(l,m)$ and such that $\mu_j-\mu_k-\mu_j+\mu_m=0,$ we have
\begin{equation*}\begin{split}
&\langle \varphi_j,B\varphi_j\rangle_{L^2_p}-\langle\varphi_k,B\varphi_k\rangle_{L^2_p}-\langle\varphi_j,B\varphi_j\rangle_{L^2_p}+\langle\varphi_m,B\varphi_m\rangle_{L^2_p} \neq0.
\end{split}\end{equation*} \end{osss}
\noindent {\bf Data availability.} Data sharing is not applicable to this article as no new data were created or analyzed in this study.
\end{document} | arXiv |
Giang Nguyen
Giang Thu Nguyen (born Nguyễn Thu Giang on 2 October 1985 in Hanoi) is a Vietnamese-Australian chess player and mathematician. She is a senior lecturer in Applied Mathematics at the University of Adelaide. In chess, she is a Woman FIDE Master (WFM),[1] and has represented Australia in seven Chess Olympiads. She has won the South Australian women's championship four times, and the South Australian (open) championship once.
Giang Nguyen
Giang Nguyen at the 2013 Labour Day Weekender in Adelaide
Country Australia
Born2 October 1985
TitleWoman FIDE Master (WFM) (2012)
FIDE rating2119 (Apr 2022)
Peak rating2186 (Sep 2015)[1]
This is a Vietnamese name; The family name is Nguyen.
Chess
Nguyen began playing chess at the age of nine. She won the gold medal in Vietnamese Junior Girls Under 13 Championship in 1998, and the silver medal in Vietnamese Junior Girls Under 15 Championship in 1999.
Nguyen first represented Vietnam in the World Rapid Girls Under 14 Championship (Disneyland, Paris) in 1998. Representing Vietnam, she won a gold medal at the Asian Girls Under 14 Championship in 1999, a silver medal at the 2nd Children of Asia International Children Sports Games in 2000, and a silver medal in the ASEAN Girls Under 16 Championship in 2000.
In 2001, Nguyen moved to Adelaide, Australia. She came equal third in the Australian Junior Championship in Sydney in 2002.
Since 2008, Nguyen has been playing chess under the Australian flag. She has represented Australia in seven Chess Olympiads: in 2008, 2010, 2012, 2014, 2016, 2018 and 2022.[2] In 2012, she scored 6/9 and was awarded the Woman FIDE Master (WFM) title for her result.[1]
She won the (open) South Australian championship in 2014, and the women's championship in 2003, 2006, 2007 and 2008.[3]
Mathematics
Nguyen completed her PhD in Mathematics from the University of South Australia (UniSA) in 2009 at the age of 23.[4] She is the youngest PhD graduate of UniSA and the second youngest from a South Australian university.[4] Her dissertation, Hamiltonian Cycle Problem, Markov Decision Processes and Graph Spectra, was jointly supervised by Jerzy Filar and Vlad Ejov.[5]
Nguyen is a senior lecturer in Applied Mathematics at the University of Adelaide, and a 2019 South Australian Tall Poppy Science Award recipient.[6]
References
1. Nguyen, Thu Giang FIDE Player Profile, www.fide.com
2. Nguyen Thu Giang Chess Olympiad statistics, www.olimpbase.org
3. Honour Boards, South Australian Chess Association
4. Our young Dr of Maths Archived 11 June 2020 at the Wayback Machine Alexandra Brown, UniSANews
5. Giang Nguyen at the Mathematics Genealogy Project
6. "Dr Giang Nguyen".
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• Official website
• Giang Nguyen at the Mathematics Genealogy Project
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| Wikipedia |
\begin{document}
\title{Boundary Regularity for Asymptotically Hyperbolic Metrics with Smooth Weyl Curvature} \author{Xiaoshang Jin \thanks{The author's research is partially supported by China Scholarship Council (File No.201706190079)}} \date{} \maketitle \begin{abstract} In this paper, we study the regularity of asymptotically hyperbolic metrics with Einstein condition near boundary and Weyl curvature smooth enough in arbitrary dimension. Following Michael Anderson's method, we show that $C^{m,\alpha}$ conformally compact Riemannian metrics with Einstein equation vanishing to finite order near boundary have conformal compactifications that are $C^{m+2,\alpha}$ up to the boundary when Weyl curvature is in $C^{m,\alpha}$ and the boundary metric is in $C^{m+2,\alpha}$ where $m\geq 3.$ \end{abstract}
\section{Introduction} \label{sect1} \par It is well known that there are very close connections between the hyperbolic space $\mathbb{H}^{n+1}$ and its boundary, which we know as a sphere $\mathbb{S}^n.$ In recent years, many mathematicians are more interested in conformally compact Einstein manifolds with negative scalar curvature instead of hyperbolic space. The physics community has also become interested in the compact Einstein manifolds since the introduction of the AdS/CFT correspondence proposed by Maldacena in the theory of quantum gravity in theoretic physics. In this paper, we mainly discuss the boundary regularity problem when the Weyl curvature of the compactification has some regularity. \par Let $M$ be the interior of a compact $(n+1)$-dimensional manifold $\overline{M}$ with non-empty boundary ${\partial M}$. We call a complete metric $g_+$ on $M$ is $C^{m,\alpha}$(or $W^{k,p}$) conformally compact if there exits a defining function $\rho$ on $\overline{M}$ such that the conformally equivalent metric $$g=\rho^2g_+$$ can extend to a $C^{m,\alpha}$(or $W^{k,p}$) Riemannian metric on $\overline{M}.$ The defining function is smooth on $\overline{M}$ and satisfies
\begin{equation}\label{1.1}
\left\{
\begin{array}{l}
\rho>0\ \ in \ M
\\\rho=0\ \ on\ {\partial M}
\\d\rho\neq 0\ \ on\ {\partial M}
\end{array}
\right.
\end{equation}
Here $C^{m,\alpha}$ and $W^{k,p}$ are usual H\"older space and the Sobolev space.
\par The induced metric $h=g|_{{\partial M}}$ is called the boundary metric associated to the compactification $g.$ The defining function is unique up to a multiplication by a positive function on $\overline{M}$. So the conformal class $[g]$ is uniquely determined by $g_+,$ and the conformal class $[h]$ is uniquely determined by $(M,g_+)$. We call $[h]$ the conformal infinity of $g_+.$ We are interested in Einstein manifolds, which means the metric $g_+$ also satisfies
\begin{equation}\label{1.2} Ric_{g_+}=-ng_+.
\end{equation} \par The boundary regularity problem was first raised by Fefferman and Graham in 1985. Namely, given a conformally compact Einstein manifold $(M,g_+)$ and its compactification $g,$ if the boundary metric $h$ is in $C^{m,\alpha},$ is there a $C^{m,\alpha}$ compactification of $g_+$? In fact, Fefferman and Graham noticed that if $dim M=n+1$ is odd, the boundary regularity in general breaks down at the order $n.$ When $dim M=n+1$ is even, the $C^{m,\alpha}$ compactification should exit. In \cite{A03} and \cite{A08}, M.T.Anderson solved the problem in dimension 4 by using the Bach equation in dimension 4. He only assumed the original compactification $g$ is in $W^{2,p}$ for some $p>4.$ I'm not sure whether the $W^{2,p}$ condition is good enough for the manifold to improve the boundary regularity. In \cite{Hell}, Helliwell solved the issue in all even dimensions by following Anderson's method. He considered the Fefferman-Graham ambient obstruction tensor instead of Bach tensor in higher dimensions. Helliwell assumed the compactification $g$ is at least in $C^{n,\alpha}$ for a $(n+1)$ smooth manifold. It means the original compactification is $C^{3,\alpha}$ for a smooth manifold of dimension 4. \par In this paper, we follow Anderson's approach to study the boundary regularity in general dimensions. As \cite{FG} pointed that when $dim M=n+1$ is odd, there are log terms in the asymptotic expansion of $g$ near ${\partial M}$ at the order $n$. If we add a condition of $g,$ the log term may be ruled out. By studying the equation (\ref{3.8}), we find that if the Weyl curvature and scalar curvature of $g$ is smooth enough near boundary, the log term may not exist. In \cite{J} and \cite{ST}, the equation (\ref{2.5}) tells us that in harmonic coordinates, the regularity of a metric can be improved to two orders higher than the regularity of its Weyl curvature locally in the sense of conformal transformation. The main idea of this paper is to extend the result to the manifolds with boundary. \par In the end of this paper, we prove that the regularity of defining function is the same as the new structure on $\overline{M}$ for Einstein case. It extends Helliwell's result in \cite{Hell}, where he obtained that the the regularity of defining function is the same as the original compactification. \par The equation (\ref{2.5}) holds for all manifolds, not necessary Einstein manifolds. We don't use the Einstein equation in the interior of $\overline{M},$ so we focus on metrics that satisfy the condition of that Einstein equation vanishes to finite order near boundary, that is \begin{equation}\label{1.3} Ric_{g_+}+ng_+=o(\rho^2).
\end{equation} The main result is as follows: \begin{theo}\label{theo1.1} Let $(M,g_+)$ be a conformally compact $(n+1)$-manifold with a $C^{m,\alpha}$ conformal compactification $g=\rho^2g_+$ in a given $C^\infty$ atlas $\{y^\beta\}_{\beta=0}^n$ of $\overline{M}$ near ${\partial M}$ ($m\geq 3,0<\alpha<1$). $Ric_{g_+}+ng_+=o(\rho^2).$ $\rho$ is a $C^\infty$ defining function of $y^\beta.$
If the boundary metric $h=g|_{\partial M}\in C^{m+2,\alpha}(\partial M)$ and the Weyl curvature $W$ of $g$ is in $C^{m,\alpha}(\overline{M})$ in the atlas $y^\beta,$ then there exits atlas $\{x^\beta\}$ of $\overline{M}$ near ${\partial M}$ and in the atlas $\{x^\beta\},$ $g_+$ has a $C^{m+2,\alpha}$ compactification $\tilde{g}=\tilde{\rho}^2g_+$ with boundary metric $h$. The atlas $\{x^\beta\}$ form a $C^{m+3,\alpha}$ structure of $\overline{M}.$ Further more, if $g_+$ is Einstein, $\tilde{\rho}$ is a $C^{m+2,\alpha} $ function in x-coordinates. \end{theo} \begin{rema}\label{rema1.2} If the $\dim M=n+1$ is even, $(M,g_+)$ is Einstein and $m+2\geq n,$ then the defining function in theorem(\ref{theo1.1}) $\tilde{\rho}$ is a $C^{m+3,\alpha} $ function in x-coordinates. \end{rema} \par In \cite{CDLS}, Chru\'sciel, Delay, Lee and Skinner showed a good result of the boundary regularity of conformal compact Einstein manifolds. They proved that when the boundary metrics are smooth, the $C^2$ conformally compact Einstein metrics have conformal compactifications that are smooth up to the boundary in the sense of $C^{1,\lambda}$ diffeomorphism in dimension 3 and all even dimensions, and polyhomogeneous in odd dimensions greater than 3. The $C^2$ condition is of course weaker than the $C^{n,\alpha}$ condition in Helliwell's paper. I think the $C^2$ condition should be the sharp condition. However, their result only holds for the smooth case. It is unknown whether their method can be used for proving the finite boundary regularity. In this paper, by assuming a condition of Weyl tensor, we solve the finite regularity problem for a conformal compact manifold, which need not to be Einstein, only need to satisfy (\ref{1.3}). Besides, by observing a calculation in section 3.2, we find that the log term of the formal power series of Weyl tensor vanishes if and only if the obstruction tensor of the metric vanishes. So if we assume the Weyl tensor are in $C^{n-2}$ in Theorem A in \cite{CDLS} when $n+1$ is odd and greater than 3, we can obtain an extended smooth result. That is: \begin{rema}\label{rema1.3}
Let $(M,g_+)$ be a conformally compact Einstein $(n+1)$-manifold with a $C^2$ conformal compactification $g=\rho^2g_+.$ If the boundary metric $h=g|_{\partial M}$ is smooth, then for any $\lambda>0,$ there exists $R>0$ and a $C^{1,\lambda}$ diffeomorphism $\Phi:{\partial M}_R\rightarrow {\partial M}$ such that $$\Phi^*g_+=\rho^{-2}(d\rho^2+G(\rho)).$$ Where ${\partial M}_R=\partial M\times [0,R],$ $\{G(\rho): 0<\rho\leq R\}$ is a one-parameter family of smooth Riemannian metrics on $\partial M.$ \par If dim M is even or equal to 3, then $\Phi^*g_+$ is conformally compact of class $C^\infty.$
\par If dim M is odd and greater than 3, the Weyl tensor of $\rho^2\Phi^*g_+$ are of class $C^{n-2},$ then $\Phi^*g_+$ is conformally compact of class $C^\infty.$ \end{rema}
\par The outline of this paper is as follows. In section 2,we introduce the constant scalar curvature compactification and construct a kind of harmonic coordinates near ${\partial M}.$ The regularity of the metric isn't changed in the above two steps. We also study the relationship between Ricci curvature and Weyl curvature when the scalar curvature is a constant. \par In section 3, we review some background for studying conformally compact Einstein manifolds and asymptotically hyperbolic metrics satisfying (\ref{1.3}), including the change of curvature under conformal transformation, the existence and regularity of geodesic defining function. We also study the reason why the boundary regularity in general breaks down at order $n$ when $dim M=n+1$ is odd. Then after some simple calculations, we show that the Weyl curvature has an influence on the regularity in geodesic coordinates. Besides, for conformally compact Einstein manifolds, by calculating the formal power series of Weyl tensor in geodesic coordinates, we show that the obstruction tensor of the conformal metric vanishes if and only if the formal power series of Weyl tensor doesn't contain $x^{n-2}\log x$ term. Which improves Theorem A in \cite{CDLS}. \par In section 4, we study some boundary conditions, including the Dirichlet condition for $g_{ij}$ and Ricci curvature, the Neumann condition for $g^{0\beta}.$ We use the geodesic defining function as a transition tool to calculate the Dirichlet condition for Ricci curvature. The regularity would drop one order when we change the defining function to geodesic. The geodesic compactification should be at least $C^2$ so that we can calculate some curvature tensor. That's why we need $m\geq 3$ in Theorem \ref{theo1.1}. We use the property of harmonic coordinates to obtain the Neumann condition for $g^{0\beta}.$ \par In section 5, we use the theory of elliptic system to prove Theorem \ref{theo1.1}. We improve the regularity of the conformal metric and defining function in the new coordinates. \section{Basic geometry equations in harmonic coordinates} \label{sect2} \par In this section, we discuss some basic geometry equations for the manifold $(\overline{M},g).$ Before doing it, we need to make an appropriate choice of conformal compactification to let the scalar curvature be a constant near boundary and construct harmonic coordinates for the metric. These work can be done on an arbitrary manifold with boundary. \subsection{Constant scalar curvature compactification} \begin{lemm}
Let $(M,g_+)$ be a conformally compact n-manifold, $M$ has a $C^{2,\alpha}$ conformal compactification $g=\rho^2g_+.$ $h=g|_{{\partial M}}$ is the boundary metric. Then there exits a $C^{2,\alpha}$ constant scalar curvature compactification $\hat{g}=\hat{\rho}^2g_+$ with boundary metric $h.$ \end{lemm} \begin{proof} We only need to solve a Yamabe problem with Dirichlet data. Let $\hat{g}=u^{\frac{4}{n-2}}g$, then we consider the equation
\begin{equation}\label{2.1}
\left\{
\begin{array}{l}
\Delta_gu-\frac{n-2}{4(n-1)}Su+\frac{n-2}{4(n-1)}\lambda u^{\frac{n+2}{n-2}}=0
\\ u>0\ \ in\ \overline{M}
\\ u\equiv 1\ \ on\ {\partial M}
\end{array}
\right.
\end{equation}
When we choose $\lambda=-1,$ From \cite{Ma} we know the equation always has a $C^{2,\alpha}$ solution. So $\hat{g}=u^{\frac{4}{n-2}}g$ is also in $C^{2,\alpha}.$ Since $ u\equiv 1$ on ${\partial M},$ the boundary metric $h$ is not changed. \end{proof} \par From the standard theory for elliptic equations, if $g$ is in $C^{m,\alpha}(\overline{M})$ for some $\alpha\in (0,1),$ then $\hat{\rho}=u^{\frac{2}{n-2}}\rho$ is also in $C^{m,\alpha},$ and $\hat{g}$ is also in $C^{m,\alpha}.$ The Weyl tensor $\hat{W}=u^{\frac{4}{n-2}}W$ of $\hat{g}$ are in $C^{m,\alpha}.$ So in the following, we don't distinguish $g$ and $\hat{g}.$ When we refer to the compactification $g,$ we mean the scalar curvature $S$ of $g$ is a constant. \subsection{The harmonic coordinates near boundary} \par The coordinates $\{x^\beta\}_{\beta=0}^n$ are called harmonic coordinates with respect to $g$ when $\Delta_gx^\beta=0$ for $0\leq\beta\leq n.$ We are now going to construct harmonic coordinate near ${\partial M}$ which is also harmonic when restricted on $({\partial M},h).$ In the following, if there are no special instructions, any use of indices will follow the convention that Roman indices will range from 1 to n, while Greek indices range from 0 to n. \par Firstly, for any point $p\in{\partial M},$ there are smooth structure $\{y^\beta\}.$ It is easy to construct new coordinates $\{x^i\}$ on $\ M.$
$\{x^i\}$ are harmonic coordinates on $({\partial M},h).$ When $h$ is in $C^{m+2,\alpha}$, $x^i$ are $C^{m+3,\alpha}({\partial M})$ functions of $y^\beta.$ Then $$h_{ij}=h(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j})\in C^{m+2,\alpha}(\partial M).$$
\par Then by solving a local Dirichlet problem: $\Delta_gx^i=0,$ with the boundary condition $x^i$ as above,we can extend $x^i$ to $\overline{M}.$ Similarly, we can choose a harmonic defining function $x^0$ satisfies $\Delta_gx^0=0,x^0|_{{\partial M}}=0.$ It is easy to see $\{x^\beta\}_{\beta=0}^n$ form harmonic coordinates with respect to $g$ in a neighborhood of ${\partial M}.$ When $g$ is in $C^{m,\alpha},$ $x^\beta$ are $C^{m+1,\alpha}$ functions of $y^\beta.$ Then $$g_{\alpha\beta}=g(\frac{\partial}{\partial x^\alpha},\frac{\partial}{\partial x^\beta})\in C^{m,\alpha}(\overline{M}).$$ \par So when we change the coordinates $\{y^\beta\}$ to $\{x^\beta\}$, the regularities of $g$ and Weyl tensor on $\overline{M}$ and $h$ on ${\partial M}$ are unchanged. \subsection{Basic geometry equations} \par In this section, we mainly study the relationship between Weyl tensor and Ricci tensor. For any $(n+1)$-manifold, $0\leq i,j,k,l,h,m\leq n.$ We have \begin{equation}\label{2.2} R_{ijkl}=\frac{1}{n-1}(R_{ik}g_{jl}+R_{jl}g_{ik}-R_{jk}g_{il}-R_{il}g_{jk})-\frac{S}{n(n-1)}(g_{ik}g_{jl}-g_{il}g_{jk})+W_{ijkl}. \end{equation} On the other hand, we have the second Bianchi identity for $g.$ $$0=Bian(g,Ric)=g^{jh}(R_{jk,h}-\frac{1}{2}R_{jh,k})$$ When the scalar curvature $S$ is constant ,we get \begin{equation}\label{2.3} g^{jh}R_{jk,h}=0. \end{equation} From (\ref{2.2}) and (\ref{2.3}), we have $$R_{ik,l}-R_{il,k}=\frac{n-1}{n-2}g^{jh}W_{ijkl,h}.$$ It follows $$g^{lt}R_{ik,lt}-g^{lt}R_{il,kt}=\frac{n-1}{n-2}g^{lt}g^{jh}W_{ijkl,ht}.$$ As $$R_{il,kt}=R_{il,tk}+Ric*Rm,\ \ g^{lt}R_{il,tk}=(g^{lt}R_{il,t})_k=0.$$ Where $Ric*Rm$ refers to a bilinear form of $R_{ij}$ and $R_{ijkl}.$ We finally get \begin{equation}\label{2.4} g^{lt}R_{ik,lt}=g*Ric*Rm+g*g*(\partial^2W+\Gamma*\partial W+\Gamma*\Gamma*W+\partial\Gamma*W). \end{equation} When $g\in C^{m,\alpha},$ in harmonic coordinates$\{x^\beta\}$, the above equation can be written as \begin{equation}\label{2.5} g^{\gamma\tau}\frac{\partial}{\partial x^\gamma}\frac{\partial}{\partial x^\tau}R_{\alpha\beta}=Q(g,\partial g,\partial^2g,W,\partial W,\partial^2W). \end{equation} Where $Q$ is a polynomial. Even when $m=3,$ we can define the first and second covariant derivatives of curvature in the sense of distribution (see \cite{J} and \cite{ST}). It can be shown (\ref{2.5}) still holds. \section{The conformal infinity of asymptotically hyperbolic metrics } \par We will discuss some background material for conformally compact metrics in this section. As the definition in introduction, Let $\rho$ be a defining function for $M,$ set $$g=\rho^2g_+.$$ We assume $\rho$ is $C^{m,\alpha}$ on $\overline{M}$ in the initial atlas $\{y^\beta\},$ $m,\alpha$ are defined as in Theorem 1.1. $g_+$ satisfies $$Ric_{g_+}+ng_+=o(\rho^2),$$ the curvature of $g$ can be expressed as following formulas: \begin{equation}\label{3.1}
K_{ab}=\frac{K_{+ab}+|\nabla\rho|^2}{\rho^2}-\frac{1}{\rho}[D^2\rho(e_a,e_a)+D^2\rho(e_b,e_b)]. \end{equation} \begin{equation}\label{3.2}
Ric=-(n-1)\frac{D^2\rho}{\rho}+[\frac{n(|\nabla\rho|^2-1)}{\rho^2}-\frac{\Delta\rho}{\rho}]g+o(1), \end{equation} \begin{equation}\label{3.3}
S=-2n\frac{\Delta\rho}{\rho}+n(n+1)\frac{|\nabla\rho|^2-1}{\rho^2}+o(1). \end{equation}
Here $D^2$ is the Hessian. It is easy to see that when $g$ is at least $C^2$ in $\overline{M},$ $|\nabla\rho|\rightarrow 1$ as $\rho\rightarrow 0,$
and $|K_{+ab}+1|=O(\rho^2).$ Hence a $C^2$ conformally compact Einstein manifold is asymptotically hyperbolic. At ${\partial M},$ we have $|\nabla\rho|=1,$
Let $D^2\rho|_{{\partial M}}=A,$ where A is the second fundamental form of ${\partial M}$ in $(\overline{M},g).$ \subsection{Geodesic conformal compactification}
As noted in introduction, defining functions are not unique, but differ by multiplication by positive functions which equal 1 when restricted on ${\partial M}.$ When the defining function $r$ and its compactification $\bar{g}=r^2g_+$ satisfies $$|\bar{\nabla} r|_{\bar{g}}\equiv1$$ in a neighborhood of ${\partial M},$ we call $r$ geodesic defining function. We show that defining function always exists. \begin{lemm}
Let $g$ be a $C^2$ conformal compactification of $(M,g_+),$ $g=\rho^2g_+.$ $h=g|_{{\partial M}}$ is the boundary metric. Then $g_+$ has a unique geodesic conformal compactification with the same boundary metric $h.$ \end{lemm} \begin{proof} Let $r=u\rho,$ $\bar{g}=r^2g_+.$ The lemma is equivalent to the equation:
\begin{equation}\label{3.4}
\left\{
\begin{array}{l}
2(\nabla\rho)(\log u)+\rho|\nabla \log u|_g^2=\frac{1-|\nabla\rho|_g^2}{\rho}
\\ u\equiv 1 \ on \ {\partial M}.
\end{array}
\right.
\end{equation} By general theory of first order partial differential equations, we know that it has a unique positive solution in a collar neighborhood $U$ of $\ {\partial M}.$ \end{proof} Further more, When $g$ is in $C^{m,\alpha},$ we know that the right hand of the equation is in $C^{m-1,\alpha}$ on the boundary. So we only have $u\in C^{m-1,\alpha},$ which means $\bar{g}$ is a $C^{m-1,\alpha}$ conformal compactification. \par It is easy to see that when $Ric_{g_+}+ng_+=o(\rho^2)$ for the defining function $\rho,$ we also have $Ric_{g_+}+ng_+=o(r^2)$ for the geodesic defining function. \\ \par In the following of this section, we assume $g_+$ is Einstein, i.e. $$Ric_{g_+}+ng_+=0.$$ Then the term $o(1)$ in (\ref{3.2}) and (\ref{3.3}) can be removed.\\ When $M$ has a $C^2$ geodesic conformal compactification $\bar{g}$, it is very convenient for us to do some calculation. From (\ref{3.2}) and (\ref{3.3}), we know that the second fundamental form $\bar{A}$ given by $\bar{A}=\bar{D}^2r$ vanishes on ${\partial M}.$ By Gauss lemma, ${\bar{g}}$ can split in $U,$ $${\bar{g}}=dr^2+g_r$$ for a 1-parameter family $g_r$ of metrics on ${\partial M}.$ \par Now we choose the local coordinates $(r,x^1,x^2,\cdots,x^n)$ on ${\partial M}$ to study the regularity of ${\bar{g}}$ near the boundary ${\partial M}.$ Using the equation (\ref{3.2}) and Gauss-Codazzi equation, we finally get \begin{equation}\label{3.5} r\partial_r^2{\bar{g}}_{ij}-(n-1)\partial_r{\bar{g}}_{ij}-{\bar{g}}^{kl}\partial_r{\bar{g}}_{kl}{\bar{g}}_{ij}-r{\bar{g}}^{kl}\partial_r{\bar{g}}_{kl}\partial_r{\bar{g}}_{ij}+ \frac{r}{2}{\bar{g}}^{kl}\partial_r{\bar{g}}_{ik}\partial_r{\bar{g}}_{jl}-2r\bar{R}ic(g_r)_{ij}=0. \end{equation} Here ${\bar{g}}_{ij}$ denotes the tensor $g_r$ on $M$ and $\bar{R}ic(g_r)_{ij}$ denotes the Ricci tensor for the induced metric on level sets of $r.$ We assume that ${\bar{g}}$ is smooth enough so that we could calculate its expansion from (\ref{3.5}). Let $r=0,$ we can derive that $\partial_r{{\bar{g}}}(0)=0.$ By using mathematical induction, we differentiate (\ref{3.5}) $p-1$ times with respect to $r$ \begin{equation}\label{3.6} \begin{aligned} r\partial_r^{p+1}{\bar{g}}_{ij}+(p-n)\partial_r^p{\bar{g}}_{ij}-{\bar{g}}^{kl}(\partial_r^p{\bar{g}}_{kl}){\bar{g}}_{ij}=Q_1({\bar{g}}^{-1},\partial_r^q{\bar{g}},\partial_r^q{\bar{g}}\partial_s^2{\bar{g}}) \\ +rQ_2({\bar{g}}^{-1},\partial_r^q{\bar{g}}, \partial_r^p{\bar{g}},\partial_r^{p-2}\partial_s^2{\bar{g}}). \end{aligned} \end{equation} Here $q<p,$ $\partial_s{\bar{g}}$ is the differential of ${\bar{g}}$ with respect to $x^i (1\leq i\leq n).$ $Q_1, Q_2$ are the third polynomials. \par Setting $r=0,$ we can calculate that $\partial_r^p{\bar{g}}(0)=0$ when $p$ is odd and $\partial_r^p{\bar{g}}(0)$ is uniquely determined by each step when $p$ is even. However, this will break down when $p=n$ if $n+1$ is odd. In that case we only have ${\bar{g}}^{kl}(\partial_r^n{\bar{g}}_{kl})=0$ at $r=0.$ This give no further information at this order. \par Now we know that $\partial_r{\bar{g}}(0)=\partial_r^3{\bar{g}}(0)=\cdots=\partial_r^{n-1}{\bar{g}}(0)=0,$ that is $$\partial_r^{n-1}{\bar{g}}(r)=O(1).$$
Considering (\ref{3.6}) when $p=n-1,$we have \begin{equation}\label{3.7} \partial_r^n{\bar{g}}_{ij}=\frac{\partial_r^{n-1}{\bar{g}}_{ij}-{\bar{g}}^{kl}(\partial_r^{n-1}{\bar{g}}_{kl}){\bar{g}}_{ij}+Q_1}{r}+Q_2. \end{equation} If $\partial_r^{n-1}{\bar{g}}(r)\neq O(r),$ there will be the log term in the expansion for $g_r.$ $\partial_r^{n-1}{\bar{g}}(r)=O(r)$ is the necessary condition to ensure ${\bar{g}}$ $n$-th differentiable. \subsection{Weyl tensor in Geodesic coordinates}
Now we show that if the Weyl tensor ${\bar{W}}$ of ${\bar{g}}$ is $(n-2)$-th differentiable, then $\partial_r^{n-1}{\bar{g}}(r)=O(r)$ holds. \par We begin with (\ref{2.4}) by taking $x^l=r,1\leq i,k\leq n.$ \par In the coordinates $(r,x^1,x^2,\cdots,x^n),$ we have $${\bar{R}}_{ik,r}=\partial_r^3{\bar{g}}+\partial_r{\bar{g}}\ast\partial_r^2{\bar{g}}\ast{\bar{g}}+\partial_r{\bar{g}}\ast\partial_r{\bar{g}}\ast\partial_r{\bar{g}}\ast{\bar{g}}+\partial_rP+\partial_r{\bar{g}}\ast P,$$ $${\bar{R}}_{ir,k}=\partial_rP\ast P+\partial_r{\bar{g}}\ast\partial_r^2{\bar{g}}\ast{\bar{g}}+\partial_r{\bar{g}}\ast\partial_r{\bar{g}}\ast\partial_r{\bar{g}}\ast{\bar{g}}+\partial_r{\bar{g}}\ast{\bar{g}}\ast{\bar{R}} ic.$$ Here $P=P({\bar{g}},\partial_{x^i}{\bar{g}},\partial_{x^i}^2{\bar{g}})$ is a polynomial for $1\leq i\leq n$. So from (\ref{2.4})we have: \begin{equation}\label{3.8} \partial_r^3{\bar{g}}=\partial_r{\bar{g}}\ast\partial_r^2{\bar{g}}\ast{\bar{g}}+\partial_r{\bar{g}}\ast\partial_r{\bar{g}}\ast\partial_r{\bar{g}}\ast{\bar{g}}+\partial_rP\ast P+\partial_r{\bar{g}}\ast{\bar{g}}\ast{\bar{R}} ic+{\bar{g}}\ast \nabla {\bar{W}}. \end{equation} We already know that $\partial_rg=\partial_r^3g=\cdots\partial_r^{n-3}=O(r).$ Differentiating (\ref{3.8}) $n-4$ times with respect to $r,$ Each term in the right hand is $O(r)$ besides ${\bar{g}}\ast \nabla {\bar{W}}.$ $\partial_r^{n-1}{\bar{g}}=O(1).$ Then we have $\partial_r^{n-3}{\bar{W}}=O(1).$ When ${\bar{W}}\in C^{n-2}$, $\partial_r^{n-3}{\bar{W}}=O(r).$ So the left hand $\partial_r^{n-1}{\bar{g}}=O(r).$ It gives the necessary condition that ${\bar{g}}$ is in $C^n$ or has higher regularity. \par In the following, we will calculate the formal power series of Weyl tensor in geodesic coordinates. $g=dr^2+g_r.$ From (\ref{3.6}),(\ref{3.7}) and \cite{G}, when n is odd, \begin{equation}\label{3.9} g_r=h+g^{(2)}r^2+(even\ powers\ of\ r) + g^{(n-1)}r^{n-1}+g^{(n)}r^n+\cdots. \end{equation} When n is even, \begin{equation}\label{3.10} g_r=h+g^{(2)}r^2+(even\ powers\ of\ r) + g^{(n)}r^{n}+fr^n\log r+\cdots, \end{equation} where $g^{(2i)}$ and $f$ are 2 tensors on $\partial M$ and $f$ is trace free and determined by $h$ locally. We are now considering the case when n is even. If $f=0,$ the n-th regularity of $g$ exists. Let $1\leq i,j\leq n.$ We know the Weyl tensor $W_{irjr}$ are \begin{equation}\label{3.11} W_{irjr}=R_{irjr}-\frac{1}{n-1}(R_{ik}+R_{rr}g_{ij})+\frac{S}{n(n-1)}g_{ij} \end{equation} The formal power series of Weyl tensor contains $r$ and $\log r$ and we only need to check the coefficients of $r^{n-2}\log r.$ By a simple calculation, we get the coefficients of $r^{n-2}\log r$ of $R_{irjr}$ is $$coeff(R_{irjr})=-\frac{n(n-1)}{2}f_{ij}.$$ And $$coeff(R_{ij})=-\frac{n(n-1)}{2}f_{ij},$$ $$coeff(R_{rr})=-\frac{n(n-1)}{2}h^{st}f_{st},$$ $$coeff(S)=-\frac{n(n-1)}{2}2h^{st}f_{st}.$$ So when $coeff(W_{irjr})=0,$ by (\ref{3.11}) we finally derive \begin{equation}\label{3.12} f_{ij}-\frac{1}{n}(h^{st}f_{st})h_{ij}=0 \end{equation} As $f$ is trace free, $f\equiv 0.$ This gives the n-th regularity of $g.$ From Theorem A in \cite{CDLS}, we know the log term is the only obstruction of the smoothness, then Remark 1.3 holds. \section{The boundary condition} \par In this section, we derive a boundary problem for $g$ and Ricci curvature of a conformal compact Einstein manifold in the harmonic coordinates as defined in section 2. We do it locally, that is, for any $p\in {\partial M},$ there is a neighborhood $V$ contains $p$ and local atlas $\{x^\beta\}.$ Let $D=V\cap {\partial M}$ be the boundary portion. $g\in C^{m,\alpha}(V), W\in C^{m,\alpha}(V), h\in C^{m+2,\alpha}(D).$ We will give the Dirichlet and Neumann boundary conditions of $g$ and $Ric(g)$ on D. \subsection{Dirichlet boundary conditions on $g_{ij}$} \begin{equation}\label{4.1} g_{ij}=h_{ij}. \end{equation} \subsection{Dirichlet boundary conditions on $R_{ij}$} \par We claim that \begin{equation}\label{4.2} R_{ij}=\frac{n-1}{n-2}(Ric_h)_{ij}+(\frac{1}{2n}S-\frac{1}{2(n-2)}S_h)h_{ij}+\frac{n-1}{2n^2}H^2h_{ij}. \end{equation} Here $Ric_h$ and $S_h$ are Ricci curvature and scalar curvature of $({\partial M},h),$ H is the mean curvature, $H=g^{ij}A_{ij}.$ We use the following three lemmas to prove (\ref{4.2}). \begin{lemm}\label{lemm 4.1} Let ${\bar{g}}=r^2g_+$ be a $C^2$ geodesic compactification of $(M,g_+)$ with boundary metric $h$ on ${\partial M}.$ Then on ${\partial M},$ \begin{equation}\label{4.3} {\bar{S}}=\frac{n}{n-1}S_h. \end{equation}
\begin{equation}\label{4.4} {\bar{R}}_{ij}=\frac{n-1}{n-2}(Ric_h)_{ij}-\frac{1}{2(n-1)(n-2)}S_hh_{ij}. \end{equation} \end{lemm} \begin{proof} Since we only need to study the Ricci curvature on $T{\partial M},$ we can choose the coordinates $(r,x^1,\cdots,x^n)$ in $V$ where $(x^1,\cdots,x^n)$ are harmonic with respect to $h$ when restricted on $D.$ So ${\bar{g}}=dr^2+g_r.$ i.e. $$g_{ri}=g^{ri}=0, g_{rr}=g^{rr}=1.$$
Since the second fundamental form ${\bar{A}}={\bar{D}}^2r$ vanishes on ${\partial M}.$ We have: \begin{equation}\label{4.5} \begin{aligned} {\bar{R}}_{ij}&={\bar{g}}^{\alpha\beta}{\bar{R}}_{i\alpha\beta j}\\
&={\bar{g}}^{kl}((R_h)_{iklj}+{\bar{A}}_{il}{\bar{A}}_{kj}-{\bar{A}}_{ij}{\bar{A}}_{kl})+{\bar{R}}_{irrj}\\
&=(R_h)_{ij}+{\bar{R}}_{irrj}. \end{aligned} \end{equation} Taking trace for $i,j,$ we get \begin{equation}\label{4.6} {\bar{R}}_{rr}=\frac{1}{2}({\bar{S}}-S_h). \end{equation} For ${\bar{R}}_{irrj},$ we have: \begin{equation}\label{4.7} \begin{aligned} {\bar{R}}_{irrj}&={\bar{g}}(\bar{\nabla}_{\partial_i}\bar{\nabla}_{\partial_r}\partial_r,\partial_j)-{\bar{g}}(\bar{\nabla}_{\partial_r}\bar{\nabla}_{\partial_i}\partial_r,\partial_j)-
{\bar{g}}(\bar{\nabla}_{[\partial_r,\partial_i]}\partial_r,\partial_j)\\
&=-\partial_r{\bar{g}}(\bar{\nabla}_{\partial_i}\partial_r,\partial_j)+{\bar{g}}(\bar{\nabla}_{\partial_i}\partial_r,\bar{\nabla}_{\partial_r}\partial_j)\\
&=-\partial_r{\bar{A}}_{ij}. \end{aligned} \end{equation} From (\ref{3.2}) and (\ref{3.3}) We have: \begin{equation}\label{4.8}
{\bar{R}}_{ij}=-(n-1)\frac{{\bar{A}}_{ij}}{r}-\frac{\bar{\Delta}r}{r}{\bar{g}}_{ij}+o(1), \end{equation} \begin{equation}\label{4.9}
{\bar{S}}=-2n\frac{\bar{\Delta}r}{r}+o(1). \end{equation} So on ${\partial M}$ we have: \begin{equation}\label{4.10}
{\bar{S}}=-2n\partial_r\bar{\Delta}r. \end{equation} In V, we have:
$${\bar{A}}_{ij}=-\frac{1}{n-1}(r{\bar{R}}_{ij}+\bar{\Delta}r{\bar{g}}_{ij})+o(r),$$ Taking trace on (4.8)
$$\bar{\Delta}r=-\frac{1}{n}(r({\bar{S}}-{\bar{R}}_{rr})+(n-1){\bar{H}})+o(r).$$ Then \begin{equation}\label{4.11}
\partial_r{\bar{A}}_{ij}|_{r=0}=-\frac{1}{n-1}({\bar{R}}_{ij}+\partial_r\bar{\Delta}r{\bar{g}}_{ij})|_{r=0}, \end{equation} \begin{equation}\label{4.12}
\partial_r\bar{\Delta}r|_{r=0}=-\frac{1}{n}({\bar{S}}-{\bar{R}}_{rr}+(n-1)\partial_r{\bar{H}})|_{r=0}. \end{equation}
At last, we only need to calculate $\partial_r{\bar{H}}|_{r=0}.$ \begin{equation}\label{4.13} \begin{aligned}
\partial_r{\bar{H}}|_{r=0}&=\partial_r({\bar{g}}^{ij}{\bar{A}}_{ij})|_{r=0}=(\partial_r{\bar{g}}^{ij}){\bar{A}}_{ij}|_{r=0}+{\bar{g}}^{ij}(\partial_r{\bar{A}}_{ij})|_{r=0}\\
&=0+{\bar{g}}^{ij}(-{\bar{R}}_{irrj})=-{\bar{R}}_{rr}. \end{aligned} \end{equation} Combining the formulas above ,we finally get (\ref{4.3}) and (\ref{4.4}). \end{proof} \begin{lemm}\label{lemm 4.2} Let $g=\rho^2g_+$ be a $C^{3,\lambda}$ conformal compactification and ${\bar{g}}=r^2g_+$ be the $C^{2,\lambda}$ geodesic conformal compactification of
$(M,g_+)$ with the same boundary metric $g|_{{\partial M}}={\bar{g}}|_{{\partial M}}=h.$ If $r=u\rho, A=D^2\rho,$ then $A|_{{\partial M}}=-u_rh.$ \end{lemm} \begin{proof} In the coordinates $(r,x^1,x^2,\ldots,x^n),$ on ${\partial M}, 0={\bar{A}}_{ij}=-\bar{\Gamma}_{ij}^r.$ So the connection $\nabla$ and $\bar{\nabla}$of $g$ and ${\bar{g}}$ have the relationship: $$\Gamma_{ij}^r=\bar{\Gamma}_{ij}^r-\frac{1}{u}(\delta^r_ju_i+\delta^r_iu_j-g_{ij}u_r)=\frac{1}{u}u_rh_{ij}.$$ As $g=u^{-2}{\bar{g}}, grad_g=u^2 grad_{{\bar{g}}}.$ \begin{equation}\label{4.14} \begin{aligned}
A_{ij}&=D^2\rho(\partial_i,\partial_j)=g(\nabla_{\partial_i}\nabla\rho,\partial_j)=-g(\nabla\rho,\nabla_{\partial_i}\partial_j)\\
&=-\Gamma_{ij}^rg(\nabla\rho,\partial_r)=-\Gamma_{ij}^r{\bar{g}}(\bar{\nabla}\rho,\partial_r)\\
&=-\Gamma_{ij}^r{\bar{g}}(\bar{\nabla}(\frac{r}{u}),\partial_r)=-\Gamma_{ij}^r{\bar{g}}(\frac{u\bar{\nabla}r-r\bar{\nabla}u}{u^2},\partial_r)\\
&=-\Gamma_{ij}^r{\bar{g}}(\bar{\nabla}r,\bar{\nabla}r)=-u_rh_{ij} \end{aligned} \end{equation} \end{proof}
Lemma \ref{lemm 4.2} tells us $u_r=-\frac{H}{n}.$ Since $u|_{{\partial M}}\equiv 1,$ we get $$\bar{\nabla}u=-\frac{H}{n}\bar{\nabla}r.$$ \begin{lemm}\label{lemm 4.3} $g,{\bar{g}}$ are defined as in Lemma \ref{lemm 4.2}. Then on ${\partial M},$ $$R_{ir}=\frac{n-1}{n}\frac{\partial H}{\partial x_i},$$ $$R_{rr}=\frac{1}{2}(S-S_h)-\frac{n-1}{2n}H^2,$$ \begin{equation}\label{4.15} R_{ij}={\bar{R}}_{ij}+(\frac{1}{2n}(S-{\bar{S}}))h_{ij}+\frac{n-1}{2n^2}H^2h_{ij}. \end{equation} \end{lemm} \begin{proof} By the standard formulas for conformal changes of the metric $g=u^{-2}{\bar{g}},$ the Ricci curvature of $g$ and ${\bar{g}}$ are related by
$$Ric={\bar{R}} ic+(n-1)\frac{\bar{D}^2u}{u}+(\frac{\bar{\Delta}u}{u}+\frac{n|\bar{\nabla}u|^2_{{\bar{g}}}}{u^2}){\bar{g}}.$$ Since $$\bar{\Delta}u=div\bar{\nabla}u=div(-\frac{H}{n}\bar{\nabla}r)=-\frac{\partial_rH}{n},$$ $${\bar{D}}^2u(\partial_i,\partial_j)=0,$$ $${\bar{D}}^2u(\partial_i,\partial_r)=-u_{ir}=\frac{1}{n}\frac{\partial H}{\partial x_i},$$ $${\bar{D}}^2u(\partial_r,\partial_r)=-\frac{\partial_rH}{n}=\bar{\Delta}u.$$ On ${\partial M},$ we have $$R_{ir}={\bar{R}}_{ir}+\frac{n-1}{n}\frac{\partial H}{\partial x_i}=\frac{n-1}{n}\frac{\partial H}{\partial x_i},$$ $$R_{rr}={\bar{R}}_{rr}+n\bar{\Delta}u+\frac{H^2}{n},$$ \begin{equation}\label{4.16} R_{ij}={\bar{R}}_{ij}+(\bar{\Delta}u+\frac{H^2}{n})h_{ij}. \end{equation} So $$S={\bar{S}}+2n\bar{\Delta}u+\frac{n+1}{n}H^2.$$ Which implies \begin{equation}\label{4.17}
\bar{\Delta}u=\frac{1}{2n}(S-{\bar{S}}-\frac{n+1}{n}H^2). \end{equation} From (\ref{4.16}) and (\ref{4.17}), the lemma \ref{lemm 4.3} holds. \end{proof} At last, the lemma \ref{lemm 4.1} and lemma \ref{lemm 4.3} implies (\ref{4.2}). \subsection{Neumann boundary conditions on $g^{0\alpha}$}
\par In this section, we use the harmonic coordinates $\{x^\beta\}_{\beta=0}^n$ as defined in section 2. Let $N=\frac{\nabla x_0}{|\nabla x_0|}$ be the unit norm vector on ${\partial M}.$ In coordinates $\{x^\beta\}_{\beta=0}^n,$ $$N=(g^{00})^{-\frac{1}{2}}g^{0\beta}\partial_\beta.$$ Then these are of the form \begin{equation}\label{4.18}
N(g^{00})=-2Hg^{00}. \end{equation} \begin{equation}\label{4.19}
N(g^{0i})=-Hg^{0i}+\frac{1}{2}(g^{00})^{-\frac{1}{2}}g^{i\beta}\partial_\beta g^{00}. \end{equation} \begin{proof} Let $\{e_i\}_{i=1}^n$ be the orthonormal basis at a given point $p\in{\partial M}.$ So we have: \begin{equation}\label{4.20}
0=\Delta x^\alpha=div \nabla x^\alpha=\sum\limits_{i=1}^ng(\nabla_{e^i}\nabla x^\alpha,e_i)+g(\nabla_N\nabla x^\alpha,N) \end{equation} Write $\nabla x^\alpha=(\nabla x^\alpha)^T+(\nabla x^\alpha)^N,$ where $$(\nabla x^\alpha)^T=\nabla_hx^\alpha,(\nabla x^\alpha)^N=g(\nabla x^\alpha,N)N=(g^{00})^{-\frac{1}{2}}g^{0\alpha}N.$$ Since $$0=\Delta_h x^j=div \nabla_h x^j=\sum\limits_{i=1}^nh(\nabla^h_{e^i}(\nabla x^j)^T,e_i)=\sum\limits_{i=1}^ng(\nabla_{e^i}(\nabla x^j)^T,e_i)$$ It also holds for $j=0$ as $x^0\equiv 0$ on ${\partial M}.$ Then (\ref{4.20}) turns into \begin{equation}\label{4.21} \sum\limits_{i=1}^ng(\nabla_{e^i}((g^{00})^{-\frac{1}{2}}g^{0\alpha} N),e_i)+g(\nabla_N\nabla x^\alpha,N)=0 \end{equation} The first term is just $(g^{00})^{-\frac{1}{2}}g^{0\alpha}\sum\limits_{i=1}^nA(e_i,e_i)=H(g^{00})^{-\frac{1}{2}}g^{0\alpha}$ and the second term is \begin{equation}\label{4.22} \begin{aligned}
g(\nabla_N\nabla x^\alpha,N)&=\frac{1}{|\nabla x^0|^2}g(\nabla_{\nabla x^0}\nabla x^\alpha,\nabla x^0)\\
&=\frac{1}{|\nabla x^0|^2}[\nabla x^0 (g(\nabla x^\alpha,\nabla x^0))-g(\nabla x^\alpha,\nabla_{\nabla x^0 }\nabla x^0)]\\
&=\frac{1}{|\nabla x^0|^2}[\nabla x^0(g^{0\alpha})-g(\nabla x^0,\nabla_{\nabla x^\alpha}\nabla x^0)]\\
&=\frac{1}{|\nabla x^0|}N(g^{0\alpha})-\frac{1}{2|\nabla x^0|^2}\nabla x^\alpha(g^{00}) \end{aligned} \end{equation}
We know $|\nabla x^0|=(g^{00})^{\frac{1}{2}},$ then (4.21) is just \begin{equation}\label{4.23} Hg^{0\alpha}+N(g^{0\alpha})-\frac{1}{2}(g^{00})^{-\frac{1}{2}}\nabla x^\alpha(g^{00})=0 \end{equation} So when $\alpha=0,$ we have $N(g^{00})=-2Hg^{00}.$ When $\alpha=i,$ we have
$$N(g^{0i})=-Hg^{0i}+\frac{1}{2}(g^{00})^{-\frac{1}{2}}g^{i\beta}\partial_\beta g^{00}.$$ \end{proof} \subsection{Dirichlet boundary conditions on $R_{\alpha\beta}$} In section 4.2, we already know the formulas of $R_{ij}$ on ${\partial M}$ and the mixed components $R_{ri}$ and $R_{rr}$ of Ricci Curvature in the coordinates $(r,x^1,x^2,\cdots,x^n)$. That is, $$R_{ir}=\frac{n-1}{n}\frac{\partial H}{\partial x_i},$$ $$R_{rr}=\frac{1}{2}(S-S_h)-\frac{n-1}{2n}H^2.$$
\par Now we study the the mixed components $R_{0i}$ and $R_{00}$ of Ricci Curvature in the harmonic coordinates $(x^0,x^1,\cdots,x^n).$ In fact, as the vector $N=\frac{\nabla x_0}{|\nabla x_0|}$ is also the unit norm vector on ${\partial M}$ with respect to $g,$ we have $N=\bar{\nabla}r=\nabla r.$ i.e. $\nabla r=(g^{00})^{-\frac{1}{2}}g^{0\beta}\partial_\beta.$ Then \begin{equation}\label{4.24} R_{0i}=(g^{00})^{-\frac{1}{2}}\frac{n-1}{n}\frac{\partial H}{\partial x_i}-\frac{g^{0j}}{g^{00}}R_{ij} \end{equation} \begin{equation}\label{4.25} R_{00}=\frac{1}{(g^{00})^2}(g^{0i}g^{0j}R_{ij}+g^{00}(\frac{1}{2}(S-S_h)-\frac{n-1}{2n}H^2)) \end{equation} \subsection{Neumann boundary conditions on $R_{0i}$} The Dirichlet condition for $R_{0i}$ in (\ref{4.24} is not good because there are second order differential terms of metric in the right side. Now we consider the differential terms of Ricci curvature. \par Since the scalar curvature is a constant, by the second Bianchi identity, we have $$0=\frac{1}{2}S_{,\alpha}=g^{\eta\beta}R_{\eta\alpha,\beta}=g^{\eta\beta}\partial_\beta R_{\alpha\eta}-g^{\eta\beta}\Gamma_{\alpha\beta}^\tau R_{\eta\tau}.$$ Then $$g^{0\beta}\partial_\beta R_{0\alpha}=-g^{j\beta}\partial_\beta R_{j\alpha}+g^{\eta\beta}\Gamma_{\alpha\beta}^\tau R_{\eta\tau}.$$ Let $\alpha=i,$ we get \begin{equation}\label{4.26} N(R_{0i})=(g^{00})^{-\frac{1}{2}}(-g^{j\beta}\partial_\beta R_{ji}+g^{\eta\beta}\Gamma_{i\beta}^\tau R_{\eta\tau}) \end{equation} \section{Proof of the main theorem} In this section, we prove the main theorem. Suppose $m\geq 3$ and $\alpha\in(0,1).$ For any point $p\in {\partial M},$ choose the harmonic coordinates $\{x^\beta\}_{\beta=0}^n$ in its neighborhood $V.$ Let $D=V\cap{\partial M}$ be the boundary portion. Now we have $g\in C^{m,\alpha}(V), h\in C^{m+2,\alpha}(D), W\in C^{m,\alpha}(V).$ \subsection{Regularity of the metric} \par \textbf{Step 1}: regularity of the Ricci curvature.\\ \par We begin with (\ref{2.5}) and the right side of (\ref{2.5}) are in $C^{m-2,\alpha}.$ \par On ${\partial M}$, we already derive the formulas of $R_{\alpha\beta}$ in section 3. As $H=g^{ij}A_{ij},$ and $A_{ij}=\frac{1}{2}(g^{00})^{\frac{1}{2}}g^{0\beta}(\partial_\beta g_{ij}-\partial_i g_{\beta j}-\partial_j g_{\beta i})\in C^{m-1,\alpha}(D).$ Then the Dirichlet condition of $R_{ij}$ and $R_{00}$ given by (\ref{4.4} and (\ref{4.25} shows that $$R_{ij}\in C^{m-1,\alpha}(D),R_{00}\in C^{m-1,\alpha}(D).$$ By standard elliptic regularity theory,
$$R_{ij}\in C^{m-1,\alpha}(V),R_{00}\in C^{m-1,\alpha}(V).$$
Then by the Neumann boundary conditions on $R_{0i}$ given by (\ref{4.26}), we also have $R_{0i}\in C^{m-1,\alpha}(V)$ since $N(R_{0i})\in C^{m-2,\alpha}(D).$ \\ In the following, we prove that $g_{\alpha\beta}\in C^{m+1,\alpha}(V)$ when $R_{\alpha\beta}\in C^{m-1,\alpha}(V).$ If it holds, we can get that $g_{\alpha\beta}\in C^{m+2,\alpha}(V)$ by repeating the steps. \par \textbf{Step 2}: regularity of $g_{ij}.$ \\ \par In harmonic coordinates, we have $$\Delta g_{ij}=-2R_{ij}+Q(g,\partial g)$$ Here $Q$ is a $1^{st}$ term of $g.$ So $\Delta g_{ij}\in C^{m-1,\alpha}(V),$ together with the boundary condition $g_{ij}=h_{ij}\in C^{m+2,\alpha}(D).$ We get $g_{ij}\in C^{m+1,\alpha}(V).$ \\ \par \textbf{Step 3}: regularity of $g_{0\beta}.$ \\ \par In section 4, we obtain the Neumann boundary condition of $g^{0\beta}$ which contain $H.$ Sense $H\in C^{m-1,\alpha},$ we can't improve the regularity of $g^{0\beta}$ in this condition. Now we are going to calculate the oblique derivative of $g_{0\beta}$ on ${\partial M}.$ As $g^{0\alpha}g_{\alpha\beta}=\delta_\beta^0,$ \begin{equation}\label{5.1} \begin{aligned} 0&=N(g^{0\alpha}g_{\alpha\beta})=N(g^{00}g_{0\beta})+N(g^{0j}g_{j\beta})\\ &=g^{00}N(g_{0\beta})+g_{0\beta}N(g^{00})+g_{j\beta}N(g^{0j})+g^{0j}N(g_{j\beta})\\ &=g^{00}N(g_{0\beta})+g_{0\beta}(-2Hg^{00})+g_{j\beta}(\frac{1}{2}(g^{00})^{-\frac{1}{2}}g^{j\tau}\partial_\tau g^{00}-Hg^{0j})+g^{0j}N(g_{j\beta})\\ &=g^{00}N(g_{0\beta})-2Hg_{0\beta}g^{00}+\frac{1}{2}(g^{00})^{-\frac{1}{2}}(\delta_\beta^\tau-g_{0\beta}g^{0\tau})\partial_\tau g^{00}-H(-g_{0\beta}g^{00})+g^{0j}N(g_{j\beta})\\ &=g^{00}N(g_{0\beta})+\frac{1}{2}(g^{00})^{-\frac{1}{2}}\partial_\beta g^{00}+g^{0j}N(g_{j\beta})-H\delta_\beta^0 \end{aligned} \end{equation} When $\beta=0,$ we have \begin{equation}\label{5.2} g^{00}N(g_{00})+\frac{1}{2}(g^{00})^{-\frac{1}{2}}\partial_0 g^{00}+g^{0j}N(g_{0j})-H=0 \end{equation} When $\beta=i,$ we have \begin{equation}\label{5.3} g^{00}N(g_{0i})+\frac{1}{2}(g^{00})^{-\frac{1}{2}}\partial_i g^{00}+g^{0j}N(g_{ij})=0 \end{equation} Now we consider the elliptic system of $g^{00},g_{01},g_{02},\cdots,g_{0n}$:
\begin{equation}\label{5.4}
\left\{
\begin{array}{l}
\Delta g^{00}=Q(g,\partial g,Ric)
\\ \Delta g_{01}=-\frac{1}{2}R_{01}+Q(g,\partial g)
\\ \ \ \vdots
\\\Delta g_{0n}=-\frac{1}{2}R_{0n}+Q(g,\partial g)
\end{array}
\right.
\end{equation} And from (\ref{4.18}), (\ref{5.3}) and the expression of $H,$ the regularities of $g_{ij}$ we obtain the boundary condition:
\begin{equation}\label{5.5}
\left\{
\begin{array}{l}
N(g^{00})-2Pg^{ij}\partial_i g_{0j}\in C^{m,\alpha}(D)
\\ N(g_{01})+\frac{1}{2P}\partial_1g^{00}\in C^{m,\alpha}(D)
\\ \ \ \vdots
\\N(g_{0n})+\frac{1}{2P}\partial_ng^{00}\in C^{m,\alpha}(D)
\end{array}
\right.
\end{equation} Where $P=(g^{00}(x))^{\frac{3}{2}}.$ We are going to prove the lemma: \begin{lemm}\label{lemm 5.1} Let $u^0=g^{00}, u^i=g_{0i}, i=1,2,\cdots,n.$ Then (\ref{5.4}) has the form $$L_{\alpha\beta}u^\beta(x)=f_\alpha(x), \ \ x\in V. $$ The boundary condition (\ref{5.5}) has the form $$B_{\alpha\beta}u^\beta(x)=g_\alpha(x), \ \ x\in D.$$ Then the operator $L$ is proper elliptic and the boundary operator $B$ satisfies the complementing condition with respect to the system $(L,B).$ \end{lemm} \begin{proof} For any $n+1$ vector $\xi=(\xi_0,\xi_1,\cdots,\xi_n),$ we consider the principal part of L \begin{equation}\label{5.6} L'_{\alpha\beta}(x,\xi)=\left[ \begin{matrix} g^{\alpha\beta}\xi_\alpha\xi_\beta&0&\cdots&0&\\ 0&g^{\alpha\beta}\xi_\alpha\xi_\beta&\cdots&0&\\ \vdots&\vdots&\ddots&\vdots&\\ 0&0&\cdots&g^{\alpha\beta}\xi_\alpha\xi_\beta& \end{matrix} \right] \end{equation}
Then$$L_{\alpha\beta}(x,\xi)=\det(L'_{\alpha\beta}(x,\xi))=|\xi|_g^{2(n+1)}.$$ For any $\xi\neq 0, L_{\alpha\beta}(x,\xi)\neq 0,$ so L is elliptic. \\ For each pair of $\xi$ and $\xi'$ of linearly independent vectors, the equation $$L_{\alpha\beta}(x,\xi+z\xi')=0$$ is equivalent to
$$(z^2\cdot|\xi'|_g^2+2<\xi,\xi'>_gz+|\xi|_g^2)^{n+1}=0.$$ It has $n+1$ roots with positive imaginary part and $n+1$ with negative imaginary part. So $L$ is proper elliptic. \\ \\
For any $x_0\in D,$ let $n=(1,0,\cdots,0)$ denote the unit normal at $x_0$ and $\xi=(0,\xi_1,\cdots,\xi_n)$ denote any nonzero real vector tangent to $D$ at $x_0.$ Let $z_s^+(x_0,\xi),s=0,1,\cdot,n$ be the roots of $L_{\alpha\beta}(x_0,\xi+zn)=0$ with positive imaginary. \\Define $$L_0^+(x_0,\xi;z)=\prod_{s=0}^n(z-z_s^+(x_0,\xi).$$ Let $L^{\alpha\beta}(x_0,\xi+zn)$ be matrix adjoint to $L'_{\alpha\beta}(x_0,\xi+zn).$ Now we define $$Q_{r\beta}=B'_{r\alpha}(x_0,\xi+zn)\cdot L^{\alpha\beta}(x_0,\xi+zn)$$ as polynomials in z, where $B'_{r\alpha}$ is the principal part of $B.$ \\ Then $B$ satisfies the complementing condition with respect to the system $(L,B)$ if and only if the rows of the $Q$ matrix are linearly independent modulo $L_0^+(x_0,\xi;z),$ that is, the polynomial $$\sum_{r=0}^nC_rQ_{r\beta}(x_0,\xi;z)\equiv 0 \ \ (mod L_0^+)$$ only if $C_r$ are all 0. \\
By a simple calculation, we can get $z_s=\sqrt{-1}\frac{|\xi|_h}{\sqrt{g^{00}}}$ for $s=0,1,\cdots,n.$ Then
$$L_0^+(x_0,\xi;z)=(z-\sqrt{-1}\frac{|\xi|_h}{\sqrt{g^{00}}})^{n+1}.$$
From the above, we know $$L'_{\alpha\beta}(x_0,\xi+zn)=|\xi+zn|_g^2\cdot\delta_{\alpha\beta}=(z^2g^{00}+|\xi|_h^2)\cdot\delta_{\alpha\beta}.$$ Its adjoint matrix is
$$L^{\alpha\beta}(x_0,\xi+zn)=(z^2g^{00}+|\xi|_h^2)^n\cdot\delta^{\alpha\beta}.$$ The principal part of B is \begin{equation}\label{5.7} B'_{\alpha\beta}(x,\xi)=\left[ \begin{matrix} z&-2Pg^{i1}\xi_i&-2Pg^{i2}\xi_i&\cdots&-2Pg^{in}\xi_i&\\ \frac{1}{2P}\xi_1&z&0&\cdots&0&\\ \frac{1}{2P}\xi_2&0&z&\cdots&0&\\ \vdots&\vdots&\vdots&\ddots&\vdots&\\ \frac{1}{2P}\xi_n&0&0&\cdots&z&\\ \end{matrix} \right] \end{equation} Then \begin{equation}\label{5.8} \begin{aligned} \sum_{r=0}^nC_rQ_{r\beta}&=\sum_{r=0}^nC_rB'_{r\alpha}\cdot L^{\alpha\beta}\\
&=\sum_{r=0}^nC_rB'_{r\alpha}\cdot(z^2g^{00}+|\xi|_h^2)^n\cdot\delta^{\alpha\beta}\\
&=\sum_{r=0}^nC_rB'_{r\beta}\cdot(z^2g^{00}+|\xi|_h^2)^n\\
&\equiv 0\ \ (mod (z-\sqrt{-1}\frac{|\xi|_h}{\sqrt{g^{00}}})^{n+1}) \end{aligned} \end{equation} Which implies \begin{equation}\label{5.9}
z-\sqrt{-1}\frac{|\xi|_h}{\sqrt{g^{00}}}\mid\sum_{r=0}^nC_rB'_{r\beta} \end{equation} for any $0\leq\beta\leq n.$\\
When $\beta\geq 1,(\ref{5.9})\Rightarrow z-\sqrt{-1}\frac{|\xi|_h}{\sqrt{g^{00}}}\mid \frac{C_0}{2P}\xi_\beta+C_\beta z.$ Then \begin{equation}\label{5.10}
C_\beta =-\frac{C_0\sqrt{g^{00}}\xi_\beta}{2P|\xi|_h\cdot\sqrt{-1}} \end{equation} When $\beta=0,$
$$(\ref{5.9})\Rightarrow z-\sqrt{-1}\frac{|\xi|_h}{\sqrt{g^{00}}}\mid C_0z-2PC_1g^{i1}\xi_i-2PC_2g^{i2}\xi_i-\cdots-2PC_ng^{in}\xi_i$$ With (\ref{5.10}), we have \begin{equation}\label{5.11}
z-\sqrt{-1}\frac{|\xi|_h}{\sqrt{g^{00}}}\mid C_0z+\frac{C_0\sqrt{g^{00}}}{2P|\xi|_h\cdot\sqrt{-1}} \end{equation} By a linear transformation, we can make $g^{00}(x_0)\neq 1, \forall x_0\in D.$ Then (\ref{5.11}) shows that $C_0=0,$ and (\ref{5.10}) implies $C_r=0,r=1,2,\cdots,n.$
\end{proof} Lemma \ref{lemm 5.1} and theorem 6.3.7 in \cite{Mo} tell us that $g^{00}, g_{01},\cdots,g_{0n}$ are all in $C^{m+1,\alpha}.$ Then the boundary condition (\ref{5.2}) can be written as $$g^{00}N(g_{00})=Q(g,\partial g^{00},\partial g_{0i},\partial g_{ij})\in C^{m,\alpha}(D)$$ With the elliptic equation $$\Delta g_{00}=-2R_{00}+Q(g,\partial g)\in C^{m-1,\alpha}(V),$$ we finally derive $g_{00}\in C^{m+1,\alpha}.$
\par Now we know that $g\in C^{m+1,\alpha}.$ Back to step 1, we have $A_{ij}\in C^{m,\alpha}(D),$ then $R_{\alpha\beta}\in C^{m,\alpha}(V).$ Repeating the steps above, we can get $g_{\alpha\beta}\in C^{m+2,\alpha}(V).$ Then we complete the proof. \subsection{Regularity of the structure and the defining function} We have already proved that $g$ is in $ C^{m+2,\alpha}$ in structure $\{x^\beta\}.$ It is trivial that $\{x^\beta\}$ is a $ C^{m+3,\alpha}$ structure of $\overline{M}.$ \par In section 2, when we make constant scalar compactification, we obtain that $u$ is in $C^{m,\alpha}(y).$ When we change the y-coordinates to harmonic coordinates x, we know that $x^\beta$ are $C^{m+1,\alpha}$ functions of $y^{\beta}.$ So the defining function $\rho\in C^{m,\alpha}(x).$\par Since the initial compactification $g$ is smooth in $M$ and the initial defining function is smooth in y-coordinates, then $\rho\in C^\infty(x)$ in $M.$ \par For any $p\in{\partial M},$ consider the neighborhood $V$ of $p$ and $D={\partial M}\cap V.$ By a linear transformation, we can assume that at $p,$ $g_{\alpha\alpha}=1, g_{ij}=g_{02}=g_{03}=\cdots=g_{0n}=0 (i\neq j), g_{01}=\delta$ for some $\delta>0$ satisefying that $1-\delta$ is a very small posivive number. When $g_+$ is Einstein, from (\ref{3.2}) and (\ref{3.3}), we have $$Ric-\frac{Sg}{n+1}=-(n-1)\frac{D^2\rho}{\rho}+\frac{n-1}{n+1}\frac{\Delta\rho}{\rho}g.$$ In local coordinates, when acting on $(\frac{\partial}{\partial x^0},\frac{\partial}{\partial x^1}),$ we have \begin{equation}\label{5.12} \Delta\rho-(n+1)\cdot g_{01}^{-1}\cdot D^2\rho(\frac{\partial}{\partial x^0},\frac{\partial}{\partial x^1})=\frac{n+1}{n-1}\cdot g_{01}^{-1}\cdot\rho(Ric_{01}-\frac{Sg_{01}}{n+1}) \end{equation}
When $1-\delta$ is small enough, we can find that the left hand of (\ref{5.12}) is a uniformly elliptic operator on $\rho$ locally. As $\rho R_{01}\in C^{m,\alpha}(\overline{M}),$ $\rho|_D\equiv 0,$ so in $V,$ $\rho\in C^{m+2,\alpha}(x).$ \\ \par To prove Remark(\ref{rema1.2}), we only need to show that $\rho R_{01}\in C^{m+1,\alpha}(\overline{M}).$ When $\dim M=n+1$ is even, we can define the obstruction tensor $\mathcal{O}_{ij}.$ In local coordinates: \begin{equation}\label{5.13} \mathcal{O}_{ij}=\Delta^{\frac{n+1}{2}-2}(P_{ij,k}^{\ \ \ \ k}-P_{ik,j}^{\ \ \ \ k})+Q_n. \end{equation} Where the $P_{ij}$ is defined by $P_{ij}=\frac{1}{2}R_{ij}-\frac{S}{12}g_{ij}$ and $Q_n$ denotes quadratic and higher terms in metric involving at most n-th derivatives. $\mathcal{O}_{ij}$ is conformally invariant of weight $2-n$ and if $g_{ij}$ is conformal to an Einstein metric, then $\mathcal{O}_{ij}=0$ (see more in \cite{GH}). \\ Since the scalar curvature of $(\overline{M},g)$ is constant, (\ref{5.13}) can be written in the following form: \begin{equation}\label{5.14} \Delta^{\frac{n+1}{2}-1}R_{ij}=Q_n. \end{equation} Now we consider the function $\rho R_{01}.$ Through a direct calculation, $\Delta(\rho R_{ij})=\rho\Delta R_{ij}+Q_3^2.$ Here $Q_3^2$ denotes quadratic and higher terms in metric involving at most 3-th derivatives and in $\rho$ involving at most 2-th derivatives. \\ Then we use iterative method to obtain that $\Delta^k(\rho R_{ij})=\rho\Delta^k R_{ij}+Q_{2k+1}^{2k}$ for $1\leq k\leq\frac{n+1}{2}-1.$ Let $k=\frac{n+1}{2}-1,$ we have an elliptic equation of second order with Dirichlet boundary condition: \begin{equation}\label{5.15}
\left\{
\begin{array}{l}
\Delta(\Delta^{\frac{n+1}{2}-2}(\rho R_{01}))=Q_n^{n-1} \ in \ \overline{M}
\\ \Delta^{\frac{n+1}{2}-2}(\rho R_{01})|_{{\partial M}}=Q_{n-2}^{n-3}
\end{array}
\right.
\end{equation}
Since $g$ and $\rho$ are all in $C^{m+2,\alpha},m+2\geq n,$ we have
$$\Delta^{\frac{n+1}{2}-2}(\rho R_{01})\in C^{m+2-(n-2),\alpha}(\overline{M}).$$
Then we consider the equation
\begin{equation}\label{5.16}
\left\{
\begin{array}{l}
\Delta(\Delta^{\frac{n+1}{2}-3}(\rho R_{01}))\in C^{m+2-(n-2),\alpha}(\overline{M})
\\ \Delta^{\frac{n+1}{2}-3}(\rho R_{01})|_{{\partial M}}=Q_{n-4}^{n-5}\in C^{m+2-(n-4),\alpha}({\partial M})
\end{array}
\right.
\end{equation}
So we have $\Delta^{\frac{n+1}{2}-3}(\rho R_{01})\in C^{m+2-(n-4),\alpha}(\overline{M})$
\\
Keep using the equation, we finally get $\rho R_{01}\in C^{m+1,\alpha}(\overline{M}),$ Which implies $\rho\in C^{m+3,\alpha}(\overline{M})$ by (\ref{5.12}). $\{x^\beta\}$ is a $C^{m+3,\alpha}$ structure of $\overline{M},$ so $C^{m+3,\alpha}$ regularity of $\rho$ is the best result we can get.
\addcontentsline{toc}{part}{References}
\noindent{Xiaoshang Jin}\\
Department of Mathematics, Nanjing University, Nanjing, 210093, P.R. China.
\\Email address:{[email protected]}
\end{document} | arXiv |
\begin{document}
\begin{frontmatter}
\title{On contact pseudo-metric manifolds satisfying a nullity condition}
\author[mymainaddress]{Narges Ghaffarzadeh}
\ead{[email protected]}
\author[mymainaddress, mycorrespondingauthor]{Morteza Faghfouri} \cortext[mycorrespondingauthor]{Corresponding author} \ead{[email protected]} \address[mymainaddress]{Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.}
\begin{abstract} In this paper, we aim to introduce and study $(\kappa, \mu)$-contact pseudo-metric manifold and prove that if the $\varphi$-sectional curvature of any point of $M$ is independent of the choice of $\varphi$-section at the point, then it is constant on $M$ and accordingly the curvature tensor. Also, we introduce generalized $(\kappa, \mu)$-contact pseudo-metric manifold and prove for $n>1$, that a non-Sasakian generalized $(\kappa, \mu)$-contact pseudo-metric manifold is a $(\kappa, \mu)$-contact pseudo-metric manifold. \end{abstract}
\begin{keyword} contact pseudo-metric manifold\sep $(\kappa, \mu)$-contact pseudo-metric structure. \MSC[2010] 53C25\sep 53C50\sep 53C15. \end{keyword}
\end{frontmatter}
\linenumbers
\section{Introduction}
Contact pseudo-metric structures were first introduced by Takahashi \cite{Takahashi:SasakianManifoldWithPseud}. He defined Sasakian manifold with pseudo-metric and the classification of Sasakian manifolds of constant $\phi$-sectional curvatures. Next, K. L. Duggal \cite{Duggal:SpaceTimeManifoldsContactStructures} and A. Bejancu \cite{Bejancu1993} studied contact pseudo-metric structures as a generalization of contact Lorentzian structures and contact Riemannian structures. Recently, contact pseudo-metric manifolds and curvature of $K$-contact pseudo-Riemannian manifolds have been studied by Calvaruso and Perrone \cite{Calvaruso.Perrone:ContactPseudoMtricManifolds} and Perrone \cite{Perrone:CurvatureKcontact}, respectively. Also Perrone \cite{Perrone2014}, Perrone investigated contact pseudo-metric manifolds of constant curvature and CR manifolds.
In \cite{Blair:Contactmetricmanifoldssatisfyingnullitycondition}, D. E. Blair et al. introduced $(\kappa,\mu)$-contact Riemannian manifold. Since then, many researchers have studied the structure \cite{koufogiorgos:Contact.constant.curvature,boeckx:a.full.classification.of.contact.metric,Koufogiorgos:OnTheExistenceContactMetricManifolds, faghfouriGhaffarzade:doublywarped,faghfouriGhaffarzade:invariant,faghfouriGhaffarzade:CtotallyReal}.
In this paper, we introduce and study $(\kappa, \mu)$-contact pseudo-metric manifold. The paper is organized as follows. Section $2$ contains some necessary background on contact pseudo-metric manifolds. After introducing $(\kappa,\mu)$-contact pseudo-metric manifold in section 3, we prove some relationships. In this section, we also prove if the $\varphi$-sectional curvature of any point of $M$ is independent of the choice of $\varphi$-section at the point, then it is constant on $M$ and we find the curvature tensor. In fact, our main purpose in this paper is to find the curvature tensor of $(\kappa,\mu)$-contact pseudo-metric manifolds. In addition, we show that $M$ has constant $\varphi$-sectional curvature if and only if $\mu=\varepsilon\kappa+1$ when $\kappa\neq\varepsilon$. In section $4$, we introduce generalized $(\kappa, \mu)$-contact pseudo-metric manifold. In this section, we also prove for $n>1$, that a non-Sasakian generalized $(\kappa, \mu)$-contact pseudo-metric manifold is a $(\kappa, \mu)$-contact pseudo-metric manifold.
\section{Preliminaries} A $(2n+1)$-dimensional differentiable manifold $M$ is called an almost contact pseudo-metric manifold if there is an almost contact pseudo-metric structure $(\varphi,\xi,\eta,g)$ consisting of a $(1, 1)$ tensor field $\varphi$, a vector field $\xi$, a $1$-form $\eta$ and a compatible pseudo-Riemannian metric $g$ satisfying \begin{gather} \eta (\xi)= 1,\varphi^2(X)=-X+\eta(X)\xi, \label{001}\\ g(\varphi X,\varphi Y)=g(X,Y)-\varepsilon\eta(X)\eta(Y),\label{002} \end{gather} where $\varepsilon=\pm1$ and $X,Y\in\Gamma(TM)$. Remark that, by (\ref{001}) and (\ref{002}), we have \begin{gather} \varphi\xi =0,\eta\circ\varphi=0,\\ \eta(X) =\varepsilon g(\xi, X),\\ g(\varphi X,Y)=-g(X,\varphi Y), \end{gather} and $\varphi$ has rank $2n$.
In particular, $g(\xi,\xi)=\varepsilon$ and so, the characteristic vector field $\xi$ is either space-like or time-like, but cannot be light-like and the signature of an associated metric is either $(2p+1,2n-2p)$ or $(2p,2n-2p-1)$. An almost contact pseudo-metric structure becomes a contact pseudo-metric structure if $d\eta=\Phi$, where $\Phi(X,Y)=g( X,\varphi Y)$ is the fundamental $2$-form of $M$.
An almost contact pseudo-metric structure of $M$ is called a normal structure if $[\varphi,\varphi]+2d\eta\otimes\xi=0$. A normal contact pseudo-metric structure is called a Sasakian structure. It can be proved that an almost contact pseudo-metric manifold is Sasakian iff \begin{align} (\nabla_X\varphi)Y=g(X,Y)\xi-\varepsilon\eta(Y)X, \end{align} for any $X, Y\in \Gamma(TM)$ or equivalently, a contact pseudo-metric structure is a Sasakian structure iff $R$ satisfies \begin{align}\label{eq:sasakicurvature} R(X,Y)\xi=\eta(Y)X-\eta(X)Y, \end{align} for $X,Y\in\Gamma(TM)$, where $R(X,Y)=[\nabla_X,\nabla_Y]-\nabla_{[X,Y]}$ is the curvature tensor and $\nabla$ is the Levi-Civita connection\cite{kumar.rani.nagachi:Onsectionalcurvatures,Calvaruso.Perrone:ContactPseudoMtricManifolds}. In a contact pseudo-metric manifold $M^{2n+1}(\varphi,\xi,\eta,g)$, we define the $(1, 1)$-tensor fields $\ell$ and $h$ by \begin{align}\label{050} \ell X=R(X,\xi)\xi,\quad hX=\frac{1}{2}(\mathcal{L}_{\xi}\varphi)(X), \end{align} where $\mathcal{L}$ denotes the Lie derivative. The tensors $h$ and $\ell$ are self-adjoint operators satisfying(\cite{Calvaruso.Perrone:ContactPseudoMtricManifolds,Perrone:CurvatureKcontact}) \begin{gather} \tr(h)=\tr( h\varphi)=0,\label{051}\\ \eta\circ h=0, \quad\ell\xi =0,\label{055}\\ h\varphi=-\varphi h,\label{013}\\ h\xi =0,\label{052}\\ \nabla_{X}\xi =-\varepsilon\varphi X-\varphi hX,\label{033}\\ (\nabla_{X}\varphi)Y=\varepsilon g(\varepsilon X+hX,Y)\xi-\eta(Y)(\varepsilon X+hX),\label{031}\\ \nabla_{\xi}\varphi =0.\label{072} \end{gather} Due to the relation of (\ref{013}), if $X$ is an eigenvector of $h$ corresponding to the eigenvalue $\lambda$, then $\varphi X$ is also an eigenvector of $h$ corresponding to the eigenvalue $-\lambda$.
Let $M^{2n+1}(\varphi,\xi,\eta,g)$ be a contact pseudo-metric manifold and $X\in\ker\eta$, either space-like or time-like. We put \begin{gather} K(X,\xi)=\dfrac{\mathcal{R}(X,\xi ,X,\xi)}{\varepsilon g(X,X)}=\dfrac{g(\ell X,X)}{\varepsilon g(X,X)},\label{060}\\ K(X,\varphi X)=\dfrac{\mathcal{R}(X,\varphi X,X,\varphi X)}{g(X,X)^{2}}. \end{gather} We call $K(X,\xi)$ the $\xi$-sectional curvature determined by $X$, and $K(X,\varphi X)$ the $\varphi$-sectional curvature determined by $X$, where $\mathcal{R}(X,Y,Z,W)=g(R(Z,W)Y,X)$. A Sasakian manifold with constant $\varphi$-sectional curvature $c$ is called a Sasakian space form and is denoted by $M(c).$ \begin{lemma}[\cite{Calvaruso.Perrone:ContactPseudoMtricManifolds}] Let $M^{2n+1}(\varphi,\xi,\eta,g)$ be a contact pseudo-metric manifold. Then: \begin{gather} \nabla_{\xi}h=\varphi-\varphi\ell-\varphi h^{2},\label{073}\\ \varphi\ell\varphi -\ell =2(\varphi^{2}+h^{2}),\label{027}\\ \Ric(\xi,\xi)=2n-\tr(h^{2}),\label{035}\\
\mathcal{R}(\xi ,X,Y,Z)=\varepsilon(\nabla _{X}\Phi)(Y,Z)+g((\nabla_{Y}\varphi h)Z,X)-g((\nabla_{Z}\varphi h)Y,X).\label{032} \end{gather} \end{lemma} \section{$(\kappa, \mu)$-contact pseudo-metric manifold}
A $(\kappa ,\mu)$-nullity distribution of a contact pseudo-metric manifold $M^{2n+1}(\varphi ,\xi,\eta, g)$ is a distribution \begin{align}\label{61}\begin{split} N_{p}(\kappa ,\mu)=\lbrace Z\in T_{p}M:R(X,Y)Z=&\kappa(g(Y,Z)X-g(X,Z)Y)\\ &+\mu(g(Y,Z)hX-g(X,Z)hY)\rbrace,\end{split} \end{align} where $(\kappa ,\mu)\in \mathbb{R}^2$.
Thus, the characteristic vector field $\xi$ belongs to the $(\kappa ,\mu)$-distribution iff \begin{align}\label{62} R(X,Y)\xi= \varepsilon\kappa(\eta(Y )X -\eta(X)Y ) +\varepsilon \mu(\eta(Y )hX -\eta(X)hY ). \end{align}
If a contact pseudo-metric manifold satisfying (\ref{62}), we call $(\kappa, \mu)$-contact pseudo-metric manifold. The class of $(\kappa, \mu)$-contact pseudo-metric manifold contains the class of Sasakian manifolds, which we get for $\kappa=\varepsilon$ (and hence $h=0$, by (\ref{eq:sasakicurvature})). \begin{lemma}\label{071} Let $M^{2n+1}(\varphi,\xi,\eta,g)$ be a $(\kappa, \mu)$-contact pseudo-metric manifold. Then, we have \begin{gather} \ell\varphi-\varphi\ell=2\varepsilon\mu h\varphi,\label{053}\\
h^2=(\varepsilon\kappa-1)\varphi ^2,\quad \varepsilon\kappa \leq 1, \quad \text{ and }\quad \kappa=\varepsilon \text{ iff } M^{2n+1} \text{ is Sasakian }, \label{030}\\ R(\xi ,X)Y=\kappa(g(X,Y)\xi-\varepsilon\eta(Y)X)+\mu(g(hX,Y)\xi -\varepsilon\eta(Y)hX),\label{023}\\ Q\xi=2n\kappa\xi,\text{\quad $Q$ is the Ricci operator},\label{054}\\ (\nabla_{X}h)Y-(\nabla _{Y}h)X=(1-\varepsilon\kappa)\{2\varepsilon g(X,\varphi Y)\xi+\eta(X)\varphi Y-\eta(Y)\varphi X\}\nonumber\\ \quad \quad \quad \quad\quad \quad \quad \quad\quad\quad+\varepsilon(1-\mu)\{\eta(X)\varphi hY-\eta(Y)\varphi hX\},\label{48}\\ \xi\kappa =0,\label{070} \end{gather} where $X,Y\in\Gamma(TM)$. \end{lemma} \begin{proof} Using (\ref{052}), we obtain $$\ell X=\varepsilon\kappa(X-\eta(X)\xi)+\varepsilon\mu hX,$$ for $X\in\Gamma(TM)$. Replacing $X$ by $\varphi X$ and at the same time applying $\varphi$, we obtain \begin{align}\label{026} \ell\varphi=\varepsilon\{\kappa\varphi+\mu h\varphi\}\quad\text{and}\quad\varphi\ell=\varepsilon\{\kappa\varphi+\mu\varphi h\}. \end{align} Subtracting (\ref{026}) and using (\ref{013}), we get (\ref{053}).\\ By using the relations (\ref{027}), (\ref{013}), (\ref{026}), (\ref{052}) and (\ref{001}), we deduce the first part of (\ref{030}). Now since $h$ is symmetric, from the second part of (\ref{001}), we have $\varepsilon\kappa\leq1$. Moreover, $\kappa=\varepsilon$ iff $h=0$. Using (\ref{62}) and (\ref{eq:sasakicurvature}), the proof of (\ref{030}) is completed.
Using (\ref{62}), we get (\ref{023}) and $g(R(\xi,X)Y,Z)=g(R(Y,Z)\xi,X)$.\\ For the relation of (\ref{054}), let $\{E_{1},\ldots, E_{n},E_{n+1}=\varphi E_{1}, \ldots,E_{2n}=\varphi E_{n},E_{2n+1}=\xi\}$ be a (local)$\varphi$-basis of $M$. For any index $i =1,\ldots,2n$, $\{\xi, E_{i}\}$ spans a non-degenerate plane on the tangent space at each point where the basis is defined. Then the definition of the Ricci operator $Q$, (\ref{023}), (\ref{051}) and (\ref{052}) give \begin{align*} \Ric (\xi,X)=&\sum_{i=1}^{2n+1}\varepsilon_{i}g(R(E_{i},\xi)X,E_{i})\\ =&\sum_{i=1}^{2n+1}\varepsilon_{i}\{\kappa[\varepsilon\eta(X)g(E_{i},E_{i})-\varepsilon g(E_{i},X)\eta(E_{i})]\\ &+\mu[\varepsilon\eta(X)g(hE_{i},E_{i})-\varepsilon g(hE_{i},X)\eta(E_{i})]\}\\ =&\varepsilon\kappa\eta(X)\sum_{i=1}^{2n+1}\varepsilon_{i}^{2}-\varepsilon\kappa\eta(X)+\mu\varepsilon\eta(X)\tr(h)\\ =&\varepsilon\kappa\eta(X)(2n+1)-\varepsilon\kappa\eta(X)=(2n+1-1)\varepsilon\kappa\eta(X)=2n\varepsilon\kappa\eta(X), \end{align*} so, we have (\ref{054}). Now with using (\ref{031}) and the symmetry of $h$, we get \begin{align*} (\nabla_{X}\varphi h)Y-(\nabla _{Y}\varphi h)X=\varphi((\nabla_{X}h)Y-(\nabla _{Y}h)X), \end{align*} for any vector fields $X,Y$ on $M$ and hence (\ref{032}) is reduced to $$R(Y,X)\xi=\eta(X)(Y +\varepsilon hY) - \eta(Y)(X+\varepsilon hX) + \varphi((\nabla_{X}h)Y - (\nabla_{Y}h)X).$$ Comparing this equation with (\ref{62}), we have \begin{align}\label{056}\begin{split} \varphi((\nabla_{X}h)Y-(\nabla_{Y}h)X)=&(\varepsilon\kappa-1)(\eta(X)Y-\eta(Y)X)\\ &+\varepsilon(\mu-1)(\eta(X)hY-\eta(Y)hX).\end{split} \end{align} Using (\ref{033}), the symmetry of $h$ and $\nabla_{X}h$, we obtain \begin{align}\label{057} g((\nabla_{X}h)Y-(\nabla_{Y}h)X,\xi)=2(\varepsilon\kappa-1)g(Y,\varphi X). \end{align} Acting now by $\varphi$ on (\ref{056}) and using (\ref{057}), we get \eqref{48}. \end{proof} \begin{lemma} Let $M^{2n+1}(\varphi,\xi,\eta,g)$ be a $(\kappa, \mu)$-contact pseudo-metric manifold. Then for all $X,Y,Z\in\Gamma(TM)$, we have \begin{align}\label{006}\begin{split} R(X,Y)\varphi Z=&\varphi R(X,Y)Z+\{(1-\varepsilon\kappa)[\eta(X)g(\varphi Y,Z)-\eta(Y)g(\varphi X,Z)]\\ &+\varepsilon(1-\mu)[\eta(X)g(\varphi hY,Z)-\eta(Y)g(\varphi hX,Z)]\}\xi\\ &-g(Y+\varepsilon hY,Z)(\varepsilon \varphi X+\varphi hX)+g(X+\varepsilon hX,Z)(\varepsilon \varphi Y+\varphi hY)\\ &-g(\varepsilon\varphi Y+\varphi hY,Z)(X+\varepsilon hX)+g(\varepsilon\varphi X+\varphi hX,Z)(Y+\varepsilon hY)\\ &-\eta(Z)\{(1-\varepsilon\kappa)[\eta(X)\varphi Y-\eta(Y)\varphi X]+\varepsilon(1-\mu)[\eta(X)\varphi hY-\eta(Y)\varphi hX]\}. \end{split} \end{align} \end{lemma} \begin{proof} Assume that $p\in M$ and $X, Y, Z$ local vector fields on a neighborhood of $p$, such that $$(\nabla X)_{p}=(\nabla Y)_{p}=(\nabla Z)_{p}=0.$$ The Ricci identity for $\varphi$: \begin{align} R( X, Y)\varphi Z-\varphi R( X, Y)Z=(\nabla_{X}\nabla_{Y}\varphi)Z-\nabla_{Y}(\nabla_{X}\varphi)Z-(\nabla_{[X,Y]}\varphi)Z, \end{align} at the point $p$, takes the form \begin{align}\label{034} R(X,Y)\varphi Z-\varphi R(X,Y)Z=\nabla_{X}(\nabla_{Y}\varphi)Z-\nabla_{Y}(\nabla_{X}\varphi)Z. \end{align} On the other hand, combining (\ref{033}) and (\ref{031}), we have at $p$ \begin{align}\label{058}\begin{split} \nabla_{X}(\nabla_{Y}\varphi)Z-\nabla_{Y}(\nabla_{X}\varphi)Z=&\varepsilon g((\nabla_{X}h)Y-(\nabla_{Y}h)X,Z)\xi-g(Y+\varepsilon hY,Z)(\varepsilon\varphi X+\varphi hX)\\ &+\varepsilon g(\varepsilon\varphi X+\varphi hX,Z)(\varepsilon Y+hY)+g(X+\varepsilon hX,Z)(\varepsilon\varphi Y+\varphi hY)\\ &-\varepsilon g(Z,\varepsilon \varphi Y+\varphi hY)(\varepsilon X+hX)-\eta(Z)((\nabla_{X}h)Y-(\nabla_{Y}h)X) \end{split} \end{align} Now equation (\ref{006}) is a straightforward combination of the (\ref{058}), (\ref{034}) and (\ref{48}). \end{proof} \begin{theorem}\label{037} Let $M^{2n+1}(\varphi,\xi,\eta,g)$ be a $(\kappa, \mu)$-contact pseudo-metric manifold. Then $\varepsilon\kappa\leq 1$. If $\kappa= \varepsilon$, then $h=0$ and $M^{2n+1}$ is a Sasakian-space-form and if $\varepsilon\kappa<1$, then $M^{2n+1}$ admits three mutually orthogonal and integrable distributions $\mathcal{D}(0)=\Span\{\xi\}, \mathcal{D}(\lambda)$ and $\mathcal{D}(-\lambda)$, defined by the eigenspaces of $h$, where $\lambda=\sqrt{1-\varepsilon\kappa}$. \end{theorem} \begin{proof} By $\xi\in N(\kappa,\mu)$, we can verify $\Ric(\xi,\xi)=2n\varepsilon\kappa$. Then, (\ref{035}) implies $\varepsilon\kappa\leq1$. Now, we suppose $\varepsilon\kappa<1$. Then since $h$ is symmetric, the relations (\ref{052}) and (\ref{031}) imply that the restriction $h\vert\mathcal{D}$ of $h$ to the contact distribution $\mathcal{D}$ has eigenvalues $\lambda=\sqrt{1-\varepsilon\kappa}$ and $-\lambda$. By $\mathcal{D}(\lambda)$ and $\mathcal{D}(-\lambda)$, we denote the distributions defined by the eigenspaces of $h$ corresponding to $\lambda$ and $-\lambda$, respectively. By $\mathcal{D}(0)$, we denote the distribution defined by $\xi$. Then these three distributions are mutually orthogonal. Let $X\in\mathcal{D}(\lambda)$, Then $hX=\lambda X$ and the relation of (\ref{013}) imply $h(\varphi X)=-\lambda(\varphi X)$. Hence, we have $\varphi X\in\mathcal{D}(-\lambda)$. This means that the dimension of $\mathcal{D}(\lambda)$ and $\mathcal{D}(-\lambda)$ are equal to $n$. We prove that $\mathcal{D}(\lambda)$ ( $\mathcal{D}(-\lambda)$, resp.) is integrable. Let $X,Y\in\mathcal{D}(\lambda)$ ($\mathcal{D}(-\lambda)$, resp.). Then $$\nabla_{X}\xi=-\varepsilon\varphi X-\varphi hX=-(\varepsilon\pm\lambda)\varphi X,$$ and $\nabla_{Y}\xi=-(\varepsilon\pm\lambda)\varphi Y$. So, $g(\nabla_{X}\xi,Y)=g(\nabla_{Y}\xi,X)$ holds. Thus, $d\eta(X,Y)=0$ and $\eta([X,Y])=0$ follow. $X,Y\in\mathcal{D}(\lambda)$ and $\xi\in N(\kappa,\mu)$ imply $R(X,Y)\xi=0$. On the other hand, \begin{align}\label{059}\begin{split} 0&=\nabla_{X}\nabla_{Y}\xi-\nabla_{Y}\nabla_{X}\xi-\nabla_{[X,Y]}\xi\\ &=-(\varepsilon\pm\lambda)\nabla_{X}(\varphi Y)+(\varepsilon\pm\lambda)\nabla_{Y}(\varphi X)+\varepsilon\varphi([X,Y])+\varphi h([X,Y])\\ &=-(\varepsilon\pm\lambda)\{(\nabla_{X}\varphi)Y-(\nabla_{Y}\varphi)X\}\mp\lambda\varphi([X,Y])+\varphi h([X,Y]). \end{split} \end{align} By (\ref{031}), the first term of the last line (\ref{059}) vanishes. And so, we obtain $$\varphi h([X,Y])=\mp\lambda\varphi([X,Y]),$$ which together with $\eta([X,Y])=0$ implies $[X,Y]\in\mathcal{D}(\lambda)$ ($\mathcal{D}(-\lambda)$, resp.). \end{proof}
\begin{pro}\label{47} Let $M^{2n+1}(\varphi,\xi,\eta,g)$ be a $(\kappa, \mu)$-contact pseudo-metric manifold with $\varepsilon\kappa <1$, then \begin{itemize}
\item If $X,Y\in \mathcal{D}(\lambda)$ (resp. $\mathcal{D}(-\lambda)$), then $\nabla_{X}Y\in \mathcal{D}(\lambda)$ (resp. $\mathcal{D}(-\lambda)$).
\item If $X\in \mathcal{D}(\lambda), Y\in \mathcal{D}(-\lambda)$, then $\nabla_{X}Y (resp. \nabla_{Y}X )$ has no component in $\mathcal{D}(\lambda)$ (resp. $\mathcal{D}(-\lambda)$). \end{itemize} \end{pro}
\begin{lemma}\label{042} Let $M^{2n+1}(\varphi,\xi,\eta,g)$ be a $(\kappa, \mu)$-contact pseudo-metric manifold. Then for any vector fields $X,Y$ on $M$, we have \begin{align}\label{041}\begin{split} (\nabla_{X}h)Y=&\{(\varepsilon-\kappa)g(X,\varphi Y)+g(X,h\varphi Y)\}\xi\\ &+\eta(Y)[h(\varepsilon\varphi X+\varphi hX)]-\varepsilon\mu\eta(X)\varphi hY. \end{split} \end{align} \end{lemma} \begin{proof} Let $\varepsilon\kappa< 1$ and $X,Y\in\mathcal{D}(\lambda)$(resp., $\mathcal{D}(-\lambda)$). Then from Proposition \ref{47}, we have $\nabla_{X}Y\in\mathcal{D}(\lambda)$ (resp., $\mathcal{D}(-\lambda)$). Then one easily proves that \begin{align}\label{038} (\nabla_{X}h)Y=0. \end{align} Suppose now that $X\in\mathcal{D}(\lambda)$, $Y\in\mathcal{D}(-\lambda)$ and $\{E_{i},\varphi E_{i},\xi\}$ be a (local) $\varphi$-basis of vector fields on $M$ with $E_{i}\in\mathcal{D}(\lambda)$ and so $\varphi E_{i}\in\mathcal{D}(-\lambda)$. For any index $i =1,\cdots,2n$, $\{\xi, E_{i}\}$ spans a non-degenerate plane on the tangent space at each point where the basis is defined. Then using Proposition \ref{47} and the relations (\ref{052}), (\ref{001}) and (\ref{033}), we calculate \begin{align*} h\nabla_{X}Y&=h\{\sum_{i=1}^{n}\varepsilon_{i}g(\nabla_{X}Y,\varphi E_{i})\varphi E_{i}+\varepsilon g(\nabla_{X}Y,\xi)\xi\}\\ &=\sum_{i=1}^{n}\varepsilon_{i}g(\nabla_{X}Y,\varphi E_{i})h\varphi E_{i}\\ &=\lambda\varphi\sum_{i=1}^{n}\varepsilon_{i}g(\varphi\nabla_{X}Y,E_{i})E_{i}\\ &=\lambda\varphi^{2}(\nabla_{X}Y)\\ &=\lambda(-\nabla_{X}Y+\varepsilon g(\nabla_{X}Y,\xi)\xi)\\ &=\lambda(-\nabla_{X}Y-\varepsilon g(Y,\nabla_{X}\xi)\xi)\\ &=\lambda(-\nabla_{X}Y-\varepsilon g(Y,-\varepsilon\varphi X-\varphi hX)\xi)\\ &=\lambda(-\nabla_{X}Y+g(Y,\varphi X+\varepsilon\varphi hX)\xi)\\ &=\lambda(-\nabla_{X}Y-g(\varphi Y,X+\varepsilon hX)\xi)\\ &=\nabla_{X}hY-\lambda(1+\varepsilon\lambda)g(X,\varphi Y)\xi, \end{align*} and so \begin{align}\label{039} (\nabla_{X}h)Y=\lambda(1+\varepsilon\lambda)g(X,\varphi Y)\xi, \end{align} Similarly, we obtain \begin{align}\label{040} (\nabla_{Y}h)X=\lambda(\varepsilon\lambda -1)g(Y,\varphi X)\xi, \end{align} Suppose now that $X,Y$ are arbitrary vector fields on $M$ and write $$X=X_{\lambda}+X_{-\lambda}+\eta(X)\xi,$$ and $$Y=Y_{\lambda}+Y_{-\lambda}+\eta(Y)\xi,$$ where $X_{\lambda}$ (resp., $X_{-\lambda}$) is the component of $X$ in $\mathcal{D}(\lambda)$ (resp., $\mathcal{D}(-\lambda)$). Then using (\ref{038}), (\ref{039}), (\ref{040}) and $\nabla_{\xi}h=\varepsilon\mu h\varphi$, which follows from (\ref{48}), we get by a direct computation \begin{align}\label{063} \begin{split} (\nabla_{X}h)Y=&\varepsilon\lambda^{2}\{g(X_{\lambda},\varphi Y_{-\lambda})+g(X_{-\lambda},\varphi Y_{\lambda})\}\xi\\ &+\lambda\{g(X_{\lambda},\varphi Y_{-\lambda})-g(X_{-\lambda},\varphi Y_{\lambda})\}\xi\\ &+\eta(Y)h(\varepsilon\varphi X+\varphi hX)-\varepsilon\mu\eta(X)\varphi hY. \end{split} \end{align} On the other hand, we easily find that \begin{gather} g(hX,\varphi Y)=\lambda\{g(X_{\lambda},\varphi Y_{-\lambda})-g(X_{-\lambda},\varphi Y_{\lambda})\},\label{064}\\ g(hX,h\varphi Y)=\lambda^{2}\{g(X_{-\lambda},\varphi Y_{\lambda})+g(X_{\lambda},\varphi Y_{-\lambda})\}.\label{065} \end{gather} The relations (\ref{064}) and (\ref{065}) with (\ref{063}), give the required equation (\ref{041}). Note that for $\kappa=\varepsilon$ (and so $h=0$), (\ref{041}) is valid identically and the proof is completed. \end{proof} \begin{lemma} Let $M^{2n+1}(\varphi,\xi,\eta,g)$ be a $(\kappa, \mu)$-contact pseudo-metric manifold. Then for any vector fields $X,Y,Z$ on $M$. We have \begin{align}\label{045}\begin{split} R(X,Y)hZ-hR(X,Y)Z=&\{\kappa[\eta(X)g(hY,Z)-\eta(Y)g(hX,Z)]\\ &+\mu(\varepsilon\kappa-1)[\eta(Y)g(X,Z)-\eta(X)g(Y,Z)]\}\xi\\ &+\kappa\{g(Y,\varphi Z)\varphi hX-g(X,\varphi Z)\varphi hY+g(Z,\varphi hY)\varphi X\\ &-g(Z,\varphi hX)\varphi Y+\varepsilon\eta(Z)[\eta(X)hY-\eta(Y)hX]\}\\ &-\mu\{\eta(Y)[(\varepsilon-\kappa)\eta(Z)X+\mu\eta(X)hZ]\\ &-\eta(X)[(\varepsilon-\kappa)\eta(Z)Y+\mu\eta(Y)hZ]+2\varepsilon g(X,\varphi Y)\varphi hZ\}. \end{split} \end{align} \end{lemma} \begin{proof} The Ricci identity for $h$ is \begin{align}\label{043} R(X,Y)hZ-hR(X,Y)Z=(\nabla_{X}\nabla_{Y}h)Z-(\nabla_{Y}\nabla_{X}h)Z-(\nabla_{[X,Y]}h)Z. \end{align} Using Lemma \ref{042}, the relations (\ref{030}), (\ref{013}) and the fact that $\nabla_{X}\varphi$ is antisymmetric, we obtain \begin{align*} (\nabla_{X}\nabla_{Y}h)Z=&\{(\varepsilon-\kappa)[g(\nabla_{X}Y,\varphi Z)-g((\nabla_{X}\varphi)Y,Z)]\\ &+g(\nabla_{X}Y,h\varphi Z)+g(\nabla_{X}(h\varphi)Y,Z)\}\xi\\ &+\{(\varepsilon-\kappa)g(Y,\varphi Z)+g(Y,h\varphi Z)\}\nabla_{X}\xi\\ &+\varepsilon g(Z,\nabla_{X}\xi)[\varepsilon h\varphi Y+(\varepsilon\kappa-1)\varphi Y]\\ &+\eta(Z)\{\varepsilon[(\nabla_{X}h\varphi)Y+h\varphi(\nabla_{X}Y)]+(\varepsilon\kappa-1)[(\nabla_{X}\varphi)Y+\varphi(\nabla_{X}Y)]\}\\ &-\varepsilon\mu\{\eta(\nabla_{X}Y)\varphi hZ+\varepsilon g(Y,\nabla_{X}\xi)\varphi hZ+\eta(Y)(\nabla_{X}\varphi h)Z\}. \end{align*} So, using also (\ref{041}), (\ref{033}), (\ref{031}) and Lemma \ref{042}, equation (\ref{043}) yields \begin{align}\label{044}\begin{split} R(X,Y)hZ-h&R(X,Y)Z\\ =&\{(\kappa-\varepsilon)g((\nabla_{X}\varphi)Y-(\nabla_{Y}\varphi)X,Z)+g((\nabla_{X}h\varphi)Y-(\nabla_{Y}h\varphi)X,Z)\}\xi\\ &+\{(\varepsilon-\kappa)g(Y,\varphi Z)+g(Y,h\varphi Z)\}\nabla_{X}\xi\\ &-\{(\varepsilon-\kappa)g(X,\varphi Z)+g(X,h\varphi Z)\}\nabla_{Y}\xi\\ &+g(Z,\nabla_{X}\xi)[h\varphi Y+(\kappa-\varepsilon)\varphi Y]\\ &-g(Z,\nabla_{Y}\xi)[h\varphi X+(\kappa-\varepsilon)\varphi X]\\ &+\eta(Z)\{\varepsilon[(\nabla_{X}h\varphi)Y-(\nabla_{Y}h\varphi)X]+(\varepsilon\kappa-1)[(\nabla_{X}\varphi)Y-(\nabla_{Y}\varphi)X]\}\\ &-\varepsilon\mu\{\eta(Y)(\nabla_{X}\varphi h)Z-\eta(X)(\nabla_{Y}\varphi h)Z+2g(X,\varphi Y)\varphi hZ\}. \end{split} \end{align} Using now (\ref{031}), (\ref{052}) and Lemma \ref{042}, we have \begin{align*} (\nabla_{X}\varphi h)Y=&\{g(X,hY)+(\kappa-\varepsilon)g(X,-Y+\eta(Y)\xi)\}\xi\\ &+\eta(Y)[\varepsilon hX+(\varepsilon\kappa-1)(-X+\eta(X)\xi)]+\varepsilon\mu\eta(X)hY. \end{align*} Therefore, equation (\ref{044}), by using (\ref{031}) again, is reduced to (\ref{045}) and the proof is completed. \end{proof} \begin{theorem}\label{049} Let $M^{2n+1}(\varphi,\xi,\eta,g)$be a $(\kappa, \mu)$-contact pseudo-metric manifold. If $\varepsilon\kappa< 1$, then for all $X_{\lambda},Z_{\lambda},Y_{\lambda}\in \mathcal{D}(\lambda)$ and $X_{-\lambda},Z_{-\lambda},Y_{-\lambda}\in \mathcal{D}(-\lambda)$, we have \begin{align} R(X_{\lambda},Y_{\lambda})Z_{-\lambda}&=(\kappa-\varepsilon\mu)[g(\varphi Y_{\lambda},Z_{-\lambda})\varphi X_{\lambda}-g(\varphi X_{\lambda},Z_{-\lambda})\varphi Y_{\lambda}],\label{41}\\ R(X_{-\lambda},Y_{-\lambda})Z_{\lambda}&=(\kappa-\varepsilon\mu)[g(\varphi Y_{-\lambda},Z_{\lambda})\varphi X_{-\lambda}-g(\varphi X_{-\lambda},Z_{\lambda})\varphi Y_{-\lambda}],\label{42}\\ R(X_{-\lambda},Y_{\lambda})Z_{-\lambda}&=-\kappa g(\varphi Y_{\lambda},Z_{-\lambda}) \varphi X_{-\lambda}-\varepsilon\mu g(\varphi Y_{\lambda},X_{-\lambda})\varphi Z_{-\lambda} ,\label{43}\\ R(X_{-\lambda},Y_{\lambda})Z_{\lambda}&=\kappa g(\varphi X_{-\lambda},Z_{\lambda}) \varphi Y_{\lambda}+\varepsilon\mu g(\varphi X_{-\lambda},Y_{\lambda})\varphi Z_{\lambda} ,\label{44}\\ R(X_{\lambda},Y_{\lambda})Z_{\lambda}&=[2(\varepsilon+\lambda)-\varepsilon\mu][g(Y_{\lambda},Z_{\lambda})X_{\lambda}-g(X_{\lambda},Z_{\lambda})Y_{\lambda}],\label{45}\\ R(X_{-\lambda},Y_{-\lambda})Z_{-\lambda}&=[2(\varepsilon-\lambda)-\varepsilon\mu][g(Y_{-\lambda},Z_{-\lambda})X_{-\lambda}-g(X_{-\lambda},Z_{-\lambda})Y_{-\lambda}],\label{46} \end{align} \end{theorem} \begin{proof} The first part of the Theorem follows from (\ref{030}) and Lemma \ref{037}.\\ Let $\{E_{1},\cdots,E_{n},E_{n+1}=\varphi E_{1},\cdots,E_{2n}=\varphi E_{n},E_{2n+1}=\xi\}$ be a (local) $\varphi$-basis of vector fields on $M$ with $E_{i}\in\mathcal{D}(\lambda)$ and so $\varphi E_{i}\in\mathcal{D}(-\lambda)$. For any index $i =1,\cdots,2n$,$\{\xi, E_{i}\}$ spans a non-degenerate plane on the tangent space at each point, where the basis is defined. Then, we have \begin{align}\label{048}\begin{split} R(X_{\lambda},Y_{\lambda})Z_{-\lambda}=&\sum_{i=1}^{n}\varepsilon_{i}\{g(R(X_{\lambda},Y_{\lambda})Z_{-\lambda},E_{i})E_{i}+g(R(X_{\lambda},Y_{\lambda})Z_{-\lambda},\varphi E_{i})\varphi E_{i}\}\\ &+\varepsilon g(R(X_{\lambda},Y_{\lambda})Z_{-\lambda},\xi)\xi. \end{split} \end{align} But since $\xi$ belonging to the $(\kappa ,\mu)$-nullity distribution, using (\ref{62}), we easily have $$g(R(X_{\lambda},Y_{\lambda})Z_{-\lambda},\xi)=-g(R(X_{\lambda},Y_{\lambda})\xi,Z_{-\lambda})=0.$$ By Proposition \ref{47}, we get $$g(R(X_{\lambda},Y_{\lambda})Z_{-\lambda},E_{i})=-g(R(X_{\lambda},Y_{\lambda})E_{i},Z_{-\lambda})=0.$$ On the other hand, if $X\in\mathcal{D}(\lambda)$ and $Y,Z\in\mathcal{D}(-\lambda)$, then applying (\ref{045}), we get $$hR(X,Y)Z+\lambda R(X,Y)Z=-2\lambda\{\kappa g(X,\varphi Z)\varphi Y+\varepsilon\mu g(X,\varphi Y)\varphi Z\},$$ and taking the inner product with $W\in\mathcal{D}(\lambda)$, we obtain \begin{align}\label{047} g(R(X,Y)Z,W)=-\kappa g(X,\varphi Z)g(\varphi Y,W)-\varepsilon\mu g(X,\varphi Y)g(\varphi Z,W), \end{align} for any $X,W\in\mathcal{D}(\lambda)$ and $Y,Z\in\mathcal{D}(-\lambda)$. Using (\ref{047}) and the first Bianchi identity, we calculate \begin{align*} \sum_{i=1}^{n}&\varepsilon_{i}g(R(X_{\lambda},Y_{\lambda})Z_{-\lambda},\varphi E_{i})\varphi E_{i}\\ =&-\sum_{i=1}^{n}\varepsilon_{i}g(R(Y_{\lambda},Z_{-\lambda})X_{\lambda},\varphi E_{i})\varphi E_{i}-\sum_{i=1}^{n}\varepsilon_{i}g(R(Z_{-\lambda},X_{\lambda})Y_{\lambda},\varphi E_{i})\varphi E_{i}\\ =&\sum_{i=1}^{n}\varepsilon_{i}g(R(Y_{\lambda},Z_{-\lambda})\varphi E_{i},X_{\lambda})\varphi E_{i}-\sum_{i=1}^{n}\varepsilon_{i}g(R(X_{\lambda},Z_{-\lambda})\varphi E_{i},Y_{\lambda})\varphi E_{i}\\ =&\sum_{i=1}^{n}\varepsilon_{i}\{-\kappa g(Y_{\lambda},\varphi^{2}E_{i})g(\varphi Z_{-\lambda},X_{\lambda})\varphi E_{i}-\varepsilon\mu g(Y_{\lambda},\varphi Z_{-\lambda})g(\varphi^{2}E_{i},X_{\lambda})\varphi E_{i}\}\\ &-\sum_{i=1}^{n}\varepsilon_{i}\{-\kappa g(X_{\lambda},\varphi^{2}E_{i})g(\varphi Z_{-\lambda},Y_{\lambda})\varphi E_{i}-\varepsilon\mu g(X_{\lambda},\varphi Z_{-\lambda})g(\varphi^{2}E_{i},Y_{\lambda})\varphi E_{i}\}\\ =&\kappa g(\varphi Z_{-\lambda},X_{\lambda})\varphi\sum_{i=1}^{n}\varepsilon_{i}g(Y_{\lambda},E_{i})E_{i}+\varepsilon\mu g(Y_{\lambda},\varphi Z_{-\lambda})\varphi\sum_{i=1}^{n}\varepsilon_{i}g(E_{i},X_{\lambda})E_{i}\\ &+\kappa g(Z_{-\lambda},\varphi Y_{\lambda})\varphi\sum_{i=1}^{n}\varepsilon_{i}g(X_{\lambda},E_{i})E_{i}+\varepsilon\mu g(\varphi X_{\lambda},Z_{-\lambda})\varphi\sum_{i=1}^{n}\varepsilon_{i}g(E_{i},Y_{\lambda})E_{i}\\ =&\kappa\{g(Z_{-\lambda},\varphi Y_{\lambda})\varphi X_{\lambda}-g(Z_{-\lambda},\varphi X_{\lambda})\varphi Y_{\lambda}\}\\ &+\varepsilon\mu\{g(\varphi X_{\lambda},Z_{-\lambda})\varphi Y_{\lambda}-g(\varphi Y_{\lambda},Z_{-\lambda})\varphi X_{\lambda}\}\\ =&(\kappa-\varepsilon\mu)\{g(Z_{-\lambda},\varphi Y_{\lambda})\varphi X_{\lambda}-g(Z_{-\lambda},\varphi X_{\lambda})\varphi Y_{\lambda}\}. \end{align*} Therefore, (\ref{048}) gives $$R(X_{\lambda},Y_{\lambda})Z_{-\lambda}=(\kappa-\varepsilon\mu)\{g(Z_{-\lambda},\varphi Y_{\lambda})\varphi X_{\lambda}-g(Z_{-\lambda},\varphi X_{\lambda})\varphi Y_{\lambda}\}.$$ The proof of the remaining cases are similar and will be omitted. \end{proof} Then they showed the following. \begin{theorem} Let $M^{2n+1}(\varphi,\xi,\eta,g)$be a $(\kappa, \mu)$-contact pseudo-metric manifold. If $\varepsilon\kappa<1$, then for any $X$ orthogonal to $\xi$\\ $(i)$ the $\xi$-sectional curvature $K(X,\xi)$ is given by \begin{equation*} K(X,\xi)=\kappa+\mu\dfrac{g(hX,X)}{g(X,X)} = \left\{ \begin{array}{lr} \kappa+\lambda\mu,\quad\text{if $X\in\mathcal{D}(\lambda)$}, \\ \\ \kappa-\lambda\mu,\quad\text{if $X\in\mathcal{D}(-\lambda)$}, \end{array}\right. \end{equation*} $(ii)$ the sectional curvature of a plane section $(X,Y)$ normal to $\xi$ is given by \begin{align}\label{025} K(X,Y)= \left\{ \begin{array}{lr} 2(\varepsilon+\lambda)-\varepsilon\mu,\quad\quad\quad\quad\text{ for any $X,Y\in\mathcal{D}(\lambda), n>1$},\\ \\ -(\kappa+\varepsilon\mu)\dfrac{g(X,\varphi Y)^{2}}{g(X,X)g(Y,Y)},\quad\text{ for any unit vectors $X\in\mathcal{D}(\lambda), Y\in\mathcal{D}(-\lambda)$},\\ \\ 2(\varepsilon-\lambda)-\varepsilon\mu,\quad\quad\quad\quad\text{ for any $X,Y\in\mathcal{D}(-\lambda), n>1$}, \end{array}\right. \end{align} $(iii)$\quad The Ricci operator is given by \begin{align} \begin{split}\label{024} QX=&\varepsilon[2(n-1)-n\mu]X+(2(n-1)+\mu)hX+[2(1-n)\varepsilon +2n\kappa +n\varepsilon\mu]\eta(X)\xi. \end{split} \end{align} \end{theorem} \begin{proof} $(i)$ From (\ref{060}), if we set $Y=\xi$ in the relation of (\ref{62}), for $X$ orthogonal to $\xi$ from which, taking the inner product with $X$, we get $$K(X,\xi)=\dfrac{\varepsilon\{\kappa g(X,X)+\mu g(hX,X)\}}{\varepsilon g(X,X)}.$$ So, we have \begin{align*} K(X,\xi)&=\kappa+\mu \dfrac{g(hX,X)}{g(X,X)}\\ &=\kappa+\mu \dfrac{\lambda g(hX_{\lambda},X_{\lambda})-\lambda g(hX_{-\lambda},X_{-\lambda})}{g(X_{\lambda},X_{\lambda})+g(X_{-\lambda},X_{-\lambda})}, \end{align*} which is the required result. \begin{itemize}
\item[(ii)] This follows immediately from Theorem \ref{049}.
\item[(iii)] The first consider a $\varphi$-basis $\{E_{1}, \ldots, E_{n}, E_{n+1}=\varphi E_{1}, \ldots,$ $ E_{2n}=\varphi E_{n}, E_{2n+1}=\xi \}$ of vector fields on $M$. \end{itemize} For any index $i =1,\ldots, 2n$,$\{\xi, E_{i}\}$ spans a non-degenerate plane on the tangent space at each point, where the basis is defined. Putting $Y=Z=E_{i}$ in $R(X,Y)Z$, adding with respect to index of $i$ and using (\ref{001}), (\ref{002}) and (\ref{013}), we get the following formula, for the Ricci operator, at any point of $M$: $$QX=\sum_{i=1}^{n}\varepsilon_{i}\{R(X,E_{i})E_{i}+R(X,\varphi E_{i})\varphi E_{i}\}+\varepsilon R(X,\xi)\xi.$$ Suppose now that $X$ is arbitrary vector fields and write $$X=X_{\lambda}+X_{-\lambda}+\eta(X)\xi,$$ On the other hand, from Theorem \ref{049}, we have \begin{align*} QX=&\sum_{i=1}^{n}\varepsilon_{i}\{R(X_{\lambda},E_{i})E_{i}+R(X_{-\lambda},E_{i})E_{i}+\eta(X)R(\xi,E_{i})E_{i}+R(X_{\lambda},\varphi E_{i})\varphi E_{i}\\ &+\eta(X)R(\xi,\varphi E_{i})\varphi E_{i}+R(X_{-\lambda},\varphi E_{i})\varphi E_{i}\}+\varepsilon R(X_{\lambda},\xi)\xi+\varepsilon R(X_{-\lambda},\xi)\xi\\ =&[2(\varepsilon+\lambda)-\varepsilon\mu](n-1)X_{\lambda}-(\kappa+\varepsilon\mu)X_{-\lambda}+n\kappa\eta(X)\xi-(\kappa+\varepsilon\mu)X_{\lambda}\\ &+[2(\varepsilon-\lambda)-\varepsilon\mu](n-1)X_{-\lambda}+(\kappa+\varepsilon\mu)X_{\lambda}+n\kappa\eta(X)\xi+(\kappa+\mu h)X_{-\lambda}\\ =&\varepsilon[(2-\mu)(n-1)-\mu](X_{\lambda}+X_{-\lambda})\\ &+[2(n-1)+\mu]h(X_{\lambda}+X_{-\lambda})+2n\kappa\eta(X)\xi. \end{align*} So, the relation of (\ref{024}) is obtained. \end{proof}
\begin{theorem}\label{087}
Let $M^{2n+1}(\eta, \xi, \varphi, g)$ be a $(\kappa, \mu)$-contact pseudo-metric manifold and $n>1$. If the $\varphi$-sectional curvature of any point of $M$ is independent of the choice of $\varphi$-section at the point, then it is constant on $M$ and the curvature tensor is given by \begin{align}\label{022} \begin{split} R(X, Y)Z=&(\frac{c+3\varepsilon}{4})\{g(Y, Z)X-g(X, Z)Y\}\\ &+(\dfrac{c-\varepsilon}{4})\{2g(X, \varphi Y)\varphi Z+g(X, \varphi Z)\varphi Y-g(Y, \varphi Z)\varphi X\}\\ &+(\dfrac{c+3\varepsilon}{4}-\kappa)\{\varepsilon\eta(X)\eta(Z)Y-\varepsilon\eta(Y)\eta(Z)X+\eta(Y)g(X, Z)\xi-\eta(X)g(Y, Z)\xi\}\\ &+\{-g(X, Z)hY-g(hX, Z)Y+g(Y, Z)hX+g(hY, Z)X\}\\ &+\dfrac{\varepsilon}{2}\{-g(hX, Z)hY+g(hY, Z)hX+g(\varphi hX, Z)\varphi hY-g(\varphi hY, Z)\varphi hX\}\\ &+(1-\mu)\{\varepsilon\eta(X)\eta(Z)hY+\eta(Y)g(hX, Z)\xi-\varepsilon\eta(Y)\eta(Z)hX-\eta(X)g(hY, Z)\xi\}, \end{split} \end{align} where $c$ is the constant $\varphi$-sectional curvature. Moreover if $\kappa\neq\varepsilon$, then $\mu=\varepsilon\kappa+1$ and $c=-2\kappa-\varepsilon$. \end{theorem} \begin{proof}
For the Sasakian case $\kappa=\varepsilon$, the proof is known (\cite{Takahashi:SasakianManifoldWithPseud}). So, we have to prove the theorem for $\kappa\neq \varepsilon$. Let $p\in M$ and $X,$ $Y\in T_{p}M$ orthogonal to $\xi$. Using the first identity of Bianchi, the basic properties of the curvature tensor, $\varphi$ is antisymmetric, $h$ is symmetric, (\ref{001}) and (\ref{002}), we obtain from (\ref{006}), successively: \begin{align}\label{007}\begin{split} g(R(X,\varphi X)Y,\varphi Y)=&g(R(X,\varphi Y)Y,\varphi X)+g(R(X,Y)X,Y)-\varepsilon g(X, Y)^{2}-\varepsilon g(hX, Y)^{2} \\ &-2g(X, Y)g(hX, Y)+\varepsilon g(X, X)g(Y, Y)+g(X, X)g(hY, Y) \\ &+g(Y, Y)g(hX, X)+\varepsilon g(hX, X)g(hY, Y)-\varepsilon g(\varphi X, Y)^{2} \\ &+\varepsilon g(\varphi hX, Y)^{2}-\varepsilon g(\varphi hX, X)g(\varphi hY,Y), \end{split} \end{align} \begin{align}\label{008}\begin{split} g(R(X, \varphi Y)X,\varphi Y)=&g(R(X, \varphi Y)Y,\varphi X)+\varepsilon g(X, Y)^{2}-\varepsilon g(hX, Y)^{2} \\ &-\varepsilon g(\varphi hX, X)g(\varphi hY, Y)-\varepsilon g(X, X)g(Y, Y)-g(Y, Y)g(hX, X) \\ &+g(X, X)g(hY, Y)+\varepsilon g(hX, X)g(hY, Y)+\varepsilon g(\varphi X, Y)^{2} \\ &+\varepsilon g(\varphi hX, Y)^{2}+2g(\varphi X, Y)g(\varphi hX,Y),\end{split} \end{align} \begin{align}\label{009}\begin{split} g(R(Y, \varphi X)Y,\varphi X)=&g(R(X, \varphi Y)Y,\varphi X)+\varepsilon g(X, Y)^{2}-\varepsilon g(hX, Y)^{2} \\ &-\varepsilon g(\varphi hX, X)g(\varphi hY, Y)+\varepsilon g(\varphi X, Y)^{2}+\varepsilon g(\varphi hX, Y)^{2} \\ &-2g(\varphi X, Y)g(\varphi hX, Y)-\varepsilon g(X, X)g(Y, Y)-g(X, X)g(hY, Y) \\ &+g(Y, Y)g(hX, X)+\varepsilon g(hX, X)g(hY, Y),\end{split} \end{align} \begin{align}\label{010}\begin{split} g(R(X, Y)\varphi X,\varphi Y)=&g(R(X, Y)X,Y)-\varepsilon g(X, Y)^{2}-\varepsilon g(hX, Y)^{2} \\ &-2g(X, Y)g(hX, Y)+\varepsilon g(X, X)g(Y, Y)+g(X, X)g(hY, Y) \\ &+g(Y, Y)g(hX, X)+\varepsilon g(hX, X)g(hY, Y)-\varepsilon g(\varphi X, Y)^{2} \\ &+\varepsilon g(\varphi hX, Y)^{2}-\varepsilon g(\varphi hX, X)g(\varphi hY, Y).\end{split} \end{align} We now suppose that the $\varphi$-sectional curvature at $p$ is independent of the $\varphi$-section at $p$, i.e. $K(X, \varphi X)=c(p)$ for any $X\in T_{p}M$ orthogonal to $\xi$. Let $X,Y\in T_{p}M$ and $X,Y$ orthogonal to $\xi$ . From \begin{align*} g(R(X+Y, \varphi X+\varphi Y)(X+Y),\varphi X+\varphi Y)&=-c(p) g(X+Y, X+Y)^{2},\\ g(R(X-Y, \varphi X-\varphi Y)(X-Y),\varphi X-\varphi Y)&=-c(p)g(X-Y, X-Y)^{2}, \end{align*} we get by a straightforward calculation \begin{align}\label{011}\begin{split} 2g(R(X,\varphi X)Y,\varphi Y)&+g(R(X, \varphi Y)X,\varphi Y)+2g(R(X, \varphi Y)Y,\varphi X)+g(R(Y, \varphi X)Y,\varphi X) \\ &=-2c(p)\{2g(X, Y)^{2}+g(X, X)g(Y, Y)\}.\end{split} \end{align} Thus with combining (\ref{007}), (\ref{008}), (\ref{009}) and (\ref{011}), we get \begin{align}\label{012}\begin{split} &3g(R(X, \varphi Y)Y,\varphi X)+g(R(X, Y)X,Y)-2\varepsilon g(hX, Y)^{2} \\ &-2g(X, Y)g(hX, Y)+g(X, X)g(hY, Y)+g(Y, Y)g(hX, X) \\ &+2\varepsilon g(hX, X)g(hYY)+2\varepsilon g(\varphi hX, Y)^{2}-2\varepsilon g(\varphi hX, X)g(\varphi hY, Y) \\ &=-c(p)\{2g(X, Y)^{2}+g(X, X)g(Y, Y)\}.\end{split} \end{align} Now, we replace $Y$ by $\varphi Y$ in (\ref{012}), then using (\ref{009}) and (\ref{013}), we have \begin{align}\label{014}\begin{split} &-3g(R(X, Y)\varphi Y,\varphi X)+g(R(X, \varphi Y)X,\varphi Y)-2g(\varphi hX, Y)^{2} \\ &+2g(X, \varphi Y)g(\varphi hX, Y)-g(X, X)g(hY, Y)+g(Y, Y)g(hX, X) \\ &-2g(hX, X)g(hY, Y)+2g(hX, Y)^{2}+2g(\varphi hX, X)g(\varphi hY, Y) \\ &=-c(p)\{2g(X, \varphi Y)^{2}+g(X, X)g(Y, Y)\}.\end{split} \end{align} On the other hand, with combining the relation of (\ref{014}) with (\ref{008}) and (\ref{010}), we get \begin{align}\label{015}\begin{split} &3g(R(X, Y)X,Y)+g(R(X, \varphi Y)Y,\varphi X)-2\varepsilon g(X, Y)^{2} \\ &-2\varepsilon g(hX, Y)^{2}-6g(X, Y)g(hX, Y)+2\varepsilon g(X, X)g(Y, Y) \\ &+3g(X, X)g(hY, Y)+3g(Y, Y)g(hX, X)+2\varepsilon g(hX, X)g(hY, Y) \\ &-2\varepsilon g(X, \varphi Y)^{2}+2\varepsilon g(\varphi hX, Y)^{2}-2\varepsilon g(\varphi hX,X)g(\varphi hY, Y) \\ &=-c(p)\{2g(X, \varphi Y)^{2}+g(X, X)g(Y, Y)\}.\end{split} \end{align} Now for any $X,Y\in T_{p}M$ and $X,Y$ orthogonal to $\xi$, (\ref{015}) together with (\ref{012}) yield \begin{align}\label{016}\begin{split} 4g(R(X,Y)Y,X)=&(c(p)+3\varepsilon)\{g(X,X)g(Y,Y)-g(X,Y)^{2}\}+3(c(p)-\varepsilon)g(X,\varphi Y)^{2} \\ &-2\{\varepsilon g(hX,Y)^{2}+4g(X,Y)g(hX,Y)-2g(X,X)g(hY,Y)-2g(Y,Y)g(hX,X) \\ &-\varepsilon g(hX,X)g(hY,Y)-\varepsilon g(\varphi hX,Y)^{2}+\varepsilon g(\varphi hX,X)g(\varphi hY,Y)\}.\end{split} \end{align} Let $X,Y,Z\in T_{p}M$ and $X,Y,Z$ orthogonal to $\xi$. Applying (\ref{016}) in $$g(R(X+Z, Y)Y,X+Z)=g(R(X, Y)Y,X)+g(R(Z, Y)Y,Z)+2g(R(X, Y)Y,Z).$$ Finally, we get \begin{align}\label{017}\begin{split} 4g(R(X, Y)Y,Z)=&(c(p)+3\varepsilon)\{g(X, Z)g(Y, Y)-g(X,Y)g(Y, Z)\} \\ &+3(c(p)-\varepsilon)g(X, \varphi Y)g(Z, \varphi Y)-2\{\varepsilon g(hX, Y)g(hZ,Y)+2g(X,Y)g(hZ, Y) \\ &+2g(Z, Y)g(hX, Y)-2g(X, Z)g(hY, Y)-2g(Y, Y)g(hX, Z) \\ &-\varepsilon g(hX, Z)g(hY, Y)-\varepsilon g(\varphi hX, Y)g(\varphi hZ, Y)+\varepsilon g(\varphi hX, Z)g(\varphi hY, Y)\}.\end{split} \end{align} Moreover, using (\ref{013}), (\ref{62}) and $ h\varphi$ is symmetric, it is easy to check that (\ref{017}) is valid for any $Z$ and for $X,Y$ orthogonal to $\xi$. Hence for any $X,Y$ orthogonal to $\xi$, the relation of (\ref{017}) is reduced to \begin{align}\label{018}\begin{split} 4R(X, Y)Y=&(c(p)+3\varepsilon)\{g(Y, Y)X-g(X, Y)Y\}+3(c(p)-\varepsilon)g(X, \varphi Y)\varphi Y \\ &-2\{\varepsilon g(hX, Y)hY+2g(X, Y)hY+2g(hX, Y)Y-2g(hY, Y)X \\ &-2g(Y, Y)hX-\varepsilon g(hY, Y)hX-\varepsilon g(\varphi hX, Y)\varphi hY+\varepsilon g(\varphi hY, Y)\varphi hX\}.\end{split} \end{align} Now, let $X,Y,Z$ be orthogonal to $\xi$. Replacing $Y$ by $Y+Z$ in (\ref{018}). Then from $$R(X,Y+Z)(Y+Z)=R(X,Y)Y+R(X,Z)Z+R(X,Y)Z+R(X,Z)Y,$$ we get \begin{align}\label{019}\begin{split} 4\{R(X, Y)Z+R(X, Z)Y\}=&(c(p)+3\varepsilon)\{2g(Y, Z)X-g(X, Y)Z-g(X, Z)Y\} \\ &+3(c(p)-\varepsilon)\{g(X, \varphi Y)\varphi Z+g(X, \varphi Z)\varphi Y\} \\ &-2\{\varepsilon g(hX, Y)hZ+g(hX, Z)hY+2g(X, Y)hZ \\ &+2g(X, Z)hY+2g(hX, Y)Z+2g(hX, Z)Y-4g(hY, Z)X \\ &-4g(Y, Z)hX-2\varepsilon g(hY, Z)hX-\varepsilon g(\varphi hX, Y)\varphi hZ \\ &-\varepsilon g(\varphi hX, Z)\varphi hY+2\varepsilon g(\varphi hY, Z)\varphi hX\}.\end{split} \end{align} Replacing $X$ by $Y$ and $Y$ by $-X$ in (\ref{019}), we have \begin{align}\label{020}\begin{split} 4\{R(X, Y)Z+R(Z, Y)X\}=&(c(p)+3\varepsilon)\{-2g(X, Z)Y+g(X, Y)Z+g(Y, Z)X\} \\ &+3(c(p)-\varepsilon)\{-g(\varphi X, Y)\varphi Z-g(\varphi Z, Y)\varphi X\} \\ &-2\{-\varepsilon g(hY, X)hZ-\varepsilon g(hY, Z)hX-2g(X, Y)hZ \\ &-2g(Y, Z)hX-2g(X, hY)Z-2g(hY, Z)X \\ &+4g(hX, Z)Y+4g(X, Z)hY+2\varepsilon g(hX, Z)hY \\ &+\varepsilon g(\varphi hY, X)\varphi hZ+\varepsilon g(\varphi hY, Z)\varphi hX-2\varepsilon g(\varphi hX, Z)\varphi hY\}.\end{split} \end{align} Adding (\ref{019}) and (\ref{020}) and using Bianchi first identity, $\varphi$ is antisymmetric and $\varphi h$ is symmetric, we get \begin{align}\label{021}\begin{split} 4R(X, Y)Z=&(c(p)+3\varepsilon)\{g(Y, Z)X-g(X, Z)Y\} \\ &+(c(p)-\varepsilon)\{2g(X, \varphi Y)\varphi Z+g(X, \varphi Z)\varphi Y-g(Y, \varphi Z)\varphi X\} \\ &-2\{\varepsilon g(hX, Z)hY+2g(X, Z)hY+2g(hX, Z)Y-2g(hY, Z)X \\ &-2g(Y, Z)hX-\varepsilon g(hY, Z)hX-\varepsilon g(\varphi hX, Z)\varphi hY+\varepsilon g(\varphi hY, Z)\varphi hX\},\end{split} \end{align} for any $X,Y,Z$ orthogonal to $\xi$. Moreover, using (\ref{62}), (\ref{052}) and the first part of (\ref{001}), we conclude that (\ref{021}) is valid for any $Z$ and for $X,Y$ orthogonal to $\xi$. Now, let $X,Y,Z$ be arbitrary vectors of $T_{p}M$. Writing $$X=X_{T}+\eta(X)\xi,\quad Y=Y_{T}+\eta(Y)\xi,$$ where $g(X_{T}, \xi)=g(Y_{T}, \xi)=0$ , and using (\ref{62}), (\ref{023}) and (\ref{052}), then (\ref{021}) gives (\ref{022}) after a straightforward calculation.\\ Now, we will prove that the $\varphi$-sectional curvature is constant. Consider a (local) $\varphi$-basis $\{E_{1},\cdots, E_{n},E_{n+1}=\varphi E_{1}, \cdots,E_{2n}=\varphi E_{n},E_{2n+1}=\xi\}$ of vector fields on $M$. For any index $i = 1,\cdots,2n$,$\{\xi, E_{i}\}$ spans a non-degenerate plane on the tangent space at each point, where the basis is defined. Putting $Y=Z=E_{i}$ in (\ref{022}), adding with respect to $i$ and using (\ref{001}), (\ref{002}) and (\ref{013}), we get the following formula, for the Ricci operator, at any point of $M$: \begin{align*}\begin{split} 2Q=&\{(n+1)c+3\varepsilon(n-1)+2\kappa\}I \\ &-\{(n+1)c+3\varepsilon(n-1)-2\kappa(2n-1)\}\eta\otimes\xi \\ &+2\{2(n-1)+\mu\}h.\end{split} \end{align*} Comparing this with (\ref{024}), which is valid on any $(\kappa, \mu)$-contact pseudo-metric manifold with $\kappa\neq\varepsilon$, we get \begin{align}\label{061} (n+1)c=(n-1)\varepsilon-2n\varepsilon\mu-2\kappa, \end{align} i.e. $c$ is constant. On the other hand, from (\ref{025}), we have \begin{align}\label{062}
c=-(\kappa+\varepsilon\mu). \end{align} Comparing (\ref{061}) and (\ref{062}), we get $(n-1)(\varepsilon\mu-\kappa-\varepsilon)=0$. Moreover, since $n>1$, we have $\mu=\varepsilon\kappa+1$ and so $c=-2\kappa-\varepsilon$. This completes the proof of the theorem. \end{proof} \begin{theorem}\label{088} Let $M^{2n+1}(\varphi,\xi,\eta,g)$ be a $(\kappa, \mu)$-contact pseudo-metric manifold with $\varepsilon\kappa<1$ and $n>1$. Then $M$ has constant $\varphi$-sectional curvature if and only if $\mu=\varepsilon\kappa+1$ . \end{theorem} \begin{proof} In Theorem \ref{087}, we proved that $\mu=\varepsilon\kappa+1$, in the case where the non Sasakian, $(\kappa, \mu)$-contact pseudo-metric manifold has constant $\varphi$-sectional curvature. Now, we will prove the inverse, i.e. supposing $M(\varphi,\xi,\eta,g)$ is a $(2n+1)$-dimensional $(n>1)$, non Sasakian, $(\kappa, \mu)$-contact pseudo-metric manifold with \begin{align}\label{a} \mu=\varepsilon\kappa+1. \end{align} We will prove that $M$ has constant $\varphi$-sectional curvature. Let $X\in T_{p}M$ be a unit vector orthogonal to $\xi$. By Theorem \ref{037}, we can write \begin{center} $X=X_{\lambda}+X_{-\lambda}\qquad\text{ where}\quad X_{\lambda}\in \mathcal{D}(\lambda) \quad\text{and}\quad X_{-\lambda}\in \mathcal{D}(-\lambda)$. \end{center} Using Theorem \ref{037}, Theorem \ref{049} and a long straightforward calculation, we get $$K(X, \varphi X)=-(\kappa+\varepsilon\mu)+4(\kappa-\varepsilon\mu+\varepsilon)(g(X_{\lambda}, X_{\lambda})g(X_{-\lambda}, X_{-\lambda})-g(X_{\lambda}, \varphi X_{-\lambda})^{2})$$ and hence by (\ref{a}), we have $K(X, \varphi X)=-(\kappa+\varepsilon\mu)=\text{const}$. \end{proof} \begin{example}[\cite{GhaffarzadehFaghfouri2}, Theorem 4.1]
The tangent sphere bundle $T_{\varepsilon}M$ is $(\kappa, \mu)$-contact pseudo-metric manifold
if and only if the base manifold $M$ is of constant sectional curvature $\varepsilon$ and $\kappa=3\varepsilon-2, \mu=-2\varepsilon$.
\end{example}
\begin{theorem} Let $M$ be an $n$-dimensional pseudo-metric manifold, $n>2$, of constant sectional curvature $c$. The tangent sphere bundle $T_{\varepsilon}M$ has constant $\varphi$-sectional curvature $(-4c(\varepsilon-1)+ c^{2})$ if and only if $c=2\varepsilon\pm\sqrt{4+\varepsilon}$. \end{theorem} \begin{example} Let $M(\varphi,\xi,\eta,g)$ be a contact pseudo-metric manifold of dimension $2n+1$, with $g(\xi,\xi)=\varepsilon$. Then, it is easy to check that, for any real constant $a>0$ and by choosing the tensors $$\overline{\eta}=a\eta,\quad\overline\xi=\frac{1}{a}\xi,\quad\overline{\varphi}=\varphi,\quad\overline{g}=ag+\varepsilon a(a-1)\eta\otimes\eta,$$
$M(\overline{\varphi},\overline{\xi},\overline{\eta},\overline{g})$ is a new $(\overline{\kappa},\overline{\mu})$-contact pseudo-metric manifold with \begin{align}\label{089} \overline{\kappa}=\frac{\kappa+\varepsilon a^{2}-\varepsilon}{a^{2}}, \quad\quad \overline{\mu}=\frac{\mu+2a-2}{a}. \end{align} Now, we find the value of $a$, so that $M$ has constant $\varphi$-sectional curvature. Using Theorem \ref{088}, we must have $\overline{\mu}=\varepsilon\overline{\kappa}+1$. So, we get
$a=\dfrac{(\varepsilon\kappa-1)}{(\mu-2)}$.
In fact with choosing $a=\dfrac{(\varepsilon\kappa-1)}{(\mu-2)}>0$, $M(\overline{\varphi},\overline{\xi},\overline{\eta},\overline{g})$ has constant $\varphi$-sectional curvature $\overline{c}=-\overline{\kappa}-\varepsilon\overline{\mu}=\varepsilon(1-2\overline{\mu})=\dfrac{(2(\mu-2)^{2}-3(1-\varepsilon\kappa))}{(\varepsilon-\kappa)}$. \end{example} \section{generalized $(\kappa, \mu)$-contact pseudo-metric manifold}
In \eqref{61} and \eqref{62}, if $\kappa$ and $\mu$ are real smooth functions on $M$, we call generalized $(\kappa, \mu)$-contact pseudo-metric manifold.
\begin{example}
We consider the $3$-dimensional manifold $M =\{(x,y,z) \in \mathbb{R}^3 | z\neq 0\}$, where $(x,y,z)$ are the standard coordinates in $\mathbb{R}^3$. The vector fields $$e_{1}=\frac{\partial}{\partial x},\quad e_{2}=\frac{1}{z^{2}}\frac{\partial}{\partial y},\quad e_{3}=2yz^{2}\frac{\partial}{\partial x}+\frac{2x}{z^{6}}\frac{\partial}{\partial y}+\frac{1}{z^{6}}\frac{\partial}{\partial z}$$ are linearly independent at each point of $M$. Let $g$ be the pseudo-Riemannian metric defined by $g(e_{i},e_{j}) = \varepsilon_{i}\delta_{ij},$ where $ i, j = 1, 2, 3$ and $\varepsilon_{1}=\varepsilon,\quad\varepsilon_{2}=\varepsilon_{3}=1$. Let $\nabla$ be the Levi-Civita connection and $R$ the curvature tensor of $g$. We easily get $$[e_{1},e_{2}]=0,\quad [e_{1},e_{3}]=\frac{2}{z^{4}}e_{2},\quad [e_{2},e_{3}]=2\left(e_{1}+\frac{1}{z^{7}}e_2\right)$$ Let $\eta$ be the $1$-form defined by $\eta(X) =\varepsilon g(X, e_{1})$ for $X\in\Gamma(TM)$. $\eta$ is a contact form. Let $\varphi$ be the $(1,1)$-tensor field, defined by $\varphi e_{1} = 0,\quad\varphi e_{2} =e_{3},\quad\varphi e_{3} =-e_{2}$.
So, $(\varphi,\xi=e_{1},\eta,g)$ defines a contact pseudo-metric structure on $M$. Now using the Koszul formula, we calculate \begin{align*} &\nabla_{e_{1}}e_{2}=-(\varepsilon+\frac{1}{z^{4}})e_{3},\quad \nabla_{e_{2}}e_{1}=-(\varepsilon+\frac{1}{z^{4}})e_{3},\\ &\nabla_{e_{1}}e_{3}=(\varepsilon+\frac{1}{z^{4}})e_{2},\quad \nabla_{e_{3}}e_{1}=(\varepsilon-\frac{1}{z^{4}})e_{2},\\ &\nabla_{e_{2}}e_{3}=(1+\frac{\varepsilon}{z^{4}})e_{1}+\frac{2}{z^{7}}e_{2},\quad\nabla_{e_{3}}e_{2}=(\frac{\varepsilon}{z^{4}}-1)e_{1},\\ &\nabla_{e_{2}}e_{2}=-\frac{2}{z^{7}}e_{3},\quad \nabla_{e_{1}}e_{1}=\nabla_{e_{3}}e_{3}=0. \end{align*} Also Using \eqref{050}, we obtain $he_{1}=0$, $he_{2}=\lambda e_{2},he_{3}=-\lambda e_{3}$, where $\lambda=\dfrac{1}{z^{4}}$. Now, putting $\mu=2(1+\varepsilon\lambda)$ and $\kappa=\varepsilon(1-\lambda^{2})$, we finally get \begin{align*} R(e_{2},e_{1})e_{1}&=\varepsilon(\kappa+\lambda\mu)e_{2},\\ R(e_{3},e_{1})e_{1}&=\varepsilon(\kappa-\lambda\mu)e_{3},\\ R(e_{2},e_{3})e_{1}&=0. \end{align*} These relations yield the following, by direct calculations, $$R(X,Y)\xi=\varepsilon\kappa\{\eta(Y)X-\eta(X)Y\}+\varepsilon\mu\{\eta(Y)hX-\eta(X)hY\},$$ where $\kappa$ and $\mu$ are non-constant smooth functions. Hence $M$ is a generalized $(\kappa,\mu)$-contact pseudo-metric manifold. \end{example} \begin{theorem}\label{th:3.6} On a non Sasakian, generalized $(\kappa,\mu)$-contact pseudo-metric manifold $M^{2n+1}$ with $n>1$, the functions $\kappa, \mu$ are constant, i.e., $M^{2n+1}$ is a $(\kappa,\mu)$-contact pseudo-metric manifold. \end{theorem}
\begin{theorem} Let $M$ be a non Sasakian, generalized $(\kappa,\mu)$-contact pseudo-metric manifold. If $\kappa, \mu$ satisfy the condition $a\kappa+b\mu =c$, where $a,b$ and $c$ are constant. Then $\kappa, \mu$ are constant. \end{theorem} The proof of the two previous theorems is the same with Theorem 3.5 and Theorem 3.6 in \cite{Koufogiorgos:OnTheExistenceContactMetricManifolds} for contact metric case. Therefore, we omit them here.
\end{document} | arXiv |
Birkhäuser Mathematics
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Advances in Commutative Algebra
Dedicated to David F. Anderson
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This book highlights the contributions of the eminent mathematician and leading algebraist David F. Anderson in wide-ranging areas of commutative algebra. It provides a balance of topics for experts and non-experts, with a mix of survey papers to offer a synopsis of developments across a range of areas of commutative algebra and outlining Anderson's work. The book is divided into two sections—surveys and recent research developments—with each section presenting material from all the major areas in commutative algebra. The book is of interest to graduate students and experienced researchers alike.
AYMAN BADAWI is Professor at the Department of Mathematics and Statistics, the American University of Sharjah, the United Arab Emirates. He earned his Ph.D. in Algebra from the University of North Texas, USA, in 1993. He is an active member of the American Mathematical Society and honorary member of the Middle East Center of Algebra and its Applications. His research interests include commutative algebra, pi-regular rings, and graphs associated to rings.
JIM COYKENDALL is Professor of Mathematical Sciences at Clemson University, South Carolina, USA. He earned his Ph.D. from Cornell University in 1995, and has held various academic positions at the California Institute of Technology, the University of Tennessee, Cornell University, Lehigh University, and North Dakota State University. He has successfully guided 12 Ph.D. students. His research interests include commutative algebra and number theory.
David Anderson and His Mathematics
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On $$\star $$-Semi-homogeneous Integral Domains
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t-Local Domains and Valuation Domains
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Strongly Divided Pairs of Integral Domains
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Finite Intersections of Prüfer Overrings
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Strongly Additively Regular Rings and Graphs
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On t-Reduction and t-Integral Closure of Ideals in Integral Domains
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Local Types of Classical Rings
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How Do Elements Really Factor in $$\mathbb {Z}[\sqrt{-5}]$$?
Chapman, Scott T. (et al.)
David Anderson's Work on Graded Integral Domains
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Divisor Graphs of a Commutative Ring
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Commutative Rings and Algebras | CommonCrawl |
Dynamical analysis of a giving up smoking model with time delay
Zizhen Zhang1,
Ruibin Wei1 &
Wanjun Xia1
Advances in Difference Equations volume 2019, Article number: 505 (2019) Cite this article
In this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.
In China and around the world, one of the public health problems that has been recognized in recent years is smoking addiction, which has developed into an epidemic causing many deaths. Taking China for example, the data from the Global Tobacco Epidemic Report published on 26 July 2019 by the World Health Organization shows that smoking-related diseases kill one million people in China every year and 100,000 non-smokers die from exposure to second-hand smoke [1]. From the global perspective, according to the survey, smoking kills about six million persons each year, and ten million persons will pass away every year because of smoking-related diseases by 2030 [2,3,4]. Consequently, it is very essential to help people quit smoking and reduce tobacco use and related deaths.
In order to reduce the future effects of smoking on the health of people, the World Health Organization has suggested a set of control policy measures since 2008, known as Framework Convention on Tobacco Control (FCTC). As stated in the Global Tobacco Epidemic Report (2019), about five billion people all over the world have been covered by at least one comprehensive tobacco control measure, although there are still 59 countries in which none of the tobacco control measures has reached the highest level of implementation [1]. On the other hand, the mathematicians have been also in effort to inform people about control of smoking by using mathematical models considering that smoking can spread through social contact since the pioneering work of Castullo-Garsow et al. in [5]. In [5], Castullo-Garsow et al. formulated a giving up smoking model including three population classes: the potential smokers (P), the smokers (S), and the quit smokers (Q). Then Sharomi and Gumel [6] developed a model taking into account the temporarily quit smokers (\(Q_{t}\)) and the permanently quit smokers (\(Q_{p}\)) in the model formulated by Castullo-Garsow et al. [5]. Afterwards, some scholars [4, 7,8,9,10,11,12,13] proposed different forms of giving up smoking models including the occasional smoker class. Rahman et al. [14] proposed a giving up smoking model with the continuous age-structure in the chain smokers and studied local and global stability of the model, and the optimal control strategy on potential smokers is also presented. Fei and Liu [15] presented a giving up smoking model with birth and death rates on complex heterogeneous networks. They examined the stability and attractivity of the proposed model. For the analytical study of stochastic giving up smoking models or some other giving up smoking models, we can refer to [16,17,18,19,20].
As stated in [12], smoking contributes to a number of human diseases such as lung cancer, heart disease, alimentary canal effect, and so on. Thus, it is reasonable to consider the smokers associated with some illness compartment in giving up smoking model. Based on this point, the following smoking model has been proposed by Din et al. [21]:
$$ \textstyle\begin{cases} \frac{dP(t)}{dt}=\alpha-\beta\sqrt{P(t)S(t)}-\gamma P(t), \\ \frac{dS(t)}{dt}=\beta\sqrt{P(t)S(t)}-(\gamma+\delta+\varepsilon )S(t)+\zeta X(t), \\ \frac{dX(t)}{dt}=\delta(1-\eta)S(t)-(\gamma+\zeta)X(t), \\ \frac{dY(t)}{dt}=\delta\eta S(t)-\gamma Y(t), \\ \frac{dZ(t)}{dt}=\varepsilon S(t)-(\gamma+\vartheta)Z(t), \end{cases} $$
where \(P(t)\), \(S(t)\), \(X(t)\), \(Y(t)\), and \(Z(t)\) denote the numbers of the potential smokers, smokers, temporarily quit smokers, permanently quit smokers, and smokers associated with some illness at time t, respectively. α is the recruitment rate of the potential smoker; β is the transmission coefficient; γ is the natural death rate; \(\delta(1-\eta)\) is the temporarily quit rate of smoking; δη is the permanently quit rate of smoking; ε is the developing rate of the smokers associated with some illness; ζ is the relapse rate from the temporarily quit smokers to the smokers; ϑ is the death rate related to smoking illness. Din et al. [21] investigated stability of system (1).
In fact, there is usually a fixed duration of temporary immunity due to self-control, after which the temporarily quit smokers return to the class of smokers. That is, the temporarily quit smokers begin to quit smoking at \(t-\tau\) and they start smoking again at t. On the other hand, it is worthy to notice that delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause the population to fluctuate [22,23,24]. Yuan et al. demonstrated that time delay can affect stability of a dynamical system and cause nonlinear phenomena such as Hopf bifurcation and periodic solutions [25, 26]. For some other works about dynamical systems, one can refer to [27,28,29,30]. Therefore, it is very crucial to examine the effect of the time delay τ on system (1). To this end, we incorporate the time delay due to the immunity period, after which the temporarily quit smokers return to the class of smokers, and investigate the following delayed system:
$$ \textstyle\begin{cases} \frac{dP(t)}{dt}=\alpha-\beta\sqrt{P(t)S(t)}-\gamma P(t), \\ \frac{dS(t)}{dt}=\beta\sqrt{P(t)S(t)}-(\gamma+\delta+\varepsilon )S(t)+\zeta X(t-\tau), \\ \frac{dX(t)}{dt}=\delta(1-\eta)S(t)-\gamma X(t)-\zeta X(t-\tau), \\ \frac{dY(t)}{dt}=\delta\eta S(t)-\gamma Y(t), \\ \frac{dZ(t)}{dt}=\varepsilon S(t)-(\gamma+\vartheta)Z(t), \end{cases} $$
where τ is the length of immunity period after which the temporarily quit smokers return to the class of smokers. The flow diagram of system (2) is shown as in Fig. 1.
The flow diagram of system (2)
The outline of this article is arranged as follows. In Sect. 2, local stability and existence of Hopf bifurcation are discussed in detail. In Sect. 3, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are determined. In order to validate the theoretical analysis and the effect of some crucial parameters on behaviors of the model, some numerical simulations are carried out in Sect. 4. Finally, conclusions are offered in Sect. 5.
Local stability and existence of Hopf bifurcation
In view of [21], we can conclude that system (2) has a unique positive equilibrium \(E^{*}(P^{*}, S^{*}, X^{*}, Y^{*}, Z^{*})\), where
$$ \begin{gathered} P^{*}=\frac{\alpha(\gamma^{2}+\gamma(\delta+\zeta+\varepsilon )+\zeta(\delta\eta+\varepsilon))}{\beta^{2}(\gamma+\zeta)+\gamma (\gamma^{2}+\gamma(\delta+\zeta+\varepsilon)+\zeta(\delta\eta +\varepsilon))}, \\ S^{*}=\frac{\alpha\beta^{2}(\gamma+\zeta)^{2}}{(\gamma^{2}+\gamma (\delta+\zeta+\varepsilon)+\zeta(\delta\eta+\varepsilon))(\beta ^{2}(\gamma+\zeta)+\gamma(\gamma^{2}+\gamma(\delta+\zeta+\varepsilon )+\zeta(\delta\eta+\varepsilon)))}, \\ X^{*}=\frac{\alpha\beta^{2}\delta(1-\eta)(\gamma+\zeta)}{(\gamma ^{2}+\gamma(\delta+\zeta+\varepsilon)+\zeta(\delta\eta+\varepsilon ))(\beta^{2}(\gamma+\zeta)+\gamma(\gamma^{2}+\gamma(\delta+\zeta +\varepsilon)+\zeta(\delta\eta+\varepsilon)))}, \\ Y^{*}=\frac{\alpha\beta^{2}\delta\eta(\gamma+\zeta)^{2}}{\gamma (\gamma^{2}+\gamma(\delta+\zeta+\varepsilon)+\zeta(\delta\eta +\varepsilon))(\beta^{2}(\gamma+\zeta)+\gamma(\gamma^{2}+\gamma (\delta+\zeta+\varepsilon)+\zeta(\delta\eta+\varepsilon )))}, \\ Z^{*}=\frac{\alpha\beta^{2}\varepsilon(\gamma+\zeta)^{2}}{(\gamma +\vartheta)(\gamma^{2}+\gamma(\delta+\zeta+\varepsilon)+\zeta (\delta\eta+\varepsilon))(\beta^{2}(\gamma+\zeta)+\gamma(\gamma ^{2}+\gamma(\delta+\zeta+\varepsilon)+\zeta(\delta\eta+\varepsilon )))}.\end{gathered} $$
The linear system of system (2) at \(E^{*}\) is
$$ \textstyle\begin{cases} \frac{dP(t)}{dt}=g_{11}P(t)+g_{12}S(t), \\ \frac{dS(t)}{dt}=g_{21}P(t)+g_{22}S(t)+h_{23}X(t-\tau), \\ \frac{dX(t)}{dt}=g_{32}S(t)+g_{33}X(t)+h_{33}X(t-\tau), \\ \frac{dY(t)}{dt}=g_{42}S(t)+g_{44}Y(t), \\ \frac{dZ(t)}{dt}=g_{52}S(t)+g_{55}Z(t), \end{cases} $$
$$\begin{aligned}& g_{11}=-\frac{\beta\sqrt{S^{*}}}{2\sqrt{P^{*}}}-\gamma,\qquad g_{12}=- \frac{\beta\sqrt{P^{*}}}{2\sqrt{S^{*}}}, \\& g_{21}=\frac{\beta\sqrt{S^{*}}}{2\sqrt{P^{*}}},\qquad g_{22}=\frac{\beta \sqrt{P^{*}}}{2\sqrt{S^{*}}}-( \gamma+\delta+\varepsilon),\qquad h_{23}=\zeta, \\& g_{32}=\delta(1-\eta),\qquad g_{33}=-\gamma,\qquad h_{33}=- \zeta, \\& g_{42}=\delta\eta, g_{44}=-\gamma,\qquad g_{52}= \varepsilon,\qquad g_{55}=-(\gamma+\vartheta). \end{aligned}$$
The characteristic equation of system (3) is given by
$$ \text{det} \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \lambda-g_{11} & -g_{12} & {0} & {0} & {0} \\ -g_{21} & \lambda-g_{22} & -h_{23}e^{-\lambda\tau} & {0} & {0} \\ {0} & -g_{32} & \lambda-g_{33}-h_{33}e^{-\lambda\tau} & {0} & {0} \\ {0} & -g_{42} & {0} & \lambda-g_{44} & {0} \\ {0} & -g_{52} & {0} & {0} & \lambda-g_{55} \end{array}\displaystyle \right ]=0, $$
which leads to
$$ \lambda^{5}+G_{4}\lambda^{4}+G_{3} \lambda^{3}+G_{2}\lambda^{2}+G_{1} \lambda+G_{0}+\bigl(H_{4}\lambda^{4}+H_{3} \lambda^{3}+H_{2}\lambda^{2}+H_{1} \lambda+H_{0}\bigr)e^{-\lambda\tau}=0, $$
$$\begin{aligned}& G_{0}=g_{33}g_{44}g_{55}(g_{12}g_{21}-g_{11}g_{22}), \\& \begin{aligned}G_{1}={}&(g_{33}g_{44}+g_{33}g_{55}+g_{44}g_{55}) (g_{11}g_{22}-g_{12}g_{21}) \\ &+g_{33}g_{44}g_{55}(g_{11}+g_{22}),\end{aligned} \\& \begin{aligned}G_{2}={}&(g_{33}+g_{44}+g_{55}) (g_{12}g_{21}-g_{11}g_{22}) \\ &-(g_{11}+g_{22}) (g_{33}g_{44}+g_{33}g_{55}+g_{44}g_{55}), \end{aligned} \\& \begin{aligned}G_{3}={}&(g_{11}+g_{22}) (g_{33}+g_{44}+g_{55})+g_{33}g_{44}+g_{33}g_{55} \\ &+g_{44}g_{55}-g_{12}g_{21},\end{aligned} \\& G_{4}=-(g_{11}+g_{22}+g_{33}+g_{44}+g_{55}), \\& H_{0}=g_{44}g_{55}(g_{11}g_{32}h_{23}-g_{11}g_{22}h_{33}+g_{12}g_{21}h_{33}), \\& \begin{aligned}H_{1}={}&g_{55}h_{33}(g_{11}g_{22}+g_{11}g_{44}+g_{22}g_{44})+g_{11}g_{22}g_{44}h_{33} \\ &-g_{32}h_{23}(g_{11}g_{44}+g_{11}g_{55}+g_{44}g_{55})-g_{12}g_{21}h_{33}(g_{44}+g_{55}), \end{aligned} \\& \begin{aligned}H_{2}={}&g_{32}h_{23}(g_{11}+g_{44}+g_{55})-g_{55}h_{33}(g_{11}+g_{22}+g_{44}) \\ &+g_{12}g_{21}h_{33}-h_{33}(g_{11}g_{22}+g_{11}g_{44}+g_{22}g_{44}), \end{aligned} \\& H_{3}=h_{33}(g_{11}+g_{22}+g_{44}+g_{55})-g_{32}h_{23},\qquad G_{44}=-h_{33}. \end{aligned}$$
When \(\tau=0\), Eq. (5) becomes
$$ \lambda^{5}+G_{04}\lambda^{4}+G_{03} \lambda^{3}+G_{02}\lambda^{2}+G_{01} \lambda+G_{00}=0, $$
$$\begin{aligned}& G_{00}=G_{0}+H_{0},\qquad G_{01}=G_{1}+H_{1},\qquad G_{02}=G_{2}+H_{2}, \\& G_{03}=G_{3}+H_{3},\qquad G_{04}=G_{4}+H_{0}. \end{aligned}$$
Based on the discussion in [21], it can be concluded that all the roots of Eq. (6) have negative real parts. Thus, according to the Hurwitz criterion, we have the following result.
Lemma 1
([21])
The unique positive equilibrium \(E^{*}(P^{*}, S^{*}, X^{*}, Y^{*}, Z^{*})\)of system (2) is locally asymptotically stable when \(\tau=0\).
For \(\tau>0\), let \(\lambda=i\omega\) (\(\omega>0\)) be a root of Eq. (5). Then
$$ \textstyle\begin{cases} (H_{1}\omega-H_{3}\omega^{3})\sin\tau\omega+(H_{4}\omega^{4}-H_{2}\omega ^{2}+H_{0})\cos\tau\omega=G_{2}\omega^{2}-G_{4}\omega^{4}-G_{0}, \\ (H_{1}\omega-H_{3}\omega^{3})\cos\tau\omega-(H_{4}\omega^{4}-H_{2}\omega ^{2}+H_{0})\sin\tau\omega=G_{3}\omega^{3}-\omega^{5}-G_{1}\omega. \end{cases} $$
It follows from Eq. (8) that
$$ \omega^{10}+J_{4}\omega^{8}+J_{3} \omega^{6}+J_{2}\omega^{4}+J_{1} \omega^{2}+J_{0}=0, $$
$$\begin{aligned}& J_{0}=G_{0}^{2}-H_{0}^{2}, \\& J_{1}=2G_{0}G_{2}-H_{1}^{2}+2H_{0}H_{2}, \\& J_{2}=G_{2}^{2}-2G_{0}G_{4}-2G_{1}G_{3}+2H_{1}H_{3}-H_{2}^{2}-2H_{0}H_{4}, \\& J_{3}=2G_{1}-2G_{2}G_{4}-H_{3}^{2}+2H_{2}H_{4}, \\& J_{4}=G_{4}^{2}-H_{4}^{2}. \end{aligned}$$
Let \(\omega^{2}=\nu\), Eq. (8) becomes
$$ \nu^{5}+J_{4}\nu^{4}+J_{3} \nu^{3}+J_{2}\nu^{2}+J_{1} \nu+J_{0}=0. $$
In order to establish the main results of this paper, we make the following necessary assumption:
\((S_{1})\):
Eq. (9) has at least one positive root.
\(f^{\prime}(\nu_{0})\neq0\), where \(f(\nu)=\nu^{5}+J_{4}\nu ^{4}+J_{3}\nu^{3}+J_{2}\nu^{2}+J_{1}\nu+J_{0}\).
It follows from \((S_{1})\) that Eq. (9) has at least one positive root, and without loss of generality we assume that Eq. (9) has five positive roots denoted by \(\nu_{1}\), \(\nu_{2}\), \(\nu_{3}\), \(\nu_{4}\), and \(\nu_{5}\). Thus, \(\omega_{l}=\sqrt{\nu_{l}}\), (\(l=1, 2, 3, 4, 5\)) is the roots of Eq. (8). Based on Eq. (7), one can obtain
$$ \tau_{l}^{j}=\frac{1}{\omega_{l}}\times \arccos\biggl\{ \frac{H_{3}\omega_{l}^{8}-(G_{3}H_{3}+H_{1})\omega _{l}^{6}+(G_{1}H_{3}+G_{3}H_{1})\omega_{l}^{4}-G_{1}H_{1}\omega_{l}^{2}}{(H_{1}\omega _{l}-H_{3}\omega_{l}^{3})^{2}+(H_{4}\omega_{l}^{4}-H_{2}\omega_{l}^{2}+H_{0})^{2}}+2n\pi \biggr\} $$
with \(l=1, 2, 3, 4, 5\); \(n=0, 1, 2, \dots\). Denote
$$\tau_{0}=\tau_{{j}_{0}}^{0}=\min\bigl\{ \tau_{l}^{0}|l=1, 2, 3, 4, 5\bigr\} , \qquad\omega_{0}= \omega|_{\tau=\tau_{0}}. $$
Let \(\lambda(\tau)=\tilde{\alpha}(\tau)+i\tilde{\beta}(\tau)\)be the root of Eq. (5) at \(\tau=\tau_{0}\)satisfying \(\tilde{\alpha}(\tau_{0})=0\), \(\tilde{\beta}(\tau_{0})=\omega_{0}\), then \(\operatorname{Re}[d\lambda/d\tau]_{\tau=\tau_{0}}\neq0\).
Differentiating Eq. (5) with respect to τ leads to
$$ \begin{aligned}[b]\biggl[\frac{d\lambda}{d\tau} \biggr]^{-1}={}&{-} \frac{5\lambda^{4}+4G_{4}\lambda^{3}+3G_{3}\lambda^{2}+2G_{2}\lambda +G_{1}}{\lambda(\lambda^{5}+G_{4}\lambda^{4}+G_{3}\lambda^{3}+G_{2}\lambda ^{2}+G_{1}\lambda+G_{0})} \\ &+\frac{4H_{4}\lambda^{3}+3H_{3}\lambda^{2}+2H_{2}\lambda+H_{1}}{\lambda (H_{4}\lambda^{4}+H_{3}\lambda^{3}+H_{2}\lambda^{2}+H_{1}\lambda+H_{0})}-\frac {\tau}{\lambda}.\end{aligned} $$
$$ \operatorname{Re} \biggl[\frac{d\lambda}{d\tau} \biggr]^{-1}_{\tau=\tau_{0}}= \frac{f^{\prime}(\nu_{0})}{(H_{1}\omega_{0}-H_{3}\omega_{0}^{3})^{2}+(H_{4}\omega _{0}^{4}-H_{2}\omega_{0}^{2}+H_{0})^{2}}. $$
It follows from \((S_{2})\) that \(\operatorname{Re}[d\lambda/d\tau ]_{\tau=\tau_{0}}\neq0\). This ends the proof of Lemma 2. Based on the discussion above and Lemmas 1 and 2, one has the following result. □
Theorem 1
For system (2), if \((S_{1})\)–\((S_{2})\)hold, then \(E^{*}(P^{*}, S^{*}, X^{*}, Y^{*}, Z^{*})\)is locally asymptotically stable when \(\tau\in [0, \tau_{0})\); system (2) undergoes a Hopf bifurcation at \(E^{*}(P^{*}, S^{*}, X^{*}, Y^{*}, Z^{*})\)when \(\tau=\tau_{0}\)and a family of periodic solutions bifurcate from \(E^{*}(P^{*}, S^{*}, X^{*}, Y^{*}, Z^{*})\). \(\tau_{0}\)is defined as in Eq. (10).
Direction and stability of Hopf bifurcation
In this section, we investigate the direction and stability of Hopf bifurcation. By Hassard et al. [31], we have the following theorem for system (2).
The Hopf bifurcation exhibited by system (2) can be determined by the parameters \(\mu_{2}\), \(\beta_{2}\), and \(T_{2}\). (i) If \(\mu_{2}>0\) (\(\mu_{2}<0\)), then the Hopf bifurcation is supercritical (subcritical); (ii) if \(\beta_{2}<0\) (\(\beta_{2}>0\)), then the bifurcating periodic solutions are stable (unstable); (iii) if \(T_{2}>0\) (\(T_{2}<0\)), then the period of the bifurcating periodic solutions increases (decrease).
The parameters \(\mu_{2}\), \(\beta_{2}\), and \(T_{2}\) can be found using the following formulas:
$$ \begin{gathered} C_{1}(0)=\frac{i}{2\tau_{0}\omega_{0}} \biggl(v_{11}v_{20}-2 \vert v_{11} \vert ^{2}-\frac{ \vert v_{02} \vert ^{2}}{3} \biggr)+\frac{v_{21}}{2}, \\ \mu_{2} =-\frac{\operatorname{Re}\{C_{1}(0)\}}{\operatorname{Re}\{ \lambda^{\prime}(\tau_{0})\}}, \\ \beta_{2}=2{\operatorname{Re}\bigl\{ C_{1}(0)\bigr\} }, \\ T_{2}=-\frac{\operatorname{Im}\{C_{1}(0)\}+\mu_{2}\operatorname{Im}\{ \lambda^{\prime}(\tau_{0})\}}{\tau_{0}\omega_{0}}, \end{gathered} $$
in which the expressions of \(v_{20}\), \(v_{11}\), \(v_{02}\), and \(v_{21}\) can be found in the following.
Proof of Theorem 2
Introduce a new perturbation parameter \(\tau=\tau_{0}+\mu\) with \(\mu \in R\), then \(\mu=0\) is the Hopf bifurcation value of system (2). Let \(u_{1}(t)=P(t)-P^{*}\), \(u_{2}(t)=S(t)-S^{*}\), \(u_{3}(t)=X(t)-X^{*}\), \(u_{4}(t)=Y(t)-Y^{*}\), \(u_{5}(t)=Z(t)-Z^{*}\), and \(u_{i}(t)=u_{i}(\tau t)\), \(i=1,2,\ldots, 5\). Then system (2) can be written as a functional differential equation in \(C=C([-1,0],R^{5})\) as follows:
$$ \dot{u}(t)=L_{\mu}(u_{t})+F(\mu, u_{t}), $$
where \(L_{\mu}: C\rightarrow R^{5}\), \(F: R\times C\rightarrow R^{5}\), and
$$\begin{aligned}& L_{\mu}\phi=(\tau_{0}+\mu) \bigl(G_{\text{max}}\phi(0)+H_{\text{max}}\phi(-1) \bigr), \end{aligned}$$
$$\begin{aligned}& F(\mu,\phi)=(\tau_{0}+\mu) (F_{1}, F_{2}, 0, 0, 0)^{T}, \end{aligned}$$
$$G_{\text{max}}=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} g_{11} &g_{12} &{0} &{0} &{0}\\ g_{21} &g_{22} &{0} &{0} &{0}\\ {0} &g_{32} &g_{33} &{0} &{0}\\ {0} &g_{42} &{0} &g_{44} &{0}\\ {0} &g_{52} &{0} &{0} &g_{55} \end{array}\displaystyle \right ),\qquad H_{\text{max}}=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} {0} &{0} &{0} &{0} &{0}\\ {0} &{0} &h_{23} &{0} &{0}\\ {0} &{0} &h_{33} &{0} &{0}\\ {0} &{0} &{0} &{0} &{0}\\ {0} &{0} &{0} &{0} &{0} \end{array}\displaystyle \right ), $$
$$\begin{aligned}& \begin{aligned} F_{1}={}&g_{13}\phi_{1}^{2}(0)+g_{14} \phi_{1}(0)\phi_{2}(0)+g_{15}\phi_{2}^{2}(0)+g_{16} \phi_{1}^{3}(0)+g_{17}\phi_{1}^{2}(0) \phi_{2}(0) \\ &+g_{18}\phi_{1}(0)\phi_{2}^{2}(0)+g_{19} \phi_{2}^{3}(0)+\cdots,\end{aligned} \\& \begin{aligned}F_{2}={}&g_{23}\phi_{1}^{2}(0)+g_{24} \phi_{1}(0)\phi_{2}(0)+g_{25}\phi_{2}^{2}(0)+g_{26} \phi_{1}^{3}(0)+g_{27}\phi_{1}^{2}(0) \phi_{2}(0) \\ &+g_{28}\phi_{1}(0)\phi_{2}^{2}(0)+g_{29} \phi_{2}^{3}(0)+\cdots,\end{aligned} \end{aligned}$$
$$\begin{aligned}& g_{13}=\frac{\beta\sqrt{S^{*}}}{8P^{*}\sqrt{P^{*}}},\qquad g_{14}=-\frac {\beta}{2\sqrt{P^{*}S^{*}}},\qquad g_{15}=\frac{\beta\sqrt{P^{*}}}{8S^{*}\sqrt{S^{*}}},\qquad g_{16}=-\frac{\beta \sqrt{S^{*}}}{16(P^{*})^{2}\sqrt{P^{*}}}, \\& g_{17}=\frac{\beta}{16P^{*}\sqrt{P^{*}S^{*}}},\qquad g_{18}=\frac{\beta }{16S^{*}\sqrt{P^{*}S^{*}}},\qquad g_{19}=-\frac{\beta\sqrt{P^{*}}}{16(S^{*})^{2}\sqrt{S^{*}}}, \\& g_{23}=-\frac{\beta\sqrt{S^{*}}}{8P^{*}\sqrt{P^{*}}},\qquad g_{24}=\frac {\beta}{2\sqrt{P^{*}S^{*}}},\qquad g_{25}=-\frac{\beta\sqrt{P^{*}}}{8S^{*}\sqrt{S^{*}}},\qquad g_{26}=\frac{\beta \sqrt{S^{*}}}{16(P^{*})^{2}\sqrt{P^{*}}}, \\& g_{27}=-\frac{\beta}{16P^{*}\sqrt{P^{*}S^{*}}},\qquad g_{28}=-\frac{\beta }{16S^{*}\sqrt{P^{*}S^{*}}},\qquad g_{29}=\frac{\beta\sqrt{P^{*}}}{16(S^{*})^{2}\sqrt{S^{*}}}. \end{aligned}$$
By using the Riesz representation theorem, let \(\eta(\theta, \mu ):[-1,0]\rightarrow R^{5\times5}\) be a function of bounded variation.
For \(\phi\in C([-1,0], R^{5})\), let
$$ L_{\mu}\phi= \int_{-1}^{0}d\eta(\theta, \mu)\phi(\theta). $$
Moreover, we can choose
$$\eta(\theta, \mu)= \textstyle\begin{cases} (\tau_{0}+\mu)G_{\text{max}},& \theta=0,\\ 0, &\theta\in(-1,0),\\ (\tau_{0}+\mu)H_{\text{max}}, &\theta=-1. \end{cases} $$
$$A(\mu)\phi= \textstyle\begin{cases} \frac{d\phi(\theta)}{d\theta},& -1\leq\theta< 0, \\ \int_{-1}^{0}d\eta(\theta,\mu)\phi(\theta),& \theta=0, \end{cases} $$
$$R(\mu)\phi= \textstyle\begin{cases} 0,& -1\leq\theta< 0, \\ F(\mu,\phi), &\theta=0. \end{cases} $$
Then system (14) can be written as follows:
$$ \dot{u}(t)=A(\mu)u_{t}+R(\mu)u_{t}. $$
For \(\varphi\in C^{1}([0,1],(R^{5})^{*})\), define the adjoint operator of \(A(0)\)
$$A^{*}(\varphi)= \textstyle\begin{cases} -\frac{d\varphi(s)}{ds},& 0< s\leq1, \\ \int_{-1}^{0}d{\eta}^{T}(s,0)\varphi(-s),& s=0, \end{cases} $$
and a bilinear product
$$ \bigl\langle \varphi(s),\phi(\theta)\bigr\rangle =\bar{\varphi}(0) \phi(0)- \int_{\theta=-1}^{0} \int_{\xi=0}^{\theta}\bar{\varphi}(\xi-\theta)\,d\eta(\theta) \phi(\xi)\,d\xi, $$
where \(\eta(\theta)=\eta(\theta, 0)\).
According to the analysis in Sect. 2, \(\pm i\tau_{0}\omega_{0}\) are eigenvalues of \(A(0)\), so they are also eigenvalues of \(A^{*}\). Then \(A(0)q(\theta)=i\tau _{0}\omega_{0}q(\theta)\) and \(A^{*}q^{*}(s)=-i\tau_{0}\omega_{0}q^{*}(s)\). Suppose that \(q(\theta)=(1,q_{2},q_{3},q_{4},q_{5})^{T}e^{i\tau_{0}\omega _{0}\theta}\) and \(q^{*}(s)=D(1,q_{2}^{*},q_{3}^{*},q_{4}^{*},q_{5}^{*})e^{i\tau_{0}\omega _{0}s}\) are the corresponding eigenvectors. By calculation we can obtain
$$\begin{gathered} q_{2}=\frac{g_{21}(i\omega_{0}-g_{33}-h_{33}e^{-i\tau_{0}\omega _{0}})}{(i\omega_{0}-g_{22})(i\omega_{0}-g_{33}-h_{33}e^{-i\tau_{0}\omega _{0}})-g_{32}h_{23}e^{-i\tau_{0}\omega_{0}}}, \\ q_{3}=\frac{g_{32}q_{2}}{i\omega_{0}-g_{33}-h_{33}e^{-i\tau_{0}\omega _{0}}},\qquad q_{4}=\frac{g_{42}q_{2}}{i\omega_{0}-g_{44}},\qquad q_{5}=\frac{g_{52}q_{2}}{i\omega_{0}-g_{55}}, \\ q_{2}^{*}=-\frac{i\omega_{0}+g_{11}}{g_{21}},\qquad q_{3}^{*}=\frac{h_{23}e^{i\tau _{0}\omega_{0}}(i\omega_{0}+g_{11})}{g_{21}(i\omega _{0}+g_{33}+h_{33}e^{i\tau_{0}\omega_{0}})},\qquad q_{4}^{*}=0,\qquad q_{5}^{*}=0.\end{gathered} $$
From \(\langle q^{*}(s),q(\theta)\rangle=1\), we have
$$\bar{D}=\Biggl[1+\sum_{i=1}^{5}\bar{q}_{i}^{*}q_{i}+ \tau_{0}e^{-i\tau_{0}\omega_{0}}q_{3}\bigl(h_{23} \bar{q}_{2}^{*}+h_{33}\bar{q}_{3}^{*}\bigr) \Biggr]^{-1}. $$
In the following, according to the algorithm given in [31] and the computation process as that in [24, 32,33,34], we can obtain
$$\begin{aligned}& v_{20}=2\tau_{0}\bar{D}\bigl[g_{13}+g_{14}q_{2}+g_{15}q_{2}^{2}+ \bar{q}_{2}^{*}\bigl(g_{23}+g_{24}q_{2}+g_{25}q_{2}^{2} \bigr)\bigr], \\& v_{11}=\tau_{0}\bar{D}\bigl[2g_{13}+g_{14}(q_{2}+ \bar{q}_{2})+2g_{15}q_{2}\bar{q}_{2}+ \bar{q}_{2}^{*}\bigl(2g_{23}+g_{24}(q_{2}+ \bar{q}_{2})+2g_{25}q_{2}\bar{q}_{2}\bigr) \bigr], \\& v_{02}=2\tau_{0}\bar{D}\bigl[g_{13}+g_{14} \bar{q}_{2}+g_{15}\bar{q}_{2}^{2}+ \bar{q}_{2}^{*}\bigl(g_{23}+g_{24} \bar{q}_{2}+g_{25}\bar{q}_{2}^{2}\bigr) \bigr], \\& \begin{aligned}v_{21}={}&2\tau_{0}\bar{D} \biggl[g_{13} \bigl(2W_{11}^{(0)}+W_{20}^{(1)}(0)\bigr) \\ &+g_{14} \biggl(W_{11}^{(1)}(0)q_{2}+ \frac{1}{2}W_{20}^{(1)}(0)\bar{q}_{2}+W_{11}^{(2)}(0)+ \frac{1}{2}W_{20}^{(2)}(0) \biggr) \\ &+g_{15}\bigl(2W_{11}^{(2)}(0)q_{2}+W_{20}^{(2)}(0) \bar{q}_{2}\bigr) \\ &+3g_{16}+g_{17}(\bar{q}_{2}+2q_{2})+g_{18} \bigl(q_{2}^{2}+2q_{2}\bar{q}_{2} \bigr)+3g_{19}q_{2}^{2}\bar{q}_{2} \\ &+\bar{q}_{2}^{*} \biggl(g_{23}\bigl(2W_{11}^{(0)}+W_{20}^{(1)}(0) \bigr) \\ &+g_{24} \biggl(W_{11}^{(1)}(0)q_{2}+ \frac{1}{2}W_{20}^{(1)}(0)\bar{q}_{2}+W_{11}^{(2)}(0)+ \frac{1}{2}W_{20}^{(2)}(0) \biggr) \\ &+g_{25}\bigl(2W_{11}^{(2)}(0)q_{2}+W_{20}^{(2)}(0) \bar{q}_{2}\bigr) \\ &+3g_{16}+g_{17}(\bar{q}_{2}+2q_{2})+g_{18} \bigl(q_{2}^{2}+2q_{2}\bar{q}_{2} \bigr)+3g_{19}q_{2}^{2}\bar{q}_{2} \biggr) \biggr]\end{aligned} \end{aligned}$$
$$\begin{aligned}& W_{20}(\theta)=\frac{iv_{20}q(0)}{\tau_{0}\omega_{0}}e^{i\tau _{0}\omega_{0}\theta}+ \frac{i\bar{v}_{02}\bar{q}(0)}{3\tau_{0}\omega_{0}}e^{-i\tau_{0}\omega _{0}\theta}+E_{1}e^{2i\tau_{0}\omega_{0}\theta}, \\& W_{11}(\theta)=-\frac{iv_{11}q(0)}{\tau_{0}\omega_{0}}e^{i\tau _{0}\omega_{0}\theta}+ \frac{i\bar{v}_{11}\bar{q}(0)}{\tau_{0}\omega_{0}}e^{-i\tau_{0}\omega _{0}\theta}+E_{2}, \end{aligned}$$
$$\begin{aligned}& E_{1}=2\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} g_{11}^{*} &-g_{12} &{0} &{0} &{0}\\ -g_{21}e^{-2i\tau_{0}\omega_{0}} &g_{22}^{*} &-g_{23}e^{-2i\tau_{0}\omega _{0}} &{0} &{0}\\ {0} &-g_{32} &g_{33}^{*} &{0} &{0}\\ {0} &-g_{42} &{0} &g_{44}^{*} &{0}\\ {0} &-g_{52} &{0} &{0} &g_{55}^{*} \end{array}\displaystyle \right )^{-1}\times \left ( \textstyle\begin{array}{c} g_{13}+g_{14}q_{2}+g_{15}q_{2}^{2}\\ g_{23}+g_{24}q_{2}+g_{25}q_{2}^{2}\\ {0}\\ {0}\\ {0} \end{array}\displaystyle \right ), \\& E_{2}=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} g_{11} &g_{12} &{0} &{0} &{0}\\ g_{21} &g_{22} &h_{23} &{0} &{0}\\ {0} &g_{32} &g_{33}+h_{33} &{0} &{0}\\ {0} &g_{42} &{0} &g_{44} &{0}\\ {0} &g_{52} &{0} &{0} &g_{55} \end{array}\displaystyle \right )^{-1}\times \left ( \textstyle\begin{array}{c} 2g_{13}+g_{14}(q_{2}+\bar{q}_{2})+2g_{15}q_{2}\bar{q}_{2}\\ 2g_{23}+g_{24}(q_{2}+\bar{q}_{2})+2g_{25}q_{2}\bar{q}_{2}\\ {0}\\ {0}\\ {0} \end{array}\displaystyle \right ), \end{aligned}$$
$$\begin{aligned}& g_{11}^{*}=2i\omega_{0}-g_{11},\qquad g_{22}^{*}=2i\omega_{0}-g_{22}, \qquad g_{33}^{*}=2i\omega_{0}-g_{33}-h_{33}e^{-2i\tau_{0}\omega_{0}}, \\& g_{44}^{*}=2i\omega_{0}-g_{44},\qquad g_{55}^{*}=2i\omega_{0}-g_{55}. \end{aligned}$$
Thus, we can conclude that \(v_{20}\), \(v_{11}\), \(v_{02}\), and \(v_{21}\) in Eq. (13) can be obtained. The proof is completed. □
Numerical simulations
In this section, we verify the correctness of the obtained theoretical results by using numerical simulations. Choosing \(\alpha=0.8\), \(\beta=0.005\), \(\gamma=0.0000391\), \(\delta=0.00913\), \(\varepsilon=0.00458\), \(\zeta=0.02\), \(\eta=0.001\), \(\vartheta =0.0457\), we obtain the following specific case of system (2):
$$ \textstyle\begin{cases} \frac{dP(t)}{dt}=0.8-0.005\sqrt{P(t)S(t)}-0.0000391 P(t), \\ \frac{dS(t)}{dt}=0.005\sqrt{P(t)S(t)}-0.0137491 S(t)+0.02 X(t-\tau), \\ \frac{dX(t)}{dt}=0.009121 S(t)-0.0000391 X(t)-0.02 X(t-\tau), \\ \frac{dY(t)}{dt}=9.1300e-006 S(t)-0.0000391 Y(t), \\ \frac{dZ(t)}{dt}=0.00458 S(t)-0.0457391 Z(t). \end{cases} $$
Thus, the unique positive equilibrium is \(E^{*}(147.6003, 170.9480, 77.8076, 39.9170, 17.1176)\). By calculating, we can obtain that \(\nu _{0}=0.00023516\), \(\omega_{0}=0.01533623\), and \(\tau_{0}=118.1368\), \(f^{\prime}(\nu_{0})=0.00052229>0\). Obviously, the parameters in system (21) fulfill assumptions \(S_{1}\) and \(S_{2}\). From Theorem 1, when \(\tau\in(0, \tau_{0})\), \(E^{*}(147.6003, 170.9480, 77.8076, 39.9170,17.1176)\) is locally asymptotically stable, which can be illustrated in Figs. 2–3. While as τ is increased to pass \(\tau_{0}\), we can see the effect of time delay that destabilizes system (21) and a Hopf bifurcation occurs and a periodic oscillation appears around \(E^{*}(147.6003, 170.9480, 77.8076, 39.9170,17.1176)\). This can be shown as in Figs. 4–5.
The equilibrium \(E^{*}\) of system (21) is asymptotically stable for \(\tau=114.685<\tau_{0}\)
The phase plots of system (2) for \(\tau=114.685<\tau_{0}\)
The equilibrium \(E^{*}\) of system (21) is unstable for \(\tau=122.905>\tau_{0}\)
The phase plots of system (2) for \(\tau=122.905>\tau_{0}\)
Now, we are interested in studying the effect of some other parameters on the dynamics of system (21). (i) The number of smokers associated with some illness decreases as the value of β decreases, whereas the value of η increases, which can be demonstrated by Figs. 6–7. (ii) The number of smokers associated with some illness decreases when the value of ζ decreases, which can be depicted by Fig. 8. In addition, it is easy to check in Fig. 9 that system (21) shows the limit cycle behavior from the stable state due to increase in ζ.
Time plot of Z for different β at \({\tau=114.685}\). The rest of the parameters are taken as given in the text
Time plot of Z for different η at \({\tau=114.685}\). The rest of the parameters are taken as given in the text
Time plot of Z for different ζ at \({\tau=114.685}\). The rest of the parameters are taken as given in the text
Dynamic behavior of system (21): projection on S–X–Z for different ζ at \({\tau=122.905}\). The rest of the parameters are taken as given in the text
In the current paper, a delayed smoking model in which the population is divided into five classes is investigated by incorporating the time delay due to the immunity period, after which the temporarily quit smokers return to the class of smokers, into the proposed model by Din et al. [21]. It is found that the delayed smoking model is locally asymptotically stable when the time delay is suitably small under some certain conditions. In this case, it is easy to control smoking. However, once the value of the time delay passes through the critical value \(\tau_{0}\), a Hopf bifurcation occurs and smoking will be out of control. Particularly, properties such as direction and stability of the Hopf bifurcation are examined with the aid of the center manifold theorem and normal form theory.
It has been observed from our simulations that the number of smokers associated with some illness decreases as we decrease the value of β or increase the value of η. Therefore, it can be concluded that we should actively propagandize the harm of smoking, so that more and more people can stay away from tobacco and quit smoking timely and permanently. It has also been shown that the number of smokers associated with some illness decreases when we decrease the value of ζ, and the model changes its behavior from stable focus to limit cycle as we increase the value of ζ. Thus, it is strongly recommended that the smokers who have quitted smoking should have strong will and resolutely prevent relapse, which is also meaningful for controlling tobacco epidemic.
At last, it should be noted that similar to smoking addiction, the other public health problem is excessive drinking, which is not only harmful to personal health, but also leads to a range of negative social effects [35,36,37]. Therefore, we will try to complete some work about drink modeling in the near future.
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Xiang, H., Wang, Y., Huo, H.F.: Analysis of the binge drinking models with demographics and nonlinear infectivity on networks. J. Appl. Anal. Comput. 8, 1535–1554 (2018)
Huo, H.F., Zhang, X.M.: Complex dynamics in an alcoholism model with the impact of Twitter. Math. Biosci. 281, 24–35 (2016)
The authors are very thankful to the anonymous reviewers for their insightful comments and suggestions, which helped us to improve the manuscript considerably and further open doors for future work.
All of the authors declare that all the data can be accessed in our manuscript in the numerical simulation section.
This research was supported by the Project of Support Program for Excellent Youth Talent in Colleges and Universities of Anhui Province (No. gxyqZD2018044) and the Natural Science Foundation of the Higher Education Institutions of Anhui Province (Nos. KJ2019A0655, KJ2019A0656, KJ2019A0662).
School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu, China
Zizhen Zhang, Ruibin Wei & Wanjun Xia
Zizhen Zhang
Ruibin Wei
Wanjun Xia
All authors read and approved the final manuscript.
Correspondence to Zizhen Zhang.
The authors declare that there is no conflict of interests.
Zhang, Z., Wei, R. & Xia, W. Dynamical analysis of a giving up smoking model with time delay. Adv Differ Equ 2019, 505 (2019). https://doi.org/10.1186/s13662-019-2450-4
Hopf bifurcation
Smoking model | CommonCrawl |
Results for 'Maxim V. Vorobiev'
Increasing Η ‐Representable Degrees.Andrey N. Frolov & Maxim V. Zubkov - 2009 - Mathematical Logic Quarterly 55 (6):633-636.details
In this paper we prove that any Δ30 degree has an increasing η -representation. Therefore, there is an increasing η -representable set without a strong η -representation.
Representation in Philosophy of Mind
Inflation Due to Quantum Potential.Maxim V. Eingorn & Vitaliy D. Rusov - 2015 - Foundations of Physics 45 (8):875-882.details
In the framework of a cosmological model of the Universe filled with a nonrelativistic particle soup, we easily reproduce inflation due to the quantum potential. The lightest particles in the soup serve as a driving force of this simple, natural and promising mechanism. It is explicitly demonstrated that the appropriate choice of their mass and fraction leads to reasonable numbers of e-folds. Thus, the direct introduction of the quantum potential into cosmology of the earliest Universe gives ample opportunities of successful (...) reconsideration of the modern inflationary theory. (shrink)
Quantum Mechanics in Philosophy of Physical Science
The Early Universe in Philosophy of Physical Science
N atalia G. S ukhova & E rki T ammiksaar, Aleksandr Fedorovich Middendorf: K dvukhsotletiyu so dnia rozhdeniya [Alexander Theodor von Middendorff: On the Bicentenary of His Birthday], 2nd edition, revised and expanded, St. Petersburg: Nestor-Istoriya, 2015, 380 pp., price 300 roubles [In Russian]. [REVIEW]Maxim V. Vinarski & Tatiana I. Yusupova - 2017 - History and Philosophy of the Life Sciences 40 (1):14.details
Revisiting the Maxim-Law Dynamic in the Light of Kant's Theory of Action.V. K. Radhakrishnan - 2019 - Kantian Journal 38 (2):45-72.details
A stable classification of practical principles into mutually exclusive types is foundational to Kant's moral theory. Yet, other than a few brief hints on the distinction between maxims and laws, he does not provide any elaborate discussion on the classification and the types of practical principles in his works. This has led Onora O'Neill and Lewis Beck to reinterpret Kant's classification of practical principles in a way that would clarify the conceptual connection between maxims and laws. In this paper I (...) argue that the revised interpretations of O'Neill and Beck stem from a mistaken reading of the fundamental basis of the classification of practical principles. To show this, I first argue that Kant distinguishes between maxims and laws on the bases of validity and reality. I then argue that although a practical principle necessarily has the feature of validity, its reality in actually moving the agents to action sufficiently makes a principle a practical principle. If this is so, I argue that the classification of practical principles must be based on the extent to which they are effective in human agents. Such a classification yields us three exhaustive and mutually exclusive types namely, "maxims that are not potential laws", "maxims that are potential laws" and "laws that are not maxims". (shrink)
Kant: Ethics, Misc in 17th/18th Century Philosophy
Kant: Maxims in 17th/18th Century Philosophy
Moral Principles, Misc in Meta-Ethics
Auditory Mismatch Negativity Response in Institutionalized Children.Irina Ovchinnikova, Marina A. Zhukova, Anna Luchina, Maxim V. Petrov, Marina J. Vasilyeva & Elena L. Grigorenko - 2019 - Frontiers in Human Neuroscience 13.details
Maximal Kripke-Type Semantics for Modal and Superintuitionistic Predicate Logics.D. P. Skvortsov & V. B. Shehtman - 1993 - Annals of Pure and Applied Logic 63 (1):69-101.details
Recent studies in semantics of modal and superintuitionistic predicate logics provided many examples of incompleteness, especially for Kripke semantics. So there is a problem: to find an appropriate possible- world semantics which is equivalent to Kripke semantics at the propositional level and which is strong enough to prove general completeness results. The present paper introduces a new semantics of Kripke metaframes' generalizing some earlier notions. The main innovation is in considering "n"-tuples of individuals as abstract "n"-dimensional vectors', together with some (...) transformations of these vectors. Soundness of the semantics is proved to be equivalent to some non- logical properties of metaframes; and thus we describe the maximal semantics of Kripke- type. (shrink)
Modal and Intensional Logic in Logic and Philosophy of Logic
Semantics for Modal Logic in Logic and Philosophy of Logic
Maximizing Students' Retention Via Spaced Review: Practical Guidance From Computational Models of Memory.Mohammad M. Khajah, Robert V. Lindsey & Michael C. Mozer - 2014 - Topics in Cognitive Science 6 (1):157-169.details
During each school semester, students face an onslaught of material to be learned. Students work hard to achieve initial mastery of the material, but when they move on, the newly learned facts, concepts, and skills degrade in memory. Although both students and educators appreciate that review can help stabilize learning, time constraints result in a trade-off between acquiring new knowledge and preserving old knowledge. To use time efficiently, when should review take place? Experimental studies have shown benefits to long-term retention (...) with spaced study, but little practical advice is available to students and educators about the optimal spacing of study. The dearth of advice is due to the challenge of conducting experimental studies of learning in educational settings, especially where material is introduced in blocks over the time frame of a semester. In this study, we turn to two established models of memory—ACT-R and MCM—to conduct simulation studies exploring the impact of study schedule on long-term retention. Based on the premise of a fixed time each week to review, converging evidence from the two models suggests that an optimal review schedule obtains significant benefits over haphazard (suboptimal) review schedules. Furthermore, we identify two scheduling heuristics that obtain near optimal review performance: (a) review the material from μ-weeks back, and (b) review material whose predicted memory strength is closest to a particular threshold. The former has implications for classroom instruction and the latter for the design of digital tutors. (shrink)
The Works of George Berkeley, Bishop of Cloyne.The Works of George Berkeley, Bishop of Cloyne: Vol. IV. De Motu: The Analyst, Defence of Free-Thinking in Mathematics, Reasons for Not Replying to Walton's Full Answer, Arithmetica, Miscellanea Mathematica, Of Infinites, Letters on Vesuvius, on Petrifactions, on Earthquakes, Description of Cave of Dunmore.The Works of George Berkeley, Bishop of Cloyne: Vol. V. Siris, Letters to Thomas Prior and Dr. Hales, Farther Thoughts on Tar-Water, Varia.The Works of George Berkeley, Bishop of Cloyne: Vol. VI. Passive Obedience, Advice to Tories Who Have Taken the Oaths, Essay Towards Preventing the Ruin of Great Britain, The Querist, Letter on a National Bank, The Irish Patriot, Discourse to Magistrates, Letters on the Jacobite Rebellion, A Word to the Wise, Maxims Concerning Patriotism.William T. Parry - 1953 - Philosophy and Phenomenological Research 14 (2):263-263.details
Berkeley: Philosophy of Science in 17th/18th Century Philosophy
Berkeley: Value Theory in 17th/18th Century Philosophy
Changes in Functional Connectivity Within the Fronto-Temporal Brain Network Induced by Regular and Irregular Russian Verb Production.Maxim Kireev, Natalia Slioussar, Alexander D. Korotkov, Tatiana V. Chernigovskaya & Svyatoslav V. Medvedev - 2015 - Frontiers in Human Neuroscience 9.details
Which Scoring Rule Maximizes Condorcet Efficiency Under Iac?Davide P. Cervone, William V. Gehrlein & William S. Zwicker - 2005 - Theory and Decision 58 (2):145-185.details
Consider an election in which each of the n voters casts a vote consisting of a strict preference ranking of the three candidates A, B, and C. In the limit as n→∞, which scoring rule maximizes, under the assumption of Impartial Anonymous Culture (uniform probability distribution over profiles), the probability that the Condorcet candidate wins the election, given that a Condorcet candidate exists? We produce an analytic solution, which is not the Borda Count. Our result agrees with recent numerical results (...) from two independent studies, and contradicts a published result of Van Newenhizen (Economic Theory 2, 69–83. (1992)). (shrink)
Condorcet in 17th/18th Century Philosophy
Maximally Consistent Sets of Instances of Naive Comprehension.Luca Incurvati & Julien Murzi - 2017 - Mind 126 (502).details
Paul Horwich (1990) once suggested restricting the T-Schema to the maximally consistent set of its instances. But Vann McGee (1992) proved that there are multiple incompatible such sets, none of which, given minimal assumptions, is recursively axiomatizable. The analogous view for set theory---that Naïve Comprehension should be restricted according to consistency maxims---has recently been defended by Laurence Goldstein (2006; 2013). It can be traced back to W.V.O. Quine(1951), who held that Naïve Comprehension embodies the only really intuitive conception of set (...) and should be restricted as little as possible. The view might even have been held by Ernst Zermelo (1908), who,according to Penelope Maddy (1988), subscribed to a 'one step back from disaster' rule of thumb: if a natural principle leads to contra-diction, the principle should be weakened just enough to block the contradiction. We prove a generalization of McGee's Theorem, anduse it to show that the situation for set theory is the same as that for truth: there are multiple incompatible sets of instances of Naïve Comprehension, none of which, given minimal assumptions, is recursively axiomatizable. This shows that the view adumbrated by Goldstein, Quine and perhaps Zermelo is untenable. (shrink)
Liar Paradox in Logic and Philosophy of Logic
Russell's Paradox in Philosophy of Mathematics
The Nature of Sets, Misc in Philosophy of Mathematics
Seismic Imaging and Statistical Analysis of Fault Facies Models.Dmitriy R. Kolyukhin, Vadim V. Lisitsa, Maxim I. Protasov, Dongfang Qu, Galina V. Reshetova, Jan Tveranger, Vladimir A. Tcheverda & Dmitry M. Vishnevsky - 2017 - Interpretation: SEG 5 (4):SP71-SP82.details
Interpretation of seismic responses from subsurface fault zones is hampered by the fact that the geologic structure and property distributions of fault zones can generally not be directly observed. This shortcoming curtails the use of seismic data for characterizing internal structure and properties of fault zones, and it has instead promoted the use of interpretation techniques that tend to simplify actual structural complexity by rendering faults as lines and planes rather than volumes of deformed rock. Facilitating the correlation of rock (...) properties and seismic images of fault zones would enable active use of these images for interpreting fault zones, which in turn would improve our ability to assess the impact of fault zones on subsurface fluid flow. We use a combination of 3D fault zone models, based on empirical data and 2D forward seismic modeling to investigate the link between fault zone properties and seismic response. A comparison of spatial statistics from the geologic models and the seismic images was carried out to study how well seismic images render the modeled geologic features. Our results indicate the feasibility of extracting information about fault zone structure from seismic data by the methods used. (shrink)
Maximizing the Predictive Value of Production Rules.Sholom M. Weiss, Robert S. Galen & Prasad V. Tadepalli - 1990 - Artificial Intelligence 45 (1-2):47-71.details
Diffusion Centrality: A Paradigm to Maximize Spread in Social Networks.Chanhyun Kang, Sarit Kraus, Cristian Molinaro, Francesca Spezzano & V. S. Subrahmanian - 2016 - Artificial Intelligence 239:70-96.details
Rotations and Pattern Formation in Granular Materials Under Loading.Elena Pasternak, Arcady V. Dyskin, Maxim Esin, Ghulam M. Hassan & Cara MacNish - 2015 - Philosophical Magazine 95 (28-30):3122-3145.details
Constrained Maximization.Jordan Howard Sobel - 1991 - Canadian Journal of Philosophy 21 (1):25 - 51.details
This paper is about David Gauthier's concept of constrained maximization. Attending to his most detailed and careful account, I try to say how constrained maximization works, and how it might be changed to work better. In section I, that detailed account is quoted along with amplifying passages. Difficulties of interpretation are explained in section II. An articulation, a spelling out, of Gauthier's account is offered in section III to deal with these difficulties. Next, in section IV, constrained maximization thus articulated (...) is tested on several choice problems and shown to be seriously wanting. It appears that there are prisoners' dilemmas in which constrained maximizers would not cooperate to mutual advantage, but would interact sub-optimally just as straight-maximizers would. 'Coordination problems' are described with which constrained maximizers might, especially if transparent to one another, not be able to cope–problems in which they might not be able to make up their minds to do anything at all. And I prove that there are prisoners' dilemmas that, though possible for real agents and for straight maximizers, are not possible for constrained maximizers, so that agents' internalising dispositions of constrained maximization could not be of help in connection with such possibly impending dilemmas. Taking constrained maximization as it stands, there are many problems for which it does not afford the 'moral solutions' with which Gauthier would have it replace Hobbesian political ones. After displaying these shortcomings of constrained maximization as presently designed, I sketch, in section V, possible revisions that would reduce them, stressing that these revisions would not be cost-free. Whether finishing the job of fixing up and making precise constrained maximization would be worth the considerable trouble it would involve lies beyond the issues taken up in this paper. So, of course, do substantive comparisons of constrained maximization, perfected and made precise, and straight maximization. (shrink)
Game Theory in Philosophy of Action
Prisoner's Dilemma in Philosophy of Action
A Simple Maximality Principle.Joel David Hamkins - 2003 - Journal of Symbolic Logic 68 (2):527-550.details
In this paper, following an idea of Christophe Chalons. I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence varphi holding in some forcing extension $V^P$ and all subsequent extensions $V^{P\ast Q}$ holds already in V. It follows, in fact, that such sentences must also hold in all forcing extensions of V. In modal terms, therefore, the Maximality Principle is expressed by the scheme $(\lozenge \square \varphi) \Rightarrow \square \varphi$ , and is equivalent to (...) the modal theory S5. In this article. I prove that the Maximality Principle is relatively consistent with ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in φ, is equiconsistent with the scheme asserting that $V_\delta \prec V$ for an inaccessible cardinal δ, which in turn is equiconsistent with the scheme asserting that ORD is Mahlo. The strongest principle along these lines is $\square MP\!_{\!\!\!\!\!\!_{\!\!_\sim}}$ , which asserts that $MP\!_{\!\!\!\!\!\!_{\!\!_\sim}}$ holds in V and all forcing extensions. From this, it follows that $0^\#$ exists, that $x^\#$ exists for every set x, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain. (shrink)
Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic
A Simple Maximality Principle.Joel Hamkins - 2003 - Journal of Symbolic Logic 68 (2):527-550.details
In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence φ holding in some forcing extension $V\P$ and all subsequent extensions V\P*\Qdot holds already in V. It follows, in fact, that such sentences must also hold in all forcing extensions of V. In modal terms, therefore, the Maximality Principle is expressed by the scheme $\implies\necessaryφ$, and is equivalent to the modal theory S5. In this article, (...) I prove that the Maximality Principle is relatively consistent with \ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in φ, is equiconsistent with the scheme asserting that $Vδ\elesub V$ for an inaccessible cardinal δ, which in turn is equiconsistent with the scheme asserting that $\ORD$ is Mahlo. The strongest principle along these lines is $\necessary\MPtilde$, which asserts that $\MPtilde$ holds in V and all forcing extensions. From this, it follows that 0# exists, that x# exists for every set x, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain. (shrink)
Transformation of Person and Society in the Anthropotechnical Turn: Educational Aspect.V. N. Vashkevich & O. V. Dobrodum - 2018 - Anthropological Measurements of Philosophical Research 13:112-123.details
Introduction. Anthropotechnical turn in culture is based on educational practices that characterize a person as a subject and at the same time as an object of educational and corrective influence. Theoretical basis. We use the method of categorical analysis, which allows revealing the main outlook potentials of anthropotechnical turn as an essential transformation of modern socio-culture. Originality. For the first time, we conducted a categorical analysis of the glossary of anthropotechnical turn as dialectic of active and passive in the personal (...) and social modes such as education. Conclusions. The anthropotechnical turn of modern socio-culture means the actualization of the dialectic of active and passive in the process of socialization and formation of a person in a modern society. The world-view potential of the anthropotechnical turn is producing a new maxim and stratagem of person's behaviour through the formation of a new way of self-identification and self-esteem. The modern educational system, given the theory of anthropotechnical rotation, should change the content of timological energies from obedience to self-actualization and self-improvement. A prerequisite for this task is the change in the motivation of the education sector and the improvement of the social status of the teacher as an intellectual and leader of opinion. The analysis of the specificity of the information society and its determinatory impact on the individual provides grounds for identifying modern culture as a culture of lost opportunities. Thus, the main cause of disorientation and ignorance of a person is not the lack of information, but the lack of motivation. Therefore, the fundamental principles of anthropotechnical turn are productive in solving pressing problems of our time. (shrink)
V = L and Intuitive Plausibility in Set Theory. A Case Study.Tatiana Arrigoni - 2011 - Bulletin of Symbolic Logic 17 (3):337-360.details
What counts as an intuitively plausible set theoretic content (notion, axiom or theorem) has been a matter of much debate in contemporary philosophy of mathematics. In this paper I develop a critical appraisal of the issue. I analyze first R. B. Jensen's positions on the epistemic status of the axiom of constructibility. I then formulate and discuss a view of intuitiveness in set theory that assumes it to hinge basically on mathematical success. At the same time, I present accounts of (...) set theoretic axioms and theorems formulated in non-strictly mathematical terms, e.g., by appealing to the iterative concept of set and/or to overall methodological principles, like unify and maximize, and investigate the relation of the latter to success in mathematics. (shrink)
Axiomatic Truth in Philosophy of Mathematics
The Nature of Sets in Philosophy of Mathematics
Universism and Extensions of V.Carolin Antos, Neil Barton & Sy-David Friedman - forthcoming - Review of Symbolic Logic:1-50.details
A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that (...) seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in `ideal' outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist. (shrink)
New Axioms in Set Theory in Philosophy of Mathematics
Set-Theoretic Constructions in Philosophy of Mathematics
Mohammed Abdellaoui/Editorial Statement 1–2 Mohammed Abdellaoui and Peter P. Wakker/The Likelihood Method for Decision Under Uncertainty 3–76 AAJ Marley and R. Duncan Luce/Independence Properties Vis--Vis Several Utility Representations 77–143. [REVIEW]Davide P. Cervone, William V. Gehrlein, William S. Zwicker, Which Scoring Rule Maximizes Condorcet, Marcello Basili, Alain Chateauneuf & Fulvio Fontini - 2005 - Theory and Decision 58:409-410.details
Scoring Rules in Philosophy of Probability
There Exist Exactly Two Maximal Strictly Relevant Extensions of the Relevant Logic R.Kazimierz Swirydowicz - 1999 - Journal of Symbolic Logic 64 (3):1125-1154.details
In [60] N. Belnap presented an 8-element matrix for the relevant logic R with the following property: if in an implication A → B the formulas A and B do not have a common variable then there exists a valuation v such that v(A → B) does not belong to the set of designated elements of this matrix. A 6-element matrix of this kind can be found in: R. Routley, R.K. Meyer, V. Plumwood and R.T. Brady [82]. Below we prove (...) that the logics generated by these two matrices are the only maximal extensions of the relevant logic R which have the relevance property: if A → B is provable in such a logic then A and B have a common propositional variable. (shrink)
Polish Philosophy in European Philosophy
Relevance Logic in Logic and Philosophy of Logic
A Co-Analytic Maximal Set of Orthogonal Measures.Vera Fischer & Asger Törnquist - 2010 - Journal of Symbolic Logic 75 (4):1403-1414.details
We prove that if V = L then there is a $\Pi _{1}^{1}$ maximal orthogonal (i.e., mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.
Model Theory in Logic and Philosophy of Logic
Theologie patristique grecque (suite). IV. Le V^ e sibcle: Isidore de Peluse, Cyrille d'Alexandrie, Theodoret. V. Du VII^ e au VIII^ e siecle: Maxime le Confesseur, Damascene. VI. La Bible des Peres. VII. Ouvrages generaux. Theologie des Peres. [REVIEW]B. Sesboue - 1998 - Recherches de Science Religieuse 86:221-248.details
Hereditarily Structurally Complete Modal Logics.V. V. Rybakov - 1995 - Journal of Symbolic Logic 60 (1):266-288.details
We consider structural completeness in modal logics. The main result is the necessary and sufficient condition for modal logics over K4 to be hereditarily structurally complete: a modal logic λ is hereditarily structurally complete $\operatorname{iff} \lambda$ is not included in any logic from the list of twenty special tabular logics. Hence there are exactly twenty maximal structurally incomplete modal logics above K4 and they are all tabular.
Explaining Maximality Through the Hyperuniverse Programme.Sy-David Friedman & Claudio Ternullo - 2018 - In Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality. Birkhäuser. pp. 185-204.details
The iterative concept of set is standardly taken to justify ZFC and some of its extensions. In this paper, we show that the maximal iterative concept also lies behind a class of further maximality principles expressing the maximality of the universe of sets V in height and width. These principles have been heavily investigated by the first author and his collaborators within the Hyperuniverse Programme. The programme is based on two essential tools: the hyperuniverse, consisting of all countable transitive models (...) of ZFC, and V -logic, both of which are also fully discussed in the paper. (shrink)
$63.00 used $91.10 new (collection) Amazon page
Maximality Principles in the Hyperuniverse Programme.Sy-David Friedman & Claudio Ternullo - forthcoming - Foundations of Science:1-19.details
In recent years, one of the main thrusts of set-theoretic research has been the investigation of maximality principles for V, the universe of sets. The Hyperuniverse Programme has formulated several maximality principles, which express the maximality of V both in height and width. The paper provides an overview of the principles which have been investigated so far in the programme, as well as of the logical and model-theoretic tools which are needed to formulate them mathematically, and also briefly shows how (...) optimal principles, among those available, may be selected in a justifiable way. (shrink)
Profit and More: Catholic Social Teaching and the Purpose of the Firm. [REVIEW]Andrew V. Abela - 2001 - Journal of Business Ethics 31 (2):107 - 116.details
The empirical findings in Collins and Porras'' study of visionary companies, Built to Last, and the normative claims about the purpose of the business firm in Centesimus Annus are found to be complementary in understanding the purpose of the business firm. A summary of the methodology and findings of Built to Lastand a short overview of Catholic Social Teaching are provided. It is shown that Centesimus Annus'' claim that the purpose of the firm is broader than just profit is consistent (...) with Collins and Porras empirical finding that firms which set a broader objective tend to be more successful than those which pursue only the maximization of profits. It is noted however that a related finding in Collins and Porras, namely that the content of the firm''s objective is not as important as internalizing some objective beyond just profit maximization, can lead to ethical myopia. Two examples are provided of this: the Walt Disney Company and Philip Morris. Centesimus Annus offers a way to expose such myopia, by providing guidance as to what the purpose of the firm is, and therefore as to what kinds of objectives are appropriate to the firm. (shrink)
Business Ethics and Religion in Applied Ethics
Isols and Maximal Intersecting Classes.Jacob C. E. Dekker - 1993 - Mathematical Logic Quarterly 39 (1):67-78.details
In transfinite arithmetic 2n is defined as the cardinality of the family of all subsets of some set v with cardinality n. However, in the arithmetic of recursive equivalence types 2N is defined as the RET of the family of all finite subsets of some set v of nonnegative integers with RET N. Suppose v is a nonempty set. S is a class over v, if S consists of finite subsets of v and has v as its union. Such a (...) class is an intersecting class over v, if every two members of S have a nonempty intersection. An IC over v is called a maximal IC , if it is not properly included in any IC over v. It is known and readily proved that every MIC over a finite set v of cardinality n ≥ 1 has cardinality 2n-1. In order to generalize this result we introduce the notion of an ω-MIC over v. This is an effective analogue ot the notion of an MIC over v such that a class over a finite set v is an ω-MIC iff it is an MIC. We then prove that every ω-MIC over an isolated set v of RET N ≥ 1 has RET 2N-1. This is a generalization, for while there only are χ0 finite sets, there are ϰ isolated sets, where c denotes the cardinality of the continuum, namely all the finite sets and the c immune sets. MSC: 03D50. (shrink)
Areas of Mathematics in Philosophy of Mathematics
Condorcet's Paradox and the Likelihood of its Occurrence: Different Perspectives on Balanced Preferences.William V. Gehrlein - 2002 - Theory and Decision 52 (2):171-199.details
Many studies have considered the probability that a pairwise majority rule (PMR) winner exists for three candidate elections. The absence of a PMR winner indicates an occurrence of Condorcet's Paradox for three candidate elections. This paper summarizes work that has been done in this area with the assumptions of: Impartial Culture, Impartial Anonymous Culture, Maximal Culture, Dual Culture and Uniform Culture. Results are included for the likelihood that there is a strong winner by PMR, a weak winner by PMR, and (...) the probability that a specific candidate is among the winners by PMR. Closed form representations are developed for some of these probabilities for Impartial Anonymous Culture and for Maximal Culture. Consistent results are obtained for all cultures. In particular, very different behaviors are observed for odd and even numbers of voters. The limiting probabilities as the number of voters increases are reached very quickly for odd numbers of voters, and quite slowly for even numbers of voters. The greatest likelihood of observing Condorcet's Paradox typically occurs for small numbers of voters. Results suggest that while examples of Condorcet's Paradox are observed, one should not expect to observe them with great frequency in three candidate elections. (shrink)
Condorcet's Paradox in Social and Political Philosophy
The Discussion on the Principle of Universalizability in Moral Philosophy in the 1970s and 1980s: An Analysis.E. V. Loginov - 2018 - Russian Journal of Philosophical Sciences 10:65-80.details
In this paper, I analyzed the discussion on the principle of universalizability which took place in moral philosophy in 1970–1980s. In short, I see two main problems that attracted more attention than others. The first problem is an opposition of universalizability and generalization. M.G. Singer argued for generalization argument, and R.M. Hare defended universalizability thesis. Hare tried to refute Singer's position, using methods of ordinary language philosophy, and claimed that in ethics generalization is useless and misleading. I have examined Singer's (...) defense and concluded that he was right and Hare was mistaken. Consequently, generalization argument is better in clarification of the relationship between universality and morality than Hare's doctrine of universalizability, and hence the universality of moral principles is not incompatible with the existence of exclusions. The second problem is the substantiation of the application of categorical imperative in the theory of relevant act descriptions and accurate understanding of the difference between maxims and non-maxims. In Generalization in Ethics, Singer drew attention to this theme and philosophers have proposed some suggestions to solve this problem. I describe ideas of H.J. Paton, H. Potter, O. O'Neill and M. Timmons. Paton coined the teleological-law theory. According to Potter, the best criterion for the relevant act descriptions is causal one. O'N eill suggested the inconsistency-of-intention theory. Timmons defended the causal-law theory. My claim is that the teleological-law theory and the causal-law theory fail to solve the relevant act descriptions problem and the causal criterion and the inconsistency-of-intention theory have their limits. From this, I conclude that these approaches cannot be the basis for clarifying the connection between universality and morality, in contrast to Singer's approach, which, therefore, is better than others to clarify the nature of universality in morality. (shrink)
On the Temporal Boundaries of Simple Experiences.Michael V. Antony - 1998 - Twentieth World Congress of Philosophy.details
I have argued elsewhere that our conception of phenomenal consciousness commits us to simple phenomenal experiences that in some sense constitute our complex experiences. In this paper I argue that the temporal boundaries of simple phenomenal experiences cannot be conceived as fuzzy or vague, but must be conceived as instantaneous or maximally sharp. The argument is based on an account of what is involved in conceiving fuzzy temporally boundaries for events generally. If the argument is right, and our conception of (...) phenomenal consciousness is assumed to reflect the facts about consciousness, then since the temporal boundaries of neurophysiological events can be conceived as fuzzy, considerable pressure can be applied to neurophysiological identity theories, as well as to dualist accounts that posit temporal correspondence with neurophysiological events. (shrink)
Temporal Experience, Misc in Philosophy of Mind
Gonzales V. Oregon and Physician-Assisted Suicide: Ethical and Policy Issues.Ken Levy - 2007 - Tulsa Law Review 42:699-729.details
The euthanasia literature typically discusses the difference between "active" and "passive" means of ending a patient's life. Physician-assisted suicide differs from both active and passive forms of euthanasia insofar as the physician does not administer the means of suicide to the patient. Instead, she merely prescribes and dispenses them to the patient and lets the patient "do the rest" – if and when the patient chooses. One supposed advantage of this process is that it maximizes the patient's autonomy with respect (...) to both her decision to die and the dying process itself. Still, despite this supposed advantage, Oregon is the only state to have legalized physician-assisted suicide. After summarizing the most important Supreme Court opinions on euthanasia (namely, Cruzan v. Director, Missouri Dep't of Health; Vacco v. Quill; Washington v. Glucksberg; and Gonzales v. Oregon), this paper argues that while there are no strong ethical reasons against legalizing physician-assisted suicide, there are some very strong policy reasons for keeping it criminal in the other forty-nine states. (shrink)
Assisted Suicide in Applied Ethics
Autonomy in Applied Ethics in Applied Ethics
Euthanasia in Applied Ethics
Suicide in Applied Ethics
The Corporate Objective After eBay V. Newmark.John R. Boatright - 2017 - Business and Society Review 122 (1):51-70.details
The Delaware court's decision in eBay v. Newmark has been viewed by many commentators as a decisive affirmation of shareholder wealth maximization as the only legally permissible objective of a for-profit corporation. The implications of this court case are of particular concern for the emerging field of social enterprise, in which some organizations, such as, in this case, Craigslist, choose to pursue a social benefit mission in the for-profit corporate form. The eBay v. Newmark decision may also threaten companies that (...) seek to be socially responsible by serving constituencies other than shareholders at the expense of some profit. This examination of the court decision concludes that a legal requirement to maximize shareholder value may not preclude a commitment to social responsibility and may even permit the pursuit of a social benefit objective, such as the preservation of the culture developed by Craigslist. In particular, the court's decision in eBay v. Newmark reflects unique features of the case that could have been avoided by Craigslist and by other similar companies. (shrink)
Co-Immune Subspaces and Complementation in V∞.R. Downey - 1984 - Journal of Symbolic Logic 49 (2):528 - 538.details
We examine the multiplicity of complementation amongst subspaces of V ∞ . A subspace V is a complement of a subspace W if V ∩ W = {0} and (V ∪ W) * = V ∞ . A subspace is called fully co-r.e. if it is generated by a co-r.e. subset of a recursive basis of V ∞ . We observe that every r.e. subspace has a fully co-r.e. complement. Theorem. If S is any fully co-r.e. subspace then S has (...) a decidable complement. We give an analysis of other types of complements S may have. For example, if S is fully co-r.e. and nonrecursive, then S has a (nonrecursive) r.e. nowhere simple complement. We impose the condition of immunity upon our subspaces. Theorem. Suppose V is fully co-r.e. Then V is immune iff there exist M 1 , M 2 ∈ L(V ∞ ), with M 1 supermaximal and M 2 k-thin, such that $M_1 \oplus V = M_2 \oplus V = V_\infty$ . Corollary. Suppose V is any r.e. subspace with a fully co-r.e. immune complement W (e.g., V is maximal or V is h-immune). Then there exist an r.e. supermaximal subspace M and a decidable subspace D such that $V \oplus W = M \oplus W = D \oplus W = V_\infty$ . We indicate how one may obtain many further results of this type. Finally we examine a generalization of the concepts of immunity and soundness. A subspace V of V ∞ is nowhere sound if (i) for all Q ∈ L(V ∞ ) if $Q \supset V$ then Q = V ∞ , (ii) V is immune and (iii) every complement of V is immune. We analyse the existence (and ramifications of the existence) of nowhere sound spaces. (shrink)
The Problem of Searching the Meaning of Human Existence: Contemporary Context.V. M. Petrushov & V. M. Shapoval - 2020 - Anthropological Measurements of Philosophical Research 17:55-64.details
Purpose. The purpose of the article is the analysis of the reasons and grounds of the crisis in the sphere of meaning-making, as well as searching answers to the questions about the meaning of human life in the contemporary world, which are maximally relevant in connection with the escalation of global problems, revealing the points of convergence between various theoretical positions, evaluation of their heuristic potential. Theoretical basis of the research is the historical-philosophical, comparative and system approaches, as well as (...) the analysis of philosophical insights in the field of global studies. Originality. Originality lies in the fact that this article is the first attempt to conduct comprehensive analysis in the problem of the sense of the Existence as it is presented in the first quarter of the 21st century and to relate it with the modern social situation that is characterized by a complex range of interconnected and interdependent anthropological problems of our time. Authors emphasize that the main reason in the crisis of meaning is that a man has lost touch with his roots, which is wildlife. He has created an artificial structure, civilization to satisfy his needs and finds no way to the transcendental, which is the true House of his being. Conclusions. A human must refuse from false self-conceit concerning his potential omniscience and omnipotence, cease dictating his own rules to the Existence, determine the boundaries of his freedom and try to clearly realize his place in the objective structure of being. The global situation can change for the better only if a dramatic change in the area of meaning-making happens. The decisive force, which may encourage nudging to the positive changes, can be either the free will of people who have realized the criticality of the situation or external natural and social circumstances that will make people reorganize radically. The proper prioritizing, a deep awareness of universal goals and solidarity between people could be the value basis that will become the foundation to find the meaning and create a more favorable future. (shrink)
The Other Machiavelli.V. D. Vinogradov & D. V. Ivanov - 1996 - Russian Studies in Philosophy 34 (4):36-50.details
The term 'Machiavellianism', used to designate a tough politics knowing no ethical barriers, entered firmly into circulation as far back as the sixteenth century. It was the negative reaction to the maxims in The Prince that defined the initial attitude toward Machiavelli's doctrine, and the internal polemic with this initial assessment has spawned an endless stream of literature endeavoring to justify in one way or other the ill-starred secretary of the Florentine Republic. In sheer number of publications, pro-Machiavelli views exceed (...) anti-Machiavelli views by many times. And yet questions remain; the original negative reaction is not eradicated, just as the striving for apologetics is not eradicated. (shrink)
Niccolo Machiavelli in Medieval and Renaissance Philosophy
Plotinus's Treatise On the Virtues (I.2) and Its Interpretation by Porphyry and Marinus.D. V. Bugai - 2003 - Russian Studies in Philosophy 42 (1):84-95.details
As is well known, Plotinus's philosophy served as the starting point for the development of all Neoplatonism. It created the basic schema that set the framework for the thought of all later representatives of this tendency from Porphyry to Damascius. The doctrine of the transcendence of the One, of the three original hypostases, the application of the categories of Plato's Parmenides in the construction of ontology—all this and much else besides became the property of the Neoplatonic schools, which were scattered (...) throughout the Roman empire, and was incorporated partly into the Christian theology, which was then in the process of formation. Naturally, as a result of its wide dissemination and the change in its cultural and social environment, Plotinus's legacy appeared in a different light and took on new forms; through these changes in form we can try to understand the difference in content. For this purpose Plotinus's treatise on the virtues is of special interest. The point is that Porphyry relies precisely on this treatise and at times even literally borrows large fragments from it in setting out the doctrine concerning the virtues in the thirty-second maxim . The same treatise serves as a basis for Marinus in his biography of Proclus. The aim of my lecture is to analyze the content of the treatise and its interpretations. (shrink)
Plotinus in Ancient Greek and Roman Philosophy
Porphyry in Ancient Greek and Roman Philosophy
The Ambitious Idea of Kant's Corollary.Susan V. H. Castro - 2018 - In Violetta L. Waibel, Margit Ruffing & David Wagner (eds.), Natur Und Freiheit. Akten des Xii. Internationalen Kant-Kongresses. De Gruyter. pp. 1779-1786.details
Misrepresentations can be innocuous or even useful, but Kant's corollary to the formula of universal law appears to involve a pernicious one: "act as if the maxim of your action were to become by your will a universal law of nature". Humans obviously cannot make their maxims into laws of nature, and it seems preposterous to claim that we are morally required to pretend that we can. Given that Kant was careful to eradicate pernicious misrepresentations from theoretical metaphysics, the (...) imperative to act as if I have this supernatural power has typically been treated as an embarrassment meriting apology. The wording of the corollary may be vindicated, however, by recognizing that "as if" (als ob) is a technical term both in the Critique of Pure Reason and here. It signals a modal shift from the assertoric to the problematic mode of cognition, one that is necessitated by the attempt to incorporate the natural effects of a free will into a universal moral imperative that is philosophically practical. In this paper I sketch how the modal shift makes sense of the corollary as a subjectively necessary, philosophically practical idealization of the extension of human freedom into nature, one that accurately represents a necessary parameter of moral conduct: moral ambition. (shrink)
Kant: Epistemology, Misc in 17th/18th Century Philosophy
Kant: Formula of Universal Law in 17th/18th Century Philosophy
$460.40 new $480.00 from Amazon $598.25 used (collection) Amazon page
Decision-Making in the Critically Ill Neonate: Cultural Background V Individual Life Experiences.C. Hammerman, E. Kornbluth, O. Lavie, P. Zadka, Y. Aboulafia & A. I. Eidelman - 1997 - Journal of Medical Ethics 23 (3):164-169.details
OBJECTIVES: In treating critically ill neonates, situations occasionally arise in which aggressive medical treatment prolongs the inevitable death rather than prolonging life. Decisions as to limitation of neonatal medical intervention remain controversial and the primary responsibility of the generally unprepared family. This research was designed to study response patterns of expectant mothers towards treatment of critically ill and/or malformed infants. DESIGN/SETTING: Attitudes were studied via comprehensive questionnaires divided into three sections: 1-Sociodemographic data and prior personal experience with perinatal problems; 2-Theoretical (...) philosophical principles used in making medical ethical decisions; and 3-Hypothetical case scenarios with choices of treatment options. SUBJECTS AND RESULTS: Six hundred and fifty pregnant women were studied. Maternal birthplace (p = 0.005) and level of religious observance (p = 0.02) were strongly associated with the desire for maximally aggressive medical intervention in the hypothetical case scenario. Specific personal experiences such as infertility problems, previous children with serious mental or physical problems were not correlated with the selection of different treatment choices. Of the theoretical principles studied, only the desire to preserve life at all costs was significantly associated with the choice for maximal medical treatment (p = 0.003). CONCLUSIONS: Maternal ethnocultural background and philosophical principles more profoundly influenced medical ethical decision-making than did specific personal life experiences. (shrink)
Regular Embeddings of the Stationary Tower and Woodin's Σ 2 2 Maximality Theorem.Richard Ketchersid, Paul B. Larson & Jindřich Zapletal - 2010 - Journal of Symbolic Logic 75 (2):711-727.details
We present Woodin's proof that if there exists a measurable Woodin cardinal δ, then there is a forcing extension satisfying all $\Sigma _{2}^{2}$ sentences ϕ such that CH + ϕ holds in a forcing extension of V by a partial order in V δ . We also use some of the techniques from this proof to show that if there exists a stationary limit of stationary limits of Woodin cardinals, then in a homogeneous forcing extension there is an elementary embedding (...) j: V → M with critical point $\omega _{1}^{V}$ such that M is countably closed in the forcing extension. (shrink)
Axioms of Set Theory in Philosophy of Mathematics
R&D Cooperation in Emerging Industries, Asymmetric Innovative Capabilities and Rationale for Technology Parks.Vivekananda Mukherjee & Shyama V. Ramani - 2011 - Theory and Decision 71 (3):373-394.details
Starting from the premise that firms are distinct in terms of their capacity to create innovations, this article explores the rationale for R&D cooperation and the choice between alliances that involve information sharing, cost sharing or both. Defining innovative capability as the probability of creating an innovation, it examines firm strategy in a duopoly market, where firms have to decide whether or not to cooperate to acquire a fixed cost R&D infrastructure that would endow each firm with a firm-specific innovative (...) capability. Furthermore, since emerging industries are often characterized by high technological uncertainty and diverse firm focus that makes the exploitation of spillovers difficult, this article focuses on a zero spillover context. It demonstrates that asymmetry has an impact on alliance choice and social welfare, as a function of ex-post market competition and fixed costs of R&D. With significant asymmetry no alliance may be formed, while with similar firms the cost sharing alliance is dominant. Finally, it ascertains the settings under which the equilibrium outcome is distinct from that maximizing social welfare, thereby highlighting some conditions under which public investment in a technology park can be justified. (shrink)
Nanotechnology in Applied Ethics
Dostoevsky and Mendeleev: An Antispiritist Dialogue.I. L. Volgin & V. L. Rabinovich - 1972 - Russian Studies in Philosophy 11 (2):170-194.details
The sources of the real conflict between science and various kinds of undertakings in occultism pretending to be science date back to the end of the 16th and beginning of the 17th centuries, when modern scientific method was barely taking shape. The natural philosophy of the 16th century, which put forth natural magic in place of divine magic, was the ideological antipode of the new science in process of formation. The pantheistic reinterpretation of monotheistic Christian creationism is a characteristic feature (...) of constructs in natural philosophy with their striving toward maximal substantialization of the nonmaterial. Thus, for example, the rationalist mystic of natural philosophy, Girolamo Cardano, returns in his work to the medieval notion of the world soul, but understands by it an entirely material substance which he identifies with light and heat. (shrink)
Russian Philosophy in European Philosophy
Marginalia to Kant's Essay "On the Alleged Right to Lie".Vadim V. Vasilyev - 2009 - Russian Studies in Philosophy 48 (3):82-89.details
The author argues that despite universal and formal character of the foundation of Kant's ethics, its principles appear to be compatible with recognition of the possibility of lying for philanthropic reason. To have an effect in the world, our obligations must necessarily have empirical components that point to specific conditions, under which the maxim will have a moral worth. One of such condition may be the requirement that probable consequences of the action will not clash with other obligations.
Kant's Works in Practical Philosophy, Misc in 17th/18th Century Philosophy
A Class of {Sigma {3}^{0}} Modular Lattices Embeddable as Principal Filters in {Mathcal{L}^{Ast }(V{Infty })}.Rumen Dimitrov - 2008 - Archive for Mathematical Logic 47 (2):111-132.details
Let I 0 be a a computable basis of the fully effective vector space V ∞ over the computable field F. Let I be a quasimaximal subset of I 0 that is the intersection of n maximal subsets of the same 1-degree up to *. We prove that the principal filter ${\mathcal{L}^{\ast}(V,\uparrow )}$ of V = cl(I) is isomorphic to the lattice ${\mathcal{L}(n, \overline{F})}$ of subspaces of an n-dimensional space over ${\overline{F}}$ , a ${\Sigma _{3}^{0}}$ extension of F. As a (...) corollary of this and the main result of Dimitrov (Math Log 43:415–424, 2004) we prove that any finite product of the lattices ${(\mathcal{L}(n_{i}, \overline{F }_{i}))_{i=1}^{k}}$ is isomorphic to a principal filter of ${\mathcal{ L}^{\ast}(V_{\infty})}$ . We thus answer Question 5.3 "What are the principal filters of ${\mathcal{L}^{\ast}(V_{\infty}) ?}$ " posed by Downey and Remmel (Computable algebras and closure systems: coding properties, handbook of recursive mathematics, vol 2, pp 977–1039, Stud Log Found Math, vol 139, North-Holland, Amsterdam, 1998) for spaces that are closures of quasimaximal sets. (shrink)
The Ontology of Intentionality I: The Dependence Ontological Account of Order: Mediate and Immediate Moments and Pieces of Dependent and Independent Objects.Gilbert T. Null - 2007 - Husserl Studies 23 (1):33-69.details
This is the first of three essays which use Edmund Husserl's dependence ontology to formulate a non-Diodorean and non-Kantian temporal semantics for two-valued, first-order predicate modal languages suitable for expressing ontologies of experience (like physics and cognitive science). This essay's primary desideratum is to formulate an adequate dependence-ontological account of order. To do so it uses primitive (proper) part and (weak) foundation relations to formulate seven axioms and 28 definitions as a basis for Husserl's dependence ontological theory of relating moments. (...) The essay distinguishes between dependence v. independence, pieces v. moments, mediate v. immediate pieces and moments, maximal v. non-maximal pieces, founded v. unfounded qualities, integrative v. disintegrative dependence, and defines the concepts of the completion of an object, the adumbrational equivalence relation of objects, moments of unity which unify objects, and relating moments which relate objects. The eight theorems [CUT90]-[CUT97] show that relating moments of unity provide an adequate account of order in terms of primitive (proper) part and (weak) foundation relations. (shrink)
Husserl: Intentionality, Misc in Continental Philosophy
Husserl: Ontology in Continental Philosophy
The Search for New Axioms in the Hyperuniverse Programme.Claudio Ternullo & Sy-David Friedman - 2016 - In Andrea Sereni & Francesca Boccuni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics. Berlin: Springer. pp. 165-188.details
The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the `maximal iterative concept', and the programme identi fies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements (...) (H-axioms) in countable transitive models, the collection of which constitutes the `hyperuniverse' (H), has remarkable 1st-order consequences, some of which we review in section 5. (shrink)
Independence Results in Set Theory in Philosophy of Mathematics
Indeterminacy in Mathematics in Philosophy of Mathematics
The Iterative Conception of Set in Philosophy of Mathematics
Simple and Hyperhypersimple Vector Spaces.Allen Retzlaff - 1978 - Journal of Symbolic Logic 43 (2):260-269.details
Let $V_\propto$ be a fixed, fully effective, infinite dimensional vector space. Let $\mathscr{L}(V_\propto)$ be the lattice consisting of the recursively enumerable (r.e.) subspaces of $V_\propto$ , under the operations of intersection and weak sum (see § 1 for precise definitions). In this article we examine the algebraic properties of $\mathscr{L}(V_\propto)$ . Early research on recursively enumerable algebraic structures was done by Rabin [14], Frolich and Shepherdson [5], Dekker [3], Hamilton [7], and Guhl [6]. Our results are based upon the more (...) recent work concerning vector spaces of Metakides and Nerode [12], Crossley and Nerode [2], Remmel [15], [16], and Kalantari [8]. In the main theorem below, we extend a result of Lachlan from the lattice E of r.e. sets to $\mathscr{L}(V_\propto)$ . We define hyperhypersimple vector spaces, discuss some of their properties and show if $A, B \in \mathscr{L}(V_\propto)$ , and A is a hyperhypersimple subspace of B then there is a recursive space C such that A + C = B. It will be proven that if $V \in \mathscr{L}(V_\propto)$ and the lattice of superspaces of V is a complemented modular lattice then V is hyperhypersimple. The final section contains a summary of related results concerning maximality and simplicity. (shrink)
Symposium on "Cognition and Rationality: Part I" Minimal Rationality. [REVIEW]Isaac Levi - 2006 - Mind and Society 5 (2):199-211.details
An argument is advanced to show why E-admissibility should be preferred over maximality as a principle of rational choice where rationality is understood as minimal rationality. Consideration is given to the distinction between second best and second worst options in three way choice that is ignored according to maximality. It is shown why the behavior exhibited in addressing the problems posed by Allais (Econometrica 21:503–546, 1952) and by Ellsberg (Q Econ 75:643–669, 1961) do not violate the independence postulate according to (...) minimal rationality. (shrink) | CommonCrawl |
\begin{document}
\begin{abstract} We prove $L^p\times L^q\rightarrow L^r$ bounds for certain lacunary bilinear maximal averaging operators with parameters satisfying the H\"older relation $1/p+1/q=1/r$. The boundedness region that we get contains at least the interior of the H\"older boundedness region of the associated single scale bilinear averaging operator. In the case of the lacunary bilinear spherical maximal function in $d\geq 2$, we prove boundedness for any $p,q\in (1,\infty]^2$, which is sharp up to boundary; we then show how to extend this result to a more degenerate family of surfaces where some curvatures are allowed to vanish. For the lacunary triangle averaging maximal operator, we have results in $d\geq 7$, and the description of the sharp region will depend on a sharp description of the H\"older bounds for the single scale triangle averaging operator, which is still open. \end{abstract}
\title{Bounds for lacunary bilinear spherical and triangle maximal functions}
\section{Introduction and statements of results}
Given $t>0$, and $f,g\in C_{0}^{\infty}(\mathbb{R}^d)$, define the bilinear spherical maximal average at scale $t$ by \begin{equation}
\mathcal{A}_{t}(f,g)(x)=\int_{S^{2d-1}}f(x-ty)g(x-tz)\,d\sigma_{2d-1}(y,z),\,x\in \mathbb{R}^d. \end{equation}
Here, $d\sigma_{2d-1}$ denotes the normalized surface measure in the unit sphere $S^{2d-1}$ in $\mathbb{R}^{2d}$. The lacunary spherical maximal function is then given by \begin{equation}
\mathcal{M}_{lac}(f,g)(x)=\sup_{l \in \mathbb{Z}}|\mathcal{A}_{2^{-l}}(f,g)(x)|. \end{equation}
We may also sometimes refer to the (full) bilinear spherical maximal function \begin{equation}
\mathcal{M}(f,g)(x)=\sup_{t>0} |\mathcal{A}_{t}(f,g)(x)| \end{equation} and the localized bilinear spherical maximal function \begin{equation}
\tilde{\mathcal{M}}(f,g)(x)=\sup_{1\leq t\leq 2} |\mathcal{A}_{t}(f,g)(x)|. \end{equation}
For $d\geq 2$, the study of the Lebesgue bounds for $\mathcal{M}$ got a lot of attention in the recent years, that is, people were interested in the description of all the parameters $1\leq p,q\leq \infty $ and $r>0$ such that $\mathcal{M}:L^p\times L^q\rightarrow L^r$ is bounded. For both $\mathcal{M}$ and $\mathcal{M}_{lac}$, it is necessary to impose the H\"older relation $p^{-1}+q^{-1}=r^{-1}$ because of scaling. The operator $\mathcal{M}$ first appeared in \cite{GGIS}, and subsequently, its Lebesgue bounds were studied by multiple authors (\cite{BGHHO},\cite{GHH},\cite{HHY}) until the description of the sharp boundedness region for $\mathcal{M}$ in $d\geq 2$ was finally settled by \cite{JL} (see Theorem \ref{boundsjl} below for the statement of these sharp bounds). In that paper, they exploited an interesting coarea formula which allowed them to slice the integral in $S^{2d-1}$ into integrals over lower dimensional spheres. From that, they got a pointwise inequality of the form $\mathcal{M}(f,g)\leq C Mf(x)\mathcal{S}g(x)$ where $M$ is the Hardy-Littlewood maximal function and $\mathcal{S}$ is the spherical maximal function, and they were able to use the known boundedness properties of these (sub)linear operators. The spherical maximal function
$$\mathcal{S}(f)(x)=\sup_{t>0} \int_{S^{d-1}} |f(x-ty)|d\sigma_{d-1}(y)$$ was first studied by Stein in \cite{stein}. He showed that $\mathcal{S}$ is bounded in $L^{p}$ if $d\geq 3$ and $\frac{d}{d-1}< p\leq \infty$ and that boundedness fails for $d\geq 2$ and $p\leq \frac{d}{d-1}$. The remaining case $d=2$ was proven later by Bourgain \cite{bourgaind2}, and it was much more complicated, since in that case the operator fails to be bounded in $L^2(\mathbb{R}^2)$. Even though the question was settled in \cite{JL} for $\mathcal{M}$, the sharp boundedness region for the lacunary case in $d\geq 2$ was still open and did not follow from the slicing strategy exploited in their paper. As we are studying the lacunary (sub)bilinear spherical maximal function, we will also naturally make use of facts about the lacunary (sub)linear spherical maximal function, defined as
$$\mathcal{S}(f)(x)=\sup_{l\in\mathbb{Z}} \int_{S^{d-1}} |f(x-2^{-l}y)|d\sigma_{d-1}(y).$$ This has better boundedness properties than the full spherical maximal function; it was shown by C. Calderón \cite{calderon} and Coifman and Weiss \cite{coifmanweiss} that $\mathcal{S}_{lac}$ is bounded in $L^p$ for any $1<p\leq \infty$.
Observe that the definition of $\mathcal{A}_t$ makes sense for any $d\geq 1$. In \cite{MCZZ} the authors studied $\mathcal{M}$ and $\mathcal{M}_{lac}$ in the case $d=1$ and they proved the sharp bounds up to the boundary. The sharp boundedness region for $\mathcal{M}$ in $d=1$ was also obtained independently by \cite{dosidisramos}.
In this article, we almost entirely settle the question of Lebesgue bounds for $\mathcal{M}_{lac}$ in $d\geq 2$.
\begin{thm}[Bounds for the lacunary bilinear spherical maximal function]\label{boundsspherical} Assume $d\geq 2$. Let $p,q\in (1,\infty]$ and $r\in (0, \infty)$ satisfy the H\"older relation $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. Then
\begin{equation}
\|\mathcal{M}_{lac}(f,g)\|_r= \|\sup_{l \in \mathbb{Z}}|\mathcal{A}_{2^{-l}}(f,g)|\|_r\lesssim \|f\|_p\|g\|_q.
\end{equation}
Moreover, $\mathcal{M}_{lac}$ does not satisfy strong bounds $L^1\times L^{\infty}\rightarrow L^{1}$ or $L^{\infty}\times L^1\rightarrow L^1$, but it satisfies weak bounds $\mathcal{M}_{lac}:L^1\times L^{\infty}\rightarrow L^{1,\infty}$ and $\mathcal{M}_{lac}:L^{\infty}\times L^{1}\rightarrow L^{1,\infty}$. \end{thm}
\begin{rem}
There are certain pieces of the boundary $p=1$ or $q=1$ that one can include in the theorem above by observing that since $\mathcal{M}_{lac}(f,g)(x)\leq \mathcal{M}(f,g)(x) $ the known bounds from \cite{JL} are also true for $\mathcal{M}_{lac}$ (see Figure \ref{figure1}). Namely, for $p=1$ for example, let $q\in (\frac{d}{d-1}, \infty)$ and $1/r=1/p+1/q$, then
$$\|\mathcal{M}_{lac}(f,g)\|_{r}\lesssim \|f\|_1 \|g\|_{q}$$
We have hope the bound above holds at least for $q\in (1,\infty)$, but the methods in this paper were not enough to prove that. We leave the remaining pieces of the boundary for future exploration.
\end{rem}
\begin{figure}
\caption{H\"older bounds for the lacunary bilinear spherical maximal function in $d\geq 2$.}
\label{figure1}
\end{figure}
\begin{rem}
$\mathcal{S}_{lac}$ fails to be bounded in $L^1(\mathbb{R}^d)$, and there are some interesting results about the endpoint behavior of $\mathcal{S}_{lac}$ in the literature (see \cite{cladek2017} and references therein). An interesting open question is what kind of bounds can be proved for $\mathcal{M}_{lac}$ in the missing pieces of the boundary and if any of the linear methods in these papers can be adapted to our bilinear setting. \end{rem}
We will also be interested in the lacunary triangle averaging maximal operator \begin{equation}
\mathcal{T}_{lac}(f,g)(x)=\sup_{l\in \mathbb{Z}} |\mathcal{T}_{2^l}(f,g)(x)| \end{equation} where $\mathcal{T}_t$ is the triangle averaging operator of radius $t>0$, namely, \begin{equation}
\mathcal{T}_{t}(f,g)(x)=\int_{\mathcal{I}} f(x-ty)g(x-tz)d\mu(y,z), \end{equation} where $\mu$ is the natural surface measure on the submanifold of $\mathbb{R}^{2d}$ given by
$$\mathcal{I}=\{(y,z)\in \mathbb{R}^{2d}\colon |y|=|z|=|y-z|=1 \}.$$
By fixing an equilateral triangle in $\mathbb{R}^d$, that is, $u,v\in \mathbb{R}^d$ with $|u|=|v|=|u-v|=1$, one can also write \begin{equation}
\mathcal{T}_{t}(f,g)(x)=\int_{O(d)} f(x-tRu)g(x-tRv)d\mu(R), \end{equation} where $\mu$ is the normalized Haar measure on the group $O(d)$, the orthogonal group in $\mathbb{R}^d$, a fact which was already exploited in \cite{IPS}, for example. The triangle averaging operator has been well studied in the last few years (\cite{triangleaveraging}, \cite{IPS}, \cite{sparsetriangle}) and it is closely related to Falconer type results for three point configurations of points in compact sets of $\mathbb{R}^d$ with large enough Hausdorff dimension (\cite{threepointconf}, \cite{GGIP}, \cite{IL}).
Let $d\geq 2$. Denote by $\mathcal{R}_1$ the region inside $[0,1]\times [0,1]$ given by the convex closure of the points \begin{equation}\label{regionr1}
(0,0),\,(0,1),\,(1,0)\text{ and }\left(\frac{d}{d+1},\frac{d}{d+1}\right). \end{equation}
\begin{thm}[Bounds for the lacunary triangle averaging maximal function]\label{boundstriangle}
Assume $d\geq 7$. For any $(\frac{1}{p},\frac{1}{q})\in \text{int}{(\mathcal{R}_1)}$, and $r\in (0, \infty)$ given by the H\"older relation $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$, it holds that
\begin{equation}
\|\mathcal{T}_{lac}(f,g)\|_{r}\lesssim \|f\|_p\|g\|_q
\end{equation}
Moreover, for any $d\geq 2$ boundedness holds for $(1/p,1/q)$ in the half open segment connecting $(0,0)$ to $(0,1)$, or in the half open segment connecting $(0,0)$ to $(1,0)$. That is, for any $q\in (1,\infty]$, we have
$$ \|\mathcal{T}_{lac}(f,g)\|_q\lesssim \|f\|_{\infty} \|g\|_q$$ and
$$\|\mathcal{T}_{lac}(f,g)\|_q\lesssim \|f\|_q \|g\|_{\infty}$$
\end{thm}
\begin{rem}
One can also state the theorem above in terms of $(1/p,1/q,1/r)$ in the interior of the H\"older boundedness region of $\mathcal{T}_1$, namely
$$\mathcal{B}_{\mathcal{T}_1}=:\{(1/p,1/q,1/r)\colon (p,q,r)\in [1,\infty]^2\times (0,\infty],\,1/p+1/q=1/r\,\text{ and } \|\mathcal{T}_1\|_{L^p\times L^q\rightarrow L^r}<\infty\}$$
One difficulty is that the sharp description of $\mathcal{B}_{\mathcal{T}_1}$ is not known, and all we know so far is that it contains the H\"older exponents $(1/p,1/q,1/r)$ with $(1/p,1/q)$ in the quadrilateral $\mathcal{R}_1$. \end{rem}
In the last section of this paper, we generalize Theorem \ref{boundsspherical} to cover a more general class of averaging operators which appeared in a recent work of Lee and Shuin \cite{leeshuin}. We refer to section \ref{leeshuinsection} for the definition of the region $\mathcal{R}^{\mathbf{a}}$ and the bilinear averaging operators $\mathcal{A}_{t}^{\mathbf{a}}(f,g)$, for $\mathbf{a}\in [1,\infty)^2$.
\begin{thm}[Bounds for the lacunary maximal operators associated to degenerate surfaces]\label{lacunaryleeshuin}
Assume $d\geq 2$ and let $\mathbf{a}\in [1,\infty)^2$. For any $(\frac{1}{p},\frac{1}{q})\in \text{int}{(\mathcal{R}^{\mathbf{a}})}$, and $r\in (0, \infty)$ given by the H\"older relation $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$, it holds that
\begin{equation}
\|\sup_{l\in\mathbb{Z}}|\mathcal{A}_{2^{-l}}^{\mathbf{a}}(f,g)|\|_{r}\lesssim \|f\|_p\|g\|_q
\end{equation} \end{thm}
There are several common features to deducing the desired bounds for the lacunary maximal operators. We generally follow a scheme of extending boundedness estimates for single scale averages to the lacunary maximal functions by localizing the Fourier support in both variables separately. If either of the localizations is low frequency, we are often able to get a pointwise domination by a product of (sub)linear maximal operators. For the remaining case (the so-called high-high part of the operator), we do a further dyadic decomposition and show that we can get bounds that decay exponentially in the scale for the operator norm on $L^2\times L^2\rightarrow L^1$. This extends to the entire boundedness region via interpolation with a trivial estimate.
\textbf{Plan of the article.} We first introduce in Section \ref{preliminaries} some preliminary facts that will be useful along the paper. In Section \ref{sectiondecayforpieces}, we describe how to break down the bilinear spherical averaging operator into pieces and we obtain decay estimates for those pieces. The crucial idea will be to combine the slicing strategy from \cite{JL} with polar coordinates in the integral over the unit ball, and exploit known bounds for linear spherical averages. Those estimates will be key in the proof of Theorem \ref{boundsspherical}, which is given in Section \ref{prooffirstthm}. In Section \ref{proofsecondthm}, we prove Theorem \ref{boundstriangle}. In that section we will get decay estimates for the pieces of the triangle averaging operator by making use of a bilinear multiplier boundedness criteria that we recall in Section \ref{preliminaries}. In section \ref{leeshuinsection}, we show how the methods we used to prove the decay of the pieces of the bilinear spherical maximal function can be generalized for a more general class of bilinear averaging operators, as stated in Theorem \ref{lacunaryleeshuin}.\\
\textbf{Notation.}
Throughout this paper for $p\in (0,\infty]$, $\|\cdot\|_p$ stands for the usual norm in $L^p(\mathbb{R}^d)$, namely
$$\|f\|_p=\left(\int_{\mathbb{R}^d} |f(x)|^p dx\right)^{1/p}, \,\text{if } p<\infty.$$
When $0<p<1$, even though $\|\cdot\|_p$ is not a norm, $\|\cdot\|_p^p$ satisfies the triangle inequality, which we will make use of later. For two quantities $A,B\geq 0$, $A\lesssim B$ means that there exists $C>0$ such that $A\leq C B$. For a measurable set $E\subset \mathbb{R}^d$, we let $\chi_E$ denote the indicator function of the set $E$. We will denote by $Mf$ the Hardy-Littlewood maximal function function of $f$, and by $\mathcal{S}_{lac}$ and $\mathcal{S}$ the lacunary and full (sub)linear spherical maximal operators.\\
\textbf{Acknowledgements.} The first author would like to thank Professor Jill Pipher for her support and for helpful discussions throughout the process of writing this paper. We also would like to thank Professor Yumeng Ou for introducing us to some of the questions in this paper.
\section{Preliminary Facts}\label{preliminaries}
\subsection{Relevant facts about $\mathcal{A}_1$}
\begin{thm}[\cite{IPS}] Let $d\geq 2$, then \begin{equation}\label{boundA1}
\mathcal{A}_1:L^1(\mathbb{R}^d)\times L^1(\mathbb{R}^d)\rightarrow L^{1/2}(\mathbb{R}^d) \text{ is bounded.} \end{equation} \end{thm}
For $p,q,r\in [1,\infty]$ such that $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$ (that is, we are in the Banach case when $r\geq 1$), it is immediate from Minkowski's inequality combined with H\"older's inequality that $$\mathcal{A}_1:L^{p}(\mathbb{R}^d)\times L^{q}(\mathbb{R}^d)\rightarrow L^{r}(\mathbb{R}^d)\text{ is bounded.}$$
Interpolating the bound in (\ref{boundA1}) with the trivial bounds above, one gets the following corollary.
\begin{cor} \label{holderboundsforA1}Let $d\geq 2$. Then for all $1\leq p,q \leq \infty$ and $r\in (0,\infty]$ given by the H\"older relation $1/r=1/p+1/q$, one has that \begin{equation}
\mathcal{A}_1:L^{p}(\mathbb{R}^d)\times L^{q}(\mathbb{R}^d)\rightarrow L^r(\mathbb{R}^d) \text{ is bounded.} \end{equation}
\end{cor}
\begin{rem}
Due the single scale nature of $\mathcal{A}_1$, one can also prove $L^p$ improving bounds for $\mathcal{A}_1$, that is, boundedness from $L^p\times L^q$ into $L^r$ for $1/r<1/p+1/q$. However, the sharp region of parameters for which this holds is still not known. So far, all it is known is that it contains the known boundedness region of the localized bilinear maximal function $\tilde{\mathcal{M}}=\sup_{t\in [1,2]}|\mathcal{A}_{t}(\cdot, \cdot)|$ (see \cite{JL} or \cite{BFOPZ} for the description of that region), and also the points $(1,1,1)$ and $(1,1,1/2)$ \cite{IPS}. For $d=1$ there is a better understanding of the $L^p$ improving region given by the results in \cite{SS},\cite{DOberlin}, and \cite{BS}. For the purposes of this article we will not need $L^p$ improving estimates and Corollary \ref{holderboundsforA1} will be enough.
\end{rem}
The sharp boundedness region for the full bilinear maximal function in $d\geq 2$ is known due to Jeong and Lee \cite{JL}: \begin{thm}[\cite{JL}]\label{boundsjl} Let $d\geq 2$, $p,q\in [1,\infty]$ and $r\in (0,\infty]$. Then the estimate \begin{equation}
\|\mathcal{M}(f,g)\|_r\lesssim \|f\|_{p}\|g\|_q \end{equation} holds if and only if $1/r=1/p+1/q$ and $1/r<(2d-1)/d$, except for the case $(p,q,r)=(1,\infty, 1)$ and $(p,q,r)=(\infty,1,1)$ where the boundedness fails. Also one has weak type bounds
$$\|\mathcal{M}(f,g)\|_{L^{1,\infty}}\lesssim \|f\|_1\|g\|_{\infty}$$ and
$$\|\mathcal{M}(f,g)\|_{L^{1,\infty}}\lesssim \|f\|_{\infty}\|g\|_{1}.$$
\end{thm}
\begin{figure}
\caption{Jeong and Lee boundedness region for the bilinear spherical maximal function when $d\geq 2$.}
\end{figure}
When studying the boundedness properties of the bilinear spherical maximal function $\mathcal{M}$, Jeong and Lee \cite{JL} showed that if $F$ is a continuous function defined in $\mathbb{R}^{2d}$ and $(y,z)\in \mathbb{R}^d\times \mathbb{R}^d$, then
$$\int_{S^{2d-1}}F(y,z)d\sigma (y,z)=\int_{B^{d}(0,1)}\int_{S^{d-1}} F(y,\sqrt{1-|y|^2}z)d\sigma_{d-1} (z) (1-|y|^2)^{\frac{d-2}{2}}\,dy$$
In particular, for any $f,g\in C^{\infty}_{0}(\mathbb{R}^d)$, \begin{equation}
\begin{split}
\mathcal{A}_t(f,g)(x)=&\int_{B^{d}(0,1)}f(x-ty)\int_{S^{d-1}} g(x-t\sqrt{1-|y|^2}z)d\sigma_{d-1} (z) (1-|y|^2)^{\frac{d-2}{2}}\,dy \\
=&\int_{B^{d}(0,1)}f(x-ty)A_{t(\sqrt{1-|y|^2})}(g)(x)(1-|y|^2)^{\frac{d-2}{2}}\,dy
\end{split} \end{equation} where $A_t f(x):=\int_{S^{d-1}}f(x-ty)d\sigma(y)$, and they used that to show that $$\mathcal{M}(f,g)(x)\lesssim Mf(x)\mathcal{S}g(x).$$
If we restrict ourselves to $t\in 2^{\mathbb{Z}}$ in the hope of getting some control by the lacunary linear spherical maximal function $\mathcal{S}_{lac}$, we soon run into trouble since even when $t= 2^{-k}$, with $k\in \mathbb{Z}$, one can observe that in this slicing formula, one needs to take into account spherical averages of $g$ in all scales $2^{-k}\sqrt{1-|y|^2}$ for any $|y|<1$. As a result, we cannot get a larger boundedness region for the lacunary maximal function than what we already have for the full maximal function, and we must instead turn to other methods.
\subsection{Relevant facts about $\mathcal{T}_1$}
\begin{thm}{\cite{IPS}} The triangle averaging operator satisfies the bound $$\mathcal{T}_1:L^{\frac{d+1}{d}}(\mathbb{R}^d)\times L^{\frac{d+1}{d}}(\mathbb{R}^d) \rightarrow L^{s}(\mathbb{R}^d), \text{ for all }s\in [\frac{d+1}{2d},1],\,d\geq 2.$$ Moreover, when the target space is $L^1(\mathbb{R}^d)$ one has a sharp description of the boundedness region, namely $$\mathcal{T}_1: L^{p}(\mathbb{R}^d)\times L^{q}(\mathbb{R}^d)\rightarrow L^{1}(\mathbb{R}^d)$$ if and only if $(\frac{1}{p},\frac{1}{q})$ lies in the convex hull of the points ${(0,1),(1,0)}$ and $(\frac{d}{d+1},\frac{d}{d+1})$. \end{thm}
Again, for H\"older exponents $1/r=1/p+1/q$, with $r\geq 1$, the boundedness of $\mathcal{T}_1$ follows directly from Minkowski's inequality for integrals and H\"older's inequality. Interpolating this with the bound $\mathcal{T}_1:L^{\frac{d+1}{d}}\times L^{\frac{d+1}{d}}\rightarrow L^{\frac{d+1}{2d}} $ above, one has the following corollary, where $\mathcal{R}_1$ is the region defined in (\ref{regionr1}).
\begin{cor} \label{holderboundsforT1}Let $d\geq 2$. Then for all $1\leq p,q \leq \infty$ and $r\in (0,\infty]$ given by the H\"older relation $1/r=1/p+1/q$, and $(1/p,1/q)\in \mathcal{R}_1$, one has that \begin{equation}
\mathcal{T}_1:L^{p}(\mathbb{R}^d)\times L^{q}(\mathbb{R}^d)\rightarrow L^r(\mathbb{R}^d) \text{ is bounded.} \end{equation} \end{cor}
\begin{figure}
\caption{Region of pairs $(1/p,1/q)$ with $(1/p,1/q,1/r)\in\mathcal{R}_1$, which is the region where we know the H\"older bounds for $\mathcal{T}_1$ are satisfied.}
\end{figure}
\begin{rem} It is not clear that the corollary above is sharp. Any bound for $\mathcal{T}_1$ of the form $L^p\times L^{q}\rightarrow L^1$, $1/p+1/q>1$, can be lifted to the H\"older bound $L^p\times L^q\rightarrow L^r$, where $1/r=1/p+1/q$ by using Proposition 4.1 in \cite{IPS}), and the bounds for $\mathcal{T}_1$ arriving in $L^1$ are sharp. So Corollary 6 is the best one can get from the lifting bounds into $L^1$ to H\"older bounds, but there could be some $r<1$ for which a H\"older bound $(p,q,r)$ holds but the bound $(p,q,1)$ is false. In the proposition below we include some necessary conditions for the boundedness of $\mathcal{T}_1$ that follow by adapting some examples in \cite{radonplane}. \end{rem}
\begin{prop}(Necessary conditions for the boundedness of $\mathcal{T}_1$ in $d\geq 2$) \label{necessarytriangle} Let $1\leq p,q\leq \infty$ and $r\in (0,\infty]$. If $$\mathcal{T}_1:L^p\times L^q \rightarrow L^r\text{ is bounded, }$$ then \begin{equation}
\begin{cases}
\frac{1}{r}\leq \frac{1}{p}+\frac{1}{q}\\
\frac{1}{p}+\frac{1}{q}\leq \frac{d}{r}\\
\frac{d}{p}+\frac{1}{q}\leq d-1+\frac{1}{r}\\
\frac{1}{p}+\frac{d}{q}\leq d-1+\frac{1}{r}
\end{cases} \end{equation}
\end{prop} \begin{rem}
Observe that the necessary conditions above do not impose restrictions for H\"older bounds for $\mathcal{T}_1$, that is, for any $p,q\in [1,\infty]$ and $1/r=1/p+1/q$ the conditions above are trivially satisfied. \end{rem}
\begin{proof}[Proof of Proposition \ref{necessarytriangle}]
We will give an example for each necessary condition to demonstrate them in order. First take $f_M=\chi_{B(0,2M)}$ where $M>1$. For all $|x|<M$, $\mathcal{T}_1(f_M,f_M)(x)= 1$, so
$$M^{d/r}\lesssim \|\mathcal{T}_1(f_M,f_M)\|_r\lesssim\|f_M\|_p \|f_M\|_q\lesssim M^{d(\frac{1}{p}+\frac{1}{q})}\,\text{ for all }M>1,$$ which implies $1/r\leq 1/p+1/q$.
Next take $f_{\delta}=\chi_{A_{\delta}}$ where $A_{\delta}=\{x\in \mathbb{R}^d\colon 1-\delta\leq|x|\leq 1+\delta\}$. Then for any $|x|<\delta$, \begin{equation}
\begin{split}
\mathcal{T}_1(f_{\delta}, f_{\delta})(x)=\int_{O(d)}\chi_{A_{\delta}}(x-Ru)\chi_{A_{\delta}}(x-Rv)d\mu(R)=\int_{O(d)}d\mu (R)=1
\end{split} \end{equation}
since $1-\delta<1-|x|\leq |x-Ru|\leq 1+|x|< 1+\delta$, for any $R\in O(d)$ since $|u|=1$.
That gives
$$\delta^{d/r}\lesssim \|\mathcal{T}_1(f_{\delta},f_{\delta})\|_r\lesssim\|f_{\delta}\|_p \|f_{\delta}\|_q=|A_{\delta}|^{\frac{1}{p}+\frac{1}{q}}\sim{\delta}^{\frac{1}{p}+\frac{1}{q}}\,\text{ for all }\delta\ll 1,$$
so $1/p+1/q\leq d/r$.
Take $f_{\delta}=\chi_{B(0,\delta)}$ and $g_{\delta}=\chi_{A_{\delta}}$. Suppose $||x|-1|\leq \frac{1}{2}\delta$, then \begin{equation}
\begin{split} \mathcal{T}_1(f_{\delta},g_{\delta})(x)=&\int_{O(d)} \chi_{B(0,\delta)}(x-Ru)\chi_{A_{\delta}}(x-Rv)d\mu(R)\\
\geq& \int_{O(d)} \chi_{|x-Ru|< \delta} d\mu(R)
\end{split} \end{equation}
since for any $|x-Ru|<\delta$, one has $x-Rv\in A_{\delta}$. Therefore \begin{equation}
\begin{split} \mathcal{T}_1(f_{\delta},g_{\delta})(x)\geq \int_{O(d)} \chi_{B(x,\delta)}(Ru)d\mu(R)=\int_{S^{d-1}} \chi_{B(x,\delta)}(y)d\sigma(y)\gtrsim \delta^{d-1}
\end{split} \end{equation} and from
$$\delta^{d-1+1/r}\lesssim \|\mathcal{T}_1(f_{\delta},g_{\delta})\|_r\lesssim \|\chi_{B(0,\delta)}\|_p\|g_{A_{\delta}}\|_q=\delta^{d/p+1/q}$$ we get $d/p+1/q\leq d-1+1/r$. The last necessary condition follows from the symmetry $\mathcal{T}_1(f,g)=\mathcal{T}_1(g,f)$. \end{proof}
\subsection{Bilinear multiplier boundedness criteria}
For a multiplier $m(\xi, \eta)$ in $\mathbb{R}^{2d}$ we denote by $T_m$ the bilinear operator given by $$T_m(f,g)(x)= \int_{\mathbb{R}^{2d}} \hat{f}(\xi)\hat{g}(\eta)m(\xi, \eta)e^{2\pi i x\cdot(\xi +\eta)}d\xi d\eta.$$ These will appear naturally throughout the paper as Fourier transforms of single-scale bilinear averaging operators.
\begin{thm}[\cite{GHS}]\label{ghscriteria} Let $1\leq q<4$ and set $M_q=\left\lfloor \frac{2d}{4-q} \right\rfloor+1$. Let $m(\xi, \eta)$ be a function in $L^{q}(\mathbb{R}^{2d})\cap \mathcal{C}^{M_q}(\mathbb{R}^{2d})$ satisfying \begin{equation}
\|\partial^{\alpha}m\|_{L^{\infty}}\leq C_0<\infty
\text{ for all multiindices }\alpha \text{ with }|\alpha|\leq M_q. \end{equation} Then there is a constant $A$ depending on $d$ and $q$ such that the associated operator $T_m$ with multiplier $m$ satisfies \begin{equation}
\|T_m\|_{L^2\times L^2\rightarrow L^1}\leq A C_0^{1-\frac{q}{4}}\|m\|_{L^q}^{q/4} \end{equation}
\end{thm}
As an immediate corollary of the above theorem, one has the following. \begin{cor}\label{corl2l2l1} Let $m(\xi, \eta )$ be a smooth compactly supported function in $\mathbb{R}^{2d}$ such that
$$\|\partial^{\alpha}m\|_{L^{\infty}}\leq C_0\,\text{ for all multi-indices } \alpha, \text{ with } |\alpha|\leq \left\lfloor \frac{2d}{3}\right\rfloor+1$$ Then there exists a constant $A$ depending on $d$ such that the associated operator $T_m$ with multiplier $m$ satisfies
$$\|T_m\|_{L^{2}\times L^{2}\rightarrow L^{1}}\leq A C_0|\text{supp}(m)|^{1/4}.$$
\end{cor}
\section{Decomposition of the operator and bounds for the pieces in the H\"older range}\label{sectiondecayforpieces}
We start by describing the decomposition used in \cite{HHY}. Choose a radial function $\varphi\in \mathcal{S}(\mathbb{R}^d)$ such that \begin{equation}\label{phi def} \hat{\varphi}(\xi)= \begin{cases}
1, \text{ if }|\xi|\leq 1\\
0, \text{ if }|\xi|\geq 2
\end{cases}. \end{equation}
Let $\hat{\psi}(\xi)=\hat{\varphi}(\xi)-\hat{\varphi}(2\xi)$, which is supported in $\{1/2<|\xi|<2\}$. Then \begin{equation}\label{psi def} \hat{\varphi}(\xi)+\sum_{j=1}^{\infty} \hat{\psi}(2^{-j}\xi)\equiv 1. \end{equation}
Define for all $i,j\geq 1$. \begin{equation} \mathcal{A}_1^{i,j}(f,g)(x)=\int_{\mathbb{R}^{2d}}\hat{f}(\xi)\hat{g}(\eta)\hat{\sigma}(\xi, \eta)\hat{\psi}(2^{-i}\xi)\hat{\psi}(2^{-j}\eta)e^{2\pi i x\cdot (\xi+ \eta)}d\xi d\eta \end{equation}
If $i=0$ replace $\hat{\psi}(2^{-i}\xi)$ by $\hat{\varphi}(\xi)$ in the expression above, and similarly if $j=0$. Then one has
$$\mathcal{A}_1(f,g)(x)=\sum_{i,j\geq 0} \mathcal{A}_1^{i,j}(f,g)(x).$$
\begin{prop}\label{nodecay} Let $p,q\in [1,\infty]$ and $r\in (0,\infty]$, such that $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. Then there exists $C>0$ such that for all $i,j\geq 0$ \begin{equation}
\|\mathcal{A}^{i,j}_1(f,g)\|_{r}\leq C\|f\|_p\|g\|_q. \end{equation} \end{prop}
\begin{proof} Denote by $\psi_{i}(x)=2^{id}\psi(2^{i}x)$. Assume $i,j\geq 1$. One can easily see that \begin{equation}
\|\mathcal{A}^{i,j}_1(f,g)\|_{r}=\|\mathcal{A}_1(\psi_i*f,\psi_j*g)\|_{r}. \end{equation} Now using Corollary \ref{holderboundsforA1}, \begin{equation} \begin{split}
\|\mathcal{A}_1(\psi_i*f,\psi_j*g)\|_{r}\leq &C \|\psi_i*f\|_p \|\psi_j*g\|_q\\
\leq &C \|f\|_p\|g\|_q \end{split} \end{equation}
where we used Young's inequality and the fact that $\|\psi_i\|_1=\|\psi\|_1\lesssim 1$. The case where $i=0$ or $j=0$ follows similarly. \end{proof}
\begin{prop}\label{keyestimate} Let $d\geq 2$. Then for every $i,j\geq 1$ \begin{equation}
\|\mathcal{A}_1^{i,j}(f,g)\|_1\lesssim 2^{-(i+j)\frac{d-1}{2}}\|f\|_2\|g\|_2 \end{equation} \end{prop}
\begin{rem} The exact value of the negative power of $2^{i+j}$ that we get in the proposition above is not very important to run our argument later, so one could also use the estimates for the pieces of $\tilde{\mathcal{M}}$ from \cite{HHY}, or the bilinear multiplier theorem in \cite{GHS} to get the proposition above with a potentially different negative power. The nice thing about the proof below is that it reduces things to spherical averages in $\mathbb{R}^d$ and it can be adapted to more general averaging operators, like the ones in the last section of this paper. \end{rem}
\begin{proof}
Let us use the notation $D_t(f)(x)=f(tx)$. Also we denote $$A_{t}(f)(x)=\int_{S^{d-1}}f(x-ty)d\sigma(y)$$ as the linear spherical average in $\mathbb{R}^d$ at scale $t$. Observe that using the same coarea formula exploited in \cite{JL}, followed by polar coordinates in the ball $B^{d}(0,1)$ one gets that \begin{equation*}
\begin{split}
\mathcal{A}_1(f,g)(x)=&\int_{B^{d}(0,1)} f(x-y) \int_{S^{d-1}}g(x-\sqrt{1-|y|^2}z)d\sigma(z) (1-|y|^2)^{\frac{d-2}{2}}dy\\
=&\int_{0}^{1} \lambda^{d-1}(1-\lambda^2)^{\frac{d-2}{2}}\left\{\int_{S^{d-1}} f(x-\lambda w)d\sigma(w)\right\}\left\{\int_{S^{d-1}}g(x-\sqrt{1-\lambda^2}z)d\sigma(z)\right\} d\lambda\\
=&\int_{0}^{1}\lambda^{d-1}(1-\lambda^2)^{\frac{d-2}{2}}A_{\lambda}(f)(x)A_{\sqrt{1-\lambda^2}}(g)(x)d\lambda
\end{split} \end{equation*}
Denote $\langle \lambda\rangle=\sqrt{1-\lambda^2} $, and $f^{i}=f*\psi_i$, where $\psi_i$ is like in Proposition \ref{nodecay}. Then $$\mathcal{A}_1^{i,j}(f,g)(x)=\mathcal{A}_1(f^{i},g^{j})(x)=\int_{0}^{1}\lambda^{d-1}\langle\lambda\rangle^{d-2}A_{\lambda}(f^{i})(x)A_{\langle\lambda\rangle}(g^{j})(x)d\lambda$$
Using Minkowski's inequality for integrals followed by H\"older's inequality gives \begin{equation}
\begin{split}
\|\mathcal{A}_1(f^{i},g^{j})\|_1\leq&\int_{0}^{1} \lambda^{d-1}\langle\lambda\rangle^{d-2} \|A_{\lambda}(f^{i})\|_2\|A_{\langle\lambda\rangle}(g^{j})\|_2 \,d\lambda.
\end{split} \end{equation}
For any $\lambda>0$, \begin{equation}\label{linearpiecedecay}
\begin{split}
\|A_{\lambda}(f^{i})\|_2=&\|f*\psi_i*d\sigma_{\lambda}\|_2\\
=&\|\hat{f}(\xi)\hat{\psi}(2^{-i}\xi)\hat{\sigma}(\lambda\xi)\|_2\\
\lesssim & (\lambda 2^{i})^{-\frac{d-1}{2}}\|f\|_2
\end{split} \end{equation}
Therefore \begin{equation}
\begin{split}
\|\mathcal{A}_1(f^{i},g^{j})\|_1\lesssim & 2^{-(i+j)\frac{d-1}{2}}\|f\|_2\|g\|_2\int_{0}^{1} \lambda^{d-1}\langle\lambda\rangle^{d-2} \lambda^{-\frac{d-1}{2}} \langle\lambda\rangle^{-\frac{d-1}{2}}d\lambda\\
=& 2^{-(i+j)\frac{d-1}{2}}\|f\|_2\|g\|_2\int_{0}^{1} \lambda^{\frac{d-1}{2}}\langle\lambda\rangle^{\frac{d-3}{2}} d\lambda\lesssim 2^{-(i+j)\frac{d-1}{2}}\|f\|_2\|g\|_2.
\end{split} \end{equation} \end{proof}
The following corollary will have a key role in the proof of the bounds for $\mathcal{M}_{lac}$.
\begin{cor}\label{decayforpieces}
Assume $d\geq 2$. Let $p,q\in (1,\infty)$ and $r\in (0,\infty)$ given by $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. Then there exists $\delta(p,q,r,d)$ such that
\begin{equation}\label{decayestimate}
\|\mathcal{A}_{1}^{i,j}(f,g)\|_{r}\lesssim 2^{-(i+j)\delta}\|f\|_p\|g\|_q.
\end{equation}
Moreover, if $d\geq 5$ then for all $q\in (1,\infty)$ and $1/r=1+1/q$,
\begin{equation}\label{dg5}
\begin{split}
\|\mathcal{A}_{1}^{i,j}(f,g)\|_{r}\lesssim 2^{-(i+j)\delta}\|f\|_1\|g\|_q. \\
\|\mathcal{A}_{1}^{i,j}(f,g)\|_{r}\lesssim 2^{-(i+j)\delta}\|f\|_q\|g\|_1.
\end{split}
\end{equation}
\end{cor}
\begin{proof}
Estimate (\ref{decayestimate}) follows immediately from interpolation of Propositions \ref{nodecay} and \ref{keyestimate}.
To prove the last part of this corollary, we recall that it was shown in \cite{HHY} that for
$$\tilde{\mathcal{M}}^{ij}(f,g)(x)=\sup_{t\in[1,2]}|\int_{\mathbb{R}^{2d}} \hat{f}(\xi)\hat{g}(\eta)\hat{\psi}(2^{-i}\xi)\hat{\psi}(2^{-j}\eta)\hat{\sigma}(t\xi,t\eta)e^{2\pi i x\cdot(\xi+\eta)}d\xi d\eta|,$$ one has
$$\|\tilde{\mathcal{M}}^{i,j}(f,g)\|_{L^{2/3}}\lesssim (\max\{2^{i},2^{j}\})^{-\frac{d-4}{2}}\|f\|_{p_1}\|g\|_{p_2}$$
for all $1\leq p_1,p_2\leq 2 $ with $3/2=1/p_1+1/p_2$.
In particular, since $\text{max}\{i,j\}\geq \frac{i+j}{2}$
\begin{equation}
\begin{split}
\|\mathcal{A}_1^{i,j}(f,g)\|_{L^{2/3}}&\leq \|\tilde{\mathcal{M}}^{ij}(f,g)\|_{L^{2/3}}\lesssim 2^{-(i+j)\frac{d-4}{4}}\|f\|_{1}\|g\|_{2}\\
\|\mathcal{A}_1^{i,j}(f,g)\|_{L^{2/3}}&\leq \|\tilde{\mathcal{M}}^{ij}(f,g)\|_{L^{2/3}}\lesssim 2^{-(i+j)\frac{d-4}{4}}\|f\|_{2}\|g\|_{1}\\
\end{split}
\end{equation}
Then interpolation of these bounds with the bounds in Proposition \ref{nodecay} give the estimates in \eqref{dg5}. \end{proof} \begin{rem}
One could think that the second part of the estimate \eqref{dg5} could potentially allow us to include the missing boundary pieces in Theorem \ref{boundsspherical} for $d\geq 5$. Unfortunately, with the methods used in this paper one runs into some technical issues trying to do that. \end{rem}
\section{Proof of the bounds for the lacunary bilinear spherical maximal operator}\label{prooffirstthm}
\begin{proof}[Proof of Theorem \ref{boundsspherical}]
$\mathcal{M}_{lac}$ satisfies a weak type bound $L^1\times L^{\infty}\rightarrow L^{1,\infty}$ (from \cite{JL} since $\mathcal{M}_{lac}$ is dominated by $\mathcal{M}$). To see that $\mathcal{M}_{lac}$ can not be bounded from $L^1\times L^{\infty} \rightarrow L^{1}$ one can adapt the example from \cite{JL} to show that for $g\equiv 1$ and $0\leq f\in L^1$ $$\sup_{l\in \mathbb{Z}}\mathcal{A}_{2^{-l}}(f,g)(x)\geq C \sup_{l\in \mathbb{Z}} \frac{1}{2^{-ld}}\int_{B(x,2^{-l})}f(y)dy=:M^{lac}f(x),$$
so the linear spherical maximal function $M^{lac}$ would be bounded in $L^1$ which is false (for any $f\in L^1$ which is not identically zero, $\frac{1}{|x|^d}\lesssim M^{lac}f(x)$ for $|x|>>1$, so $M^{lac}f\notin L^1$).
The case $p=\infty$ or $q=\infty$ follows immediately from the known bounds for $\mathcal{M}$ as stated in Theorem \ref{boundsjl}. So we can assume $p,q<\infty$.
The proof basically follows from Corollary \ref{decayforpieces} and the ideas in section 4 of \cite{HHY}. For completeness and to point out some obstructions in including the missing pieces of the boundary that force $d\geq 5$, we will write the details below.
Let
\begin{equation}
\mathcal{A}_{2^{-l}}(f,g)(x)=\int_{\mathbb{R}^{2d}} \hat{f}(\xi)\hat{g}(\eta) \hat{\sigma}_{2d-1}(2^{-l}\xi,2^{-l}\eta)e^{2\pi i x\cdot (\xi+\eta)}d\xi d\eta.
\end{equation}
One can break down $\mathcal{A}_{2^{-l}}$ into low/low, low/high, high/low and high/high frequencies as follows
\begin{equation}
\mathcal{A}_{2^{-l}}(f,g)(x)=-\mathcal{A}_{2^{-l}}^{00}(f,g)(x)+\mathcal{A}_{2^{-l}}^{0\infty}(f,g)(x)+\mathcal{A}_{2^{-l}}^{\infty 0}(f,g)(x)+\sum_{i,j\geq 1}\mathcal{A}_{2^{-l}}^{i,j}(f,g)(x),
\end{equation}
where
\begin{equation}\label{sphere HL decomp}
\begin{split}
\mathcal{A}_{2^{-l}}^{00}(f,g)(x)=&\int_{\mathbb{R}^{2d}}\hat{f}(\xi)\hat{g}(\eta)\hat{\sigma}_{2d-1}(2^{-l}\xi,2^{-l}\eta)\hat{\varphi}(2^{-l}\xi)\hat{\varphi}(2^{-l}\eta)e^{2\pi i x\cdot (\xi+\eta)}d\xi d\eta, \\
\mathcal{A}_{2^{-l}}^{0\infty}(f,g)(x)=&\int_{\mathbb{R}^{2d}}\hat{f}(\xi)\hat{g}(\eta)\hat{\sigma}_{2d-1}(2^{-l}\xi,2^{-l}\eta)\hat{\varphi}(2^{-l}\xi)e^{2\pi i x\cdot (\xi+\eta)}d\xi d\eta, \\
\mathcal{A}_{2^{-l}}^{\infty 0}(f,g)(x)=&\int_{\mathbb{R}^{2d}}\hat{f}(\xi)\hat{g}(\eta)\hat{\sigma}_{2d-1}(2^{-l}\xi,2^{-l}\eta)\hat{\varphi}(2^{-l}\eta)e^{2\pi i x\cdot (\xi+\eta)}d\xi d\eta, \\
\mathcal{A}_{2^{-l}}^{i,j}(f,g)(x)=&\int_{\mathbb{R}^{2d}}\hat{f}(\xi)\hat{g}(\eta)\hat{\sigma}_{2d-1}(2^{-l}\xi,2^{-l}\eta)\hat{\psi}(2^{-i-l}\xi)\hat{\psi}(2^{-j-l}\eta)e^{2\pi i x\cdot (\xi+\eta)}d\xi d\eta.
\end{split}
\end{equation}
Denote $N(i,j;p,q,r)=\|\mathcal{A}_{1}^{i,j}\|_{L^p\times L^q\rightarrow L^r}$, which we know from Corollary \ref{decayforpieces} satisfies
$$N(i,j;p,q,r)\lesssim C(p,q,r)2^{-(i+j)\delta}$$
for some $\delta=\delta(d,p,q,r)>0.$
Another important fact is that for all $l\in \mathbb{Z}$, and H\"older exponents $1/r=1/p+1/q$, one has
\begin{equation}\label{normij}
\|\mathcal{A}_{2^{-l}}^{i,j}\|_{L^p\times L^q\rightarrow L^r}=\|\mathcal{A}_{1}^{i,j}\|_{L^p\times L^q\rightarrow L^r}=:N(i,j;p,q,r).
\end{equation} which as observed in \cite{HHY} follows from the fact that \begin{equation}
\mathcal{A}_{2^{-l}}^{i,j}(f,g)(x)=\mathcal{A}_{1}^{i,j}(D_{2^{-l}}f, D_{2^{-l}}g)(2^lx), \text{ for all }i,j\geq 0, \end{equation} where $D_{2^{-l}}f(x)=f(2^{-l}x)$.
\begin{prop}\label{lowfrequency} For $d\geq 2$, one has that \begin{equation} \begin{split}
\mathcal{M}^{LL}_{lac}(f,g)(x):=\sup_{l\in\mathbb{Z}}|\mathcal{A}_{2^{-l}}^{00}(f,g)(x)|&\lesssim Mf(x)Mg(x),\\
\mathcal{M}^{LH}_{lac}(f,g)(x):= \sup_{l\in\mathbb{Z}} |\mathcal{A}_{2^{-l}}^{0\infty}(f,g)(x)|&\lesssim Mf(x)\sup_{t\in 2^\mathbb{Z}} \left(\int_{S^{2d-1}}|g(x-tz)|d\sigma_{2d-1}(y,z)\right),\\
\mathcal{M}^{HL}_{lac}(f,g)(x):= \sup_{l\in\mathbb{Z}}| \mathcal{A}_{2^{-l}}^{\infty 0}(f,g)(x)|&\lesssim Mg(x)\sup_{t\in 2^\mathbb{Z}}\left(\int_{S^{2d-1}}|f(x-ty)|d\sigma_{2d-1}(y,z)\right), \end{split} \end{equation} where $M$ stands for the Hardy-Littlewood maximal operator. Since \begin{equation}
\sup_{t>0} \left(\int_{S^{2d-1}}|g(x-tz)|d\sigma_{2d-1}(y,z)\right)\lesssim Mg(x), \end{equation} one has that \begin{equation} [\mathcal{M}^{LL}_{lac}+\mathcal{M}^{LH}_{lac}+\mathcal{M}^{HL}_{lac}](f,g)(x)\lesssim Mf(x)Mg(x). \end{equation} \end{prop}
Using H\"older's inequality and the boundedness of $M$ in $L^p$ for all $1<p\leq \infty,$ one gets the following corollary. \begin{cor}\label{non HH sphere bounds} Let $d\geq2$. Then
\begin{equation}
\|\mathcal{M}^{LL}_{lac}+\mathcal{M}^{LH}_{lac}+\mathcal{M}^{HL}_{lac}\|_{L^p\times L^q\rightarrow L^r}<\infty \end{equation} for all $p,q\in (1,\infty]$, $r\in (0,\infty]$ and $1/r=1/p+1/q$. \end{cor}
\begin{proof}[Proof of Proposition \ref{lowfrequency}]
Let us check the inequality for $\mathcal{M}_{lac}^{LH}$, since the others follow similarly. We follow the arguments in \cite{HHY} most closely.
Since $(f*\varphi)(x-y)\lesssim Mf(x)$ for all $|y|\leq 1$, one has
$$|\mathcal{A}_1^{0\infty}(f,g)(x)|\leq \int_{S^{2d-1}}|f*\varphi(x-y)||g(x-z)|d\sigma(y,z)\lesssim Mf(x)\cdot \int_{S^{2d-1}}|g(x-z)|d\sigma(y,z).$$
Therefore \begin{equation*} \begin{split}
\sup_{l\in \mathbb{Z}}|\mathcal{A}_{2^{-l}}^{0\infty}(f,g)(x)|=&\sup_{l\in \mathbb{Z}}|\mathcal{A}_{1}^{0\infty}(D_{2^{-l}}f,D_{2^{-l}}g)(2^lx)|\\
\lesssim&M(D_{2^{-l}}f)(2^lx)\cdot \sup_{l\in\mathbb{Z}}\int_{S^{2d-1}}|D_{2^{-l}}g(2^lx-z)|d\sigma(y,z)\\
\lesssim&Mf(x)\cdot\sup_{l\in\mathbb{Z}}\int_{S^{2d-1}}|g(x-2^{-l}z)|d\sigma(y,z). \end{split} \end{equation*}
Then one just need to observe that by the coarea formula used in \cite{JL}, for any $t>0$ one has \begin{equation*} \begin{split}
\int_{S^{2d-1}} |g(x-tz)|d\sigma(y,z)=&\int_{B^d(0,1)}|g(x-tz)||S^{d-1}|(1-|z|^2)^{\frac{d-2}{2}}dz\\
\lesssim &\int_{B^d(0,1)}|g(x-tz)|dz\lesssim Mg(x). \end{split} \end{equation*}
\end{proof}
Define $\mathbb{A}_{\mathcal{N}}(f,g)(x)=\sup_{|l|\leq \mathcal{N}}|\mathcal{A}_{2^{-l}}(f,g)(x)|$
Here we observe that since $\|\mathcal{A}_{2^{-l}}\|_{L^p\times L^q\rightarrow L^r}=\|\mathcal{A}_{1}\|_{L^p\times L^q\rightarrow L^r}<\infty$, it is clear that for all $\mathcal{N}\in \mathbb{N}$,
$$A_{\mathcal{N}}(p,q,r):=\|\mathbb{A}_{\mathcal{N}}\|_{L^p\times L^q\rightarrow L^r}<\infty.$$
Our goal is to show that $A_{\mathcal{N}}(p,q,r)$ is bounded by a constant independent of $\mathcal{N}$. Moreover, in view of Corollary \ref{non HH sphere bounds}, it suffices to prove this for the high-high frequency part of the operator. Thus, for every $i,j\geq 1$, define the vector valued operator \begin{equation}
\mathbb{A}_{\mathcal{N}}^{i,j}:\{f_l\}_{|l|\leq \mathcal{N}}\times \{g_l\}_{|l|\leq \mathcal{N}}\longmapsto \{\mathcal{A}_{2^{-l}}(f_l*\psi_{i+l},g_l*\psi_{j+l})(x)\}_{|l|\leq \mathcal{N}}. \end{equation}
By the definition of $A_{\mathcal{N}}(p,q,r)$ and the same computation as in \cite{HHY} (page 430), if $p,q>1$ then \begin{equation}\label{firstbound}
\|\mathbb{A}_{\mathcal{N}}^{i,j}(\{f_l\}_{|l|\leq \mathcal{N}}\times \{g_l\}_{|l|\leq \mathcal{N}})\|_{L^r(\ell^{\infty})}\lesssim A_{\mathcal{N}}(p,q,r) \|\{f_l\}_{|l|\leq \mathcal{N}}\|_{L^{p}(\ell^{\infty})} \| \{g_l\}_{|l|\leq \mathcal{N}}\|_{L^{q}(\ell^{\infty})} \end{equation}
The second computation in \cite{HHY}, which exploits the equality (\ref{normij}), gives for $p,q\in (1,\infty)$ and $1/r=1/p+1/q$ \begin{equation}\label{secondbound} \begin{split}
\|\mathbb{A}_{\mathcal{N}}^{i,j}(\{f_l\}_{|l|\leq \mathcal{N}}\times \{g_l\}_{|l|\leq \mathcal{N}})\|_{L^r(\ell^{r})}\lesssim& N(i,j;p,q,r) \|\{f_l\}_{|l|\leq \mathcal{N}}\|_{L^{p}(\ell^{p})} \| \{g_l\}_{|l|\leq \mathcal{N}}\|_{L^{q}(\ell^{q})}\\
\lesssim& N(i,j;p,q,r) \|\{f_l\}_{|l|\leq \mathcal{N}}\|_{L^{p}(\ell^{1})} \| \{g_l\}_{|l|\leq \mathcal{N}}\|_{L^{q}(\ell^{1})} \end{split} \end{equation}
Interpolation of the bounds in (\ref{firstbound}) and (\ref{secondbound}) will then give for any $$p,q\in (1,\infty), \text{ and }1/r=1/p+1/q $$ one has \begin{equation}\label{thirdbound} \begin{split}
\|\mathbb{A}_{\mathcal{N}}^{i,j}(\{f_l\}_{|l|\leq \mathcal{N}}\times &\{g_l\}_{|l|\leq \mathcal{N}})\|_{L^r(\ell^{2r})}\\
\lesssim &A_{\mathcal{N}}(p,q,r)^{1/2} N(i,j;p,q,r)^{1/2} \|\{f_l\}_{|l|\leq \mathcal{N}}\|_{L^{p}(\ell^{2})} \| \{g_l\}_{|l|\leq \mathcal{N}}\|_{L^{q}(\ell^{2})}. \end{split} \end{equation}
Using bound (\ref{thirdbound}) and the Littlewood-Paley Theorem (Theorem 6.1.2 in \cite{classicalgrafakos}), for any $p,q>1$ \begin{equation}
\begin{split}
\left\|\sup_{|l|\leq \mathcal{N}} |\mathcal{A}_{2^{-l}}^{i,j}(f,g)|\right\|_{L^{r}(\mathbb{R}^d)}=& \left\|\sup_{|l|\leq \mathcal{N}} |\mathcal{A}_{2^{-l}}(f*\psi_{i+l},g*\psi_{j+l})|\right\|_{L^r(\mathbb{R}^d)}\\
=&\left\|\mathbb{A}_{\mathcal{N}}^{i,j} (\{f*\psi_{i+l}\}_{|l|\leq \mathcal{N}}\times \{g*\psi_{j+l}\}_{|l|\leq \mathcal{N}})\right\|_{L^r(\ell^{\infty})}\\
\le&\left\|\mathbb{A}_{\mathcal{N}}^{i,j} (\{f*\psi_{i+l}\}_{|l|\leq \mathcal{N}}\times \{g*\psi_{j+l}\}_{|l|\leq \mathcal{N}})\right\|_{L^r(\ell^{2r})}\\ \lesssim& A_{\mathcal{N}}(p,q,r)^{1/2}N(i,j;p,q,r)^{1/2}\\
&\,\,\left\|\{f*\psi_{i+l}\}_{|l|\leq \mathcal{N}}\right\|_{L^p(\ell^2)}\left\|\{g*\psi_{j+l}\}_{|l|\leq \mathcal{N}}\right\|_{L^q(\ell^2)}\\
\lesssim&A_{\mathcal{N}}(p,q,r)^{1/2}N(i,j;p,q,r)^{1/2}\|f\|_p\|g\|_q
\end{split} \end{equation}
\begin{rem} The argument fails for $p=1$ since with Littlewood-Paley theory we can only control $\|\{f*\psi_{i+l}\}_{|l|\leq \mathcal{N}}\|_{L^{1,\infty}(\ell^2)}\lesssim \|f\|_1$ but not $\|\{f*\psi_{i+l}\}_{|l|\leq \mathcal{N}}\|_{L^{1}(\ell^2)}$. Even if one tries to use this weak type control of the Littlewood-Paley pieces, one runs into the issue there are no bounds of the form
$\|\mathcal{A}_{1}^{i,j}(f,g)\|_{L^{r,\infty}}\leq N(i,j;1,q,r)\|f\|_{L^{1,\infty}} \|g\|_q$. \end{rem}
To finish the proof for the interior points $p,q\in (1,\infty)$, we split into two cases for $r$. When $r<1$, we use that $\|\cdot\|_{r}^{r}$ satisfies the triangle inequality, and for $r\geq 1$, we use that $\|\cdot\|_r$ is a norm.
\textbf{Case $r<1$}: \begin{equation}
\begin{split}
A_{\mathcal{N}}(p,q,r)^r=&\|\sup_{|l|\leq \mathcal{N}}|\mathcal{A}_{2^{-l}}|\|^r_{L^p\times L^q\rightarrow L^r}\\
\leq&\|\mathcal{M}_{lac}^{LL}+\mathcal{M}_{lac}^{LH}+\mathcal{M}_{lac}^{HL}\|_{L^p\times L^q\rightarrow L^r}^{r}+\|\sum_{i,j\geq 1} \sup_{|l|\leq \mathcal{N}}|\mathcal{A}_{2^{-l}}^{i,j}|\|_{L^p\times L^q\rightarrow L^r}^r\\
\lesssim &1+\sum_{i,j\geq 1} \|\sup_{|l|\leq \mathcal{N}}|\mathcal{A}_{2^{-l}}^{i,j}|\|_{L^p\times L^q\rightarrow L^r}^r \\ \lesssim & 1+A_{\mathcal{N}}(p,q,r)^{r/2}\sum_{i,j\geq 1}N(i,j;p,q,r)^{r/2} \end{split} \end{equation} Hence, we deduce \begin{equation}
A_{\mathcal{N}}(p,q,r)\lesssim 1 +\left\{\sum_{i,j\geq 1} N(i,j;p,q,r)^{r/2}\right\}^{2/r}. \end{equation}
By Corollary \ref{decayforpieces} one has $$\sum_{i,j\geq 1} N(i,j;p,q,r)^{r/2}\lesssim \sum_{i,j\geq 1} 2^{-(i+j)\delta r/2}\lesssim 1.$$
The case $r\geq 1$ is similar but we use the triangle inequality for the norm $\|\cdot\|_r$ to care of the sum in $i,j$.
Therefore for any $p,q\in (1,\infty)$ and $1/r=1/p+1/q$ $$\sup_{\mathcal{N\in \mathbb{N}}} A_{\mathcal{N}}(p,q,r)\leq C$$ and this finishes the proof.
\end{proof}
\section{Proof of the bounds for the lacunary triangle averaging maximal operator}\label{proofsecondthm}
In analogy with what we had for the bilinear spherical averaging operator define \begin{equation}
\begin{split}
\mathcal{T}_{l}^{00}(f,g)(x)=&\int_{\mathbb{R}^{2d}}\hat{f}(\xi)\hat{g}(\eta)\hat{\mu}(2^{-l}\xi,2^{-l}\eta)\hat{\varphi}(2^{-l}\xi)\hat{\varphi}(2^{-l}\eta)e^{2\pi i x\cdot (\xi+\eta)}d\xi d\eta \\
\mathcal{T}_{l}^{0\infty}(f,g)(x)=&\int_{\mathbb{R}^{2d}}\hat{f}(\xi)\hat{g}(\eta)\hat{\mu}(2^{-l}\xi,2^{-l}\eta)\hat{\varphi}(2^{-l}\xi)e^{2\pi i x\cdot (\xi+\eta)}d\xi d\eta \\
\mathcal{T}_{l}^{\infty 0}(f,g)(x)=&\int_{\mathbb{R}^{2d}}\hat{f}(\xi)\hat{g}(\eta)\hat{\mu}(2^{-l}\xi,2^{-l}\eta)\hat{\varphi}(2^{-l}\eta)e^{2\pi i x\cdot (\xi+\eta)}d\xi d\eta \\
\mathcal{T}_{l}^{i,j}(f,g)(x)=&\int_{\mathbb{R}^{2d}}\hat{f}(\xi)\hat{g}(\eta)\hat{\mu}(2^{-l}\xi,2^{-l}\eta)\hat{\psi}(2^{-i-l}\xi)\hat{\psi}(2^{-j-l}\eta)e^{2\pi i x\cdot (\xi+\eta)}d\xi d\eta
\end{split}
\end{equation} and let $\mathcal{T}_{lac}^{LL},\,\mathcal{T}_{lac}^{LH},\,\mathcal{T}_{lac}^{HL}$ be
\begin{equation} \begin{split}
\mathcal{T}^{LL}_{lac}(f,g)(x):=&\sup_{l\in\mathbb{Z}}|\mathcal{T}_{l}^{00}(f,g)(x)|\\
\mathcal{T}^{LH}_{lac}(f,g)(x):=&\sup_{l\in\mathbb{Z}} |\mathcal{T}_{l}^{0\infty}(f,g)(x)|\\
\mathcal{T}^{HL}_{lac}(f,g)(x):=&\sup_{l\in\mathbb{Z}} |\mathcal{T}_{l}^{\infty 0}(f,g)(x)| \end{split} \end{equation}
Recall that we are assuming $d\geq 2$ and $\mathcal{R}_1$ is the region inside $[0,1]\times [0,1]$ given by the closure of the points $(0,0),\,(0,1),\,(1,0)$ and $(\frac{d}{d+1},\frac{d}{d+1})$.
With similar arguments as the lacunary bilinear maximal function the proof will reduce to some key ingredients. First we will need to guarantee that the low-low, low-high, and high-low parts of the operator are bounded. Secondly, it is immediate that the analogue of the estimate in Proposition \ref{nodecay} holds for the pieces of the triangle averaging operator $\mathcal{T}_1$ for exponents $(1/p,1/q)\in \mathcal{R}_1$, where we know $\mathcal{T}_1$ is bounded by Corollary \ref{holderboundsforT1}. We state this as a proposition.
\begin{prop}\label{nodecaytriangle} Let $p,q\in [1,\infty]$ and $r\in (0,\infty]$, such that $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. If $(1/p,1,q)\in \mathcal{R}_1$, then there exists $C>0$ such that for all $i,j\geq 0$ \begin{equation}
\|\mathcal{T}^{i,j}_1(f,g)\|_{r}\leq C\|f\|_p\|g\|_q. \end{equation} \end{prop}
Thirdly, if we prove a decay bound for $\mathcal{T}^{i,j}$ in the triple $(p,q,r)=(2,2,1)$, then by interpolation with the bound in Proposition \ref{nodecaytriangle} we will have decay in $i,j$ throughout the boundedness region $\mathcal{R}_1$. Hence, we still need to the control of the low-low, low-high, high-low terms and establish the decay bound at the exponent $ (p,q,r)=(2,2,1)$.
\begin{prop}[Controlling LL, LH, HL parts]\label{lowtriangle}
For any $d\geq 2$, one has
\begin{equation} [\mathcal{T}_{lac}^{LL}+\mathcal{T}_{lac}^{LH}+\mathcal{T}_{lac}^{HL}](f,g)(x)\lesssim Mf(x)\mathcal{S}_{lac}(g)(x)
\end{equation}
where $\mathcal{S}_{lac}$ stands for the lacunary spherical maximal function. \end{prop}
\begin{proof} For any $l\in \mathbb{Z}$, \begin{equation}\label{trianglelhatl}
\begin{split}
\mathcal{T}_l^{0\infty}(f,g)(x)=&\int_{\mathbb{R}^{2d}} \hat{f}(\xi)\hat{g}(\eta) \hat{\varphi}(2^{-l}\xi)\hat{\sigma}(2^{-l}\xi,2^{-l}\eta)e^{2\pi ix\cdot(\xi+\eta)}d\xi d\eta\\
=&\mathcal{T}_{0}^{0\infty}(D_{2^{-l}}f,D_{2^{-l}}g)(2^lx)
\end{split} \end{equation} by a simple change of variables. So let us look at the case $l=0$. \begin{equation*} \begin{split}
|\mathcal{T}_{0}^{0\infty}(f,g)(x)|\leq& \int_{\mathcal{I}} |f*\varphi(x-y)||g(x-z)|d\mu(y,z)\\
\lesssim &Mf(x)\int_{\mathcal{I}} |g(x-z)|d\mu(y,z) \end{split} \end{equation*}
Recall that (see \cite{IPS}) fixing $u,v\in S^{d-1}$ with $|u-v|=1$, one has
$$\int_{\mathcal{I}}h(x-y)g(x-z)d\mu(y,z)=\int_{O(d)}h(x-Ru)g(x-Rv)dR$$
In particular, for $h\equiv 1$, \begin{equation}
\begin{split}
\int_{\mathcal{I}} |g(x-z)|d\mu(y,z)=&\int_{O(d)}|g(x-Rv)|dR\\
= &A_1(|g|)(x)
\end{split} \end{equation} where $A_t$ is the spherical average $A_tg(x)=\int_{S^{d-1}}g(x-ty)d\sigma(y)$. Hence,
$$|\mathcal{T}_0^{0\infty}(f,g)(x)|\lesssim Mf(x)\cdot A_1(|g|)(x)$$
Going back to equality (\ref{trianglelhatl}) one gets that \begin{equation}
\begin{split}
\sup_{l\in\mathbb{Z}} |\mathcal{T}_l^{0\infty}(f,g)(x)|
\lesssim& \sup_{l\in\mathbb{Z}}M(D_{2^{-l}}f)(2^lx)\cdot A_1(|D_{2^{-l}}g|)(2^lx)\\
\lesssim & Mf(x)\cdot \sup_{l\in\mathbb{Z}}A_{2^{-l}}(|g|)(x)=Mf(x)\cdot \mathcal{S}_{lac}g(x)
\end{split} \end{equation} \end{proof}
Recall that $\mathcal{S}_{lac}$ is bounded in $L^p$ for any $1<p\leq \infty$. Combining that with the boundedness properties of the Hardy-Littlewood maximal function $M$, the following corollary follows immediately from Proposition \ref{lowtriangle}.
\begin{cor} For any $p,q\in(1,\infty]$, $r\in (0,\infty]$ and $1/r=1/p+1/q$, one has
$$\|\mathcal{T}_{lac}^{LL}+\mathcal{T}_{lac}^{LH}+\mathcal{T}_{lac}^{HL}\|_{L^p\times L^q\rightarrow L^r}<\infty$$ \end{cor}
Define for all $i,j\geq 1$. \begin{equation} \mathcal{T}_1^{i,j}(f,g)(x)=\int_{\mathbb{R}^{2d}}\hat{f}(\xi)\hat{g}(\eta)\hat{\mu}(\xi, \eta)\hat{\psi}(2^{-i}\xi)\hat{\psi}(2^{-j}\eta)e^{2\pi i x\cdot (\xi+ \eta)}d\xi d\eta \end{equation} We have the following result which follows directly from the methods in \cite{triangleaveraging}. We defer the details to the appendix.
\begin{prop} \label{PS ij estimate} Let $d\geq 7$. Then there exists $\delta=\delta(d)=\dfrac{3d-20}{32}>0$ such that for all $i,j\geq 1$
\begin{equation}
\|\mathcal{T}_{1}^{i,j}(f,g)\|_1\leq C 2^{-(i+j)\delta} \|f\|_2\|g\|_2.
\end{equation} \end{prop}
This is summable, so repeating the argument in section \ref{prooffirstthm} gives the claim.
\section{Lee-Shuin class of averaging operators associated to degenerate surfaces}\label{leeshuinsection}
Let $d\geq 2$ and $\mathbf{a}=(\mathbf{a}_1,\mathbf{a}_2)\in [1,\infty)^2$. Denote by $\mathcal{A}_1^{\mathbf{a}}$ the average corresponding to the compact surface in Lee and Shuin paper \cite{leeshuin}, which we denote by $\mathcal{S}^{\mathbf{a}}$: \begin{equation}
\mathcal{S}^{\mathbf{a}}=\{(y,z)\in \mathbb{R}^d\times \mathbb{R}^d\colon |y|^{\mathbf{a}_1}+|z|^{\mathbf{a}_2}=1\} \end{equation} and \begin{equation}
\mathcal{A}_{t}^{\mathbf{a}}(f,g)(x)=\int_{\mathcal{S}^{\mathbf{a}}} f(x-ty)g(x-tz)d\mu^{\mathbf{a}}(y,z) \end{equation} where $\mu^{\mathbf{a}}$ is the surface measure in $\mathcal{S}^{\mathbf{a}}$.
Our goal will be to state the bound for the associated lacunary maximal function in terms of the H\"older boundedness region of $\mathcal{A}_1^{\mathbf{a}}$, namely,
$$\mathcal{R}^{\mathbf{a}}=\{\left(\frac{1}{p},\frac{1}{q},\frac{1}{r}\right)\colon (p,q,r) \in[1,\infty]^2\times (0,\infty]\colon 1/r=1/p+1/q\text{ and } \|\mathcal{A}^{\mathbf{a}}_1\|_{L^p\times L^{q}\rightarrow L^r}<\infty \}$$
Fix $\mathbf{a}_1,\mathbf{a}_2\geq1$ and $\mathbf{a}=(\mathbf{a}_1,\mathbf{a}_2)$. Let $\Phi^{\mathbf{a}}(y,z)=|y|^{\mathbf{a}_1}+|z|^{\mathbf{a}_2}-1$. One has $|\nabla \Phi^{\mathbf{a}}(y,z)|^2=\mathbf{a}_1^2|y|^{2(\mathbf{a}_1-1)}+\mathbf{a}_2^2|z|^{2(\mathbf{a}_2-1)}\neq 0$ for all $(y,z)\in \mathcal{S}^{\mathbf{a}}=(\Phi^{\mathbf{a}})^{-1}(\{0\})$ so there exists $c^{\mathbf{a}}>0$ and $C^{\mathbf{a}}>0$ such that $c^{\mathbf{a}}\leq |\nabla \Phi^{\mathbf{a}}(y,z)|\leq C^{\mathbf{a}}$ for $(y,z)\in \mathcal{S}^{\mathbf{a}}$.
The $L^p\times L^q \rightarrow L^r$ bounds for \begin{equation}
\mathcal{M}^{\mathbf{a}}_{lac}(f,g)(x):=\sup_{t\in 2^\mathbb{Z}} |\mathcal{A}_t^{\mathbf{a}}(f,g)(x)| \end{equation} are the same as those of the operator \begin{equation}
\mathcal{N}^{\mathbf{a}}_{lac}(f,g)(x):=\sup_{t\in 2^\mathbb{Z}} |\mathcal{B}_t^{\mathbf{a}}(f,g)(x)|, \end{equation} where \begin{equation}
\mathcal{B}_{t}^{\mathbf{a}}(f,g)(x)=\int_{\mathcal{S}^{\mathbf{a}}} f(x-ty)g(x-tz)\dfrac{d\mu^{\mathbf{a}}(y,z)}{|\nabla\Phi^{\mathbf{a}}(y,z)|}. \end{equation}
For any $i,j\geq 0$, define \begin{equation}
(\mathcal{B}_{1}^{\mathbf{a}})^{i,j}(f,g)=\mathcal{B}_1^{\mathbf{a}} (f^{i}, g^{j}) \end{equation} where for $i\geq 1$, $\hat{f^{i}}(\xi)=\hat{f}(\xi)\hat{\psi}(2^{-i}\xi)$ and for $i=0$, $\hat{f^{0}}(\xi)=\hat{f}(\xi)\hat{\varphi}(\xi)$. In other words, $f^{i}=f*\psi_i$ where $\psi_i(x)=2^{id}\psi(2^{i}x)$ for $i\geq 1$, and $f^{0}=f*\varphi$. \begin{prop}
Assume $d\geq 2$. Let $(\frac{1}{p},\frac{1}{q},\frac{1}{r})\in \text{int}(\mathcal{R}^{\mathbf{a}})$. Then there exists $\delta(p,q,r,d)$ such that for all $i,j\geq 1$
\begin{equation}\label{decayestimate2}
\|(\mathcal{B}_1^{\mathbf{a}})^{i,j}(f,g)\|_{r}\leq C 2^{-(i+j)\delta}\|f\|_p\|g\|_q.
\end{equation} \end{prop}
\begin{proof}
By interpolation with the bound
$$\|(\mathcal{B}_{1}^{\mathbf{a}})^{i,j}(f,g)\|_{r}\leq C\|f\|_p\|g\|_q,\,\forall\, \left(\frac{1}{p},\frac{1}{q},\frac{1}{r}\right)\in \mathcal{R}^{\mathbf{a}}$$
it is enough to prove that there exists $\delta>0$ such that
\begin{equation}
\|(\mathcal{B}_{1}^{\mathbf{a}})^{i,j}(f,g)\|_{1}\leq C 2^{-(i+j)\delta}\|f\|_2\|g\|_2
\end{equation}
We do it by generalizing the proof given in Proposition \ref{keyestimate}, which corresponds to the case $\mathbf{a}_1=\mathbf{a}_2=2$.
For $y\in \mathbb{R}^d$ fixed, let \begin{equation*}
\begin{split}
\Phi^{\mathbf{a}}_y(z)=&|y|^{\mathbf{a}_1}+|z|^{\mathbf{a}_2}-1 \\ \Omega_y=&(\Phi_y^{\mathbf{a}})^{-1}(0)
\end{split} \end{equation*}
For all $z\in \Omega_y$, $|\nabla\Phi_y^{\mathbf{a}} (z)|=\mathbf{a}_2|z|^{\mathbf{a}_2-1}=\mathbf{a}_2(1-|y|^{\mathbf{a}_1})^{\frac{\mathbf{a}_2-1}{\mathbf{a}_2}}=:\mathbf{a}_2\omega_{\mathbf{a}}(y)^{\mathbf{a}_2-1}$
\begin{equation}
\begin{split}\label{slicingleeshuin}
\mathcal{B}_1^{\mathbf{a}}(f,g)(x)=&\int_{\mathcal{S}^{\mathbf{a}}} f(x-y)g(x-z)\dfrac{d\mu_{\mathbf{a}}(y,z)}{|\nabla \Phi^{\mathbf{a}}(y,z)|}\\
=& \int_{\mathbb{R}^{2d}} f(x-y)g(x-z)\delta(\Phi^{\mathbf{a}})dydz\\
=&\int_{|y|<1 } f(x-y)\left(\int g(x-z)\delta(\Phi_y^{\mathbf{a}})dz\right)dy\\
=&\int_{|y|< 1} f(x-y)\left(\int_{\Omega_y} g(x-z)\dfrac{d\sigma_y(z)}{|\nabla\Phi_y^{\mathbf{a}}(z)|}\right) dy\\
=&\int_{|y|< 1} f(x-y)\left(\frac{1}{\omega_{\mathbf{a}}(y)^{d-1}}\int_{|z|=\omega_{\mathbf{a}}(y)} g(x-z)d\sigma_y(z)\right) \frac{\omega_{\mathbf{a}}(y)^{d-\mathbf{a}_2}}{\mathbf{a}_2}dy
\end{split} \end{equation}
Denote $$Af(x,t):=A_tf(x)=\int_{S^{d-1}}f(x-ty)d\sigma(y).$$ Then \begin{equation}
\begin{split}
\mathcal{B}_{1}^{\mathbf{a}}(f,g)(x)= \frac{1}{\mathbf{a}_2}\int_{|y|\leq 1} f(x-y) Ag(x,\omega_{\mathbf{a}}(y)){\omega_{\mathbf{a}}(y)^{d-\mathbf{a}_2}}dy.
\end{split} \end{equation}
If we use polar coordinates in the unit ball this says that
\begin{equation}
\begin{split}
\mathcal{B}_{1}^{\mathbf{a}}(f,g)(x)=& \frac{1}{\mathbf{a}_2}\int_{0}^{1} t^{d-1} \int_{S^{d-1}}f(x-t\omega)d \sigma(\omega) Ag(x,(1-t^{\mathbf{a}_1})^{1/\mathbf{a}_2}){(1-t^{\mathbf{a}_1})^{\frac{d-\mathbf{a}_2}{\mathbf{a}_2}} }dt\\
=&\frac{1}{\mathbf{a}_2}\int_{0}^{1} Af(x,t) Ag(x,(1-t^{\mathbf{a}_1})^{1/\mathbf{a}_2})t^{d-1}{(1-t^{\mathbf{a}_1})^{\frac{d-\mathbf{a}_2}{\mathbf{a}_2}} }dt.
\end{split} \end{equation}
Applying this equality for $f^{i}$ and $g^{j}$ and using Minkowski's inequality, one gets \begin{equation}
\begin{split}
\|\mathcal{B}_{1}^{\mathbf{a}}(f^{i},g^{j})\|_{1}\leq & \frac{1}{\mathbf{a}_2} \int_{0}^{1} \|A(f^{i})(x,t)\|_{L^{2}_x} \|A(g^{j})(x,(1-t^{\mathbf{a}_1})^{1/\mathbf{a}_2})\|_{L^{2}_x}t^{d-1}{(1-t^{\mathbf{a}_1})^{\frac{d-\mathbf{a}_2}{\mathbf{a}_2}}}dt \\
\lesssim& 2^{-(i+j)\frac{d-1}{2}}\|f\|_2\|g\|_2 \int_{0}^{1} t^{d-1-\frac{d-1}{2}}\left\{(1-t^{\mathbf{a}_1})^{1/\mathbf{a}_2}\right\}^{d-\mathbf{a}_2-\frac{d-1}{2}} dt\\
\leq &2^{-(i+j)\frac{d-1}{2}}\|f\|_2\|g\|_2 \int_{0}^{1} \left\{1-t^{\mathbf{a}_1}\right\}^{\frac{d-2\mathbf{a}_2+1}{2\mathbf{a}_2}} dt.\\
\end{split} \end{equation}
Observe that in the estimate above we used estimate (\ref{linearpiecedecay}) again, which gives \begin{equation}
\|A(f^{i})(\cdot,t)\|_2
\lesssim (t 2^{i})^{-\frac{d-1}{2}}\|f\|_2. \end{equation}
One can check that for $\mathbf{a}_1\geq 1$, $\int_{0}^{1}(1-t^{\mathbf{a}_1})^{p}dt<\infty$ if $p>-1$.
Since $\frac{d-2\mathbf{a}_2+1}{2\mathbf{a}_2}>-1$, we get
$$ \|\mathcal{B}_{1}^{\mathbf{a}}(f^{i},g^{j})\|_{1}\lesssim 2^{-(i+j)\frac{d-1}{2}}\|f\|_2\|g\|_2.$$
The argument is similar as before, for $i,j\geq 1$ one considers pieces like \begin{equation} \begin{split}
(\mathcal{B}_{2^{-l}}^{\mathbf{a}})^{i,j}(f,g)(x):=&\mathcal{B}_{2^{-l}}(f*\psi_{i+l},g*\psi_{j+l})(x)\\
=&\mathcal{B}_1^{\mathbf{a}}(D_{2^{-l}}(f*\psi_{i+l}),D_{2^{-l}}(g*\psi_{j+l}))(2^l x).
\end{split} \end{equation}
The previous proposition allow us to control the high-high pieces of the maximal operator, but we also need to make sure we can control the low-low, low-high, and high-low parts as well.
Let us illustrate with the high-low part \begin{equation}
\begin{split}
\mathcal{N}_{lac}^{HL}(f,g)(x)\mathcal:=\sup_{l\in \mathbb{Z}}|\mathcal{B}^{\mathbf{a}}_{2^{-l}}(f,g*\varphi_l)|(x).
\end{split} \end{equation}
For any $l\in \mathbb{Z}$, using that $D_{2^{-l}}(g*\varphi_l)=(D_{2^{-l}}g)*\varphi$
\begin{equation}
\begin{split}
|\mathcal{B}_{2^{-l}}^{\mathbf{a}}(f, g*\varphi_l)(x)|=&|\mathcal{B}_{1}^{\mathbf{a}}(D_{2^{-l}}f,D_{2^{-l}}(g*\varphi_l))(2^lx)|\\
=&\int_{\mathcal{S}^{\mathbf{a}}}f(x-2^{-l}y)|(D_{2^{-l}}g*\varphi)(2^lx-z)| \dfrac{d\mu^{\mathbf{a}}(y,z)}{|\nabla \Phi^{\mathbf{a}}(y,z)|}\\
\lesssim & M(D_{2^{-l}}g)(2^lx) \int_{\mathcal{S}^{\mathbf{a}}} f(x-2^{-l}y)\dfrac{d\mu^{\mathbf{a}}(y,z)}{|\nabla \Phi^{\mathbf{a}}(y,z)|}\\
\lesssim & M(g)(x) \mathcal{B}_{2^{-l}}(f,1)(x).
\end{split}
\end{equation} With the slicing in equation (\ref{slicingleeshuin}), one can see that
$$\mathcal{B}_{2^{-l}}(f,1)(x) \lesssim \int_{|y|\leq 1} f(x-2^{-l}y)dy\lesssim Mf(x).$$
Therefore, $$\mathcal{N}^{HL}_{lac}(f,g)(x)\lesssim Mf(x)Mg(x).$$ \end{proof}
\section*{Appendix: Proof of Proposition \ref{PS ij estimate}} For the reader's convenience, we present the details as to how the argument from \cite{triangleaveraging} implies Proposition \ref{PS ij estimate}.
\begin{proof} We are going to adapt some of the arguments in \cite{triangleaveraging} but using a slightly different decomposition to better match our notation in this paper. We separately localize the variables $|\xi|\sim 2^{i}$ and $|\eta|\sim 2^{j}$. The main tools will be Theorem \ref{ghscriteria} and Corollary \ref{corl2l2l1}.
We start by further decomposing the operator according to the angle $\theta=\theta(\xi,\eta)$ between $\xi$ and $\eta$, which is given by $\cos(\theta)=\frac{\xi\cdot\eta}{|\xi||\eta|}$ for every $\xi\neq 0$ and $\eta \neq 0$. \begin{equation} \mathcal{T}_1^{i,j}=\sum_{k\geq 0} \mathcal{T}_1^{i,j,k} \end{equation}
where the bilinear multiplier associated to $\mathcal{T}_1^{i,j,k}$ is localized at $|\sin(\theta)|\sim 2^{-k}$. More precisely, $$\mathcal{T}_{1}^{i,j,k}=\mathcal{T}_{m_{i,j,k}}$$ where for $i,j\geq 1$ \begin{equation}
m_{i,j,k}(\xi, \eta)=\hat{\mu}(\xi,\eta)\hat{\psi}(2^{-i}\xi)\hat{\psi}(2^{-j}\eta)\rho_k(\xi, \eta). \end{equation} Here, $\psi$ is the same as defined in \eqref{phi def}, \eqref{psi def} and $\rho_k$ are smooth functions with $\sum_{k\geq 0}\rho_k(\xi,\eta)\equiv1$ except at the origin and \begin{equation} \begin{split}
\text{supp}(\rho_k)&\subseteq\{(\xi, \eta)\colon 2^{-k-1}\leq |\sin(\theta)|\leq 2^{-k+1}\},\,\text{if }k\geq1;\\
\text{supp}(\rho_0)&\subseteq\{(\xi, \eta)\colon |\sin(\theta)|\geq 1/2\}. \end{split} \end{equation} We can choose such a partition in such a way that when one defines \begin{equation}
\rho^{l}(\xi,\eta)=\sum_{k=l}^{\infty} \rho_k(\xi,\eta), \end{equation}
then $\text{supp}(\rho^l)$ is contained in $\{(\xi, \eta)\colon |\sin(\theta)|\leq 2^{-l+1}\}$ and $\rho^l\equiv 1$ in $\{(\xi, \eta)\colon |\sin(\theta)|\leq 2^{-l}\}$. In particular, that implies that the support of any derivative of $\rho^l$ is contained in $\{(\xi,\eta)\colon 2^{-l}\leq|\sin(\theta)|\leq 2^{-l+1}\}$. The interested reader may find more details on how to construct such a partition of unity $\{\rho_k\}$ in \cite{triangleaveraging}.
One can observe that for $i,j,k\geq 1$ \begin{equation*}
\text{supp}(m_{i,j,k})
\subseteq
\{(\xi,\eta)\in \mathbb{R}^{2d}\colon 2^{i-1}\leq |\xi|\leq 2^{i+1},2^{j-1}\leq |\eta|\leq 2^{j+1},2^{-k-1}\leq |\sin(\theta)|\leq 2^{-k+1}\}. \end{equation*}
This allow us to estimate $|\text{supp}(m_{i,j,k})|$. \begin{equation}\label{support} \begin{split}
|\text{supp}(m_{i,j,k})|\leq & |\{(\xi,\eta)\in \mathbb{R}^{2d}\colon |\xi|\leq 2^{i+1}, |\eta|\leq 2^{j+1},|\sin(\theta)|\leq 2^{-k+1}\}|\\
\lesssim &2^{(i+j)d}2^{-k(d-1)} \end{split} \end{equation}
The final estimate above follows in a straightforward manner by noticing that for a fixed $\xi$ with $|\xi|\le 2^{i+1}$, the admissible points $\eta$ lie in a sector of a ball of radius $2^{j+1}$ and angle at most $C2^{-k}$; for the details (with a slightly modified definition of the multiplier), see \cite{triangleaveraging}.
For $k=0$ it is also trivially true because
$$|\text{supp}(m_{i,j,0})|\leq |\{(\xi,\eta)\in \mathbb{R}^{2d}\colon |\xi|\leq 2^{i+1};|\eta|\leq 2^{j+1}\}|\lesssim 2^{(i+j)d}$$
Also define $m_{i,j}^{k}(\xi,\eta)=\hat{\mu}(\xi,\eta)\hat{\psi}(2^{-i}\xi)\hat{\psi}(2^{-j}\eta)\rho^{k}(\xi,\eta)$
We recall that in \cite{triangleaveraging} they showed that for any multi-indices $\alpha,\beta$ \begin{equation}
|\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}\hat{\mu}(\xi,\eta)|\leq C_{\alpha,\beta} \left(1+\min\{|\xi|,|\eta|\}|\sin(\theta)|\right)^{-\frac{d-2}{2}} (1+|(\xi,\eta)|)^{-\frac{d-2}{2}} \end{equation}
Assume $i\geq j$. In this case
$$\min\{|\xi|,|\eta|\}|\sin(\theta)|\sim 2^{j-k}$$
As in \cite{triangleaveraging}, when $k\leq \lfloor{j/2}\rfloor$, the derivatives of the cutoff functions $\rho_k$ are all bounded, and so are the derivatives of $\rho^{\lfloor{j/2}\rfloor}$. Hence, for any multi-indices $\alpha,\beta$ \begin{equation}
|\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}m_{i,j,k}|\leq C_{\alpha,\beta} 2^{-(j-k)\frac{d-2}{2}} 2^{-i\frac{d-2}{2}} \end{equation} for all $(\xi,\eta)\in\text{supp}(m_{i,j,k}) $ and \begin{equation}
|\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}m_{i,j}^{\lfloor j/2\rfloor}|\leq C_{\alpha,\beta} 2^{-i\frac{d-2}{2}} \end{equation} for all $(\xi,\eta)\in\text{supp}(m_{i,j}^{\lfloor j/2\rfloor}) $.
An application of Corollary \ref{corl2l2l1} will give us the control of $\|T_{m_{i,j,k}}\|_{L^2\times L^2 \rightarrow L^1}$ for $k\leq \lfloor j/2 \rfloor$ \begin{equation} \begin{split}
\|\mathcal{T}_{1}^{i,j,k}\|_{L^2\times L^2\rightarrow L^1}\lesssim &2^{-(j-k)\frac{d-2}{2}} 2^{-i\frac{d-2}{2}} |\text{supp}(m_{i,j,k})|^{1/4}\\
\lesssim & 2^{-(i+j)\frac{d-2}{2}} 2^{k\frac{d-2}{2}}2^{(i+j)\frac{d}{4}}2^{-k\frac{d-1}{4}}\\
\lesssim & 2^{-(i+j)\frac{d-4}{4}} 2^{k\frac{d-3}{4}} \end{split} \end{equation}
For $\|T_{m_{i,j}^{\lfloor j/2 \rfloor}}\|_{L^2\times L^2\rightarrow L^1}$, Theorem \ref{ghscriteria} will give us \begin{equation}
\|T_{m_{i,j}^{\lfloor j/2 \rfloor}}\|_{L^2\times L^2\rightarrow L^1}\lesssim (2^{-\frac{i(d-2)}{2}})^{\frac{3}{4}} \|(m_{i,j}^{\lfloor j/2 \rfloor})\|_{1}^{1/4}
\end{equation}
One can check that \begin{equation} \begin{split}
\|(m_{i,j}^{\lfloor j/2 \rfloor})\|_{1}\lesssim & 2^{-i(\frac{d-2}{2})}\int_{\text{supp}(m_{i,j}^{\lfloor j/2 \rfloor})} (1+2^{j}|\sin(\theta(\xi,\eta))|)^{-\frac{d-2}{2}}d\xi d\eta\\
\lesssim &2^{-i(\frac{d-2}{2})}|\{|\xi|\sim 2^i,|\eta|\sim 2^j, |\sin(\theta)|\leq 2^{-j}\}|\\
&+2^{-i(\frac{d-2}{2})}\int_{|\xi|\sim 2^i,\,|\eta|\sim 2^j, 2^{-j}\leq |\sin(\theta)|\leq 2^{-j/2}} (1+2^{j}|\sin(\theta(\xi,\eta))|)^{-\frac{d-2}{2}}d\xi d\eta\\
\lesssim &2^{-i(\frac{d-2}{2})} \left(2^{(i+j)d}2^{-j(d-1)}+2^{-j(\frac{d-2}{2})}2^{(i+j)d}2^{-jd/4}\right)\\
\lesssim &2^{-i(\frac{d-2}{2})}2^{-j(\frac{3d-4}{4})} 2^{(i+j)d}\\ \end{split} \end{equation}
Then
$$\|T_{m_{i,j}^{\lfloor j/2\rfloor}}\|_{L^2\times L^2\rightarrow L^1}\lesssim 2^{-i(\frac{d-2}{2})}2^{(i+j)\frac{d}{4}}2^{-j(\frac{3d-4}{16})}$$
Hence, for $i\geq j$ \begin{equation}
\begin{split} \|\mathcal{T}_{1}^{i,j}\|_{L^2\times L^2\rightarrow L^1}\leq &\sum_{k\geq 0}^{\lfloor j/2\rfloor-1} \|\mathcal{T}_{1}^{i,j,k}\|_{L^2\times L^2\rightarrow L^1} +\|\mathcal{T}_{m_{i,j}^{\lfloor j/2\rfloor}}\|_{L^2\times L^2\rightarrow L^1} \\ \lesssim&\left(\sum_{k=0}^{\lfloor j/2\rfloor-1} 2^{-(i+j)\frac{d-4}{4}}2^{k\frac{d-3}{4}}\right)+2^{-i(\frac{d-2}{2})}2^{(i+j)\frac{d}{4}}2^{-j(\frac{3d-4}{16})}\\ \lesssim & 2^{-(i+j)\frac{d-4}{4}}2^{\frac{j}{2}\frac{d-3}{4}}+2^{-i(\frac{d-4}{4})}2^{j(\frac{d+4}{16})}\\ \lesssim &2^{-i\frac{d-4}{4}}2^{-j(\frac{d-5}{8})}+2^{-i(\frac{d-4}{4})}2^{j(\frac{d+4}{16})}\lesssim 2^{-i(\frac{d-4}{4})}2^{j(\frac{d+4}{16})} \\ \end{split} \end{equation}
When $j\geq i$ one can get instead,
$$\|\mathcal{T}_1^{i,j}\|_{L^{2}\times L^2\rightarrow L^1} \lesssim 2^{-j(\frac{d-4}{4})}2^{i(\frac{d+4}{4})}$$
In any case, \begin{equation}
\begin{split}
\|\mathcal{T}_1^{i,j}\|_{L^{2}\times L^2\rightarrow L^1} \lesssim& 2^{-\max\{i,j\}(\frac{d-4}{4})}2^{\min\{i,j\}(\frac{d+4}{16})}\\ \leq& 2^{-\frac{(i+j)}{2}(\frac{d-4}{4})}2^{\frac{(i+j)}{2}\frac{d+4}{16}}=2^{-\frac{(i+j)}{2}(\frac{3d-20}{16})}
\end{split} \end{equation}
which finishes the proof of the proposition with $\delta=\frac{3d-20}{32}$. \end{proof}
\end{document} | arXiv |
Gamas's Theorem
Gamas's theorem is a result in multilinear algebra which states the necessary and sufficient conditions for a tensor symmetrized by an irreducible representation of the symmetric group $S_{n}$ to be zero. It was proven in 1988 by Carlos Gamas.[1] Additional proofs have been given by Pate[2] and Berget.[3]
Statement of the theorem
Let $V$ be a finite-dimensional complex vector space and $\lambda $ be a partition of $n$. From the representation theory of the symmetric group $S_{n}$ it is known that the partition $\lambda $ corresponds to an irreducible representation of $S_{n}$. Let $\chi ^{\lambda }$ be the character of this representation. The tensor $v_{1}\otimes v_{2}\otimes \dots \otimes v_{n}\in V^{\otimes n}$ symmetrized by $\chi ^{\lambda }$ is defined to be
${\frac {\chi ^{\lambda }(e)}{n!}}\sum _{\sigma \in S_{n}}\chi ^{\lambda }(\sigma )v_{\sigma (1)}\otimes v_{\sigma (2)}\otimes \dots \otimes v_{\sigma (n)},$
where $e$ is the identity element of $S_{n}$. Gamas's theorem states that the above symmetrized tensor is non-zero if and only if it is possible to partition the set of vectors $\{v_{i}\}$ into linearly independent sets whose sizes are in bijection with the lengths of the columns of the partition $\lambda $.
See also
• Algebraic combinatorics
• Immanant
• Schur polynomial
References
1. Carlos Gamas (1988). "Conditions for a symmetrized decomposable tensor to be zero". Linear Algebra and Its Applications. Elsevier. 108: 83–119. doi:10.1016/0024-3795(88)90180-2.
2. Thomas H. Pate (1990). "Immanants and decomposable tensors that symmetrize to zero". Linear and Multilinear Algebra. Taylor & Francis. 28 (3): 175–184. doi:10.1080/03081089008818039.
3. Andrew Berget (2009). "A short proof of Gamas's theorem". Linear Algebra and Its Applications. Elsevier. 430 (2): 791–794. arXiv:0906.4769. doi:10.1016/j.laa.2008.09.027. S2CID 115172852.
| Wikipedia |
Alternated octagonal tiling
In geometry, the tritetragonal tiling or alternated octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of {(4,3,3)} or h{8,3}.
Alternated octagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(3.4)3
Schläfli symbol(4,3,3)
s(4,4,4)
Wythoff symbol3 | 3 4
Coxeter diagram
Symmetry group[(4,3,3)], (*433)
[(4,4,4)]+, (444)
DualAlternated octagonal tiling#Dual tiling
PropertiesVertex-transitive
Geometry
Although a sequence of edges seem to represent straight lines (projected into curves), careful attention will show they are not straight, as can be seen by looking at it from different projective centers.
Triangle-centered
hyperbolic straight edges
Edge-centered
projective straight edges
Point-centered
projective straight edges
Dual tiling
In art
Circle Limit III is a woodcut made in 1959 by Dutch artist M. C. Escher, in which "strings of fish shoot up like rockets from infinitely far away" and then "fall back again whence they came". White curves within the figure, through the middle of each line of fish, divide the plane into squares and triangles in the pattern of the tritetragonal tiling. However, in the tritetragonal tiling, the corresponding curves are chains of hyperbolic line segments, with a slight angle at each vertex, while in Escher's woodcut they appear to be smooth hypercycles.
Related polyhedra and tiling
Uniform (4,3,3) tilings
Symmetry: [(4,3,3)], (*433) [(4,3,3)]+, (433)
h{8,3}
t0(4,3,3)
r{3,8}1/2
t0,1(4,3,3)
h{8,3}
t1(4,3,3)
h2{8,3}
t1,2(4,3,3)
{3,8}1/2
t2(4,3,3)
h2{8,3}
t0,2(4,3,3)
t{3,8}1/2
t0,1,2(4,3,3)
s{3,8}1/2
s(4,3,3)
Uniform duals
V(3.4)3 V3.8.3.8 V(3.4)3 V3.6.4.6 V(3.3)4 V3.6.4.6 V6.6.8 V3.3.3.3.3.4
Uniform (4,4,4) tilings
Symmetry: [(4,4,4)], (*444) [(4,4,4)]+
(444)
[(1+,4,4,4)]
(*4242)
[(4+,4,4)]
(4*22)
t0(4,4,4)
h{8,4}
t0,1(4,4,4)
h2{8,4}
t1(4,4,4)
{4,8}1/2
t1,2(4,4,4)
h2{8,4}
t2(4,4,4)
h{8,4}
t0,2(4,4,4)
r{4,8}1/2
t0,1,2(4,4,4)
t{4,8}1/2
s(4,4,4)
s{4,8}1/2
h(4,4,4)
h{4,8}1/2
hr(4,4,4)
hr{4,8}1/2
Uniform duals
V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V8.8.8 V3.4.3.4.3.4 V88 V(4,4)3
See also
• Circle Limit III
• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
Wikimedia Commons has media related to Uniform tiling 3-4-3-4-3-4.
• Douglas Dunham Department of Computer Science University of Minnesota, Duluth
• Examples Based on Circle Limits III and IV, 2006:More “Circle Limit III” Patterns, 2007:A “Circle Limit III” Calculation, 2008:A “Circle Limit III” Backbone Arc Formula
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery Archived 2013-03-24 at the Wayback Machine
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Lowndean Professor of Astronomy and Geometry
The Lowndean chair of Astronomy and Geometry is one of the two major Professorships in Astronomy (alongside the Plumian Professorship) and a major Professorship in Mathematics at Cambridge University. It was founded in 1749 by Thomas Lowndes, an astronomer from Overton in Cheshire.
The original bequest stated that the holder must give two courses of twenty lectures each year, one in astronomy, and the other in geometry, and spend at least six weeks making astronomical observations.
Originally the holder was elected by a committee consisting of the Lord Chancellor, the Lord President of the Privy Council, the Lord Privy Seal, the Lord Steward of the Household, and the Lord High Treasurer of the First Lord of the Treasury.
By the 20th century, the electors had changed to comprise the most senior scientists in the United Kingdom: the President of the Royal Society, the President of the Royal Astronomical Society, the Astronomer Royal, the Vice Chancellor of the University of Cambridge, and the Lucasian, Sadleirian, and Plumian Professors.
Notwithstanding the title, a professor can be chosen who specializes solely or chiefly in only one, rather than both, of the subjects of astronomy and geometry.[1]
Lowndean Professors
• 1750–1771 Roger Long
• 1771–1795 John Smith
• 1795–1837 William Lax
• 1837–1859 George Peacock
• 1859–1892 John Couch Adams
• 1892–1913 Robert Stawell Ball
• 1914–1936 H. F. Baker
• 1936–1970 W. V. D. Hodge
• 1970–1989 J. Frank Adams
• 1990–1999 Graeme Segal
• 2000–2014 Burt Totaro[2]
• 2015-present Mihalis Dafermos[3]
References
1. "The Lowndean Professorship of Astronomy and Geometry". Statutes and Ordinances of the University of Cambridge. Statute E, Chapter XXXV. Retrieved 2011-03-09.
2. "Elections and appointments". Cambridge University Reporter (5808). 4 May 2000. Retrieved 5 September 2019.
3. "Elections, appointments, reappointment, and grants of title". Cambridge University Reporter (6396). 23 September 2015. Retrieved 13 September 2019.
| Wikipedia |
\begin{document}
\title{Singularities of pairs via jet schemes}
\author[M. Musta\c{t}\v{a}]{Mircea~Musta\c{t}\v{a}} \address{Department of Mathematics, University of California, Berkeley, CA, 94720 and Institute of Mathematics of the Romanian Academy} \email{{\tt [email protected]}}
\date{\today} \maketitle
\section*{Introduction}
Let $X$ be a smooth complex variety and $Y$ a closed subscheme of $X$. The study of the singularities of the pair $(X,Y)$ is a topic which received a lot of atention, due mainly to applications to the classification of higher dimensional algebraic varieties. Our main goal in this paper is to propose a new point of view in this study, based on the properties of the jet schemes of $Y$.
For an arbitrary scheme $W$, the $m$th jet scheme $W_m$ parametrizes morphisms $\Spec\,{\mathbb C}[t]/(t^{m+1})\longrightarrow W$. Our main result is the
following theorem.
\begin{theorem}\label{main_result} If $X$ is a smooth variety, $Y\subset X$ a closed subscheme, and $q>0$ a rational number, then: \begin{enumerate} \item The pair $(X, q\cdot Y)$ is log canonical if and only if $\dim\,Y_m\leq (m+1)(\dim\,X-q)$, for all $m$. \item The pair $(X, q\cdot Y)$ is Kawamata log terminal if and only if $\dim\,Y_m<(m+1)(\dim\,X-q)$, for all $m$. \end{enumerate} \end{theorem}
In fact, it is enough to check
the conditions in Theorem~\ref{main_result} for one value of $m$, depending on a log resolution of $(X,Y)$ (see Theorem~\ref{main} for the precise statement). As a consequence of the above result, we obtain a formula for the log canonical threshold.
\begin{corollary}\label{main_corollary} If $X$ is a smooth variety and $Y\subset X$ is a closed subscheme, then the log canonical threshold of the pair $(X,Y)$ is given by $$c(X,Y)=\dim\,X-\sup_{m\geq 0}{{\dim\,Y_m}\over{m+1}}.$$ \end{corollary}
In the same spirit with the previous remark, the above supremum can be obtained for specific values of $m$. A similar formula holds for the log canonical threshold around a closed subset $Z\subset X$. Note that as a consequence of the above formula, the log canonical threshold depends only on $Y$ and $\dim\,X$, but not on the particular ambient variety or the embedding.
We apply Corollary~\ref{main_corollary} to give simpler proofs of some results on the log canonical threshold proved by Demailly and Koll\'{a}r in \cite{demailly} using analytic techniques. For example, we use a semicontinuity statement about the dimension of the jet schemes to deduce the semicontinuity of the log canonical threshold.
The main technique we use in the proof of Theorem~\ref{main_result} is motivic integration, technique due to Kontsevich, Batyrev, and Denef and Loeser. The idea is similar to the one used in our previous paper \cite{mustata}. Here is a brief description of the idea of the proof.
Let $X_{\infty}=\projlim_m X_m$ be the space of arcs of $X$. A ${\mathbb C}$-valued point of $X_{\infty}$ corresponds to a morphism $\Spec\,{\mathbb C}[[t]] \longrightarrow X$. Let $F_Y\,:\,X_{\infty}\longrightarrow{\mathbb N}\cup\{\infty\}$ be the function such that for an arc $\gamma$ over $x\in X$ which corresponds to a morphism $\widetilde{\gamma}\,:\,{\mathcal O}_{X,x} \longrightarrow{\mathbb C}[[t]]$, we have $F_Y(\gamma)={\rm ord}\,(\widetilde{\gamma}({\mathcal I}_{Y,x}))$. For a suitable function $f\,:\,{\mathbb N} \longrightarrow{\mathbb N}$, the motivic integral of $f\circ F_Y$ on $X_{\infty}$ encodes the information about the dimension of $Y_m$, for all $m$.
On the other hand, if we take a log resolution $\pi\,:\,X' \longrightarrow X$, then by the change of variable formula of Batyrev (\cite{batyrev}) and Denef and Loeser (\cite{denef}), this integral can be
expressed as the integral on $X'_{\infty}$ of $f\circ F_{\pi^{-1}(Y)} +F_W$. Here $F_{\pi^{-1}(Y)}$ and $F_W$ are the functions associated as above with the effective divisors $\pi^{-1}(Y)$ and $W$, where $W$ is the relative canonical divisor of $\pi$. Since both $W$ and $\pi^{-1}(Y)$ are divisors with simple normal crossings, this last integral can be
explicitely computed by a formula involving the coefficients of $W$ and $\pi^{-1}(Y)$. By an inspection of the monomials which enter in the expression of the two integrals we deduce the equivalence in Theorem~\ref{main_result}.
The paper is organized as follows. In the first section we review the
definitions of log canonical and Kawamata log terminal pairs.
In the
algebraic context the usual setting in the literature
is that of a pair $(X,Y)$, where $Y$ is a divisor on $X$. Therefore we give the extension to the case of pairs of arbitrary codimension, making reference for the case of divisors to Koll\'{a}r (\cite{kollar}). Note, however, that we will always stick to the case of a smooth ambient variety, as our main results are valid only in this context.
The second section contains an overview of the basic definitions and notions related to jet schemes and motivic integration. We give also some further properties of jet schemes. In particular, we prove a semicontinuity result about the dimensions of the jet schemes.
The third section is the heart of the paper. It gives the proof of Theorem~\ref{main_result} along the lines described above. In the last section we show how Corollary~\ref{main_corollary} can be used to prove properties of the log canonical threshold. For example, the canonical isomorphism $(Y\times Y')_m\simeq Y_m\times Y'_m$ implies that $c(X\times X',Y\times Y')=c(X,Y)+c(X'.Y')$. This can be used to prove the inequality in \cite{demailly}: $c_z(X,Y_1\cap Y_2)\leq c_z(X,Y_1)+c_z(X,Y_2)$, for every $z\in Y_1\cap Y_2$. Another application is to the semicontinuity theorem in \cite{demailly} for log canonical thresholds. We conclude with a concrete example: we use the formula in Corollary~\ref{main_corollary} to compute the log canonical threshold of monomial ideals, recovering in this way a result from \cite{agv} (see also \cite{howald} for a proof via multiplier ideals).
\section{Log terminal and log canonical pairs}
All schemes are of finite type over ${\mathbb C}$. A variety is a reduced, irreducible scheme. We use the theory of singularities of pairs for which our main reference is \cite{kollar}. However, as we work with pairs of arbitrary codimension, we review in this section some extensions to this setting of the definitions we need. For an analytic approach to singularities of pairs of arbitrary codimension, see \cite{demailly}.
We consider pairs $(X,Y)$, where $X$ is a smooth variety and $Y\hookrightarrow X$ is a closed subscheme with $Y\neq X$.
A log resolution of $(X, Y)$ is a proper, birational morphism $\pi\,:\,X'\longrightarrow X$ such that $X'$ is smooth,
$\pi^{-1}(Y)=D$ is an effective divisor and the union $D\cup Ex(\pi)$ has simple normal crossings. Here $Ex(\pi)$ denotes the exceptional locus of $\pi$. The results in \cite{hironaka} show that log resolutions exist (in fact, one can assume in addition that $\pi$ is an isomorphism over $X\setminus Y$.)
For an arbitrary proper, birational morphism $\pi\,:\,X'\longrightarrow X$, with $X'$ and $X$ smooth, the relative canonical divisor $K_{X'/X}$ is the unique effective divisor supported on $Ex(\pi)$ such that $\mathcal{O}_{X'}(K_{X'/X})\simeq \omega_{X'}\otimes\pi^*(\omega_X^{-1})$.
\begin{definition} Let $(X,Y)$ be a pair as above and $q>0$ a rational number. For a log resolution $\pi\,:\,X'\longrightarrow X$ of $(X,Y)$, with $\pi^{-1}(Y)=D$, we write $K_{X'/X}-q\cdot D =\sum_i\alpha_iE_i$. The total discrepancy of
$(X, q\cdot Y)$ is defined by \begin{equation} {\rm totdiscrep}\,(X,q\cdot Y)= \begin{cases} -\infty, &\text{if $\alpha_i<-1$, for some $i$;}\\ \min_i\{0,\alpha_i\}, &\text{otherwise.}\\ \end{cases} \end{equation} The pair $(X, q\cdot Y)$ is called Kawamata log terminal or log canonical if ${\rm totdiscrep}\,(X, q\cdot Y)>-1$ or ${\rm totdiscrep}\,(X,q\cdot Y)\geq -1$, respectively. \end{definition}
\begin{remark}\label{connection} Since $q\cdot D-K_{X'/X}$ is a divisor with simple normal crossings, it follows from Lemma 3.11 in \cite{kollar}, that we could have defined the total discrepancy of $(X,q\cdot Y)$ as the total discrepancy of $q\cdot D-K_{X'/X}$. In particular, $(X,q\cdot Y)$ is Kawamata log terminal (log canonical) if and only if $(X', q\cdot D-K_{X'/X})$ is Kawamata log terminal (log canonical). \end{remark}
\begin{proposition}\label{no_dependance} The total discrepancy does not depend on the particular log resolution, and therefore neither do the notions of log canonical and Kawamata log terminal pairs. \end{proposition}
\begin{proof} By the above remark, it is enough to show that if ${\pi'}\,:\,X''\longrightarrow X'$ is a log resolution of $(X',D\cup Ex(\pi))$, with ${\pi'}^{-1}(D)=F$, then $$\totdiscrep\,(q\cdot D-K_{X'/X})=\totdiscrep\,(q\cdot F-K_{X''/X}).$$ This follows from the fact that $K_{X''/X}=K_{X''/X'}+{\pi'}^*(K_{X'/X})$ and the invariance of the total discrepancy under proper birational morphisms in the case of divisors (see \cite{kollar}, Lemma 3.10). \end{proof}
\begin{definition} Let $(X, Y)$ be a pair as above, $Z\subset X$ a nonempty closed subset,
and $q>0$ a rational number. The log canonical threshold of $(X, q\cdot Y)$ around $Z$ is defined by $c_Z(X, q\cdot Y)=\sup\{q'\in{\mathbb Q}_+\mid (X, qq'\cdot Y)\,\text{is log canonical in an open neighbourhood of}\,Z\}$. When $Z=X$, we omit it from the notation. \end{definition}
As it is obvious that $c_Z(X, q\cdot Y)=(1/q)\,c(X, Y)$, it follows that when computing log canonical thresholds, there is no loss of generality in assuming that $q=1$. The proposition below is the analogue of Proposition 8.5 in \cite{kollar}. It shows
how to compute the log canonical threshold from a log resolution.
Let $(X, Y)$ be a pair and $Z\subseteq X$ a nonempty closed subset. Let $\pi\,:\,X'\longrightarrow X$ be a log resolution of $(X, Y)$ which is an isomorphism over $X\setminus Y$. We write $\pi^{-1}(Y)=D= \sum_ia_iE_i$, with $a_i\geq 1$ and $K_{X'/X}=\sum_ib_iE_i$.
\begin{proposition}(\,\cite{demailly} 1.7, \cite{kollar} 8.5)\label{lc_formula} With the above notation, the log canonical threshold of $(X,Y)$ around $Z$ is given by \begin{equation} c_Z(X,Y)=\min_i\{(b_i+1)/a_i\,\mid\,\pi(E_i)\cap Z\neq\emptyset\}. \end{equation} In particular, either $c_Z(X,Y)$ is a positive rational number, or $\,\,\,\,$
$c_Z(X,Y)=\infty$. \end{proposition}
\begin{proof} We have $(X, q\cdot Y)$ log canonical if and only if $b_i-qa_i\geq -1$, or equivalently $q\leq (b_i+1)/a_i$, for all $i$. By replacing $X$ with the complement of the union of those images of $E_i$ which do not intersect $Z$, we get our result. \end{proof}
\begin{remark} The log canonical threshold $c_Z(X,Y)$ is infinite if and only if $Z\cap Y=\emptyset$. \end{remark}
\begin{remark} We assumed above that for every pair $(X,Y)$, we have $Y\neq X$. We will sometimes find convenient to include also the case of the pair $(X,X)$, so that we make the convention $c_Z(X,X)=0$, for every nonempty closed subset $Z\subseteq X$. \end{remark}
Though unnecessary for our needs, for the sake of completeness we mention the translation of the above notions into the language of multiplier ideals (see \cite{ein} or \cite{lazarsfeld}). Recall that if $\pi$ is a log resolution of $(X, Y)$ such that $\pi^{-1}(Y)=D$, then the multiplier ideal of the pair $(X,q\cdot Y)$ is defined by \begin{equation} I(X, q\cdot\mathcal{I}_{Y/X})=\pi_*(\mathcal{O}_{X'}(K_{X'/X}-[q\cdot D])), \end{equation} where $[q\cdot D]$ denotes the integral part of the ${\mathbb Q}$--divisor $q\cdot D$.
\begin{proposition} The pair $(X,q\cdot Y)$ is Kawamata log terminal if and only if $I(X, q\cdot\mathcal{I}_{Y/X})=\mathcal{O}_X$ and it is log canonical if and only if $I(X, q'\cdot \mathcal{I}_{Y/X})=\mathcal{O}_X$ for every $q'<q$ . In particular, if $Z\subseteq X$ is a nonempty closed subset, then \begin{equation} c\,(X,Y)=\sup\{q>0 \mid I(X,q\cdot\mathcal{I}_{Y/X})=\mathcal{O}_X\,{\rm around}\,Z\}. \end{equation} \end{proposition}
\begin{proof} It is enough to prove the first assertion. This follows from the fact that $I(X,q\cdot\mathcal{I}_{Y/X})=\mathcal{O}_X$ if and only if $K_{X'/X}-[q\cdot D]$ is effective or equivalently, all the coefficients in $K_{X'/X}-q\cdot D$ are $>-1$. \end{proof}
\section{An overview of jet schemes and motivic integration}
In this section we collect the definitions and basic properties of jet schemes. We review also motivic integration on smooth varieties, technique which plays a major role in the proof of our main result in the next section.
For every scheme $W$ (of finite type over ${\mathbb C}$) and every $m\in{\mathbb N}$, the jet scheme $W_m$ is a scheme of finite type over ${\mathbb C}$ characterized by \begin{equation} \Hom(\Spec\,A,W_m)\simeq\Hom(\Spec\,A[t]/(t^{m+1}), W), \end{equation} for every ${\mathbb C}$--algebra $A$.
In particular, the closed points of $W_m$ correspond to ${\mathbb C}[t]/(t^{m+1})$--valued points of $W$. The correspondence $W\longrightarrow W_m$ is a functor. In fact, from the definition it follows that it is the right adjoint of the functor $Z\longrightarrow Z\times\Spec\,{\mathbb C}[t]/(t^{m+1})$.
It is clear that we have $W_0\simeq W$ and $W_1\simeq T\,W$, the tangent space of $W$. In general, we have projections $\phi_m\,:\,W_m\longrightarrow W_{m-1}$ which are induced by the projections $A[t]/(t^{m+1}) \longrightarrow A[t]/(t^m)$, for a ${\mathbb C}$--algebra $A$. By composing these morphisms we get natural projections $\rho_m\,:\,W_m\longrightarrow W$. Whenever the variety is not clear from the context, we will add a superscript: for example, $\rho_m^W$. The projective limit of the schemes $W_m$ is a scheme $W_{\infty}$, in general not of finite type over ${\mathbb C}$, called the space of arcs of $W$. Its ${\mathbb C}$--valued points correspond to ${\mathbb C}[[t]]$--valued points of $W$. We have canonical morphisms $\psi_m\,:\,W_{\infty}\longrightarrow W_m$.
This construction is local in the sense that if $U\subseteq W$ is an open subset, then $U_m\simeq \rho_m^{-1}(U)$, for all $m$. More generally, if $f\,:\,W'\longrightarrow W$ is an \'{e}tale morphism, then $W'_m\simeq W_m\times_WW'$.
We can therefore reduce the description of $W_m$ to the case when $W$ is affine. In this case one can write down explicit equations for $W_m$ as follows. Suppose that $W\subseteq\Spec\,{\mathbb C}[X_i;i\in I]$ is defined by a system of polynomials $(f_{\alpha})_{\alpha}$. Consider the ring $R_m={\mathbb C}[X_i,X'_i,\ldots X_i^{(m)}; i\in I]$ and $D$ the unique ${\mathbb C}$--derivation of $R_m$ such that $D(X^{(j)}_i)=X^{(j+1)}_i$ for all $i$ and $j$, where $X_i^{(0)}=X_i$ and $X_i^{(m+1)}=0$. For a polynomial $f\in {\mathbb C}[X_i; i\in I]$, we put $f^{(j)}=D^j(f)$. With this notation, $W_m\subseteq\Spec\,R_m$ is defined by $(f_{\alpha},f'_{\alpha},\ldots,f^{(m)}_{\alpha})_{\alpha}$. In follows from this description that if $W'\hookrightarrow
W$ is a closed subscheme of $W$, then the induced morphism $W'_m\longrightarrow W_m$ is also a closed immersion.
If $W$ is a smooth variety of dimension $n$,
then the morphisms $\phi_m$ are locally trivial with fiber ${\mathbb A}^n$. In particular, $W_m$ is a smooth variety of dimension $(m+1)n$.
We discuss some more properties of jet schemes which we will apply in the last section to deduce corresponding properties of the log canonical threshold.
\begin{proposition}\label{product_of_jets} For every two schemes $W$ and $Z$ and every $m$, there is a natural isomorphism $(W\times Z)_m\simeq W_m\times Z_m$. \end{proposition}
\begin{proof} The assertion follows from the fact that the functor $W\longrightarrow W_m$ has a left adjoint, and therefore it commutes with direct products. \end{proof}
For every scheme $W$ and every $m\geq 1$ there is a natural ``action'': $$\Phi_m\,:\,{\bf A}^1\times W_m\longrightarrow W_m$$ over $W$ defined at the level of $A$-valued points as follows. For an algebra $A$, an $A$-valued point in ${\bf A}^1\times W_m$ consists of a pair $(a,f)$, for some
$a\in A$ and a morphism $f\,:\,\Spec\,A[t]/(t^{m+1}) \longrightarrow W$.
$\Phi_m(a,f)$ is the composition $f\circ g_a$, where $g_a$ is induced by the $A$-algebra homomorphism $A[t]/(t^{m+1}) \longrightarrow A[t]/(t^{m+1})$ which maps $t$ to $at$.
It is clear that these ``actions'' are compatible with the projection $\phi_m$. The restriction of $\Phi_m$ to ${\mathbb C}^*\times W_m$ induces an action of the torus on $W_m$. On the other hand, note that $\Phi_m(\{0\}\times W_m)$ is the image of the ``zero section'' $\sigma_m$ of $\rho_m$. This is defined by composition with the scheme morphism induced by the inclusion $A\hookrightarrow A[t]/(t^{m+1})$. It is clear that $W_m\setminus \sigma_m(W)$ is ${\mathbb C}^*$-invariant. With this notation, we have
\begin{lemma}\label{cone} For every $w\in W$, the fiber $\rho^{-1}_m(w)$ is the cone over the (possibly empty) projective scheme $(\rho^{-1}_m(w)\setminus\{\sigma _m(w)\})/{{\mathbb C}^*}$. \end{lemma}
\begin{proof} By restricting to an open neighbourhood of $w$, we may assume that $W\subseteq {\bf A}^N$ and that $w=0$ is the origin. We have an embedding $W_m\subseteq {\bf A}^{(m+1)N}$ which induces an embedding $\rho_m^{-1}(0)\subseteq {\bf A}^{mN}$ such that $\sigma_m(0)$ corresponds to the origin. The action of ${\mathbb C}^*$ on $\rho_m^{-1}(0)\setminus\{0\}$ extends to an action on ${\bf A}^{mN}=\Spec[X'_i,\ldots,X^{(m)}_i;i]$ induced by $\lambda\cdot X_i^{(j)}=\lambda^jX_i^{(j)}$. Therefore $\rho_m^{-1}(0)\setminus\{0\}/ {\mathbb C}^*$ is a subscheme of a weighted projective space, hence it is projective, and $\rho_m^{-1}(w)$ is the cone over it. \end{proof}
Our next goal is the proof of the following semicontinuity result. For a scheme $\pi\,:\,{\mathcal W}\longrightarrow S$ over $S$ and a closed point $s\in S$, we denote the fiber of $\pi$ over $s$ by ${\mathcal W}_s$.
\begin{proposition}\label{semicontinuity1} Let $\pi\,:\,{\mathcal W}\longrightarrow S$ be a family of schemes and $\tau\,:\,S\longrightarrow {\mathcal W}$ a section of $\pi$. For every $m\geq 1$, the function $$f(s)=\dim\,(\rho_m^{{\mathcal W}_s})^{-1}(\tau(s))$$ is upper semicontinuous on the set of closed points of $S$. \end{proposition}
\eject
The key to the proof of Proposition~\ref{semicontinuity1} is to use a relative version of the jet schemes. Suppose we work over a fixed scheme $S$. If ${\mathcal W}\longrightarrow S$ is a scheme over $S$, then the $m$th relative jet scheme $({\mathcal W}/S)_m$ is characterized by $${\rm Hom}_S({\mathcal Z}\times\Spec\,{\mathbb C}[t]/(t^{m+1}), {\mathcal W}) \simeq {\rm Hom}_S({\mathcal Z}, ({\mathcal W}/S)_m),$$ for every scheme ${\mathcal Z}$ over $S$. Therefore the functor ${\mathcal W}\longrightarrow ({\mathcal W}/S)_m$ is the right adjoint of the functor ${\mathcal Z}\longrightarrow {\mathcal Z}\times \Spec\,{\mathbb C}[t]/(t^{m+1})$ between schemes over $S$.
The existence of $({\mathcal W}/S)_m$ can be settled as in the absolute case by giving local equations. More precisely, since the construction is local on ${\mathcal W}$, we may assume that both $S$ and ${\mathcal W}$ are affine: $S=\Spec\,A$ and ${\mathcal W}\hookrightarrow\Spec\,A[X_i;i]$ is defined by $(f_{\alpha})_{\alpha}$. Then $({\mathcal W}/S)_m \hookrightarrow\Spec\,A[X_i,X'_i,\ldots,X_i^{(m)};i]$ is defined by $(f_{\alpha}, f'_{\alpha},\ldots,f^{(m)}_{\alpha})_{\alpha}$. Here $f^{(j)}=D^j(f)$, where $D$ is the unique derivation over $A$ of $A[X_i,X'_i,\ldots, X^{(m)}_i;i]$ such that $D(X_i^{(p)})= X_i^{(p+1)}$ for all $p$ (we put $X_i^{(0)}=X_i$ and $X_i^{(m+1)}=0$).
As in the absolute case, we have canonical projections $\rho_m\,:\,({\mathcal W}/S)_m\longrightarrow {\mathcal W}$. We have also a ``zero section'' $\sigma_m\,:\,{\mathcal W} \longrightarrow ({\mathcal W}/S)_m$.
The following lemma follows immediately from the functorial definition of relative jet schemes.
\begin{lemma} For every scheme morphism $S'\longrightarrow S$, if ${\mathcal W'}\simeq {\mathcal W}\times_SS'$, then we have a canonical isomorphism $({\mathcal W'}/S')_m\simeq ({\mathcal W}/S)\times_SS'$, for every $m$. In particular, for every closed point $s\in S$, the fiber of $({\mathcal W}/S)_m$ over $s$ is isomorphic with $({\mathcal W}_s)_m$. \end{lemma}
Over every closed point $s\in S$, the projection $\rho_m$ and the ``zero section'' $\sigma_m$ are the ones we defined in the
absolute case. On the other hand, the actions of ${\mathbb C}^*$ on each fiber globalize to an action on $({\mathcal W}/S)_m$ such that $({\mathcal W}/S)_m\setminus\sigma_m({\mathcal W})$ is invariant. After these preparations we can give the proof of Proposition~\ref{semicontinuity1}.
\begin{proof}[Proof of Proposition~\ref{semicontinuity1}] Consider the projection $({\mathcal W}/S)_m\setminus\sigma_m({\mathcal W}) \longrightarrow {\mathcal W}$ and let ${\mathcal Z}$ be the inverse image by this morphism of $\tau(S)$. We have a natural action of ${\mathbb C}^*$ on ${\mathcal Z}$ and the quotient by this action is a scheme $\overline {\mathcal Z}$ which is proper over $S$. The properness follows from the fact that it is locally projective, an assertion which is just the globalization of Lemma~\ref{cone} and can be proved in the same way.
But for every closed point $s\in S$ we have $f(s)=\dim\,\overline{\mathcal Z} _s+1$ (or $f(s)=0$ if $\overline{\mathcal Z}_s=\emptyset$)
and our assertion follows from the semicontinuity of the dimension of the fibers of a proper morphism. \end{proof}
We review now the basic facts about motivic integration on smooth varieties. This technique was developed by Kontsevich \cite{kontsevich}, Batyrev \cite{batyrev} and Denef and Loeser \cite{denef}. There are several possible extensions (see \cite{denef} for the general treatment), but we use only Hodge realizations of
motivic integrals on the space of arcs of a smooth variety. For a nice introduction to these ideas we refer to Craw \cite{craw}.
Suppose from now on that $X$ is a smooth variety of dimension $n$. On $X_{\infty}$ the theory provides an algebra of functions ${\mathcal M}$ and a finitely additive measure on $\mathcal M$ with values in the Laurent power series ring in $u^{-1}$ and $v^{-1}$: $$\mu\,:\,{\mathcal M}\longrightarrow S={\mathbb Z}[[u^{-1},v^{-1}]][u,v].$$ On $S$ we consider the topology defined by the descending sequence of subgroups $\{\oplus_{i+j\geq l}{\mathbb Z} u^{-i}v^{-j}\}_l$.
We will be interested in the subalgebra ${\mathcal Cyl}$ of ${\mathcal M}$ consisting of cylinders of the form: $\psi_m^{-1}(C)$, for some $m\geq 1$ and some constructible subset $C\subset X_m$. The measure of such a cylinder is given by $\mu(\psi_m^{-1}(C))=E(C;u,v)(uv)^{-(m+1)n}$, where $E(C;u,v)$ is the Hodge-Deligne polynomial of $C$. What is important for us is that if $C\subseteq X_m$ is a locally closed subset, then $E(C; u,v)$ is a polynomial of degree $2\,(\dim\, C)$ and the term of degree $2\,(\dim\,C)$ is $l(C)(uv)^{\dim\,C}$, where $l(C)$ is the number of irreducible components of $C$ of maximal dimension.
Besides the cylinders, some sets of measure zero appear in the definition of measurable functions. If $T\subset X_{\infty}$ is such that there is a sequence of cylinders $W_r\in {\mathcal Cyl}$ with $T\subset W_r$ for all $r$ and $\mu(W_r)\longrightarrow 0$, then $T\in {\mathcal M}$ and $\mu(T)=0$.
A function $F\,:\,X_{\infty}\longrightarrow {\mathbb N}\cup\{\infty\}$
is called measurable if $F^{-1}(s)\in {\mathcal M}$ for every $s\in{\mathbb N}\cup\{\infty\}$ and if $\mu(F^{-1}(\infty))=0$. If the sum $\,\,\,\,$ $\sum_{s\in{\mathbb N}}\mu(F^{-1}(s)) (uv)^{-s}$ is convergent in $S$, then $F$ is caled integrable, and the sum is called the motivic integral of $F$ and denoted by $\int_{X_{\infty}}e^{-F}$.
Every subscheme $Y\hookrightarrow X$ defines a function $F_Y$ on $X_{\infty}$, as follows. If $\gamma\in X_{\infty}$ is an arc over $x\in X$, then it can be identified with a ring homomorphism $\widetilde{\gamma}\,:\,\mathcal{O}_{X,x}\longrightarrow{\mathbb C}[[t]]$. We define $F_Y(\gamma):={\rm ord} (\widetilde{\gamma}(\mathcal{I}_{Y,x}))$. It follows from this definition that $$F_Y^{-1}(s)=\psi_{s-1}^{-1}(Y_{s-1})\setminus \psi_s^{-1}(Y_s),$$ for every integer $s\geq 0$ and $F_Y^{-1}(\infty)=Y_{\infty}$. Here we make the convention $Y_{-1}=Y$ and $\psi_{-1}=\psi_0$. It follows that $F_Y^{-1}(s)\in {\mathcal Cyl}$ for every $s\geq 0$. Moreover, the following lemma shows that if $Y\neq X$, then $F_Y^{-1}(\infty)\in {\mathcal M}$ and $\mu(F_Y^{-1}(\infty))=0$. Therefore $F_Y$ is measurable.
\begin{lemma}(\cite{mustata}, 3.7)\label{measurable} If $D\subset X$ is a divisor and $x\in X$ is a point such that ${\rm mult}_xD=a$, then \begin{equation} \dim\,(\rho_m^D)^{-1}(x)\leq nm-[m/a], \end{equation} where $[y]$ denotes the integral part of $y$. In particular, for every closed subscheme $Y$ of $X$, with $Y\neq X$, we have $\dim\,Y_m-n(m+1)\longrightarrow -\infty$, so that $F_Y^{-1}(\infty)\in {\mathcal M}$ and $\mu(F_Y^{-1}(\infty))=0$. \end{lemma}
One of the main results of the theory is the change of variable formula for motivic integrals.
\begin{proposition}\label{change_of_variable}[\,\cite{batyrev} 6.27, \cite{denef} 3.3] Let $\pi\,:\,X'\longrightarrow X$ be a proper, birational morphism of smooth varieties. Let $W=K_{X'/X}$ be the relative canonical divisor and $\pi_{\infty}\,:\,X'_{\infty} \longrightarrow X_{\infty}$ the morphism induced by $\pi$. For every measurable function $F\,:\,X_{\infty}\longrightarrow {\mathbb N}\cup\{\infty\}$, we have $$\int_{X_{\infty}}e^{-F}=\int_{X'_{\infty}}e^{-(F\circ\pi_{\infty}+F_W)},$$ meaning that one integral exists if and only if the other one does, and in this case they are equal. \end{proposition}
\begin{remark} In connection with the change of variable formula, note that if $F=F_Y$, for some proper subscheme $Y\hookrightarrow X$, then $F\circ\pi_{\infty}=F_{\pi^{-1}(Y)}$. \end{remark}
\section{Jet schemes of log terminal and log canonical pairs}
The main goal of this section is to prove the characterization of log canonical and Kawamata log terminal pairs in terms of jet schemes. First let us fix the notation.
Consider a pair $(X,Y)$ with $\dim\,X=n$ and fix a log resolution of
this pair $\pi\,:\,X'\longrightarrow X$ such that $\pi$ is an isomorphism over $X\setminus Y$.
Let $D=\pi^{-1}(Y)$ and $W=K_{X'/X}$ and write $D=\sum_{i=1}^ra_iE_i$, with $a_i\geq 1$ for all $i$
and $W=\sum_{i=1}^rb_iE_i$.
\begin{theorem}\label{main} With the above notation, if $q>0$, we have the following equivalences. \begin{enumerate} \item $(X, q\cdot Y)$ is log canonical if and only if $\dim\,Y_m\leq (m+1)(n-q)$, for every $m\geq 0$. Moreover, it is enough to have this inequality for some $m\geq 0$ such that $a_i\mid (m+1)$ for all $i$. \item $(X, q\cdot Y)$ is Kawamata log terminal if and only if $\dim\,Y_m < (m+1)(n-q)$, for every $m\geq 0$. As above, it is enough to have this inequality for some $m$ such that $a_i\mid (m+1)$ for all $i$. \end{enumerate} \end{theorem}
\begin{proof} The idea of the proof is similar to that of Theorem~3.1 in \cite{mustata}, so that we refer to that paper for some of the details.
With the notation in the theorem, note that we have $(X, q\cdot Y)$ log canonical (Kawamata log terminal) if and only if $b_i+1\geq (>) qa_i$,
for all $i$. Since the case $Y=\emptyset$ is trivial (we follow the convention that $\dim(\emptyset)=-\infty$), we assume from now on $Y$ nonempty.
We fix a function $f\,:\,{\mathbb N}\longrightarrow{\mathbb N}$, such that for every $s\geq 0$, \begin{equation} (\star)\,f(s+1)>f(s)+\dim\,Y_s+C(s+1), \end{equation} where $C\in{\mathbb N}$ is a constant such that $C>\mid N-(b_i+1)/a_i\mid$, for all $i$. We extend this function by defining $f(\infty)=\infty$. For the proof of the ``if'' part, we will add later an extra condition of the same type on $f$.
We integrate over $X_{\infty}$ the function $F=f\circ F_Y$.
Computing the integral from the definition, one can see that $\int_{X_{\infty}}e^{-F}=S_1-S_2$, where $$S_1=\sum_{s\geq 0}E(Y_{s-1};u,v)(uv)^{-sn-f(s)},$$ $$S_2=\sum_{s\geq 0}E(Y_s;u,v)(uv)^{-(s+1)n -f(s)}.$$
It is clear that every monomial which appears in the $s$th term of $S_1$ has degree bounded above by $2P_1(s)$ and below by $2P_2(s)$, where $P_1(s)=\dim\,Y_{s-1}-sn-f(s)$, and $P_2(s) =-sn-f(s)$, for every $s\geq 0$ (recall our convention that $Y_{-1}=X$). We have precisely one monomial of degree $2P_1(s)$, namely $(uv)^{P_1(s)}$ whose coefficient is $l(Y_{s-1})$, the number of irreducible components of $Y_{s-1}$ of maximal dimension.
Similarly, every monomial which appears in the $s$th term of $S_2$ has degree bounded above by $2Q_1(s)$ and below by $2Q_2(s)$, where $$Q_1(s)=\dim\,Y_s-(s+1)n-f(s),$$ $$Q_2(s)=-(s+1)n-f(s).$$ There is exactly one monomial of degree $2Q_1(s)$, namely $(uv)^{Q_1(s)}$ with coefficient $l(Y_s)$.
From the above evaluation of the terms in $S_1$ and $S_2$ and Lemma~\ref{measurable} we see that $F$ is integrable. Moreover, it follows from condition $(\star)$ that $P_1(s+1)<\min\{P_2(s), Q_2(s)\}$ for every $s$. We have also $Q_1(s)\leq P_1(s)$ for every $s$, with equality if and only if $s\geq 1$ and $\dim\,Y_s=\dim\,Y_{s-1}+n$. Lemma~\ref{measurable} implies therefore that we have strict inequality for infinitely many $s$. This shows that in $\int_{X_{\infty}}e^{-F}$ we have monomials of degree $2P_1(s)$ for infinitely many values of $s$.
We apply now the change of variable formula in Proposition~\ref{change_of_variable} for the morphism $\pi$ to get \begin{equation} \int_{X_{\infty}}e^{-F}=\int_{X'_{\infty}}e^{-(F\circ\pi_{\infty}+F_W)}. \end{equation}
Using the fact that $F\circ\pi_{\infty}=f\circ F_{\pi^{-1}(Y)}$ and that $W\cup\pi^{-1}(Y)$ has simple normal crossings, one can compute
explicitely the integral. For a subset $J\subseteq\{1,\ldots,r\}$ let $E_J^{\circ}=\cap_{i\in J}\setminus \cup_{i\not\in J}E_i$. With this notation we have $$\int_{X_{\infty}}e^{-F}=\sum_{J\subset\{1,\ldots,r\}}S_J,\,{\rm where}$$
$$ S_J=\sum_{\alpha_i\geq 1,i\in J}E(E_J^{\circ};u,v) (uv-1)^{|J|} \cdot (uv)^{-n-\sum_{i\in J}\alpha_i(b_i+1)-f(\sum_{i\in J}\alpha_ia_i)}.$$
Every monomial in the term of $S_J$ corresponding to $(\alpha_i)_{i\in J}$ has degree bounded above by $2R_1(\alpha_i;i\in J)$ and below by $2R_2(\alpha_i;i\in J)$, where $$R_1(\alpha_i;i\in J)=-\sum_{i\in J}\alpha_i(b_i+1)-f(\sum_{i\in J} \alpha_ia_i)$$ and $R_2(\alpha_i;i\in J)=R_1(\alpha_i;i\in J)-n$. Note that
$R_1(\emptyset)= -f(0)$.
Let us introduce the notation $\tau(s)=\dim\,Y_s-(s+1)(n-q)$. We see that if $J\neq\emptyset$, then $$R_1(\alpha_i; i\in J)= P_1(\sum_{i\in J}\alpha_ia_i)- \tau(\sum_{i\in J}\alpha_i a_i-1) +\sum_{i\in J}\alpha_i(qa_i-b_i-1).$$ Moreover, property $(\star)$ implies that if $J\neq\emptyset$, then \begin{equation}\label{eq1} P_1(\sum_{i\in J}\alpha_ia_i+1)<R_2(\alpha_i;i\in J) < R_1(\alpha_i;i\in J)< \end{equation} $$\min\{P_2(\sum_{i\in J}\alpha_ia_i-1), Q_2(\sum_{i\in J}\alpha_ia_i-1)\},$$ and $P_1(1)<R_2(\emptyset)$. This shows that the only monomial of the form $(uv)^{P_1(s)}$ which could appear in the part of $S_J$ corresponding to $(\alpha_j)_j$ is for $s=\sum_{j\in J}\alpha_ja_j$.
We prove now $(1)$.
Suppose first that $(X, q\cdot Y)$ is log canonical,
so that $b_i+1\geq qa_i$, for all $i$. Assume that for some $m\geq 0$, we have $\tau(m)>0$. It follows from the above discussion that $(uv)^{P_1(m+1)}$ does not appear in $\int_{X_{\infty}}e^{-F}$. As we have seen, this implies $\dim\,Y_{m+1}=\dim\,Y_m+n$. In particular, we have $\tau(m+1)>0$. Continuing in this way, we deduce $\dim\,Y_{s+1}=\dim\,Y_s+n$ for all $s\geq m$, which is impossible.
Conversely, suppose that for some fixed $m$ such that $a_i\mid (m+1)$ for all $i$ we have $\tau(m)\leq 0$, and that $b_j+1<ca_j$, for some $j$.
We choose the function $f$ such that in addition to $(\star)$ it satisfies the following condition. For every $p$, let ${\mathcal J}_p$ be the finite set consisting of all the pairs $(J, (\alpha_i)_{i\in J})$ such that $\sum_{i\in J}a_i\alpha_i=p$. The extra condition we require for $f$ is that for every $(J, (\alpha_i)_{i\in J})\in {\mathcal J}_{m+1}$ and every $(J', (\alpha'_i)_{i\in J'})\in {\mathcal J}_p$, with $p\leq m$, we have \begin{equation}\label{requirement} f(m+1)> f(p)-\sum_{i\in J}\alpha_i(b_i+1)+\sum_{i\in J'}\alpha'_i(b_i+1)+n. \end{equation} This means that $R_1(\alpha_i;i\in J)<R_2(\alpha'_i;i\in J')$.
Note that from $(\ref{eq1})$ we can deduce that if $(J',(\alpha'_i)_{i\in J'})\in {\mathcal J}_p$, with $p\geq m+2$, then \begin{equation}\label{eq2} P_1(m+1)> R_1(\alpha'_i;i\in J'). \end{equation}
On the other hand, the top degree monomials which appear in different terms of the sums $S_J$ (for possible different $J$) don't cancel each other, as they have positive coefficients. Let $d$ be the highest degree of a monomial which appears in a term corresponding to some $(J, (\alpha_i)_{i\in J})\in {\mathcal J}_{m+1}$. The previous remark shows that the corresponding monomial does not cancel with the other monomials which appear in terms corresponding to
$(J', (\alpha'_i)_{i\in J'})\in {\mathcal J}_{m+1}$.
Obviously we have $(\{j\}, (m+1)/a_j)\in {\mathcal J}_{m+1}$ and therefore our assumption that $\tau(m)\leq 0$ and $b_j+1<qa_j$ gives $d>2P_1(m+1)$. We also deduce from $(\ref{eq1})$ that $d<\min\{2P_2(m), 2Q_2(m)\}$.
We see from $(\ref{requirement})$ and $(\ref{eq2})$ that this monomial of
degree $d$ does not cancel with other monomials in terms corresponding to $(J', (\alpha'_i)_{i\in J'})\in {\mathcal J}_p$, if $p\neq m+1$. Therefore in the integral
$\int_{X_{\infty}}e^{-F}$ we have a monomial of degree $d$, with $2P_1(m+1)<d<\min\{2P_2(m), 2Q_2(m)\}$, a contradiction.
The proof of $(2)$ is entirely similar. \end{proof}
In order to state the formula for the log canonical threshold which follows from Theorem~\ref{main}, we make the following definition. Consider a pair $(X,Y)$ as before and $Z\subseteq X$
a nonempty closed subset. We allow also the case $Y=X$.
\begin{definition} With the above notation, if $m\in{\mathbb N}$, we define $\dim_ZY_m$ to be the dimension of $Y_m$ along $Y_m\cap\rho_m^{-1}(Z)$ i.e. the maximum dimension of an irreducible component $T$ of $Y_m$ such that $\rho_m(T)\cap Z\neq\emptyset$. \end{definition}
\begin{remark}\label{existence_U} It follows from our discussion of jet schemes in the previous section that
for every irreducible component $T$ of $Y_m$, the image $\rho_m(T)$ is closed in $Y$ (and therefore in $X$). Indeed, since $T$ must be invariant with respect to the ``action'' of ${\bf A}^1$ on $Y_m$, it follows that $\rho_m(T)=(\sigma_m^Y)^{-1}(T)$.
Therefore there is an open neighbourhood $U$ of $Z$ such that for every irreducible component $T$ of $Y_m$ with $\rho_m(T)\cap Z=\emptyset$, $U$ does not intersect $\rho_m(T)$. For every such $U$, we have $\dim_ZY_m=\dim\,(Y\cap U)_m$. \end{remark}
In the corollary below, we keep the previous notation for a log resolution of the pair $(X,Y)$.
\begin{corollary}\label{compute_lc} For every pair $(X,Y)$ and $Z\subseteq X$ as above, $$c_Z(X,Y)=n-\sup_{m\geq 0}{{\dim_ZY_m}\over {m+1}}.$$ Moreover, if $m\geq 0$ is such that $a_i\mid(m+1)$, for all $i$ such that $\pi(E_i)\cap Z\neq\emptyset$, then $$c_Z(X,Y)=n-{{\dim_ZY_m}\over {m+1}}.$$ \end{corollary}
\begin{proof} Note first that the formula is trivial when $Y=X$, since in this case for every $m\geq 1$, we know that $Y_m=X_m$ is a smooth variety of dimension $(m+1)\,\dim\,X$. Suppose from now on that $Y\neq X$.
For a fixed $m$, we can find an open neighbourhood $U$ of $Z$ such that $\dim_ZY_m=\dim\,(Y\cap U)_m$ and $c_Z(X,Y)=c(U,Y\cap U)$. Therefore, in order to prove the second assertion of the corollary we may assume $Z=X$, in which case the formula follows from Theorem~\ref{main}.
In order to prove the first assertion, it remains to be proved that $c_Z(X,Y)\geq n-(\dim_ZY_m)/(m+1)$, for every $m$. We reduce again to the case when $Z=X$, when we conclude by applying Theorem~\ref{main}. \end{proof}
\begin{corollary}\label{independance} If $(X,Y)$ is a pair and $Z$ a nonempty closed subset of $X$, then $c_Z(X,Y)$ depends only on $Y$, $\dim\,X$ and the closed subset $Y\cap Z\subseteq Y$, but not on the particular ambient variety $X$ or the embedding $Y\hookrightarrow X$. \end{corollary}
\begin{proof} The statement is a consequence of the formula in Corollary~\ref{compute_lc}. \end{proof}
We give now a version of the formula in Corollary~\ref{compute_lc} involving only the fibers of $\rho_m^Y$
over $Y\cap Z$. The drawback is that in this case, the supremum which appears in the formula can not be obtained, in general, for a specific integer.
\begin{corollary}\label{compute_lc2} If $(X,Y)$ is a pair and $Z\subset X$ a closed subset as before, then $$c_Z(X,Y)=\dim\,X-\sup_{m\geq 0} {{\dim\,(\rho^Y_m)^{-1}(Y\cap Z)}\over{m+1}}.$$ \end{corollary}
\begin{proof} Since the cases $Y=X$ and $Y\cap Z=\emptyset$ are trivial, we may assume that we are in neither of them. We obviously have $\dim\,(\rho^Y_m)^{-1}(Z\cap Y)\leq\dim_ZY_m$, for every $m$, so that the formula in Corollary~\ref{compute_lc} gives $$c_Z(X,Y)\leq\dim\,X-\sup_{m\geq 0} {{\dim\,(\rho^Y_m)^{-1}(Z\cap Y)}\over{m+1}}.$$
On the other hand, the formula giving the lower bound for the dimension of the fibers of a
morphism implies that $$\dim_ZY_m\leq\dim\,Y+\dim\,(\rho^Y_m)^{-1}(Y\cap Z).$$
Note that by Corollary~\ref{compute_lc}, there is a positive integer $N$ such that $c_Z(X,Y)=\dim\,X-{{\dim_ZY_m}\over
{m+1}}$ if $N\,|(m+1)$. Therefore we deduce $$c_Z(X,Y)\geq\dim\,X-{{\dim\,Y}\over{m+1}}- {{\dim\,(\rho^Y_m)^{-1}(Y\cap Z)}\over{m+1}},$$
for every $m$, such that $N\,|(m+1)$. This implies that for every $p\geq 1$, $$c_Z(X,Y)\geq \dim\,X- {{\dim\,Y}\over {pN}}-\sup_{m\geq 0} {{\dim\,(\rho^Y_m)^{-1}(Y\cap Z)}\over{m+1}},$$ so that in fact $$c_Z(X,Y)\geq \dim\,X-\sup_{m\geq 0} {{\dim\,(\rho_m^Y)^{-1}(Y\cap Z)}\over{m+1}},$$ which completes the proof of the corollary. \end{proof}
We end this section with the following \begin{example} Consider the case of a cusp: $$Y=Z(u^2-v^3)\hookrightarrow X={\mathbb A}^2.$$ $Y$ is a curve and $Y_{\rm sing}=\{0\}$, so that $\dim\,Y_m=\dim(\rho_m^Y) ^{-1}(0)$.
In order to describe $(\rho_m^Y)^{-1}(0)$, note that it consists of ring homomorphisms $$\phi\,:\,{\mathbb C}[u,v]/(u^2-v^3)\longrightarrow {\mathbb C}[t]/(t^{m+1})$$ such that ${\rm ord}(\phi(u))$, ${\rm ord}(\phi(v))\geq 1$. If $m>5$, this implies that $\phi(u)=t^3f$ and $\phi(v)=t^2g$, such that the classes $\overline{f}$, $\overline{g}$ of $f$ and $g$ in ${\mathbb C}[t]/(t^{m-5})$ satisfy $\overline{f}^2-\overline{g}^3=0$. In this way we get an isomorphism $(\rho_m^Y)^{-1}(0)\simeq Y_{m-6}\times{\mathbb A}^7$, so that $\dim\,Y_m=\dim\,Y_{m-6}+7$. This easily gives ${\rm sup}_m\{(\dim\,Y_m)/(m+1)\}=7/6$, so that $c({\mathbb A}^2,Y)=5/6$. \end{example}
\section{The log canonical threshold via jets}
In this section we apply Corollary~\ref{compute_lc} to deduce properties of the log canonical threshold which appear in the analytic context in \cite{demailly}. In the codimension one case, some of these properties are proved also in \cite{kollar} using log resolutions. We believe that our treatment, using properties of the jet schemes, simplifies many of the proofs.
In this section, when we consider pairs $(X,Y)$ we allow also the case $Y=X$. Recall that we follow the convention that $\dim(\emptyset)=-\infty$.
\begin{proposition}(\,\cite{demailly} 1.4)\label{inclusion} If $X$ is a smooth variety, $Y'$, $Y''$ are two closed subschemes such that $Y'\hookrightarrow Y''$ and $Z\subseteq X$ is a nonempty closed subset, then $c_Z(X,Y')\geq c_Z(X, Y'')$. \end{proposition}
\begin{proof} Since $Y'$ is a subscheme of $Y''$, it follows that $Y'_m$ is a subscheme of $Y''_m$, so that $\dim_ZY'_m\leq\dim_ZY''_m$ for every $m$.
The assertion now follows from Corollary~\ref{compute_lc}. \end{proof}
\begin{proposition}(\,\cite{demailly} 1.4)\label{bound1} For every pair $(X,Y)$
and every nonempty closed subset $Z\subseteq X$, we have $c_Z(X,Y)\leq {\rm codim}_Z(Y,X)$, where ${\rm codim}_Z(X,Y)$ denotes the smallest codimension of an irreducible component of $Y$ meeting $Z$. \end{proposition}
\begin{proof}
Using Corollary~\ref{compute_lc}, we have $$c_Z(X,Y)=\dim\,X-\sup_{m\geq 0}{{\dim_ZY_m}\over{m+1}}$$ $$\leq\dim\,X-\dim_ZY_0={\rm codim}_Z(Y,X).$$ \end{proof}
For a pair $(X,Y)$ and a point $x\in X$, we define ${\rm mult}_xY$ as follows. If $\mathcal{O}_{X,x}$ is the local ring of $X$ at $x$, with maximal ideal $\underline{m}_{X,x}$, and if ${\mathcal I}_{Y,x}\subseteq\mathcal{O}_{X,x}$ is the ideal of $Y$ at $x$, then ${\rm mult}_xY$ is the largest $q\in{\mathbb N}$ such that $I_{Y,x}\subseteq\underline{m}_{X,x}^q$. Note that if $Y$ is a divisor, then this is the same with the Hilbert-Samuel multiplicity, but in general the two notions are different.
\begin{proposition}(\,\cite{demailly} 1.4, \cite{kollar} 8.10)\label{bound2} Let $(X,Y)$ be a pair and $Z\subseteq X$ a nonempty closed subset. If $q$ is the smallest $p\in{\mathbb N}$ such that ${\rm mult}_xY\leq p$ for every $x\in Z$, then \begin{equation} 1/q\leq c_Z(X,Y)\leq (\dim\,X)/q. \end{equation} \end{proposition}
\begin{proof} The cases $Y=X$ (when $q=\infty$) and $Y\cap Z=\emptyset$ (when $q=0$) are trivial, so that we may suppose that we are in neither of them.
There is $y\in Z$ such that ${\rm mult}_{y}Y=q$. If $\rho_{q-1}\,:\,X_{q-1}\longrightarrow X$ is the canonical projection, then $\rho_{q-1}^{-1}(y)\subseteq Y_{q-1}$. Indeed, every local morphism $\mathcal{O}_{X,y}\longrightarrow{\mathbb C}[t]/(t^q)$ factors through $I_{Y,y}$, as $I_{Y,y}\subseteq\underline{m}_{X,y}^q$.
This gives $\dim_ZY_{q-1}\geq\dim\,\rho_{q-1}^{-1}(y)=(q-1)\dim\,X$. Corollary~\ref{compute_lc} implies that $c_Z(X,Y)\leq\dim\,X-(1/q)(q-1)\dim\,X=(\dim\,X)/q$.
For the other inequality we use Lemma~\ref{measurable}. This shows that for every $x\in X$ and every $m\in{\mathbb N}$, we have $\dim(\rho_m^{-1}(x)\cap Y_m)\leq m\dim\,X-[m/{\rm mult}_xY]$. Since ${\rm mult}_xY\leq q$ for every $x\in Z$, the same is true on an open neighbourhood $U$ of $Z$. We deduce $$\dim\,(U\cap Y)_m\leq \dim\,Y+m\cdot\dim\,X-[m/q], $$ for every $m$, and an easy computation gives $\dim\,(Y\cap U)_m/(m+1)\leq \dim\,X-1/q$. Corollary ~\ref{compute_lc} implies $c_Z(X,Y)\geq c(U, U\cap Y) \geq 1/q$. \end{proof}
\begin{proposition}(\,\cite{demailly} 2.7)\label{product} If $(X',Y')$ and $(X'',Y'')$ are two pairs and $Z'\subseteq X'$, $Z''\subseteq X''$ are two nonempty closed subsets, then \begin{equation} c_{Z'\times Z''}(X'\times X'', Y'\times Y'')= c_{Z'}(X',Y')+c_{Z''}(X'',Y''). \end{equation} \end{proposition}
\begin{proof} Proposition~\ref{product_of_jets} gives a canonical isomorphism $(Y'\times Y'')_m\simeq Y'_m\times Y''_m$, for every $m$. Therefore we have $\dim_{Z'\times Z''}(Y'\times Y'')_m= \dim_{Z'}Y'_m+\dim_{Z''}Y''_m$. We pick $m$ such that $a'_i$, $a''_j$, $a_k\mid (m+1)$ for all $i$, $j$ and $k$, where $a'_i$, $a''_j$ and $a_k$ are the coefficients appearing in log resolutions of $(X',Y')$, $(X'',Y'')$ and $(X'\times X'', Y'\times Y'')$.
We conclude by applying the second assertion in Corollary~\ref{compute_lc}. \end{proof}
\begin{proposition}(\,\cite{demailly} 2.2)\label{inverse_adjunction} Let $(X,Y)$ be a pair. If $H\subset X$ is a smooth irreducible divisor
and $Z\subseteq H$ is a nonempty closed subset, then \begin{equation} c_Z(X,Y)\geq c_Z(H, Y\cap H). \end{equation} \end{proposition}
\begin{proof} Let $U$ be an open neighbourhood of $Z$ in $X$ such that $c_Z(H, H\cap Y)=c\,(U\cap H, U\cap H\cap Y)$. Since $c_Z(X,Y)\geq c_{U\cap H} (U, U\cap Y)$, by restricting everything to $U$, we may assume that $Z=H$.
From the local description of the jet schemes, it follows that since $H\cap Y$ is defined locally in $Y$ be one equation, we have $(H\cap Y)_m$ defined locally in $Y_m$ by $m+1$ equations, for every $m\in{\mathbb N}$. Moreover, if $\rho_m\,:\,X_m\longrightarrow X$ is the canonical projection, then for every irreducible component $T$ of $Y_m$ such that $\rho_m(T)\cap H\neq\emptyset$, we have $T\cap (Y\cap H)_m\neq \emptyset$.
Indeed, it follows from our discussion in Section 2 that if $\sigma_m^Y\,:\,Y\longrightarrow Y_m$ is the ``zero section'', then $\sigma^Y_m(\rho_m(T))\subseteq T$. Moreover, since the ``zero section'' is functorial, we have $\sigma^Y_m(Y\cap H)\subset (Y\cap H)_m$, and we deduce that $(Y\cap H)_m\cap T$ contains $\sigma^Y_m(\rho_m(T)\cap H)$.
It follows that $\dim_HY_m\leq\dim\,(Y\cap H)+m+1$, so that
$$\sup_{m\geq 0}{{\dim_HY_m}\over {m+1}} \leq\sup_{m\geq 0}{{\dim\,(Y\cap H)_m}\over {m+1}}+1.$$ Applying Corollary~\ref{compute_lc}, we deduce the inequality in the proposition. \end{proof}
\begin{corollary}\label{gen_codimension} The statement of Proposition~\ref{inverse_adjunction} remains true if we replace $H$ with a smooth variety of arbitrary codimension. \end{corollary}
\begin{proof} If we cover $X$ with open subsets $U_i$, then we obviously have $c_Z(X,Y)=\inf_i\{c_{Z\cap U_i}(U_i, Y\cap U_i)\}$ and a similar relation for the restrictions to $H$. Since $H$ is smooth, it is locally a complete intersection, so that we can find an open cover of $X$ such that on each subset $U_i$, $H\cap U_i$ is an intersection of smooth divisors. It is therefore enough to apply Proposition~\ref {inverse_adjunction} inductively on each of these open subsets. \end{proof}
This corollary can be used to deduce a formula for the log canonical threshold of an intersection $Y'\cap Y''$ in terms of the dimensions of the jet schemes of $Y'$ and $Y''$. We give below the case when the closed subset $Z\subseteq X$ is a point, when the formula has a particularly nice form.
\begin{proposition}(\,\cite{demailly} 2.9)\label{intersection} If $X$ is a smooth variety, $Y'$, $Y''$ two proper closed subschemes and $z\in Y'\cap Y''$ a point, then: \begin{equation} c_z(X,Y')+c_z(X,Y'')\geq c_z(X,Y'\cap Y''). \end{equation} \end{proposition}
\begin{proof}
Consider the following cartesian diagram: \[ \begin{CD} Y'\cap Y'' @>>> X \\ @VVV @VV{\Delta}V \\ Y'\times Y'' @>>>X\times X \end{CD} \] where the horizontal maps
are the natural embeddings and $\Delta$ is the diagonal embedding. Applying Corollary~\ref{gen_codimension} to the embedding $\Delta$, the pair $(X\times X, Y'\times Y'')$, and the closed subset $\{(z,z)\}\subset \Delta(X)$, we deduce $$c_{(z,z)}(X\times X, Y'\times Y'')\geq c_z(X, Y'\cap Y'').$$
On the other hand, Proposition~\ref{product} implies $$c_{(z,z)}(X\times X,Y'\times Y'')=c_z(X,Y')+c_z(X,Y''),$$ and these two relations prove the proposition. \end{proof}
We use now Corollary~\ref{compute_lc} and Proposition~\ref{semicontinuity1} to prove a result of Demailly and Koll\'{a}r (\,\cite{demailly}) about the semicontinuity of the log canonical threshold. We first need a preliminary result. For the notation concerning the coefficients of the divisors in a log resolution, see the beginning of the previous section.
\begin{lemma}\label{finite} Let $\pi\,:\,{\mathcal X}\longrightarrow S$ be a smooth morphism and ${\mathcal Y}\hookrightarrow {\mathcal X}$ a closed subscheme. It is possible to find log resolutions for each fiber $({\mathcal X}_s, {\mathcal Y}_s)$ such that the coeficients $a_i$, $b_i$ which appear in all these resolutions form a finite set. \end{lemma}
\begin{proof} We make induction on $\dim\,S$. It is enough to find a locally closed cover of $S$ such that the restriction of $\pi$ over each member of the cover has the desired property. Therefore we may assume that $S$ is a smooth variety and it is enough to find a nonempty open subset $U$ of $S$ over which the restriction of $\pi$ has this property.
By a theorem of Nagata, we can embed ${\mathcal X}$ as an open subscheme in a scheme ${\mathcal X}'$ proper over $S$. By replacing ${\mathcal X}$ with ${\mathcal X}'$ and ${\mathcal Y}$ by its closure in ${mathcal X}'$, we may assume that $\pi$ is proper. After restricting to an open subset of $S$, we may assume that $\pi$ is also smooth.
Let $\pi'\,:\, \widetilde{\mathcal X}\longrightarrow {\mathcal X}$ be a log resolution for the pair $({\mathcal X}, {\mathcal Y})$. It is easy to see that after further restricting over an open subset of $S$ we can assume that for every $s\in S$, the restriction to the fiber $\pi_s\,:\, \widetilde{{\mathcal X}} _s\longrightarrow {\mathcal X}_s$ is a log resolution of $({\mathcal X}_s, {\mathcal Y}_s)$, in which case the assertion is obvious. \end{proof}
We can give now the proof of the semicontinuity result.
\begin{theorem}(\,\cite{demailly} 3.1)\label{semicontinuity2} Let $\pi\,:\,{\mathcal X}\longrightarrow S$ be a smooth morphism, ${\mathcal Y}\hookrightarrow {\mathcal X}$ a closed subscheme, and
$\tau\,:\,S\longrightarrow {\mathcal Y}$ a section of $\pi|_{\mathcal Y}$. The function defined by $$f(s)=c_{\tau(s)}({\mathcal X}_s,{\mathcal Y}_s),$$ for every closed point $s\in S$, is lower semicontinuous. \end{theorem}
\begin{proof} Lemma~\ref{finite} and Proposition~\ref{lc_formula} show that the set
$\{c_{\tau(s)}({\mathcal X}, {\mathcal Y})\,|\,s\in S\}$
is finite. Moreover, it follows from Lemma~\ref{finite} and Corollary~\ref{compute_lc} that we can find $N\geq 1$ such that for every $s\in S$ and every $m$ with $N| (m+1)$, we have $$c_{\tau(s)}({\mathcal X}_s, {\mathcal Y}_s)= \dim\,{\mathcal X}_s-{{\dim_{\tau(s)}({\mathcal Y}_s)_m}\over{m+1}}.$$
Fix $s_0\in S$. Since there are only finitely many log canonical thresholds to consider,
it is enough to show that for every $\epsilon>0$, there is an open neighbourhood $U$ of $s_0$ such that $c_{\tau(s)}({\mathcal X}_s, {\mathcal Y}_s)\geq c_{\tau(s_0)}({\mathcal X}_{s_0},{\mathcal Y}_{s_0}) -\epsilon$, for every $s\in U$. Since $\pi$ is flat, by restricting to an open neighbourhood of $s_0$ we may assume that $\dim\,{\mathcal X}_s$ is constant on $S$.
Therefore it is enough to find $m$ with $N|\,(m+1)$ and $U$ such that $$\dim_{\tau(s)}({\mathcal Y}_s)_m/(m+1)\leq \dim _{\tau(s_0)}({\mathcal Y}_{s_0})_m/(m+1)+\epsilon,$$
for all $s\in U$. We fix $m$ such that $N|\,(m+1)$ and $\dim\,{\mathcal X}_s/(m+1) \leq\epsilon$ for all $s$. Using Proposition~\ref{semicontinuity1}, we choose an open neighbourhood $U$ of $s_0$ such that $$\dim\,(\rho_m^{{\mathcal Y}_s})^{-1}(\tau(s))\leq \dim\,(\rho ^{{\mathcal Y}_{s_0}}_m)^{-1}(\tau(s_0)),$$ for all $s\in U$. The following inequalities show that $U$ satisfies our requirement: $$\dim_{\tau(s)}({\mathcal Y}_s)_m/(m+1)\leq \dim\,(\rho_m^{{\mathcal Y}_s})^{-1}(\tau(s))/(m+1)+\dim\,{\mathcal X}_s/ (m+1)\leq$$ $$\dim\,(\rho_m^{{\mathcal Y}_{s_0}})^{-1}(\tau(s_0))/(m+1)+\epsilon \leq \dim_{\tau(s_0)}({\mathcal Y}_{s_0})_m/(m+1)+\epsilon.$$ \end{proof}
We conclude by showing how the formula in Corollary~\ref{compute_lc} can be used to explicitely compute the log canonical threshold. We consider the case of monomial ideals and derive a formula from \cite{agv} (see also \cite{howald} for a formula in this context for all the multiplier ideals).
Let us fix the notation. $X={\bf A}^n$ is the affine space and $Y=V(I)$, where $I\subset R=k[X_1,\ldots, X_n]$ is a monomial ideal such that $(0)\neq I\neq R$. For ${\bf a}\in{\mathbb Z}^n$, we use the notation $X^{\bf a}=\prod_iX_i^{a_i}$. The vector $(1,\ldots,1)\in{\mathbb Z}^n$ is denoted by ${\bf e}$.
The Newton polytope of $I$, denoted by $P_I$, is the convex hull (in ${\mathbb R}^n$) of the set $\{{\bf a}\in{\mathbb Z}^n\,|\, X^{\bf a}\in I\}$.
\begin{proposition}(\cite{agv}, \cite{howald})\label{monomial} With the above notation, the log canonical threshold of the pair $(X,Y)$ is given by
$$c(X,Y)=\sup\{r>0\,|\,{\bf e}\in rP_I\}.$$ \end{proposition}
\begin{proof} We will use the formula in Corollary~\ref{compute_lc}, so that we estimate first $\dim\,Y_m$, for every $m\geq 1$. Note that $Y_m$ can be covered by the locally closed subsets $Z_{a_1,\ldots,a_n}$, with $0\leq a_i\leq m+1$ for all $i$, where $Z_{a_1,\ldots,a_n}$ is the set of ring homomorphisms $\phi\,:\,R/I\longrightarrow k[t]/(t^{m+1})$, with ${\rm ord}\,(\phi(X_i)) =a_i$, for every $i$. We put ${\rm ord}(\phi(X_i))=m+1$ if $\phi(X_i)=0$.
It is clear that if $Z_{a_1,\ldots,a_n}\neq\emptyset$, then $\dim\,Z_{a_1,\ldots,a_n}=(m+1)n-\sum_i a_i$. On the other hand, $Z_{a_1,\ldots,a_n}\neq\emptyset$ if and only if for every ${\bf b}\in{\mathbb N}^n$ such that $X^{\bf b}\in I$, we have $\sum_ia_ib_i \geq (m+1)$.
Let $P_I^{\circ}$ be the polar polyhedron of $P_I$, defined by
$$P_I^{\circ}=\{{\bf u}\in{\mathbb R}^n\,|\,\sum_iu_iv_i\geq 1, {\rm for}\,{\rm all}\, {\bf v}\in P_I\}.$$ We see that $Z_{a_1,\ldots,a_n}\neq\emptyset$ if and only if $(a_1/(m+1),\ldots, a_n/(m+1))\in P_I^{\circ}$.
From the above discussion, we deduce that $${{\dim\,X_m}\over{m+1}}=n-{1\over{(m+1)}}\inf_{\bf a}\sum_ia_i,$$ where the infimum is taken over all ${\bf a}\in{\mathbb N}^n\cap (m+1)P_I ^{\circ}$ such that $a_i\leq m+1$ for all $i$.
The formula in Corollary~\ref{compute_lc} gives $$c\,(X,Y)=\inf_{\bf a}\sum_ia_i,$$ the infimum being taken over all ${\bf a}\in P_I^{\circ}\cap{\mathbb Q}^n\cap [0,1]^n$.
We clearly have $P_I^{\circ}\subset{\mathbb R}_+^n$. Note that if ${\bf a}\in P_I^{\circ}$ and ${\bf a}'$ is defined by $a'_i={\rm min}\{a_i,1\}$ for all $i$, then ${\bf a}'\in P_I^{\circ}$. We deduce that
$c\,(X,Y)=\inf_{{\bf a}\in P_I^{\circ}}\sum_ia_i$. In order to complete the proof it is enough to note that ${\bf e}\in rP_I$ if and only if for every ${\bf a}\in P_I^{\circ}$ we have $\sum_i a_i\geq r$ (this comes from the fact that $P_I$ is the polar polyhedron of $P_I^{\circ}$. \end{proof}
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\end{document} | arXiv |
The 5G candidate waveform race: a comparison of complexity and performance
Robin Gerzaguet1,
Nikolaos Bartzoudis2,
Leonardo Gomes Baltar3,
Vincent Berg1,
Jean-Baptiste Doré1,
Dimitri Kténas1,
Oriol Font-Bach2,
Xavier Mestre2,
Miquel Payaró2,
Michael Färber3 &
Kilian Roth3
EURASIP Journal on Wireless Communications and Networking volume 2017, Article number: 13 (2017) Cite this article
5G will have to cope with a high degree of heterogeneity in terms of services and requirements. Among these latter, the flexible and efficient use of non-contiguous unused spectrum for different network deployment scenarios is considered a key challenge for 5G systems. To maximize spectrum efficiency, the 5G air interface technology will also need to be flexible and capable of mapping various services to the best suitable combinations of frequency and radio resources. In this work, we propose a comparison of several 5G waveform candidates (OFDM, UFMC, FBMC and GFDM) under a common framework. We assess spectral efficiency, power spectral density, peak-to-average power ratio and robustness to asynchronous multi-user uplink transmission. Moreover, we evaluate and compare the complexity of the different waveforms. In addition to the complexity analysis, in this work, we also demonstrate the suitability of FBMC for specific 5G use cases via two experimental implementations. The benefits of these new waveforms for the foreseen 5G use cases are clearly highlighted on representative criteria and experiments.
The Next Generation Mobile Networks (NGMN) Alliance highlights in [1] the necessity to make more spectrum available in the existing sub-6 GHz radio bands and introduce new agile waveforms that exploit the existing underutilized fragmented spectrum, in order to satisfy specific fifth-generation (5G) operating scenarios. The goal of the waveform symbiosis will therefore be to flexibly optimize the use of existing underutilized spectrum resources, guarantee interference-free coexistence with legacy transmissions and provide an improved spectral containment compared to the orthogonal frequency division multiplexing (OFDM) modulation that is widely used in broadband wireless systems operating below 6 GHz. The need for a new 5G waveform has also been discussed in the context of asynchronous multi-user 5G operating scenarios [2], which envision sporadic access of mobile nodes that rapidly enter in a dormant state after a data transaction. This feature, called fast dormancy, has been identified as the root cause of significant signaling overhead on cellular networks [3]. Relaxed synchronization schemes have been considered to limit the amount of required signaling. This is the case, for instance, when the mobile node carries only a coarse knowledge of time synchronization. The massive number of devices and the support of multi-point transmissions in 5G use cases will imply the use of relaxed synchronization, potentially leading to strong inter-user interference.
OFDM is a multicarrier communication scheme that has been widely adopted in a number of different wired and wireless communication systems. Among others, 3GPP adopted it as the underlying physical layer (PHY) technology in mobile broadband systems denoted as 4G long-term evolution (LTE). It exhibits some intrinsic drawbacks including frequency leakage caused by its rectangular pulse shape, spectral efficiency loss due to the use of a cyclic prefix (CP) and need for fine time and frequency synchronization in order to preserve the carrier orthogonality, which guarantees a low level of intra and inter-cell interferences. To overcome these limitations, several alternative candidates have been intensively studied in the literature over the past few years, such as universal filtered multi carrier (UFMC) [2], generalized frequency division multiplexing (GFDM) [4] and filter bank multicarrier (FBMC) [5].
This paper presents these popular candidate 5G waveforms and compares them in terms of specific performance features such as spectral efficiency, power spectral density and peak-to-average power ratio (PAPR). In addition, we also analyze multi-user interference scenarios and compare the performance of candidate waveforms for several delays and carrier frequency offsets, accounting for a different number of guard carriers and according to different waveform parameters. We also compare their baseband computational complexity using as a baseline reference the current waveforms used in 4G LTE downlink (DL) and uplink (UL). Finally, we present practical implementations of FBMC-based waveforms demonstrating the feasibility of adopting such PHY-layer schemes and verifying their superior performance when compared to CP-OFDM, under shared licensed spectrum use cases (i.e. a driving technology of several 5G use cases).
The 3GPP is in the process of studying and eventually adopting proposals for the new 5G air interface, which eventually will be standardized within 2017. Depending on the end use and specific operation band (i.e. sub 6 GHz and millimeter wave frequencies), it is expected that two versions of 5G radio access waveforms will be standardized. Previous works from other researchers have focused either on a specific 5G candidate waveform [6, 7] or on comparing different performance features (or target applications) [8, 9] from the ones presented in this paper. Details related to real-time implementation of 5G waveforms [10] and laboratory-based experimental validation [11, 12] are very scarce in the literature and typically provide benchmarking of a particular use case [13]. The work presented in this paper is in this sense more transversal covering key performance aspects of the most popular 5G waveform candidates, along with a computational complexity analysis and practical real-time implementations targeting field programmable gate array (FPGA) devices under realistic spectrum cohabitation scenarios (including experimental validation). Other sources related to the work presented in this paper are encountered in white papers [14] or application notes [15] describing add-on software libraries that target arbitrary waveform generation instruments. Such software pre-products are used to underpin the market readiness of the instrumentation and measurements sector and its ability to timely provide test solutions once the 5G air interface will be finally standardized; typically, the non-academic references do not enter in a fine-grain analysis of the computational complexity and do not present hardware implementation details of the different candidate 5G waveforms.
The main objectives of this paper, in addition to be a comprehensive introduction and comparison of the most promising multicarrier waveforms are to (i) provide a unified comparison framework where waveform performance are assessed wrt representative criteria, (ii) perform a baseband complexity analysis of these aforementioned waveforms and (iii) propose implementation examples for FBMC as well as to describe a use case example where FBMC shows its interest. This proposed analysis shows the interest in designing, studying and implementing alternatives to classic CP-OFDM.
This paper is organized as follows: the main 5G waveforms candidates are presented in Section 2. A comparison regarding several criteria (spectral density, power spectral density, PAPR and robustness in asynchronous multi-user scenario) and a complexity analysis are described in Section 3. Two practical implementations of FBMC are presented in Section 4. Finally, the last section draws some conclusions.
5G candidate waveforms background
In this section, we briefly introduce the main 5G waveform candidates that we will compare and study in Sections 3 and 4.
CP-OFDM
In CP-OFDM, a block of complex symbols is mapped onto a set of orthogonal carriers (see Fig. 1). Due to the use of inverse fast Fourier transform (IFFT) (resp. FFT) process of size N FFT, CP-OFDM architecture has a low complexity. The principle of OFDM is to divide the total bandwidth into N FFT carriers, so that channel equalization can often be reduced as a one tap coefficient per carrier. Finally, a cyclic prefix (CP) is inserted. It guarantees circularity of the OFDM symbol, if the delay spread of the multipath channel is lower than the CP length. It, however, leads to a loss of spectral efficiency, as the CP is used to transmit redundant data. To limit the PAPR, an additional discrete Fourier transform (DFT) (resp. IDFT) a precoding stage can be inserted before the IFFT (resp. after FFT), leading to the so-called single carrier frequency division multiple access (SC-FDMA) used in the uplink of 3GPP-LTE.
CP-OFDM transceiver (additional SC-FDMA stages in dash). S/P and P/S, respectively, denote parallel to serial and serial to parallel
FBMC
FBMC waveform consists in a set of parallel data that are transmitted through a bank of modulated filters. The prototype filter, parametrized by the overlapping factor K, can be chosen to have very low adjacent channel leakage. One may differentiate between two main variants of FBMC: one based on complex (QAM) signaling, also referred to as filtered multi tone (FMT), and another based on real valued offset QAM (OQAM) symbols, also referred to as FBMC/OQAM. The latter ensure orthogonality in real domain to maximize spectral efficiency. The first variant (FMT) is currently employed in standards like Telecommunication Equipment Distribution Service (TEDS), and achieves orthogonality among subcarriers by physically reducing their frequency domain overlapping, thus reducing the SE in a similar proportion as CP-OFDM.
FBMC/OQAM, on the contrary, is able to achieve the maximum SE [16] by imposing the orthogonality in the real domain only. Given the SE optimality of FBMC/OQAM, this variant is universally considered as the baseline FBMC modulation. Multiple alternative ways of implementing FBMC-OQAM in a computationally efficient manner are existing, although the most important are polyphase networks (PPN) and frequency spreading (FS) implementations. In PPN architecture (see Fig. 2), OQAM symbols feed an FFT of size N FFT and then into a PPN. The receiver applies matched filtering before a FFT of size N FFT and multi-tap equalization is performed in a per carrier basis [17].
PPN-FBMC transceiver
In FS-FBMC (see Fig. 3), OQAM symbols are filtered in frequency domain [5]. The result then feeds an IFFT of size K N FFT, followed by an overlap and sum operation. At the receiver side, a sliding window selects K N FFT points every N FFT/2 samples [18]. A FFT of size K N FFT is applied followed by filtering by the prototype matched filter.
FS-FBMC transceiver
UFMC
UFMC waveform is a derivative of OFDM waveform combined with post-filtering, where a group of carriers is filtered by using a frequency domain efficient implementation [2]. This subband filtering operation is motivated by the fact that the smallest unit used by the scheduling algorithm in frequency domain in 3GPP LTE is a resource block (RB), which is a group of 12 carriers. The filtering operation leads to a lower out-of-band leakage than for OFDM. The UFMC transmitter (see Fig. 4) is composed of B subband filtering that modulate the B data blocks. The transmitted signal uses no CP, but there is still a spectral efficiency loss due to the time transient (tails) of the shaping filter. The Rx stage is composed of a 2 N FFT point FFT, which is then decimated by a factor 2 to recover the data. A windowing stage can also be inserted before the FFT. It introduces interference between carriers but is interesting to consider for asynchronous uplink transmissions as it helps to separate contiguous users.
UFMC transceiver
GFDM
GFDM waveform is based on the time-frequency filtering of a data block, which leads to a flexible, non-orthogonal waveform [4]. A data block is composed of K carriers and M time slots, and transmits N=K M complex modulated data. In this paper, we consider that the data is cyclic filtered by a root-raised-cosine (RRC) filter that is translated into both frequency and time domains (see Fig. 5) as it is customarily done [4, 6]. To avoid inter-symbol interference, a CP is added at the end of each block of symbols. To further improve the spectral location, a windowing process can be added in the transmitter.
GFDM transceiver
Several receiver architectures have been investigated in the literature for GFDM, and we consider in this paper a matched filter (MF) receiver scheme: each received block is filtered by the same time and frequency translated filters as in the transmission stage [4]. As the modulation is non-orthogonal, it is necessary to implement an interference cancellation (IC) scheme [19], which improves the performance but severely increases the complexity of the receiver. More recently, OQAM was also considered in GFDM to allow the use of less complex linear receivers instead of IC [20].
Performance comparison and complexity
In Section 2, 5G candidate waveforms have been introduced, and their main parameters and architectures have been described. In this section, we compare the waveforms regarding several criteria: their power spectral density, their spectral efficiency and their PAPR. Besides, a performance comparison of the waveform candidates in a typical multi-user asynchronous access scheme is done. We eventually compare the computational complexity of the different waveforms.
Spectral efficiency, power spectral density and PAPR comparison
We consider the parameters based on LTE 10 MHz with QPSK modulation, a FFT size of 1024 (and a CP size of 72 samples) and a sampling frequency of 15.36 MHz. For FBMC-OQAM, the overlapping factor is set to 4 using the PHYDYAS filter [16]. For GFDM, the number of symbols per carrier M is set to 15, and the roll-off factor of the RRC filter is set to 0.1. For UFMC, we use a Dolph-Chebyshev filter of length L=73 (with 40 dB attenuation) to have the same SE as OFDM [2]. In the following, we compare the SE (Fig. 6), the power spectral density (Fig. 7), and the complementary cumulative distribution function (CCDF) of the PAPR (Fig. 8) of selected 5G candidate waveforms, which constitute a representative set of performance metrics.
Spectral efficiency vs burst duration for QPSK
Power spectral density of the waveforms
CCDF of PAPR
We first consider the spectral efficiency on Fig. 6. In OFDM, SC-FDMA, GFDM and UFMC, the spectral efficiency does not depend on the burst duration and it is a function of the modulation parameters. But for FBMC-OQAM, it depends on the frame duration, and the spectral efficiency loss is due to the transient state of the shaping filter if assumed that no transmission takes place during this period. Thus, there is no constant loss per symbol and the spectral efficiency increases with the burst duration to reach an asymptotic level equal to the modulation order. For GFDM, the spectral efficiency is higher compared to OFDM as a GFDM symbol is M times longer compared to an OFDM one. Indeed, for GFDM, the spectral efficiency loss due to the CP insertion is limited as there is one CP per GFDM symbol (i.e. 1 CP per M equivalent OFDM symbols).
We now consider the power spectral density in Fig. 7. To better stress the impact of the adjacent channel leakage, we consider two users that occupy 36 carriers (3 RBs), with 12 guard carriers (1 RB) as guard band. The best spectral localization is obtained with FBMC-OQAM. GFDM has a slightly lower out-of-band leakage compared to OFDM but is clearly outperformed by UFMC. With the addition of the windowing process, GFDM becomes comparable to the UFMC.
We compute on Fig. 8 the CCDF of the PAPR for the considered waveforms, for a burst duration of 3 ms. SC-FDMA, due to its (quasi) single carrier property, offers the best performance. The other modulations, which are multicarrier, have a higher PAPR and none of the multicarrier candidates with the chosen parametrization offers better performance than OFDM. However, it should also be noted that the gap is small, around 0.5 dB.
Multi-user access scheme
In this section, we compare the performance of the 5G waveform candidates in a typical multi-user asynchronous access scheme [21]. We consider two users, user equipment (UE) 1 and UE 2. The first user occupies three RBs and is assumed to be perfectly synchronized in time and frequency domains with its serving base station. The secondary user occupies nine RBs and suffers from a delay error (i.e. a timing offset) and a potential carrier frequency offset due to a synchronization mismatch with downlink channel. Due to the timing and frequency errors, the secondary user interferes with the first one. The data stream of the first user is decoded (assuming no channel and no noise), and the performance in terms of mean square error (MSE) on the decoded constellation is evaluated. The interference only comes from the interferer user. The spacing in terms of guard carriers between the two users is variable: no guard carrier (contiguous allocation), one guard carrier and two guard carriers.
We consider the previously introduced waveforms. OFDM and SC-FDMA have the same performance, so only OFDM curves are plotted. For GFDM and UFMC, we consider the impact of additional windowing (denoted, respectively, by wGFDM and wUFMC). The performance results are depicted on Fig. 9, on the left without carrier frequency offset and on the right with a carrier frequency offset of 10%.
Performance of the different candidate 5G waveforms in asynchronous multi-user scenario
We have shown in Fig. 7 that windowing for GFDM lowers the out-of-band leakages, as it improves the spectral isolation between users. For the multi-user scenario, it is shown that the performance without windowing is better in case of low delay error value (as the interference introduced by the windowing effect is not negligible), but that the performance with windowing is better when the delay error does not belong to the CP interval. This is due to the trade-off between the self-interference introduced by the windowing and the isolation gain between users offered by the windowing.
The windowing effect for UFMC is different from GFDM as the windowing is applied on the receiver side, and has no consequences on the power spectral density of the transmitted signal. It however improves the performance in the multi-user scenario. These results are very similar to the results presented in [2] and validate the positive impact of the windowing scheme for UFMC. Due to the very good spectral location of the FBMC prototype filter, the MSE reaches its lower bound as soon as a guard carrier is inserted. Besides that, the performance is independent from the delay error value.
We now consider the performance with an additional carrier frequency offset of 10%. Due to the additional interference introduced by the frequency error, the MSE is higher for all the waveforms, except for FBMC-OQAM with at least one guard carrier. For OFDM, the orthogonality cannot be preserved anymore and a strong interference level is present even if the delay error belongs to the CP interval. Besides that, without guard band, the performance of GFDM and FBMC-OQAM becomes very similar, and is slightly better than UFMC out of CP. If the guard carrier number is non-null, FBMC exhibits no interference, and the hierarchy between the other candidates is the same as without carrier frequency offset.
As a conclusion, GFDM, UFMC and FBMC-OQAM are promising candidates for the multi-user asynchronous access scheme and outperform classic CP-OFDM. UFMC waveform is an interesting option as the SE is comparable to OFDM and the pulse shaping function gives robustness to asynchronous access. Backward compatibility with well-known OFDM algorithms (e.g. channel estimation, massive-input massive-output (MIMO) detectors) is also preserved. FBMC and GFDM go further since the well-localized frequency response enables the use of fragmented spectrum with minor interference on adjacent bands. Very good performances are demonstrated in non-synchronous access as well. However, transceiver complexity should be managed and some concepts should be revisited (e.g. MIMO schemes, short packet adaptation) for a future deployment.
Computational complexity comparison
In this section, we perform a comparison of the computational complexity for the different waveform schemes in a single antenna configuration. We quantify the complexity in terms of the total number of real multiplications per symbol. We consider the signal processing operations involved in the generation of the MC and single-carrier (SC) signals, as well as the recovery of the subcarrier/subchannel signals and equalization in the presence of multipath propagation. Here, we do not consider the operations involved in channel estimation or calculation of the equalizer coefficients. The first reason is because those signal processing tasks are not in the user data chain, which is the one that concentrates the processing burden, and the second is because of the many existing algorithms for those tasks making the choice of one not trivial. Moreover, we assume that all systems are perfectly synchronized.
Cyclic prefix OFDM
We assume that the total of M subcarriers are available out of which M f are occupied with symbols. We will consider first the number of real valued multiplications to transmit one block of M f symbols. Starting with the fast Fourier transform (FFT), the number of real multiplications of a M-point FFT/ inverse FFT (IFFT) using a split-radix algorithm is given by
$$ C_{\text{FFT}}(M) = M\left(\log_{2}(M) - 3\right) + 4. $$
Since the transmitter (Tx) of a CP-OFDM system is basic built with one single IFFT and by including the windowing operation we get
$$ C^{\text{Tx}}_{\text{OFDM}} = C_{\text{FFT}}(M)+ 4 (M+L_{\text{CP}}). $$
For the demodulation and recovery of the subcarrier signals, two processing tasks are necessary: FFT and single-tap equalization per subcarrier. The complexity is given by
$$ C^{\text{Rx}}_{\text{OFDM}} = C_{\text{FFT}}(M) + 4 M_{f}, $$
for the MC demodulation and equalization.
FBMC-OQAM
Assuming an FBMC-OQAM system where the prototype filter has length KM, two approaches can be adopted for the generation and recovery of the MC signal: the polyphase-based and the frequency spread-based structures. We consider first the complexity of FBMC-OQAM implemented with a structure based on the polyphase decomposition of the prototype filter and using a direct form realization of the polyphase components (PC) [22]. The Tx is composed of three steps after the OQAM modulation:
Phase rotations to get linear phase filters in each subcarrier
IFFT
Polyphase filtering followed by block overlapping of 50%
At the Rx side, similar operations in the inverted order are implemented including one more step: polyphase filtering, FFT, multitap channel equalization per sub-carrier with an equalizer of length L eq and the OQAM demodulation. The phase rotations at the receiver side can be embedded in the equalizer coefficients. The total number of real valued multiplications is then given by
$$\begin{array}{@{}rcl@{}} C^{\text{Tx}}_{\text{PC-FB}} &=& 2C_{\text{FFT}}(M) + 4MK + 4M_{f}, \end{array} $$
$$\begin{array}{@{}rcl@{}} C^{\text{Rx}}_{\text{PC-FB}} &=& 2C_{\text{FFT}}(M) + 4MK + 4L_{\text{eq}}M_{f}, \end{array} $$
where we have taken into account that the IFFT and the polyphase network work with double of the QAM symbol rate and that the coefficients of the prototype are real valued.
The second approach is a frequency domain filtering, a.k.a. frequency spread [23] based FBMC, featuring also a general prototype with length KM and designed using the frequency sampling approach with only 2(K−1) non-zero coefficients. In this case, the structure changes drastically. The subcarrier signals have to be spread over K+1 frequency domain samples and each of them multiplied by one of the prototype frequency domain coefficients. The overlapping parts in frequency domain are all added and then transformed with the IFFT of size KM and finally an overlap and add of dimension M/2 is performed to generate the time domain signal. At the Rx side, the inverse operations are done resulting in the following complexity
$$\begin{array}{@{}rcl@{}} C^{\text{Tx}}_{\text{FS-FB}} &=& 2C_{\text{FFT}}(KM) + 8M_{f}(K - 1). \end{array} $$
$$\begin{array}{@{}rcl@{}} C^{\text{Rx}}_{\text{FS-FB}} &=& 2C_{\text{FFT}}(KM) + 16M_{f}(K - 1), \end{array} $$
where we have taken into account that the equalizer coefficients can be incorporated in the frequency domain coefficients of the filters.
UFMC/UF-OFDM/filtered CP-OFDM
The UFMC system can be parametrized between two extremes: in one end, one single CP-OFDM signal is filtered by one filter to reduce the out-of-band radiation. At the other end, each or a minimum number of resource blocks is transformed with the IFFT and filtered with its own filter. In an UFMC system with maximum granularity, B resource blocks each with M B subcarriers require B FFTs of size M NB, where each of them has only M B non-zero inputs. The modulation is performed in the following steps: First, the signal of each subband is spread over the whole symbol length and transformed into the frequency domain. Then, the filtering is performed in the frequency domain and the sum of all subbands is converted into the time domain [24].
Instead of filtering and then transforming, a non-matched filtering is applied in the frequency domain [25]. The Rx has then three steps:
Windowing in the time domain
FFT transformation of size 2M with zero padding and half of the outputs thrown away
Frequency domain filtering and equalization
The total number of multiplications is then given by
$$\begin{array}{@{}rcl@{}} C^{\text{Tx}}_{\text{UFMC}}\! &\,=\,& \!B\left[C_{\text{FFT}}(2M_{\text{NB}}) + C_{\text{FFT}}(M_{\text{NB}})\right. \end{array} $$
$$\begin{array}{@{}rcl@{}} & \quad +& \left. 8M_{\text{NB}}\right] + C_{\text{FFT}}(2M), \\ C^{\text{Rx}}_{\text{UFMC}}\! &\,=\,&\! C_{\text{FFT}}(2M) + 8M + 4M_{f}. \end{array} $$
Generalized frequency division multiplexing
The generalized frequency division multiplexing (GFDM) modulation scheme is based on circular convolving each subcarrier in a block of data with a filter kernel. In contrast to OFDM, a cyclic prefix is added per block and not per symbol [6, 26]. Since a circular convolution can be calculated as a multiplication of two vectors in frequency domain, the transmitter and receiver can be efficiently implemented using the FFT. Out of a total number of subcarriers M, only M f are used. N symbols per subcarrier are combined to form one transmission block. In total, N M f data symbols can be transmitted per block. The prototype filter is designed to overlap with M a adjacent subcarriers and it is typically chosen to be an RRC filter M a =2. As described in [6], excluding the trivial operations like reordering, the following signal processing tasks need to be performed at the transceiver:
Transformation of the data signal of each subcarrier into the frequency domain
Filtering in the frequency domain
Transformation of the signal into the time domain
The complexity at Tx is given by
$$ C^{\text{Tx}}_{\text{GFDM}} = M_{f} C_{\text{FFT}}(N) + 4M_{f} M_{a} N + C_{\text{FFT}}(NM). $$
The details of the corresponding receiver are described in [26]. It is important to mention that since the subcarriers are overlapping, it is necessary to cancel this interference to achieve a sufficient performance. In [26], the authors use the detected symbols to subtract the interference to adjacent subcarriers in an iterative fashion. For a constellation as large as 64QAM it was shown that J=8 iterations are sufficient. The receiver can be divided into the following signal processing tasks:
Transformation of the signal into the frequency domain
Channel equalization
Iterative interference cancellation
The complexity at the receiver (Rx) is then given by
$$ \begin{aligned} C^{\text{Rx}}_{\text{GFDM}} &= C_{\text{FFT}}(NM) + 4(M_{f} + 2 (M_{a} -1))N \\ & \quad + 4M_{f} M_{a} N + J M_{f} \left(2C_{\text{FFT}}(N) + 4N\right). \end{aligned} $$
Numerical evaluation
In this section, we perform a comparison of the computational complexity for the different waveform schemes in a single antenna configuration. We quantify the complexity in terms of the total number of real multiplications per symbol. We consider the signal processing operations involved in the generation of the MC and SC signals, as well as the recovery of the subcarrier/subchannel signals and equalization in the presence of multipath propagation. Here, we do not consider the operations involved in channel estimation or calculation of the equalizer coefficients. The first reason is because those signal processing tasks are not in the user data chain, which is the one that concentrates the processing burden, and the second is because of the many existing algorithms for those tasks making the choice of one not trivial. Moreover, we assume that all systems are perfectly synchronized.
For the numerical evaluation, we evaluate the complexity in the base and mobile stations (BS, MS) separately and consider the number of multiplications and additions normalized by the number of transmitted QAM symbols. We assume a similar overhead in terms of training or reference signals for all waveforms. Moreover, we consider the following four scenarios:
Downlink with narrowband allocation per mobile station
Downlink with broadband allocation
Uplink with narrowband allocation per mobile station
Uplink with broadband allocation
In Scenarios 1 and 3, for each MS, six resource blocks are allocated and 17 mobiles are served simultaneously. In Scenarios 2 and 4, only one MS is served and all 1320 available frequency bins are allocated to it. The parameters are described in Table 1.
Table 1 Simulation parameters
In Figs. 10 and 11, we have the complexity results for the different waveforms in different scenarios in the BS and MS.
Computational complexity in terms of number of real valued multiplications per Tx/Rx symbol in the base station
Computational complexity in terms of number of real valued multiplications per Tx/Rx symbol in the base mobile station
For UFMC, we have used the most efficient structure to the best of our knowledge and the filter impulse response is set in order to get the same overhead as in CP-OFDM. For GFDM, we consider four symbols per carrier and an IC receiver with eight iterations.
We can see that PPN FBMC and GFDM involve less than three times the number of operations than SC-FDMA, while FS-FBMC involves seven times more operations and UFMC more than nine times. It should besides be noted that, in case of FBMC, UFMC and GFDM, a filtering process is embedded in the waveform generation stage. One can note that if an agile access to fragmented spectrum is needed, a filtering process should be added to OFDM transmitter (with the granularity of a RB) and then the complexity of OFDM-based waveform increases exponentially.
Practical implementations
The benefits of adopting new agile waveforms in 5G wireless communication systems has also been evaluated in the context of two practical FPGA-based implementations that reproduce two different coexistence scenarios that are envisioned to be highly relevant in 5G. After carefully considering the conclusions drawn in Sections 3 and 4 related to the coexistence of 5G waveforms with legacy ones in fragmented spectrum use cases and the associated computational complexity under fair comparison conditions, we have selected to implement a waveform based on the FBMC scheme. These real-time implementations allow to address the inherent digital design challenges of FBMC waveforms and also, in one of the cases, to experimentally validate the prime spectral efficiency and spectral characteristics of this 5G candidate waveform.
A flexible radio transceiver for TVWS based on FBMC
Dynamic spectrum sharing has been proposed to improve spectrum utilization. The digital switch over (DSO) in TV bands, which has resulted in making the so-called TV white space (TVWS) UHF spectrum available, was the first actual example where such a mechanism has been allowed. In 2009, the US radio regulator—the Federal Communication Commission (FCC)—authorized opportunistic unlicensed operation in the TV bands [27]. The initiative has recently been followed by the UK regulator (Ofcom) [28] and by the Ministry of Internal Affairs and Communications of Japan. In this context, opportunistic communication systems have to coexist with incumbent systems, i.e. TV broadcast signals. The coexistence scheme is enforced with a priority mechanism where opportunistic systems must guarantee that no harmful interference will be incurred to the incumbents. Harmful interference is defined in a twofold way. Firstly, co-channel communication between incumbent and opportunistic systems is prohibited. This means that opportunistic systems must be able to assess the presence of incumbent signals and access only channels vacant from any incumbent. Besides, opportunistic systems have a limited amount of time to evacuate the channel when an incumbent is switched on. Secondly, the adjacent channel leakage ratio (ACLR) is limited in order to prevent an opportunistic system from interfering with an incumbent operating in another channel, and in particular in adjacent channels. In [27], ACLR is restricted to be at least 55 dB. Such a high ACLR requirement is specific to the TVWS context and similar requirements are considered in other countries (e.g. in the UK [28]). These requirements of flexibility and stringent ACLR have led IEEE DYSPAN Standard Committee to identify the necessity to develop a new standard defining radio interface for white space radio systems: IEEE 1900.7 standard [29]. The standard is based on FBMC PHY. Through an implementation on a flexible hardware TVWS transmitter, [30] showed that FBMC modulation can meet ACLR levels prescribed by the FCCs coexistence requirements. The actual implementation was aimed at assessing the performance under real hardware impairment conditions, such as limited dynamic range digital-to-analog converters (DAC). One of the main shortcomings of FBMC was supposed to be its implementation complexity. However, recent results have shown that a flexible approach was possible with very limited complexity overhead [30].
The complexity has been evaluated for a Xilinx Kintex-7 FPGA and is given by the amount of slice registers, look-up tables (LUT) and DSP48E1 cells used by the different modules of the receiver design. Slice registers correspond to the amount of register cells used, while LUT to the amount of combinatorial logic in the design. DSP48E1 cells are digital signal processing (DSP) cells dedicated to multiplication and accumulation (MAC) operations. The results have shown that the computational complexity of the FBMC transmitter is very similar to the OFDM transmitter complexity in this context. Furthermore, the receiver complexity is only around 30% higher than the one of the CP-OFDM receiver (see Table 2). In addition, the proposed block-wise processing approach requires FBMC symbols to be stored, which impacts the size of the memory (2.5x the one of an equivalent CP-OFDM RX). However, such memory sizes can be implemented at a very limited footprint and cost on current silicon technology nodes.
Table 2 Complexity comparison for FBMC implementation on Xilinx FGPA
An agile FBMC waveform for fragmented spectrum use cases
In this section, we present the real-time FPGA implementation of an agile FBMC DL transmitter, designed to optimally exploit unused fragmented spectrum. The transmitter has been validated in a waveform cohabitation scenario that includes a real-life professional mobile radio (PMR) system operating in the 400 MHz band. The PMR terminals use the terrestrial trunked radio for police (TETRAPOL) air interface. The benefits of the FBMC waveform have been benchmarked versus an LTE system, and for this reason, the DL FBMC frame features high similarity with the LTE standard specifications (release 9). Each FBMC symbol comprises 72 active carriers with 15 kHz spacing within the 1.4-MHz signal bandwidth. This results in a 10-ms radio frame organized in ten subframes, containing 150 FBMC symbols. The first three symbols in each frame include a preamble which enables synchronization under non-contiguous spectrum. The pilot pattern is based on the reference signal structure of LTE with additional "auxiliary pilots" that compensate the contribution from surrounding symbols. The FBMC waveform uses a fast convolution scheme [31] with a short transform length of eight FFT bins per carrier spacing (i.e. 16 points) and a long transform length of 1024 points.
The DL FBMC and LTE transmitters feature a single- and a two-antenna configuration (i.e. based on the open-loop spatial multiplexing scheme of LTE) and they have been jointly implemented in an FPGA-based system-on-chip (SoC) device. The baseband design optimizes the utilization of processing resources in the different digital signal processing (DSP) stages. Time division multiplexing allows reusing only two 16-point FFT and two 1024-point IFFT engines to implement the MIMO FBMC scheme (i.e. 128 16-point FFTs are combined to provide the inputs to each large IFFT). The latter is based on an overlap and save convolution, which results in a variable number of samples in the input of each DSP stage and also a variable number of bits per sample for the fixed-point arithmetic operations. A latency-aware memory plane helps to minimize the embedded memory utilization and addresses the variable storage needs of the pipelined DSP architecture. Moreover, a centralized control unit has been designed to govern the synchronous communication of the diverse DSP stages. Finally, clock-gating techniques minimize the dynamic power consumption. Figure 12 shows the baseband design of the DL MIMO FBMC transmitter and Table 3 the overall implementation results in the target Xilinx XC7Z045 device.
Optimized digital baseband design of the MIMO FC-FBMC transmitter and detailed implementation metrics of the four different transmitters (SISO and MIMO, FC-FBMC and CP-OFDM) hosted in the FPGA-based SoC device
Table 3 FPGA implementation metrics of the LTE and FBMC DL PHY-layer for the SISO and MIMO (open-loop spatial multiplexing) antenna schemes
The hardware setup described in [31] has been used to assess the TETRAPOL terminal performance when coexisting in the same band either with a MIMO FBMC or LTE transmission. A configurable spectral hole of 30 kHz has been left at the FBMC or LTE DL signal to accommodate the 12.5 kHz TETRAPOL signal. The performance of the TETRAPOL terminal was evaluated by calculating the BER for different received signal powers and carrier power ratios between the coexisting signals, under an ITU vehicular-A mobile channel (50 km/h). Each curve shown in Fig. 13 averages 10.000 TETRAPOL frames for each measurement configuration. As it can be observed, the FBMC waveform offers superior interference protection to the coexisting TETRAPOL transmission (around 29 dB) when compared to LTE.
Performance of the TETRAPOL terminal evaluated in relation to the interference received from the in-band MIMO broadband transmissions
Flexible and efficient use of non-contiguous unused spectrum targeting heterogeneous mobile network deployment scenarios is one of the key challenges that future 5G systems would need to tackle. To maximize SE, the 5G air interface technologies will need to be flexible and capable of mapping various services to the best suitable combinations of frequency and radio resources. Alternatives to classic CP-OFDM have thus been intensively studied in the past few years. In this work, a fair comparison of several 5G multicarrier waveform candidates (OFDM, UFMC, FBMC, GFDM) has been conducted under a common framework. SE, power spectral density, PAPR and computational complexity have been assessed for the different waveforms. Resistance of the waveforms in a typical asynchronous multi-user uplink scenario, for different parametrisation and configuration, has also been addressed. A synthesis chart is depicted on Fig. 14. We have shown that UFMC waveform is an interesting option as the SE is comparable to that of OFDM and the pulse shaping function enhances the performance in the asynchronous multi-user scenario. UFMC also preserves backward compatibility with well-known OFDM algorithms (channel estimation, MIMO detectors). FBMC-OQAM and GFDM go a step further: interference between adjacent bands is minor, making these waveforms particularly interesting for 5G scenarios, at a price of slight complexity increase. FBMC-OQAM is a promising solution even when it comes to practical implementations: in this paper, we have presented results that reveal the feasibility of the FBMC-OQAM waveform. We have demonstrated the relevance of FBMC especially when targeting the deployment of secondary systems in existing underutilized (and spectrally fragmented) bands, where interference protection of primary transmissions is mandatory. Two different DL FBMC systems were implemented and validated when coexisting at the same band either with primary PMR or TVWS transmissions. The efficient and non-interfering shared utilization of licensed spectrum (either between primary or primary and secondary transmissions) is a enabler of 5G systems [1] the benefits of which can also be applied on unlicensed shared spectrum access (or even combinations of the two).
Waveform comparison: synthesis
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The work of Intel and CEA-Leti is partially supported by the European Commission under Horizon 2020 projects FANTASTIC-5G (GA 671660) and Flex5Gware (GA 671563). The work of CTTC was partially supported by the European Commission under the project EMPhAtiC (GA 318362) and currently partially supported by the Generalitat de Catalunya (2014 SGR 1551) and by the Spanish Government under project TEC2014-58341-C4-4-R.
CEA-Leti, Minatec Campus, Grenoble, France
Robin Gerzaguet, Vincent Berg, Jean-Baptiste Doré & Dimitri Kténas
Centre Tecnológic de Telecomunicacions de Catalunya, Barcelona, Spain
Nikolaos Bartzoudis, Oriol Font-Bach, Xavier Mestre & Miquel Payaró
Intel Deutschland GmbH, Feldkirchen, Germany
Leonardo Gomes Baltar, Michael Färber & Kilian Roth
Robin Gerzaguet
Nikolaos Bartzoudis
Leonardo Gomes Baltar
Vincent Berg
Jean-Baptiste Doré
Dimitri Kténas
Oriol Font-Bach
Xavier Mestre
Miquel Payaró
Michael Färber
Kilian Roth
Correspondence to Robin Gerzaguet.
Gerzaguet, R., Bartzoudis, N., Baltar, L.G. et al. The 5G candidate waveform race: a comparison of complexity and performance. J Wireless Com Network 2017, 13 (2017). https://doi.org/10.1186/s13638-016-0792-0
Emerging Air Interfaces and Management Technologies for the 5G Era | CommonCrawl |
Spherical 3-manifold
In mathematics, a spherical 3-manifold M is a 3-manifold of the form
$M=S^{3}/\Gamma $
where $\Gamma $ is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere $S^{3}$. All such manifolds are prime, orientable, and closed. Spherical 3-manifolds are sometimes called elliptic 3-manifolds or Clifford-Klein manifolds.
Properties
A spherical 3-manifold $S^{3}/\Gamma $ has a finite fundamental group isomorphic to Γ itself. The elliptization conjecture, proved by Grigori Perelman, states that conversely all compact 3-manifolds with finite fundamental group are spherical manifolds.
The fundamental group is either cyclic, or is a central extension of a dihedral, tetrahedral, octahedral, or icosahedral group by a cyclic group of even order. This divides the set of such manifolds into 5 classes, described in the following sections.
The spherical manifolds are exactly the manifolds with spherical geometry, one of the 8 geometries of Thurston's geometrization conjecture.
Cyclic case (lens spaces)
The manifolds $S^{3}/\Gamma $ with Γ cyclic are precisely the 3-dimensional lens spaces. A lens space is not determined by its fundamental group (there are non-homeomorphic lens spaces with isomorphic fundamental groups); but any other spherical manifold is.
Three-dimensional lens spaces arise as quotients of $S^{3}\subset \mathbb {C} ^{2}$ by the action of the group that is generated by elements of the form
${\begin{pmatrix}\omega &0\\0&\omega ^{q}\end{pmatrix}}.$
where $\omega =e^{2\pi i/p}$. Such a lens space $L(p;q)$ has fundamental group $\mathbb {Z} /p\mathbb {Z} $ for all $q$, so spaces with different $p$ are not homotopy equivalent. Moreover, classifications up to homeomorphism and homotopy equivalence are known, as follows. The three-dimensional spaces $L(p;q_{1})$ and $L(p;q_{2})$ are:
1. homotopy equivalent if and only if $q_{1}q_{2}\equiv \pm n^{2}{\pmod {p}}$ for some $n\in \mathbb {N} ;$ ;}
2. homeomorphic if and only if $q_{1}\equiv \pm q_{2}^{\pm 1}{\pmod {p}}.$
In particular, the lens spaces L(7,1) and L(7,2) give examples of two 3-manifolds that are homotopy equivalent but not homeomorphic.
The lens space L(1,0) is the 3-sphere, and the lens space L(2,1) is 3 dimensional real projective space.
Lens spaces can be represented as Seifert fiber spaces in many ways, usually as fiber spaces over the 2-sphere with at most two exceptional fibers, though the lens space with fundamental group of order 4 also has a representation as a Seifert fiber space over the projective plane with no exceptional fibers.
Dihedral case (prism manifolds)
A prism manifold is a closed 3-dimensional manifold M whose fundamental group is a central extension of a dihedral group.
The fundamental group π1(M) of M is a product of a cyclic group of order m with a group having presentation
$\langle x,y\mid xyx^{-1}=y^{-1},x^{2^{k}}=y^{n}\rangle $
for integers k, m, n with k ≥ 1, m ≥ 1, n ≥ 2 and m coprime to 2n.
Alternatively, the fundamental group has presentation
$\langle x,y\mid xyx^{-1}=y^{-1},x^{2m}=y^{n}\rangle $
for coprime integers m, n with m ≥ 1, n ≥ 2. (The n here equals the previous n, and the m here is 2k-1 times the previous m.)
We continue with the latter presentation. This group is a metacyclic group of order 4mn with abelianization of order 4m (so m and n are both determined by this group). The element y generates a cyclic normal subgroup of order 2n, and the element x has order 4m. The center is cyclic of order 2m and is generated by x2, and the quotient by the center is the dihedral group of order 2n.
When m = 1 this group is a binary dihedral or dicyclic group. The simplest example is m = 1, n = 2, when π1(M) is the quaternion group of order 8.
Prism manifolds are uniquely determined by their fundamental groups: if a closed 3-manifold has the same fundamental group as a prism manifold M, it is homeomorphic to M.
Prism manifolds can be represented as Seifert fiber spaces in two ways.
Tetrahedral case
The fundamental group is a product of a cyclic group of order m with a group having presentation
$\langle x,y,z\mid (xy)^{2}=x^{2}=y^{2},zxz^{-1}=y,zyz^{-1}=xy,z^{3^{k}}=1\rangle $
for integers k, m with k ≥ 1, m ≥ 1 and m coprime to 6.
Alternatively, the fundamental group has presentation
$\langle x,y,z\mid (xy)^{2}=x^{2}=y^{2},zxz^{-1}=y,zyz^{-1}=xy,z^{3m}=1\rangle $
for an odd integer m ≥ 1. (The m here is 3k-1 times the previous m.)
We continue with the latter presentation. This group has order 24m. The elements x and y generate a normal subgroup isomorphic to the quaternion group of order 8. The center is cyclic of order 2m. It is generated by the elements z3 and x2 = y2, and the quotient by the center is the tetrahedral group, equivalently, the alternating group A4.
When m = 1 this group is the binary tetrahedral group.
These manifolds are uniquely determined by their fundamental groups. They can all be represented in an essentially unique way as Seifert fiber spaces: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 3.
Octahedral case
The fundamental group is a product of a cyclic group of order m coprime to 6 with the binary octahedral group (of order 48) which has the presentation
$\langle x,y\mid (xy)^{2}=x^{3}=y^{4}\rangle .$
These manifolds are uniquely determined by their fundamental groups. They can all be represented in an essentially unique way as Seifert fiber spaces: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 4.
Icosahedral case
The fundamental group is a product of a cyclic group of order m coprime to 30 with the binary icosahedral group (order 120) which has the presentation
$\langle x,y\mid (xy)^{2}=x^{3}=y^{5}\rangle .$
When m is 1, the manifold is the Poincaré homology sphere.
These manifolds are uniquely determined by their fundamental groups. They can all be represented in an essentially unique way as Seifert fiber spaces: the quotient manifold is a sphere and there are 3 exceptional fibers of orders 2, 3, and 5.
References
• Peter Orlik, Seifert manifolds, Lecture Notes in Mathematics, vol. 291, Springer-Verlag (1972). ISBN 0-387-06014-6
• William Jaco, Lectures on 3-manifold topology ISBN 0-8218-1693-4
• William Thurston, Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, New Jersey, 1997. ISBN 0-691-08304-5
| Wikipedia |
Equity impact of a choice reform and change in reimbursement system in primary care in Stockholm County Council
Janne Agerholm1,2,
Daniel Bruce1,2,
Antonio Ponce de Leon1,2,3 &
Bo Burström1,2
In 2008 reforms were introduced in primary care in Stockholm County Council to increase patient choice. These reforms included changes to the reimbursement system from one that was primarily based on need-weighted capitation system (75 %) to a system largely based on fee-for-service (60 %) and freedom of establishment of primary care clinics. The new reimbursement system created incentives for producing many visits and additional primary care clinics were established, particularly in areas that were already well served. This study analyses if and how the choice reform and change of reimbursement system has affected equity in primary care consumption by investigating whether the increase in visits reflects levels of need and to what extent the reform have affected equity in health care between areas.
Cross-sectional data from the public health survey in Stockholm County 2006 (n = 34,707) and 2010 (n = 30,767) were linked to individual register data on socio-demographic characteristics and health care utilization in 2007 and 2011. Information on self-reported health status and disability pension was used as indicators of need of health care. Negative binomial regression was used to analyse the differences in GP visits between the two years.
The total number of visits to GPs increased by 46 % from 2007 to 2011 and the proportion visiting a GP increased by 17 %. Both men and women reporting poor mental health and women with limiting longstanding illness and poor self-rated health had significantly smaller increase in number of visits than healthy women and men. Men with poor health status living in disadvantaged areas had a smaller increase than men with poor health status living in other areas of Stockholm County.
The reform did not particularly benefit those with greater health care needs, and there are indications of a negative impact on equity in primary care after the introduction of the reform. There were signs of a lesser increase in total number of visits to GPs among those with poor mental health, among women with poor self-rated health and limiting longstanding illness, and among men living in disadvantaged areas.
The Swedish health care system is universal and primarily financed through general taxation to minimize financial barriers for access. Co-payments do exist for most types of health care, approximately 22€ for a visit to the general practitioner (GP) and 38€ for a visit to a specialist. The same level of co-payments apply to all adults, but when the yearly cost of outpatient health care services exceeds 119€ all patients have the right to free outpatient services for additional visits. The health care system is regulated by the Swedish Health Care Act (1982) and as equity in health and health care has high priority, both horizontal and vertical equity are emphasized in the act: "Health and medical services are aimed at assuring the entire population of good health and of care on equal terms. Care shall be provided with due respect for the equal worth of all people and the dignity of the individual. Priority should be given to those who are in greatest need of health and medical care" [1]. Despite these intentions, health care services are not always made available to all social groups in an equitable way.
There is an inverse social gradient in health, both on individual level and on area level: the lower the socioeconomic group, the higher mortality and morbidity [2, 3]. Consequently, the need of health care services is higher in lower socioeconomic groups and in more disadvantaged areas as well.
Equity in health care implies that health services should match needs [4]; that populations with greater needs should have more services than those with lesser needs. How to assess and measure whether this principle of equity actually applies is not straightforward. Often the number of visits to doctors [5–7] or cost of care [8, 9] are used as outcomes to assess equity in health care. However, different quality measures including patient's experience and assessment of quality of care or access to health care have also been used [10]. For certain diseases differences between socioeconomic groups in the attainment of target levels of certain quality indicators of care such as HbA1c in diabetes, and of target levels of blood pressure have been used to assess equity in health care [11–13]. All measures address different aspects of equity in health care, and presumably more than one measure is needed for a deeper understanding of equity in health care.
Several studies on Swedish data have demonstrated inequity in utilization of health care services despite a long tradition of universal health care coverage [5, 14–17], and higher socioeconomic groups in some cases receive more complex and more expensive treatment than do lower socioeconomic groups [9].
Although the health care system might not be thought of as a main determinant of inequalities in health, it has an important role in tackling inequalities in the consequences of ill health and could potentially promote equity in health by providing care to groups in need, and by protecting lower income groups from further impoverishment due to ill health [18].
It is particularly in the first tier of the health care system that there may be a potential for health care services to help reduce inequity in health [19]. It is therefore of interest to know whether primary care services are accessible and offered in an equitable way.
The choice reform in primary care
In 2010 a choice reform was legislated and introduced in primary care in Sweden. The focus of the reform was on giving patients free choice of provider and freedom of establishment for providers. Many county councils also changed their reimbursement systems in primary care with the introduction of the reform. In Stockholm, where the reform was introduced already in 2008, the reimbursement system was changed from a need-based resource allocation system, based primarily on need-weighted capitation (75 %) with age and area specific socioeconomic indicators used as proxies for need; to a system based more on fee-for-service (60 %), less on capitation (40 %) and now only age-weighted capitation, letting patient choice and demand direct the resource allocation to a much higher greater degree than previously [20].
The intention of the primary care choice reform was primarily to increase access to primary care. In Stockholm the total number of visits in primary care increased from 4.8 million visits in 2007 to 5.8 million visits in 2012 [21] and the number of primary care clinics increased following the reform. New clinics were established in many areas, but according to a national report, most clinics were established in areas where the service level was already high and the needs of health care were lesser [22].
The former need-based resource allocation system benefitted primary care units operating in socioeconomically disadvantaged areas. The current reimbursement system does not take into account the fact that health is poorer and disease strikes at younger ages in more deprived areas and resources are now shifting from areas with greater health care needs to areas with smaller needs [21]. This means that primary care units in areas with a population with greater health care needs now have to produce more visits in order to maintain the same amount of resources as before the reform. This could lead to lower quality of care or to prioritizing less demanding patients in order to achieve the production needed to sustain the unit's income.
In view of the strong emphasis on equity in the Swedish Health Care Act, it is relevant to assess how the reform and the change in reimbursement system have affected groups with different levels of need of health care. To date, no scientific studies have investigated how the reform has affected equity in health care in Stockholm County. However, there have been several reports on the effect of the reform on health care utilization where also issues about equity have been partly addressed [20–24]. One of these, a survey among doctors in charge of primary health care clinics in Stockholm County, showed that only 1 % of the responders believed that the present system favoured groups with greater need and 78 % believed that the system discriminated against these groups [23].
One explanation for these findings is in the construction of the reimbursement system, which creates incentives to produce many visits, as short visits are reimbursed the same as long visits. Hence, there is a risk that doctors might prioritise patients with less complex health problems, at the expense of patients with more complex health problems.
This study investigates if and how the choice reform and change of reimbursement system has affected equity in primary care consumption, through addressing the following research questions
Did the visits increase more in groups with greater health care needs?
Did the visits increase more in disadvantaged areas?
We used cross-sectional data from the Stockholm County Council's Public Health Survey (PHS) from 2006 and 2010, a survey sent to randomly chosen individuals in Stockholm County above 18 years of age [25]. In total 65,474 participated (34,707 in 2006 (61 %) and 30,767 in 2010 (56 %)). This study was restricted to individuals aged 25–84 years (n = 59,065). The lower age limit was chosen to allow the use of income and educational level as a proxy for socioeconomic position. The upper age limit was chosen because the upper age limit in the 2006 survey was 84 years of age.
We obtained register data on health care utilization in 2007 for the responders of the 2006 survey and from 2011 for the responders of the 2010 survey. The health care data was obtained from the Stockholm County Council's administrative database for analysis and follow-up of health care utilization, which contains information on all registered outpatient and inpatient care financed by Stockholm County Council. The data are anonymized through encrypted personal identity numbers.
Data on socio-demographic background characteristics and disability pension were obtained from the Longitudinal Integration Database for Health Insurance and Labour Market Studies from Statistics Sweden. This is a collection of variables from different population registers linked individually. We used the variables: age, sex, disability pension, and educational level.
All participants were informed at baseline about the survey data being linked with register data. Ethical approval for this study was obtained from the Central Ethical Review Board in Stockholm, Sweden (Dnr.:2008/1542-32).
Health care utilization in primary care was measured by the number of visits to GPs and by the proportion of people having one or more visits to a GP.
Health status measures used as indicators for need of health care
Self-rated health and limiting longstanding illness
In the analysis of horizontal equity in health care two health status measures from the PHS were used to control for need: Self-rated health (SRH) and Limiting longstanding illness (LLI) (18). The question on self-rated health was phrased: "How do you assess your overall health? Is it: Excellent, Good, Fair, Poor or Very poor? In the analysis this variable was dichotomized into: good health (very good and good) and less than good health (fair, poor and very poor).
Participants were asked if they had a longstanding illness, problems after an accident, a handicap or another longstanding health problem. Those responding affirmatively were asked if the problem caused any difficulties in relation to the ability to work and perform other everyday activities (yes, to a high degree; yes, to some degree; not at all). Participants responding affirmatively to both questions were categorized as having a LLI.
SRH and LLI were also used to differentiate between groups with high and low needs of health care, to assess the vertical aspect of equity in health care. The two variables were combined and individuals with LLI and less than good SRH were compared to individuals with no LLI and good SRH.
Individuals with disability pension were compared to individuals with no disability pension; these analyses were restricted to individuals aged 25–64 years.
General Health Questionnaire-12 (GHQ12)
To differentiate between groups with and without mental health problems we used the GHQ-12, which is a screening instrument used to detect diagnosable psychiatric disorders [26]. We used the GHQ-scoring, rating each problem as either present or absent [27] and set the threshold to 2/3, where 3 or more was coded as having mental health problems and 2 or less as having no mental health problems [27, 28].
Disability pension
Individuals, aged 25–64, who had obtained disability pension during the year were compared to individuals aged 25–64 who did not receive disability pension during the year of interest.
Disadvantaged areas
In 1998 disadvantaged residential areas with high levels of unemployment, high proportion of foreign-born residents, low level of education, in the larger Swedish cities were identified for a Metropolitan Development Initiative, a programme which increased resources from state and municipal level during the period 1998–2004 to decrease segregation and improve living conditions. In these areas health is poorer and disease strikes at younger ages [29] and could therefore be regarded as areas with greater health care needs. In this study respondents living in a disadvantaged area in Stockholm County were compared to respondents living in other areas of the county.
The association between age and number of visits was not linear across all ages and the analysis with the total study population was adjusted for age - mean age, age2 and age3. In the age group 25–64, age had a linear association with the number of visits and the analysis of relative changes in visits among those with disability pension (aged 25–64) was therefore only adjusted for age as a continuous variable.
Educational level was categorized into 3 different levels: Primary school (9–10 years of schooling or less), Secondary school (at least one year of secondary school) and Post-secondary school (at least one year of post-secondary education).
We calculated the mean number of visits and the proportion visiting a doctor separately for men and women in different age, education and income groups, and among groups with different health care needs. When analysing the change in proportion of individuals visiting the doctor, the statistic is a ratio of two random variables (the proportion going to the doctor 2011 and the proportion going to the doctor 2007). Such an expression does not have a closed form solution for the variance. We therefore derived an approximation for the variance based on a Taylor expansion [30].
The outcome, number of visits to GPs, is a discrete variable that has a very non-normal distribution. Among different count data regression models we chose the negative binomial regression model based on goodness of fit measures, reliable estimates, and comparisons of loglikelihoods and AIC. The negative binomial regression model was used specifically to analyse the differences in GP visits between groups over time adjusted for covariates [31]. Estimates from the regression model had to be compared in a somewhat complex setting which is illustrated below using an example.
We want to compare the difference in number of visits between disadvantaged areas and the rest of the Stockholm County between 2007 and 2011. Let μdis, 2007 denote the average number of visits among individuals in disadvantaged areas in 2007. An expression for the increase in number of visits from 2007 to 2011 for disadvantaged areas vs the rest of Stockholm County is then
$$ \frac{\frac{\mu_{dis,\kern0.5em 2011}}{\mu_{rest,\kern0.5em 2011}}}{\frac{\mu_{dis,\kern0.5em 2007}}{\mu_{rest,\kern0.5em 2007}}} $$
This expression of interest is a relative comparison of the gradient 'dis' vs 'rest' 2011 with the same gradient in 2007. It can be shown that using the estimates of the negative binomial regression model the expression becomes:
$$ \frac{\frac{{\widehat{\mu}}_{dis,\kern0.5em 2011}}{{\widehat{\mu}}_{rest,\kern0.5em 2011}}}{\frac{{\widehat{\mu}}_{dis,\kern0.5em 2007}}{{\widehat{\mu}}_{rest,\kern0.5em 2007}}}={e}^{b_{dis,\kern0.1em 2011}-{b}_{rest,\kern0.1em 2011}-{b}_{dis,\kern0.1em 2007}} $$
In the above coding individuals living in the rest of Stockholm County in 2007 is the reference category.
To obtain a confidence interval for the expression of interest we first calculate a confidence interval for bdis, 2011 − brest, 2011 − bdis, 2007. By taking the exp-function of this confidence interval, in analogy with logistic regression models, we then obtain the confidence interval for the expression of interest [32].
Change in number of visits
The mean number of GP visits increased in all groups from 2007 to 2011. The mean number of visits for the 2006 study population was 1.82 and 2.66 for the study population from 2010, an increase of 0.83 visits or 46 %. The increase in number of visits was greater among men (50 %) than among women (43 %) (Table 1 and 2), but this difference was not significant.
Table 1 Mean number of visits, proportion having one or more visits to the GP in 2007 and in 2011 and the relative changes in 2011 compared to 2007 among women
Table 2 Mean number of visits, proportion having one or more visits to the GP in 2007 and in 2011 and the relative changes in 2011 compared to 2007 among men
Women with mental health problems had significantly smaller increase in the number of visits than women with no mental health problems. Otherwise there were no significant differences in the change in number of visits between groups with different health care needs and between groups in disadvantaged areas compared to the rest of Stockholm County.
Change in the proportion of individuals visiting the doctor in each group
Overall the proportion of people making one or more visits to the doctor increased from 56 % in 2007 to 65 % in 2011, a relative increase of 1.17. Each subgroup had a significant increase in the proportion making one or more visits to the doctor. Men had a significantly greater increase than women (1.19 vs 1.15) and the oldest age group (75–84 years) had a significantly smaller increase compared to all the other age groups, among both men and women.
Groups with greater health care needs had a smaller increase from 2007 to 2011 in the proportion making one or more visits to the doctor, compared to groups with lesser health care needs.
Changes in equity in health care
Women with poor health had a significantly lower increase in number of visits than women with good health. This was true for all health care need indicators except disability pension when controlling for age and educational level. There was the same tendency for most health care need indicators among men; however, this was only significant for men with poor mental health compared to men with no good mental health (Table 3).
Table 3 Relative change in relative differences in number of visits between groups with different health care needs in 2011 compared with 2007, using negative binomial regression
Regarding equity between areas with different socio-economic characteristics, men living in disadvantaged areas had a significantly smaller increase in number of visits compared to men living in other areas of Stockholm County (Table 3). These differences were apparent primarily among those with poor health status. When stratifying by health status area differences were significant only among individuals with poor health status. Men with poor health status in disadvantaged areas had 0.68 (95 % CI: 0.50–0.92) times lower increase in visits than men with poor health status in other areas of Stockholm County (data not shown). These area differences were not significant among women.
The results showed that the number of visits to GPs had increased between 2007 and 2011 in all groups regardless of health status or area of residence. This was also true for the proportion of people making one or more visits to the doctor, but while there were significant differences in the increase of visits only between women with and without mental health problems, there was a tendency for all groups with greater health care needs to have a smaller increase in the proportion of people making one or more visits to the doctor. This could be due to these groups having an already high proportion of people making one or more visits to the doctor.
The results of the negative binomial analysis of changes in equity in health care showed that especially women with poor health status, both physically and mentally, and men with poor mental health had smaller increase in number of visits than the comparison groups. This is in line with a previous report on the effect of primary care reforms in Sweden, where the authors found that groups with specific health care demanding diseases had had a smaller increase in the total number of visits to GPs [24] and indicates that the general increase in number of visits might not benefit those with greater needs to the same extent as those with lesser needs.
Also men in disadvantaged areas, where levels of unemployment is higher and the proportion of individuals with low educational level greater, had a smaller increase in number of visits than men in the rest of Stockholm County, suggesting that men in disadvantaged areas did not benefit from the reform as expected. When stratifying by health status these differences were only significant for individuals with poor health status, indicating some interaction between the effect of area and health status on the rate of change in visits to the doctor (data not shown).
This could be due to lower access to primary care in more disadvantaged areas as new primary care clinics primarily have opened in inner city and well served areas and to the fact that primary care resources have been shifting from areas with greater health care needs to areas with lesser needs [21].
Strengths and limitations
There is a lack of scientific studies about how health care reforms and especially changes in reimbursement systems affect equity in health and health care utilization. This study contributes to bridging that knowledge gap and is, to our knowledge, the first scientific study to investigate the equity perspective of the primary care reform in Stockholm County. Effects of policy changes may be very dependent on the context in which they are implemented. Nevertheless, the conclusions of this study may be useful to policy makers when changing reimbursement systems in other contexts. As most health policy documents underline the importance of equity and providing services according to need, health care reforms should be evaluated from this aspect.
Another strength of this study was the use of individually linked survey and register data, enabling the analysis of changes in GP visits by level of need of health care, to address different aspects of equity in health care. Further, the use of register data for utilization of health care in terms of number of GP visits avoids the potential recall problems associated with using survey data on consumption of health care [33].
We also had the opportunity to include data on health care utilization from the year after the survey, which avoids the bias in distinguishing between initial health status and health outcome [19].
A problem with using only two measure points in time is that it is not possible to infer that the effect observed is only due to the reform. It could also be part of a time trend, but as the increase in visits has previously been shown to be linked to the reform [21, 24] it is plausible that most of the increase in visits and the differences between groups observed in this study is due to the introduction of the reform and the new reimbursement system.
When using survey data with a response rate of 60 % or less, the findings may not reflect the entire population surveyed. Non-responders are over-represented among socially and economically disadvantaged groups, as well as in groups with greater health care needs such as individuals on sick leave [34]. Therefore this study may not correctly represent these groups.
Another limitation is the outcome measure used. The number of visits may not be the optimal indicator when studying how changes in primary care affect equity in health care. The number of visits does not necessarily show whether a person has received the care needed - sometimes one longer visit may be more beneficial to a patient than several shorter visits. One reason for the observed increase in the number of visits could be that visits which previously were longer in time, because of the fee-for service reimbursement system have been divided into several shorter visits. This might not be an improvement for the patients, as they now need to make more visits in order to meet the same health care need, this would especially be important for individuals with more complex health issues and higher health care needs. With the available data it was, however, not possible to investigate changes in the quality of care for different groups of patients. Nevertheless, the fact that resources have been distributed away from areas with high proportion of people with low socioeconomic position to areas with high proportion of people with high socioeconomic position [21] indicates that quality of care might have deteriorated in disadvantaged areas after the reform. Therefore, changes in the number of visits might mainly reflect changes in the reimbursement system and may not fully explain the effects of the reform on equity in health care.
Other studies investigating how GPs perceive the impact of the reform and the reimbursement systems on providing care according to need suggest that the reform and the change in reimbursement systems are not seen to support the intentions of equitable care according to need (25), as stated in the Swedish Health Care Act. Further studies, using other methodology and measures, are warranted of how the reform and the changes in reimbursement systems have affected the way primary care is provided in different clinics, and on how to ensure that primary care is provided based on need.
We discuss the results under the assumption that the observed changes were an effect of the reform. However, other factors, that we have not been able to control for, may have contributed. Nevertheless, our analyses focus on changes in relative differences between subgroups and we do not have reason to believe that any such factor would have differential effect on specific subgroups to a degree that could affect changes in relative differences between subgroups.
This study was restricted to studying changes in visits to GPs, as the reform was only introduced in primary care. Some of the changes observed might be compensated by visits to other specialist doctors. Further analyses taking such factors into account are needed in order to fully understand the impact of the reform.
This study found no evidence that the reform particularly benefitted those with greater health care needs. On the contrary individuals with mental health problems and women with poor health status had a significantly smaller increase in primary care visits than their respective reference group, indicating that the reform had a negative impact on equity in primary care. Also men living in more disadvantaged areas had had a lower increase in number of visits than men in more affluent areas. This could reflect the fact that resources have been shifted from areas with higher health care needs to areas with lower health care needs and should be further investigated to ensure equitable primary care services in the Stockholm County Council.
GP:
LLI:
Limiting Longstanding Illness
PHS:
Public Health Survey
SRH:
Self-rated Health
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A special thanks to the 'Equity and Health Policy' research group, Department of Public Health Sciences, Karolinska Institutet and to Ben Barr at the Department of Public Health and Policy, University of Liverpool for helpful comments and suggestions on the different drafts of this paper. This study was partly supported by a grant from Swedish Research Council for Health, Working Life and Welfare.
Department of Public Health Sciences, Karolinska Institutet, Tomtebodavägen 18a, 171 77, Stockholm, Sweden
Janne Agerholm, Daniel Bruce, Antonio Ponce de Leon & Bo Burström
Centre for Epidemiology and Community Medicine, Stockholm County Council, Stockholm, Sweden
Institute of Social Medicine, University of Rio de Janeiro State, Rio de Janeiro, Brazil
Antonio Ponce de Leon
Janne Agerholm
Daniel Bruce
Bo Burström
Correspondence to Janne Agerholm.
JA, BB and DB have conceived of the study, and participated in its design and coordination. JA performed the statistical analysis and drafted the manuscript. JA, BB, DB and APL were all involved in the interpretation of data and the revising of the manuscript. All authors read and approved the final manuscript.
Agerholm, J., Bruce, D., Ponce de Leon, A. et al. Equity impact of a choice reform and change in reimbursement system in primary care in Stockholm County Council. BMC Health Serv Res 15, 420 (2015). https://doi.org/10.1186/s12913-015-1105-8
Area differences | CommonCrawl |
The Product of Two Nonsingular Matrices is Nonsingular
Prove that if $n\times n$ matrices $A$ and $B$ are nonsingular, then the product $AB$ is also a nonsingular matrix.
(The Ohio State University, Linear Algebra Final Exam Problem)
Find an Orthonormal Basis of the Range of a Linear Transformation
Let $T:\R^2 \to \R^3$ be a linear transformation given by
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2
\end{bmatrix} \,\right)
x_1-x_2 \\
x_1+ x_2
\end{bmatrix}.\] Find an orthonormal basis of the range of $T$.
Diagonalize a 2 by 2 Matrix if Diagonalizable
Determine whether the matrix
\[A=\begin{bmatrix}
\end{bmatrix}\] is diagonalizable.
If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
1 & 2 & 1 \\
-1 &4 &1 \\
2 & -4 & 0
\end{bmatrix}.\] The matrix $A$ has an eigenvalue $2$.
Find a basis of the eigenspace $E_2$ corresponding to the eigenvalue $2$.
Find All the Eigenvalues of 4 by 4 Matrix
Find all the eigenvalues of the matrix
0 & 1 & 0 & 0 \\
0 &0 & 1 & 0 \\
1 & 0 & 0 & 0
\end{bmatrix}.\]
Every Group of Order 72 is Not a Simple Group
Prove that every finite group of order $72$ is not a simple group.
Click here if solved 146
The Determinant of a Skew-Symmetric Matrix is Zero
Prove that the determinant of an $n\times n$ skew-symmetric matrix is zero if $n$ is odd.
A Linear Transformation Preserves Exactly Two Lines If and Only If There are Two Real Non-Zero Eigenvalues
Let $T:\R^2 \to \R^2$ be a linear transformation and let $A$ be the matrix representation of $T$ with respect to the standard basis of $\R^2$.
Prove that the following two statements are equivalent.
(a) There are exactly two distinct lines $L_1, L_2$ in $\R^2$ passing through the origin that are mapped onto themselves:
\[T(L_1)=L_1 \text{ and } T(L_2)=L_2.\]
(b) The matrix $A$ has two distinct nonzero real eigenvalues.
Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$
Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$.
(a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$.
(b) Let
\[A^{100}=aA^2+bA+cI,\] where $I$ is the $3\times 3$ identity matrix.
Using the Cayley-Hamilton theorem, determine $a, b, c$.
(Kyushu University, Linear Algebra Exam Problem)
A Subgroup of Index a Prime $p$ of a Group of Order $p^n$ is Normal
Let $G$ be a finite group of order $p^n$, where $p$ is a prime number and $n$ is a positive integer.
Suppose that $H$ is a subgroup of $G$ with index $[G:P]=p$.
Then prove that $H$ is a normal subgroup of $G$.
(Michigan State University, Abstract Algebra Qualifying Exam)
If Squares of Elements in a Group Lie in a Subgroup, then It is a Normal Subgroup
Let $H$ be a subgroup of a group $G$.
Suppose that for each element $x\in G$, we have $x^2\in H$.
(Purdue University, Abstract Algebra Qualifying Exam)
If $A$ is a Skew-Symmetric Matrix, then $I+A$ is Nonsingular and $(I-A)(I+A)^{-1}$ is Orthogonal
Let $A$ be an $n\times n$ real skew-symmetric matrix.
(a) Prove that the matrices $I-A$ and $I+A$ are nonsingular.
(b) Prove that
\[B=(I-A)(I+A)^{-1}\] is an orthogonal matrix.
Example of Two Groups and a Subgroup of the Direct Product that is Not of the Form of Direct Product
Give an example of two groups $G$ and $H$ and a subgroup $K$ of the direct product $G\times H$ such that $K$ cannot be written as $K=G_1\times H_1$, where $G_1$ and $H_1$ are subgroups of $G$ and $H$, respectively.
Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$
4& 3
(a) Find eigenvalues of the matrix $A$.
(b) Find eigenvectors for each eigenvalue of $A$.
(c) Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
(d) Diagonalize the matrix $A^3-5A^2+3A+I$, where $I$ is the $2\times 2$ identity matrix.
(e) Calculate $A^{100}$. (You do not have to compute $5^{100}$.)
(f) Calculate
\[(A^3-5A^2+3A+I)^{100}.\] Let $w=2^{100}$. Express the solution in terms of $w$.
The Symmetric Group is a Semi-Direct Product of the Alternating Group and a Subgroup $\langle(1,2) \rangle$
Prove that the symmetric group $S_n$, $n\geq 3$ is a semi-direct product of the alternating group $A_n$ and the subgroup $\langle(1,2) \rangle$ generated by the element $(1,2)$.
Every Sylow 11-Subgroup of a Group of Order 231 is Contained in the Center $Z(G)$
Let $G$ be a finite group of order $231=3\cdot 7 \cdot 11$.
Prove that every Sylow $11$-subgroup of $G$ is contained in the center $Z(G)$.
Are Linear Transformations of Derivatives and Integrations Linearly Independent?
Let $W=C^{\infty}(\R)$ be the vector space of all $C^{\infty}$ real-valued functions (smooth function, differentiable for all degrees of differentiation).
Let $V$ be the vector space of all linear transformations from $W$ to $W$.
The addition and the scalar multiplication of $V$ are given by those of linear transformations.
Let $T_1, T_2, T_3$ be the elements in $V$ defined by
T_1\left(\, f(x) \,\right)&=\frac{\mathrm{d}}{\mathrm{d}x}f(x)\\[6pt] T_2\left(\, f(x) \,\right)&=\frac{\mathrm{d}^2}{\mathrm{d}x^2}f(x)\\[6pt] T_3\left(\, f(x) \,\right)&=\int_{0}^x \! f(t)\,\mathrm{d}t.
Then determine whether the set $\{T_1, T_2, T_3\}$ are linearly independent or linearly dependent.
Every Group of Order 20449 is an Abelian Group
Prove that every group of order $20449$ is an abelian group.
The Group of Rational Numbers is Not Finitely Generated
(a) Prove that the additive group $\Q=(\Q, +)$ of rational numbers is not finitely generated.
(b) Prove that the multiplicative group $\Q^*=(\Q\setminus\{0\}, \times)$ of nonzero rational numbers is not finitely generated.
Every Finitely Generated Subgroup of Additive Group $\Q$ of Rational Numbers is Cyclic
Let $\Q=(\Q, +)$ be the additive group of rational numbers.
(a) Prove that every finitely generated subgroup of $(\Q, +)$ is cyclic.
(b) Prove that $\Q$ and $\Q \times \Q$ are not isomorphic as groups.
Page 15 of 38« First«...1213141516171819...30...»Last »
A Condition that a Linear System has Nontrivial Solutions
Determine Whether a Set of Functions $f(x)$ such that $f(x)=f(1-x)$ is a Subspace
Galois Group of the Polynomial $x^2-2$
A Group of Order $pqr$ Contains a Normal Subgroup of Order Either $p, q$, or $r$
Determine When the Given Matrix Invertible
Eigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is Even
Orthonormal Basis of Null Space and Row Space
Express the Eigenvalues of a 2 by 2 Matrix in Terms of the Trace and Determinant
Find the Inverse Matrix Using the Cayley-Hamilton Theorem
Eigenvalues of a Matrix and its Transpose are the Same
Eigenvalues of Orthogonal Matrices Have Length 1. Every $3\times 3$ Orthogonal Matrix Has 1 as an Eigenvalue | CommonCrawl |
Elementary matrix
In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GLn(F) when F is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations.
Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form. They are also used in Gauss–Jordan elimination to further reduce the matrix to reduced row echelon form.
Elementary row operations
There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations):
Row switching
A row within the matrix can be switched with another row.
$R_{i}\leftrightarrow R_{j}$
Row multiplication
Each element in a row can be multiplied by a non-zero constant. It is also known as scaling a row.
$kR_{i}\rightarrow R_{i},\ {\mbox{where }}k\neq 0$
Row addition
A row can be replaced by the sum of that row and a multiple of another row.
$R_{i}+kR_{j}\rightarrow R_{i},{\mbox{where }}i\neq j$
If E is an elementary matrix, as described below, to apply the elementary row operation to a matrix A, one multiplies A by the elementary matrix on the left, EA. The elementary matrix for any row operation is obtained by executing the operation on the identity matrix. This fact can be understood as an instance of the Yoneda lemma applied to the category of matrices.
Row-switching transformations
See also: Permutation matrix
The first type of row operation on a matrix A switches all matrix elements on row i with their counterparts on a different row j. The corresponding elementary matrix is obtained by swapping row i and row j of the identity matrix.
$T_{i,j}={\begin{bmatrix}1&&&&&&\\&\ddots &&&&&\\&&0&&1&&\\&&&\ddots &&&\\&&1&&0&&\\&&&&&\ddots &\\&&&&&&1\end{bmatrix}}$
So Ti,j A is the matrix produced by exchanging row i and row j of A.
Coefficient wise, the matrix Ti,j is defined by :
$[T_{i,j}]_{k,l}={\begin{cases}0&k\neq i,k\neq j,k\neq l\\1&k\neq i,k\neq j,k=l\\0&k=i,l\neq j\\1&k=i,l=j\\0&k=j,l\neq i\\1&k=j,l=i\\\end{cases}}$
Properties
• The inverse of this matrix is itself: $T_{i,j}^{-1}=T_{i,j}.$
• Since the determinant of the identity matrix is unity, $\det(T_{i,j})=-1.$ It follows that for any square matrix A (of the correct size), we have $\det(T_{i,j}A)=-\det(A).$
• For theoretical considerations, the row-switching transformation can be obtained from row-addition and row-multiplication transformations introduced below because $T_{i,j}=D_{i}(-1)\,L_{i,j}(-1)\,L_{j,i}(1)\,L_{i,j}(-1).$
Row-multiplying transformations
The next type of row operation on a matrix A multiplies all elements on row i by m where m is a non-zero scalar (usually a real number). The corresponding elementary matrix is a diagonal matrix, with diagonal entries 1 everywhere except in the ith position, where it is m.
$D_{i}(m)={\begin{bmatrix}1&&&&&&\\&\ddots &&&&&\\&&1&&&&\\&&&m&&&\\&&&&1&&\\&&&&&\ddots &\\&&&&&&1\end{bmatrix}}$
So Di(m)A is the matrix produced from A by multiplying row i by m.
Coefficient wise, the Di(m) matrix is defined by :
$[D_{i}(m)]_{k,l}={\begin{cases}0&k\neq l\\1&k=l,k\neq i\\m&k=l,k=i\end{cases}}$
Properties
• The inverse of this matrix is given by $D_{i}(m)^{-1}=D_{i}\left({\tfrac {1}{m}}\right).$
• The matrix and its inverse are diagonal matrices.
• $\det(D_{i}(m))=m.$ Therefore for a square matrix A (of the correct size), we have $\det(D_{i}(m)A)=m\det(A).$
Row-addition transformations
The final type of row operation on a matrix A adds row j multiplied by a scalar m to row i. The corresponding elementary matrix is the identity matrix but with an m in the (i, j) position.
$L_{ij}(m)={\begin{bmatrix}1&&&&&&\\&\ddots &&&&&\\&&1&&&&\\&&&\ddots &&&\\&&m&&1&&\\&&&&&\ddots &\\&&&&&&1\end{bmatrix}}$
So Lij(m)A is the matrix produced from A by adding m times row j to row i. And A Lij(m) is the matrix produced from A by adding m times column i to column j.
Coefficient wise, the matrix Li,j(m) is defined by :
$[L_{i,j}(m)]_{k,l}={\begin{cases}0&k\neq l,k\neq i,l\neq j\\1&k=l\\m&k=i,l=j\end{cases}}$
Properties
• These transformations are a kind of shear mapping, also known as a transvections.
• The inverse of this matrix is given by $L_{ij}(m)^{-1}=L_{ij}(-m).$
• The matrix and its inverse are triangular matrices.
• $\det(L_{ij}(m))=1.$ Therefore, for a square matrix A (of the correct size) we have $\det(L_{ij}(m)A)=\det(A).$
• Row-addition transforms satisfy the Steinberg relations.
See also
• Gaussian elimination
• Linear algebra
• System of linear equations
• Matrix (mathematics)
• LU decomposition
• Frobenius matrix
References
See also: Linear algebra § Further reading
• Axler, Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0-387-98259-0
• Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978-0-321-28713-7
• Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0-89871-454-8, archived from the original on 2009-10-31
• Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, ISBN 0-534-99845-3
• Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
• Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall
• Strang, Gilbert (2016), Introduction to Linear Algebra (5th ed.), Wellesley-Cambridge Press, ISBN 978-09802327-7-6
Matrix classes
Explicitly constrained entries
• Alternant
• Anti-diagonal
• Anti-Hermitian
• Anti-symmetric
• Arrowhead
• Band
• Bidiagonal
• Bisymmetric
• Block-diagonal
• Block
• Block tridiagonal
• Boolean
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• Conference
• Complex Hadamard
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• Diagonal
• Discrete Fourier Transform
• Elementary
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• Metzler
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Conditions on eigenvalues or eigenvectors
• Companion
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• Defective
• Definite
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• Hurwitz
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• Stieltjes
Satisfying conditions on products or inverses
• Congruent
• Idempotent or Projection
• Invertible
• Involutory
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• Normal
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• Unimodular
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With specific applications
• Adjugate
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Used in statistics
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Related terms
• Jordan normal form
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| Wikipedia |
Resources Aops Wiki 2015 AMC 10A Problems
2015 AMC 10A Problems
Revision as of 14:51, 4 February 2015 by Nsheth12 (talk | contribs) (→Problem 3)
1 Problem 1
10 Problem 10
What is the value of
$\textbf{(A)}\ -125\qquad\textbf{(B)}\ -120\qquad\textbf{(C)}\ \frac{1}{5}\qquad\textbf{(D)}}\ \frac{5}{24}\qquad\textbf{(E)}\ 25$ (Error compiling LaTeX. ! Extra }, or forgotten $.)
A box contains a collection of triangular and square tiles. There are tiles in the box, containing edges total. How many square tiles are there in the box?
$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}}\ 9\qquad\textbf{(E)}\ 11$ (Error compiling LaTeX. ! Extra }, or forgotten $.)
Ann made a 3-step staircase using 18 toothpicks. How many toothpicks does she need to add to complete a 5-step staircase?
(A) 9 (B) 18 (C) 20 (D) 22 (E) 24
Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia?
$\textbf{(A)}\ \frac{1}{12}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)
Mr. Patrick teaches math to students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was . After he graded Payton's test, the test average became . What was Payton's score on the test?
$\textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}}\ 94\qquad\textbf{(E)}\ 95$ (Error compiling LaTeX. ! Extra }, or forgotten $.)
The sum of two positive numbers is times their difference. What is the ratio of the larger number to the smaller number?
$\textbf{(A)}\ \frac{5}{4}\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ \frac{9}{5}\qquad\textbf{(D)}}\ 2 \qquad\textbf{(E)}\ \frac{5}{2}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)
How many terms are there in the arithmetic sequence , , , . . ., , ?
How many rearrangements of are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either or .
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 3\qquad\textbf{(E)}\ 4$ (Error compiling LaTeX. ! Extra }, or forgotten $.)
A rectangle has area and perimeter , where and are positive integers. Which of the following numbers cannot equal ?
The zeros of the function are integers. What is the sum of the possible values of ?
AMC Problems and Solutions
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2015_AMC_10A_Problems&oldid=67319" | CommonCrawl |
Search SpringerLink
Feasibility study of measuring \(b\rightarrow s\gamma \) photon polarisation in \(D^0\rightarrow K_1(1270)^- e^+\nu _e\) at STCF
Yu-Lan Fan ORCID: orcid.org/0000-0001-9616-97051,
Xiao-Dong Shi2,3,
Xiao-Rong Zhou2,3 &
Liang Sun1
The European Physical Journal C volume 81, Article number: 1068 (2021) Cite this article
We report a sensitive study of measuring \(b\rightarrow s\gamma \) photon polarisation in \(D^{0}\rightarrow K_1(1270)^-e^+\nu _e\) with an integrated luminosity of \(\mathscr {L}\) = 1 ab\(^{-1}\) at a center-of-mass energy of 3.773 GeV at a future Super Tau Charm Facility. More than 61,000 signals of \(D^{0}\rightarrow K_1(1270)^-e^+\nu _e\) are expected. Based on a fast simulation software package, the statistical sensitivity for the ratio of up-down asymmetry is estimated to be \(1.5\times 10^{-2}\) by performing a two-dimensional angular analysis in \(D^{0}\rightarrow K_1(1270)^-e^+\nu _e\). Combining with measurements of up-down asymmetry in \(B\rightarrow K_1\gamma \), the photon polarisation in \(b\rightarrow s\gamma \) can be determined model-independently.
The new physics (NP) and related phenomena beyond the Standard Model (SM) could be explored by indirect searches in \(b\rightarrow s\gamma \) processes. The photon emitted from the electroweak penguin loop in \(b\rightarrow s\gamma \) transitions is predominantly polarised in SM. New sources of chirality breaking can modify the \(b\rightarrow s\gamma \) transition strongly as suggested in several theories beyond the SM [1,2,3]. A representative example is the left-right symmetric model (LRSM) [4, 5], in which the photon can acquire a significant right-handed component. An observation of right-handed photon helicity would be a clear indication for NP [4].
The effective Hamiltonian of \(b\rightarrow s\gamma \) is
$$\begin{aligned} \mathscr {H}_{eff} = -\frac{4G_{F}}{\sqrt{2}}V_{tb}V^*_{ts}(C_{7L}\mathscr {O}_{7L}+C_{7R}\mathscr {O}_{7R}), \end{aligned}$$
where \(C_{7L}\) and \(C_{7R}\) are the Wilson coefficients for left- and right-handed photons, respectively. In the SM, the chiral structure of \(W^{\pm }\) couplings to quarks leads to a dominant polarisation photon and a suppressed right-handed configuration, and the photon from radiative \(\bar{B}\) (B) decays is predominantly left- (right-) handed, i.e., \(|C_{7L}|^2 \gg |C_{7R}|^2\) (\(|C_{7L}|^2 \ll |C_{7R}|^2\)).
Various methods have been proposed to determine the photon polarisation of the \(b\rightarrow s\gamma \). The first method [6] suggests that the CP asymmetries which depend on the photon helicity could be measured in time dependent asymmetry in the charged and neutral \(B(t)\rightarrow \)X\(^{CP}_{s/d}\gamma \) decays. The second method [7] to determine the photon polarisation is based on \(b\rightarrow s ~l^+ l^-\) transition where the dilepton pair originates from a virtual photon. The third method in \(\varLambda _b\rightarrow \varLambda \gamma \) also could be used to measure the photon polarisation directly. The forward–backward asymmetry defined in [8, 9] is proportional to the photon polarisation.
Measuring the photon polarisation in radiative B decays into K resonance states, \(K_\mathrm{res}(\rightarrow K\pi \pi )\), is proposed in [10, 11]. The photon polarisation parameter \(\lambda _{\gamma }\) could be described by an up-down asymmetry (\(A_\mathrm{UD}\)) of the photon momentum relative to the \(K\pi \pi \) decay plane in \(K_\mathrm{res}\) rest frame. The photon polarisation in \(B\rightarrow K_\mathrm{res}\gamma \) is given in terms of Wilson coefficients [11]:
$$\begin{aligned} \mathscr {\lambda }_{\gamma } = \frac{|C_{7R}|^2-|C_{7L}|^2}{|C_{7R}|^2+|C_{7L}|^2}, \end{aligned}$$
with \(\lambda _{\gamma } \simeq -1\) for \(b\rightarrow s \gamma \) and \(\lambda _{\gamma } \simeq +1\) for \(\bar{b}\rightarrow \bar{s} \gamma \). The integrated up-down asymmetry which is proportional to photon polarisation parameter \(\lambda _{\gamma }\) for the radiative process proceeding through a single resonance \(K_\mathrm{res}\) is defined [10, 11]
$$\begin{aligned} \begin{aligned} {A}_\mathrm{UD}&= \frac{\varGamma _{K_\mathrm{res}\gamma }[\cos \theta _K>0]-\varGamma _{K_\mathrm{res}\gamma }[\cos \theta _K<0]}{\varGamma _{K_\mathrm{res}\gamma }[\cos \theta _K>0]+\varGamma _{K_\mathrm{res}\gamma }[\cos \theta _K<0]}\\&= \lambda _{\gamma }\frac{3~\mathrm{Im}[\mathbf {n} \cdot (\mathbf {J}\times \mathbf {J^{*}})]}{4~|\mathbf {J}|^2}. \end{aligned} \end{aligned}$$
where \(\theta _K\) is defined as the relative angle between the normal direction \(\mathbf {n}\) of the \(K_\mathrm{res}\) decay plane and the opposite flight direction of the photon in the \(K_\mathrm{res}\) rest frame, and \(\mathbf {J}\) denotes the \(K_\mathrm{res}\rightarrow K\pi \pi \) decay amplitude [10]. In charm sector, radiative \(D^0-\) decays into CP eigenstate are expected to determine the photon polarization by means of the charm meson's finite width difference [12]. Recently, LHCb collaboration reported the direct observation of the photon polarisation with a significance of 5.2\(\sigma \) in \(B^+\rightarrow K^+\pi ^-\pi ^+\gamma \) decay [13]. In the \(K\pi \pi \) mass interval [1.1,1.3] GeV/\(c^2\) which is dominated by \(K_1(1270)\), \(A_\mathrm{UD}\) is extracted to be (6.9 ± 1.7) \(\times 10^{-2}\). However, the currently limited knowledge of the structure of the \(K\pi \pi \) mass spectrum, which includes interfering kaon resonances, prevents the translation of a measured asymmetry into an actual value for \(\lambda _{\gamma }\).
To solve this dilemma, three methods are proposed [14,15,16]. In Ref. [14], by using \(B\rightarrow J/\psi K_1\rightarrow J/\psi K\pi \pi \) channel, the hadronic information of \(K\pi \pi \) can be determined. Along the lines of the method known to \(B\rightarrow K_1(\rightarrow K\pi \pi )\gamma \) decays, the extraction of photon polarization in \(D_(s)\rightarrow K_1(\rightarrow K\pi \pi )\gamma \) decays is introduced in the \(K\pi \pi \) system [15]. A novel method is proposed in Ref. [16] to determine the photon helicity in \(b\rightarrow s\gamma \) by combining the \(B\rightarrow K_1\gamma \) and semi-leptonic decay \(D\rightarrow K_1l^+\nu _l(l=\mu ^+,e^+)\) model-independently. A ratio of up-down asymmetries is introduced in [16]. Reference [16] introduces two angles \(\theta _K\) and \(\theta _l\) in \(D^0\rightarrow K_1^-e^+\nu _e\) shown in Fig. 1 and the ratio of up-down asymmetry \(A^{'}_\mathrm{UD}\) is defined as [16]
$$\begin{aligned} \begin{aligned} {A}^{'}_\mathrm{UD}&= \frac{\varGamma _{K_1^-e^+\nu _e}[\cos \theta _K>0]-\varGamma _{K_1^-e^+\nu _e}[\cos \theta _K<0]}{\varGamma _{K_1^-e^+\nu _e}[\cos \theta _l>0]-\varGamma _{K_1^-e^+\nu _e}[\cos \theta _l<0]}\\&= \frac{\mathrm{Im}[\mathbf {n}\cdot (\mathbf {J}\times \mathbf {J^{*}})]}{|\mathbf {J}|^2}. \end{aligned} \end{aligned}$$
The kinematics for \(D^0\rightarrow K_1^-(K^-\pi ^+\pi ^-) e^+\nu _e\). The relative angle between the normal direction of \(K_1^-\) decay plane and the opposite of the \(D^0\) flight direction in the \(K_1^-\) rest frame is denoted as \(\theta _K\), where the normal direction of \(K_1^-\) decay plane is defined as \(\mathbf {p}_{\pi ,\mathrm slow}\times \mathbf {p}_{\pi ,\mathrm fast}\) in which \(\mathbf {p}_{\pi ,\mathrm slow}\) and \(\mathbf {p}_{\pi ,\mathrm fast}\) corresponding to the momenta of the lower and higher momentum pions, respectively. The \(\theta _l\) is introduced as the relative angle between the flight direction of \(e^+\) in the \(e^+\nu _e\) rest frame and \(e^+\nu _e\) in the \(D^0\) rest frame [16]
Here the definition of the normal direction of \(K_1\) decay plane is the same as in \(B\rightarrow K_\mathrm{res}\gamma \) in LHCb [13].
Then photon helicity parameter of \(b\rightarrow s\gamma \) could be extracted by [16]
$$\begin{aligned} \mathscr {\lambda }_{\gamma } = \frac{4~A_\mathrm{UD}}{3~A^{'}_\mathrm{UD}}. \end{aligned}$$
So the photon polarisation of \(b\rightarrow s\gamma \) could be determined model-independently by the combination of \(A_\mathrm{UD}^{'}\) in \(D^0\rightarrow K_1^-(\rightarrow K^-\pi ^+\pi ^-)e^+\nu _e\) and \(A_\mathrm{UD}\) in \(B^+\rightarrow K_1^+(\rightarrow K^+\pi ^-\pi ^+)\gamma \).
Experimentally, the semileptonic decay of \(D^0\rightarrow K_1(1270)^-\) \(e^+ \nu _e\) has been observed for the first time with a statistical significance greater than 10\(\sigma \) by using 2.93 fb\(^{-1}\) of \(e^+e^-\) collision data at \(\sqrt{s}\) = 3.773 GeV by BESIII [17]. About 109 signals are observed, and the measured branching fraction is
$$\begin{aligned}&\mathscr {B}(D^0\rightarrow K_1(1270)^- e^+\nu _e) = (1.09\pm 0.13_{-0.16}^{+0.09}\pm 0.12)\\&\quad \times 10^{-3}. \end{aligned}$$
where the first and second uncertainties are the statistical and systematic uncertainties, respectively, and the third uncertainty is the external uncertainty from the assumed branching fractions (BFs) of \(K_1\) subdecays.
Still, the statistics of the current BESIII data set are insufficient to measure the ratio of up-down asymmetry in \(D^0\rightarrow K_1(1270)^- e^+\nu _e\). A much larger data sample with similarly low background level is urgently needed for performing the angular analysis in \(D^0\rightarrow K_1(1270)^- e^+\nu _e\), which calls for the construction of a next generation \(e^+e^-\) collider operating at the \(\tau \)-charm energy region with much higher luminosity.
The Super Tau Charm Facility (STCF) is a scientific project proposed in China for high energy physics frontier [18]. The STCF plans to produce charmed hadron pairs near the charm threshold which allow for exclusive reconstruction of their decay products with well-determined kinematics. Such samples at the threshold allow for a double-tag technique [19] to be employed where the full events can be reconstructed and provide a unique environment to measure \(A^{'}_\mathrm{UD}\) in \(D^0\rightarrow K_1(1270)^-e^+\nu _e\) with very low background level.
In this work, we present a feasibility study of a ratio of up-down asymmetry in \(D^0\rightarrow K_1(1270)^-e^+\nu _e\) at STCF. Throughout this paper, charged conjugated modes are always implied. This paper is organised as follows: in Sect. 2, detector concept for STCF is introduced as well as the Monte Carlo (MC) samples used in this feasibility study. In Sect. 3, the event selection and analysis method are described. The optimisation of detector response is elaborated in Sect. 4 and the results are presented in Sect. 5. Finally, we conclude in Sect. 6.
Detector and MC simulation
The proposed STCF is a symmetric electron-positron beam collider designed to provide \(e^+e^-\) interactions at a center-of-mass (c.m.) energy \(\sqrt{s}\) from 2.0 to 7.0 GeV. The peaking luminosity is expected to be \(0.5\times 10^{35}\) cm\(^{-2}\)s\(^{-1}\) at \(\sqrt{s}=\) 4.0 GeV, and the integrated luminosity per year is 1 ab\(^{-1}\). Such an environment will be an important low-background playground to test the SM and probe possible new physics beyond the SM. The STCF detector is a general purpose detector designed for \(e^+e^-\) collider which includes a tracking system composed of the inner and outer trackers, a particle identification (PID) system with 3\(\sigma \) charged \(K/\pi \) separation up to 2 GeV/c, and an electromagnetic calorimeter (EMC) with an excellent energy resolution and a good time resolution, a super-conducting solenoid and a muon detector (MUD) that provides good charged \(\pi /\mu \) separation. The detailed conceptual design for each sub-detector, the expected detection efficiency and resolution can be found in [18, 20, 21].
Currently, the STCF detector and the corresponding offline software system are under active development. A reliable fast simulation tool for STCF has been developed [21], which takes the most common event generators as input to perform a fast and realistic simulation. The simulation includes resolution and efficiency responses for tracking of final state particles, PID system and kinematic fit related variables. Besides, the fast simulation also provide some functions for adjusting performance of each sub-system which can be used to optimise the detector design according to physical requirement.
This study uses MC simulated samples corresponding to 1 ab\(^{-1}\) of integrated luminosity at \(\sqrt{s}\) = 3.773 GeV. The simulation includes the beam-energy spread and initial-state radiation (ISR) in the \(e^+e^-\) annihilations modeled with the generator kkmc [22, 23]. The inclusive MC samples consist of the production of the \(D\bar{D}\) pairs, the non-\(D\bar{D}\) decays of the \(\psi (3770)\), the ISR production of the \(J/\psi \) and \(\psi (3686)\) states, and the continuum process incorporated in kkmc [22, 23]. The known decay modes are modeled with evtgen [24, 25] using BFs taken from the Particle Data Group [26], and the remaining unknown decays from the charmonium states with lundcharm [27]. Final-state radiation (FSR) from charged final-state particles is incorporated with the photos package [28].
Included in the inclusive \(D\bar{D}\) MC sample, the \(D^0\) \(\rightarrow \) \(K_1(1270)^-\) \(e^+\) \(\nu _e\) decay is generated with the ISGW2 model [29] with BF comes from Ref. [26] and \(K_1(1270)^-\) meson is allowed to decay into all intermediate processes that result in a \(K^-\pi ^+\pi ^-\) final state. The resonance shape of the \(K_1(1270)^-\) meson is parameterised by a relativistic Breit-Wigner function. The mass and width of \(K_1(1270)^-\) meson are fixed at the known values as shown in Table 1, and the BFs of \(K_1(1270)\) subdecays measured by Belle [30] are input to generate the signal MC events, since they give better consistency [17] between data and MC simulation than those reported in [26].
Table 1 Mass, width [26] and ratios of subdecays of \(K_1(1270)^-\) (Fit2) [30] used in this analysis
a The \(M_\mathrm{miss}^2\) vs. \(M_{K\pi \pi }\) distribution of semi-leptonic candidate events. b, c the \(M_\mathrm{miss}^2\)/\(M_{K\pi \pi }\) distribution of the semi-leptonic candidate events, where the red part denotes the signal events and other parts denote the remaining background events
Event section and analysis
The feasibility study employs the \(e^+e^-\rightarrow \psi (3770)\rightarrow D^0 \bar{D}^0\) decay chain. The \(\bar{D}^0\) mesons are reconstructed by three channels with low background level, \(\bar{D}^0\rightarrow K^+\pi ^-\), \(K^+\pi ^-\pi ^0\) and \(K^+\pi ^-\pi ^+\pi ^-\). These inclusively selected events are referred to as single-tag (ST) \(\bar{D}^0\) mesons. In the presence of the ST \(D^0\) mesons, candidates for \(D^0\rightarrow K_1(1270)^-e^+\nu _e\) are selected to form double-tag (DT) events.
Each charged track is required to satisfy the vertex requirement and detector acceptance in fast simulation. The combined confidence levels under the positron, pion and kaon hypotheses (\(CL_e\), \(CL_{\pi }\) and \(CL_{K}\), respectively) are calculated. Kaon (pion) candidates are required to satisfy \(CL_K > CL_{\pi }\) (\(CL_{\pi } > CL_K\)). Positron candidates are required to satisfy \(CL_e\) / (\(CL_e\) + \(CL_K\) + \(CL_{\pi }\)) > 0.8. To reduce the background from hadrons and muons, the positron candidate is further required to have a deposit energy in the EMC greater than 0.8 times its momentum in the MDC. The \(\pi ^0\) meson is reconstructed via \(\pi ^0\rightarrow \gamma \gamma \) decay. The \(\gamma \gamma \) combination with an invariant mass in the range (0.115, 0.150) GeV/\(c^2\) are regarded as a \(\pi ^0\) candidates, and a kinematic fit by constraining the \(\gamma \gamma \) invariant mass to the \(\pi ^0\) nominal mass [26] is performed to improve the mass resolution.
The ST \(\bar{D}^0\) mesons are identified by the energy difference \(\varDelta E \equiv E_{\bar{D}^0}-E_\mathrm{beam}\) and the beam-constrained mass \(M_\mathrm{BC}\) \(\equiv \) \(\sqrt{E^2_\mathrm{beam}-|\mathbf {p}_{\bar{D}^0}|^2}\), where \(E_\mathrm{beam}\) is the beam energy, and \(E_{\bar{D}^0}\) and \(\mathbf {p}_{\bar{D}^0}\) are the total energy and momentum of the ST \(\bar{D}^0\) in the \(e^+e^-\) rest frame. If there are multiple combinations in an event, the combination with the smallest \(\varDelta E\) is chosen for each tag mode. The combinatorial backgrounds in the \(M_\mathrm{BC}\) distributions are suppressed by requiring \(\varDelta E\) within (− 29, 27), (− 69, 38) and (− 31, 28) MeV for \(\bar{D}^0\rightarrow K^+\pi ^-\), \(K^+\pi ^-\pi ^0\) and \(K^+\pi ^-\pi ^+\pi ^-\), respectively, which correspond to about 3.5\(\sigma \) away from the fitted peak.
Particles recoiling against the ST \(\bar{D}^0\) mesons candidates are used to reconstruct candidates for \(D^0\rightarrow K_1(1270)^- e^+\!\nu _e\) decay, where the \(K_1(1270)^-\) meson is reconstructed using its dominant decay \(K_1(1270)^-\rightarrow K^-\pi ^+\pi ^-\). It is required that there are only four good unused charged tracks available for this selection. The charge of the lepton candidate is required to be the same as that of the charged kaon of the tag side. The other three charged tracks are identified as a kaon and two pions, based on the same PID criteria used in the ST selection. The kaon candidate must have charge opposite to that of the positron.
The main peaking background comes from misidentifying a pion to a positron, and additional criteria as in [17] are used to improve the \(\pi \)/e separation.
Information concerning the undetectable neutrino inferred by the kinematic quantity \(M^2_\mathrm{miss} \equiv E^2_\mathrm{miss} - |\mathbf {p}_\mathrm{miss}|^2\), where \(E_\mathrm{miss}\) and \(\mathbf {p}_\mathrm{miss}\) are the missing energy and momentum of the signal candidate, respectively, calculated by \(E_\mathrm{miss} \equiv \mathbf {E}_\mathrm{beam} - \sum _{j}E_j\) and \(\mathbf {p}_\mathrm{miss}\equiv -\mathbf {p}_{\bar{D}^0} - \sum _j\mathbf {p}_{j}\) in the \(e^+e^-\) center-of-mass frame. The index j sums over the \(K^-\), \(\pi ^+\), \(\pi ^-\) and \(e^+\) of the signal candidate, and \(E_j\) and \(\mathbf {p}_j\) are the energy and momentum of the j-th particle, respectively. To partially recover the energy lost to the FSR and bremsstrahlung, the four-momenta of photon(s) within 5\(^\circ \) of the initial positron direction are added to the positron four-momentum.
Figure 2 shows the distribution of \(M_{K^-\pi ^+\pi ^-}\) vs. \(M^2_\mathrm{miss}\) of the accepted \(D^0\rightarrow K^-\pi ^+\pi ^-e^+\nu _e\) candidate events in the MC sample after combining all tag modes. A clear signal, which concentrates around the \(K_1(1270)^-\) nominal mass in the \(M_{K^-\pi ^+\pi ^-}\) distribution and around zero in the \(M^2_\mathrm{miss}\) distribution, can be seen. The selection efficiency of signal candidates with the ST modes \(\bar{D}^0\rightarrow K^+\pi ^-\), \(K^+\pi ^-\pi ^0\) and \(K^+\pi ^-\) \(\pi ^-\pi ^+\) are 12.11\(\%\), 6.93\(\%\) and 6.25\(\%\), respectively. In order to determine the angular distributions of \(\cos \theta _K\) and \(\cos \theta _l\), a two-dimensional (2-D) fit to \(M^2_\mathrm{miss}\) and \(M_{K^-\pi ^+\pi ^-}\) is performed to extract the signal yield in each angle bin. The 2-D fit projections to the \(M^2_\mathrm{miss}\) and \(M_{K^-\pi ^+\pi ^-}\) distributions are shown in Fig. 3. In the fit, the 2-D signal shape is described by the MC-simulated shape extracted from the signal MC events while the 2-D background shape is modeled by those derived from the inclusive MC sample. The smooth 2-D probability density functions of signal and background are modeled by using RooNDKeysPdf [31, 32].
Projection of the \(M^2_\mathrm{miss}\)(left) and \(M_{K^-\pi ^+\pi ^-}\)(right) of the DT candidate events of all three tag channels. The point with error bar are MC sample; the blue solid red dotted and green dashed curves are total fit, signal and background, respectively
The reconstructed efficiencies of signal candidates in each \(\cos \theta _K\) and \(\cos \theta _l\) interval are shown in Fig. 4. The signal reconstruction efficiency shows a clear trend of increasing monotonically with \(\cos \theta _l\), which is due to strong correlation between \(\cos \theta _l\) and electron momentum, and \(D^0\) candidates with lower momentum electrons are less likely to satisfy electron tracking and PID requirements.
The signal reconstruction efficiencies (in percentage) in bins of \(\cos \theta _K\) and \(\cos \theta _l\). In each bin j, the signal reconstruction efficiency is obtained by \(\epsilon _\mathrm{DT}^{j} = \frac{\sum _{i}\mathscr {B}_\mathrm{ST}^{i}\epsilon _\mathrm{DT}^{ij}}{\sum _{i}\mathscr {B}_\mathrm{ST}^i}\), where the \(\mathscr {B}_{ST}^i\) denotes the known BF of each tag mode i, the \(\epsilon _{DT}^{ij}\) represents the DT efficiency in each bin j for tag mode i
The signal yields in each \(\cos \theta _K\) and \(\cos \theta _l\) interval corrected by the signal reconstruction efficiency are fitted with a polynomial function [16]
$$\begin{aligned} \begin{aligned}&f(\cos \theta _K, \cos \theta _l; A_\mathrm{UD}^{'}, d_{+}, d_{-})= (4+d_+ + d_-)\\&\quad [1+\cos ^2\theta _{K}\cos ^2\theta _l]\\&\quad + 2(d_+-d_-) [1+\cos ^2\theta _K]\cos \theta _l\\&\quad + 2A_\mathrm{UD}^{'}(d_+-d_-) \cos \theta _K[1+\cos ^2\theta _l]\\&\quad + 4A_\mathrm{UD}^{'}(d_++d_-) \cos \theta _K\cos \theta _l\\&\quad - (4-d_+-d_-) [\cos ^2\theta _K+\cos ^2\theta _l]. \end{aligned} \end{aligned}$$
where the \(d_{\pm }\) are the angular coefficients, defined as:
$$\begin{aligned} \begin{aligned} d_+ =\frac{|c_+|^2}{|c_0|^2}, d_- =\frac{|c_-|^2}{|c_0|^2} \end{aligned} \end{aligned}$$
The coefficients \(c_{\pm }\) and \(c_0\) correspond to the nonperturbative amplitudes for D decays into \(K_1\) with transverse and longitudinal polarisations, respectively. The ratio of up-down asymmetry \(A_\mathrm{UD}^{'}\) can be extracted directly. Besides, the fraction of longitudinal polarisation \(\frac{|c_0|^2}{|c_0|^2+|c_+|^2+|c_-|^2}\) can be derived from the fitted \(d_{\pm }\) values. Form factor calculations based on different approaches such as covariant light-front quark model(LFQM) and light-cone QCD sum rules(LCSR) obtain the different results significantly [33, 34].
Efficiency corrected signal yields in bins of \(\cos \theta _K\) (left) and \(\cos \theta _l\) (right). The curve is the result of fit using polynomial function
The optimisation of DT efficiency for charged tracks of reconstructed efficiency (a); the optimisation of figure-of-merit for neutral tracks of reconstructed efficiency (b); the optimisation of figure-of-merit for misidentification from \(\pi ^+\) to \(e^+\) (c). And the red star denotes the default result
The possibility of some events migrating from a bin to its neighbor caused by the detection resolution is considered by calculating the full width at half maximum(FWHM) of cos\(\theta _{l}\) and cos\(\theta _K\), respectively. The value of FWHM is 0.115 and 0.05, which indicates that the bin migration effects can be ignored, due to the larger bin width of 0.5.
A 2-D \(\chi ^2\) fit to the \(\cos \theta _l\) and \(\cos \theta _K\) distributions allows to extract \(A_\mathrm{UD}^{'}\), and the fit projections are shown in Fig. 5. The statistical sensitivity of \(A_\mathrm{UD}^{'}\) based on 1 ab\(^{-1}\) MC sample is thus determined to be in the order of 1.8\(\times \)10\(^{-2}\). As a cross-check, \(A_\mathrm{UD}^{'}\) is also determined with a counting method according to Eq. (4) and the corresponding result is compatible with the angular fit method. However, the angular fit method yields a more precise result on \(A_\mathrm{UD}^{'}\) and is taken as the nominal result.
Optimization of detector response
The main loss of the signal efficiency comes from the effects of charged tracking selection, neutral selection and identification of electron at low momentum. These effects correspond to the sub-detectors of the tracking system, the EMC and the PID system. By studying the DT efficiencies or signal-to-background ratios for this process with variation of the sub-detector's responses, the requirement of detector design can be optimised accordingly. With the help of fast simulation software package, three kinds of detector responses are studied as introduced below:
a. Tracking efficiency The tracking efficiency in fast simulation is characterised by two dimensions: transverse momentum \(P_T\) and polar angle cos\(\theta \), which are correlated with the level of track bending and hit positions of tracks in the tracker system. For low-momentum tracks (\(P_T\) < 0.2 GeV/c), it is difficult to reconstruct efficiently due to stronger electromagnetic multiple scattering, electric field leakage, energy loss etc.. However, with different technique in the tracking system design at STCF, or with advanced track finding algorithm, the efficiency is expected to be improved for low-momentum tracks.
Benefiting from the flexible approach to change the response of charged track, the detection efficiency is scaled with a factor from 1.1 to 1.5 in the fast simulation. The figure-of-merit, defined by DT efficiency for characterising the performance of tracking efficiency is shown in Fig. 6a. From Fig. 6a, it is found that the DT efficiency can be significantly improved with the given scale factors. The resolution of momentum and position can also be optimised in fast simulation with proper functions. Insignificant improvement is found among optimisation of absolute \(\sigma _{xy}\) from 30 to 150 \(\upmu \)m and absolute \(\sigma _z\) from 500 to 2500 \(\upmu \)m yet, where \(\sigma _{xy}\) and \(\sigma _z\) are the resolution of the tracking system in the xy plane and z direction, respectively. This can be understood since the main source that affects the momentum resolution comes from electromagnetic multiple scattering on the material in the detector, instead of the position resolution. Therefore, material with low atomic number Z is required in the tracking system.
b.Detection efficiency for photon In this analysis, \(\pi ^0\)s are selected as part of the tag mode \(\bar{D}^0\rightarrow K^+\pi ^-\pi ^0\) and the \(\pi ^0\) selection also helps to suppress the main background of \(D^0\rightarrow K^-\pi ^+\pi ^-\pi ^+\pi ^0\) in signal side. The figure-of-merit, defined by \(\frac{S}{\sqrt{S+B}}\), to characterise the effect of photon detection efficiency on signal significance, in which S denotes the expected signal yield of \(D^0\rightarrow K_1(1270)^-(\rightarrow K^-\pi ^+\pi ^-) e^+\nu _e\) while B denotes the background yield. The value of \(\frac{S}{\sqrt{S+B}}\) versus the scale factor of photon detection efficiency scanned from 1.1 to 1.5 is shown in Fig. 6b.
c. \(\pi /e\) identification Misidentification from a pion to electron in the momentum smaller than 0.6 GeV/c forms the main peaking background \(D^0\rightarrow K^-\pi ^+\pi ^-\pi ^+\) and \(D^0\rightarrow K^-\pi ^+\pi ^-\pi ^+\pi ^0\). As the fast simulation provides the function for optimising the \(\pi \)/e identification which allows to vary the misidentification rate for \(\pi \)/e, the \(\pi \)/e misidentification rate at 0.2 GeV/c is scanned from 5.7 to 0.64\(\%\), shown in Fig. 6c. And \(\frac{S}{\sqrt{S+B}}\) defined before is used to characterise the effect of misidentification of \(\pi ^+\) to \(e^+\) on the signal significance.
In summary, three sets of optimization factors for different sub-detector responses are calculated separately: compared with our fast simulation with default settings, the DT efficiency is improved by \(\sim \)27% if the reconstructed efficiency for charged track is scaled by the factor of 1.1, and the value of \(\frac{S}{\sqrt{S+B}}\) is improved by 4% if the photon detection efficiency is scaled by a factor of 1.1, or 7% if the \(\pi \)/e misidentification rate is lowered by half to 3.2%, as reasonable assumptions in real case scenarios. With the above three factors applied altogether, the DT efficiency is improved by a factor of 33\(\%\). The corresponding angular 2-D \(\chi ^2\) fit based on updated efficiency-corrected signal yields in different angular bins is performed. From the fit, the statistical uncertainty of the ratio of up-down asymmetry is extracted to be 1.5\(\times 10^{-2}\), that is, improved by 17\(\%\) compared with the no optimisation scenario.
With the above selection criteria and optimisation procedure, the 2-D simultaneous fit to \(M_\mathrm{miss}^2\) vs. \(M_{K\pi \pi }\) in the different interval of \(\cos \theta _K\) vs. \(\cos \theta _l\) is performed, the semi-leptonic decay signal yields produced are used for fitting the angular distribution. Therefore, the sensitivity of the ratio of up-down asymmetry in \(D^0\rightarrow K_1(1270)^-e^+\nu _e\) with an integrated luminosity of 1 ab\(^{-1}\) is extracted as 1.5\(\times \)10\(^{-2}\). Besides, the selection efficiency for this process at \(\sqrt{s}\) = 3.773 GeV where the cross section for \(e^+e^-\rightarrow D^0\bar{D}^0\) = 3.6 nb [35] is studied by a large MC sample, with a negligible error.
Equation (2) indicates that the Wilson coefficients can be constrained by measuring the uncertainty of the photon polarisation parameter \(\lambda _{\gamma }\). Combining the uncertainty of \(A_\mathrm{UD}\) measurement [13] with the uncertainty of \(A_\mathrm{UD}^{'}\) measurement in this analysis, the sensitivity of \(\lambda _{\gamma }\) can be determined using Eq. (5). Thus, the Wilson coefficients can be translated by the sensitivity of \(A_\mathrm{UD}^{'}\). Figure 7 depicts the dependency of Wilson coefficients on ratio of \(A_\mathrm{UD}^{'}\), using the \(A_\mathrm{UD}\) measured in the \(K\pi \pi \) mass range of (1.1,1.3) GeV/\(c^2\) [13] as the input, shown in the blue solid line. Considering the uncertainty of \(A_\mathrm{UD}\), the corresponding constraints shown in the green parts. The photon polarisation parameter \(\lambda _{\gamma }\) is predicted to be \(\lambda _{\gamma } \simeq \) 1 for \(\bar{b}\rightarrow \bar{s}\gamma \) in SM, which translated to \(A_\mathrm{UD}^{'} \simeq \) (9.2 ± 2.3)\(\times \)10\(^{-2}\) shown in the red and black solid line in Fig. 7.
Dependence of Wilson coefficient on ratio of up-down asymmetry, shown in the blue line, the green parts denote the consideration of uncertainties of \(A_\mathrm{UD}\), the red solid line denotes \(A_\mathrm{UD}^{'}\) corresponding to the photon polarisation parameter predicted in SM, with the consideration of uncertainties of \(A_\mathrm{UD}\) shown between the black solid lines
For the systematic uncertainty on \(A_\mathrm{UD}^{'}\), possible sources include the electron tracking and PID efficiencies as functions of electron momentum which cannot cancel out in the \(\cos \theta _l\) distribution due to strong correlation between \(\cos \theta _l\) and electron momentum as mentioned before. With the current binning scheme as shown in Fig. 4, the possibility of some events migrating from an angular bin to its neighbor because of the detector resolution effects on \(\cos \theta _K\) and \(\cos \theta _l\) is expected to be small and the related systematic uncertainty should be manageable. Moreover, as in the BESIII analysis [17], the signal and background shape modeling would affect the signal yields considerably in different angular bins, due to imprecise knowledge on the \(K_1(1270)\) line shape, and background events such as \(D^0\rightarrow K^-\pi ^+\pi ^-\pi ^+\pi ^0\).
Our simulation does not include non-\(K_1(1270)^-\) sources of \(K^-\pi ^+\pi ^-\) in the \(D^0 \rightarrow K^- \pi ^+ \pi ^- e^+ \nu _e\) decay, which are estimated to be at least one order of magnitude lower than our signal decay of \(K_1(1270)^-\) [17]. We expect the systematic effect of the non-\(K_1(1270)^-\) sources on \(A^{'}_\mathrm{UD}\) to be small, although detailed studies on the \(K_1(1400)^-\) contribution are needed when more data become available.
Summary and prospect
In this work, the statistical sensitivity of a ratio of up-down asymmetry \(D^0\rightarrow K_1(1270)^-e^+\nu _e\) with an integrated luminosity of \(\mathscr {L}\) = 1 ab\(^{-1}\) at \(\sqrt{s}\) = 3.773 GeV and the optimised efficiency with the fast simulation, is determined to be 1.5\(\times 10^{-2}\) by performing an angular analysis. The hadronic effects in \(K_1\rightarrow K\pi \pi \) can be quantified by \(A_\mathrm{UD}^{'}\), therefore, combined with the measured up-down asymmetry \(A_\mathrm{UD}\) in \(B^+\rightarrow K_1^+(\rightarrow K^+\pi ^-\pi ^+)\gamma \) [13], the photon polarisation in \(b\rightarrow s\gamma \) can be measured to probe the new physics.
This manuscript has no associated data or the data will not be deposited. [Authors' comment: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.]
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The authors are grateful to Wei Wang, Fu-Sheng Yu, Hai-Long Ma and Xiang Pan for useful discussions. We express our gratitude to the supercomputing center of USTC and Hefei Comprehensive National Science Center for their strong support. This work is supported by the Double First-Class university project foundation of USTC and the National Natural Science Foundation of China under Projects No. 11625523.
School of Physics and Technology, Wuhan University, Wuhan, 430072, People's Republic of China
Yu-Lan Fan & Liang Sun
State Key Laboratory of Particle Detection and Electronics, Hefei, 230026, People's Republic of China
Xiao-Dong Shi & Xiao-Rong Zhou
School of Physical Sciences, University of Science and Technology of China, Hefei, 230026, People's Republic of China
Yu-Lan Fan
Xiao-Dong Shi
Xiao-Rong Zhou
Liang Sun
Correspondence to Xiao-Rong Zhou or Liang Sun.
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Fan, YL., Shi, XD., Zhou, XR. et al. Feasibility study of measuring \(b\rightarrow s\gamma \) photon polarisation in \(D^0\rightarrow K_1(1270)^- e^+\nu _e\) at STCF. Eur. Phys. J. C 81, 1068 (2021). https://doi.org/10.1140/epjc/s10052-021-09841-y
DOI: https://doi.org/10.1140/epjc/s10052-021-09841-y
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DSP Illustrations
The complex Fourier Series and its relation to the Fourier Transform¶
In two recent articles we have talked about the Fourier Series and an application in harmonic analysis of instrument sounds in terms of their Fourier coefficients. In this article, we will analyze the relation between the Fourier Series and the Fourier Transform.
The Fourier Series as sums of sines and cosines¶
To recap, the Fourier series of a signal $x(t)$ with period $P$ is given by
$$\begin{align}x(t)=\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos(2\pi nt/P)+b_n\sin(2\pi nt/P)\end{align}$$
where the coefficients are given by
$$\begin{align}a_n&=\frac{2}{P}\int_{-\frac{P}{2}}^\frac{P}{2}x(t)\cos(2\pi nt/P)dt\\b_n&=\frac{2}{P}\int_{-\frac{P}{2}}^\frac{P}{2}x(t)\sin(2\pi nt/P)dt\end{align}.$$
As we see, the Fourier series is a sum of sines and cosines with different amplitudes. Let us first look at the sum of a sine and cosine with different amplitudes:
Sum of a sine and cosine with equal frequency¶
Run code interactively Hide
Fs = 100 # the sampling frequency for the discrete analysis
T = 3 # time duration to look at
P = 1 # signal period
t = np.arange(0, T, 1/Fs)
a_n = 1
b_n = 0.6
s = lambda t: a_n*np.cos(2*np.pi*t/P)
c = lambda t: b_n*np.sin(2*np.pi*t/P)
plt.plot(t, s(t), 'b', label='$a_n\cos(2\pi t)$')
plt.plot(t, c(t), 'g', label='$b_n\sin(2\pi t)$')
plt.plot(t, s(t)+c(t), 'r', label='$a_n\cos(2\pi t)+b_n\sin(2\pi t)$')
As it appears, the sum of a sine and cosine of different amplitudes but same frequency equals another harmonic function with different amplitude and some phase shift. Hence, we can write $$a_n\cos(2\pi nt/P)+b_n\sin(2\pi nt/P) = A_n\cos(2\pi nt/P+\phi_n)$$ where $A_n$ is the amplitude and $\phi_n$ is the phase of the resulting harmonic. In the following, we will calculate the values of $A_n$ and $\phi_n$ from $a_n, b_n$. Let us start with the following identities:
$$\begin{align}\cos(x)&=\frac{1}{2}(\exp(jx)+\exp(-jx))\\ \sin(x)&=-\frac{j}{2}(\exp(jx)-\exp(-jx))\end{align}.$$
Then, we can write the sine and cosine and their sum as
$$ \begin{align} a_n\cos(2\pi nt/P)&=\frac{a_n}{2}(\exp(j2\pi nt/P)+\exp(-j2\pi nt/P))\\ b_n\sin(2\pi nt/P)&=-\frac{jb_n}{2}(\exp(j2\pi nt/P)-\exp(-j2\pi nt/P))\\ a_n\cos(2\pi nt/P)+b_n\sin(2\pi nt/P)&=(a_n-jb_b)\frac{1}{2}\exp(j2\pi nt/P)+(a_n+jb_n)\frac{1}{2}\exp(-j2\pi nt/P) \end{align} $$
We can now convert the cartesian expression for $a_n-jb_n$ into the polar form by $$\begin{align}&&a;_n-jb_n&=A_n\exp(j\phi_n)\\ \text{with }A_n&=\sqrt{a_n^2+b_n^2} &\text{and}&&\phi_n&=\tan^{-1}(-b_n/a_n)\end{align}$$
Accordingly, we can reformulate the sum of sine and cosine as $$\begin{align}a_n\cos(2\pi nt/P)+b_n\sin(2\pi nt/P)&=A_n\frac{1}{2}(\exp(j2\pi nt/P+\phi_n)+\exp(-j(2\pi nt/P+\phi_n))\\ &=A_n\cos(2\pi nt/P+\phi_n).\end{align}$$
This statement eventually confirms that the sum of a sine and cosine of same frequency but different amplitude is indeed another harmonic function. Let us verify this numerically:
def sumSineCosine(an, bn):
Fs = 100
T = 3
P = 1
A = np.sqrt(an**2+bn**2)
phi = np.arctan2(-bn, an)
f1 = an*np.cos(2*np.pi*t/P)
f2 = bn*np.sin(2*np.pi*t/P)
overall = A*np.cos(2*np.pi*t/P + phi)
plt.plot(t, f1, 'b', label='$x(t)=a_n\cos(2\pi nft)$')
plt.plot(t, f2, 'g', label='$y(t)=b_n\sin(2\pi nft)$')
plt.plot(t, f1+f2, 'r', label='$x(t)+y(t)$')
plt.plot(t, overall, 'ro', lw=2, markevery=Fs//(10), label='$A_n\cos(2\pi nft+\phi)$')
As we can see, the result perfectly holds.
The Fourier Series with amplitude and phase¶
Now, let us express the Fourier Series in terms of our new formulation $$x(t)=\frac{a_0}{2}+\sum_{n=1}^\infty A_n\cos(2\pi nt/P+\phi_n)$$
Here we see, that $x(t)$ is consisting of different harmonics, with the $n$th one having the amplitude $A_n$. Since a harmonic function wave with amplitude $A$ has power $A^2/2$, the $n$th harmonic of $x(t)$ has the power $A_n^2/2=\frac{1}{2}(a_n^2+b_n^2)$.
Fourier Series with complex exponential¶
Let us now write the Fourier Series even in a different form. By replacing the sum of sine and cosine with exponential terms, we get
$$\begin{align}x(t)&=\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos(2\pi nt/P)+b_n\sin(2\pi nt/P)\\ &=\frac{a_0}{2}+\sum_{n=1}^\infty \frac{a_n-jb_n}{2}\exp(j2\pi nt/P) + \frac{a_n+jb_n}{2}\exp(-j2\pi nt/P)\end{align}$$
Let us now set $$c_n=\begin{cases}\frac{a_n-jb_n}{2} & n > 0\\\frac{a_0}{2} & n=0 \\ \frac{a_n+jb_n}{2} & c < 0\end{cases},$$
such that we can alternatively write the Fourier series as
$$x(t)=\sum_{n=-\infty}^{\infty}c_n\exp(j2\pi nt/P).$$
Even, the calculation of the coefficients $c_n$ is very straight-forward, as we have
$$\begin{align}c_n = \frac{a_n-jb_n}{2}&=\frac{1}{2}\left[\frac{2}{P}\int_{-\frac{P}{2}}^{\frac{P}{2}}x(t)\cos(2\pi nt/P)dt-j\frac{2}{P}\int_{-\frac{P}{2}}^{\frac{P}{2}}x(t)\sin(2\pi nt/P)dt\right]\\&=\frac{1}{P}\int_{-\frac{P}{2}}^{\frac{P}{2}}x(t)[\cos(2\pi nt/P)-j\sin(2\pi nt/P)]dt\\&=\frac{1}{P}\int_{-\frac{P}{2}}^{\frac{P}{2}}x(t)\exp(-j2\pi nt/P)dt\end{align}$$
for $n>0$. We get exactly the same expression for $n\leq 0$.
So, to summarize, the formulation for the Fourier series is given by
$$\begin{align}x(t)&=\sum_{n=-\infty}^{\infty}c_n\exp(j2\pi nt/P)\\ \text{with }c_n&=\int_{-\frac{P}{2}}^{\frac{P}{2}}x(t)\exp(-j2\pi nt/P)dt.\end{align}$$
We can again verify this numerically. First, let us implement the two different possibilities to calculate the Fourier series coefficients $a_n,b_n$ or $c_n$:
def fourierSeries_anbn(period, N):
"""Calculate the Fourier series coefficients an, bn up to the Nth harmonic"""
result = []
T = len(period)
t = np.arange(T)
for n in range(N+1):
an = 2/T*(period * np.cos(2*np.pi*n*t/T)).sum()
bn = 2/T*(period * np.sin(2*np.pi*n*t/T)).sum()
result.append((an, bn))
return np.array(result)
def fourierSeries_cn(period, N):
c_plusn = 1/T * (period * np.exp(-2j*np.pi*n*t/T)).sum()
c_minusn = 1/T * (period * np.exp(2j*np.pi*n*t/T)).sum()
result.append((c_plusn, c_minusn))
Then, let's calculate the coefficients for some function $x(t)$ with both methods and compare them.
x = lambda t: (abs(t % 1)<0.05).astype(float) # define a rectangular function
t = np.arange(-1.5, 1.5, 0.001)
plt.plot(t, x(t))
t_period = np.arange(0, 1, 0.001)
period = x(t_period)
anbn = fourierSeries_anbn(period, 100)
cn = fourierSeries_cn(period, 100)
plt.plot(anbn[:,0], label='$a_n$')
plt.plot(anbn[:,1], label='$b_n$')
plt.plot(cn[:,0].real, label='$Re(c_n)$')
plt.plot(cn[:,0].imag, label='$Im(c_n)$')
As shown, the relation $c_n=\frac{a_n-jb_n}{2}, n>0$ exactly holds.
The relation between the Fourier Series and Fourier Transform¶
Let us first repeat the Fourier series and Fourier transform pairs:
$$\begin{align}x(t)&=\sum_{n=-\infty}^{\infty}c_n\exp(j2\pi \frac{n}{P}t) &c;_n&=\int_{-\frac{P}{2}}^{\frac{P}{2}}x(t)\exp(-j2\pi \frac{n}{P}t)dt&\text{Fourier Series}\\ x(t)&=\int_{-\infty}^{\infty}X(f)\exp(j2\pi ft)dt&X;(f)&=\int_{-\infty}^{\infty}x(t)\exp(-j2\pi ft)dt&\text{Fourier Transform}\end{align}$$
We already see, that there is quite some similarity between the expressions for the series and transform. Let us investigate their relations:
We know that the Fourier transform can be applied for an aperiodic signal, whereas the Fourier series is used for a periodic signal with period $P$. Furthermore, we see that the Fourier transform allows the signal $x(t)$ to consist of arbitrary frequencies $f$, whereas the periodic signal $x(t)$ in the Fourier series is consisting only of harmonics of discrete frequency $f_n=\frac{n}{P}$. Let us reformulate the Fourier series with using the Dirac filter property
$$\int_{-\infty}^{\infty}x(t)\delta(t-\tau)dt=x(\tau)$$
to become
$$x(t)=\int_{-\infty}^{\infty}X(f)\exp(j2\pi \frac{n}{P}t)df \text{ with }X(f)=\sum_{n=-\infty}^{\infty}c_n\delta(f-\frac{n}{P}).$$
The expression for $x(t)$ is now equal to the inverse Fourier transform, and we can already identify $X(f)$ as the spectrum of the periodic $x(t)$. We see that $X(f)$ of the periodic signal is discrete, i.e. it is nonzero at only the harmonic frequencies $\frac{n}{P}$. The difference between the discrete frequencies is $\frac{1}{P}$, i.e. it decreases with larger period lengths. If we now eventually assume $P\rightarrow\infty$, i.e. we let the period duration of the signal become infinite, we directly end up with the expression for the Fourier transform, because
$$\lim_{P\rightarrow\infty}\sum_{n=-\infty}^{\infty}c_n\delta(f-\frac{n}{P})$$
becomes a continuous function of $f$, since the Diracs get closer and closer together, eventually merging to a smooth function (intuitively; mathematical rigorous treatment is omitted here).
Let us eventually verify this relation numerically: We take a single rectangular pulse and increase its period's length, i.e. we keep the length of the rect pulse constant, but increase the distance between the pulses, eventually leading to a single, aperiodic pulse, when the period duration becomes infinite:
def compareSeriesAndTransform(P):
Fs = 1000
t = np.arange(0, 100, 1/Fs)
t_period = np.arange(0, P, 1/Fs)
x_p = lambda t: (abs((t % P)-0.5) <= 0.5).astype(float)
x = lambda t: (abs(t-0.5) <= 0.5).astype(float)
plt.plot(t, x_p(t))
cn = fourierSeries_cn(x_p(t_period), 100)[:,0]
f_discrete = np.arange(len(cn))/P
f = np.linspace(0, Fs, len(t), endpoint=False)
X = np.fft.fft(x(t))/Fs
plt.plot(f, abs(X), label='Fourier Tr. of rect')
plt.stem(f_discrete, abs(cn*P), label='Fourier Series $c_n$')
As we have expected, the Fourier series provides a discrete spectrum of the periodic signal. The value of the discrete samples is equal to the value of the Fourier transform of the aperiodic signal.
Summary¶
The Fourier Series can be formulated in 3 ways:
$$\begin{align}1)\quad x(t)&=\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos(2\pi nt/P)+b_n\sin(2\pi nt/P)&a;_n&=\frac{2}{P}\int_{-\frac{P}{2}}^\frac{P}{2}x(t)\cos(2\pi nt/P)dt&b;_n&=\frac{2}{P}\int_{-\frac{P}{2}}^\frac{P}{2}x(t)\sin(2\pi nt/P)dt \\ 2)\quad x(t)&=\frac{a_0}{2}+\sum_{n=1}^\infty A_n\cos(2\pi nt/P+\phi_n)&A;_n&=\sqrt{a_n^2+b_n^2}&\phi_n&=\tan^{-1}(-b_n/a_n)\\3)\quad x(t)&=\sum_{n=-\infty}^{\infty}c_n\exp(j2\pi nt/P)&c;_n&=\int_{-\frac{P}{2}}^{\frac{P}{2}}x(t)\exp(-j2\pi nt/P)dt\end{align}$$
The Fourier Transform can be understood as the limiting case of the complex Fourier series, when the period grows to infinity.
Do you have questions or comments? Let's dicuss below!
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The role of mPFC and MTL neurons in human choice under goal-conflict
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Two opposing hippocampus to prefrontal cortex pathways for the control of approach and avoidance behaviour
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Tal Gonen1,3 na1,
Guy Gurevitch ORCID: orcid.org/0000-0001-9245-68381,4 na1,
Noa Cohen1,2,
Ido Strauss2,5,
Yoav Zeevi6,7,
Hagar Yamin1,
Firas Fahoum ORCID: orcid.org/0000-0003-2262-55302,8,
Talma Hendler ORCID: orcid.org/0000-0002-4182-43351,2,4,6 na1 &
Itzhak Fried1,2,5,9 na1
Nature Communications volume 11, Article number: 3192 (2020) Cite this article
An Author Correction to this article was published on 10 August 2020
This article has been updated
Resolving approach-avoidance conflicts relies on encoding motivation outcomes and learning from past experiences. Accumulating evidence points to the role of the Medial Temporal Lobe (MTL) and Medial Prefrontal Cortex (mPFC) in these processes, but their differential contributions have not been convincingly deciphered in humans. We detect 310 neurons from mPFC and MTL from patients with epilepsy undergoing intracranial recordings and participating in a goal-conflict task where rewards and punishments could be controlled or not. mPFC neurons are more selective to punishments than rewards when controlled. However, only MTL firing following punishment is linked to a lower probability for subsequent approach behavior. mPFC response to punishment precedes a similar MTL response and affects subsequent behavior via an interaction with MTL firing. We thus propose a model where approach-avoidance conflict resolution in humans depends on outcome value tagging in mPFC neurons influencing encoding of such value in MTL to affect subsequent choice.
Humans often find themselves facing a choice involving conflicting emotions. Spinoza defined such conflicting emotions as those which draw a man in different directions (Part IV of the Ethics, on Human Bondage). Indeed approach-avoidance behavioral choices are resolved by the human capacity to adapt goal-directed behaviors to the emotional value of prospective outcomes. Rewarding outcome serves to strengthen or reinforce context-behavior associations, thereby increasing the likelihood of future approach behavior1. Aversive outcomes, on the other hand, are encoded so as to avoid similar future punishment, thus encouraging avoidance behavior2.
Animal studies have investigated the neural mechanism responsible for encoding the effects of various outcomes on subsequent behavior, mostly in the context of reinforcement learning. Accumulating evidence points to the striatum as an important region involved in signaling prediction errors (PEs)3 and to the medial prefrontal cortex (mPFC)4, and medial temporal lobe (MTL) as processing outcome values and valence5,6. Of particular importance is the known role of the hippocampus and the amygdala in forming, respectively, contextual and emotional associations7 that guide future behavior in reinforcement learning procedures. However, less is known regarding the effects of outcome valence on the probability of subsequent behavior in situations of goal conflict.
Goal conflicts arise when we encounter potential gains and losses simultaneously within the same context8. Such conflicts are thought to be central to the generation of anxiety; a state of high arousal and negative outcome bias that often leads to disadvantageous dominance of choosing avoidance behavior9,10. Classical animal studies using goal-conflict paradigms such as the elevated plus maze (EPM)11,12 have implicated the amygdala13, hippocampus9,14, and mPFC15,16 as being crucial in triggering avoidance behavior in goal conflict situations. For example, in Kimura et al.12, rats were punished with a delivery of an electrical shock as they consumed food (avoidance training). Over time, control animals increased their latency to enter the target box, while rats with hippocampal lesions presented impaired acquisition of such passive avoidance behavior. However, classical animal studies have not clearly differentiated the neural substrates involved in using information regarding the valence of outcomes (reward vs. punishment) for subsequent adaptation of approach behavior, from those that mediate the actual resolution of the goal conflict17. Schumacher et al.17 showed that the ventral hippocampus (vHPC) is involved in the resolution of approach-avoidance conflict at the moment of decision making rather than in learning about the value of outcomes for future decisions. On the other hand, further studies showed that the hippocampus, as well as the amygdala, seems to support learning from outcomes and thus affect future behavior. For example, Davidow et al.5 showed that adolescents were better than adults at learning from outcomes to adapt subsequent decisions, and that this was related to heightened PE-related BOLD activity in the hippocampus. Using lesions to macaque amygdalae, Costa et al.18 present evidence that the amygdala plays an important role in learning from outcomes to influence subsequent choice behavior. With relation to psychopathology, it has been suggested that patients suffering from depression are unable to exploit affective information to guide behavior19. For example, Kumar et al.20 found reduced reward learning signals in the hippocampus and anterior cingulate in patients suffering from major depression. Disruption of prediction-outcome associations in the bilateral amygdala–hippocampal complex was found in patients with schizophrenia21. Yet, it remains to be seen whether these results, pointing to the significance of the MTL in the processing of outcomes and adapting behavior, are relevant to outcomes that appear in the context of an approach-avoidance conflict.
To investigate these processes, we use a rare opportunity to perform intracranial recordings from multiple sites in the MTL and mPFC of 14 patients with epilepsy (Table 1). We apply a previously validated game-like computerized task22 that enables the measurement of goal-directed behavior (the tendency to approach) under high or low goal conflict and evaluate the neural response to the outcome of this behavior (reward or punishment, Fig. 1). During the game participants control the movement of a cartoon avatar across the screen in order to approach and capture falling coins (Reward, marked by a new Israeli shekel sign) while attempting to avoid being hit by balls that fall simultaneously (Punishment). The simultaneous appearance of these potentially rewarding and punishing cues introduces a goal-conflict behavioral decision of either approaching the reward or forfeiting it to minimize the risk of consequent punishment. To account for behavioral choice effects, the game also includes events in which outcomes occur independently of behavior (Uncontrolled condition). In this condition participants receive rewarding coins or were hit by punishing balls, regardless of their management of the avatar's movement on the screen. Reward trials during the Controlled condition are classified as either high goal conflict (HGC; more than one ball between the avatar and the reward cue) or low goal conflict (LGC; zero or one ball between the avatar and the rewarding cue). Here we focus on behavior during HGC, since previous results from this task showed differential activation in reward circuitry (ventral striatum) during this condition as well as an effect for individual differences22.
Table 1 Clinical data.
Fig. 1: The paradigm.
The goal of the game was to earn virtual money by catching shekel signs and avoiding balls. A small avatar on a skateboard was located at the bottom of the screen and subjects had to move the avatar right and left using right and left arrow keys, in order to catch the money and avoid the balls falling from the top of the screen. There were two ways to gain or lose money—a "controlled" condition, where players actively approached green money signs (marked here as dollar signs) and avoided red balls, and an "uncontrolled" condition, where although cues appeared on the top of the screen (reward—cyan dollar sign, punishment—magenta ball), they always hit the avatar with no relation to the players' action (they chase the avatar during their fall). Each money catch resulted in a five-point gain and each ball hit resulted in a loss of five points, regardless of controllability (the outcome was shown on the screen after each trial). All four outcome event types occurred roughly at the same frequency, adaptive to the player's behavior. Each money trial was separated by a jittered interstimulus interval (ISI), which varied randomly between 550 and 2050 ms.
Responses to motivational outcomes (Rewards or Punishments) from single and multi-units are recorded in the MTL, from the Amygdala (N = 79) and Hippocampus (N = 61) and in the mPFC, from the dorsomedial prefrontal cortex (dmPFC, N = 63) and cingulate cortex (CC, N = 107). Unit activity is analyzed with respect to outcome occurrence, evaluated for outcome valence specificity, controllability effect and the relation to subsequent behavioral choice (i.e., approach probability when facing a reward under HGC). We find that when players have control over the outcome, units in mPFC and MTL areas demonstrate a complementary role in the encoding of punishment and the affect on subsequent behavioral choice toward a reward cue. Specifically, while mPFC neurons selectively encode the negativity of motivational outcomes, relating neural responses to subsequent behavioral choices under high-conflict seemed to be the responsibility of neurons in the MTL (hippocampus and amygdala). Intriguingly, this cross-region outcome selective effect does not appear when participants had no control over the motivational outcome.
Similar to our previous findings in healthy populations22, subjects showed an overall tendency to approach the rewarding cues on most trials (89.4% of 3285 trials) but less so when they were presented under HGC (83.4 ± 10.4% approach, mean Ntrials = 104 per patient) compared with LGC trials (94.6 ± 2.7% approach, mean Ntrials = 115 per patient, Supplementary Fig. 1) [t(14) = 4.78, p = 0.0003, mean difference = 0.11, CI = (0.06, 0.16), Cohen's d = 4.9]. Mean response times were found to be significantly lower for HGC trials (807.05 ± 151.2 ms) compared with LGC trials (902.08 ± 160.3 ms) [t(14) = 3.52, p = 0.003, mean difference = 95, CI = (37.2, 152.9), Cohen's d = 0.92] (see Supplementary Fig. 1). Shorter reaction time during the HGC condition may be a result of task-related demands, as a faster response is necessary to avoid punishment when facing multiple threats.
Approach tendencies did not differ between patients with an MTL seizure onset zone (SOZ) (five patients, 84.7% and 91.1% approach for HGC and LGC, respectively) and patients with an outside MTL SOZ (nine patients, 80.8% and 92.7% approach for HGC and LGC, respectively) [Mann–Whitney test, U = 19.5, Z = 0.61, p > 0.05 for HGC and U = 19, Z = 0.07, p > 0.05 for LGC].
A generalized linear mixed model (GLMM) with subsequent HGC behavior as the dependent variable found no effect of: behavior on the current trial, movement, outcome (achieved or missed coin), and number of balls hitting the avatar on the way to the coin. Similarly, we did not find a significant behavioral- or paradigm-related effect of Punishment outcomes on behavior in subsequent HGC trials. Furthermore, no effect was found for the time lag between Punishment and subsequent HGC trials or movement in a period of 1 s before or after Punishment outcomes.
Neuronal response selectivity to outcomes
Neurons were considered responsive to a specific outcome condition (i.e., Controlled Reward, Uncontrolled Reward, Controlled Punishment, and Uncontrolled Punishment) if they significantly changed their firing rate (FR) following that outcome (between 200 and 800 ms post outcome occurrence, evaluated using a bootstrapping approach, see "Methods"). We found 31 of 79 (39%), 26 of 61 (43%), 26 of 63 (41%), and 46 of 107 (43%) neurons that significantly responded to at least one of the four outcome conditions in the MTL; Amygdala, Hippocampus, and mPFC: dmPFC, CC, respectively (see Fig. 2 for examples of neuronal selective FR).
Fig. 2: Example neural responses to the different outcome conditions.
a Sagittal slices show the location of active electrode contacts in mPFC and MTL areas (yellow markers, registered to an MNI atlas) that refer to the raster plots and peri-stimulus time histograms (PSTH), per condition (see legend for color codes). b, c PSTH from two different amygdala neurons showing significant increase in firing following reward outcome in both controlled and uncontrolled conditions. d–f PSTH from three different mPFC neurons showing significant increase in firing following punishment outcome in controlled but not in uncontrolled condition. Time 0 on the x-axis represents the timing of outcome (coin or ball hit the avatar). FR firing rate.
To assess the sensitivity of neurons to the ability of players to control the outcomes, we examined their response probability to Controlled and Uncontrolled outcomes across valence type (Rewards and Punishments). A higher probability of responding to the Controlled outcomes over Uncontrolled outcomes was apparent in neurons from all four areas (Fig. 3a) [McNemar's exact test: MTL; Amygdala χ2 = 9.3, p = 0.0088; Hippocampus χ2 = 3.3, p = 0.07 (χ2 = 13.8, p = 0.0004 for MTL combined), mPFC; dmPFC χ2 = 7.7, p = 0.011; CC χ2 = 4.3, p = 0.049 (χ2 = 12.25, p = 0.0005 for mPFC combined), p values were corrected for false-discovery rate (FDR)]. No main effect was found for valence in any of the recording areas (Fig. 3b). During the Controlled condition, mPFC neurons appeared more responsive to Controlled Punishments over Controlled Rewards (17 vs. 4 in the dmPFC and 20 vs. 10 in the CC), while in the MTL the response probability was similar for both types of valence (Fig. 3c; 12 vs. 12 in the Amygdala and 8 vs. 10 in the Hippocampus) [χ2 = 7.2, p = 0.065 for the four areas and χ2 = 6.04, p = 0.014 comparing MTL to mPFC]. This was not observed during the Uncontrolled condition (Fig. 3d). No such selectivity was observed for neurons in both MTL regions, even when removing neurons within the MTL SOZ (see Supplementary Note 2).
Fig. 3: Neurons' response probability in different regions and outcome conditions.
Percent of neurons per region presenting a significant change in firing rate (FR) between 200 and 800 ms in response to: a Controlled (black) or Uncontrolled (gray) outcomes (across valence); N = 79, 61, 63, and 107 independently sampled neurons for the amygdala, hippocampus, dmPFC, and CC, respectively. A two-sided McNemar's exact test found effects at p = 0.088, p = 0.011, and p = 0.049 for the amygdala, dmPFC, and CC, respectively, FDR corrected. Asterisks represent significant at p < 0.05. b Reward (black) or Punishment (gray) outcomes (across controllability); N is similar to (a). c Controlled rewards or punishments (N = 93 independenty sampled neurons from four or two brain regions, p = 0.065 and p = 0.014 using χ2 test, respectively). Diamond denotes significant valence preference between MTL and mPFC at p < 0.05. d Uncontrolled rewards or punishments (N = 39 independenty sampled neurons from four or two brain regions, p = 0.97 and p = 0.99 using χ2 test, respectively). Amy amygdala, Hip hippocampus, dmPFC dorsomedial prefrontal cortex, CC cingulate cortex.
To examine the magnitude of neuronal selective responses we further calculated normalized FR changes separately for neurons with a significant increase in FR and neurons with a significant decrease in FR in at least one of the four outcome conditions, excluding neurons with a mixed response (13% of responsive neurons: 6 in MTL and 11 in mPFC). Figure 4 presents the results obtained from this analysis for neurons with a significant increase in FR in response to outcomes. Overall, in line with the probability of FR, this analysis shows that there was greater selectivity in average response to outcome valence under Controlled conditions, more so in mPFC regions than in MTL regions (Fig. 4a). In light of the similarity in response selectivity for electrodes situated in areas within mPFC and within MTL, in further analyses we combined neurons from the amygdala and hippocampus to form an MTL neural group (140 neurons) and neurons from the dmPFC and CC to form the mPFC neural group (170 neurons).
Fig. 4: Selectivity of neural time-course responses per outcome types.
Average normalized FR for positively responsive neurons shown for (a) each of the four recording regions N = 12, 9, 11, and 22 for Amygdala, Hippocampus, dmPFC, and CC, respectively. b Combined regions in MTL and mPFC groups, N = 24 MTL and 34 mPFC neurons. Shaded area corresponds to standard error of mean (SEM). Source data are provided as a Source Data file. c Responsivity profile projected on two sagittal atlas slices for mPFC (upper panel) and MTL (lower panel) region groups. Coloring is according to the averaged normalized FR change for each condition (coloring key is presented in lower square). Circle size corresponds to the number of neurons from each contact groups, from 1 (smallest) to 5 (largest) (see key). Time 0 on the x-axis represents the timing of outcome (coin or ball hit the avatar). Amy amygdala, Hip hippocampus, dmPFC dorsomedial prefrontal cortex, CC cingulate cortex, MTL medial temporal lobe, mPFC medial prefrontal cortex, STDs standard deviations, Rew reward, Pun punishment, Uncon uncontrol, Con control.
A repeated measures ANOVA with normalized FR increase following outcome (200–800 msec) as the dependent variable and region groups [MTL, mPFC], controllability (Controlled/Uncontrolled) and outcome valence (Reward/Punishment) as the independent factors, revealed a greater response to Controlled Punishment outcomes, specifically in the mPFC region group (three-way interaction [F(1, 56) = 13.6, p = 0.001, η2 = 0.2] demonstrated in time-course graphs in Fig. 4b). The ANOVA further showed that the negative bias in response to outcome was more pronounced in mPFC neurons (two-way interaction of valence and region [F(1, 56) = 5.6, p = 0.021, η2 = 0.09]), and that the preferred response to Controlled outcomes was more pronounced for negative valence (two-way interaction of valence and control [F(1, 56) = 7.72, p = 0.007, η2 = 0.12]). A main effect for controllability showed higher FR in response to Controlled [mean = 3, CI = (2.2, 3.8)] compared with Uncontrolled outcomes [mean = 1.1, CI = (0.6, 1.5)] in both region groups [F(1, 56) = 22.44, p < 0.001, η2 = 0.29]. The results were still significant when removing neurons within the SOZ from the analysis. Analyzing each region group separately, we found that MTL neurons displayed higher FR to Controlled [mean = 3.2, CI = (1.9, 4.6)] over Uncontrolled [mean = 1.6, CI = (0.7, 2.4)] outcomes [F(1, 23) = 6.3, p = 0.02, η2 = 0.22]. In mPFC neurons, we found a significantly higher FR in response to Controlled [mean = 2.8, CI = (1.7, 3.8)] vs. Uncontrolled [mean = 0.6, CI = (0.2, 1.1)] outcomes [F(1, 33) = 19.2, p = 0.001, η2 = 0.37], Punishment [mean = 2.9, CI = (1.9, 3.9)] vs. Reward [mean = 0.4, CI = (−0.7, 1.5)], [F(1, 33) = 9.37, p = 0.004, η2 = 0.22], as well as a significant interaction [F(1, 33) = 18.8, p < 0.001, η2 = 0.36] resulting from a higher response to Controlled Punishment over the other three conditions (p < 0.001). Altogether these results suggest that although all neurons showed greater responsivity to Controlled outcomes, mPFC neurons exhibited significant selectivity to negative outcomes when players had control over the trial (shown graphically per region group and recording site in Fig. 4c). Supplementary Fig. 2 presents the results for FR decreases. A repeated measures ANOVA (with similar variables and factors as above) revealed a main effect of controllability [F(1, 51) = 11.6, p = 0.001, η2 = 0.19], where Controlled trials evoked stronger decreases in FR [mean = −2.2, CI = (−2.8, −1.6)] compared with Uncontrolled trials [mean = −0.8, CI = (−1.4, −0.2)] in both region groups. No other main effect or interaction was significant.
To account for players' movements during the game, we performed a separate analysis while balancing trials across conditions according to the amount of key presses during each trial. The mPFC sensitivity to Punishment and Controllability did not seem to result from motion planning or artifacts, as evident by the similar results (Supplementary Fig. 4). mPFC neurons responded more to Control Punishment outcomes and MTL neurons more to Control Reward (χ2 = 4.23, p = 0.04). An increased FR was observed following Controlled Punishment vs. Uncontrolled Punishment in the mPFC, and was maintained following movement balancing [sign test, Z = 2.6, p = 0.009, FDR corrected]. An increased FR was also observed following Controlled Punishment vs. Controlled Reward in the mPFC [sign test, Z = 3.36, p = 0.003, FDR corrected]. In contrast, a higher MTL FR during the Controlled as compared with the Uncontrolled condition was not significant after controlling for movements.
mPFC selectivity to Controlled Punishment over Controlled Reward and Uncontrolled Punishment seems to be a general phenomenon regardless of whether punishments were obtained when a reward was chased (an unsuccessful approach trial) or when there was no reward present at all (Supplementary Fig. 5). For example, 21 mPFC neurons exclusively responded to Punishments without Rewards on the screen compared with 8 neurons exclusively responsive to Rewards, and 14 mPFC neurons exclusively responded to Punishments during an unsuccessful approach trial compared with 8 neurons exclusively responsive to Rewards (the same result was found when comparing to Uncontrolled Punishment, see Supplementary Table 3). In contrast, there was no such difference observed in MTL neurons (it should be noted that Punishments can also be obtained during failed avoidance trials, but such events were rare and could not be evaluated).
Lastly, we examined the relative timing of response to outcome in each region group per outcome type. Overlapping the time courses revealed earlier responses in mPFC compared with MTL neurons following outcome (Fig. 5a). These were only evident in the Controlled conditions, where responses were already significantly above baseline at 0–200 ms following Punishment for mPFC neurons [signed rank, Z = 346, p < 0.05, FDR corrected] and only 200–400 ms for MTL neurons [signed rank Z = 78, p < 0.05, FDR corrected]. Responses to Reward were significantly above baseline already at 0–200 ms for mPFC neurons [signed rank, Z = 43, p < 0.01, FDR corrected] and only 400–600 ms for MTL neurons [signed rank Z = 102, p < 0.01, FDR corrected]. In the Uncontrolled trials (Fig. 5a) responses were overall not significantly above baseline at any of the 200 ms epochs we measured following outcome with the exception of the 200–400 ms window in mPFC neurons during the Uncontrolled Reward condition [signed rank Z = 36, p = 0.02, FDR corrected].
Fig. 5: Neural firing change and their effect on subsequent behavioral choice.
a Time courses for mean normalized FR change in mPFC (brown trace) and MTL (blue trace) sites for each outcome type. Asterisks mark for which 200 ms window the FR is significantly above baseline (two-sided signed rank test, p < 0.05 FDR corrected, Control Reward N = 14 and 9 for MTL and mPFC, respectively). Control Punishment N = 12 and 29 for MTL and mPFC, respectively. Uncontrol Reward N = 8 and 8 for MTL and mPFC, respectively. Uncontrol Punishment N = 8 and 2 for MTL and mPFC, respectively. Note that for both controlled outcomes mPFC neurons fire slightly before MTL neurons (left) but not for uncontrolled outcomes (right). Time 0 on the x-axis represents the timing of outcome (coin or ball hit the avatar). Shaded area corresponds to SEM. Source data are provided as a Source Data file. b, c The effect of neural responses following controlled outcomes on subsequent behavioral choice under HGC condition. b The mean probability for approaching a coin, subsequent to trials where a neuron fired 200–800 ms following a controlled outcome (black bars) vs. trials where a neuron did not fire (white bars), shown for MTL (b, top) and mPFC (b, bottom) neuron groups per outcome type. Note that only MTL neuron showed a consistent pattern of subsequent behavior of less approach after punishment outcome (b, top right). c Approach probability change following neural firing to controlled rewards (green markers) and controlled punishments (red markers) in MTL (c, top) and mPFC (c, bottom). c (top) Asterisk denotes a significant two-sided Mann–Whitney test, p = 0.01, FDR corrected, N = 21 neurons (10 punishment, 11 reward). STDs standard deviations, Delta Prob difference in probability.
Neuronal response to outcome effects on subsequent behavioral choice
To test for brain-behavior interactions we assessed behavioral approach tendency with respect to neuronal firing in the previous trial. When tested with respect to HGC Controlled trials we found a distinct effect in the MTL neurons. We concentrated on evaluating the effect on behavior during HGC trials because approach probability during LGC trials was very high (92%). Neuronal firing following Controlled Punishment outcomes correlated with decreased probability for approach behavior in the next Controlled trials, whereas firing following Controlled Reward outcomes correlated with increased probability for approach behavior in subsequent Controlled trials [Mann–Whitney test, U = 13, Z = −2.9, p = 0.01, FDR corrected]. Neurons in the mPFC did not present such a differential effect (Fig. 5b, c). Neural responses to both types of Uncontrolled outcomes in the MTL and mPFC were not predictive of subsequent approach behavior in the following Controlled HGC trials (see Supplementary Fig. 3).
To further evaluate the complex interaction between the observed phenomenon and other paradigm-related variables we performed six GLMM (binomial), with behavior in subsequent HGC trials as the dependent variable and evaluating each of: Punishment temporal, Punishment frontal, Reward temporal, Reward frontal, Punishment interaction, and Reward interaction as independent variables. We found that only MTL firing following Punishment outcomes significantly correlated with behavior in subsequent HGC reward trials [beta = 1.1, t = 4.22, p < 0.0001, FDR corrected], even after accounting for movement and time between Punishment and subsequent HGC trials. Even when removing MTL neurons (two responsive neurons from the left amygdala of patient 6) that were within the SOZ, this finding remained significant [beta = 1.2, t = 4.3, p < 0.0001, FDR corrected]. This result was not replicated for the LGC trials; MTL response to punishment did not predict subsequent behavior under LGC. Breaking this result down into the different structures, we found that this was significant in the Hippocampus [beta = 1.25, t = 3.2, p = 0.006, FDR corrected] but not in the other regions: Amygdala, dmPFC, and CC.
A similar analysis on neural firing following Reward compared subsequent HGC behavior with the previous HGC trial, after accounting for the previous HGC-related variables: movement, behavior outcome (achieved or missed the coin), and number of ball hits. However, we did not find that mPFC or MTL response correlated with behavior in subsequent HGC trials. A similar analysis for LGC also failed to show an effect of neural response to outcome on subsequent trials.
To evaluate the association between mPFC responsivity to Control Punishment on the one hand and the subsequent behavioral effect of MTL neurons to Punishment on the other hand, we focused on four sessions that had increased firing neurons in both region groups. We found that the interaction between regions' firing following Controlled Punishment was predictive of subsequent HGC behavior [beta = 12.19, t = 3.14, p = 0.0018, FDR corrected].
The present study applied intracranial recordings from neurons in the mPFC (dmPFC and CC) and MTL (amygdala and hippocampus) of humans while they participated in an ecological goal-conflict game situation. We now present evidence of timed involvement of MTL and mPFC neurons in integrating outcome valence and its effect on subsequent goal-conflict resolution. Our results show that neurons in the mPFC areas were more sensitive to Punishment than Reward outcomes, but only in the game periods in which participants had a choice and could control the outcome with their behavior (i.e., Controlled condition). Compared with mPFC, the MTL showed smaller preference to Controlled outcomes, but without bias to outcome valence (Figs. 3 and 4). Yet, despite this apparent valence blindness, MTL firing following Controlled Punishment outcomes was associated with decreased approach probability when faced with a high-conflict situation in the next trial (HGC trials, Fig. 5). Although mPFC neurons alone did not show such a direct association with behavioral choice, the interaction of mPFC with MTL neuronal responses to Controlled Punishment outcomes decreased subsequent approach probability.
The bias of neurons in the dmPFC and CC to encode negative outcome is in line with previous studies showing the involvement of these regions in processing pain23 and economic loss24. As these regions are also involved in motion planning and inhibition25; such negative bias in the context of goal-directed behavioral choice could be explained by the critical need to reduce false negatives for survival26. This evolutionary rationale is supported by our finding that negative neuronal bias was only apparent when players had control over the outcome. This, however, stands in contrast to results reported by Hill et al.27 showing neural sensitivity to positive over negative outcomes (wins vs. losses) in the human mPFC. In this study, participants had to choose between two decks of cards with positive or negative reward probabilities and values. Thus, in contrast to our task, they were not directly faced with conflicting goals but rather had to learn the probability of positive and negative outcomes. Moreover, they did not have to actively move toward or away from a goal. These differences represent a major contrast between reinforcement learning tasks and anxiety-related approach-avoidance paradigms that are more similar to our task. the immersive nature and call for action of approach-avoidance scenarios in our study may bias sensitivity to punishing threats over desired rewards. One could argue that the negative outcome sensitivity shown by the mPFC neurons could be accounted for by an inhibition of prior approach behavior rather than to the response to punishment itself. To refute this possibility we show that the valence selectivity of FR in the mPFC is unrelated to the different types of punishments; occurring either with or without the presence of a reward (see Supplementary Fig. 5).
This finding that MTL were responsive to both reward and punishment outcomes is consistent with known involvement of amygdala and hippocampus in both positive and negative emotion processing in a motivation-related context6,28,29. For example, Paton et al.6 found that distinct amygdala neurons respond preferentially to positive or negative value.
A central finding in this study is the role of MTL neurons in reduced approach choice following negative outcome. Further examination of this finding in the various MTL structures showed that this result primarily stems from the hippocampal neurons, as their effect on subsequent behavioral choices was significant. The Reinforcement Sensitivity Theory (RST) of Gray and McNaughton9 proposed that the hippocampus, as part of the behavioral inhibition system, is in charge of resolving goal-conflict situations mediating the selection of more adaptive behaviors according to the acquired motivational significance. More recently, fMRI studies in humans supported such a role of the hippocampus in goal-directed gambling tasks30,31. For example, Gonen et al.31 applied dynamic causal modeling to fMRI data showing that the hippocampus received inputs regarding both positive and negative reinforcements, while participants decided to take a risk or play safely in a computerized gambling game.
However, diverging from the RST model, our results point to the significance of the MTL, not only in the online processing of positive and negative reinforcements but also in the use of such information to influence future motivation behavior. This fits well with the hippocampus' known role in association learning and extinction32. Unfortunately, our design did not allow an objective evaluation of neural response directly following cue appearance due to unbalanced trials across the different conditions (see Supplementary Note 1), resulting in confounding saliency effects. Future studies with a similar design but balanced cue trials are warranted to evaluate the MTL's role during the decision-making phase.
In addition, classical reinforcement learning studies have highlighted the important role of PEs, in the striatum, in providing a learning signal to update subsequent behavior. Recent studies, however, have shown that the hippocampus and the striatum interact cooperatively to support both episodic encoding and reinforcement learning33,34,35. Thus, it is interesting to observe that converging evidence from different study cohorts, including goal conflict, reinforcement learning and memory, seem to point to the important role of the MTL, and hippocampus in particular, in learning from outcomes to update behavior.
In a subsample of our data, the interaction of mPFC and MTL neuronal activity following punishment was significantly predictive of subsequent avoidance. This finding corresponds with a line of recent animal studies showing that inputs from the hippocampus and/or amygdala to the mPFC underlie anxiety state and avoidance behavior36,37. For example, theta (4–12 Hz) synchrony emerged between the vHPC and mPFC during rodents' exposure to anxiogenic environments38. Moreover, single units in the mPFC that synchronized with the vHPC theta bursts, preferentially represented arm type in the EPM15. Further analysis in humans could test for the relation between hippocampus-mPFC theta synchrony and unit activity in the hippocampus.
The combined evidence from animal studies and our findings in humans, suggest that the MTL, and the hippocampus in particular, play an important role in updating approach tendencies after receiving a signal of negative outcome value from the mPFC. This temporal sequence though seems to contradict the anxiety-related animal models described previously. However, these studies often apply the EPM and related paradigms that cannot dissociate in time the acquisition and updating of approach tendencies following the outcome phase from the behavioral decision-making phase.
It has been widely acknowledged that the hippocampus, amygdala, and mPFC share anatomical and functional connectivity as a distributed network that supports anxiety behavior in an interdependent manner, and that mPFC to MTL innervation exists and is related to approach-avoidance tendencies15,37. The evidence also shows that the leading direction of such connections is context dependent39,40. We speculate that in the outcome evaluation phase and before the next behavioral choice, the MTL is responsible for storing their motivational significance for future decisions under goal conflict, using inputs received from a number of cortical and subcortical nodes, including negative value signals from the mPFC following punishments. Conversely, during the actual goal-conflict behavior, an inhibiting approach could be a more direct product of mPFC activity, dependent on MTL updated inputs36,41. Neural dynamics during the decision phase in our paradigm was difficult to assess due to excessive movements and rapidly changing contexts (balls falling continuously) and further studies are warranted.
Remarkably, both the valence selectivity of mPFC neural responses and the effect of MTL outcome responses on subsequent behavior were evident only during Controlled condition (see Fig. 2). It has been argued that the neural response to outcome value and valence, as well as subsequent goal-directed behavior, is influenced by one's sense of control over a given situation—often referred to as the process of agency estimation42,43,44. Thus, one's sense of control may play a role in future motivational behavior45. Specifically, it has been suggested that a sense of control can bias the organism toward a proactive response, encouraging it to optimize approach-related decisions while giving more weight to certain outcomes. For example, a diminished sense of control as seen in depression may prevent one from learning adaptive behavior toward rewards, despite an intact ability to assign motivational significance to the goals46. Another example is PTSD, where an exaggerated sense of agency over a traumatic event is suggested to intensify the negative value attached to even distant reminders of the traumatic event, resulting in maladaptive avoidance behavior, even when motivational significance is realized46,47.
Intriguingly, recent imaging studies show that agency estimation relies on activations of the mPFC, particularly its dorsal aspect (e.g., the supplementary motor area (SMA), pre-SMA, and dorsomedial PFC)48. Further studies are needed to evaluate whether the findings observed here relate to agency estimation processing in the mPFC or reflect the effect of another region (such as the angular gyrus49) on motivational processing in mPFC.
Our data were obtained from patients with epilepsy and therefore the generalization of these results to other populations should be considered with caution. It has been previously suggested that patients with temporal lobe lesions present more approach behavior compared with controls30. In addition, studies using Stroop-related paradigms showed that patients with MTL lesions present impaired performance on conflict tasks50,51. Other studies, however, found no major difference in the Stroop task between MTL patients and healthy controls52. We believe our findings are not specific to MTL lesions or epilepsy for several reasons. First, only 5 of the 14 patients had seizures originating from the MTL, and these five patients did not exhibit different approach tendencies compared with the extra-MTL patient group. Second, removing the few neurons from within the epileptic SOZ in the MTL did not change the significance of the results, either neuronally or behaviorally. Lastly, approach probabilities and reaction times obtained from our group of patients were similar to those obtained from a control group of 20 healthy participants (see Supplementary Note 2). To evaluate this further, future studies should adopt similar ecological procedures using noninvasive imaging methods (e.g., EEG, NIRS, or fMRI).
Unfortunately, the ecological nature of our paradigm does not allow the evaluation of neural responses at timings prior to outcome (i.e., anticipation), since this time period is contaminated by movements and simultaneous occurrence of different events (rewards and punishment).
In addition, neurons from different substructures, such as the ventral and dorsal hippocampi, basolateral and central amygdala, were aggregated. It is known that these substructures play a different and sometimes contradictory role in motivational processes53.
Lastly, in an attempt to increase participants' engagement, players were encouraged to obtain the rewards, which resulted in a high approach probability. Due to the sparseness of avoidance behavioral choices, avoidance trials were not analyzed by themselves for neural responsivity. It would be of interest to evaluate neural response to avoidance in future studies. However, despite approach dominance, in a previous fMRI study with this paradigm22, we found marked differences in behavior and brain responses between HGC and LGC conditions, which lead us to conclude that there is significant goal conflict under the HGC condition. First, there was less approach behavior under HGC than LGC trials. Furthermore, brain mapping analysis during approach under HGC vs. LGC conditions showed greater mesolimbic BOLD activity and functional connectivity under HGC. Lastly, individual differences in approach/avoidance personality tendencies (indicated by standard personality questionnaires) revealed that individuals with approach personality tendency showed more approach behavior during HGC trials than individuals with avoidance-oriented personality. Altogether, these findings support our operationalization of the HGC condition.
In summary, our findings suggest differential process specificity for the MTL and mPFC following goal-directed behavior under conflict. The mPFC showed response sensitivity to the integration of negative outcomes under a controlled condition, possibly reflecting the importance of a sense of agency on assigning value to outcomes. In contrast to mPFC, the MTL neurons showed minimal response selectivity to valence of outcome, yet following punishment their responses modified approach behavioral choices under high conflicts. These findings point to the important role of MTL, and the hippocampus in particular, in learning from outcomes in order to update our behavior, a major issue in mental disorders such as addiction and borderline personality disorders. Future studies should evaluate how this differential processing could assist in computational modeling of psychiatric disorders as well as assigning process-specific targets for brain-guided interventions.
Fourteen patients with pharmacologically intractable epilepsy (nine males, 35.2 ± 14.6 years old) participated in this study. Ten patients were recorded at the Tel Aviv Sourasky Medical Center (TASMC) and four at the University of California Los Angeles (UCLA) with similar experimental protocols and recording systems. One patient (patient 4) underwent two separate implantations with a time lag of 6 months. A total of 20 sessions were recorded. Patients were implanted with chronic depth electrodes for 1–3 weeks to determine the seizure focus for possible surgical resection. The number and specific sites of electrode implantation were determined exclusively on clinical grounds. Patients volunteered for the study and gave written informed consent. The study conformed to the guidelines of the Medical Institutional Review Boards of TASMC and UCLA. An additional group of 20 healthy participants performed the task in a laboratory room (Supplementary Note 2). These participants volunteered for the study and gave written informed consent. The healthy participant study was approved by the Tel Aviv University Ethics Committee.
Through the lumen of the clinical electrodes, nine Pt/Ir microwires were inserted into the tissue, eight active recording channels and one reference. The differential signal from the microwires was amplified and sampled at 30 kHz using a 128-channel BlackRock recording system (Blackrock Microsystems) and recorded using the Neuroport Central software up to version 6.05. The extracellular signals were band-pass filtered (300 Hz to 3 kHz) and later analyzed offline. Spikes were detected and sorted using the wave_clus toolbox (version 1.1)54 and MATLAB (mathworks, version 2018a). Units were classified by one of the authors (T.G.) based on spike shape, variance, and the presence of a refractory period for the single units55. Units were classified as putative single units and multi-unit clusters based on the presence of a refractory period (single unit had to present at least 99% of action potentials which were separated by an inter-spike interval of 3 ms or more) and based on spike shape. To anatomically localize single-unit recording sites we registered computerized tomography images acquired postimplantation to high-resolution T1-weighted magnetic resonance imaging data acquired preimplantation using SPM12 (http://www.fil.ion.ucl.ac.uk/spm). Micro Electrode locations with units can be seen in Fig. 2e. In total, we recorded 79 Amygdala, 61 hippocampus, 63 dmPFC, and 107 cingulate units. Microwire locations are described in Supplementary Table 1.
Experimental design and sessions
Subjects sat in bed facing a laptop and were asked to perform the Punishment, Reward, and Incentive Motivation game (PRIMO game22; Java1.6, Oracle, Redwood-Shores, CA & Processing package, http://www.processing.org). The goal of the game was to earn money by catching coins and avoiding balls. The monetary reward was virtual, no real money was delivered at any time to the participants. A small avatar on a skateboard was located at the bottom of the screen and subjects had to move the avatar right and left using the right and left arrow keys, in order to catch the money and avoid the balls falling from the top of the screen. There were two ways to gain or lose money—a "Controlled" condition, where players actively approached coins and avoided balls (which fall in a straight to zig-zag fashion from the top of the screen) and an "Uncontrolled" condition, where although cues appeared on the top of the screen, they hit the avatar randomly without relation to the players' action (Fig. 1). During Uncontrolled Reward or Punishment trials, controlled balls can appear along with the uncontrolled cue, but they are relatively easy to avoid as there is no conflict (the subject has the ability to move the avatar throughout the game regardless of the trial's condition). Each coin catch resulted in a five-point gain and each ball hit resulted in a loss of five points, regardless of controllability. To create an ecological environment, the difficulty level of the game was modified every 10 s according to the local and global performance of the player. By dynamically adjusting the difficulty level (speed) and actively balancing the number of Uncontrolled events, the game was tailored to match each player's skills and all Outcome event types occurred roughly at the same frequency. Each Reward trial was separated by a jittered interstimulus interval, which varied randomly between 550 and 2050 ms. To construct HGC and LGC, the number of obstacles (i.e., balls) placed between the player and the falling coin changed in each trial. Trials with 0–1 balls between the player and the falling coin were defined as LGC trials, while trials with 2–6 balls between the player and the coin were defined as HGC. The game was played for three or four blocks (according to the patient's agreement) of 6 min each, starting with 1-min fixation point. Subjects received instructions prior to playing the first session. Subjects 2, 4, 6, 8, 10, and 12 played the game twice during their monitoring period at a lag of 13, 5, 4, 2, 4, and 2 days between sessions, respectively. The paradigm was identical to that used in Gonen et al.22 with one exception: the flying figure used in the previous version to signal Uncontrolled trials was removed (to avoid neural responses to its appearance). In the current version, the colors of the Rewards and Punishment were changed to signal non-Controllability. Uncontrolled rewards were in cyan color (vs. green for controlled rewards) and Uncontrolled balls were orange in color (vs. red for controlled balls). This was explained to the subjects during training.
Analysis of behavioral data
Once a Controlled Reward cue appeared at the top of the screen, subjects had to decide whether to approach it (at the risk of a possible hit by a ball) or to avoid it (and thus minimize the risk of getting hit). Controlled Reward trials were classified to approach and avoidance trials according to the player's behavior in each game session, based on a machine learning classification model22.
Analysis of neural data
Data were analyzed using MATLAB (version 2018a). Raster plots were binned to nonoverlapping windows of 200 ms length to create FR per window and summed across trials to create peri-stimulus time histograms (PSTH). PSTHs were initially calculated for a period of 8 s from 3 s before outcome stimulus to 5 s post stimulus (40 windows). For evaluating neuron responsiveness, we concentrated on the time period of 200–800 ms post outcome stimulus (similarly to Ison et al.56). As this study concentrates on neural response to outcome, time 0 relates to the moment when the ball or coin hits the avatar (time of outcome), throughout the manuscript.
Criteria for a responsive unit
To evaluate neural responsiveness to the different conditions, we adopted a bootstrapping approach. Since the PRIMO game is interactive and ongoing there is no distinct baseline period prior to each trial. We thus created a distribution of FR where each instance in the distribution is calculated as a FR average of N windows randomly selected from the entire session period, where N is the number of trials of the specific condition. Thus, the only difference between the measured FR and such an instance is that the actual measured FR was time-locked to the events of the specific condition. The distribution was built from 1000 such instances and the neuron's window was considered positively responsive to the condition if the probability of obtaining the measured FR or higher was <0.01, and negatively responsive if the probability of obtaining the measured FR or lower was <0.01. A neuron was considered responsive to the condition if it was responsive in at least one of the three windows between 200 and 800 ms post event. Neural yield per area is described in Supplementary Table 2.
Comparison between conditions
To evaluate the main effect of controllability (over emotional value) we united reward and punishment trials and evaluated which neurons significantly changed FR following controlled or uncontrolled trials (with a similar bootstrapping approach). A similar analysis was performed for evaluating emotional value (over controllability). Next, we used chi-square analysis to assess whether the probability of responding to a specific condition varied between the different anatomical groups of neurons. In an additional analysis, we evaluated which neurons were positively responsive (increased FR) to at least one of the four conditions and which were negatively responsive to at least one condition. Neurons with a positive response to one condition and a negative response to another were excluded (N = 4, 2, 0, and 12 for Amygdala, Hippocampus, dmPFC, and CC, respectively). For both the positively responsive neurons and the negatively responsive neurons we computed a repeated measures ANOVA with region as the between subject variable and controllability and valence as the within subject variables and normalized FR of the different neurons as the dependent variable. Normalized FR was calculated by:
$${\rm{F}} \bar{\rm{R}} = {\rm{mean}}\left( {\frac{{{\rm{FR}}_{{\rm{window}}} - {\rm{F}}\bar{\rm{R}}_{{\rm{random}}}}}{{\sigma _{{\rm{random}}}}}} \right),$$
where \({\rm{F}}\bar{\rm{R}}_{{\rm{random}}}\) is the average of N 200 ms long randomly selected windows, σrandom is the standard deviation of these randomly selected windows, and the mean is across the three windows (200–400, 400–600, and 600–800 ms post stimulus).
To evaluate whether the difference between frontal and temporal neurons is due to motions (either motion planning or artifact), we balanced motion (as obtained from the number of key presses) by excluding trials with high or low motion resulting in similar median motion scores (across remaining trials) between the two compared conditions. This analysis was performed separately for each condition pair.
Similarly, to evaluate the neural responses for different scenarios in which punishment was obtained, we balanced the number of trials in each paired comparison and tested the number of units that responded to each condition. The conditions for the paired comparisons included punishment without a reward present, punishment following a failed approach response, an uncontrolled punishment. These were also compared with a controlled reward outcome (see comparison results in Supplementary Table 3).
Time course of neural data
PSTHs were calculated for a period of 8 s, from 3 s before the stimulus to 5 s post stimulus (40 windows of length 200 ms each). The time course for each condition and region was created by averaging normalized FR (per window) across condition-specific positively responsive neurons during this time period. For each of the four windows between 0 and 800 ms, we evaluated whether the response to the specific outcome condition was significantly above baseline.
Outcome—behavioral link
To evaluate whether neural response to outcome affects future behavior, we analyzed each of the four outcome conditions separately. For each condition, we focused on neurons with a significant increase in FR following its outcome. For these neurons we divided outcome trials into trials with a neural response (neural firing between 200 and 800 ms following outcome) and trials without a neural response. Neurons with a high FR which resulted in less than one trial in which the neuron did not fire and followed by an approach choice or less than one trial in which the neuron did not fire and followed by an avoidance choice were omitted from this analysis. Next, for each trial, we evaluated whether the subsequent coin trial resulted in approach or avoidance behavior. Thus we could compare, for each condition, the effect of neural response following outcome on subsequent behaviors. Only outcome trials with a subsequent HGC coin trial (with more than one ball on the way to the coin) were included as LGC trials almost always resulted in approach behavior (above 94%).
To evaluate the complex interaction of neural firing, behavior- and paradigm-related variables, we performed six GLMM (binomial) with behavior in subsequent HGC trials as the dependent variable. GLMM test 1–2: for each HGC trial, we evaluated the previous punishment outcome by calculating the following variables: (1) a binary index indicating whether a temporal/frontal neuron fired in the time range 200–800 ms following punishment outcome (as before, only neurons that significantly increased FR following punishment outcome were evaluated); (2) normalized total movement ±1 s of outcome time; (3) time delay between the punishment outcome and subsequent HGC trial. The analysis was done separately for temporal (GLMM test 1: 12 neurons, 953 total trials) and frontal neurons (GLMM test 2: 29 neurons, 2187 total trials) with neuron as the grouping variable. We used these three variables for both fixed and random effects including fixed and random intercepts grouping trials by neurons.
GLMM tests 3–4: for each HGC trial, we evaluated the previous HGC trial by calculating the following variables: (1) a binary index indicating whether a temporal/frontal neuron fired in the time range 200–800 ms following reward outcome (as before, only neurons that significantly increased FR following reward outcome were evaluated); (2) outcome—a binary variable indication whether the coin was caught or missed; (3) normalized total movement between reward appearance and disappearance either by avatar catching or missing; (4) number of ball hits on the way to the coin; and (5) behavioral decision (to approach or not). The analysis was done separately for temporal (GLMM test 3: 14 neurons, 1023 total trials) and frontal neurons (GLMM test 4: 19 neurons, 748 total trials) with neuron as the grouping variable.
To evaluate the connection between frontal responsivity to controlled punishment on one hand and the subsequent behavioral effect of temporal neurons to punishment on the other hand, we concentrated on four sessions (from patients 3, 4 and two sessions from patient 7) that had neurons with punishment-related FR increase in both the temporal and frontal lobes. For each HGC trial, we evaluated previous punishment calculating the following variables: (1) average firing of temporal neurons in the time range 200–800 ms following punishment outcome; (2) average firing of frontal neurons in the time range 200–800 ms following punishment outcome; (3) interaction between the previous variables; (4) normalized total movement ±1 s of outcome time; (5) time delay between the punishment outcome and subsequent HGC trial.
Similarly, for reward trials, we concentrated on three sessions (from patients 3, 7, and 10) that had neurons with reward-related FR increase in both the temporal and frontal lobes. For each HGC trial, we evaluated the previous HGC trial by calculating the following variables: (1) average firing of temporal neurons in the time range 200–800 ms following HGC reward outcome; (2) average firing of frontal neurons in the time range 200–800 ms following HGC reward outcome; (3) interaction between the previous variables; (4) normalized total movement between reward appearance and disappearance either by avatar catching or missing; (5) number of ball hits on the way to the coin (we did not add the behavior variable since all previous HGC trials in this case turned out to be approach trials).
Statistics and reproducibility
All experiments were only performed once. Source data are provided with this paper.
A reporting summary for this article is available as a Supplementary Information file. The source data underlying Figs. 4a, b, 5 and Supplementary Figs. 1, 2, 4, and 5 are provided as a Source Data file. Additional data are available from the corresponding author upon reasonable request.
Custom Matlab scripts are available through the following URL: https://github.com/tomergazit1/mPFC-and-MTL-neuronal-response-to-outcome-affects-subsequent-choice-paper-.
An amendment to this paper has been published and can be accessed via a link at the top of the paper.
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We acknowledge financial support from the European Union Seventh Framework Program (FP7/2007-2013) under grant agreement no. 604102 (Human Brain Project). This work was also supported by the I-CORE Program of the Planning and Budgeting Committee and the Israel Science Foundation (grant no. 51/11, TH) and the Sagol Family Fund. Special thanks to Dr. Eran Eldar (Hebrew University of Jerusalem) for the inspiration and access to early versions of the PRIMO game.
These authors contributed equally: Tomer Gazit, Tal Gonen, Guy Gurevitch, Talma Hendler, Itzhak Fried.
Sagol Brain Institute Tel Aviv, Wohl Institute for Advanced Imaging, Tel Aviv Sourasky Medical Center, Tel Aviv, Israel
Tomer Gazit, Tal Gonen, Guy Gurevitch, Noa Cohen, Hagar Yamin, Talma Hendler & Itzhak Fried
Sackler Faculty of Medicine, Tel Aviv University, Tel Aviv, Israel
Tomer Gazit, Noa Cohen, Ido Strauss, Firas Fahoum, Talma Hendler & Itzhak Fried
Department of Neurosurgery, Tel Aviv Sourasky Medical Center, Tel Aviv, Israel
Tal Gonen
School of Psychological Sciences, Faculty of Social Sciences, Tel Aviv University, Tel Aviv, Israel
Guy Gurevitch & Talma Hendler
Functional Neurosurgery Unit, Tel Aviv Sourasky Medical Center, Tel Aviv, Israel
Ido Strauss & Itzhak Fried
Sagol School of Neuroscience, Tel Aviv University, Tel Aviv, Israel
Yoav Zeevi & Talma Hendler
Department of Statistics and Operation Research, Tel Aviv University, Tel Aviv, Israel
Yoav Zeevi
Epilepsy Unit, Department of Neurology, Tel Aviv Sourasky Medical Center, Tel Aviv, Israel
Firas Fahoum
Department of Neurosurgery, David Geffen School of Medicine, University of California Los Angeles, Los Angeles, CA, USA
Itzhak Fried
Tomer Gazit
Guy Gurevitch
Noa Cohen
Ido Strauss
Hagar Yamin
Talma Hendler
T.H., T.Ga., and T.Go. conceived the study and designed the experiment. T.Ga., G.G., and N.C. analyzed the data. T.Ga., H.Y., and G.G. ran the experiments. I.F. and I.S. performed the surgeries and supervised the experiments and all aspects of data collection. Y.Z. assisted with statistical analyses. G.G. contributed to electrode localization. F.F. took care of the patients at TASMC. T.H. and I.F. supervised methodology and interpretation of findings. T.Ga., G.G., and T.H. wrote the paper. I.F., F.F., and T.Go. further contributed to the writing by reviewing and editing the manuscript.
Correspondence to Talma Hendler.
Peer review informationNature Communications thanks Gabriel Kreiman and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Gazit, T., Gonen, T., Gurevitch, G. et al. The role of mPFC and MTL neurons in human choice under goal-conflict. Nat Commun 11, 3192 (2020). https://doi.org/10.1038/s41467-020-16908-z
Editors' Highlights
Nature Communications (Nat Commun) ISSN 2041-1723 (online) | CommonCrawl |
Riemann–Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion
Min Yang1 &
Haibo Gu ORCID: orcid.org/0000-0001-8408-38592
Journal of Inequalities and Applications volume 2021, Article number: 8 (2021) Cite this article
This article is devoted to the study of the existence and uniqueness of mild solution to a class of Riemann–Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion. Our results are obtained by using fractional calculus, stochastic analysis, and the fixed-point technique. Moreover, an example is provided to illustrate the application of the obtained abstract results.
Fractional calculus has gained considerable popularity during the past decades since it has been recognized as one of the best tools to model physical systems possessing long-term memory and long-range spatial interactions, which also plays an important role in diverse areas of science and engineering, as well as other applied sciences. As a result of the intensive development of fractional calculus, there has been a significant breakthrough in theoretical analysis and applications of fractional differential equations in the literature [8, 14, 15, 17, 23]. In recent decades, the existence and uniqueness of mild solutions, as well as controllability for fractional differential equations with Caputo fractional derivative, have attracted much attention. We can refer to [1–3, 11, 13, 16, 18–20, 24, 25] and the references therein.
It is worth mentioning that by applying Laplace transform and probability density functions, Zhou et al. [26] gave a suitable concept of mild solutions for a class of Riemann–Liouville fractional evolution equations with nonlocal conditions. Li et al. [10] considered the Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives by using the α-resolvent operator. Yang and Wang [22] investigated the approximate controllability of Riemann–Liouville fractional differential inclusions. For more details about the existence of mild solutions of Riemann–Liouville fractional differential equations, one can refer to [12, 20]. On the other hand, Ahmed and El-Borai [1] studied Hilfer fractional stochastic integro-differential equations. We can refer to [5–7] for more details about the existence of mild solutions and approximate controllability of Hilfer fractional differential equations and inclusions.
As an generalization of Brownian motion, fractional Brownian motion has received a lot of attention in the recent years, which as we know is a Gaussian process with self-similarity and stationary increments, as well as long-range dependence properties. When Hurst parameter \(H \in (0,\frac{1}{2})\), fractional Brownian motion is neither a semimartingale nor a Markov process. Hence, the classical stochastic analysis techniques are invalid to study it. Recently, the research of stochastic differential equations driven by fractional Brownian motion has been investigated by many authors, see [4, 9, 21, 28] and the references therein.
However, we would like to emphasize that it is natural and also important to study the existence of mild solutions for Riemann–Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion since it has not yet been sufficiently studied in contrast with the integer-order case. In this paper we are concerned with the following Riemann–Liouville fractional stochastic evolution equations with nonlocal conditions driven by both Wiener process and fractional Brownian motion:
$$ \textstyle\begin{cases} {} ^{L}\mathcal{D}^{\alpha }[x(t)-h(t,x(t))] = Ax(t)+F(t,x(t)) \frac{d\omega (t)}{dt} \\\hphantom{{} ^{L}\mathcal{D}^{\alpha }[x(t)-h(t,x(t))] =} {} +\sigma (t) \frac{dB_{Q}^{H}(t)}{dt}, \quad t\in J{'}=(0,b], \\ \mathcal{I}_{0^{+}}^{1-\alpha }[x(0)-g(x)]=x_{0}\in X, \end{cases} $$
where \({}^{L}\mathcal{D}_{t}^{\alpha }\) denotes the Riemann–Liouville fractional derivative in time defined for \(\frac{1}{2}<\alpha <1\), \(\mathcal{I}^{\alpha }\) is the temporal Riemann–Liouville fractional integral operator of order α; \(x(\cdot )\) takes values in a separable Hilbert space X, A is the infinitesimal generator of an analytic semigroup \(\{S(t)\}_{t\geq 0}\) on a real separable Hilbert space X with inner product \(\langle \cdot , \cdot \rangle \) and norm \(\Vert \cdot \Vert \). Assume K to be another separable Hilbert space with inner product \(\langle \cdot , \cdot \rangle \) and norm \(\Vert \cdot \Vert _{K}\). Let \(L(K,X)\) denote the space of all bounded linear operators from K to X, \(h: J\times X\rightarrow X\) and \(F: J\times X\rightarrow L(K,X)\) be functions satisfying some specific assumptions given in (\(\mathrm{H}_{2}\))–(\(\mathrm{H}_{3}\)); \(\{\omega (t)\}_{t\geq 0}\) is a given K-valued Wiener process with a finite trace nuclear covariance operator \(Q>0\) defined on the filtered complete probability space \((\Omega ,\mathcal{F},P)\); \(B^{H}\) is a fractional Brownian motion with Hurst parameter \(H \in (\frac{1}{2}, 1)\). The initial data \(x_{0}\) is an \(\mathcal{F}_{0}\)-measurable, stochastic process independent of the Wiener process ω and fBm \(B^{H}(t)\) with finite second moment.
A brief outline of this paper is given as follows. In Sect. 2, we recall some notations and preliminaries about fractional Brownian motion and fractional calculus. Using the fixed point theorem, Sect. 3 establishes the existence of mild solutions for system (1.1). In Sect. 4, we will give an example to illustrate the application of the obtained abstract results. Conclusions and discussions are given in the final section.
In this section, we first present some notations, definitions, and preliminary facts.
We first recall some basic facts about fBm and the Wiener integral with respect to fBm. Let \((\Omega ,\mathcal{F},P)\) be a complete probability space. We choose a time interval \(J=[0, b]\) with arbitrary fixed horizon b, assume \(\{B^{H}(t), t \in J\}\) to be a one-dimensional fractional Brownian motion with Hurst parameter \(H\in (0,1)\). The definition implies that \(B^{H}\) is a continuous and centered Gaussian process with covariance function
$$ R_{H}(s,t)=\frac{1}{2} \bigl(t^{2H}+s^{2H}- \vert t-s \vert ^{2H} \bigr). $$
In what follows, we always assume \(\frac{1}{2}< H<1\). Consider the square integrable kernel
$$ K_{H}(s,t)=c_{H}s^{\frac{1}{2}-H} \int _{s}^{t}(u-s)u^{H-\frac{1}{2}}\,du, $$
where \(c_{H}=[\frac{H(2H-1)}{\beta (2-2H,H-\frac{1}{2})}]^{\frac{1}{2}}\), \(t>s\). Then we have
$$ \frac{\partial K_{H}}{\partial t}(t,s)=c_{H} \biggl(\frac{t}{s} \biggr)^{H- \frac{1}{2}}(t-s)^{H-\frac{3}{2}}. $$
We will denote by \(\mathcal{H}\) the reproducing kernel Hilbert space of the fBm. In fact, \(\mathcal{H}\) is the closure of the linear space of indicator functions \(\{I_{[0, t]}, t \in [0, T]\}\) with respect to the scalar product
$$ \langle I_{[0,t]},I_{[0,s]}\rangle _{\mathcal{H}}=R_{H}(t,s). $$
The mapping \(I_{[0,t]}\rightarrow {B^{H}(t)}\) can be extended to an isometry between \(\mathcal{H}\) and the first Wiener chaos, and we denote by \(B^{H}(\varphi )\) the image of φ under this isometry.
Set \(K_{H}^{*}\) be the linear operator from \(\mathcal{H}\) to \(L^{2}([0,b])\) defined by
$$ \bigl(K_{H}^{*}\varphi \bigr) (s)= \int _{s}^{t}\varphi (t) \frac{\partial K_{H}}{\partial t}(t,s)\,dt. $$
Recall that the operator \(K_{H}^{*}\) is an isometry between \(\mathcal{H}\) and \(L^{2}([0,b])\).
Let the process \(\omega =\omega (t)\), \(t\in [0,b]\) be defined by
$$ \omega (t)=B^{H} \bigl( \bigl(K_{H}^{*} \bigr)^{-1}I_{[0,b])} \bigr). $$
Then ω is a Wiener process, while \(B^{H}\) has the integral representation
$$ B^{H}(t)= \int _{0}^{t}K_{H}(t,s)\,d\omega (s). $$
We recall that for φ, ψ on \([0,b]\), their scalar product in \(\mathcal{H}\) is given by
$$ \langle \varphi ,\psi \rangle _{\mathcal{H}}=\alpha _{H} \int _{0}^{b} \int _{0}^{b}\varphi (r)\psi (u) \vert r-u \vert ^{2H-2}\,du\,dr< \infty , $$
where \(\alpha _{H}=H(2H-1)\).
The embedding relationship is as follows [21]:
$$ L^{1} \bigl([0,b] \bigr)\subset L^{\frac{1}{\mathcal{H}}}\subset \vert \mathcal{H} \vert \subset \mathcal{H}. $$
Assume that Y is a real separable Hilbert space. Suppose that there exists a complete orthogonal system \(\{e_{n}\}_{n=1}^{\infty }\) in Y. Let \(Q\in L(Y,Y)\) be an operator with finite trace \(\operatorname{Tr}Q=\sum_{n=1}^{\infty }\lambda _{n}<\infty \) (\(\lambda _{n} \geq 0\)) such that \(Qe_{n}=\lambda _{n} e_{n}\). By using the covariance operator Q, the infinite-dimensional fBM on Y can be defined as
$$ B^{H}(t)=B_{Q}^{H}(t)=\sum _{n=1}^{\infty }\sqrt{\lambda _{n}} \varphi e_{n}B_{n}^{H}(t), $$
where \(B_{n}^{H}(t)\) are one-dimensional standard fractional Brownian motions mutually independent on \((\Omega ,\mathcal{F},P)\). Assume that the space \(\mathcal{L}_{2}^{0}:=\mathcal{L}_{2}^{0}(Y,X)\) consists of all Q-Hilbert–Schmidt operators \(\varphi :Y\rightarrow X\). Recall that \(\varphi \in L(Y,X)\) is called a Q-Hilbert–Schmidt operator if
$$ \Vert \varphi \Vert _{\mathcal{L}_{2}^{0}}^{2}:=\sum _{n=1}^{\infty } \Vert \sqrt{\lambda _{n}}\varphi e_{n} \Vert ^{2}< \infty , $$
and that the space \(\mathcal{L}_{2}^{0}\) equipped with the inner product \(\langle \varphi ,\psi \rangle _{\mathcal{L}_{2}^{0}}=\sum_{n=1}^{ \infty }\langle \varphi e_{n},\psi e_{n}\rangle \) is a separable Hilbert space.
Let \((\phi (s))_{s\in [0,b]}\) be a deterministic function with values in \(\mathcal{L}_{2}^{0}(Y,X)\). The stochastic integral of ϕ with respect to \(B^{H}\) is defined as
$$ \int _{0}^{t}\phi (s)\,dB^{H}(s)=\sum _{n=1}^{\infty } \int _{0}^{t} \sqrt{\lambda _{n}} \bigl(K_{H}^{*}(\phi e_{n}) \bigr) (s)\,dB_{n}(s). $$
([4])
If \(\varphi :[0,b]\rightarrow \mathcal{L}_{2}^{0}(Y,X)\) satisfies \(\int _{0}^{b}\Vert \varphi (s)\Vert ^{2}_{\mathcal{L}_{2}^{0}}\,ds<\infty \), then the aforementioned sum in (2.1) is well defined as an X-valued random variable, and we have
$$ E \biggl\Vert \int _{0}^{t}\varphi (s)\,dB^{H}(s) \biggr\Vert ^{2}\leq c_{0}H(2H-1)t^{2H-1} \int _{0}^{t} \bigl\Vert \varphi (s) \bigr\Vert _{\mathcal{L}_{2}^{0}}^{2}\,ds. $$
We next denote by \(\mathcal{L}_{2}(\Omega ,X)\) the collection of all strongly-measurable, square-integrable, X-valued random variables. Obviously, \(\mathcal{L}_{2}(\Omega ,X)\) is a Banach space equipped with norm \(\Vert x(\cdot )\Vert _{\mathcal{L}_{2}(\Omega ,X)}=(E\vert x(\cdot )\vert )^{ \frac{1}{2}}\). Let \(C(J,\mathcal{L}_{2}(\Omega ,X))\) denote the Banach space of all X-valued continuous functions from \(J=[0,b]\) into \(\mathcal{L}_{2}(\Omega ,X)\) satisfying \(\sup_{t\in J}\Vert x(t)\Vert ^{2}<\infty \). Let \(J{'}=(0,b]\). To define the mild solution of system (1.1), we also need to consider the Banach space \(C_{\alpha }(J,X)=\{x:t^{1-\alpha }x(t)\in C(J,\mathcal{L}_{2}(\Omega ,X)) \}\) with the norm
$$ \Vert x \Vert _{C_{\alpha }}= \Bigl(\sup_{t\in J} E \bigl\Vert t^{1-\alpha }x(t) \bigr\Vert ^{2} \Bigr)^{ \frac{1}{2}}. $$
Throughout this paper, we assume that \(0\in \rho (A)\) where \(\rho (A)\) denotes the resolvent set of A. Then it is possible for us to define the fractional power \(A^{\eta }\) as a closed linear operator on its domain \(D(A^{\eta })\) for \(0<\eta \leq 1\) (see [25]). For an analytic semigroup \(\{S(t)\}_{t\geq 0}\), the following properties hold:
There exists \(M\geq 1\) such that
$$\begin{aligned} M:=\sup_{t\in [0,+\infty )}S(t)< \infty ; \end{aligned}$$
For \(\forall \eta \in (0,1]\), there exists a positive constant \(C_{\eta }\) such that
$$\begin{aligned} \bigl\Vert A^{\eta }S(t) \bigr\Vert \leq \frac{C_{\eta }}{t^{\eta }}, \quad 0< t \leq b. \end{aligned}$$
For further convenience, set
$$\begin{aligned} \mathcal{S}_{\alpha }(t)x= \int _{0}^{\infty }\phi _{\alpha }(\theta )S \bigl(t^{ \alpha }\theta \bigr)x\,d\theta ,\quad\quad \mathcal{P}_{\alpha }(t)x= \int _{0}^{ \infty }\alpha \theta \phi _{\alpha }( \theta )S \bigl(t^{\alpha }\theta \bigr)x\,d \theta , \end{aligned}$$
$$\begin{aligned} \phi _{\alpha }(\theta )=\frac{1}{\alpha }\theta ^{-1-\frac{1}{\alpha }} \psi _{\alpha } \bigl(\theta ^{\frac{-1}{\alpha }} \bigr), \end{aligned}$$
which involves the Wright-type function
$$\begin{aligned} \psi _{\alpha }(\theta )=\frac{1}{\pi }\sum_{n=1}^{\infty }(-1)^{n-} \theta ^{-\alpha n-1} \frac{\Gamma (n\alpha +1)}{n!}\sin (n\pi \alpha ), \quad \theta \in (0,\infty ), \end{aligned}$$
connected with the following one-sided stable probability function: \(\phi _{\alpha }(\theta )\geq 0\), \(\theta \in (0,\infty )\) and \(\int _{0}^{\infty }\phi _{\alpha }(\theta )\,d\theta =1\).
([23])
The operators \(\mathcal{S}_{\alpha }\) and \(\mathcal{P}_{\alpha }\) have the following properties:
For any fixed \(t\geq 0,\mathcal{S}_{\alpha }(t)\) and \(\mathcal{P}_{\alpha }(t)\) are linear and bounded operators, i.e., for any \(x\in X\),
$$\begin{aligned}& \bigl\Vert \mathcal{S}_{\alpha }(t)x \bigr\Vert \leq M \Vert x \Vert \quad \textit{and}\quad \bigl\Vert \mathcal{P}_{\alpha }(t)x \bigr\Vert \leq \frac{M}{\Gamma (\alpha )} \Vert x \Vert . \end{aligned}$$
\(\{\mathcal{S}_{\alpha }(t)\}_{t\geq 0}\) and \(\{\mathcal{P}_{\alpha }(t)\}_{t\geq 0}\) are strongly continuous.
(iii)
For every \(t>0\), \(\mathcal{S}_{\alpha }(t)\) and \(\mathcal{P}_{\alpha }(t)\) are also compact operators if \(S(t), t>0\) is compact.
([25, 27])
For any \(t > 0\) and \(0 \leq \gamma < 1\), there exists a positive constant \(C_{\gamma }\) such that
$$ A\mathcal{P}_{\alpha }(t)x=A^{1-\gamma }\mathcal{P}_{\alpha }(t)A^{ \gamma }x $$
$$ \bigl\Vert A^{\gamma }\mathcal{P}_{\alpha }(t) \bigr\Vert \leq \frac{\alpha C_{\gamma }\Gamma (2-\gamma )}{t^{\alpha \gamma }\Gamma (1+\alpha (1-\gamma ))}. $$
For \(\sigma \in (0,1]\) and \(0< a\leq b\), we have \(\vert a^{\sigma }-b^{\sigma }\vert \leq (b-a)^{\sigma }\).
$$ \begin{aligned} x(t)={}& \frac{t^{\alpha -1}}{\Gamma (\alpha )}\bigl[x_{0}+h \bigl(0,x(0)\bigr)+g(x)\bigr]+h\bigl(t,x(t)\bigr)+ \frac{1}{\Gamma (\alpha )} \int _{0}^{t}(t-s)^{\alpha -1}Ax(s)\,ds \\ & {} + \frac{1}{\Gamma (\alpha )} \int _{0}^{t}(t-s)^{\alpha -1}F\bigl(s,x(s) \bigr)\,d \omega (s)+ \frac{1}{\Gamma (\alpha )} \int _{0}^{t}(t-s)^{ \alpha -1}\sigma (s)\,dB_{Q}^{H}(s), \quad t>0, \end{aligned} $$
then we have
$$ \begin{aligned} x(t)&=t^{\alpha -1}\mathcal{P}_{\alpha }(t) \bigl[x_{0}+h\bigl(0,x(0)\bigr)+g(x)\bigr]+h\bigl(t,x(t)\bigr) \\ &\quad {} + \int _{0}^{t}(t-s)^{\alpha -1} \mathcal{P}_{\alpha }(t-s)F\bigl(s,x(s)\bigr)\,d \omega (s) \\ &\quad {} + \int _{0}^{t}(t-s)^{\alpha -1}A \mathcal{P}_{\alpha }(t-s)h\bigl(s,x(s)\bigr)\,ds \\ &\quad {} + \int _{0}^{t}(t-s)^{\alpha -1} \mathcal{P}_{\alpha }(t-s) \sigma (s)\,dB_{Q}^{H}(s), \quad t>0. \end{aligned} $$
Referring to [25], we omit the proof here. □
In this section, we present and prove the existence of mild solutions for system (1.1). To develop our results, we first give the concept of mild solution for system (1.1).
Definition 3.1
An \(\mathcal{F}_{t}\)-adapted and measurable stochastic process \(x\in C_{\alpha }(J,X)\) is said to be a mild solution of system (1.1) if \(x_{0},g\in \mathcal{L}_{2}^{0}(\Omega ,X)\), for each \(s\in [0,b)\), the function \((t-s)^{\alpha -1}AT_{\alpha }(t-s)h(s,x(s))\) is integrable and the following integral equation is verified:
$$\begin{aligned} x(t) =&t^{\alpha -1}\mathcal{P}_{\alpha }(t) \bigl[x_{0}+h \bigl(0,x(0) \bigr)+g(x) \bigr]+h \bigl(t,x(t) \bigr) \\ & {} + \int _{0}^{t}(t-s)^{\alpha -1} \mathcal{P}_{\alpha }(t-s)F \bigl(s,x(s) \bigr)\,d \omega (s) \\ & {} + \int _{0}^{t}(t-s)^{\alpha -1}A \mathcal{P}_{\alpha }(t-s)h \bigl(s,x(s) \bigr)\,ds \\ & {} + \int _{0}^{t}(t-s)^{\alpha -1} \mathcal{P}_{\alpha }(t-s)\sigma (s)\,dB_{Q}^{H}(s), \quad t \in J{'}. \end{aligned}$$
For our readers' convenience, we first introduce some notations:
$$\begin{aligned}& \bigl\Vert A^{-\beta } \bigr\Vert =M_{0}, \quad\quad K( \alpha ,\beta )= \frac{C_{1-\beta }\Gamma (1+\beta )}{\beta \Gamma (1+\alpha \beta )}, \quad\quad c= \frac{\alpha -1}{1-\alpha _{1}}, \quad\quad \Lambda = \frac{b^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}}, \\& \alpha _{1} \in \bigg[\frac{1}{2}, \alpha \bigg). \end{aligned}$$
To establish the main results, we require the following hypotheses:
(\(\mathrm{H}_{1}\)):
Semigroup \(S(t)\) is compact for each \(t>0\);
for each \(x\in X\), the function \(F(\cdot ,x):J\rightarrow \mathcal{L}_{2}^{0}(X, Y)\) is strongly measurable with respect to t, and for each \(t\in J\), the function \(F(t,\cdot ):X\rightarrow \mathcal{L}_{2}^{0}(X, Y)\) is continuous with respect to x;
there exist a function \(N(t)\in L^{\frac{1}{2\alpha _{1}-1}}(J)\), \(\alpha _{1}\in [\frac{1}{2}, \alpha )\) and a continuous nondecreasing function \(\vartheta :[0,\infty )\to (0,\infty )\) such that for any \((t,x)\in J\times C_{\alpha }\), we have \(E\Vert F(t,x(t))\Vert ^{2}\leq N(t)\times \vartheta (\Vert x\Vert _{C_{\alpha }})\), \(\liminf_{r\rightarrow \infty }\frac{\vartheta (r)}{r}\,ds= \Theta <\infty \);
\(h(t,\cdot ):X\rightarrow X\) is continuous for each \(t\in J\), and for each \(x\in X\), the function \(h(\cdot ,x):J\rightarrow X\) is strongly measurable;
there exist constants \(\beta \in (0,1)\) and \(L>0\) such that \(h\in D(A^{\beta })\) and for any \(x,y\in C_{\alpha }(J,X)\), \(t\in J\), the function \(A^{\beta }h(\cdot ,x)\) is strongly measurable, and \(A^{\beta }h(t,x(t))\) satisfies
$$ E \bigl\Vert A^{\beta }h \bigl(t,x(t) \bigr)-A^{\beta }h \bigl(t,y(t) \bigr) \bigr\Vert ^{2}\leq L \Vert x-y \Vert _{C_{ \alpha }}, $$
for all \(x,y\in C_{\alpha }(J,X)\), \(t\in J\);
there exist a continuous nondecreasing function \(\zeta :[0,\infty )\rightarrow (0,\infty )\) and a constant \(r>0\) such that for a.e. \(t\in J\), \(x\in C_{\alpha }(J,X)\), we have
$$ E \bigl\Vert A^{\beta }h \bigl(t,x(t) \bigr) \bigr\Vert ^{2} \leq L \bigl(1+\zeta \bigl( \Vert x \Vert _{C_{\alpha }} \bigr) \bigr), \quad \quad \liminf_{r\rightarrow \infty }\frac{\zeta (r)}{r}\,ds= \Pi _{1}< \infty ; $$
\(g : C_{\alpha }(J,X) \rightarrow \mathcal{L}_{0}^{2} (\Omega ,X)\) is such that
there exist a continuous nondecreasing function \(\mu :[0,\infty )\to (0,\infty )\) such that \(E\Vert g(x) \Vert ^{2}\leq \mu (\Vert x\Vert _{C_{\alpha }})\) for all \(x \in C_{\alpha }(J,X)\) and a constant \(r>0\) such that for a.e. \(t\in J\), \(x\in C_{\alpha }(J,X)\), we have
$$ E \bigl\Vert g(x) \bigr\Vert ^{2}\leq L \bigl(1+\mu \bigl( \Vert x \Vert _{C_{\alpha }} \bigr) \bigr), \quad\quad \liminf _{r \rightarrow \infty } \frac{\mu (r)}{r}\,ds=\Pi _{2}< \infty ; $$
g is a completely continuous map;
the function \(\sigma : J \rightarrow \mathcal{L}_{2}^{0}(X, Y)\) satisfies
$$ \int _{0}^{b} \bigl\Vert \sigma (s) \bigr\Vert _{\mathcal{L}_{2}^{0}}^{ \frac{2}{2\alpha _{1}-1}}\,ds< \infty , \quad \forall b>0. $$
We define the operator \(\Psi :C_{\alpha }(J,X)\rightarrow C_{\alpha }(J,X)\) as follows:
$$\begin{aligned} (\Psi x) (t) =&t^{\alpha -1}\mathcal{P}_{\alpha }(t) \bigl[x_{0}+h \bigl(0,x(0) \bigr)+g(x) \bigr]+h \bigl(t,(t) \bigr) \\ & {} + \int _{0}^{t}(t-s)^{\alpha -1}A \mathcal{P}_{\alpha }(t-s)h \bigl(s,x(s) \bigr)\,ds \\ & {} + \int _{0}^{t}(t-s)^{\alpha -1} \mathcal{P}_{\alpha }(t-s)F \bigl(s,x(s) \bigr)\,d \omega (s) \\ & {} + \int _{0}^{t}(t-s)^{\alpha -1} \mathcal{P}_{\alpha }(t-s)\sigma (s)\,dB_{Q}^{H}(s), \quad t \in J{'}. \end{aligned}$$
$$\begin{aligned}& \begin{aligned} (\Psi _{1}x) (t)={}&t^{\alpha -1} \mathcal{P}_{\alpha }(t) \bigl[x_{0}+h \bigl(0,x(0) \bigr)+g(x) \bigr]+h \bigl(t,x(t) \bigr) \\ & {} + \int _{0}^{t}(t-s)^{\alpha -1}A \mathcal{P}_{\alpha }(t-s)h \bigl(s,x(s) \bigr)\,ds, \quad t \in J{'}, \end{aligned} \\& \begin{aligned} (\Psi _{2}x) (t)={}& \int _{0}^{t}(t-s)^{\alpha -1} \mathcal{P}_{\alpha }(t-s)F \bigl(s,x(s) \bigr)\,d \omega (s) \\ & {} + \int _{0}^{t}(t-s)^{\alpha -1} \mathcal{P}_{\alpha }(t-s)\sigma (s)\,dB_{Q}^{H}(s), \quad t \in J{'}. \end{aligned} \end{aligned}$$
According to assumption (\(\mathrm{H}_{3}\)) and Lemma 2.3, we obtain
$$\begin{aligned}& E \biggl\Vert \int _{0}^{t}(t-s)^{\alpha -1}A \mathcal{P}_{\alpha }(t-s)h \bigl(s,x(s) \bigr)\,ds \biggr\Vert ^{2} \\ & \quad \leq E \int _{0}^{t} \bigl\Vert (t-s)^{\alpha -1}A^{1-\beta } \mathcal{P}_{ \alpha }(t-s)A^{\beta }h \bigl(s,x(s) \bigr) \bigr\Vert ^{2}\,ds \\ & \quad \leq \int _{0}^{t} \bigl\Vert (t-s)^{\alpha -1}A^{1-\beta } \mathcal{P}_{\alpha }(t-s) \bigr\Vert \,ds \\ & \quad \quad {} \times \int _{0}^{t}(t-s)^{\alpha -1}A^{1-\beta } \mathcal{P}_{ \alpha }(t-s)E \bigl\Vert A^{\beta }h \bigl(s,x(s) \bigr) \bigr\Vert ^{2}\,ds \\ & \quad \leq \frac{\alpha ^{2}(C_{1-\beta })^{2}\Gamma ^{2}(1+\beta )}{\Gamma ^{2}(1+\alpha \beta )} \\ & \quad \quad {} \times \int _{0}^{t}(t-s)^{\alpha \beta -1}\,ds \int _{0}^{t}(t-s)^{ \alpha \beta -1}E \bigl\Vert A^{\beta }h \bigl(s,x(s) \bigr) \bigr\Vert ^{2}\,ds \\ & \quad \leq b^{2\alpha \beta } \frac{(C_{1-\beta })^{2}\Gamma ^{2}(1+\beta )}{\beta ^{2}\Gamma ^{2}(1+\alpha \beta )}L \bigl(1+ \zeta \bigl( \Vert x \Vert _{C_{\alpha }} \bigr) \bigr) \\ & \quad = b^{2\alpha \beta }K(\alpha ,\beta )L \bigl(1+\zeta \bigl( \Vert x \Vert _{C_{\alpha }} \bigr) \bigr). \end{aligned}$$
Now, from assumption (\(\mathrm{H}_{2}\)), we get
$$\begin{aligned}& E \biggl\Vert \int _{0}^{t}(t-s)^{\alpha -1} \mathcal{P}_{\alpha }(t-s)F \bigl(s,x(s) \bigr) \,d \omega (s) \biggr\Vert ^{2} \\ & \quad \leq \operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )} \int _{0}^{t}(t-s)^{2( \alpha -1)}E \bigl\Vert F \bigl(s,x(s) \bigr) \bigr\Vert ^{2}\,ds \\ & \quad \leq \operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )} \biggl( \int _{0}^{t}(t-s)^{ \frac{2(\alpha -1)}{2-2\alpha _{1}}}\,ds \biggr)^{2-2\alpha _{1}}\vartheta \bigl( \Vert x \Vert _{C_{\alpha }} \bigr) \Vert N \Vert _{L^{\frac{1}{2\alpha _{1}-1}}} \\ & \quad \leq \operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )}\vartheta \bigl( \Vert x \Vert _{C_{\alpha }} \bigr) \frac{b^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}} \Vert N \Vert _{L^{ \frac{1}{2\alpha _{1}-1}}} \\ & \quad = \operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )}\vartheta \bigl( \Vert x \Vert _{C_{ \alpha }} \bigr)\Lambda \Vert N \Vert _{L^{\frac{1}{2\alpha _{1}-1}}}. \end{aligned}$$
For the last term of the mild solution, from assumption (\(\mathrm{H}_{5}\)), we have the estimate
$$\begin{aligned}& E \biggl\Vert \int _{0}^{t}(t-s)^{\alpha -1} \mathcal{P}_{\alpha }(t-s)\sigma (s) \,dB^{H}_{Q}(s) \biggr\Vert ^{2} \\ & \quad \leq c_{0}H(2H-1)t^{2H-1}\frac{M^{2}}{\Gamma ^{2}(\alpha )} \int _{0}^{t}(t-s)^{2( \alpha -1)} \bigl\Vert \sigma (s) \bigr\Vert _{\mathcal{L}_{2}^{0}}^{2}\,ds \\ & \quad \leq c_{0}H(2H-1)t^{2H-1}\frac{M^{2}}{\Gamma ^{2}(\alpha )} \biggl( \int _{0}^{t}(t-s)^{ \frac{2(\alpha -1)}{2-2\alpha _{1}}}\,ds \biggr)^{2-2\alpha _{1}} \\ & \quad \quad {} \times \biggl( \int _{0}^{b} \bigl\Vert \sigma (s) \bigr\Vert ^{\frac{2}{2\alpha _{1}-1}}\,ds \biggr)^{2 \alpha _{1}-1} \\ & \quad \leq c_{0}H(2H-1)\frac{M^{2}}{\Gamma ^{2}(\alpha )} \frac{b^{(1+c)(2-2\alpha _{1})+2H-1}}{(1+c)^{2-2\alpha _{1}}} \biggl( \int _{0}^{b} \bigl\Vert \sigma (s) \bigr\Vert ^{\frac{2}{2\alpha _{1}-1}}\,ds \biggr)^{2\alpha _{1}-1}. \end{aligned}$$
In the following, we set to present our first existence result for system (1.1).
Suppose that hypotheses (\(\mathrm{H}_{1}\))–(\(\mathrm{H}_{5}\)) hold, then system (1.1) has at least one mild solution defined on \(J{'}\) provided that \(2b^{2(1-\alpha )}M_{0}L+2b^{2(1-\alpha +\alpha \beta )}K^{2}(\alpha , \beta )L<1\) and
$$\begin{aligned}& \frac{15M^{2}}{\Gamma ^{2}(\alpha )} \bigl[M^{2}_{0}L\Pi _{1}+L \Pi _{2} \bigr]+5b^{2(1- \alpha )}M_{0}L\Pi _{1}+5b^{2(1-\alpha +\alpha \beta )}K^{2}(\alpha , \beta )L\Pi _{1} \\& \quad {} +5b^{2(1-\alpha )}\frac{M^{2}}{\Gamma ^{2}(\alpha )}\Theta \Lambda \Vert N \Vert _{\frac{1}{2\alpha _{1}-1}}< 1. \end{aligned}$$
Denote \(B_{q}=\{x\in C_{\alpha }(J,X),\Vert x\Vert _{C_{\alpha }}\leq q\}\). Then it is obvious that \(B_{q}\) is a bounded, closed, convex set in \(C_{\alpha }(J,X)\). We demonstrate the proof in six steps.
Step 1. We shall show that there exists a constant \(r=r(a)\) such that \(\Psi (B_{r})\subset B_{r}\).
In fact, if this claim is not true, then for each positive constant r there exists some \(x^{(r)}\in B_{r}\) such that \(\Psi (x^{(r)})\notin B_{r}\), i.e.,
$$\begin{aligned} r < & \bigl\Vert \Psi \bigl(x^{(r)} \bigr) \bigr\Vert ^{2}_{C_{\alpha }} \\ \leq &\sup_{t\in J{'}}t^{2(1 -\alpha )} \biggl\{ 5E \bigl\Vert t^{ \alpha -1}\mathcal{P}_{\alpha }(t) \bigl[x_{0}+h \bigl(0,x^{(r)}(0) \bigr)+g \bigl(x^{(r)} \bigr) \bigr] \bigr\Vert ^{2} \\ & {} +5E \bigl\Vert h \bigl(t,x^{(r)}(t) \bigr) \bigr\Vert ^{2} \\ & {} +5E \biggl\Vert \int _{0}^{t}(t-s)^{\alpha -1}A \mathcal{P}_{\alpha }(t-s)h \bigl(s,x^{(r)}(s) \bigr)\,ds \biggr\Vert ^{2} \\ & {} +5E \biggl\Vert \int _{0}^{t}(t-s)^{\alpha -1} \mathcal{P}_{\alpha }(t-s)F \bigl(s,x^{(r)}(s) \bigr)\,d \omega (s) \biggr\Vert ^{2} \\ & {} +5E \biggl\Vert \int _{0}^{t}(t-s)^{\alpha -1} \mathcal{P}_{\alpha }(t-s)\sigma (s) \,dB^{H}_{Q}(s) \biggr\Vert ^{2} \biggr\} \\ \leq & \frac{5M^{2}}{\Gamma ^{2}(\alpha )} \bigl[E \bigl\Vert x_{0}+h \bigl(0,x^{(r)}(0) \bigr)+g \bigl(x^{(r)} \bigr) \bigr\Vert ^{2} \bigr] \\ & {} +5b^{2(1-\alpha )} \bigl\Vert A^{-\beta } \bigr\Vert ^{2}L \bigl(1+\zeta \bigl\Vert x^{(r)} \bigr\Vert _{C_{ \alpha }} \bigr) \\ & {} +5b^{2(1-\alpha +\alpha \beta )}K^{2}(\alpha ,\beta )L \bigl(1+\zeta \bigl\Vert x^{(r)} \bigr\Vert _{C_{\alpha }} \bigr) \\ & {} +5b^{2(1-\alpha )}\operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )} \vartheta \bigl( \bigl\Vert x^{(r)} \bigr\Vert _{C_{\alpha }} \bigr)\Lambda \Vert N \Vert _{L^{ \frac{1}{2\alpha _{1}-1}}} \\ & {} +5c_{0}H(2H-1)\frac{M^{2}}{\Gamma ^{2}(\alpha )} \frac{b^{(1+c)(2-2\alpha _{1})+2H+1-2\alpha }}{(1+c)^{2-2\alpha _{1}}} \biggl( \int _{0}^{b} \bigl\Vert \sigma (s) \bigr\Vert ^{\frac{2}{2\alpha _{1}-1}}\,ds \biggr)^{2\alpha _{1}-1} \\ \leq & \frac{5M^{2}}{\Gamma ^{2}(\alpha )} \bigl[3E \Vert x_{0} \Vert ^{2}+3M^{2}_{0}L \bigl(1+ \zeta (r) \bigr)+3L \bigl(1+\mu (r) \bigr) \bigr] \\ & {} +5b^{2(1-\alpha )}M^{2}_{0}L \bigl(1+\zeta (r) \bigr)+5b^{2(1-\alpha +\alpha \beta )}K^{2}(\alpha ,\beta )L \bigl(1+\zeta (r) \bigr) \\ & {} +5b^{2(1-\alpha )}\operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )} \vartheta (r)\Lambda \Vert N \Vert _{L^{\frac{1}{2\alpha _{1}-1}}} \\ & {} +5c_{0}H(2H-1)\frac{M^{2}}{\Gamma ^{2}(\alpha )} \frac{b^{(1+c)(2-2\alpha _{1})+2H+1-2\alpha }}{(1+c)^{2-2\alpha _{1}}} \biggl( \int _{0}^{b} \bigl\Vert \sigma (s) \bigr\Vert ^{\frac{2}{2\alpha _{1}-1}}\,ds \biggr)^{2\alpha _{1}-1}. \end{aligned}$$
Dividing both sides by r and letting \(r\rightarrow \infty \) yields
$$\begin{aligned}& \frac{15M^{2}}{\Gamma ^{2}(\alpha )} \bigl[M^{2}_{0}L\Pi _{1}+L \Pi _{2} \bigr]+5b^{2(1- \alpha )}M_{0}L\Pi _{1}+5b^{2(1-\alpha +\alpha \beta )}K^{2}(\alpha , \beta )L\Pi _{1} \\& \quad {} +5b^{2(1-\alpha )}\frac{M^{2}}{\Gamma ^{2}(\alpha )}\Theta \Lambda \Vert N \Vert _{\frac{1}{2\alpha _{1}-1}}> 1. \end{aligned}$$
This contradicts the assumption of Theorem 3.1, which implies that there exists r such that Ψ maps \(B_{r}\) into itself.
Step 2. We show that \(\Psi _{1}\) is a contraction on \(B_{r}\).
For any \(x,y\in B_{r}\), we derive
$$\begin{aligned}& \bigl\Vert (\Psi _{1}x) (t)-(\Psi _{1}y) (t) \bigr\Vert ^{2}_{C_{\alpha }} \\& \quad =\sup_{t\in J{'}}t^{2(1-\alpha )}E \bigl\Vert (\Psi _{1}x) (t)-(\Psi _{1}y) (t) \bigr\Vert ^{2} \\& \quad \leq \sup_{t\in J^{\prime }}2t^{2(1-\alpha )}E \bigl\Vert h \bigl(t,x(t) \bigr)-h \bigl(t,y(t) \bigr) \bigr\Vert ^{2} \\& \quad \quad {} +\sup_{t\in J{'}}2t^{2(1-\alpha )}E \biggl\Vert \int _{0}^{t}(t-s)^{ \alpha -1}A \mathcal{P}_{\alpha }(t-s) \bigl[h \bigl(s,x(s) \bigr)-h \bigl(s,y(s) \bigr) \bigr]\,ds \biggr\Vert ^{2} \\& \quad \leq \sup_{t\in J{'}}2t^{2(1-\alpha )} \bigl\Vert A^{-\beta } \bigr\Vert ^{2}E \bigl\Vert A^{\beta }h \bigl(t,x(t) \bigr)-A^{\beta }h \bigl(t,y(t) \bigr) \bigr\Vert ^{2} \\& \quad \quad {} +\sup_{t\in J{'}}2t^{2(1-\alpha )} \int _{0}^{t}(t-s)^{ \alpha -1}A^{1-\beta } \mathcal{P}_{\alpha }(t-s)\,ds \\& \quad \quad {} \times \int _{0}^{t}(t-s)^{\alpha -1}A^{1-\beta } \mathcal{P}_{ \alpha }(t-s)E \bigl\Vert A^{\beta }h \bigl(s,x(s) \bigr)-A^{\beta }h \bigl(s,y(s) \bigr) \bigr\Vert ^{2}\,ds \\& \quad \leq 2b^{2(1-\alpha )}M^{2}_{0}L \Vert x-y \Vert _{C_{1-\alpha }}+2b^{2(1- \alpha +\alpha \beta )}K^{2}(\alpha ,\beta )L \Vert x-y \Vert _{C_{\alpha }} \\& \quad = \bigl[2b^{2(1-\alpha )}M^{2}_{0}L+2b^{2(1-\alpha +\alpha \beta )}K^{2}( \alpha ,\beta )L \bigr] \Vert x-y \Vert _{C_{\alpha }}. \end{aligned}$$
Thus, \(\Psi _{1}\) is a contraction by assumption of Theorem 3.1.
Step 3. \(\Psi _{2}\) is completely continuous.
To prove this assertion, we subdivide Step 3 into three claims.
Claim 1. \(\Psi _{2}\) maps bounded sets into uniformly bounded sets in \(C_{\alpha }(J,X)\).
We only need to show that there exists constant \(\Delta >0\) such that for each \(\Psi _{2}x, x\in B_{r}\), \(\Vert \Psi _{2}x\Vert _{C_{\alpha }}\leq \Delta \) holds. In fact, for each \(t\in J{'}\), by using Hölder's inequality, we have
$$\begin{aligned} \Vert \Psi _{2}x \Vert _{C_{\alpha }}^{2}&=\sup _{t\in J{'}}t^{2(1- \alpha )}E \biggl\Vert t^{\alpha -1} \mathcal{P}_{\alpha }(t) \bigl[x_{0}+h \bigl(0,x(0) \bigr)+g(x) \bigr] \\ &\quad {} + \int _{0}^{t}(t-s)^{\alpha -1} \mathcal{P}_{\alpha }(t-s)F \bigl(s,x(s) \bigr)\,d \omega (s) \\ &\quad {} + \int _{0}^{t}(t-s)^{\alpha -1} \mathcal{P}_{\alpha }(t-s)\sigma (s)\,dB_{Q}^{H} \biggr\Vert ^{2} \\ &\leq 3\frac{M^{2}}{\Gamma ^{2}(\alpha )}E \bigl\Vert x_{0}+h \bigl(0,x(0) \bigr)+g(x) \bigr\Vert ^{2} \\ &\quad {} +3b^{2(1-\alpha )}\operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )} \vartheta (r) \Lambda \Vert N \Vert _{L^{\frac{1}{2\alpha _{1}-1}}} \\ &\quad {} +3b^{2H+1-2\alpha }c_{0}H(2H-1)\frac{M^{2}}{\Gamma ^{2}(\alpha )} \frac{b^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}} \biggl( \int _{0}^{b} \bigl\Vert \sigma (s) \bigr\Vert ^{\frac{2}{2\alpha _{1}-1}}\,ds \biggr)^{2\alpha _{1}-1} \\ &\leq 3\frac{M^{2}}{\Gamma ^{2}(\alpha )} \bigl[3E \Vert x_{0} \Vert ^{2}+3M^{2}_{0}L \bigl(1+ \zeta (r) \bigr)+3L \bigl(1+\mu (r) \bigr) \bigr] \\ &\quad {} +3b^{2-2\alpha }\operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )} \vartheta (r) \Lambda \Vert N \Vert _{L^{\frac{1}{2\alpha _{1}-1}}} \\ &\quad {} +3b^{2H+1-2\alpha }c_{0}H(2H-1)\frac{M^{2}}{\Gamma ^{2}(\alpha )} \frac{b^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}} \biggl( \int _{0}^{b} \bigl\Vert \sigma (s) \bigr\Vert ^{\frac{2}{2\alpha _{1}-1}}\,ds \biggr)^{2\alpha _{1}-1} \\ & :=\Delta . \end{aligned}$$
Consequently, for each \(\rho \in \Psi _{2}x\), we have \(\Vert \rho (t)\Vert _{C_{\alpha }}\leq \Delta \).
Claim 2. \(\Psi _{2}(B_{r})\) is equicontinuous on \(B_{r}\).
Denoting \(\bar{E}=\{y\in C(J,X):y(t)=t^{1-\alpha }(\Psi _{2}x)(t), y(0)=y(0^{+}), x \in B_{r}\}\), for \(t_{1}=0\), \(0 < t_{2}\leq b\), we can obtain
$$\begin{aligned}& E \bigl\Vert y(t_{2})-y(0) \bigr\Vert ^{2} \\& \quad \leq 3 \bigl\Vert \bigl[\mathcal{P}_{\alpha }(t_{2})- \mathcal{P}_{\alpha }(0) \bigr] \bigl[x_{0}+h \bigl(0,x(0) \bigr)+g(x) \bigr] \bigr\Vert ^{2} \\& \quad\quad {} +3t_{2}^{1-\alpha } \biggl\Vert \int _{0}^{t_{2}}(t_{2}-s)^{\alpha -1} \mathcal{P}_{\alpha }(t_{2}-s)F \bigl(s,x(s) \bigr)\,d\omega (s) \biggr\Vert ^{2} \\& \quad \quad {} +3t_{2}^{1-\alpha } \biggl\Vert \int _{0}^{t_{2}}(t_{2}-s)^{\alpha -1} \mathcal{P}_{\alpha }(t_{2}-s)\sigma (s)\,dB_{Q}^{H}(s) \biggr\Vert ^{2} \rightarrow 0,\quad \text{as }t_{2} \rightarrow t_{1}=0. \end{aligned}$$
For \(0< t_{1}< t_{2}\leq b\), the strong continuity of \(\{\mathcal{P}_{\alpha }(t):t\geq 0\}\) implies that there exists a constant \(\delta >0\) such that \(\vert t_{2}-t_{1}\vert <\delta \) and \(\Vert \mathcal{P}_{\alpha }(t_{1})-\mathcal{P}_{\alpha }(t_{2})\Vert <\tau \). In addition, note that \(c=\frac{\alpha -1}{1-\alpha _{1}}\), then for \(\forall x\in B_{r}\), this yields that
$$\begin{aligned}& E \bigl\Vert y(t_{2})-y(t_{1}) \bigr\Vert ^{2} \\& \quad \leq 9E \bigl\Vert \mathcal{P}_{\alpha }(t_{2}) \bigl[x_{0}+h \bigl(0,x(0) \bigr)+g(x) \bigr]- \mathcal{P}_{\alpha }(t_{1}) \bigl[x_{0}+h \bigl(0,x(0) \bigr)+g(x) \bigr] \bigr\Vert ^{2} \\& \quad \quad {} +9E \biggl\Vert \int _{0}^{t_{1}} \bigl[t_{2}^{1-\alpha }(t_{2}-s)^{\alpha -1}-t_{1}^{1- \alpha }(t_{1}-s)^{\alpha -1} \bigr] \mathcal{P}_{\alpha }(t_{2}-s)F \bigl(s,x(s) \bigr)\,d \omega (s) \biggr\Vert ^{2} \\& \quad \quad {} +9E \biggl\Vert \int _{0}^{t_{1}} \bigl[t_{2}^{1-\alpha }(t_{2}-s)^{\alpha -1}-t_{1}^{1- \alpha }(t_{1}-s)^{\alpha -1} \bigr] \mathcal{P}_{\alpha }(t_{2}-s)\sigma (s)\,dB_{Q}^{H}(s) \biggr\Vert ^{2} \\& \quad \quad {} +9E \biggl\Vert \int _{0}^{t_{1}-\varepsilon }t_{1}^{1-\alpha }(t_{1}-s)^{ \alpha -1} \bigl[\mathcal{P}_{\alpha }(t_{2}-s)-\mathcal{P}_{\alpha }(t_{1}-s) \bigr]F \bigl(s,x(s) \bigr)\,d \omega (s) \biggr\Vert ^{2} \\& \quad \quad {} +9E \biggl\Vert \int _{0}^{t_{1}-\varepsilon }t_{1}^{1-\alpha }(t_{1}-s)^{ \alpha -1} \bigl[\mathcal{P}_{\alpha }(t_{2}-s)-\mathcal{P}_{\alpha }(t_{1}-s) \bigr] \sigma (s)\,dB_{Q}^{H}(s) \biggr\Vert ^{2} \\& \quad \quad {} +9E \biggl\Vert \int _{t_{1}-\varepsilon }^{t_{1}}t_{1}^{1-\alpha }(t_{1}-s)^{ \alpha -1} \bigl[\mathcal{P}_{\alpha }(t_{2}-s)-\mathcal{P}_{\alpha }(t_{1}-s) \bigr]F \bigl(s,x(s) \bigr)\,d \omega (s) \biggr\Vert ^{2} \\& \quad \quad {} +9E \biggl\Vert \int _{t_{1}-\varepsilon }^{t_{1}}t_{1}^{1-\alpha }(t_{1}-s)^{ \alpha -1} \bigl[\mathcal{P}_{\alpha }(t_{2}-s)-\mathcal{P}_{\alpha }(t_{1}-s) \bigr] \sigma (s)\,dB_{Q}^{H}(s) \biggr\Vert ^{2} \\& \quad \quad {} +9E \biggl\Vert \int _{t_{1}}^{t_{2}}t_{2}^{1-\alpha }(t_{2}-s)^{\alpha -1} \mathcal{P}_{\alpha }(t_{2}-s)F \bigl(s,x(s) \bigr)\,d\omega (s) \biggr\Vert ^{2} \\& \quad \quad {} +9E \biggl\Vert \int _{t_{1}}^{t_{2}}t_{2}^{1-\alpha }(t_{2}-s)^{\alpha -1} \mathcal{P}_{\alpha }(t_{2}-s)\sigma (s)\,dB_{Q}^{H}(s) \biggr\Vert ^{2} \\& \quad \leq 9 \bigl\Vert \mathcal{P}_{\alpha }(t_{2})- \mathcal{P}_{\alpha }(t_{1}) \bigr\Vert ^{2}E \bigl\Vert x_{0}+h \bigl(0,x(0) \bigr)+g(x) \bigr\Vert ^{2} \\& \quad \quad {} +9\operatorname{Tr}Q \int _{0}^{t_{1}} \bigl\Vert t_{2}^{1-\alpha }(t_{2}-s)^{ \alpha -1}-t_{1}^{1-\alpha }(t_{1}-s)^{\alpha -1} \bigr\Vert ^{2} \\& \quad \quad {} \times \bigl\Vert \mathcal{P}_{\alpha }(t_{2}-s) \bigr\Vert ^{2}E \bigl\Vert F \bigl(s,x(s) \bigr) \bigr\Vert ^{2}\,ds \\& \quad \quad {} +9c_{0}H(2H-1)t_{1}^{2H-1} \int _{0}^{t_{1}} \bigl\Vert t_{2}^{1-\alpha }(t_{2}-s)^{ \alpha -1}-t_{1}^{1-\alpha }(t_{1}-s)^{\alpha -1} \bigr\Vert ^{2} \\& \quad \quad {} \times \bigl\Vert \mathcal{P}_{\alpha }(t_{2}-s) \bigr\Vert ^{2} \bigl\Vert \sigma (s) \bigr\Vert _{ \mathcal{L}^{2}_{0}}^{2}\,ds \\& \quad \quad {} +9\sup_{[0,t_{1}-\varepsilon ]} \bigl\Vert \mathcal{P}_{\alpha }(t_{2}-s)- \mathcal{P}_{\alpha }(t_{1}-s) \bigr\Vert ^{2} \operatorname{Tr}Q\vartheta (r) \Vert N \Vert _{L^{ \frac{1}{2\alpha _{1}-1}}} \\& \quad \quad {} \times \biggl[ \frac{t_{1}^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}}- \frac{\varepsilon ^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}} \biggr] \\& \quad \quad {} +9c_{0}H(2H-1)b^{2H-1}\sup_{[0,t_{1}-\varepsilon ]} \bigl\Vert \mathcal{P}_{\alpha }(t_{2}-s)-\mathcal{P}_{\alpha }(t_{1}-s) \bigr\Vert ^{2} \\& \quad \quad {} \times \biggl[ \frac{t_{1}^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}}- \frac{\varepsilon ^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}} \biggr] \biggl( \int _{0}^{b} \bigl\Vert \sigma (s) \bigr\Vert ^{\frac{2}{2\alpha _{1}-1}}\,ds \biggr)^{2\alpha _{1}-1} \\& \quad \quad {} +9\operatorname{Tr}Q\frac{4M^{2}}{\Gamma ^{2}(\alpha )} \int _{t_{1}- \varepsilon }^{t_{1}}t_{1}^{2(1-\alpha )}(t_{1}-s)^{2(\alpha -1)}E \bigl\Vert F \bigl(s,x(s) \bigr) \bigr\Vert ^{2} \,ds \\& \quad \quad {} +9c_{0}H(2H-1)t_{1}^{2H-1} \frac{4M^{2}}{\Gamma ^{2}(\alpha )} \int _{t_{1}- \varepsilon }^{t_{1}}t_{1}^{2(1-\alpha )}(t_{1}-s)^{2(\alpha -1)} \bigl\Vert \sigma (s) \bigr\Vert _{\mathcal{L}_{2}^{0}}^{2}\,ds \\& \quad \quad {} +9\operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )}b^{2-2\alpha } \int _{t_{1}}^{t_{2}}(t_{2}-s)^{2(\alpha -1)}E \bigl\Vert F \bigl(s,x(s) \bigr) \bigr\Vert ^{2}\,ds \\& \quad \quad {} +9c_{0}H(2H-1)\frac{M^{2}}{\Gamma ^{2}(\alpha )}b^{2H+1-2\alpha } \int _{t_{1}}^{t_{2}}(t_{2}-s)^{2(\alpha -1)} \bigl\Vert \sigma (s) \bigr\Vert _{ \mathcal{L}_{2}^{0}}^{2}\,ds \\& \quad =9\sum_{i=1}^{9}I_{i}, \end{aligned}$$
$$\begin{aligned}& I_{1}= \bigl\Vert \mathcal{P}_{\alpha }(t_{2})- \mathcal{P}_{\alpha }(t_{1}) \bigr\Vert ^{2}E \bigl\Vert x_{0}+h \bigl(0,x(0) \bigr)+g(x) \bigr\Vert ^{2}, \\& \begin{aligned} I_{2}={}&\operatorname{Tr}Q \int _{0}^{t_{1}} \bigl\Vert t_{2}^{1-\alpha }(t_{2}-s)^{ \alpha -1}-t_{1}^{1-\alpha }(t_{1}-s)^{\alpha -1} \bigr\Vert ^{2} \\ & {} \times \bigl\Vert \mathcal{P}_{\alpha }(t_{2}-s) \bigr\Vert ^{2}E \bigl\Vert F \bigl(s,x(s) \bigr) \bigr\Vert ^{2}\,ds, \end{aligned} \\& \begin{aligned} I_{3}={}&c_{0}H(2H-1)t_{1}^{2H-1} \int _{0}^{t_{1}} \bigl\Vert t_{2}^{1-\alpha }(t_{2}-s)^{ \alpha -1}-t_{1}^{1-\alpha }(t_{1}-s)^{\alpha -1} \bigr\Vert ^{2} \\ & {} \times \bigl\Vert \mathcal{P}_{\alpha }(t_{2}-s) \bigr\Vert ^{2} \bigl\Vert \sigma (s) \bigr\Vert ^{2}\,ds, \end{aligned} \\& \begin{aligned} I_{4}={}&\sup_{[0,t_{1}-\varepsilon ]} \bigl\Vert \mathcal{P}_{\alpha }(t_{2}-s)- \mathcal{P}_{\alpha }(t_{1}-s) \bigr\Vert ^{2}\operatorname{Tr}Q\vartheta (r) \Vert N \Vert _{L^{ \frac{1}{2\alpha _{1}-1}}} \\ & {} \times \biggl[ \frac{t_{1}^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}}- \frac{\varepsilon ^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}} \biggr], \end{aligned} \\& \begin{aligned} I_{5}={}&c_{0}H(2H-1)b^{2H-1}\sup _{[0,t_{1}-\varepsilon ]} \bigl\Vert \mathcal{P}_{\alpha }(t_{2}-s)- \mathcal{P}_{\alpha }(t_{1}-s) \bigr\Vert ^{2} \\ & {} \times \biggl[ \frac{t_{1}^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}}- \frac{\varepsilon ^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}} \biggr] \biggl( \int _{0}^{b} \bigl\Vert \sigma (s) \bigr\Vert ^{\frac{2}{2\alpha _{1}-1}}\,ds \biggr)^{2\alpha _{1}-1}, \end{aligned} \\& \begin{aligned} I_{6}={}&\operatorname{Tr}Q\frac{4M^{2}}{\Gamma ^{2}(\alpha )} \int _{t_{1}- \varepsilon }^{t_{1}}t_{1}^{2(1-\alpha )}(t_{1}-s)^{2(\alpha -1)}E \bigl\Vert F \bigl(s,x(s) \bigr) \bigr\Vert ^{2}\,ds \\ \leq{}&\operatorname{Tr}Q\frac{4M^{2}}{\Gamma ^{2}(\alpha )}t_{1}^{2-2 \alpha } \frac{\varepsilon ^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}} \vartheta (r) \Vert N \Vert _{L^{\frac{1}{2\alpha _{1}-1}}}, \end{aligned} \\& \begin{aligned} I_{7}={}&c_{0}H(2H-1)t_{1}^{2H-1} \frac{4M^{2}}{\Gamma ^{2}(\alpha )} \int _{t_{1}-\varepsilon }^{t_{1}}t_{1}^{2(1-\alpha )}(t_{1}-s)^{2( \alpha -1)} \bigl\Vert \sigma (s) \bigr\Vert _{\mathcal{L}_{2}^{0}}^{2}\,ds \\ \leq{}&c_{0}H(2H-1)t_{1}^{2H+1-2\alpha } \frac{4M^{2}}{\Gamma ^{2}(\alpha )} \frac{\varepsilon ^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}} \biggl( \int _{0}^{b} \bigl\Vert \sigma (s) \bigr\Vert ^{\frac{2}{2\alpha _{1}-1}}\,ds \biggr)^{2\alpha _{1}-1}, \end{aligned} \\& \begin{aligned} I_{8}&=\operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )}b^{2-2\alpha } \int _{t_{1}}^{t_{2}}(t_{2}-s)^{2(\alpha -1)}E \bigl\Vert F \bigl(s,x(s) \bigr) \bigr\Vert ^{2}\,ds \\ &\leq \operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )}b^{2-2\alpha } \frac{(t_{2}-t_{1})^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}} \vartheta (r) \Vert N \Vert _{L^{\frac{1}{2\alpha _{1}-1}}}, \end{aligned} \\& \begin{aligned} I_{9}&=c_{0}H(2H-1)\frac{M^{2}}{\Gamma ^{2}(\alpha )}b^{2H+1-2\alpha } \int _{t_{1}}^{t_{2}}(t_{2}-s)^{2(\alpha -1)} \bigl\Vert \sigma (s) \bigr\Vert _{ \mathcal{L}_{2}^{0}}^{2}\,ds \\ &\leq c_{0}H(2H-1)\frac{M^{2}}{\Gamma ^{2}(\alpha )}b^{2H+1-2 \alpha } \frac{(t_{2}-t_{1})^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}} \biggl( \int _{0}^{b} \bigl\Vert \sigma (s) \bigr\Vert ^{\frac{2}{2\alpha _{1}-1}}\,ds \biggr)^{2\alpha _{1}-1}. \end{aligned} \end{aligned}$$
Since \(p\in (0,\alpha )\), we have \((1+c)(2-2\alpha _{1})>0\), thus the terms from \(I_{6}\) to \(I_{9}\) tend to zero as \(t_{2}-t_{1}\rightarrow 0\) and \(\varepsilon \rightarrow 0\). The strong continuity of \(\{\mathcal{P}_{\alpha }(t): t\geq 0\}\) indicates that \(\Vert \mathcal{P}_{\alpha }(t_{2}-s)-\mathcal{P}_{\alpha }(t_{1}-s)\Vert ^{2} \rightarrow 0\) as \(\delta \rightarrow 0\). Hence \(I_{1}\), \(I_{4}\), \(I_{5}\) also tend to zero as \(t_{2}-t_{1}\rightarrow 0\).
For \(I_{2}\), by a standard calculation and for \(p\in (0,\alpha )\), we have
$$\begin{aligned} I_{2} \leq &\operatorname{Tr}Q \int _{0}^{t_{1}} \bigl\Vert t_{2}^{1-\alpha }(t_{2}-s)^{ \alpha -1}-t_{1}^{1-\alpha }(t_{1}-s)^{\alpha -1} \bigr\Vert ^{2} \\ & {} \times \bigl\Vert \mathcal{P}_{\alpha }(t_{2}-s) \bigr\Vert ^{2}E \bigl\Vert F \bigl(s,x(s) \bigr) \bigr\Vert ^{2}\,ds \\ \leq &\operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )} \int _{0}^{t_{1}} \bigl\Vert t_{2}^{1-\alpha }(t_{2}-s)^{\alpha -1}-t_{1}^{1-\alpha }(t_{1}-s)^{ \alpha -1} \bigr\Vert ^{2}E \bigl\Vert F \bigl(s,x(s) \bigr) \bigr\Vert ^{2}\,ds \\ \leq &\operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )} \frac{1}{(1+c)^{2-2\alpha _{1}}} \bigl[(t_{2}-t_{1})^{(1+c)(2-2\alpha _{1})} \\ & {} +t_{1}^{(1+c)(2-2\alpha _{1})}-t_{2}^{(1+c)(2-2\alpha _{1})} \bigr]\times \vartheta \bigl( \Vert x \Vert _{C_{\alpha }} \bigr) \Vert N \Vert _{L^{ \frac{1}{2\alpha _{1}-1}}}. \end{aligned}$$
Thus \(I_{2}\) tends to zero as \(t_{2}-t_{1}\rightarrow 0\). Similarly, we can get that \(I_{3}\) tends to zero as \(t_{2}-t_{1}\rightarrow 0\).
Therefore, the relationship of E and \(\{\Psi _{2}x:x\in B_{r}\}\) implies that \(\Psi _{2}\) is equicontinuous on \(B_{r}\).
Claim 3. \(V(t)=\{(\Psi _{2}x)(t), x\in B_{r}\}\) is a relatively compact in X.
Let \(0< t\leq b\) be fixed. Then for \(\forall \lambda \in (0,t)\) and \(\forall \delta >0\), \(x\in B_{r}\), define an operator
$$\begin{aligned} \bigl(\Psi _{2}^{\lambda ,\delta }x \bigr) (t) =&t^{1-\alpha } \mathcal{P}_{\alpha }(t) \bigl[x_{0}+h \bigl(0,x(0) \bigr)+g(x) \bigr] \\ & {} +\alpha \int _{0}^{t-\lambda } \int _{\delta }^{\infty }\theta (t-s)^{ \alpha -1}\phi _{\alpha }(\theta )S \bigl((t-s)^{\alpha }\theta \bigr)F \bigl(s,x(s) \bigr)\,d \omega (s) \\ & {} +\alpha \int _{0}^{t-\lambda } \int _{\delta }^{\infty }\theta (t-s)^{ \alpha -1}\phi _{\alpha }(\theta )S \bigl((t-s)^{\alpha }\theta \bigr)\sigma (s)\,dB^{H}_{Q}(s) \\ =&t^{1-\alpha }\mathcal{P}_{\alpha }(t) \bigl[x_{0}+h \bigl(0,x(0) \bigr)+g(x) \bigr] \\ & {} +\alpha S \bigl(\lambda ^{\alpha }\theta \bigr) \int _{0}^{t-\lambda } \int _{ \delta }^{\infty }\theta (t-s)^{\alpha -1}\phi _{\alpha }(\theta )S \bigl((t-s)^{ \alpha }\theta -\lambda ^{\alpha } \theta \bigr)F \bigl(s,x(s) \bigr)\,d\omega (s) \\ & {} +\alpha S \bigl(\lambda ^{\alpha }\theta \bigr) \int _{0}^{t-\lambda } \int _{ \delta }^{\infty }\theta (t-s)^{\alpha -1}\phi _{\alpha }(\theta )S \bigl((t-s)^{ \alpha }\theta -\lambda ^{\alpha } \theta \bigr)\sigma (s)\,dB^{H}_{Q}(s). \end{aligned}$$
From the compactness of \(S(\lambda ^{\alpha }\delta ), \lambda ^{\alpha }\delta >0\), we obtain that for \(\forall \lambda \in (0,t)\) and \(\forall \delta >0\), the set \(V^{\lambda ,\delta }(t)=\{(\Psi _{2}^{\lambda ,\delta }x)(t),x\in B_{r} \}\) is relatively compact in X.
Moreover, for each \(x\in B_{r}\), we have
$$\begin{aligned}& \bigl\Vert (\Psi _{2}x) (t)- \bigl(\Psi _{2}^{\lambda ,\delta }x \bigr) (t) \bigr\Vert _{C_{\alpha }} \\ & \quad =\sup_{t\in J{'}}t^{2(1-\alpha )}E\Vert \int _{0}^{t} \int _{0}^{ \delta }\alpha \theta (t-s)^{\alpha -1}\phi _{\alpha }(\theta )S \bigl((t-s)^{ \alpha }\theta \bigr)F(s,x(s)\,d\omega (s) \\ & \quad\quad {} + \int _{0}^{t} \int _{\delta }^{\infty }\alpha \theta (t-s)^{\alpha -1} \phi _{\alpha }(\theta )S \bigl((t-s)^{\alpha }\theta \bigr)F(s,x(s)\,d \omega (s) \\ & \quad\quad {} + \int _{0}^{t} \int _{\delta }^{\infty }\alpha \theta (t-s)^{\alpha -1} \phi _{\alpha }(\theta )S \bigl((t-s)^{\alpha }\theta \bigr)\sigma (s)\,dB^{H}_{Q}(s) \\ & \quad\quad {} - \int _{0}^{t-\lambda } \int _{\delta }^{\infty }\alpha \theta (t-s)^{ \alpha -1}\phi _{\alpha }(\theta )S \bigl((t-s)^{\alpha }\theta \bigr)F(s,x(s)\,d \omega (s) \\ & \quad\quad {} - \int _{0}^{t-\lambda } \int _{\delta }^{\infty }\alpha \theta (t-s)^{ \alpha -1}\phi _{\alpha }(\theta )S \bigl((t-s)^{\alpha }\theta \bigr)\sigma (s)\,dB^{H}_{Q}(s) \Vert ^{2} \\ & \quad \leq 4\alpha ^{2}\sup_{t\in J{'}} t^{2(1-\alpha )} E \Vert \int _{0}^{t} \int _{0}^{\delta }\theta (t-s)^{\alpha -1}\phi _{\alpha }( \theta )S \bigl((t-s)^{\alpha }\theta \bigr) F(s,x(s)\,d\omega (s) \Vert ^{2} \\ & \quad\quad {} +4\alpha ^{2}\sup_{t\in J{'}} t^{2(1-\alpha )} E \biggl\Vert \int _{0}^{t} \int _{0}^{\delta }\theta (t-s)^{\alpha -1}\phi _{\alpha }(\theta )S \bigl((t-s)^{ \alpha }\theta \bigr)\sigma (s)\,dB^{H}_{Q}(s) \biggr\Vert ^{2} \\ & \quad\quad {} +4\alpha ^{2}\sup_{t\in J{'}} t^{2(1-\alpha )} E\Vert \int _{t- \lambda }^{t} \int _{\delta }^{\infty }\theta (t-s)^{\alpha -1}\phi _{ \alpha }(\theta )S \bigl((t-s)^{\alpha }\theta \bigr) F(s,x(s)\,d\omega (s)\,ds \Vert ^{2} \\ & \quad\quad {} +4\alpha ^{2}\sup_{t\in J{'}} t^{2(1-\alpha )} E \biggl\Vert \int _{t- \lambda }^{t} \int _{\delta }^{\infty }\theta (t-s)^{\alpha -1}\phi _{ \alpha }(\theta )S \bigl((t-s)^{\alpha }\theta \bigr)\sigma (s)\,dB^{H}_{Q}(s) \biggr\Vert ^{2} \\ & \quad \leq 4\alpha ^{2} b^{2-2\alpha }M^{2} \operatorname{Tr}Q\Lambda \vartheta (r) \Vert N \Vert _{L^{ \frac{1}{2\alpha _{1}-1}}} \biggl( \int _{0}^{\delta }\theta \phi _{\alpha }( \theta )\,d \theta \biggr)^{2} \\ & \quad\quad {} +4\alpha ^{2}c_{0}H(2H-1) b^{2H+1-2\alpha }M^{2} \Lambda \biggl( \int _{0}^{ \delta }\theta \phi _{\alpha }(\theta )\,d \theta \biggr)^{2} \\ & \quad\quad {} \times \biggl( \int _{0}^{t} \bigl\Vert \sigma (s) \bigr\Vert ^{\frac{2}{2\alpha _{1}-1}}\,ds \biggr)^{2 \alpha _{1}-1} \\ & \quad\quad {} +4\alpha ^{2} b^{2-2\alpha }M^{2} \operatorname{Tr}Q \vartheta (r) \Vert N \Vert _{L^{ \frac{1}{2\alpha _{1}-1}}} \frac{1}{\Gamma ^{2}(\alpha +1)} \frac{\lambda ^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}} \\ & \quad\quad {} +4\alpha ^{2} c_{0}H(2H-1)b^{2H+1-2\alpha }M^{2} \frac{1}{\Gamma ^{2}(\alpha +1)} \frac{\lambda ^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}} \\ & \quad\quad {} \times \biggl( \int _{0}^{\lambda } \bigl\Vert \sigma (s) \bigr\Vert ^{ \frac{2}{2\alpha _{1}-1}}\,ds \biggr)^{2\alpha _{1}-1}, \end{aligned}$$
where we have used the equality
$$ \int _{0}^{\infty }\theta ^{\xi }\phi _{\alpha }(\theta )\,d\theta = \int _{0}^{ \infty }\frac{1}{\theta ^{\alpha \xi }}\psi _{\alpha }( \theta )\,d\theta = \frac{\Gamma (1+\xi )}{\Gamma (1+\alpha \xi )}, \quad \xi \in [0,1]. $$
The right-hand side of the above inequality tends to zero as \(\lambda ,\delta \rightarrow 0\). So we can deduce that \(\Vert (\Psi _{2}x)(t)-(\Psi _{2}^{\lambda ,\delta }x)(t)\Vert _{C_{\alpha }} \rightarrow 0\) as \(\lambda ,\delta \rightarrow 0^{+}\) which enables us to claim that there are relatively compact sets arbitrarily close to the set \(V(t)=\{(\Psi _{2}x)(t), x\in B_{r}\}\). Hence, \(V(t)=\{(\Psi _{2}x)(t), x\in B_{r}\}\) is relatively compact in X.
We deduce, from Claims 1–3 and the Arzola–Ascoli theorem, that \(\Psi _{2}\) is a completely continuous map. Using Krasnoselskii's fixed point theorem, we claim that the operator equation \(\Psi x = \Psi _{1}x+\Psi _{2}x\) has a fixed point on \(B_{r}\) which is a mild solution for system (1.1). The proof is complete. □
To give our last existence theorem, we require the following hypotheses:
\(S(t)\) is continuous in the uniform operator topology for \(t\geq 0\), and \(\{S(t)\}_{t\geq 0}\) is uniformly bounded, i.e., there exists \(M\geq 1\) such that \(\sup_{t\in [0,+\infty )}\vert S(t)\vert \leq M\);
there exists positive constant L such that for any \(x_{1},x_{2}\in C_{\alpha }(J,X)\), we have \(E\Vert g(x_{1})-g(x_{2})\Vert ^{2}\leq L\Vert x_{1}-x_{2}\Vert _{C_{\alpha }}\);
there exists a function \(N_{1}(t)\in L^{\frac{1}{2\alpha _{1}-1}}(J)\), \(\alpha _{1}\in [ \frac{1}{2},\alpha )\), such that for any \(x,y\in X\), \(t\in J\), we have
$$ E \bigl\Vert F \bigl(t,x(t) \bigr)-F \bigl(t,y(t) \bigr) \bigr\Vert ^{2}\leq N_{1}(t)\cdot \Vert x-y \Vert _{C_{\alpha }}. $$
For convenience, let
$$\begin{aligned} M{'} =&4b^{2(1-\alpha )}\frac{M^{2}}{\Gamma ^{2}(\alpha )}L+4b^{2(1- \alpha )}M^{2}_{0}L+4b^{2(1-\alpha +\alpha \beta )}K^{2}( \alpha , \beta )L \\ & {} +4b^{2(1-\alpha )}\operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )} \Lambda \Vert N_{1} \Vert _{L^{\frac{1}{2\alpha _{1}-1}}}. \end{aligned}$$
To end this section, we shall proceed with our last existence and uniqueness theorem for system (1.1) based on the Banach contraction principle.
Assume that hypotheses (\(\mathrm{H}_{0}\)), (\(\mathrm{H}_{2}\))–(\(\mathrm{H}_{7}\)) hold, then system (1.1) has a unique mild solution on \(B_{r}\) provided that \(M{'}<1\).
Define operator Ψ as in Theorem 3.1. Then by similar arguments employed in Theorem 3.1, we can get that operator Ψ maps \(B_{r}\) into itself, where \(B_{r}\) is defined as in Theorem 3.1.
Moreover, we have
$$\begin{aligned}& \bigl\Vert (\Psi x) (t)-(\Psi y) (t) \bigr\Vert ^{2}_{C_{\alpha }} \\& \quad =\sup_{t\in J{'}}t^{2(1-\alpha )}E \bigl\Vert (\Psi x) (t)-( \Psi y) (t) \bigr\Vert ^{2} \\& \quad \leq 4\sup_{t\in J{'}}t^{2(1-\alpha )}E \bigl\Vert \mathcal{P}_{ \alpha }(t) \bigl[g(x)-g(y) \bigr] \bigr\Vert ^{2} \\& \quad \quad {} +4\sup_{t\in J{'}}t^{2(1-\alpha )}E \bigl\Vert h \bigl(t,x(t) \bigr)-h \bigl(t,y(t) \bigr) \bigr\Vert ^{2} \\& \quad \quad {} +4\sup_{t\in J{'}}t^{2(1-\alpha )}E \biggl\Vert \int _{0}^{t}(t-s)^{ \alpha -1}A \mathcal{P}_{\alpha }(t-s) \bigl[h \bigl(s,x(s) \bigr)-h \bigl(s,y(s) \bigr) \bigr]\,ds \biggr\Vert ^{2} \\& \quad \quad {} +4\sup_{t\in J{'}}t^{2(1-\alpha )}E \biggl\Vert \int _{0}^{t}(t-s)^{ \alpha -1} \mathcal{P}_{\alpha }(t-s) \bigl[F \bigl(s,x(s) \bigr)-F \bigl(s,y(s) \bigr) \bigr]\,d \omega (s) \biggr\Vert ^{2} \\& \quad \leq 4\sup_{t\in (0,b]}t^{2(1-\alpha )} \frac{M^{2}}{\Gamma ^{2}(\alpha )}E \bigl\Vert g(x)-g(y) \bigr\Vert ^{2} \\& \quad \quad {} +4\sup_{t\in J{'}}t^{2(1-\alpha )} \bigl\Vert A^{-\beta } \bigr\Vert ^{2}E \bigl\Vert A^{ \beta }h \bigl(t,x(t) \bigr)-A^{\beta }h \bigl(t,y(t) \bigr) \bigr\Vert ^{2} \\& \quad \quad {} +4\sup_{t\in J{'}}t^{2(1-\alpha )} \int _{0}^{t} \bigl\Vert (t-s)^{ \alpha -1}A^{1-\beta }{P}_{\alpha }(t-s) \bigr\Vert \,ds \\& \quad \quad {} \times \int _{0}^{t}(t-s)^{\alpha -1}A^{1-\beta } \mathcal{P}_{ \alpha }(t-s)E \bigl\Vert A^{\beta }h \bigl(s,x(s) \bigr)-A^{\beta }h \bigl(s,y(s) \bigr) \bigr\Vert ^{2}\,ds \\& \quad \quad {} +4\sup_{t\in (0,b]}\operatorname{Tr}Q \frac{M^{2}}{\Gamma ^{2}(\alpha )} \int _{0}^{t}(t-s)^{2\alpha -2}E \bigl\Vert F \bigl(s,x(s) \bigr)-F \bigl(s,y(s) \bigr) \bigr\Vert ^{2}\,ds \\& \quad \leq 4b^{2(1-\alpha )}\frac{M^{2}}{\Gamma ^{2}(\alpha )}L \Vert x-y \Vert _{C_{ \alpha }}+4b^{2(1-\alpha )}M^{2}_{0}L \Vert x-y \Vert _{C_{\alpha }} \\& \quad \quad {} +4b^{2(1-\alpha +\alpha \beta )}K^{2}(\alpha ,\beta )L \Vert x-y \Vert _{C_{ \alpha }} \\& \quad \quad {} +4b^{2(1-\alpha )}\operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )} \frac{b^{(1+c)(2-2\alpha _{1})}}{(1+c)^{2-2\alpha _{1}}} \Vert N_{1} \Vert _{L^{ \frac{1}{2\alpha _{1}-1}}} \Vert x-y \Vert _{C_{\alpha }} \\& \quad = \biggl[4b^{2(1-\alpha )}\frac{M^{2}}{\Gamma ^{2}(\alpha )}L+4b^{2(1- \alpha )}M^{2}_{0}L+4b^{2(1-\alpha +\alpha \beta )}K^{2}( \alpha , \beta )L \\& \quad\quad {} +4b^{2(1-\alpha )}\operatorname{Tr}Q\frac{M^{2}}{\Gamma ^{2}(\alpha )} \Lambda \Vert N_{1} \Vert _{L^{\frac{1}{2\alpha _{1}-1}}} \biggr] \Vert x-y \Vert _{C_{ \alpha }} \\& \quad < \Vert x-y \Vert _{C_{\alpha }}. \end{aligned}$$
Hence, the Banach contraction principle implies that Ψ has a unique fixed point in \(B_{r}\) which is a mild solution for system (1.1). This completes the proof. □
As an application, we present an example to illustrate our results. Consider the following fractional stochastic evolution equation driven by both fBm and Wiener process:
$$\begin{aligned} \textstyle\begin{cases} {}^{L}\mathcal{D}^{\frac{3}{4}}[u(t,z)-\int _{0}^{\pi }W(z,y)u(t,y)\,dy]= u_{zz}(t,z)+ \hat{F}(t,u(t)(z)\frac{d\omega (t)}{dt}, \\ \hphantom{{}^{L}\mathcal{D}^{\frac{3}{4}}[u(t,z)-\int _{0}^{\pi }W(z,y)u(t,y)\,dy]=} {} +\hat{\sigma }(t,u(t)(z) \frac{dB_{Q}^{H}(t)}{dt}, \quad t\in (0,b], z\in [0,\pi ], \\ u(t,0)=u(t,\pi )=0, \quad t\in J=[0,b], \\ \mathcal{I}^{\frac{1}{4}}(u(t,z)-\sum_{i=1}^{n}\int _{0}^{ \pi }k(z,y)u(t_{i},y)\,dy)=u_{0}(z), \quad z\in [0,\pi ], \end{cases}\displaystyle \end{aligned}$$
where \({}^{L}\mathcal{D}^{\frac{3}{4}}\) denotes the Riemann–Liouville fractional derivative of order \(\frac{3}{4}\), \(\mathcal{I}^{\frac{1}{4}}\) is a Riemann–Liouville integral of order \(\frac{1}{4}\), n is a positive integer, \(0 < t_{0} < t_{1} < \cdots < t_{n} < b\), \(z\in [0,\pi ]\).
To write the above system (4.1) into the abstract form of (1.1), we choose the space \(X=L^{2}[0,\pi ]\). Next we define an operator A by \(Av=v{''}\) with the domain \(D(A)=\{v\in X:v,v{'}\text{ absolutely continuous}, v{''}\in X,v(0)=v(\pi )=0\}\). Then A is a generator of a strongly continuous semigroup \(\{S(t)_{t\geq 0}\}\) which, as we know, is compact, analytic, and self-adjoint.
Moveover, from the previous work [25], A has a discrete spectrum, the eigenvalues of A are \(-n^{2},n\in N\) with corresponding orthogonal eigenvectors \(e_{n}(z)=\sqrt{\frac{2}{\pi }} \sin (nz)\). Then \(Az=\sum_{n=1}^{\infty }n^{2}\langle z,e_{n}\rangle e_{n}\). On the other hand, we know that
for each \(v\in X\), \(S(t)v=\sum_{n=1}^{\infty }e^{-n^{2}t}\langle v,e_{n} \rangle e_{n}\), in addition, \(S(\cdot )\) is a uniformly stable semigroup and \(\Vert S(t)\Vert \leq e^{-t}\);
for each \(v\in X\), \(A^{-\frac{1}{2}}v=\sum_{n=1}^{\infty }\frac{1}{n} \langle v,e_{n}\rangle e_{n}\) and \(\Vert A^{-\frac{1}{2}}\Vert ^{2}=1\);
the operator \(A^{\frac{1}{2}}\) is given by \(A^{\frac{1}{2}}v=\sum_{n=1}^{\infty }n\langle v,e_{n} \rangle e_{n}\in X\) on the space \(D(A^{\frac{1}{2}})=\{v(\cdot )\in X,\sum_{n=1}^{\infty } \langle v,e_{n}\rangle e_{n}\in X\}\).
Letting \((W_{h}v)(z)=\int _{0}^{\pi }W(z,y)v(y)\,dy\), for \(v\in E=L^{2}([0,\pi ],\mathbb{R})\), \(z\in [0,\pi ]\), we assume that the following assumptions hold:
\((\bar{a})\):
\(\omega (s)\) denotes a one-dimensional standard Brownian motion;
\((\bar{b})\):
the function \(W(z,y),z,y\in [0,\pi ]\) is measurable and
$$ \int _{0}^{\pi } \int _{0}^{\pi }W^{2}(z,y)\,dy\,dz< \infty ; $$
\((\bar{c})\):
the function \(\partial _{z}W(z,y)\) is measurable, \(W(0,y)=W(\pi ,y)=0\), and
$$ \bar{\Lambda }= \biggl( \int _{0}^{\pi } \int _{0}^{\pi } \bigl(\partial _{z}W(z,y) \bigr)^{2}\,dy\,dz \biggr)^{ \frac{1}{2}}< L< \infty . $$
Note that \((\bar{c})\) enables us to show that \(W_{h}\) is a bounded linear operator on X. Furthermore, \(W_{h}(v)\in D(A^{\frac{1}{2}})\), and \(\Vert A^{\frac{1}{2}}W_{h}\Vert _{L^{2}[0,\pi ]}<\infty \). Indeed, the definition of \(W_{h}\) and (ii) yield that
$$ \bigl\langle W_{h}(v),e_{n} \bigr\rangle = \int _{0}^{\pi }e_{n}(z) \biggl( \int _{0}^{\pi }W(z,y)v(y)\,dy \biggr)\,dz= \frac{1}{n}\sqrt{\frac{2}{\pi }} \bigl\langle \bar{W}(v),\cos (nz) \bigr\rangle , $$
where W̄ is defined by
$$ \bigl(\bar{W}(v) \bigr) (z)= \int _{0}^{\pi }W_{z}(z,y)v(y)\,dy. $$
We know from \((\bar{b})\) that \(\bar{W}:X\rightarrow X\) is a bounded linear operator with \(\Vert \bar{W}\Vert _{L^{2}[0.b]}\leq \bar{\Lambda }\). Hence we can write \(\Vert A^{\frac{1}{2}}W_{h}(v)\Vert ^{2}_{L^{2}[0,\pi ]}=\Vert \bar{W}(v)\Vert ^{2}_{L^{2}[0, \pi ]}\).
(iv)
For the function \(\hat{F} : [0, b] \times [0,\pi ] \rightarrow [0, \pi ]\) the following conditions are satisfied:
for each \(s \in [0, b], \hat{F}( s,\cdot )\) is continuous, and for each \(u\in X, \hat{F}(\cdot ,u )\) is measurable;
there exist a function \(N(t)\in L^{\frac{1}{2\alpha _{1}-1}}(J)\), \(\alpha _{1}\in [\frac{1}{2}, \alpha )\) and a continuous nondecreasing function \(\vartheta :[0,\infty )\to (0,\infty )\) such that for any \((t,u)\in J\times X\), we have \(E\Vert F(t,u(t))\Vert ^{2}\leq N(t)\times \vartheta (\Vert u\Vert _{C_{1-\alpha }})\), \(\liminf_{r\rightarrow \infty }\frac{\vartheta (r)}{r}\,ds= \Theta <\infty \).
Define the fractional Brownian motion in Y by
$$ B_{Q}^{H}(t) =\sum_{n=1}^{\infty } \sqrt{(}\lambda _{n})B_{n}^{H}(t)e_{n}, $$
where \(H \in (\frac{1}{2}, 1)\) and \(\{B^{H}_{n}, n\in N\}\) is a sequence of one-dimensional fractional Brownian motions which are mutually independent. Let us assume that the function \(\hat{\sigma } : [0,+\infty ) \rightarrow \mathcal{L}_{2}^{0}(L^{2}([0, 1]), L^{2}([0, 1]))\) satisfies
$$ \int _{0}^{b} \bigl\Vert \hat{\sigma }(s) \bigr\Vert _{L_{0}^{2}}^{ \frac{2}{2\alpha _{1}-1}}\,ds< \infty , \quad \forall b>0. $$
Letting \(u(t)(z)=u(t,z)\), \(t\in J\), \(z\in [0,\pi ]\), we define \(h:[0, b ]\times X\rightarrow X \), \(F:[0, b ]\times X\rightarrow L (K,X)\) and \(g: E\rightarrow X \) by \(h(t,u) = W_{h}(u) \), \(\hat{F}(s,u)(z) = F(s,u(z))\), \(g(u) = \sum_{i=0}^{p}\bar{Q}u(t_{i})\) and \(\sigma (t)=\hat{\sigma }(t)\), respectively, where
$$ \bar{Q}(u)= \int _{0}^{\pi }k(y,z)u(y)\,dy. $$
With the above choices of A, F, h, σ, system (4.1) can be seen as an abstract form of system (1.1). On the other hand, suppose that the assumptions of Theorem 3.1 hold. Thus, using Theorem 3.1, we claim that system (4.1) admits a mild solution on \([0, b]\) under the above additional assumptions.
In this paper, we have considered a class of Riemann–Liouville fractional stochastic evolution equations. In particular, the fixed point technique, fractional calculus, and stochastic analysis were used for achieving the sufficient conditions to ensure the existence of mild solutions of Riemann–Liouville fractional stochastic evolution equations driven by Wiener process and fBm. We last gave an illustrative example. Our future work will address the explosive solutions of fractional Navier–Stokes stochastic differential equations driven by fBm and multiplicative noise.
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This work is supported by National Natural Science Foundation of China (Grant Nos. 12001393, 11961069), Natural Science Foundation of Shanxi (201901D211103), Natural Science Foundation of Xinjiang (2019D01A71), Scientific Research Programs of Colleges in Xinjiang (XJEDU2018Y033), Outstanding Young Science and technology personnel Training program of Xinjiang(2019Q022).
College of Mathematics, Taiyuan University of Technology, 030024, Taiyuan, China
Min Yang
School of Mathematics Sciences, Xinjiang Normal University, 830017, Urumqi, China
Haibo Gu
All authors have contributed equally. All authors read and approved the final manuscript.
Correspondence to Haibo Gu.
Yang, M., Gu, H. Riemann–Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion. J Inequal Appl 2021, 8 (2021). https://doi.org/10.1186/s13660-020-02541-3
Riemann–Liouville fractional derivative
Stochastic evolution equations
Fractional Brownian motion
Mild solution | CommonCrawl |
Analytic discs and uniform algebras generated by real-analytic functions
by Alexander J. Izzo PDF
Under very general conditions it is shown that if $A$ is a uniform algebra generated by real-analytic functions, then either $A$ consists of all continuous functions or else there exists a disc on which every function in $A$ is holomorphic. This strengthens several earlier results concerning uniform algebras generated by real-analytic functions.
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Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46J10, 46J15, 32E20, 30H50, 32A65
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Alexander J. Izzo
Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403
MR Author ID: 307587
Email: [email protected]
Received by editor(s): June 25, 2018
Published electronically: January 9, 2019
Dedicated: Dedicated to John Wermer on the occasion of his 90th birthday
Communicated by: Harold P. Boas
MSC (2010): Primary 46J10, 46J15, 32E20, 30H50, 32A65
DOI: https://doi.org/10.1090/proc/14311 | CommonCrawl |
1 − 1 + 2 − 6 + 24 − 120 + ⋯
In mathematics,
$\sum _{k=0}^{\infty }(-1)^{k}k!$
For a related alternating partial sum of factorials, see Alternating factorial.
is a divergent series, first considered by Euler, that sums the factorials of the natural numbers with alternating signs. Despite being divergent, it can be assigned a value of approximately 0.596347 by Borel summation.
Euler and Borel summation
This series was first considered by Euler, who applied summability methods to assign a finite value to the series.[1] The series is a sum of factorials that are alternately added or subtracted. One way to assign a value to this divergent series is by using Borel summation, where one formally writes
$\sum _{k=0}^{\infty }(-1)^{k}k!=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{\infty }x^{k}e^{-x}\,dx.$
If summation and integration are interchanged (ignoring that neither side converges), one obtains:
$\sum _{k=0}^{\infty }(-1)^{k}k!=\int _{0}^{\infty }\left[\sum _{k=0}^{\infty }(-x)^{k}\right]e^{-x}\,dx.$
The summation in the square brackets converges when $|x|<1$, and for those values equals ${\tfrac {1}{1+x}}$. The analytic continuation of ${\tfrac {1}{1+x}}$ to all positive real $x$ leads to a convergent integral for the summation:
${\begin{aligned}\sum _{k=0}^{\infty }(-1)^{k}k!&=\int _{0}^{\infty }{\frac {e^{-x}}{1+x}}\,dx\\[4pt]&=eE_{1}(1)\approx 0.596\,347\,362\,323\,194\,074\,341\,078\,499\,369\ldots \end{aligned}}$
where E1(z) is the exponential integral. This is by definition the Borel sum of the series.
Connection to differential equations
Consider the coupled system of differential equations
${\dot {x}}(t)=x(t)-y(t),\qquad {\dot {y}}(t)=-y(t)^{2}$
where dots denote derivatives with respect to t.
The solution with stable equilibrium at (x,y) = (0,0) as t → ∞ has y(t) = 1/t, and substituting it into the first equation gives a formal series solution
$x(t)=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {(n-1)!}{t^{n}}}$
Observe x(1) is precisely Euler's series.
On the other hand, the system of differential equations has a solution
$x(t)=e^{t}\int _{t}^{\infty }{\frac {e^{-u}}{u}}\,du.$
By successively integrating by parts, the formal power series is recovered as an asymptotic approximation to this expression for x(t). Euler argues (more or less) that since the formal series and the integral both describe the same solution to the differential equations, they should equal each other at $t=1$, giving
$\sum _{n=1}^{\infty }(-1)^{n+1}(n-1)!=e\int _{1}^{\infty }{\frac {e^{-u}}{u}}\,du.$
See also
• Alternating factorial
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's)
• 1 + 2 + 3 + 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 − 2 + 4 − 8 + ⋯
References
1. Euler, L. (1760). "De seriebus divergentibus" [On divergent series]. Novi Commentarii Academiae Scientiarum Petropolitanae (5): 205–237. arXiv:1202.1506. Bibcode:2012arXiv1202.1506E.
Further reading
• Kline, Morris (November 1983), "Euler and Infinite Series", Mathematics Magazine, 56 (5): 307–313, CiteSeerX 10.1.1.639.6923, doi:10.2307/2690371, JSTOR 2690371
• Kozlov, V. V. (2007), "Euler and mathematical methods in mechanics" (PDF), Russian Mathematical Surveys, 62 (4): 639–661, Bibcode:2007RuMaS..62..639K, doi:10.1070/rm2007v062n04abeh004427, S2CID 250892576
• Leah, P. J.; Barbeau, E. J. (May 1976), "Euler's 1760 paper on divergent series", Historia Mathematica, 3 (2): 141–160, doi:10.1016/0315-0860(76)90030-6
Sequences and series
Integer sequences
Basic
• Arithmetic progression
• Geometric progression
• Harmonic progression
• Square number
• Cubic number
• Factorial
• Powers of two
• Powers of three
• Powers of 10
Advanced (list)
• Complete sequence
• Fibonacci sequence
• Figurate number
• Heptagonal number
• Hexagonal number
• Lucas number
• Pell number
• Pentagonal number
• Polygonal number
• Triangular number
Properties of sequences
• Cauchy sequence
• Monotonic function
• Periodic sequence
Properties of series
Series
• Alternating
• Convergent
• Divergent
• Telescoping
Convergence
• Absolute
• Conditional
• Uniform
Explicit series
Convergent
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 4 − 8 + ⋯
• Infinite arithmetic series
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
• 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
• 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
• Taylor series
• Power series
• Formal power series
• Laurent series
• Puiseux series
• Dirichlet series
• Trigonometric series
• Fourier series
• Generating series
Hypergeometric series
• Generalized hypergeometric series
• Hypergeometric function of a matrix argument
• Lauricella hypergeometric series
• Modular hypergeometric series
• Riemann's differential equation
• Theta hypergeometric series
• Category
| Wikipedia |
\begin{definition}[Definition:Right Angle/Perpendicular]
{{EuclidDefinition|book = I|def = 10|name = Right Angle}}
:400px
In the above diagram, the line $CD$ has been constructed so as to be a '''perpendicular''' to the line $AB$.
\end{definition} | ProofWiki |
Resource-dependent branching process
A branching process (BP) (see e.g. Jagers (1975)) is a mathematical model to describe the development of a population. Here population is meant in a general sense, including a human population, animal populations, bacteria and others which reproduce in a biological sense, cascade process, or particles which split in a physical sense, and others. Members of a BP-population are called individuals, or particles.
If the times of reproductions are discrete (usually denoted by 1,2, ...) then the totality of individuals present at time n and living to time n+1 excluded are thought of as forming the nth generation. Simple BPs are defined by an initial state (number of individuals at time 0) and a law of reproduction, usually denoted by pk, k = 1,2,....
A resource-dependent branching process (RDBP) is a discrete-time BP which models the development of a population in which individuals are supposed to have to work in order to be able to live and to reproduce. The population decides on a society form which determines the rules how available resources are distributed among the individuals. For this purpose a RDBP should incorporate at least four additional model components, namely the individual demands for resources, the creation of new resources for the next generation, the notion of a policy to distribute resources, and a control option for individuals for interactions with the society.
Definition
A (discrete-time) resource-dependent branching process is a stochastic process Γ defined on the non-negative integers which is a BP defined by
• an initial state Γ0;
• a law of reproduction of individuals;
• a law of individual creation of resources;
• a law of individual resource demands (claims);
• a policy to distribute available resources to individuals which are present in the population
• a tool of interaction between individuals and the society.
History and objectives of RDBPs
RDBPs may be seen (in a wider sense) as so-called controlled branching processes. They were introduced by F. Thomas Bruss (1983)) with the objective to model different society structures and to compare the advantages and disadvantages of different forms of human societies. In these processes individuals have a means of interaction with the society which determines the rules how the current available resources should be distributed among them. This interaction (as e.g. in form of emigration) changes the effective rate of reproduction of the individuals remaining in the society. In that respect RDBPs have some parts in common with so-called population-size dependent BPs (see Klebaner (1984) and Klebaner & Jagers (2000)) in which the law of individual independent reproduction (see Galton-Watson process) is a function of the current population size.
Tractable RDBPs
Realistic models for human societies ask for a bisexual mode of reproduction whereas in the definition of an RDBP one simply speaks of a law of reproduction. However the notion of an average reproduction rate per individual (Bruss 1984) for bisexual processes shows that for all relevant questions for the long-term behavior of human societies it is justified for simplicity to assume asexual reproduction. This is why certain limiting results of Klebaner (1984) and Jagers & Klebaner (2000) bear over to RDBPs. Models for the development of a human society in time must allow for interdependence between the different components. Such models are in general very complicated and risk to become intractable. This led to the idea not to try to model the development of a society with a (single) realistic RDBP but rather by a sequence of control actions defining a sequence of relevant short-horizon RDBPs.
Two special policies stand out as guidelines for the development of any society. The two policies are the so-called weakest-first policy (wf-policy) and the so-called strongest-first policy (sf-policy).
Definition
The wf-policy is the rule to serve in each generation, as long as the accumulated resource space allows for it, with priority always the individuals with the smallest individual claims. The sf-policy is the rule to serve in each generation always with priority the largest individual resource claims, again as long as the accumulated resource space suffices. The societies adapting these policies strictly are called the wf-society, respectively the sf-society.
Survival criteria
In the theory of BPs it is of interest to know whether survival of a process is possible in the long run. For RDBPs this question depends also strongly on a feature on which individuals have a great influence, namely the policy to distribute resources.
Let:
m = mean reproduction (descendants) per individual
r = mean production (resource creation) per individual
F = the individual probability distribution of claims (resources)
Further suppose that all individuals which will not obtain their resource claim will either die or emigrate before reproduction. Then using results on expected stopping times for sums of order statistics (1991) the survival criteria can be explicitly computed for both the wf-society and the sf-society as a function of m, r and F.
The arguably strongest result known for RDBPs is the theorem of the envelopment of societies (Bruss and Duerinckx 2015). It says that, in the long run, any society which would like to survive and in which individuals prefer in general a higher standard of living to a lower one is bound to live in the long run between the wf-society and the sf-society. Intuition why this should be true, is wrong. The mathematical proof depends on the mentioned results on expected stopping times for sums of order statistics (1991) and fine-tuned balancing acts between model assumptions and different notions of Convergence of random variables.
See also
• Controlled branching process
• Bisexual Galton–Watson process
• Bruss–Duerinckx theorem
References
• Jagers, Peter (1975). Branching processes with biological applications. London: Wiley-Interscience [John Wiley & Sons].
• Bruss, F. Thomas (1983). "Resource-dependent branching processes". Stochastic Processes and Their Applications. 16: 36.
• Bruss, F. Thomas (1984). "A note on extinction criteria for bisexual Galton–Watson processes". Journal of Applied Probability. 21: 915–919. doi:10.2307/3213707.
• Klebaner, Fima C. (1984). "On population-size dependent branching processes". Advances in Applied Probability. 16: 30–55. doi:10.2307/1427223.
• Bruss, F. Thomas; Robertson, James (1991). "Wald's Lemma for the sum of order statistics of i.i.d. random variables". Advances in Applied Probability. 23: 612–623. doi:10.2307/1427625.
• Jagers, Peter; Klebaner, Fima C. (2000). "Population-size-dependent and age-dependent branching processes". Stochastic Processes and Their Applications. 87: 235–254. doi:10.1016/s0304-4149(99)00111-8.
• Bruss, F. Thomas; Duerinckx, Mitia (2015). "Resource–dependent branching processes and the envelope of societies". Annals of Applied Probability. 25: 324–372. arXiv:1212.0693. doi:10.1214/13-aap998.
| Wikipedia |
Highly abundant number
In mathematics, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number.
Highly abundant numbers and several similar classes of numbers were first introduced by Pillai (1943), and early work on the subject was done by Alaoglu and Erdős (1944). Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any N is at least proportional to log2 N.
Formal definition and examples
Formally, a natural number n is called highly abundant if and only if for all natural numbers m < n,
$\sigma (n)>\sigma (m)$
where σ denotes the sum-of-divisors function. The first few highly abundant numbers are
1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, ... (sequence A002093 in the OEIS).
For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ.
The only odd highly abundant numbers are 1 and 3.[1]
Relations with other sets of numbers
Although the first eight factorials are highly abundant, not all factorials are highly abundant. For example,
σ(9!) = σ(362880) = 1481040,
but there is a smaller number with larger sum of divisors,
σ(360360) = 1572480,
so 9! is not highly abundant.
Alaoglu and Erdős noted that all superabundant numbers are highly abundant, and asked whether there are infinitely many highly abundant numbers that are not superabundant. This question was answered affirmatively by Jean-Louis Nicolas (1969).
Despite the terminology, not all highly abundant numbers are abundant numbers. In particular, none of the first seven highly abundant numbers (1, 2, 3, 4, 6, 8, and 10) is abundant. Along with 16, the ninth highly abundant number, these are the only highly abundant numbers that are not abundant.
7200 is the largest powerful number that is also highly abundant: all larger highly abundant numbers have a prime factor that divides them only once. Therefore, 7200 is also the largest highly abundant number with an odd sum of divisors.[2]
Notes
1. See Alaoglu & Erdős (1944), p. 466. Alaoglu and Erdős claim more strongly that all highly abundant numbers greater than 210 are divisible by 4, but this is not true: 630 is highly abundant, and is not divisible by 4. (In fact, 630 is the only counterexample; all larger highly abundant numbers are divisible by 12.)
2. Alaoglu & Erdős (1944), pp. 464–466.
References
• Alaoglu, L.; Erdős, P. (1944). "On highly composite and similar numbers" (PDF). Transactions of the American Mathematical Society. 56 (3): 448–469. doi:10.2307/1990319. JSTOR 1990319. MR 0011087.
• Nicolas, Jean-Louis (1969). "Ordre maximal d'un élément du groupe Sn des permutations et "highly composite numbers"". Bull. Soc. Math. France. 97: 129–191. doi:10.24033/bsmf.1676. MR 0254130.
• Pillai, S. S. (1943). "Highly abundant numbers". Bull. Calcutta Math. Soc. 35: 141–156. MR 0010560.
Divisibility-based sets of integers
Overview
• Integer factorization
• Divisor
• Unitary divisor
• Divisor function
• Prime factor
• Fundamental theorem of arithmetic
Factorization forms
• Prime
• Composite
• Semiprime
• Pronic
• Sphenic
• Square-free
• Powerful
• Perfect power
• Achilles
• Smooth
• Regular
• Rough
• Unusual
Constrained divisor sums
• Perfect
• Almost perfect
• Quasiperfect
• Multiply perfect
• Hemiperfect
• Hyperperfect
• Superperfect
• Unitary perfect
• Semiperfect
• Practical
• Erdős–Nicolas
With many divisors
• Abundant
• Primitive abundant
• Highly abundant
• Superabundant
• Colossally abundant
• Highly composite
• Superior highly composite
• Weird
Aliquot sequence-related
• Untouchable
• Amicable (Triple)
• Sociable
• Betrothed
Base-dependent
• Equidigital
• Extravagant
• Frugal
• Harshad
• Polydivisible
• Smith
Other sets
• Arithmetic
• Deficient
• Friendly
• Solitary
• Sublime
• Harmonic divisor
• Descartes
• Refactorable
• Superperfect
Classes of natural numbers
Powers and related numbers
• Achilles
• Power of 2
• Power of 3
• Power of 10
• Square
• Cube
• Fourth power
• Fifth power
• Sixth power
• Seventh power
• Eighth power
• Perfect power
• Powerful
• Prime power
Of the form a × 2b ± 1
• Cullen
• Double Mersenne
• Fermat
• Mersenne
• Proth
• Thabit
• Woodall
Other polynomial numbers
• Hilbert
• Idoneal
• Leyland
• Loeschian
• Lucky numbers of Euler
Recursively defined numbers
• Fibonacci
• Jacobsthal
• Leonardo
• Lucas
• Padovan
• Pell
• Perrin
Possessing a specific set of other numbers
• Amenable
• Congruent
• Knödel
• Riesel
• Sierpiński
Expressible via specific sums
• Nonhypotenuse
• Polite
• Practical
• Primary pseudoperfect
• Ulam
• Wolstenholme
Figurate numbers
2-dimensional
centered
• Centered triangular
• Centered square
• Centered pentagonal
• Centered hexagonal
• Centered heptagonal
• Centered octagonal
• Centered nonagonal
• Centered decagonal
• Star
non-centered
• Triangular
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• Square triangular
• Pentagonal
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• Decagonal
• Dodecagonal
3-dimensional
centered
• Centered tetrahedral
• Centered cube
• Centered octahedral
• Centered dodecahedral
• Centered icosahedral
non-centered
• Tetrahedral
• Cubic
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pyramidal
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4-dimensional
non-centered
• Pentatope
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• Tesseractic
Combinatorial numbers
• Bell
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• Eulerian
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• Telephone number
• Wedderburn–Etherington
Primes
• Wieferich
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• Wilson
Pseudoprimes
• Carmichael number
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Arithmetic functions and dynamics
Divisor functions
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Prime omega functions
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Euler's totient function
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Aliquot sequences
• Amicable
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Primorial
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Other prime factor or divisor related numbers
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Numeral system-dependent numbers
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and dynamics
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Generated via a sieve
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| Wikipedia |
Normal form
Any equivalence relation $\sim$ on a set of objects $\mathscr M$ defines the quotient set $\mathscr M/\sim$ whose elements are equivalence classes: the equivalence class of an element $M\in\mathscr M$ is denoted $[M]=\{M'\in\mathscr M:~M'\sim M\}$. Description of the quotient set is referred to as the classification problem for $\mathscr M$ with respect to the equivalence relation. The normal form of an object $M$ is a "selected representative" from the class $[M]$, usually possessing some nice properties (simplicity, integrability etc). Often (although not always) one requires that two distinct representatives ("normal forms") are not equivalent to each other: $M_1\ne M_2\iff M_1\not\sim M_2$.
The equivalence $\sim$ can be an identical transformation in a certain formal system: the respective normal form in such case is a "canonical representative" among many possibilities, see, e.g., disjunctive normal form and conjunctive normal form for Boolean functions.
However, the most typical classification problems appear when there is a group $G$ acting on $\mathscr M$: then the natural equivalence relation arises, $M_1\sim M_2\iff \exists g\in G:~g\cdot M_1=M_2$. If both $\mathscr M$ and $G$ are finite-dimensional spaces, the classification problem is usually much easier than in the case of infinite-dimensional spaces.
Below follows a list (very partial) of the most important classification problems in which normal forms are known and very useful. For more detailed description of specific cases, follow the links indicated in the appropriate subsections.
1 Finite-dimensional classification problems
1.1 Linear maps between finite-dimensional linear spaces
1.2 Linear operators (self-maps)
1.3 Quadratic forms on linear spaces
1.4 Quadratic forms on Euclidean spaces
1.5 Quadratic forms on the symplectic spaces
1.6 Conic sections in the real affine and projective plane
1.7 Families of finite-dimensional objects
2 Singularities of differentiable mappings
2.1 Maps of full rank
2.2 Germs of maps in small dimension
2.2.1 Holomorphic curves
2.2.2 Nondegenerate critical points of functions and the Morse lemma
2.2.3 Degenerate critical points of smooth functions
2.2.4 "Elementary catastrophes"
3 Classification of dynamical systems
4 References and basic literature
Finite-dimensional classification problems
When the objects of classification form a finite-dimensional variety, in most cases it is a subvariety of matrices, with the equivalence relation induced by transformations reflecting the change of basis.
Linear maps between finite-dimensional linear spaces
Let $\Bbbk$ be a field. A linear map from $\Bbbk^m$ to $\Bbbk^n$ is represented by an $n\times m$ matrix over $\Bbbk$ ($m$ rows and $n$ columns). A different choice of bases in the source and the target space results in a matrix $M$ being replaced by another matrix $M'=HML$, where $H$ (resp., $L$) is an invertible $m\times m$ (resp., $n\times n$) matrix of transition between the bases, $$ M\sim M'\iff\exists H\in\operatorname{GL}(m,\Bbbk),\ L\in \operatorname{GL}(n,\Bbbk):\quad M'=HML. \tag{LR} $$
Obviously, this binary relation $\sim$ is an equivalence (symmetric, reflexive and transitive), called left-right linear equivalence. Each matrix $M$ is left-right equivalent to a matrix (of the same size) with $k\leqslant\min(n,m)$ units on the diagonal and zeros everywhere else. The number $k$ is a complete invariant of equivalence (matrices of different ranks are not equivalent) and is called the rank of a matrix.
A similar question may be posed about homomorphisms of finitely generated modules over rings. For some rings the normal form is known as the Smith normal form.
Linear operators (self-maps)
The matrix of a linear operator of an $n$-dimensional space over $\Bbbk$ into itself is transformed by a change of basis in a more restrictive way compared to (LR): if the source and the target spaces coincide, then necessarily $n=m$ and $L=H^{-1}$. The corresponding equivalence is called similarity (sometimes conjugacy or linear conjugacy) of matrices, and the normal form is known as the Jordan normal form, see also here. This normal form is characterized by a specific block diagonal structure and explicitly features the eigenvalues on the diagonal. Note that this form holds only over an algebraically closed field $\Bbbk$, e.g., $\Bbbk=\CC$.
Quadratic forms on linear spaces
A quadratic form $Q\colon\Bbbk^n\to\Bbbk$, $(x_1,\dots,x_n)\mapsto \sum a_{i,j}^n a_{ij}x_ix_j$ with a symmetric matrix $Q$ after a linear invertible change of coordinates will have a new matrix $Q'=HQH^*$ (the asterisk means the transpose): $$ Q'\sim Q\iff \exists H\in\operatorname{GL}(n,\Bbbk):\ Q'=HQH^*.\tag{QL} $$ The normal form for this equivalence, termed matrix congruence, is diagonal, but the diagonal entries depend on the field:
Over $\RR$, the diagonal entries can be all made $0$ or $\pm 1$. The signature gives the number of entries of each type: by Sylvester's law of inertia it is an invariant of classification.
Over $\CC$, one can keep only zeros and units (not signed). The number of units is called the rank of a quadratic form; it is a complete invariant.
Quadratic forms on Euclidean spaces
This classification deals with real symmetric matrices representing quadratic forms, yet the condition (QL) is represented by a more restrictive condition that the conjugacy matrix $H$ is orthogonal (preserves the Euclidean scalar product): $$ Q'\sim Q\iff \exists H\in\operatorname{O}(n,\RR)=\{H\in\operatorname{GL}(n,\RR):\ HH^*=E\}:\ Q'=HQH^*.\tag{QE} $$ The normal form is diagonal, with the diagonal entries forming a complete system of invariants.
A similar set of normal forms exists for self-adjoint matrices conjugated by Hermitian matrices.
Quadratic forms on the symplectic spaces
A symplectic space is an even-dimensional space $\R^{2n}$ equipped with the linear symplectic structure, a nondegenerate bilinear form denoted by the brackets $[\cdot,\cdot]\to\R$, which is antisymmetric: $[v,w]=-[w,v]$ for any $v,w\in\RR^{2n}$, [Ar74, Sect. 41]. Any such form can be brought into the normal form with the matrix $$ [e_i,e_j]=[e'_i,e'_j]=0,\qquad [e_i,e'_j]=\begin{cases}1,\quad &i=j,\\0,&i\ne j,\end{cases}\qquad \forall i,j=1,\dots,n. $$ for a suitable basis $\{e_1,\dots,e_n,e'_1,\dots,e'_n\}$ in $\R^{2n}$. If $\R^{2n}$ is equipped with the standard Euclidean structure (in which the above basis is orthonormal), then the symplectic form is generated by a linear operator $I$, $$ [v,w]=(Iv,w),\qquad I=\begin{pmatrix} 0_n&-E_n\\E_n&0_n\end{pmatrix},\quad I=-I^*,\ I^2=-E_{2n}. $$ Here $0_n$ and $E_n$ denote the zero and identity matrices of size $n\times n$ and the asterisk denotes the transposition.
A linear self-map $M:\R^{2n}\to\R^{2n}$ is called canonical, or a symplectomorphism, if it preserves the symplectic structure, $[Mv,Mw]=[v,w]$ for any $v,w$. Linear symplectomorphisms form a finite-dimensional Lie group called the symplectic group and denoted by $\operatorname{Sp}(2n,\R)$ (fields other than $\R$ can also be considered). The matrix of a symplectomorphism in the canonical basis satisfies the condition $M^*IM=I$. The characteristic polynomial $p$ of a symplectic matrix is palindromic, i.e., $\lambda^{2n}p(1/\lambda)=p(\lambda)$.
Two (symmetric) quadratic forms $\tfrac12(Qx,x)$ and $\tfrac12(Q'x,x)$ on the symplectic $\R^{2n}$ with symmetric $2n\times 2n$-matrices are called canonically equivalent, if there exists a canonical transformation $M$ conjugating them, $M^*QM=Q'$. The canonical equivalence preserves the Hamiltonian form of equations and hence conjugates also the Hamiltonian linear vector fields $v(x)=IQx$ and $v'=IQ'x$: $M^{-1}IQM=IQ'$.
The eigenvalues of a real matrix $A=IQ$ with $Q^*=Q$ are symmetric both with respect to real axis and to the change of sign, hence if nonzero, they come in pairs (real $\pm a$ or imaginary $\pm i\omega$), quadruples $\pm a\pm i\omega$. The Jordan block structure is the same for all eigenvalues in the pair (quadruple). In the simplest case when all Jordan blocks are trivial, the quadratic form $Q$ can be brought by a canonical transformation to the sum of terms of the three types[1] $$ Q_{\pm a}=-a(x_iy_i),\qquad, Q_{\pm i\omega}=\pm\tfrac12(\omega^2x_i^2+y_i^2),\qquad Q_{4}=-a(x_iy_i+x_{i+1}y_{i+1})+\omega^2(x_iy_{i+1}-x_{i+1}y_i) $$ in the canonical coordinates $(x_1,\dots,x_n,y_1,\dots,y_n)$. In the case the operator $IQ$ has nontrivial Jordan blocks, the complete list of normal forms is known but rather complicated [Ar74, Appendix 6].
↑ The terms of type $Q_{\pm i\omega}$ with different signs are not equivalent.
Conic sections in the real affine and projective plane
This problem reduces to classification of quadratic forms on $\RR^3$. An conic section is the intersection of the cone $\{Q(x,y,z)=0\}$ defined by a quadratic form on $\RR^3$, with the affine subspace $\{z=1\}$. Projective transformations are defined by linear invertible self-maps of $\RR^3$, respectively, the affine transformations consist of linear self-maps preserving the plane $\{z=0\}$ in the homogeneous coordinates (the "infinite line"). In addition, one can replace the form $Q$ by $\lambda Q$ with $\lambda\ne 0$. This defines two equivalence relations on the space of quadratic forms.
The list of normal forms for both classifications is follows from the normal form of quadratic forms:
Rank of $Q$
Projective curves
Affine curves
3 $\varnothing_1=\{x^2+y^2=-1\}$, circle $\{x^2+y^2=1\}$ $\varnothing_1=\{x^2+y^2=-1\}$, circle $\{x^2+y^2=1\}$, parabola $\{y=x^2\}$, hyperbola $\{x^2-y^2=1\}$
2 point $\{x^2+y^2=0\}$, two lines $\{x^2-y^2=0\}$ point $\{x^2+y^2=0\}$, two crossing lines $\{x^2-y^2=0\}$,
two parallel lines $\{x^2=1\}$, $\varnothing_2=\{x^2=-1\}$
1 "double" line $\{x^2=0\}$ $\varnothing_3=\{1=0\}$, "double" line $\{x^2=0\}$
Note that the three empty sets $\varnothing_i$, are different from the algebraic standpoint: $\varnothing_1$ is an imaginary cicrle, $\varnothing_2$ is a pair of parallel imaginary lines which intersect "at infinity" (if these imaginary lines intersect at a finite point, this point is real), and $\varnothing_3$ is a double line "at infinity".
Families of finite-dimensional objects
$\def\l{\lambda}$ In each of the above problems one can instead of an individual map $M$ (or a form $Q$) consider a local parametric family of objects $\{M_\lambda\}$, depending regularly (continuously, $C^k$- or $C^\infty$-differentiably, holomorphically) on finitely many real or complex parameters $\lambda$ varying near a certain point $a$ in the parameter space, $\l\in(\RR^p,a)$ or $\l\in(\CC^p,0)$ respectively. Two such local families $M_\lambda$ and $M'_\lambda$ are said to be equivalent by the action of a group $G$, if there exists a local parametric family of group elements, $\{g_\lambda\}$, also regular (although perhaps in a weaker or just different sense) that conjugates the two families: $g_\lambda\cdot M_\lambda=M_\lambda$ for all admissible values of $\lambda$.
The most instructive example is that of families of linear operators. A "generic" operator $M=M_0$ is diagonalizable with pairwise different eigenvalues $\mu_1(\lambda),\dots,\mu_n(\lambda)$ (depending, naturally, on $\lambda$). One can show that any finite-parametric family $\{M_\lambda|\lambda\in(\RR^p,0)\}$ can be diagonalized by a transformation $M_\lambda\mapsto H_\lambda M_\lambda H_\lambda^{-1}$ by the similarity transformation depending on $\l\in(\RR^p,0)$ with the same regularity. This follows from the Implicit function theorem.
However, when some of the eigenvalues tend to a collision $\mu_i(0)=\mu_j(0)$, the diagonalizing transformation $H_\lambda$ may tend to a degenerate matrix so that $H_\lambda^{-1}$ diverges to infinity, while the transformation of a matrix to its Jordan normal form is far away from the family $\{H_\lambda\}$. However, a different choice of the normal form resolves these problems.
Example. Assume that the local family of matrices $\{M_\l|\l\in(\RR^p,0)\}$ is a deformation of the matrix $M_0$ whose normal form is a single Jordan block of size $n$. Then there exists a family of invertible matrices $\{H_\l|\l\in(\RR^p,0)\}$ such that $$ H_\l M_\l H_\l^{-1}= \begin{pmatrix} \mu & 1&\\ &\mu& 1&\\ &&\mu&1&\\ &&&\ddots&\ddots\\ &&&&\mu&1\\ \alpha_1&\alpha_2&\alpha_3&\cdots&\alpha_{n-1}&\alpha_n \end{pmatrix},\tag{SF} $$ where $\mu=\mu(\l)$ and $\alpha_i=\alpha_i(\l)$, $i=1,\dots,n$ are regular (continuous, smooth, analytic,\dots) functions of the parameters $\l\in(\RR^p,0)$ of the same class as the initial family $\{M_\l\}$.
The normal form (SF) is called the Sylvester form, or sometimes the companion matrix. It is closely related to the transformation reducing a higher order linear ordinary differential equation to the system of first order equations, cf. here.
Deformation of a matrix which consists of several Jordan blocks with different eigenvalues can be reduced to a finite parameter normal form which involves $d$ constants which will depend regularly on $\l$, with $$ d=\sum_\mu (\nu_1(\mu)+3\nu_2(\mu)+5\nu_3(\mu)+\cdots). $$ Hhere $\nu_1(\mu)\geqslant n_2(\mu)\geqslant \nu_3(\mu)\geqslant\cdots~$ are the sizes of the Jordan blocks of $M_0$ with the same eigenvalue $\mu$ (arranged in the non-increasing order), and the summation is extended over all different eigenvalues of the matrix $M_0$ [A71, Theorem 4.4.].
For a systematic exposition of this subject, see [A83, Sect. 29, 30]. Normal forms for parametric families of objects (mainly dynamical systems) belong to the area of responsibility of the bifurcation theory.
Singularities of differentiable mappings
This area refers to classification of (germs of) maps $(\RR^m,0)\to(\RR^n,0)$, which constitute an infinite-dimensional space, with respect to the left-right equivalence: two germs $f,f':(\RR^m,0)\to(\RR^n,0)$ are equivalent, if there exist two germs of diffeomorphisms $h:(\RR^m,0)\to(\RR^m,0)$ and $g:(\RR^n,0)\to(\RR^n,0)$ such that $f=g^{-1}\circ f\circ h$. This left-right action corresponds to a change of local coordinates near the source and target points.
One can consider several parallel flavors of the classification theory:
holomorphic (or real analytic), when both the germ $f$ and the conjugacies $g,h$ are assumed/required to be sums of the convergent Taylor series;
smooth, more precisely, $C^\infty$-smooth;
formal theory, where all objects are represented by formal Taylor series without any assumptions on their convergence.
However, for the left-right classification, the three classifications usually coincide. In particular, if two holomorphic germs are conjugated by a pair of formal self-maps, then they also can be conjugated by a pair of holomorphic self-maps. If two $C^\infty$ germs are formally conjugated, then they are also $C^\infty$ conjugated, etc. The finite smoothness category is not as developed as the three flavors above: one could expect that the differentiability class of the conjugacies will in general be lower than that of the maps, but the sharp estimates are mostly unknown.
For more detailed exposition see Singularities of differentiable mappings. Here we give only a brief summary of available results.
Maps of full rank
With each smooth germ $f:(\RR^m,0)\to(\RR^n,0)$ one can associate a linear map $M:\RR^m\to\RR^n$ which is the linearization of $f$ ($M$ is also called the tangent map to $f$, the Jacobian matrix or the differential of $f$ at the origin). In coordinates one can write this as follows, $$ \forall x\in (\RR^m,0)\quad f(x)=Mx+\phi(x)\in (\RR^n,0),\qquad M=\biggl(\frac{\partial f_i}{\partial x_j}(0)\biggr)_{\!\!\substack{i=1,\dots,n \\ j=1,\dots, m}},\quad \|\phi(x)\|=O(\|x\|^2). $$
If the operator $M$ has the full rank, then $f$ is right-left equivalent to the linear germ $g'(x)=Mx$ [GG, Corollaries 2.5, 2.6].
These assumptions hold in two cases: where $m\le n$ and $M$ is injective, and where $m\ge n$ and $M$ is surjective. The conclusion reduces the classification of nonlinear germs to that of linear maps, which was already discussed earlier.
This result is equivalent to the Implicit function theorem. In particular, it shows that the image of an immersion locally looks like a coordinate subspace, and the preimages of points by a submersion locally look like a family of parallel affine subspaces of the appropriate dimension.
The obvious reformulation of this theorem is valid also for real-analytic and complex holomoprhic germs.
Germs of maps in small dimension
When the rank condition fails, the normal form is nonlinear and is known in small dimensions. The corresponding theory is known by the name Singularity theory of differential maps, or the Catastrophe theory.
The classification is organized along a tree: the normal forms depend on the rank of the Jacobian matrix, but also on some relationships between higher order Taylor coefficients of $f$ at the origin, introducing deeper and deeper degeneracy. Each such set of conditions is characterized by its codimension, the number of algebraically independent conditions imposed on the initial segment of the Taylor series of $f$ (in the invariant terms, on the jet of $f$). By the Thom's Transversality theorem, singularities of codimension $k$ and higher generically do not occur in generic families of maps involving less than $k$ parameters.
Holomorphic curves
A nonconstant holomorphic (or real analytic) germ $f:(\C^1,0)\to(\C^1,0)$ is biholomorphically left-right equivalent to the monomial map $g:z\mapsto z^\mu$, $\mu\in\NN$; the number $\mu=1$ corresponds to a full rank map and the normal form is linear, for $\mu>1$ nonlinear. The list of simple normal forms for holomorphic curves $f:(\CC,0)\to(\CC^2,0)$ consists[1] of 6 different series, of which the simplest two are $$ A_{2k}:\ t\mapsto (t^2,t^{2k+1}),\qquad E_{6k}:\ t\mapsto (t^3, t^{3k+1}+\delta t^{3k+p+2}), 0\leqslant p\leqslant k-s,\ \delta\in\{0,1\}. $$
↑ J. W. Bruce, T. J. Gaffney, Simple singularities of mappings $\CC,0$ to $\CC^2,0$, J. London Math. Soc. 26 (1982):3, 465-474, doi:10.1112/jlms/s2-26.3.465, MR0684560.
Nondegenerate critical points of functions and the Morse lemma
A smooth map $f:(\RR^n,0)\to(\RR,0)$ which is not of the full rank, has a critical point at the origin: $\rd f(0)=0$. In this case the quadratic approximation $Q:\RR^n\to\RR$, $(x_1,\dots,x_n)\mapsto\sum_{i,j=1}^n q_{ij}x_ix_j$ provided by the Hessian matrix $\rd ^2f(0)=\|q_{ij}\|$, $q_{ij}=\frac{\partial^2 f}{\partial x_i\partial x_j}(0)$, is the normal form for the left-right equivalence, assuming that the rank of this form is full. This assertion is famous under the name of the Morse lemma [M], [AVG]: $$ \rd f(0)=0,\ \operatorname{rank}\rd^2 f(0)=n\implies f(x)\sim Q(x). $$ The known classification of quadratic forms allows to bring $f(x)$ to the normal form $f(x)=x_1^2+\cdots+x_k^2-x_{k+1}^2-\cdots-x_n^2$. It is worth mentioning that one can transform a germ to its normal form by applying the change of variables in the source only: change of the variable in the target space is unnecessary for critical points.
Degenerate critical points of smooth functions
If the critical point of a function is degenerate and its corank $\delta=\operatorname{corank}Q=n-\operatorname{rank}Q>0$, the normal forms become more complicated, although the initial steps are still simple.
If $\delta=1$, then the classification reduces to that of (smooth or analytic) functions of one variable. Except for an "infinitely degenerate" subcase, a function with Hessian of corank 1 can be brought to the normal form denoted by "class $A_\mu$": $$ \rd f(0)=0,\ \operatorname{corank} \rd^2f(0)=1\implies f\sim x_1^{\mu+1}+\sum_{k=2}^n \pm x_k^2. $$ Singularities of corank $\delta\geqslant 2$ and small codimension also have polynomial normal forms. Among these one has to distinguish simple singularities (of critical points of functions), which appear in two series and three exceptional cases. Apart from the series $A_\mu$ mentioned above, the other series, denoted by $D_\mu$, has the normal form $$ f(x)\sim x_1^{\mu-1}+x_1x_2^2+\sum_{k=3}^n \pm x_k^2,\qquad \mu=4,5,\dots. $$ The three exceptional simple singularities also occur for the $\operatorname{corank} \rd^2f(0)=2$ and have the normal form (we omit for simplicity the quadratic Morse part) as follows: $$ E_6:\ x^3+y^4,\qquad E_7:\ x^3+xy^3,\qquad E_8:\ x^3+y^5. $$ This classification is intimately linked to the classification of simple Lie algebras [1][2].
More degenerate critical points can (to some extent) be reduced to polynomial normal forms involving one or more real parameters (thus the number of different non-equivalent critical points becomes infinite), see hundreds of cases in [AVG, Ch. II, Sect. 16-17]. Further degeneracy requires normal forms involving arbitrary functions, even more increasing the "size" of the lists.
↑ V. I. Arnold, Normal forms of functions near degenerate critical points, the Weyl groups $A_k,D_k,E_k$ and Lagrangian singularities, Funct. Anal. Appl. 6 (1972), no. 4, 3–25, MR0356124
↑ M. Entov, On the $ADE$-classification of the simple singularities of functions, Bull. Sci. Math. 121 (1997), no. 1, 37–60, MR1431099
"Elementary catastrophes"
Smooth germs between two different spaces $f:(\RR^2,0)\to(\RR^2,0)$ have polynomial normal forms for the case $\operatorname{rank}\rd f(0)<2$, if the higher order terms are not too degenerate. The rank condition means that the determinant (Jacobian) $\det \rd f(x)$ vanishes on a curve $\varSigma\subseteq(\RR^2,0)$ passing through the origin. The curve $\varSigma$, called the discriminant (the critical locus of $f$) is generically smooth at the origin and has a tangent line $\ell=T_0\varSigma\subseteq T_0\RR^2$. Position of this line can be compared with another line $\ell'=\operatorname{Ker} \rd f(0)\subseteq T_0\RR^2$.
If the two lines are transversal (cross each other by a nonzero angle), $T_0\varSigma\pitchfork \operatorname{Ker}\rd f(0)$, then the corresponding singular point is called fold and is right-left equivalent to the quadratic map $$ f:\begin{pmatrix}x\\y\end{pmatrix}\mapsto \begin{pmatrix}u\\v\end{pmatrix}=\begin{pmatrix}x^2\\y\end{pmatrix}. $$ This map is a two-fold cover of the right half-plane $\{u\geqslant0\}$ in the targer plane. The line $\{u=0\}$ is the visible contour of the map.
If the two lines coincide, one needs an additional nondegeneracy assumption[1], yet under this condition the singular point is called cuspidal singularity and is right-left equivalent to the cubic map $$ f:\begin{pmatrix}x\\y\end{pmatrix}\mapsto \begin{pmatrix}u\\v\end{pmatrix}=\begin{pmatrix}xy+x^3\\y\end{pmatrix}. $$ The image of the curve $\varSigma$, the visible contour of the map, is a semicubic parabola $4u^2-9v^3=0$, also referred to as the cusp. For the detailed exposition see [GG, Ch. VI, Sect. 2].
↑ The angle between the directions $\ell$ and $\ell'$, measured along the curve $\varSigma$, should have a simple root at the origin.
Classification of dynamical systems
Main page: Local normal forms for dynamical systems.
This is the classification of (usually invertible) self-maps $f\:(\RR^n,0)\to(\RR^m,0)$ with two self-maps $f,f'$ considered as equivalent if there exists a germ of diffeomorphism $h:(\RR^n,0)\to(\RR^n,0)$ such that $f'=h^{-1}\circ f\circ h$. This equivalence respects iteration, i.e., extends as the equivalence of cyclic subgroups of $\operatorname{Diff}(\RR^n,0)$: $$ f\sim f'\iff \underbrace{f\circ \cdots\circ f}_{n\text{ times}} \sim \underbrace{f'\circ \cdots\circ f'}_{n\text{ times}}\qquad\forall n=1,2,\dots $$ Such subgroups are naturally identified with discrete-time dynamical systems. A closely related classification of one-parametric subgroups $\{f^t:t\in\RR,\ f^{t+s}=f^t\circ f^s\}\subseteq\operatorname{Diff}(\RR^n,0)$ reduces to classification of germs of vector fields with a singular point at the origin[1]. Two vector fields $v,v'$ are called equivalent, if there exists a diffeomorphism $h$ as above, such that $$ h_*v=v'\circ h,\qquad h_*=\biggl(\frac{\partial h}{\partial x}\biggr)(0)=\rd f(0). $$
As with the left-right equivalence of maps, one could first attempt to conjugate a vector field $v$ (or a self-map $f$) to its linear part $A=\rd v(0)$ (resp., $M=\rd f(0)$) and reduce the classification to that of linear operators. However, unlike the previous theory, the possibility of such linearization depends very strongly on the arithmetic nature of eigenvalues of $A$ (resp., $M$), in particular, on the presence of resonances between them.
Besides, again in contrast with the left-right classification, the three parallel theories (analytic, $C^\infty$-smooth and formal) in the case of dynamic equivalence differ very much: the unavoidable divergence of the formal series conjugating an object with its normal form is a typical phenomenon.
↑ The vector field generating a one-parametric group (the flow of this field) is defined as the velocity, $v(x)=\left.\frac{\rd}{\rd t}\right|_{t=0}f^t(x)$
In addition to the above "general" theory, one can consider maps (and conjugacy) preserving various additional structures.
For instance, an even-dimension neighborhood $(\RR^{2n},0)$ can be equipped by the standard symplectic structure $\omega=\sum_{i=1}^n \rd x_i\land\rd y_i$. Then with any germ of a smooth critical function $H$ (Hamiltonian) one can associate the Hamiltonian vector field $v_H$ uniquely defined by the identity $\rd H=\omega(v_H,\cdot)$ between 1-forms. The equivalence relation rather naturally requires the conjugating diffeomorphism $h$ be canonic, i.e., preserve the symplectic structure: $h^*\omega=\omega$. The corresponding classification theory is important for the Hamiltonian dynamical systems. As with the "general" theory, the answers depend on the arithmetical properties of the eigenvalues of the linearization, with resonances (defined in a slightly different way) to play the central role.
Another important structure is that of a vector bundle. Consider vector fields on $(\CC^{n+1},0)$ which are linear in the last $n$ coordinates and are fibered over the 1-dimensional base: such a vector field can be always written (after a suitable change of coordinates) under the form $$ \dot x=x^{\mu+1},\quad \dot y=A(x)y,\qquad \mu\in\ZZ_+,\ A(x)=A_0+xA_1+x^2A_2+\cdots\text{ a holomorphic matrix-valued function}. $$ The natural equivalence relation on such vector fields is that of gauge equivalence, corresponding to the change of variables $y=H(x)w$ with a holomorphic invertible matrix function $H(\cdot)$. The corresponding classification differs substantially for $\mu=0$ (Fuchsian singularities) where formal normal forms are polynomial and convergent, and $\mu>0$ (irregular singularities), where the divergence of the formal transformations is a rule[1], see also Stokes phenomenon and [IY, Sect. 20].
In a different spirit, a possible ramification concerns "dynamical systems with multidimensional time": for such systems one is given a tuple of commuting vector fields $v_1,\dots,v_k$ with $[v_i,v_j]=0$ for all $i,j$ (resp., tuple of commuting self-maps $f_1,\dots,f_k\in\operatorname{Diff}(\R^n,0)$ with $f_i\circ f_j=f_j\circ f_i$ for all $i,j$), and the question is about simultaneous reduction of all fields (resp., germs) to some tuple of normal forms, see [2] and the references therein.
Smooth/holomorphic actions of groups more general than $\ZZ^k$ or $\RR^k$ are usually considered in the framework of the Group theory.
↑ Yu. Ilyashenko, Nonlinear Stokes phenomena, Nonlinear Stokes phenomena, 1–55, Adv. Soviet Math., 14, Amer. Math. Soc., Providence, RI, 1993, MR1206041.
↑ L. Stolovitch, Normalisation holomorphe d'algèbres de type Cartan de champs de vecteurs holomorphes singuliers, Ann. of Math. (2) 161 (2005), no. 2, 589–612, MR2153396
References and basic literature
[sort]
[M] J. W. Milnor, Morse theory. Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963, MR0163331.
[A71] V. I. Arnold, Matrices depending on parameters. Russian Math. Surveys 26 (1971), no. 2, 29--43, MR0301242
[GG] M. Golubitsky, V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14. Springer-Verlag, New York-Heidelberg, 1973, MR0341518.
[A83] V. I. Arnold, Geometrical methods in the theory of ordinary differential equations. Grundlehren der Mathematischen Wissenschaften, 250. Springer-Verlag, New York-Berlin, 1983, MR0695786
[Ar74] V. I. Arnold, Mathematical methods of classical mechanics. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. MR1345386
[AVG] V. I. Arnold, S. M. Guseĭn-Zade, A. N. Varchenko, Singularities of differentiable maps, Vol. I, The classification of critical points, caustics and wave fronts. Monographs in Mathematics, 82. Birkhäuser Boston, Inc., Boston, MA, 1985, ISBN: 0-8176-3187-9, MR0777682.
[IY] Yu. Ilyashenko, S. Yakovenko, Lectures on analytic differential equations. Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008 MR2363178
Normal form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_form&oldid=36204
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Tagged: linear combination
by Yu · Published 03/19/2018
Find a Spanning Set for the Vector Space of Skew-Symmetric Matrices
Let $W$ be the set of $3\times 3$ skew-symmetric matrices. Show that $W$ is a subspace of the vector space $V$ of all $3\times 3$ matrices. Then, exhibit a spanning set for $W$.
Read solution
Click here if solved 86
Add to solve later
Find a basis for $\Span(S)$, where $S$ is a Set of Four Vectors
Find a basis for $\Span(S)$ where $S=
\left\{
\begin{bmatrix}
1 \\ 2 \\ 1
\end{bmatrix}
-1 \\ -2 \\ -1
2 \\ 6 \\ -2
\right\}$.
How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix
Let $A=\begin{bmatrix}
2 & 4 & 6 & 8 \\
1 &3 & 0 & 5 \\
1 & 1 & 6 & 3
\end{bmatrix}$.
(a) Find a basis for the nullspace of $A$.
(b) Find a basis for the row space of $A$.
(c) Find a basis for the range of $A$ that consists of column vectors of $A$.
(d) For each column vector which is not a basis vector that you obtained in part (c), express it as a linear combination of the basis vectors for the range of $A$.
Click here if solved 104
Can We Reduce the Number of Vectors in a Spanning Set?
Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^3$. Is it possible that $S_2=\{\mathbf{v}_1\}$ is a spanning set for $V$?
Does an Extra Vector Change the Span?
Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^5$. If $\mathbf{v}_4$ is another vector in $V$, then is the set
\[S_2=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\] still a spanning set for $V$? If so, prove it. Otherwise, give a counterexample.
Find a Basis for Nullspace, Row Space, and Range of a Matrix
If $\mathbf{v}, \mathbf{w}$ are Linearly Independent Vectors and $A$ is Nonsingular, then $A\mathbf{v}, A\mathbf{w}$ are Linearly Independent
Let $A$ be an $n\times n$ nonsingular matrix. Let $\mathbf{v}, \mathbf{w}$ be linearly independent vectors in $\R^n$. Prove that the vectors $A\mathbf{v}$ and $A\mathbf{w}$ are linearly independent.
Compute $A^5\mathbf{u}$ Using Linear Combination
\[A=\begin{bmatrix}
-4 & -6 & -12 \\
-2 &-1 &-4 \\
\end{bmatrix}, \quad \mathbf{u}=\begin{bmatrix}
6 \\
\end{bmatrix}, \quad \mathbf{v}=\begin{bmatrix}
-2 \\
\end{bmatrix}, \quad \text{ and } \mathbf{w}=\begin{bmatrix}
\end{bmatrix}.\]
(a) Express the vector $\mathbf{u}$ as a linear combination of $\mathbf{v}$ and $\mathbf{w}$.
(b) Compute $A^5\mathbf{v}$.
(c) Compute $A^5\mathbf{w}$.
(d) Compute $A^5\mathbf{u}$.
Spanning Sets for $\R^2$ or its Subspaces
In this problem, we use the following vectors in $\R^2$.
\[\mathbf{a}=\begin{bmatrix}
\end{bmatrix}, \mathbf{b}=\begin{bmatrix}
\end{bmatrix}, \mathbf{c}=\begin{bmatrix}
\end{bmatrix}, \mathbf{d}=\begin{bmatrix}
\end{bmatrix}, \mathbf{e}=\begin{bmatrix}
\end{bmatrix}, \mathbf{f}=\begin{bmatrix}
\end{bmatrix}.\] For each set $S$, determine whether $\Span(S)=\R^2$. If $\Span(S)\neq \R^2$, then give algebraic description for $\Span(S)$ and explain the geometric shape of $\Span(S)$.
(a) $S=\{\mathbf{a}, \mathbf{b}\}$
(b) $S=\{\mathbf{a}, \mathbf{c}\}$
(c) $S=\{\mathbf{c}, \mathbf{d}\}$
(d) $S=\{\mathbf{a}, \mathbf{f}\}$
(e) $S=\{\mathbf{e}, \mathbf{f}\}$
(f) $S=\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}$
(g) $S=\{\mathbf{e}\}$
How to Obtain Information of a Vector if Information of Other Vectors are Given
Let $A$ be a $3\times 3$ matrix and let
\[\mathbf{v}=\begin{bmatrix}
\end{bmatrix} \text{ and } \mathbf{w}=\begin{bmatrix}
\end{bmatrix}.\] Suppose that $A\mathbf{v}=-\mathbf{v}$ and $A\mathbf{w}=2\mathbf{w}$.
Then find the vector
\[A^5\begin{bmatrix}
by Yu · Published 01/15/2018 · Last modified 01/16/2018
The Range and Nullspace of the Linear Transformation $T (f) (x) = x f(x)$
For an integer $n > 0$, let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis.
Let $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, for $f \in \mathrm{P}_n$,
\[T (f) (x) = x f(x).\]
Prove that $T$ is a linear transformation, and find its range and nullspace.
Prove that $\{ 1 , 1 + x , (1 + x)^2 \}$ is a Basis for the Vector Space of Polynomials of Degree $2$ or Less
Let $\mathbf{P}_2$ be the vector space of polynomials of degree $2$ or less.
(a) Prove that the set $\{ 1 , 1 + x , (1 + x)^2 \}$ is a basis for $\mathbf{P}_2$.
(b) Write the polynomial $f(x) = 2 + 3x – x^2$ as a linear combination of the basis $\{ 1 , 1+x , (1+x)^2 \}$.
A Condition that a Vector is a Linear Combination of Columns Vectors of a Matrix
Suppose that an $n \times m$ matrix $M$ is composed of the column vectors $\mathbf{b}_1 , \cdots , \mathbf{b}_m$.
Prove that a vector $\mathbf{v} \in \R^n$ can be written as a linear combination of the column vectors if and only if there is a vector $\mathbf{x}$ which solves the equation $M \mathbf{x} = \mathbf{v}$.
Write a Vector as a Linear Combination of Three Vectors
Write the vector $\begin{bmatrix} 1 \\ 3 \\ -1 \end{bmatrix}$ as a linear combination of the vectors
\[\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \, \begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix} , \, \begin{bmatrix} 2 \\ 0 \\ 4 \end{bmatrix}.\]
Prove that any Set of Vectors Containing the Zero Vector is Linearly Dependent
Prove that any set of vectors which contains the zero vector is linearly dependent.
Orthogonal Nonzero Vectors Are Linearly Independent
Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ be a set of nonzero vectors in $\R^n$.
Suppose that $S$ is an orthogonal set.
(a) Show that $S$ is linearly independent.
(b) If $k=n$, then prove that $S$ is a basis for $\R^n$.
Use Coordinate Vectors to Show a Set is a Basis for the Vector Space of Polynomials of Degree 2 or Less
Let $P_2$ be the vector space over $\R$ of all polynomials of degree $2$ or less.
Let $S=\{p_1(x), p_2(x), p_3(x)\}$, where
\[p_1(x)=x^2+1, \quad p_2(x)=6x^2+x+2, \quad p_3(x)=3x^2+x.\]
(a) Use the basis $B=\{x^2, x, 1\}$ of $P_2$ to prove that the set $S$ is a basis for $P_2$.
(b) Find the coordinate vector of $p(x)=x^2+2x+3\in P_2$ with respect to the basis $S$.
The Subspace of Linear Combinations whose Sums of Coefficients are zero
Let $V$ be a vector space over a scalar field $K$.
Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset
\[W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } a_1+a_2+\cdots+a_k=0\}.\] So each element of $W$ is a linear combination of vectors $\mathbf{v}_1, \dots, \mathbf{v}_k$ such that the sum of the coefficients is zero.
Prove that $W$ is a subspace of $V$.
Determine Whether Each Set is a Basis for $\R^3$
Determine whether each of the following sets is a basis for $\R^3$.
(a) $S=\left\{\, \begin{bmatrix}
\end{bmatrix}, \begin{bmatrix}
\end{bmatrix} \,\right\}$
(b) $S=\left\{\, \begin{bmatrix}
(c) $S=\left\{\, \begin{bmatrix}
(d) $S=\left\{\, \begin{bmatrix}
If there are More Vectors Than a Spanning Set, then Vectors are Linearly Dependent
Let $V$ be a subspace of $\R^n$.
Suppose that
\[S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_m\}\] is a spanning set for $V$.
Prove that any set of $m+1$ or more vectors in $V$ is linearly dependent.
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Rainfall drives variation in rates of change in intrinsic water use efficiency of tropical forests
Mark A. Adams ORCID: orcid.org/0000-0002-8154-00971,2,
Thomas N. Buckley3 &
Tarryn L. Turnbull1,2
Ecophysiology
Rates of change in intrinsic water use efficiency (W) of trees relative to those in atmospheric [CO2] (ca) have been mostly assessed via short-term studies (e.g., leaf analysis, flux analysis) and/or step increases in ca (e.g., FACE studies). Here we use compiled data for abundances of carbon isotopes in tree stems to show that on decadal scales, rates of change (dW/dca) vary with location and rainfall within the global tropics. For the period 1915–1995, and including corrections for mesophyll conductance and photorespiration, dW/dca for drier tropical forests (receiving ~ 1000 mm rainfall) were at least twice that of the wettest (receiving ~ 4000 mm). The data also empirically confirm theorized roles of tropical forests in changes in atmospheric 13C/12C ratios (the 13C Suess Effect). Further formal analysis of geographic variation in decade-to-century scale dW/dca will be needed to refine current models that predict increases in carbon uptake by forests without hydrological cost.
Rate(s) at which plant processes adjust, acclimate and adapt to rising atmospheric [CO2] (ca), especially processes that govern exchanges of carbon and water with the atmosphere (and their roles in 'physiological forcing' of climates1,2,3), have profound global implications for policy, practice and predictive models. This significance has been recognised by major research infrastructure (e.g., in free-air carbon enrichment (FACE) studies4), decades-long monitoring programs of atmospheric chemistry (e.g.,5) and a vast array of modelling studies (e.g. ref. 6).
For more than 30 years, theory has suggested rising ca should increase the intrinsic water-use efficiency (W) of plants7. There is also an extraordinary volume of empirical research on W (and its components-photosynthetic carbon fixation, A, and stomatal conductance, gs) across ecosystems. For temperate and boreal forests, the theory of rising W with ca is thus backed by a large body of work (e.g., refs. 8,9,10,11,12,13), and increases in W of trees are amongst the most common of global responses to rising ca, albeit with exceptions. Evidence of increases in W comes from multiple sources including large-scale flux networks8,9, tree ring (or ring proxy) isotope series10, catchment-scale studies11, model-data fusions11,12,13 and hundreds of leaf-scale analyses (e.g.,14), some based on historic herbarium samples15. However, and in contrast to the general case that W should rise with ca, the long-term rate at which W has been changing has not been rigorously examined across ecosystems or regions or climates.
Many individual studies note that W has seemingly increased more quickly since the 1960s, in concert with the faster rate of increase in ca, but formal examinations are largely restricted to leaf-level and relatively short-term studies4,16 (see also ref. 11 for a 28-year study). 13C/12C ratios of cellulose derived from annual tree rings (or from otherwise age-identified wood) provide a time-integrated measure of W that can be extended to hundreds of years in the case of long-lived trees, and used as an important complement and contrast to shorter-term (often leaf-level) data17. Such data have been widely used to test models of feedbacks amongst the biosphere, atmosphere and climate. Some of the more significant constraints to its interpretation, such as potential confounding (ontogenic) effects of tree age and size, have recently been characterised18.
We asked the broad question: what are the rates of change in W of tropical forests? We followed recent suggestions19 in focusing on isotope series (time series of abundances of stable isotopes of C, as captured in stemwood) as a means of improving our ability to predict responses of forests to global change. Our formal hypotheses were that in the long term, W increases with ca, and dW/dt or dW/dca will depend on climate. We used tropical forests to test these hypotheses, since previous theoretical predictions20,21, and medium-term (<30 years) catchment studies11, have explicitly supported our first hypothesis, albeit over shorter time scales. The significance of our hypotheses was recently demonstrated in a modelling study22, which suggested that local changes in the rate at which W adjusts to ca (physiological forcing) are responsible for the majority of precipitation change above tropical forests. Isotope series data revealed that rainfall is a significant determinant of long-term (80 year) dW/dca, as is nitrogen-fixing capability (legume vs. non-legume).
Data availability and preliminary analysis
From a global search (see the Methods section), we compiled two data sets. Data Set 1 was composed of all available isotope series23,24,25,26,27,28,29 that spanned the period 1915–1995, excluding data from heavily modified sites. Data Set 2 comprised all other isotope series for the tropics30,31,32,33,34,35,36,37,38. Using the traditional calculation (see the Methods section), analysis of all available data (Data Set 1 + Data Set 2, Supplementary Table 1, Supplementary Fig. 1b) reveals that for 32 of 42 isotope series, W had a significant positive relationship with ca (as it does in temperate and boreal forests8,9,10,12,13). For eight, mostly short-term series, there was no relationship, and two series showed negative relationships. It is instructive that one such negative relationship between W and ca34 was based on a short-term study (1925–1938) during the Great Depression (1929–1939)—a period of exceptionally slow annual rates of increase in ca. When compared, mean W for both data sets was very similar (and not significantly different) for a period when nearly all isotope series overlapped (1990–1995; Supplementary Fig. 1c), notwithstanding some isotope series in Data Set 2 showing fast rates of change in W with ca.
Analysis of long-term data
We subjected all available long-term isotope series (Data Set 1) to more detailed analysis. Positive relationships of W with ca were just as clear when data were aggregated by site (see Table 1; Supplementary Fig. 2). All long-term relationships between ca and W were linear and highly significant (Table 1; P < 0.0001)), indicating relative invariance in the ratio of intercellular (ci) to ca (as W = ca(1 − ci/ca)/1.6; see Eq. (3) in the Methods section). Across the tropics, W of canopy dominants has increased at different rates in response to increasing ca, with fourfold differences amongst sites in dW/dca or dW/dt (Fig. 1a, b). Perhaps surprisingly, mean annual precipitation (MAP) has alone accounted for half of the site-to-site variation in dW/dca (Fig. 1a). When expressed on an annual (time) basis, dW/dt declined by 0.05 μmol mol−1 year−1 for each 1000 -mm increase in rainfall (Fig. 2a). In other words, over the course of the last century the W of trees at the driest included site increased by at least 15 μmol mol−1 more than of trees at the wettest. Both dW/dca and dW/dt were even more strongly related to latitude (Fig. 1b, 2b), and were greatest for systems distinguished by a distinct dry season (see also ref. 39). The ratio of MAP to potential evapotranspiration (PET) was also significantly related to dW/dca (see Supplementary Fig. 3a), again explaining ~ 50% of the variation. Mean annual temperature was not related to dW/dca (Supplementary Fig. 3b).
Table 1 Significant bivariate relationships between inherent water-use efficiency (W μmol mol−1) and atmospheric [CO2] (ca ppm) for canopy-dominant trees in tropical biomes (minimum period = 1915–1995)
Rainfall and latitude influences on change in W per unit ca. a The relationship of dW/dca (change in intrinsic water-use efficiency (W) per unit atmospheric [CO2] (ca); as derived from tree rings) to mean annual precipitation. b Relationship of dW/dca to absolute latitude. The data are site averages of long-term isotope series (each spanning at least the period 1915–1995) for tropical biomes. Points marked in red in both (a) and (b) are the data for legumes. For both (a) and (b), data sources are: (a) Wils et al.24, (b) Nock et al.27, (c) van der Sleen et al.23, (d) Ballantyne et al.25, (e) Hietz et al.29, (f) Schollaen et al.28, (g) Loader et al.26
Rainfall and latitude influences on change in W per unit time. The relationship of dW/dt (change in intrinsic water-use efficiency (W) per year; as derived from tree rings) and mean annual precipitation (a), and absolute latitude (b). The data are site averages of long-term isotope series (each spanning at least the period 1915–1995) for tropical biomes. Points marked in red in both (a) and (b) are legumes. For both (a) and (b), data sources are: (a) Wils et al.24, (b) Nock et al.27, (c) van der Sleen et al.23, (d) Ballantyne et al.25, (e) Hietz et al.29, (f) Schollaen et al.28, (g) Loader et al.26
When we followed Keeling et al.5 in adopting a more comprehensive approach to calculating W, and included potential effects of mesophyll conductance and photorespiration on isotope discrimination, there was little change in relationships of dW/dca to MAP, MAP/PET and latitude (Fig. 3a–c). Similarly, we modelled impacts of potential changes in the ratio A/ca (where A is net photosynthesis) with increasing ca on the relationship of dW/dca to MAP and latitude, in order to test if observed relationships with rainfall might be due to other influences over the past 100 or so years. As one limit, we assumed that A/ca remains constant (i.e., A increases in proportion to ca). As the other, we assumed that A remains constant regardless of any rise in ca. If MAP had no effect on the sensitivity of A to ca then patterns shown in Figs. 1 and 2 are little changed (Fig. 3a–c). If A changed in proportion to ca at the lowest rainfall, but became increasingly insensitive to ca as rainfall increased (and was constant at the highest rainfall), then relationships were much weakened (Fig. 3i–k). Finally, if the sensitivity of A to ca increased with rainfall, then relationships strengthened (Fig. 3e–g). While arguments can be made for all scenarios, the extreme case shown in Fig. 3i–k is highly unlikely, and recent evidence from tropical forests11 suggests the scenario in Fig. 3a–c is most likely.
Modelled effects of physiology on rates of change in W. Effects of methods for computing intrinsic water-use efficiency (W), and modelled sensitivity to mean annual precipitation (MAP) of the A/ca response to ca (where A is net photosynthesis and ca is atmospheric [CO2]) for relationships of dW/dca (change in W per unit ca) to: MAP, the ratio of MAP to potential evapotranspiration, absolute latitude and mean annual temperature (MAT). a–d W includes terms to account for mesophyll conductance and photorespiration. e–h as for a–d but A/ca response to ca increases with MAP, (i–l) as for a–d but A/ca response to ca decreases as MAP increases. The data are site averages from sources as described in Figs. 1 and 2. See the Methods section for further detail
Growth of individual species responded variably to rising ca (Supplementary Table 1), although the magnitude of responses depended on both the period of measurement and site/climate (see also Supplementary Fig. 1b). Most species showed no distinct growth response to rising ca, in contrast to positive responses of W (Supplementary Table 1). We also examined other possible influences on W. Clear patterns of decreasing dW/dca (Fig. 1) or dW/dt (Fig. 2) with increasing rainfall, are the opposite of what might be expected if tree growth, rather than ca, were driving changes in W (see ref. 18).
The novel approach and related sampling adopted by van der Sleen et al.23 accounts for potential bias due to effects of ontogenic development on W. Nonetheless, the van der Sleen et al.23 data integrate well with all other available data for the tropics (e.g., Figs. 1–3), including data from single-tree studies.
Nutrient availability is another, frequently suggested, non-climatic constraint to tree responses to rising ca. The van der Sleen et al. data23 contained both legumes and non-legumes, which we have partitioned accordingly. Legumes (Supplementary Fig. 4) maintained significantly greater W than non-legumes, over the full period for which records were obtained (a difference of 7–8 μmol mol−1), in agreement with a recent synthesis of leaf-level data40. In all analyses shown in Figs. 1–3, and Supplementary Figs. 2 and 3, data for legumes are identified separately.
Collectively, the data compiled here represent tens of thousands of individual measurements of abundance of stable isotopes of carbon in wood samples from a broad range of climates and tree species. Represented ecosystems include monsoonal conifer forests in the Ethiopian highlands (Juniperus), Congo basin rainforests in Cameroon (Daniellia, Terminalia and Brachystegia), lowland dipterocarp rainforests in Indonesia and in Borneo (Tectona, Shorea and Eusideroxylon) and savannas and forests of Thailand (Melia, Toona and Chukrasia) and Brazil (Swietenia, Cedrela and Sweetia). The data show that long-term rates of change in W for trees from tropical biomes (Supplementary Table 1; range 0.10–0.43 μmol mol−1 year−1) are broadly comparable with trees from boreal (0.22 μmol mol−1 year−1), semi-arid (0.17 μmol mol−1 year−1) and temperate (0.28 μmol mol−1 year−1) forests in North America9 and Europe8,12.
Keeling et al.5 recently calculated that to account for the changing relative abundances of 13C vs. 12C in atmospheric CO2 (the 13C-Suess effect), there must have been a ~20% increase in W across the globe over the 40 years period 1975–2015, and that a significant proportion of the increase must have been due to tropical forests. The data compiled here provide an empirical confirmation of that calculation for tropical forests (Supplementary Table 1). As far as we can ascertain, our analysis is also the first evidence of rainfall-driven and systematic variation in long-term rates of change in water-use efficiency for any global forest biome. Current models of global patterns in W do not account for this variation (e.g., ref. 6).
Shorter term, more recent studies are an interesting contrast to the long-term patterns that are the focus here. For example, trees used by Nock et al.27 spanned generally shorter time periods (88 years) than other long-term isotope series (see Figs. 1–3), and their data are somewhat of an outlier. Greater dW/dca (Fig. 1a, b), and greater dW/dt recorded by Nock et al.27, is at least partially due to faster rates-of-increase in ca in recent decades. Other shorter-term (mostly post 1960) isotope series for the tropics (see Supplementary Table 1, Supplementary Fig. 1 and refs. 30,31,32,33,34,35,36,37,38), as well as leaf-level studies14,15 strongly support shown long-term patterns (including the influence of rainfall; see ref. 41 for a summary of effects of rainfall on W within individual studies of both ring-forming and ringless trees).
Obviously, rainfall alone does not define water availability to trees. In the tropics especially, seasonal distributions of rainfall and evaporative demand ensure that soil water storage plays a major role in year-round water availability, which in turn is reflected in seasonal variation in photosynthetic productivity of tropical forests39. Broadly speaking, tropical regions with pronounced dry seasons (typically savannah systems) can also be distinguished by seasonal changes in leaf area and transpiration from regions that have more uniform rainfall39. Nonetheless, rainfall is a strong predictor of dW/dca across long-term studies (e.g., Fig. 1), as well as within shorter-term and individual studies41 of tropical forests.
Donohue et al.42 used modelling and FACE data to argue that rates of carbon fixation have broadly increased with ca. They also showed that disturbance plays a significant role in the reliability of modelled predictions of vegetation responses to ca. Our analysis supports a conclusion that rates of carbon fixation have continued to rise with ca in tropical forests, irrespective of water availability. Disturbances (e.g., fires, floods, hurricanes) are frequently associated with major changes in nutrient and water availability. Evidence (Supplementary Fig. 4) of significant influence of nitrogen fixation on W (but not on dW/dca), as well as the enhanced abundance of legumes after disturbance, points to the need for a stronger focus on regenerating forest ecosystems as a means of disentangling more proximal (e.g., nutrient availability) and distal (e.g., disturbance) influences on dW/dca.
As reviewed a decade ago43, the significance of tropical forests to global carbon and hydrological cycles can scarcely be overstated. That significance has prompted many calls for increased research, for example into the notoriously unknown acclimation responses of plant physiology to rising ca21,44. Large year-to-year variation in climatic conditions, phenology and disturbance can easily render tenuous any conclusions based on shorter-term experimentation and observation, as well as models built on such foundations. Long-term rates of change in W shown here, based on integration of climatic and atmospheric information in tree stems, remain one of the best available means of validating and improving models for the tropics. Recorded dW/dca (and dW/dt) will also help guide efforts to predict future changes in W, at least for the tropical biome, over coming decades. We endorse the call19,41 for greatly increased availability of isotope series data (as collated here) for tropical forests owing to its scientific significance. However, we immediately recognise that developing such an increased availability of data (and being able to reliably apply statistical approaches such as meta-analysis) will likely take many years, if not decades.
A key unknown in the broad field of global climate change is how quickly forests adjust/acclimate/adapt to changing atmospheric and climatic conditions. Evidence here for the tropics suggests that while W has increased with ca, differences in water availability at any given site determined large shifts/variation in the rate of change (dW/dca). This helps constrain thinking and models directed towards resolving drivers of changing rainfall in the tropics (e.g., ref. 22), and elsewhere. For the tropics, recent evidence45,46,47 also suggests a slowing of growth and weakening of the tropical forest sink for carbon. At some point, carbon uptake by forests can no longer increase without commensurate increases in availability of water and nutrients. Analysis of water limitation and drought effects on physiological performance48,49 emphasises that at least parts of the tropics may be approaching limits to the rate of change in W with ca (see ref. 3; Supplementary Fig. 4)−a phenomenon for which there is also evidence from temperate forests in Europe50. The roles of water and nutrient availability as regulators of dW/dca need further elucidation.
Identification of data
We identified relevant literature by screening the Web of Science and Google Scholar search engines for keywords: dendrochron*, cellulose, tree ring, carbon isotope discrimination, δ13C, WUE, water-use efficiency, tropic* and also included relevant citations documented within these literature. We only accepted literature for forests classified as tropical (Köppen Climate Classification A). We identified 42 isotope series from 16 published studies that could provide estimates of W for tropical forests. The majority of isotope series were of short duration. Two were derived from sites that had been subject to heavy modification (clearing, fertiliser use etc.), and these were not used in this study.
Latitude and longitude of each site was used to identify MAP (mm), mean annual potential evaporation (PET), MAT, (Climatic Research Unit, University of East Anglia) and confirm sites were of Köppen classification A (tropical/megathermal). Constant vapour pressure difference between air and intercellular airspace would ensure trends in W derived from wood cellulose represent trends in water-use efficiency23, and previous studies show vapour pressure deficits (VPD) for tropical biomes (and elsewhere) have changed little during the past 150 years51.
Data inclusion and testing
Our analysis includes the multi-species, multi-site data set of van der Sleen et al.23, as well as isotope series from studies that encompass smaller numbers of sites and species (see Table 1; Supplementary Table 1, for details); including some single-tree studies. The data represent deciduous, evergreen and semi-deciduous trees, and also represents both ring-forming and ringless trees. The longest isotope series was almost 250 years, and the shortest was less than a decade (See Supplementary Fig. 1b).
In addition, the novel method of van der Sleen et al.23 (sampled wood located at a nominated diameter within each core from large numbers of trees rather than sampling all adjacent rings within a few trees; each value for W corresponds with an individual tree of a given age/size) allowed us to test for differences amongst sympatric legumes and non-legumes. We excluded data from their Bolivian site, owing to the unusual, and highly localised, terra preta soils (nutrient-enriched by centuries of human habitation).
We tested for the potentially confounding effects of century-scale changes in photosynthesis, mesophyll conductance and photorespiration (see below). Likewise, we tested if climate-related biases in photosynthetic responses to ca, influenced patterns in W (see below). We based our analysis on cellulose-δ13C derived from rings (or from wood of otherwise identified age) of trees from tropical forests around the world (Supplementary Fig. 1a).
Statistical approach and data quality
Our investigation was focused on exploring rates of change in W per unit change in atmospheric CO2 (dW/dca), and effects of climate on dW/dca. We considered a range of statistical approaches. Our research questions and available data were not well suited to meta-analytical tools (inappropriate use of these has been recently summarised52). Instead, and as far as possible, we present data as originally reported in each individual study (e.g., see Supplementary Figs. 2, 4). We used regression analysis to examine non-ca influences (e.g., climate, location) on dW/dca. Originally reported data for isotope abundances were derived from Tables and Figures, or obtained from the authors. Isotope series data are most commonly recorded in conjunction with wood age (as identified via rings or other means), measured in years, and we used all data as reported. All calculations of long-term dW/dca were based on the entire reported isotope series (provided the series met our criterion of spanning at least 1915–1995).
We created two data sets: Data Set 1 comprised 18 long-term isotope series (Supplementary Table 1) that encompassed the period 1915–1995 for 15 species from 9 different, largely undisturbed sites (two sites provided data for both legumes and non-legumes); Data Set 2 comprised 22 isotope series from a further 18 species and 16 undisturbed sites. Data Set 2 contained many studies that overlapped Data Set 1 for the 5-year period 1990–1995 (Supplementary Fig. 1).
Data Set 1 combined data for 720 individual trees across the major tropical forest biomes of Asia, Africa and South America (Table 1). While our focus was long-term rates of change in W (Data Set 1), we also sought to ensure that the long-term data reflected the broader data for tropical forests by comparing Data Sets 1 and 2 over the period 1990–1995 (for which there were the greatest number of overlapping isotope series).
Testing for other influences on c i/c a
We used the standard approach53 (as used by all tree ring studies reported here23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38) to calculate Δ from the δ13C of wood cellulose. We then used four approaches to calculate ci/ca from Δ:
(Approach #1) Ignoring effects of mesophyll conductance and photorespiration.
This is the 'standard' and most widely used approach (see also ref. 53, as follows: Eq. (1).
$$c_{\mathrm{i}}{\mathrm{/}}c_{\mathrm{a}}= (\Delta-a)/\left( {b-a} \right)$$
with b = 27‰ and a = 4.4‰.
This approach was used for all calculations apart from those presented in Fig. 3.
(Approaches #2–4) Accounting for effects of mesophyll conductance (gm) and photorespiration.
In these three approaches we used the same formulation as Keeling et al.5: Eq. (2).
$$c_i/c_{\mathrm{a}} = (\Delta-a + \left( {b-a_{\mathrm{m}}} \right)\left( {A/c_{\mathrm{a}}} \right)/g_{\mathrm{m}} + f\Gamma_ \ast /c_{\mathrm{a}})/\left( {b-a} \right)$$
with b = 30‰, a = 4.4‰, am = 1.8‰, f = 12‰, gm = 0.2 mol m−2 s−1 and Γ* (photorespiratory CO2 compensation point) = 43 ppm.
The ratio A/ca (where A is net photosynthesis) typically declines as ca increases, which can be modelled as A/ca = (A/ca)280·(ca/280)β, where (A/ca)280 is A/ca at ca = 280 ppm, β = ln(DR)/ln(2) – 1 and DR is the doubling ratio (i.e., DR = 1.45, if A increases by 45% with doubling of ca from 280 to 560 ppm).
For Approach #2, we assumed DR = 1.45, following Keeling et al.5. If we adopted a DR for the tropics of 2 (in place of the 1.45 adopted by Keeling et al.5), as suggested by the results of Yang et al.11, there was no significant change to our results.
For Approach #3, we assumed DR ∝ −MAP, declining from 2.0 at MAP = 1170 mm to 1.0 at MAP = 4000 mm; this reduces the strength of the inferred relationship between dW/dca and MAP.
For Approach #4, we assumed DR ∝ + MAP, increasing from 1.0 at MAP = 1170 mm to 2.0 at MAP = 4000 mm, which enhances the dW/dca—MAP relationship. W was calculated as: Eq. (3).
$$A/g_s = c_{\mathrm{a}} \cdot \left( {1 - c_i/c_{\mathrm{a}}} \right)/1.6$$
Where A is net photosynthesis and gs is stomatal conductance.
We used regression analyses to assess the rate of change in W with ca. Linear mixed models were initially used to test for differences between W of Data Set 1 and Data Set 2 for the period 1990–1995. We also used linear mixed models to quantify the combined influences of N-fixing status and ca on W by testing for differences between the legume and non-legume isotope series presented in van der Sleen et al.23. Site was treated as a random variable in all linear mixed model analyses.
All statistical analyses were completed with SPSS or R.
All data used in results are available from the first author, as extracted from primary sources or as provided by original authors.
Code used in calculations of model estimates of rates of change is the property of T.N.B. and is available on request ([email protected]).
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We thank the Australian Research Council and the National Science Foundation (Award #1557906) for financial assistance. This work was supported by the USDA National Institute of Food and Agriculture, Hatch project 1016439. Mana Gharun is thanked for her help in extracting climatic data.
Department of Chemistry and Biotechnology, Faculty of Science, Engineering and Technology, Swinburne University of Technology, Melbourne, VIC, Australia
Mark A. Adams & Tarryn L. Turnbull
School of Life and Environmental Sciences, University of Sydney, Sydney, NSW, Australia
Department of Plant Sciences, College of Agricultural and Environmental Sciences, University of California, Davis, CA, USA
Thomas N. Buckley
Mark A. Adams
Tarryn L. Turnbull
All authors contributed to the conceptualisation of the study, to discussions about the data and to the writing of the Ms. T.L.T. extracted the original data and prepared the presentation materials. T.N.B. conceived and prepared the models. M.A.A. prepared the draft paper. All authors contributed equally to editing drafts of the paper.
Correspondence to Mark A. Adams.
Peer review information: Nature Communications thanks Yuting Yang and other anonymous reviewer(s) for their contribution to the peer review of this work.
Adams, M.A., Buckley, T.N. & Turnbull, T.L. Rainfall drives variation in rates of change in intrinsic water use efficiency of tropical forests. Nat Commun 10, 3661 (2019). https://doi.org/10.1038/s41467-019-11679-8
CO2, nitrogen deposition and a discontinuous climate response drive water use efficiency in global forests
Nature Communications (2021)
Responses of forest carbon and water coupling to thinning treatments from leaf to stand scales in a young montane pine forest
Yi Wang
Antonio D. del Campo
Qiang Li
Carbon Balance and Management (2020)
Diminishing CO2-driven gains in water-use efficiency of global forests
Nature Climate Change (2020)
Forests in the Anthropocene | CommonCrawl |
\begin{document}
\begin{abstract} We prove global well-posedness below the charge norm (i.e., the $L^2$ norm of the Dirac spinor) for the Dirac-Klein-Gordon system of equations (DKG) in one space dimension. Adapting a method due to Bourgain, we split off the high frequency part of the initial data for the spinor, and exploit nonlinear smoothing effects to control the evolution of the high frequency part. To prove the nonlinear smoothing we rely on the null structure of the DKG system, and bilinear estimates in Bourgain-Klainerman-Machedon spaces. \end{abstract}
\maketitle
\section{Introduction}\label{Section1}
We study the Dirac-Klein-Gordon system (DKG) in one space dimension, \begin{equation}\label{DKG1} \left\{ \begin{alignedat}{2} &D_t \psi + \alpha D_x \psi + M\psi= \phi \beta\psi,& \qquad\qquad &\left(D_t=-i \partial_t, \,\, D_x=-i \partial_x\right) \\ &- \square \phi + m^2 \phi= \innerprod{\beta\psi}{\psi}_{\mathbb{C}^2}, & &\left(\square = -\partial_t^2 + \partial_x^2 \right) \end{alignedat} \right. \end{equation} with initial data \begin{equation}\label{data}
\psi \vert_{t = 0} = \psi_0 \in H^{-s}, \qquad \phi \vert_{t = 0} = \phi_0 \in H^r, \qquad \partial_t \phi \vert_{t = 0} = \phi_1 \in H^{r-1}, \end{equation} where $r,s \ge 0$ are fixed later. Here $\phi(t,x)$ is real-valued and $\psi(t,x) \in \mathbb{C}^2$ is the Dirac spinor, regarded as a column vector with components $\psi_1$, $\psi_2$. Further, $M,m \ge 0$ are constants, $H^r = (I-\partial_x^2)^{-r/2}L^2$ is the standard Sobolev space of order $r$ on $\mathbb{R}$, $\innerprod{\cdot}{\cdot}_{\mathbb{C}^2}$ is the standard inner product on $\mathbb{C}^2$, and we use the following representation of the Dirac matrices: $$
\alpha = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix},
\qquad
\beta = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}. $$
Global well-posedness (GWP) for DKG in 1d was first proved by Chadam \cite{Chadam:1973}, for data $$
(\psi_0,\phi_0,\phi_1) \in H^1 \times H^1 \times L^2. $$ In recent years, many authors have improved this result, in the sense that the required regularity on the data has been lowered: see Table \ref{Table0} for an overview (here and elsewhere we use the convention that if the regularity of $\phi_1$ is not specified, it is understood to be always one order less than that of $\phi_0$).
\begin{table} \caption{Global well-posedness for DKG in 1d} \label{Table0} \def1.3{1.3} \begin{center}
\begin{tabular}{|r|c|l|} \hline & $\psi_0 \in$ & $\phi_0 \in$ \\ \hline\hline Chadam \cite{Chadam:1973}, 1973 & $H^1$ & $H^1$ \\ \hline Bournaveas \cite{Bournaveas:2000}, 2000 & $L^2$ & $H^1$ \\ \hline Fang \cite{Fang:2004b}, 2004 & $L^2$ & $H^r$, $1/2 < r \le 1$ \\ \hline Bournaveas and Gibbeson \cite{BournaveasGibbeson:2006}, 2006 & $L^2$ & $H^r$, $1/4 \le r \le 1$ \\ \hline Machihara \cite{Machihara:2006}, Pecher \cite{Pecher:2006}, 2006 & $L^2$ & $H^r$, $0 < r \le 1$ \\ \hline \end{tabular} \end{center} \end{table}
All these GWP results rely on the conservation of charge: \begin{equation}\label{conservation_of_charge}
\norm{\psi(t)}_{L^2} = \norm{\psi(0)}_{L^2}, \end{equation} which holds for smooth solutions decaying sufficiently fast at spatial infinity, hence also for any solution evolving from initial data for which local well-posedness (LWP) holds, provided the regularity of the spinor is at least $L^2$, of course.
By combining \eqref{conservation_of_charge} with a sufficiently strong a priori estimate for $\phi$, one can reduce the problem of GWP to proving LWP, again provided the regularity of the spinor is at least $L^2$. In fact, by the energy inequality for the wave equation, \begin{equation}\label{phi_energy_estimate}
\norm{\phi(t)}_{H^r}
+ \norm{\partial_t \phi(t)}_{H^{r-1}}
\le
C_T \left( \norm{\phi_0}_{H^r}
+ \norm{\phi_1}_{H^{r-1}}
+ \int_0^t \norm{\square \phi(\sigma)}_{H^{r-1}} \, d\sigma \right), \end{equation} for $0 \le t \le T < \infty$, assuming existence up to time $T > 0$. If $r < 1/2$, one can then apply the product law for Sobolev spaces (in one space dimension): \begin{equation}\label{Sobolev_product}
\norm{fg}_{H^{-c}} \le C_{a,b,c} \norm{f}_{H^a} \norm{g}_{H^b}
\qquad \text{if}
\qquad
\left\{
\begin{aligned}
&a+b+c > 1/2,
\\
&a+b \ge 0,
\quad b+c \ge 0,
\quad a+c \ge 0.
\end{aligned}
\right. \end{equation} Thus, if $r < 1/2$, combining \eqref{Sobolev_product}, \eqref{phi_energy_estimate} and \eqref{conservation_of_charge} gives \begin{equation}\label{phi_apriori} \norm{\phi(t)}_{H^r}
+ \norm{\partial_t \phi(t)}_{H^{r-1}}
\le
C_T \left( \norm{\phi_0}_{H^r}
+ \norm{\phi_1}_{H^{r-1}}
+ T \norm{\psi_0}_{L^2}^2 \ \right), \end{equation} for $0 \le t \le T$. This together with the conservation of charge \eqref{conservation_of_charge} shows that if LWP holds with data $\psi_0 \in L^2$ and $\phi_0 \in H^r$, then GWP follows immediately, if $r < 1/2$. The same is in fact true if $1/2 \le r \le 1$; we remark on this below.
Concerning LWP, the best result is the following. Note that we use $-s$ to denote the order of the Sobolev space for the spinor, since we are primarily interested in spaces of negative order.
\begin{theorem}\label{LWP_theorem} The DKG system \eqref{DKG1} is LWP for data $\,\psi_0 \in H^{-s}$, $\phi_0 \in H^r$ provided $$
s < \frac{1}{4}, \qquad r>0,
\qquad
\abs{s}
\le r \le 1-s. $$ \end{theorem}
In the most interesting case $s \ge 0$ the attributions are as follows: Bournaveas \cite{Bournaveas:2000} proved LWP for $s = 0$ and $r=1$; Fang \cite{Fang:2004b} proved the result for $0 \le s < 1/4$ and $1/2 < r \le 1-2s$; Bournaveas and Gibbeson \cite{BournaveasGibbeson:2006} obtained $s = 0$ and $1/4 \le r \le 1$; Machihara \cite{Machihara:2006} proved LWP for $0 \le s < 1/4$ and $2\abs{s} \le r \le 1 - 2s$; Pecher \cite{Pecher:2006}, independently of Machihara, obtained $0 \le s < 1/4$ and $\abs{s} \le r < 1 - 2s$; finally, it was shown by A.\ Tesfahun and the present author in \cite{Selberg:2006e} that the upper bound on $r$ in Pecher's result can be relaxed to $r \le 1-s$. Moreover, it was shown in \cite{Selberg:2006e} that the conditions in Theorem \ref{LWP_theorem} are optimal (up to, possibly, the endpoint $r=s=0$) if one iterates in Bourgain-Klainerman-Machedon spaces.
As remarked already, LWP implies GWP when the regularity of the spinor is at least $L^2$, i.e., when $s \le 0$ in Theorem \ref{LWP_theorem}, essentially due to the conservation of charge.
\begin{theorem}\label{GWP_theorem} (Cf.\ Table \ref{Table0}.) The DKG system \eqref{DKG1} is GWP for data $\,\psi_0 \in H^{-s}$, $\phi_0 \in H^r$ if $$
s \le 0, \qquad r>0,
\qquad
\abs{s}
\le r \le 1-s. $$ \end{theorem}
\begin{remark} If $s=0$, this follows directly from Theorem \ref{LWP_theorem} if $0 < r < 1/2$, by the argument shown above, and the case $1/2 \le r \le 1$ then reduces to the case $0 < r < 1/2$ by using the propagation of higher regularity, which is discussed in the next remark. The case $s < 0$ can be reduced to $s=0$, again by propagation of higher regularity. \end{remark}
\begin{remark}\label{Remark2} Note that the conditions in Theorem \ref{LWP_theorem} describe a convex region $R$ of the $(s,r)$-plane. Suppose that $(s,r)$ and $(s',r')$ both belong to $R$, and that $-s' \le -s$ and $r' \le r$. Assume that $\psi_0 \in H^{-s}$ and $\phi_0 \in H^r$. Then $\psi_0 \in H^{-s'}$ and $\phi_0 \in H^{r'}$ also, of course. Now suppose that the solution in the latter data space exists up to some time $0 < T' < \infty$, so $$
\psi \in C\bigl([0,T'];H^{-s'}\bigr), \qquad
(\phi,\partial_t \phi) \in C\bigl([0,T'];H^{r'} \times H^{r'-1}\bigr). $$ Then in fact the solution retains the higher regularity of the initial data throughout the time interval $[0,T']$, so that $$
\psi \in C\bigl([0,T'];H^{-s}\bigr), \qquad
(\phi,\partial_t \phi) \in C\bigl([0,T'];H^{r} \times H^{r-1}\bigr). $$ This property, known as propagation of higher regularity, is included in the LWP result of Theorem \ref{LWP_theorem}, since it is proved by an iteration scheme using estimates on the iterates to get a contraction. \end{remark}
In this paper, we shall extend Theorem \ref{GWP_theorem} to a range of negative order Sobolev exponents for the spinor.
Our main result is the following.
\begin{theorem}\label{Main_theorem} The DKG system \eqref{DKG1} is GWP for data $\,\psi_0 \in H^{-s}$, $\phi_0 \in H^r$ if $$
0 < s < \frac{1}{8},
\qquad
s + \sqrt{s^2+s} < r \le 1-s. $$ \end{theorem}
Since $\psi_0$ is in a Sobolev space of negative order, the conservation law \eqref{conservation_of_charge} is not directly applicable. Bourgain \cite{Bourgain:1998} devised a method for dealing with situations where a PDE has a conservation law, but the initial data are too rough to use it directly. The essence of Bourgain's idea is to split the PDE into two parts, a `good' part and a `bad' part, by separating the low and high frequency parts of the data. For the good part one can use the conservation law, while for the `bad' part there must be a nonlinear smoothing effect in order for the method to work, so that the smoothed out nonlinear part can be fed back into the `good' part at the end of the possibly short time interval of existence of the `bad' part. Then this procedure is iterated, to reach a given, final time. This is the underlying idea, but in order to make it work, i.e., to close the finite induction, one needs sufficiently good estimates, of course.
The implementation of Bourgain's method for DKG requires an additional idea, since there is no conservation law for the field $\phi$, only for the spinor $\psi$. We overcome this difficulty by using the estimate \eqref{phi_apriori} for the inhomogeneous part, together with an additional induction argument involving a cascade of free waves; see Section \ref{Conclusion}.
To prove the nonlinear smoothing we rely on the null structure of the DKG system, which was recently completed by P.\ D'Ancona, D.\ Foschi and the present author in \cite{Selberg:2006b}, and on bilinear estimates in Bourgain-Klainerman-Machedon spaces.
The rest of this paper is organized as follows: In Section \ref{Preliminaries} we split the system into two separate systems, by a frequency cut-off on the spinor data. Then in Sections \ref{uw_estimates} and \ref{vz_estimates} we prove the main estimates for the split DKG system. In Section \ref{Conclusion} we put everything together to finish the proof of our main theorem, Theorem \ref{Main_theorem}. In Section \ref{Lemmas} we prove some lemmas that are used in earlier sections.
For simplicity we set $M = m = 0$ in the rest of the paper, but our proof can be modified to handle the massive case as well.
The following spaces of Bourgain-Klainerman-Machedon type are used throughout the paper. For $a,b \in \mathbb{R}$, we let $X^{a,b}_\pm$, $H^{a,b}$ and $\mathcal H^{a,b}$ be the completions of $\mathcal S(\mathbb{R}^{1+1})$ with respect to the norms \begin{equation}\label{Spaces} \begin{aligned}
\norm{u}_{X^{a,b}_\pm} &= \bignorm{\angles{\xi}^a \angles{\tau\pm \xi}^b \widetilde u(\tau,\xi)}_{L^2_{\tau,\xi}},
\\
\norm{u}_{H^{a,b}} &= \bignorm{\angles{\xi}^a \angles{\abs{\tau} - \abs{\xi}}^b \widetilde u(\tau,\xi)}_{L^2_{\tau,\xi}},
\\
\norm{u}_{\mathcal H^{a,b}} &= \norm{u}_{H^{a,b}} + \norm{\partial_t u}_{H^{a-1,b}}, \end{aligned} \end{equation} where $\widetilde u(\tau,\xi)$ denotes the space-time Fourier transform of $u(t,x)$, and $\angles{\cdot} = 1 + \abs{\cdot}$. The restrictions of these spaces to a time slab $$
S_T = (0,T) \times \mathbb{R} $$ are denoted $X^{a,b}_\pm(S_T)$ etc.; see \cite{Selberg:2006b} for more details about these spaces. We shall need the following basic estimates (see \cite{Selberg:2006b} for further references):
\begin{lemma}\label{X_lemma} Let $1/2 < b \le 1$, $a \in \mathbb{R}$, $0 < T \le 1$ and $0 \le \delta \le 1-b$. Then for all data $F \in X_\pm^{a,b-1+\delta}(S_T)$ and $f \in H^a$, the Cauchy problem $$
(D_t \pm D_x) u = F(t,x) \quad \text{in $\,(0,T) \times \mathbb{R}$}, \qquad u(0,x) = f(x), $$ has a unique solution $u \in X_\pm^{a,b}(S_T)$. Moreover, $$
\norm{u}_{X_\pm^{a,b}(S_T)} \le C \left( \norm{f}_{H^a}
+ T^\delta \norm{F}_{X_\pm^{a,b-1+\delta}(S_T)} \right), $$ where $C$ only depends on $b$. \end{lemma}
\begin{lemma}\label{H_lemma} \cite[Theorem 12]{Selberg:1999}. Let $1/2 < b < 1$, $s \in \mathbb{R}$, $0 < T \le 1$ and $0 \le \delta \le 1-b$. Then for all $F \in H^{a-1,b-1+\delta}(S_T)$, $f \in H^a$ and $g \in H^{a-1}$, there exists a unique $u \in \mathcal H^{a,b}(S_T)$ solving $$
\square u = F(t,x) \quad \text{in $\,(0,T) \times \mathbb{R}$}, \qquad u(0,x) = f(x),
\qquad \partial_t u(0,x) = g(x). $$ Moreover, $$
\norm{u}_{\mathcal H^{a,b}(S_T)} \le C \left( \norm{f}_{H^a} +
\norm{g}_{H^{a-1}} + T^{\delta/2}
\norm{F}_{H^{a-1,b-1+\delta}(S_T)} \right), $$ where $C$ depends only on $b$. \end{lemma} We shall also need the fact that (see \cite{Selberg:2006b}) if $b > 1/2$, then \begin{equation}\label{Basic_embedding}
\norm{u(t)}_{H^a} \le C \norm{u}_{H^{a,b}(S_T)} \le C \norm{u}_{X_\pm^{a,b}(S_T)}
\qquad \text{for $0 \le t \le T$}, \end{equation} where $C$ only depends on $b$.
\section{Preliminaries}\label{Preliminaries}
We first observe that by propagation of higher regularity (see Remark \ref{Remark2} in Section \ref{Section1}), it suffices to prove Theorem \ref{Main_theorem} for $r < 1/2$. We therefore fix $s$ and $r$ satisfying \begin{equation}\label{sr_cond}
0 < s < \frac{1}{8},
\qquad
s + \sqrt{s^2+s} < r < \frac12, \end{equation} and set \begin{equation}\label{b_def}
b=\frac12+\varepsilon, \end{equation} where $\varepsilon > 0$ will be chosen sufficiently small, depending on $s$ and $r$; $b$ will be used as the second exponent in the spaces defined by \eqref{Spaces}.
Fix an arbitrary time $0 < T < \infty$. We shall split the interval $[0,T]$ into subintervals of length $\Delta T$, where the small time $\Delta T > 0$ remains to be chosen, and prove well-posedness on each subinterval successively. Let $M$ be the number of subintervals, so $M = T/\Delta T$.
Following Bourgain's idea, we split the data into low and high frequency parts: $$
\psi_0 = \psi_0^L + \psi_0^H, $$ where $$(\psi_0^L)\,\, \widehat{} \,\,(\xi) = \mathbb{1}_{\abs{\xi} \le N} \widehat \psi_0(\xi),$$ and the cut-off parameter $N \gg 1$ will be chosen later. Then \begin{alignat}{2}
\label{LowFreq}
\norm{\psi_0^L}_{H^{-\zeta}} &\le C N^{s-\zeta}& \qquad &\text{for $\zeta \le s$},
\\
\label{HighFreq}
\norm{\psi_0^H}_{H^{-\zeta}} &\le C N^{s-\zeta}& \qquad &\text{for $\zeta \ge s$}, \end{alignat} where the constant $C$ depends on $\fixednorm{\psi_0}_{H^{-s}}$, but is independent of $N$. Note that we do not split the data for $\phi$, only for the spinor $\psi$.
Throughout the paper we use $C$ to denote a constant which can change from line to line; $C$ may depend on $s$, $r$, $T$ and $\varepsilon$, but it is always independent of $\Delta T$ and $N$.
We shall use $o(1)$ to denote a real-valued function $f(\varepsilon)$ such that $f(\varepsilon) \to 0$ as $\varepsilon \to 0$; these functions $f$ may depend implicitly on $s$ and $r$.
On each subinterval $[(n-1)\Delta T,n \Delta T]$ we split the solution of DKG into two parts: \begin{equation}\label{induction_split}
\psi = u_n + v_n, \qquad \phi = w_n + z_n \qquad \text{on} \qquad
[(n-1)\Delta T,n \Delta T] \times \mathbb{R}, \end{equation} where $(u_n,w_n)$ solves DKG: \begin{equation}\label{DKGuw} \left\{ \begin{aligned}
&\left(D_t + \alpha D_x\right)u_n = w_n \beta u_n,
\\
&\square w_n = -\innerprod{\beta u_n}{u_n}_{\mathbb{C}^2}, \end{aligned} \right. \end{equation} and hence the remaining part $(v_n,z_n) = (\psi-u_n,\phi-w_n)$ satisfies \begin{equation}\label{DKGvz} \left\{ \begin{aligned}
&\left(D_t + \alpha D_x\right)v_n = \phi\beta\psi - w_n \beta u_n
\equiv z_n \beta v_n + z_n \beta u_n + w_n \beta v_n,
\\
&\square z_n = \innerprod{\beta u_n}{u_n}_{\mathbb{C}^2} - \innerprod{\beta \psi}{\psi}_{\mathbb{C}^2} \equiv -\innerprod{\beta v_n}{v_n}_{\mathbb{C}^2}-2\re \innerprod{\beta u_n}{v_n}_{\mathbb{C}^2}. \end{aligned} \right. \end{equation}
For the first subinterval we take initial data \begin{equation}\label{First_data} \begin{alignedat}{3}
u(0) &= \psi_0^L \in L^2,&
\qquad w(0) &= \phi_0 \in H^r,&
\qquad \partial_t w(0) &= \phi_1 \in H^{r-1},
\\
v(0) &= \psi_0^H \in H^{-s},&
z(0) &= 0,&
\partial_t z(0) &= 0. \end{alignedat} \end{equation} Le $v^{(0)}$ denote the free evolution of $\psi_0^H$ by the Dirac equation: \begin{equation}\label{v_free}
(D_t + \alpha D_x ) v^{(0)} = 0, \qquad v^{(0)}(0) = \psi_0^H \in H^{-s}. \end{equation} By Lemma \ref{X_lemma} and the estimate \eqref{HighFreq}, \begin{equation}\label{v:free:a}
\bignorm{v_\pm^{(0)}}_{X_\pm^{-\zeta,b}(S_T)} \le C\norm{\psi_0^H}_{H^{-\zeta}} \le C N^{s-\zeta} \qquad \text{for $\zeta \ge s$}. \end{equation} Note that this holds on the whole time interval $[0,T]$, hence for any subinterval also.
For $(v_n,z_n)$ we then specify data, at time $t = (n-1) \Delta T$, \begin{equation}\label{vz_data}
v_n\bigl((n-1)\Delta T\bigr) = v^{(0)}\bigl((n-1)\Delta T\bigr),
\qquad z_n\bigl((n-1)\Delta T\bigr) = \partial_t z_n\bigl((n-1)\Delta T\bigr) = 0. \end{equation} Writing $V_n$ for the inhomogeneous part of $v_n$ we then have \begin{equation}\label{v_split}
v_n = v^{(0)} + V_n \qquad \text{on} \qquad
[(n-1)\Delta T,n \Delta T] \times \mathbb{R}. \end{equation} Of course, the free part $v^{(0)}$ just retains the regularity of the data. The inhomogeneous part $V_n(t)$, on the other hand, turns out to be smoother than the data, and in fact has $L^2$ regularity. This nonlinear smoothing effect is due to the null structure of the system, and it allows us to implement Bourgain's idea for the present problem, feeding $V_n$ back into the data for $u_{n+1}$, at time $n\Delta T$. We shall also feed $z_n$ into $w_{n+1}$.
Therefore, we specify the initial data for $(u_n,w_n)$ by the induction scheme \begin{equation}\label{uw_data} \begin{gathered}
u_{n+1}(n\Delta T) = u_{n}(n\Delta T) + V_{n}(n\Delta T),
\\
w_{n+1}(n\Delta T) = w_{n}(n\Delta T) + z_{n}(n\Delta T),
\quad \partial_t w_{n+1}(n\Delta T) =
\partial_t w_{n}(n\Delta T) + \partial_t z_{n}(n\Delta T) \end{gathered} \end{equation} for $1 \le n < M$. Recall that $M$ denotes the number of subintervals.
The main induction hypotheses will be: \begin{gather}
\label{u_n:data_bound}
\norm{u_n\bigl((n-1)\Delta T\bigr)}_{L^2} \le A_n N^{s},
\\
\label{w_n:data_bound}
\norm{w_n\bigl((n-1)\Delta T\bigr)}_{H^{r}} + \norm{\partial_t w_n\bigl((n-1)\Delta T\bigr)}_{H^{r-1}} \le B_n N^{2s}, \end{gather} for all large $N$, where $A_n$ and $B_n$ are independent of $N$. At the first induction step $n=1$, \eqref{u_n:data_bound} is satisfied in view of \eqref{LowFreq}, whereas the left side of \eqref{w_n:data_bound} does not depend on $N$ at all, so \eqref{w_n:data_bound} is trivially satisfied by choosing $B_1$ large enough. Note also that $A_1$ and $B_1$ do not depend $\varepsilon$, but this will not be the case for subsequent induction steps.
Thus, the underlying idea is simple, but to make it work in practice is not entirely trivial; we need quite strong estimates on $u$, $V$, $w$ and $z$ to close the induction. These estimates are derived in the next two sections.
\section{Estimates for $u$ and $w$}\label{uw_estimates}
Here we describe the general induction step from time zero to time $\Delta T$, hence we drop the subscript $n$. Thus, we shall derive estimates for $(u,w)$ solving DKG: \begin{equation}\label{DKGuw:2}
\left(D_t + \alpha D_x\right)u = w \beta u,
\qquad
\square w = -\innerprod{\beta u}{u}_{\mathbb{C}^2}, \end{equation} with \begin{equation}\label{uw:data_bounds}
\norm{u(0)}_{L^2} \le AN^{s},
\qquad
\norm{w(0)}_{H^{r}} + \norm{\partial_t w(0)}_{H^{r-1}} \le BN^{2s}, \end{equation} for all large $N$, where $A$ and $B$ are independent of $N$. In fact, the solution exists globally, by Theorem \ref{GWP_theorem}.
By the conservation of charge \eqref{conservation_of_charge}, \begin{equation}\label{u:cons}
\norm{u(t)}_{L^2} = \norm{u(0)}_{L^2} \end{equation} for all $t$.
In order to derive sufficiently strong spacetime estimates for the solution, we need to exploit the null structure inherent in DKG; to reveal this structure we decompose the spinor $\psi$ using the eigenspace projections of the Dirac operator $\alpha D_x$, following \cite{Selberg:2006b, Selberg:2006e}. The symbol $\alpha \xi$ of $\alpha D_x$ has eigenvalues $\pm \xi$ and corresponding eigenspace projections $$
P_{\pm} = \frac12
\begin{pmatrix}
1 & \pm1 \\
\pm1 & 1
\end{pmatrix}. $$ We then write $$
u = u_+ + u_-,
\qquad \text{where} \qquad
u_+ = P_+ u,
\quad
u_- = P_- u. $$ Applying $P_{\pm}$ to the first equation in (\ref{DKG1}), and using the identities $\alpha = P_+ - P_-$, $P_{\pm}^2 = P_{\pm}$ and $P_+P_-=P_-P_+=0$, and the fact that $\beta$ is hermitian, \eqref{DKGuw:2} is rewritten as \begin{equation}\label{DKGuw:3} \left\{ \begin{aligned}
&(D_t + D_x ) u_+ = P_+(w \beta u) \equiv w\beta u_-,
\\
&(D_t - D_x ) u_- = P_-(w \beta u) \equiv w\beta u_+,
\\
&\square w = -2\re \innerprod{\beta u_+}{u_-}_{\mathbb{C}^2}, \end{aligned} \right. \end{equation} where we also used $\beta P_+ = P_- \beta$ and $\beta P_- = P_+ \beta$. The intrinsic null structure of DKG manifests itself through the different signs in the right hand side of the last equation; the same structure is in fact encoded in the first two equations, as becomes apparent via a duality argument; see \cite{Selberg:2006b, Selberg:2006e}.
We write $$
\norm{u} \equiv \norm{u_+}_{X_+^{0,b}(S_{\Delta T})} + \norm{u_-}_{X_-^{0,b}(S_{\Delta T})}. $$
To estimate $w$ we shall use the following bilinear spacetime estimate for free waves in 1d, proved in \cite{Selberg:2006e}. A related estimate was proved by Bournaveas \cite[Lemma 1]{Bournaveas:2000}. The null structure, i.e., the difference of signs, is crucial here. In fact, the following is a 1d version of the null form estimate of Klainerman and Machedon \cite{Klainerman:1993}.
\begin{lemma}\label{WaveLemma} \cite{Selberg:2006e}. Suppose $u,v$ solve $$ \begin{alignedat}{2}
&(D_t + D_x ) u = 0,& \qquad &u(0,x) = f(x),
\\
&(D_t - D_x ) v = 0,& \qquad &v(0,x) = g(x), \end{alignedat} $$ where $f,g \in L^2(\mathbb{R})$. Then $$
\norm{uv}_{L^2(\mathbb{R}^{1+1})} \le \sqrt{2} \norm{f}_{L^2} \norm{g}_{L^2}. $$ \end{lemma}
By the transfer principle (see \cite[Lemma 4]{Selberg:2006b}), Lemma \ref{WaveLemma} immediately implies:
\begin{corollary}\label{WaveCorollary} For any $b > 1/2$, the estimate $$
\norm{uv}_{L^2(\mathbb{R}^{1+1})} \le C \norm{u}_{X_+^{0,b}} \norm{v}_{X_-^{0,b}} $$ holds, where $C$ depends only on $b$. \end{corollary}
By Lemma \ref{H_lemma}, the induction hypothesis \eqref{uw:data_bounds} and the above corollary, \begin{equation}\label{w:spacetime:a}
\norm{w}_{H^{r,b}(S_{\Delta T})}
\le
C \left( \norm{w(0)}_{H^{r}} + \norm{\partial_t w(0)}_{H^{r-1}}
+
\norm{\square w}_{L^2(S_{\Delta T})} \right)
\le CBN^{2s} + C \norm{u}^2, \end{equation} since $\square w = -2\re\innerprod{\beta u_+}{u_-}$ by \eqref{DKGuw:3}.
To estimate $u$ we need the following lemma, proved in Section \ref{Lemmas}. Here there is again a crucial null structure due to the difference of signs between the left and right sides in the estimate.
\begin{lemma}\label{Lemma5} Suppose $u$ solves $$
(D_t + D_x) u = \Phi \beta U, \qquad u(0) = u_0, $$ where $\Phi$ is real-valued and $U$ is a 2-spinor. Assume $0 < r < 1/2$. Then with $b=1/2+\varepsilon$ and $\varepsilon > 0$ sufficiently small, depending on $r$, we have the estimate $$
\norm{u}_{X_+^{0,b}(S_{\Delta T})} \le C\norm{u_0}_{L^2} + C (\Delta T)^{2r-7\varepsilon} \norm{\Phi}_{H^{r,b}(S_{\Delta T})} \norm{U}_{X_-^{0,b}(S_{\Delta T})} $$ for all $0 < \Delta T \le T$, where $C$ only depends on $r$, $\varepsilon$ and $T$. \end{lemma}
Applying this lemma to \eqref{DKGuw:3} and using \eqref{uw:data_bounds} and \eqref{w:spacetime:a}, we get \begin{align*}
\norm{u_+}_{X_+^{0,b}(S_{\Delta T})}
&\le C \norm{u(0)}_{L^2}
+ C(\Delta T)^{2r-7\varepsilon} \norm{w}_{H^{r,b}(S_{\Delta T})}
\norm{u_-}_{X_-^{0,b}(S_{\Delta T})}
\\
&\le CA N^s
+ C (\Delta T)^{2r-7\varepsilon} B N^{2s}\norm{u}
+ C (\Delta T)^{2r-7\varepsilon} \norm{u}^3. \end{align*} Of course, the same estimate holds for $u_-$ in $X_-^{0,b}(S_{\Delta T})$, hence \begin{equation}\label{u_est}
\norm{u}
\le CA N^s
+ C (\Delta T)^{2r-7\varepsilon} B N^{2s}\norm{u}
+ C (\Delta T)^{2r-7\varepsilon} \norm{u}^3, \end{equation} so if (with the same $C$) \begin{equation}\label{contraction}
C (\Delta T)^{2r-7\varepsilon} N^{2s} \left( 2B + 8C^2 A^2 \right)
\le 1, \end{equation} then it follows by a boot-strap argument outlined below that (again with the same $C$) \begin{equation}\label{u:spacetime}
\norm{u}
\le 2C A N^s. \end{equation} Then by \eqref{w:spacetime:a} we get also (now modifying $C$) \begin{equation}\label{w:spacetime}
\norm{w}_{H^{r,b}(S_{\Delta T})}
\le C\left( B + A^2 \right) N^{2s}. \end{equation}
Motivated by the condition \eqref{contraction}, we now make the following choice: \begin{equation}\label{time}
\Delta T \sim N^{-(2s+\varepsilon)/(2r-7\varepsilon)}. \end{equation} Then $(\Delta T)^{2r-7\varepsilon} N^{2s} \sim N^{-\varepsilon}$ tends to zero as $N$ tends to infinity, hence \eqref{contraction} holds for $N$ large enough, provided $A$ and $B$ remain bounded throughout the induction; for the time being, however, \eqref{contraction} should be considered an induction hypothesis.
\begin{remark}\label{Remark3} For the boot-strap argument referred to above we use the Picard iterates. Alternatively, one could use a continuity argument, but we want to avoid this since the continuity of, say, $\norm{u_+}_{X_+^{0,b}(S_{\Delta T})}$ as a function of $\Delta T$ is not obvious. The iterates $u^{(n)}$ are defined by setting $u^{(-1)} \equiv 0$ and then $$
(D_t \pm D_x ) u_\pm^{(n+1)} = w^{(n)} \beta u_\mp^{(n)}, \qquad u_\pm^{(n+1)}(0) = u_\pm(0),
\qquad \text{for}\quad n = -1,0,1,\dots, $$ where $\square w^{(n)} = -2\re\biginnerprod{\beta u_+^{(n)}}{u_-^{(n)}}$ with the same data as for $w$. Then \eqref{u_est} takes the form $$
y_{n+1} \le CA N^s
+ C (\Delta T)^{2r-7\varepsilon} B N^{2s}y_n
+ C (\Delta T)^{2r-7\varepsilon} y_n^3, $$ for $n \ge 0$, where $y_n = \norm{u^{(n)}}$. By Lemma \ref{X_lemma} and \eqref{uw:data_bounds}, $y_0 \le CAN^s$, so if \eqref{contraction} holds, it now follows by induction that $y_n \le 2CAN^s$ for all $n \ge 1$. Since $\norm{u^{(n)}-u}$ tends to zero as $n$ tends to infinity (see \cite{Selberg:2006e}), this proves \eqref{u:spacetime}. \end{remark}
\section{Estimates for $v,z$}\label{vz_estimates}
Again, we drop the subscript $n$, since we consider the general induction step. We first reformulate \eqref{DKGvz} using the splitting $v=v_+ + v_-$, where $v_+ = P_+v$ and $v_- = P_-v$. Thus, we prove estimates on the time interval $[0,\Delta T]$ for $(v_+,v_-,z)$ solving \begin{equation}\label{DKGvz:2} \left\{ \begin{aligned}
&(D_t + D_x ) v_+ = z \beta v_- + z \beta u_- + w \beta v_-,
\\
&(D_t - D_x ) v_- = z \beta v_+ + z \beta u_+ + w \beta v_+,
\\
&\square z = -2\re\innerprod{\beta v_+}{v_-}_{\mathbb{C}^2}-2\re \innerprod{\beta u_+}{v_-}_{\mathbb{C}^2}
-2\re \innerprod{\beta u_-}{v_+}_{\mathbb{C}^2}, \end{aligned} \right. \end{equation} with $z(0) = \partial_t z(0) = 0$. Here $u$ and $w$ are exactly as in the previous section. From now on we drop the subscript $\mathbb{C}^2$ on the inner product.
We write $v = v^{(0)} + V$, where $v^{(0)}$ is the free part, which we assume satisfies (recall \eqref{v:free:a}), \begin{equation}\label{v:free}
\bignorm{v^{(0)}}_{X_\pm^{-\zeta,b}(S_{\Delta T})}
\le C N^{s-\zeta} \qquad \text{for $\zeta \ge s$}. \end{equation}
Consider now the inhomogeneous part $V$ of $v$. We write $$
\norm{V_\pm} \equiv \norm{V_\pm}_{X_\pm^{0,b}(S_{\Delta T})},
\qquad
\norm{V} \equiv
\norm{V_+} + \norm{V_-}, $$ and $$
V_\pm = V'_\pm + V''_\pm + V'''_\pm, $$ where (see \eqref{DKGvz:2}) \begin{align*}
(D_t \pm D_x) V'_\pm &= z\beta v_\mp,
\\
(D_t \pm D_x) V''_\pm &= w\beta v_\mp,
\\
(D_t \pm D_x) V'''_\pm &= z\beta u_\mp, \end{align*} with zero data at time zero.
We need the following null form estimate, proved in Section \ref{Lemmas}.
\begin{lemma}\label{Lemma2} Let $b=1/2+\varepsilon$, where $0 < \varepsilon \le 1/2$, and assume $0 \le r \le b$. Then we have the 2-spinor estimate (note the different signs on the right) $$
\norm{\innerprod{\beta U}{V}}_{H^{-r,-b}} \le C \norm{U}_{X_+^{-r+2\varepsilon,b}} \norm{V}_{X_-^{0,1-b}}. $$ Thus, by duality, $$
\norm{\Phi \beta U}_{X_-^{0,b-1}} \le C \norm{\Phi}_{H^{r,b}} \norm{U}_{X_+^{-r+2\varepsilon,b}} $$ where $\Phi$ is real-valued and $U$ is a 2-spinor. Here $C$ depends only on $r$ and $\varepsilon$. \end{lemma}
Using Lemmas \ref{X_lemma}, \ref{Lemma5} and \ref{Lemma2}, and the estimate \eqref{v:free}, we get \begin{equation}\label{V:1} \begin{aligned}
\bignorm{V'_+}
&\le C \norm{z \beta v^{(0)}_-}_{X_+^{0,b-1}(S_{\Delta T})}
+ C(\Delta T)^{2r-7\varepsilon}
\norm{z}_{H^{r,b}(S_{\Delta T})} \norm{V}
\\
&\le C \norm{z}_{H^{r,b}(S_{\Delta T})}
\norm{v_-^{(0)}}_{X_-^{-r+2\varepsilon,b}(S_{\Delta T})}
+ C(\Delta T)^{2r-7\varepsilon}
\norm{z}_{H^{r,b}(S_{\Delta T})} \norm{V}
\\
&\le C \norm{z}_{H^{r,b}(S_{\Delta T})}
\left[ N^{s-r+2\varepsilon}
+ (\Delta T)^{2r-7\varepsilon} \norm{V} \right]. \end{aligned} \end{equation} Note that we are justified in applying \eqref{v:free}, since $r > s$ by the assumption \eqref{sr_cond}.
Similarly, using \eqref{w:spacetime}, \begin{equation}\label{V:2} \begin{aligned}
\bignorm{V''_+}
&\le C \norm{w}_{H^{r,b}(S_{\Delta T})}
\left[ N^{s-r+2\varepsilon}
+ (\Delta T)^{2r-7\varepsilon} \norm{V} \right]
\\
&\le C \left( B + A^2 \right) N^{2s}
\left[ N^{s-r+2\varepsilon}
+ (\Delta T)^{2r-7\varepsilon} \norm{V} \right]. \end{aligned} \end{equation}
By Lemma \ref{Lemma5} and \eqref{u:spacetime}, \begin{equation}\label{V:3}
\bignorm{V'''_+}
\le C(\Delta T)^{2r-7\varepsilon} \norm{z}_{H^{r,b}(S_{\Delta T})}
\norm{u}
\le CA N^s(\Delta T)^{2r-7\varepsilon} \norm{z}_{H^{r,b}(S_{\Delta T})}. \end{equation}
Next, we estimate $z$. Using \eqref{DKGvz:2} and the fact that $z$ has zero data at time zero, we write \begin{equation}\label{z:spacetime:a}
\norm{z}_{H^{r,b}(S_{\Delta T})}
\le 2\left( I_1 + I_2 + I_3 + I_4 + I_5 \right) + (\cdots), \end{equation} where \begin{align*}
I_1 &= \norm{\square^{-1} \biginnerprod{\beta v^{(0)}_+}{v^{(0)}_-}}_{H^{r,b}(S_{\Delta T})},
\\
I_2 &= \norm{\square^{-1} \biginnerprod{\beta v^{(0)}_+}{V_-}}_{H^{r,b}(S_{\Delta T})},
\\
I_3 &= \norm{\square^{-1} \biginnerprod{\beta v^{(0)}_+}{u_-}}_{H^{r,b}(S_{\Delta T})},
\\
I_4 &= \norm{\square^{-1} \biginnerprod{\beta V_+}{V_-}}_{H^{r,b}(S_{\Delta T})},
\\
I_5 &= \norm{\square^{-1} \biginnerprod{\beta V_+}{u_-}}_{H^{r,b}(S_{\Delta T})}, \end{align*} and $(\cdots)$ indicates similar terms for which we have the same estimates as for the $I_j$'s. Here we use the notation $\square^{-1} F$ for the solution of $\square Z = F$ with zero data at time zero.
To estimate $I_1$ we need the following, proved in Section \ref{Lemmas}.
\begin{lemma}\label{Lemma6} Assume $r \le 1/2$ and $b=1/2+\varepsilon$, where $0 < \varepsilon \le 1/2$. Then $$
\norm{\innerprod{\beta U}{V}}_{H^{r-1,b-1}} \le C \norm{U}_{X_+^{-1/4+\varepsilon,b}} \norm{V}_{X_-^{-1/4+\varepsilon,b}}, $$ where $C$ depends only on $r$ and $\varepsilon$. \end{lemma}
By Lemmas \ref{X_lemma}, \ref{Lemma6} and the estimate \eqref{v:free}, \begin{equation}\label{I:1} \begin{aligned}
I_1 &\le C\norm{\innerprod{\beta v^{(0)}_+}{v^{(0)}_-}}_{H^{r-1,b-1}(S_{\Delta T})}
\\
&\le C\bignorm{v_+^{(0)}}_{X_+^{-1/4+\varepsilon,b}(S_{\Delta T})}
\bignorm{v_-^{(0)}}_{X_-^{-1/4+\varepsilon,b}(S_{\Delta T})}
\\
&\le C N^{2(s-1/4+\varepsilon)}. \end{aligned} \end{equation}
For $I_2$ and $I_3$ we need the following variation of Lemma \ref{Lemma6}.
\begin{lemma}\label{Lemma7} Assume $r < 1/2$ and $b=1/2+\varepsilon$, where $0 < \varepsilon \le 1/2$. Then $$
\norm{\innerprod{\beta U}{V}}_{H^{r-1,b-1}} \le C \norm{U}_{X_+^{-1/2+\varepsilon,b}} \norm{V}_{X_-^{0,b}}, $$ where $C$ depends only on $r$ and $\varepsilon$. \end{lemma}
Combining this with Lemma \ref{X_lemma}, \eqref{v:free} and \eqref{u:spacetime}, we get \begin{align}
\label{I:2}
I_2 &\le C\bignorm{v_+^{(0)}}_{X_+^{-1/2+\varepsilon,b}(S_{\Delta T})} \norm{V}
\le CN^{s-1/2+\varepsilon} \norm{V},
\\
\label{I:3}
I_3 &\le C\bignorm{v_+^{(0)}}_{X_+^{-1/2+\varepsilon,b}(S_{\Delta T})} \norm{u} \le C AN^{2s-1/2+\varepsilon}. \end{align}
Finally, to estimate $I_4$ and $I_5$ we need the following. Note that in this estimate the signs do not matter.
\begin{lemma}\label{Lemma8} Suppose $u$ solves $$
\square u = \innerprod{\beta U}{V}, \qquad u \vert_{t = 0} = \partial_t u \vert_{t = 0} = 0, $$ where $U,V$ are 2-spinors. Assume $r < 1/2$. Then with $b=1/2+\varepsilon$ and $\varepsilon > 0$ sufficiently small, depending on $r$, we have $$
\norm{u}_{H^{r,b}(S_{\Delta T})} \le C (\Delta T)^{3/4-2\varepsilon} \norm{U}_{X_+^{0,b}(S_{\Delta T})} \norm{V}_{X_\pm^{0,b}(S_{\Delta T})} $$ for all $0 < \Delta T \le T$, where $C$ only depends on $r$, $\varepsilon$ and $T$. \end{lemma}
This lemma gives \begin{equation}\label{I:4}
I_4 \le C (\Delta T)^{3/4-2\varepsilon} \norm{V}^2, \end{equation} and, using \eqref{u:spacetime}, \begin{equation}\label{I:5}
I_5 \le C (\Delta T)^{3/4-2\varepsilon} \norm{V} \norm{u}
\le CA N^s (\Delta T)^{3/4-2\varepsilon} \norm{V}. \end{equation}
From \eqref{z:spacetime:a}--\eqref{I:5} we conclude that $$
\norm{z}_{H^{r,b}(S_{\Delta T})}
\le C AN^{2s-1/2+2\varepsilon}
+ CAN^s \left[(\Delta T)^{3/4-2\varepsilon} + N^{-1/2+\varepsilon}\right] \norm{V}
+ C (\Delta T)^{3/4-2\varepsilon} \norm{V}^2. $$ However, $N^{-1/2+\varepsilon}$ is negligible compared to $(\Delta T)^{3/4-2\varepsilon}$ if $\varepsilon > 0$ is sufficiently small and $N$ sufficiently large, since by \eqref{time}, $\Delta T \sim N^{-s/r+o(1)}$, and $r > 2s$ by \eqref{sr_cond}. Hence, \begin{equation}\label{z:spacetime}
\norm{z}_{H^{r,b}(S_{\Delta T})}
\le C AN^{2s-1/2+2\varepsilon}
+ CAN^s (\Delta T)^{3/4-2\varepsilon} \norm{V}
+ C (\Delta T)^{3/4-2\varepsilon} \norm{V}^2. \end{equation} Combining this with \eqref{V:1}, \eqref{V:2} and \eqref{V:3}, we conclude: \begin{equation}\label{V:boot_strap:1} \begin{aligned}
\norm{V}
&\le
C \left(B + A^2\right) N^{2s} \left[ N^{s-r+2\varepsilon} + N^{s-1/2+2\varepsilon}(\Delta T)^{2r-7\varepsilon} \right]
\\
&\quad
+ C\left(B + A^2\right) N^{2s} \left[ (\Delta T)^{2r-7\varepsilon} + (\Delta T)^{3/4+2r-9\varepsilon}
+ N^{-r+2\varepsilon}(\Delta T)^{3/4-2\varepsilon} \right]
\norm{V}
\\
&\quad
+ CAN^{s} (\Delta T)^{3/4-2\varepsilon}\left[ (\Delta T)^{2r-7\varepsilon}
+ N^{-r+2\varepsilon}
\right]
\norm{V}^2
\\
&\quad
+ C (\Delta T)^{3/4+2r-9\varepsilon} \norm{V}^3. \end{aligned} \end{equation} We want to simplify the right hand side. Consider first the sum inside the parentheses in the first line. Clearly, the term $N^{s-r+2\varepsilon}$ dominates, since $r < 1/2$. Now look at terms in parentheses in the third line. The term $(\Delta T)^{2r-7\varepsilon}$ dominates, since $(\Delta T)^{2r-7\varepsilon} \sim N^{-2s+2\varepsilon}$, by \eqref{time}, and since $r > 2s$; the same reasoning applies for the second line.
Thus, \begin{equation}\label{V:boot_strap:2} \begin{aligned}
\norm{V}
&\le
C \left(B + A^2\right) N^{3s-r+2\varepsilon}
+ C\left(B + A^2\right) N^{2s} (\Delta T)^{2r-7\varepsilon}
\norm{V}
\\
&\quad
+ CAN^{s} (\Delta T)^{3/4+2r-9\varepsilon} \norm{V}^2
+ C (\Delta T)^{3/4+2r-9\varepsilon} \norm{V}^3. \end{aligned} \end{equation} But in view of the induction hypothesis \eqref{contraction}, the term which is linear in $\norm{V}$ can be moved to the left hand side, yielding \begin{equation}\label{V:boot_strap:3}
\norm{V}
\le
C \left(B + A^2\right) N^{3s-r+2\varepsilon}
+ CAN^{s} (\Delta T)^{3/4+2r-9\varepsilon} \norm{V}^2
+ C (\Delta T)^{3/4+2r-9\varepsilon} \norm{V}^3. \end{equation} A boot-strap argument described below then gives (with the same $C$ as in \eqref{V:boot_strap:3}) \begin{equation}\label{V:final}
\norm{V} \le 2\Gamma,
\quad \text{where}
\quad
\Gamma = C \left(B + A^2\right) N^{3s-r+2\varepsilon}, \end{equation} provided that (still with the same $C$) the following induction hypothesis holds: \begin{equation}\label{contraction2}
C (\Delta T)^{3/4+2r-9\varepsilon} \left[ 4AN^{s}\Gamma + 8 \Gamma^2 \right] \le 1. \end{equation}
\begin{remark}\label{Remark4} The above estimates imply local well-posedness of \eqref{DKGvz:2} for data $v(0) \in H^{-s}$ satisfying $\norm{v(0)}_{H^{-\zeta}} \le CN^{s-\zeta}$ for $\zeta \ge s$, and zero data for $z$, with existence up to the time $\Delta T > 0$ determined by the conditions \eqref{contraction} and \eqref{contraction2}, by a standard argument using the iterates. To define the iterates we write $v_\pm^{(n)} = v_\pm^{(0)} + V_\pm^{(n)}$ for $n = 1,2,\dots$, where $v^{(0)}$ is the free part and $V_\pm^{(n)}$ for $n = 1,2,\dots$ are determined by the scheme $$
(D_t \pm D_x) V^{(n+1)}_\pm = z^{(n)} \beta v_\mp^{(n)} + z^{(n)}\beta u_\mp + w \beta v_\mp^{(n)}, \qquad V^{(n+1)}_\pm(0) = 0,
\qquad \text{for $n \ge 0$}, $$ where $$
\square z^{(n)} = - 2\re\innerprod{\beta v_+^{(n)}}{v_-^{(n)}}
- 2\re\innerprod{\beta u_+}{v_-^{(n)}}
- 2\re\innerprod{\beta v_+^{(n)}}{u_-} $$ with zero initial data. Then our estimates imply that $V_\pm^{(n)}$ converges to $V_\pm$ in $X_\pm^{0,b}(S_{\Delta T})$, as $n$ tends to infinity. The boot-strap procedure referred to above is included in this iteration argument; in this setting, \eqref{V:boot_strap:3} becomes \begin{equation}
y_{n+1}
\le
\Gamma
+ CAN^{s} (\Delta T)^{3/4+2r-9\varepsilon} y_n^2
+ C (\Delta T)^{3/4+2r-9\varepsilon} y_n^3,
\qquad \text{for $n \ge 0$}, \end{equation} where $y_0 = 0$ and $y_n = \norm{V^{(n)}}$ for $n \ge 1$. So if \eqref{contraction2} holds, then by induction we get $y_n \le 2\Gamma$ for all $n$. \end{remark}
From \eqref{z:spacetime} and \eqref{V:final} we conclude that $$
\norm{z}_{H^{r,b}(S_{\Delta T})}
\le
C AN^{2s-1/2+2\varepsilon}
+ C (\Delta T)^{3/4-2\varepsilon} \left( AN^s \Gamma
+ \Gamma^2 \right). $$ But since $r > 2s$, \begin{equation}\label{gamma_est}
\Gamma^2 \le C (B+A^2) N^s \Gamma, \end{equation} so we get \begin{equation}\label{z:final}
\norm{z}_{H^{r,b}(S_{\Delta T})}
\le
C AN^{2s-1/2+2\varepsilon}
+ C (B+A^2)^2 N^{4s-r-3s/4r + o(1)}, \end{equation} where we used that $\Delta T \sim N^{-s/r+o(1)}$, by \eqref{time}.
\section{Conclusion of the proof}\label{Conclusion}
We now apply the estimates proved in the last two sections to the induction scheme described in Section \ref{Preliminaries}. Recall that on the $n$-th subinterval $[(n-1)\Delta T,\Delta T]$, where $\Delta T$ satisfies \eqref{time}, $(u_n,w_n)$ solves \eqref{DKGuw} and $(v_n,z_n)$ solves \eqref{DKGvz}; the data prescribed at time $(n-1)\Delta T$ are given by \eqref{vz_data} for $(v_n,z_n)$ and by the induction scheme \eqref{uw_data} for $(u_n,w_n)$. The main induction hypotheses are \eqref{u_n:data_bound} and \eqref{w_n:data_bound}, involving the constants $A_n$ and $B_n$, which must be independent of the large parameter $N$. In addition, the boot-strap conditions \eqref{contraction} and \eqref{contraction2} must be satisfied at each step of the induction. Recall also that $M$ denotes the number of induction steps, so \begin{equation}\label{M}
M = \frac{T}{\Delta T} \sim N^{s/r+o(1)}, \end{equation} where we used \eqref{time}.
Note that the boot-strap conditions \eqref{contraction} and \eqref{contraction2} reduce to, using \eqref{time} and \eqref{gamma_est}, \begin{equation}\label{contraction3}
CN^{-\varepsilon}(B_n+A_n^2) \le 1
\qquad \text{and} \qquad
CN^{2s-r-(3s)/(4r) + o(1)} ( B_n + A_n^2 )^2 \le 1. \end{equation}
The crucial point now is to prove that $A_n$ and $B_n$ do not grow indefinitely, but remain bounded for $1 \le n \le M$. Otherwise it would be impossible to satisfy \eqref{contraction3} with $N$ chosen independently of $n$.
By \eqref{uw_data}, the conservation of charge \eqref{conservation_of_charge}, the induction hypothesis \eqref{u_n:data_bound}, the embedding \eqref{Basic_embedding} and the estimate \eqref{V:final}, assuming \eqref{contraction3} holds, \begin{equation}\label{S5:A} \begin{aligned}
\norm{u_{n+1}(n\Delta T)}_{L^2}
&\le \norm{u_{n}(n\Delta T)}_{L^2} + \norm{V_{n}(n\Delta T)}_{L^2}
\\
&= \norm{u_{n}\bigl((n-1)\Delta T\bigr)}_{L^2} + \norm{V_{n}(n\Delta T)}_{L^2}
\\
&\le A_n N^s + \norm{V_{n}(n\Delta T)}_{L^2}
\\
&\le A_n N^s + C \left(B_n + A_n^2\right) N^{3s-r+2\varepsilon}. \end{aligned} \end{equation} Therefore, \begin{equation}\label{A_induction}
A_{n+1} \le A_n + C \left(B_n + A_n^2\right) N^{2s-r+2\varepsilon}. \end{equation}
It will be convenient to use the notation $$
\norm{\phi[t]}_{H^r} \equiv \norm{\phi(t)}_{H^r} + \norm{\partial_t \phi(t)}_{H^{r-1}}. $$ By \eqref{uw_data}, \begin{equation}\label{w_naive}
\norm{w_{n+1}[n\Delta T]}_{H^{r}}
\le
\norm{w_n[n\Delta T]}_{H^{r}}
+ \norm{z_n[n\Delta T]}_{H^{r}}, \end{equation} but the problem is that we have no conservation law for $w$. Implementing Bourgain's idea for DKG is therefore not so straightforward, and requires an additional idea.
We first split $w_n$ into its free part $w_n^{(0)}$ and its inhomogeneous part $W_n$, so \begin{equation}\label{w_splitting}
w_n = w_n^{(0)} + W_n, \end{equation} where $\square W_n = \square w_n$ and $W_n$ has zero data at time $(n-1)\Delta T$. The inhomogeneous part is quite favorable. In fact, applying the estimate \eqref{phi_apriori} to $W_n$ on the interval $[(n-1)\Delta T,\Delta T]$, and using \eqref{u_n:data_bound}, we get, assuming \eqref{contraction3} holds, \begin{equation}\label{W_add}
\norm{W_n[n\Delta T]}_{H^{r}}
\le C (\Delta T) \norm{u_n\bigl((n-1)\Delta T\bigr)}_{L^2}^2
\le C (\Delta T) A_n^2 N^{2s}. \end{equation} By \eqref{Basic_embedding} and \eqref{z:final}, assuming \eqref{contraction3} holds, \begin{equation}\label{z_add} \begin{aligned}
\norm{z_n[n\Delta T]}_{H^{r}}
&\le C \norm{z_n}_{H^{r,b}([(n-1)\Delta T,n\Delta T] \times \mathbb{R})}
\\
&\le C A_n N^{2s-1/2+2\varepsilon}
+ C (B_n+A_n^2)^2 N^{4s-r-3s/4r + o(1)}. \end{aligned} \end{equation} That leaves us with the free part $w_n^{(0)}$. Certainly, by the energy inequality and \eqref{w_n:data_bound}, $$
\norm{w_n^{(0)}[n\Delta T]}_{H^{r}} \le C \norm{w_n^{(0)}[(n-1)\Delta T]}_{H^{r}} \le C B_n N^{2s}, $$ but this naive estimate is useless, since it gives at best $B_n \sim C^n$. However, things are not as bad as they may seem at a first glance. The important point is to keep accurately track of what is added to the free part at the end of each induction step. In fact, by induction, referring to the scheme \eqref{uw_data}, we have \begin{equation}\label{cascade}
w_n^{(0)} = w_1^{(0)} + \tilde w_2^{(0)} + \tilde w_3^{(0)} + \dots + \tilde w_n^{(0)}, \end{equation} for $n \ge 1$, where \begin{align*}
\square \tilde w_n^{(0)} &= 0,
\\
\tilde w_n^{(0)}\bigl( (n-1)\Delta T \bigr) &=
W_{n-1}\bigl( (n-1)\Delta T \bigr)
+ z_{n-1}\bigl( (n-1)\Delta T \bigr),
\\
\partial_t \tilde w_n^{(0)}\bigl( (n-1)\Delta T \bigr) &=
\partial_t W_{n-1}\bigl( (n-1)\Delta T \bigr)
+ \partial_t z_{n-1}\bigl( (n-1)\Delta T \bigr). \end{align*} Thus, we express $w_n^{(0)}$ as a cascade of free waves. By the energy inequality for the wave equation, and the estimates \eqref{W_add} and \eqref{z_add}, \begin{equation}\label{w_tilde}
\bignorm{\tilde w_{n+1}^{(0)}[t]}_{H^r} \le
C (\Delta T) A_n^2 N^{2s}
+ C A_n N^{2s-1/2+2\varepsilon}
+ C (B_n+A_n^2)^2 N^{4s-r-3s/4r + o(1)}, \end{equation} in the entire time interval $0 \le t \le T$.
We now replace the induction hypothesis \eqref{w_n:data_bound} by the stronger condition \begin{equation}\label{w:free_bound}
\sup_{0 \le t \le T} \norm{w_n^{(0)}[t]}_{H^r} \le B_n N^{2s}. \end{equation} In fact, by the energy inequality this condition is equivalent to \eqref{w_n:data_bound}, up to multiplication by a constant depending only on $T$.
Since $w_{n+1}^{(0)} = w_n^{(0)} + \tilde w_{n+1}^{(0)}$, we have $$
\bignorm{w_{n+1}^{(0)}[t]}_{H^r} \le \norm{w_n^{(0)}[t]}_{H^r}
+ \bignorm{\tilde w_{n+1}^{(0)}[t]}_{H^r}, $$ for all $0 \le t \le T$, and we conclude from \eqref{w_tilde} that \begin{equation}\label{B_induction}
B_{n+1}
\le B_n
+ C (\Delta T) A_n^2
+ C A_n N^{-1/2+2\varepsilon}
+ C (B_n+A_n^2)^2 N^{2s-r-3s/4r + o(1)}. \end{equation}
We claim that if $\varepsilon > 0$ is chosen small enough, and then $N$ large enough, depending on $\varepsilon$, then for $1 \le n \le M$, \begin{equation}\label{AB_bound}
A_n \le R \equiv 2A_1
\qquad \text{and} \qquad
B_n \le S \equiv 2B_1 + 4CT A_1^2 \end{equation} with the same $C$ that appears in the second term of the right hand side of \eqref{B_induction}.
We proceed by induction. Assume \eqref{AB_bound} holds for $1 \le n < m$, for some $m \le M$. Then \eqref{contraction3} reduces to \begin{equation}\label{C1}
CS N^{-\varepsilon} \le 1, \end{equation} and \begin{equation}\label{C2}
C S^2 N^{2s-r-(3r)/(4s) + o(1)} \le 1, \end{equation} for $n < m$. Since $r > 2s$, we can ensure that the exponent of $N$ in \eqref{C2} is negative by choosing $\varepsilon > 0$ small enough. Then $C$, and hence $S$, may grow, but this is not a problem, since we can now choose $N$ as large as we need, depending on $\varepsilon$, to ensure that \eqref{C1} and \eqref{C2} are satisfied. Then \eqref{A_induction} and \eqref{B_induction} are also satisfied for $n < m$, so by the assumption that \eqref{AB_bound} holds for $n < m$, we get \begin{align*}
A_{n+1} &\le A_1 + n CS N^{2s-r+o(1)},
\\
B_{n+1} &\le B_1 + n \left[ CRN^{-1/2+o(1)}+ CS^2 N^{2s-r-3s/(4r)+o(1)}
+ CR^2 \Delta T \right], \end{align*} for $n < m$. Thus, \eqref{AB_bound} will hold also for $A_m$ and $B_m$ provided that \begin{gather*}
(m-1) CS N^{2s-r+o(1)}
\le A_1,
\\
(m-1) \left[ CR N^{-1/2+o(1)} + CS^2 N^{2s-r-3s/(4r)+o(1)} + CR^2 (\Delta T) \right]
\le B_1 + 4CTA_1^2. \end{gather*} But $m \le M \le C N^{s/r+o(1)}$, by \eqref{M}, so it suffices to have \begin{gather}
\label{Cond1}
CS N^{2s-r+s/r+o(1)} \le A_1,
\\
\label{Cond2}
CR N^{-1/2+s/r+o(1)} \le B_1/2,
\\
\label{Cond3}
CS^2 N^{2s-r-3s/(4r)+s/r+o(1)} \le B_1/2,
\\
\label{Cond5}
C T R^2 \le 4CTA_1^2, \end{gather} where to get \eqref{Cond5} we used the fact that $m \Delta T \le M \Delta T = T$, by \eqref{M}. Note that \eqref{Cond5} holds with equality since $R=2A_1$. To satisfy \eqref{Cond1}--\eqref{Cond3} it suffices to have \begin{equation}\label{Conditions}
2s-r+\frac{s}{r} < 0,
\qquad
-\frac12 +\frac{s}{r} < 0,
\qquad
2s-r-\frac{3s}{4r}+\frac{s}{r} < 0. \end{equation} Indeed, if these inequalities hold, then we first choose $\varepsilon > 0$ so small that the exponents of $N$ in \eqref{Cond1}--\eqref{Cond3} are also negative. Then the constants $C$ (hence also $S$) may grow, since they depend on $\varepsilon$. But by subsequently choosing $N$ large enough we can ensure that \eqref{Cond1}--\eqref{Cond3} hold.
The first inequality in \eqref{Conditions} is equivalent to $r^2-2sr-s > 0$, i.e., $r > s + \sqrt{s^2+s}$, which holds by assumption \eqref{sr_cond}. The last two inequalities in \eqref{Conditions} are weaker than the first one.
Thus, \eqref{AB_bound} holds for $n=1,\dots,M$, and the proof of Theorem \ref{Main_theorem} is complete.
\section{Proof of Lemmas}\label{Lemmas}
\subsection{Proof of Lemma \ref{Lemma2}}
This estimate is a variation on an estimate proved in \cite{Selberg:2006e}, and we use the same method of proof as in that paper. First, by Plancherel's theorem, the estimate is equivalent to \begin{equation}\label{I_def}
I = \norm{ \int_{\mathbb{R}^{1+1}} \frac{ F(\lambda,\eta)
G(\lambda-\tau,\eta-\xi)d\lambda \, d\eta}
{\angles{\xi}^{r}\angles{\eta}^{-r+2\varepsilon} \angles{A}^{b}
\angles{B_+}^{b}
\angles{C_-}^{1-b}}}_{L^2_{\tau,\xi}}
\le C \norm{F}_{L^2} \norm{G}_{L^2}, \end{equation} for arbitrary $F, G \in L^2(\mathbb{R}^{1+1})$, where $$
A=\abs{\tau}-\abs{\xi}, \qquad
B_+=\lambda+\eta, \qquad
C_-=(\lambda-\tau)-(\eta-\xi). $$
The null structure, i.e., the fact that there are two different signs, is crucial, since it allows us to use the following estimate, proved in \cite[Lemma 1]{Selberg:2006e}: \begin{equation}\label{algebraic}
\min \left( \fixedabs{\eta}, \fixedabs{\eta-\xi} \right)
\le \frac32 \max\left( \abs{A}, \abs{B_+}, \abs{C_-} \right). \end{equation} In fact, this estimate is the 1d version of \cite[Lemma 7]{Selberg:2006b}.
Thus, we can reduce to proving \eqref{I_def} with $I$ replaced by one of the following: \begin{align*}
I_1 &= \norm{ \int_{\mathbb{R}^{1+1}} \frac{ F(\lambda,\eta)
G(\lambda-\tau,\eta-\xi)d\lambda \, d\eta}
{\angles{\xi}^{r}\angles{\eta}^{1-b-r+2\varepsilon} \angles{A}^{\theta_1}
\angles{B_+}^{\theta_2}
\angles{C_-}^{\theta_3}}}_{L^2_{\tau,\xi}},
\\
I_2 &= \norm{ \int_{\mathbb{R}^{1+1}} \frac{ F(\lambda,\eta)
G(\lambda-\tau,\eta-\xi)d\lambda \, d\eta}
{\angles{\xi}^{r}\angles{\eta}^{-r+2\varepsilon} \angles{\eta-\xi}^{1-b} \angles{A}^{\theta_1}
\angles{B_+}^{\theta_2}
\angles{C_-}^{\theta_3}}}_{L^2_{\tau,\xi}}, \end{align*} where the $\theta_j$ are nonnegative and $\theta_1+\theta_2+\theta_3 > 1/2$. Then we can apply the following product law for the spaces $H^{s,b}$.
\begin{theorem}\label{H_products} \cite[Proposition 10]{Selberg:1999}. Suppose $s_1,s_2,s_3 \in \mathbb{R}$, \,$\theta_1, \theta_2, \theta_3 \ge 0$, and $$
\theta_1+\theta_2+\theta_3 > \frac12,
\qquad
s_1+s_2+s_3 > \frac12,
\qquad s_1+s_2 \ge 0,
\qquad s_1+s_3 \ge 0,
\qquad s_2+s_3 \ge 0. $$ Then (in one space dimension) $$
\norm{uv}_{H^{-s_1,-\theta_1}} \le C \norm{u}_{H^{s_2,\theta_2}} \norm{v}_{H^{s_3,\theta_3}}, $$ where $C$ depends on the $s_j$ and the $\,\theta_j$. \end{theorem}
\begin{remark} In \cite{Selberg:1999} only the case $s_1,s_2,s_3 \ge 0$ is proved, but the general case reduces to this by Leibniz' rule (the triangle inequality in Fourier space). \end{remark}
Applying this to $I_1$ and $I_2$, and recalling that $b=1/2+\varepsilon$, we see that it suffices to have $0 \le r \le 1/2+\varepsilon = b$ and $0 < \varepsilon \le 1/2$. This completes the proof of Lemma \ref{Lemma2}.
\subsection{Proof of Lemma \ref{Lemma7}} Proceeding as in the proof of Lemma \ref{Lemma2}, we reduce to proving \eqref{I_def}, but with $I$ replaced by one of the following: \begin{align*}
I_1 &= \norm{ \int_{\mathbb{R}^{1+1}} \frac{ F(\lambda,\eta)
G(\lambda-\tau,\eta-\xi)d\lambda \, d\eta}
{\angles{\xi}^{1-r}\angles{\eta}^{1-b-1/2+\varepsilon} \angles{A}^{\theta_1}
\angles{B_+}^{\theta_2}
\angles{C_-}^{\theta_3}}}_{L^2_{\tau,\xi}},
\\
I_2 &= \norm{ \int_{\mathbb{R}^{1+1}} \frac{ F(\lambda,\eta)
G(\lambda-\tau,\eta-\xi)d\lambda \, d\eta}
{\angles{\xi}^{1-r}\angles{\eta}^{-1/2+\varepsilon} \angles{\eta-\xi}^{1-b} \angles{A}^{\theta_1}
\angles{B_+}^{\theta_2}
\angles{C_-}^{\theta_3}}}_{L^2_{\tau,\xi}}, \end{align*} where the $\theta_j$ are nonnegative and $\theta_1+\theta_2+\theta_3 > 1/2$. Applying Theorem \ref{H_products} and recalling that $b=1/2+\varepsilon$ with $0 < \varepsilon \le 1/2$, we get the sufficent condition $r < 1/2$.
\subsection{Proof of Lemma \ref{Lemma6}} This reduces to \eqref{I_def} with $I$ replaced by $$
I = \norm{ \int_{\mathbb{R}^{1+1}} \frac{ F(\lambda,\eta)
G(\lambda-\tau,\eta-\xi)d\lambda \, d\eta}
{\angles{\xi}^{1-r}\angles{\eta}^{1-b-1/4+\varepsilon}\angles{\eta-\xi}^{-1/4+\varepsilon} \angles{A}^{\theta_1}
\angles{B_+}^{\theta_2}
\angles{C_-}^{\theta_3}}}_{L^2_{\tau,\xi}}, $$ where the $\theta_j$ are nonnegative and $\theta_1+\theta_2+\theta_3 > 1/2$. Since $b=1/2+\varepsilon$ with $0 < \varepsilon \le 1/2$, Theorem \ref{H_products} yields the sufficent condition $r \le 1/2$.
\subsection{Proof of Lemma \ref{Lemma8}}
Applying Lemma \ref{H_lemma}, H\"older's inequality in time, the Sobolev product law \eqref{Sobolev_product}, and the embedding \eqref{Basic_embedding}, we get for all $r < 1/2$, \begin{align*}
\norm{u}_{H^{r,b}(S_{\Delta T})}
&\le C (\Delta T)^{(1-b)/2}
\norm{\innerprod{\beta U}{V}}_{{L_t^2H^{r-1}(S_{\Delta T})}}
\\
&\le C (\Delta T)^{(1-b)/2} (\Delta T)^{1/2}
\norm{\innerprod{\beta U}{V}}_{{L_t^\infty H^{r-1}(S_{\Delta T})}}
\\
&\le C (\Delta T)^{(1-b)/2} (\Delta T)^{1/2}
\norm{U}_{{L_t^\infty L_x^2(S_{\Delta T})}}
\norm{V}_{{L_t^\infty L_x^2(S_{\Delta T})}}
\\
&\le C (\Delta T)^{3/4-\varepsilon} \norm{U}_{X_+^{0,b}(S_{\Delta T})} \norm{V}_{X_\pm^{0,b}(S_{\Delta T})}. \end{align*}
\subsection{Proof of Lemma \ref{Lemma5}}
It suffices to prove this for $u_0 = 0$, in view of Lemma \ref{X_lemma}. Then by Lemma \ref{X_lemma}, H\"older's inequality in time, the product law \eqref{Sobolev_product}, and the embedding \eqref{Basic_embedding}, \begin{align*}
\norm{u}_{X_+^{0,b}(S_{\Delta T})}
&\le C (\Delta T)^{1/2-\varepsilon} (\Delta T)^{1/2}
\norm{\Phi\beta U}_{L_t^\infty L_x^2(S_{\Delta T})}
\\
&\le C (\Delta T)^{1-\varepsilon} \norm{\Phi}_{H^{1/2+\varepsilon,b}(S_{\Delta T})} \norm{U}_{X_-^{0,b}(S_{\Delta T})}. \end{align*} On the other hand, by Lemmas \ref{X_lemma} and \ref{Lemma2}, $$
\norm{u}_{X_+^{0,b}(S_{\Delta T})}
\le C \norm{\Phi\beta U}_{X_+^{0,b-1}(S_{\Delta T})}
\le C \norm{\Phi}_{H^{2\varepsilon,b}(S_{\Delta T})} \norm{U}_{X_-^{0,b}(S_{\Delta T})}. $$ Interpolation gives, for $0 \le \theta \le 1$, $$
\norm{u}_{X_+^{0,b}(S_{\Delta T})}
\le C (\Delta T)^{\theta(1-\varepsilon)} \norm{\Phi}_{H^{(1-\theta)2\varepsilon+\theta(1/2+\varepsilon),b}(S_{\Delta T})} \norm{U}_{X_-^{0,b}(S_{\Delta T})}. $$ If $0 < r < 1/2$, then taking $\theta = 2r-6\varepsilon$ with $\varepsilon$ small enough, we get the desired result, since then $(1-\theta)2\varepsilon+\theta(1/2+\varepsilon) \le \theta/2 + 3\varepsilon = r$, and $\theta(1-\varepsilon) \ge 2r-7\varepsilon$.
\end{document} | arXiv |
\begin{document}
\draft
\title{Formal context for cryptographic models}
\author{John M. Myers}
\address{Gordon McKay Laboratory, Division of Engineering and Applied Sciences,\\ Harvard University, Cambridge, MA 02138\\[-12pt]}
\author{and\\[-12pt]}
\author{F. Hadi Madjid}
\address{82 Powers Road, Concord, MA 01742}
\maketitle
\vspace*{-.5\baselineskip} \begin{abstract}\indent To clarify what is involved in linking models to instruments, we adapt quantum mechanics to define models that display explicitly the points at which they can be linked to statistics of results of the use of instruments. Extending an earlier proof that linking models to instruments takes guesswork, we show: Any model of cryptographic instruments can be {\em enveloped}, nonuniquely, by another model that expresses conditions of instruments that must be met if the first model is to fit a set of measured outcomes. As a result, model $\alpha$ of key distribution can be enveloped in various ways to reveal alternative models that Eve can try to implement, in conflict with model $\alpha$ and its promise of security. A different enveloping model can help Alice and Bob by expressing necessities of synchronization that they manipulate to improve their detection of eavesdropping. Finally we show that models based on pre-quantum physics are also open to envelopment. \end{abstract}
\vspace*{\baselineskip} \pacs{PACS numbers: 03.67.Dd, 03.65.Bz, 89.70.+c}
\narrowtext \section{Introduction}
A designer Diana and the users Alice and Bob of cryptographic transmitting and receiving instruments, as well as the eavesdropper Eve, all employ various equations to model how results of the use of these instruments depend on what the participants do. Between a model as a set of equations and instruments made of glass and silicon there is a great divide. In choosing a model to analyze instruments or to be employed in a feedback loop where the model helps to operate instruments, one makes a link across this divide. While one can interpret measured results as refuting some candidate models, we recently proved that neither they nor logic can uniquely determine a quantum model: linking a model to instruments requires something beyond logic and measured data, something well named by the word {\em guess} \cite{ams}.
Proofs of the security of quantum key distribution invoke inner products of quantum state vectors, and these depend on the model chosen. Here we prove that any given set of outcomes from a transmitter and receiver used to distribute a key can be fitted by many quantum-mechanical models which differ greatly among themselves in their inner products and hence in their implications for the security of a key. On one hand, this encourages Eve to invent snooping instruments even though she knows Alice and Bob have a proof of security, and, on the other hand, our findings encourage discovery and repair of ``hidden security loopholes'' \cite{lo} arising because their transmitting and receiving instruments ``violate ... assumptions [that underlie their model] in ways not immediately apparent to Alice and Bob'' \cite{slutskyPRA}.
To underpin an examination of the linking of models to instruments, in Section II we adapt quantum mechanics to define models that display explicitly the points at which they can be linked to statistics of results of the use of instruments. The models to be introduced express ``what the participants do'' in terms of {\em commands} sent to the instruments {\em via} Classical, digital Process-control Computers (CPC's) that control them and that also record results from them; we call these CPC-oriented models. Section III extends an earlier proof that linking CPC-oriented models to instruments takes guesswork: For any quantum-mechanical model of transmitting and receiving instruments there is another model (not unique) that expresses constraints in using the instruments that must be met if the first model is to fit a set of measured outcomes. We say the first model is {\em enveloped} by the second.
In Section IV we prove that for any quantum model $\alpha$ of key distribution, there exists an enveloping model $\beta$ that matches $\alpha$ with respect to measurements contemplated in $\alpha$ but that has smaller inner products and allows for other measurements, which, if Eve can implement them, allow undetected eavesdropping, in conflict with model $\alpha$ and its promise of security. For this reason, no proof can relieve Alice and Bob of the burden of making judgments about what models to link to their instruments, something implicit in \cite{lo,slutskyPRA}, but here made vivid.
They can, however, put the same burden of judgment on Eve, for she too must use models. In Section V model $\alpha$ is enveloped by another model that expresses necessities of synchronization that Alice and Bob can manipulate to improve their detection of eavesdropping. In Section VI we indicate how models based on pre-quantum physics are also open to envelopment.
In summary, we find that instruments modeled are used in a context of circumstances and intentions which no model can fully describe. In creating an enveloping model, one formally expresses (rightly or wrongly) some hitherto unexpressed feature of this context. As will be proved, there is no end of opportunities to assert features of context, because any enveloping model can in turn be enveloped.
\section{Linking instruments to models}
The central issue is the linking of uses of instruments to models. By {\em model} we mean a set of equations written in mathematical language, primarily quantum mechanics, with the intent of predicting statistics of the results of using instruments (such as transmitters and receivers made of silicon and glass fibers {\em etc.}). Some of the equations of the set act as a set of assumptions from which the rest of the equations can be derived. Quantum mechanics provides a mathematical language in which to write down a wide variety of models, constrained by a grammar of logical constraints, so within a model conclusions can be proved to follow from assumptions. Because different sets of assumptions generate different quantum-mechanical models, quantum mechanics is a language, as distinct from a particular model written in that language; it has more room in it for diverse models that accord with any given set of experimental results than has been appreciated.
Although, as we shall see, instruments cannot be discussed independently of models, we separate them as best we can by the trick supposing the instruments are operated via digital computers. This will allow us to express ``how the instruments are used'' in terms of {\em commands} sent to the instruments by a Classical, digital Process-control Computer (CPC) that controls them and that also records results from them \cite{ams}. Instruments swallow commands and give back recordable results.
As discussed in \cite{ams}, parsing a stream of data from instruments into a sequence of measurement occurrences, each with a quantum-mechanical {\em outcome}, cannot be determined from the data alone, but takes extra hypotheses, indeed a kind of stripped-down model, determined in part by guesswork, which we call a {\em parsing rule}. Parsing requires guesswork, both to assert the statistical independence of one segment of data from another, and to select criteria by which to weed out artifacts in the data attributed to instrumental imperfections, such as false and missed detections. Using a parsing rule, one parses a stream of data from the instruments into a sequence of measurement occurrences, assumed statistically independent, and one formats the data segment for each occurrence into (1) how the instruments were configured and (2) an outcome from the instruments. The parsing rule makes no statement about values of probabilities of outcomes, but does assert that such values exist; it provides a range of outcomes that are possible to record as well as set of possible command sequences. In this way it limits the models that can be tested by measured data that it parses.
To view the linking of instruments to models we postulate an analytic frame in which (a) instruments write via some parsing rule what we and other scientists interpret as numerical outcomes, (b) they write these numbers in memories of CPC's, and (c) CPC's send commands to some or all of the instruments. We view each set-up of instruments in terms of records in CPC memories of commands sent to the instruments by CPC's and of outcomes from the instruments. Notice that we make no assumption that an instrument works as the manufacturer says it does, nor that it works the way any model compatible with the parsing rule says it does, nor that it functions statistically the same on Tuesday as it does on Monday. While such assumptions are made in the models to be discussed, the analytic frame provides room to consider cases in which the instruments write numbers that conflict with any or all models.
We define a CPC-oriented model of a set-up of instruments to be a model that expresses conditional probabilities of outcomes given commands to the instruments. For instance, a model $\alpha$ to be introduced in Section IV will express a conditional probability of outcome $j$ given a command $b_A$ from Alice's CPC to her transmitter and a command $b_E$ from Eve's CPC to her eavesdropping receiver, written $\Pr_\alpha(j|b_A,b_E)$. The subscript marks it as an assertion within model $\alpha$, leaving room to consider a different model $\beta$ that asserts a different numerical value
$\Pr_\beta(j|b_A,b_E) \neq \Pr_\alpha(j|b_A,b_E)$.\footnote{This extends our earlier analysis of instruments controlled by a single CPC programmed according to some model \cite{ams,spie} to deal with a setup of instruments controlled by several CPC's.}
The same CPC's that control the instruments house in their memories CPC-oriented models and programs designed using them. These models and model-derived programs are used off-line to simulate the instruments; they are used on-line, not to simulate the instruments, but to help operate them, for example in a feedback loop of Bob's receiver, as discussed in Section~V. By considering both the instruments and the models as they are reflected in files of a CPC, we conceptually separate (as well as possible) these CPC-oriented models from the instruments modeled while allowing for interaction between models and instruments.
Like any set of equations, a CPC-oriented model can be copied, so copies of the same model can be used concurrently in different places for the same or different purposes. What can be done with a model or a program depends on where it is, for example on whether it is written in Alice's CPC or in Eve's. Because the models used in programming one CPC need not be the same as those used in programming another, several CPC's controlling interacting instruments can work from different models concurrently. Where and when and how a CPC-oriented model is used is traceable in the execution sequences of the CPC's in which copies of the model are housed, so the CPC frame allows analysis of various of CPC-oriented models and model-derived programs used to operate instruments that interact. Some or all of the instruments can be modeled by more than one model, and one model can conflict with another. Some models model other models: a component of Eve's model can be her model $\alpha'$ of Alice and Bob's model $\alpha$; this tells her (rightly or wrongly) how Alice and Bob, using their model $\alpha$, will decide on their security, distinct from how Eve decides using her model $\beta$. Conversely, Alice and Bob's model $\alpha$ contains as a component a model $\beta'$, their model of Eve's model $\beta$. There is no necessary stopping place in modeling models.
If a model $\alpha$ invokes all the assumptions of a model $\beta$ and possibly more, we say model $\alpha$ {\em specializes} model $\beta$, or that model $\beta$ {\em generalizes} model $\alpha$ (meaning it has fewer assumptions). This is the first of several types of relations among models that will be used to express interactions between the invention and the modeling of transmitting and receiving instruments used in cryptographic key distribution.
\section{Models of communication}
Ignoring eavesdropping for the moment, we focus on Alice communicating to Bob, as described quantum mechanically. Consider Alice transmitting $m$ quantum bits of raw data to Bob, with Alice using one CPC to control her transmitter and Bob using another to control his receiver. They want to jointly implement a CPC-oriented model $\gamma$ of quantum communication, which says at each of a sequence of $m$ occurrences, Alice causes her CPC to command the preparation of a state vector, choosing ${\bf u}$ for 0 or ${\bf v}$ for 1 at random. Her CPC records her choices of 0 or 1. Bob's receiver has, say, light detectors, one interpreted in model $\gamma$ as detecting ${\bf u}$ to indicate Alice's choice of 0 and another detecting ${\bf v}$ to indicate Alice's transmission of 1. Bob's CPC records the decisions of his receiver, described as deciding on 0 or 1 or, if neither detector fires, `inconclusive' \cite{B92,ekert94}.
Model $\gamma$ is composed of: \begin{enumerate} \item a set $A_\gamma$ of command strings that Alice's CPC can send to generate states, here just the set $\{0,1\}$; a set $B_\gamma$ of commands that Bob's CPC can send to his receiver, which in this case is empty; \item a Hilbert space ${\cal H}_\gamma = {\bf C}^2$ ({\em i.e.}, the vector space of complex dimension 2);
\item a function for states as functions of commands (here only Alice's commands), $|v_\gamma \rangle : A_\gamma \rightarrow
{\cal H}_\gamma$ such that $|v_\gamma(0) \rangle = {\bf u}$ and
$|v_\gamma(1) \rangle = {\bf v}$; \item a set of possible outcomes of Bob's measurement, indexed by $j$
ranging over natural numbers or some subset of natural numbers, here 0 for $|v_\gamma(0) \rangle$, 1 for $|v_\gamma(1) \rangle $, and 2 for `inconclusive'; \item a function from Bob's commands to positive operator valued measures (POVM's) on ${\cal H}_\gamma$, here simplified to the single POVM $M_\gamma$ consisting of a set of detection operators $M_\gamma(j)$ with \begin{enumerate} \item[] $\sum_j M_\gamma(j) = {\bf 1}$, \item[] $(\forall j) M_\gamma(j) \geq 0$ and $M_\gamma(j) = M_\gamma(j)^{\dag}$. \end{enumerate} \end{enumerate} Model $\gamma$ asserts the probability of outcome $j$ given a command $b_A \in A_\gamma$ for state preparation to be
\begin{equation} \mbox{$\Pr_\alpha(j|b_A)$} = \langle v_\alpha(b_A) |
M_\gamma(j)| v(b_A) \rangle. \end{equation}
In relating model $\gamma$ to results in his CPC, Bob thinks of his CPC as recording detection results of Alice's $m$-bit transmission in a sequence of $m$ memory segments, each of which can hold two bits, coded 00 for Alice's `0', 01 for Alice's `1', and 10 for `inconclusive'. We will refer to these two-bit memory segments in connection with timing, to which we now turn.
\subsection{Need for synchronization}
Model $\gamma$ is an armchair view of Bob's receiver that lacks the detail necessary to design it. To design a receiver that works according to model $\gamma$, Diana must provide for synchronizing it to Alice's transmitter within some allowed leeway. For this Diana envelops model $\gamma$ with a more detailed model $\delta$ that expresses the conditions of synchronization that must be maintained between Alice's transmitter and Bob's receiver if model $\gamma$ is to accord the records of Alice's commands and Bob's detections. Diana provides for Bob's receiver to meet these conditions by adjusting the rate of Bob's clock in response to measured results interpreted in model $\delta$. She designs this feedback loop by choosing a classical-control model $\epsilon$, to be discussed shortly. Without the gaps in synchronization defined by model $\delta$ and their containment within the allowed leeway in accord with model $\epsilon$, Bob's CPC, driven by its clock, would mistime its routing of a detector signal to the $k$-th memory segment, resulting in an erroneous record.\footnote{Although for a short transmission line of a fixed delay, Bob and Alice can use the same clock to drive their CPC's synchronously, but for variable delay, {\em e.g.} if Bob is in motion, he needs his own clock, independently adjustable \cite{hj84}. And even where the single-clock design works, Bob's receiver must adjust its phase.}
To express the effect on reception of the drift of the clock of Bob's CPC relative to Alice's clock, Diana (having learned from Einstein) defines synchrony in terms of measurements made of Alice's signal arriving at Bob's clock; however, in using a quantum model of the measurement of that signal, she must cope with quantum indeterminacy, which limits Bob's receiver's knowledge of arrival times to what it can deduce via Bayes rule from outcomes. To produce suitable outcomes, Diana invents a model $\delta$, which has a Hilbert space ${\cal H}_\delta$, of dimension higher than that of
${\cal H}_\gamma$, along with states $|v_\delta(b_A,s) \rangle$ that are functions not only of Alice's commands in $A_\delta = A_\gamma$, but also of a {\em skew} $s$ of Bob's clock relative to an imagined ideally synchronized clock. Diana designs Bob's receiver to measure the $k$-th signal from Alice when the clock of Bob's CPC reads $t_k$. When we imagine Bob's clock reads $t_k$ as the ideal clock reads $t_k - s_k$, we say Bob's clock is fast by a skew $s_k$. The state measured by Bob's receiver when his clock reads $t_k$ is then
$|v_\delta(b_A,s_k)\rangle = U_\delta (-s_k)|v_\delta(b_A,0) \rangle$, where $U_\delta$ is a unitary-operator-valued function of $s_k$ by which Diana expresses skew.
In order to allow different possible outcomes for different values of the skew $s_k$ at the reception of the $k$-th of the $m$ signals from Alice, model $\delta$ must assume more possible outcomes than the 0, 1, and `inconclusive' of model $\gamma$, so the POVM $M_\delta$ has more than the three detection operators of $M_\gamma$. When restricted to skews $s_k$ of magnitude smaller than some allowed bound $s_0$, model $\delta$ projects onto model $\gamma$ as follows:
\begin{enumerate} \item $|v_\delta(b,s_k)\rangle \mapsto |v_\gamma(b) \rangle$, and \item the detection operators of $M_\delta$ partition into three sets, such that the sum of operators for each set maps to a single detection operator $M_\gamma(j)$ with respect to probabilities of detection of $j$. \end{enumerate} But outcomes in model $\delta$ tell more than these projections. At each signal reception, Bob's receiver records in his CPC not only a decision among 0, 1, and `inconclusive' but also finer distinction from which his CPC estimates its clock skew (via Bayes rule and a prior probability distribution that Diana assumes for skew). In order to record the outcomes that help estimate skew and guide clock-rate adjustment, Bob's receiver, designed using model $\delta$, needs a memory segment for the $k$-th reception of more than two bits. Hence, the record previously discussed in connection with model $\gamma$ is extracted from a larger record required by model $\delta$.\footnote{Must there exist a quantum mechanical model that accords with experimental results of measurements of a skew-dependent state? Yes, because, any digital record can be interpreted (nonuniquely) as a record of quantum outcomes, and for any set of outcomes with their relative frequencies as functions of commands, many quantum mechanical models have probabilities that exactly fit \cite{ams}.}
To contain skews within the tolerable bound $s_0$, Diana chooses a classical model $\epsilon$ by which to design a program that, when executed by Bob's CPC, responds to estimated skews by sending a command from the CPC to set a `faster-slower' lever on the clock that drives that CPC; the command is a value of a {\em control function} $F_\epsilon$ that takes as its argument a computer file consisting of skews calculated from recently recorded detection results and recently issued commands to the clock itself. Although the quantum state to be controlled has a history that Bob's CPC can only estimate via Bayes rule from outcomes, the design of a control function $F_\epsilon$ is within the discipline of classical feedback design.\footnote{For discussion of Bayes rule in a non-quantum context of control, see \cite{jazwinski}.} If model $\delta$ is implemented and if model $\epsilon$ succeeds in generating steering commands that are adequate, the skews are held within the bound $\pm s_0$ so that Bob's detection results fulfill the intention of model $\gamma$ and, additionally, allow his CPC to make skew estimates necessary to guide clock adjustment.
\vskip\abovedisplayskip \noindent{\bf Remark 1}: Models, such as $\gamma$ and $\delta$, express desires and obstacles more flexibly than do {\em inputs} used for this purpose in control theory \cite{jazwinski,stefani}. Alice expresses what she wants by choosing model $\gamma$ altogether, not just by an `input' of 0 or 1 to her transmitter. Because Diana wants Alice and Bob's instruments to work in accord with Alice's model $\gamma$, in spite of the obstacle of clock drift, she chooses models $\delta$ and its classical companion, model $\epsilon$.
\vskip\abovedisplayskip \noindent{\bf Remark 2}: The number of bits that arrive at Bob's receiver is model-dependent: whether a detection result for a signal is seen as two bits (ignoring skew) or as more bits (allowing for skew) depends on whether the record of the signal detection is interpreted using model $\gamma$ or model $\delta$.
\vskip\abovedisplayskip Recall the freedom always present in quantum mechanical modeling to shift the boundary between the `system' modeled and the measuring instrument, for instance by counting more of the measuring instrument as part of the system \cite{peres}. In view of this freedom, we conclude:
\vskip\abovedisplayskip \noindent{\bf Remark 3}: Every quantum mechanical model is contingent in the sense that it is projected onto by a restriction of an enveloping model that shows other possibilities.
\section{Models of vulnerability to eavesdropping}
Widely discussed quantum-mechanical models of key distribution assert a nonzero inner product between quantum state vectors that Alice communicates to Bob, with the consequence that eavesdropping almost always leaves tracks in the form of errors that Bob and Alice can detect. If Alice's transmitter, Bob's receiver, and Eve's snooping instruments can be counted on to work in accord with any of these models, then Alice can send Bob a key secure against undetected eavesdropping. The models can all be translated into CPC-oriented models to make visible the points at which they can be linked to results of the use of instruments, and it is to the credit of some of these models that relative frequencies of experimental results accord reasonably well with conditional probabilities of outcomes derived from the states and operators posited by the models. But we are sloppy if we forget that quantum states are terms in models, rather than model-independent features of instruments.
In linking a CPC-oriented model $\alpha$ to instruments, one identifies commands in model $\alpha$ with commands sent from CPC's to the instruments, for example commands $b_A$, $b_B$, and $b_E$ from CPC's controlled by Alice, Bob, and Eve, respectively; one also parses results of the use of instruments in response to commands as quantum-mechanical outcomes, so that one can compare relative frequencies of these results to the conditional probabilities asserted by model $\alpha$, e.g. $\Pr_\alpha(j|b_A,b_B,b_E)$ as the conditional probability of a quantum outcome $j$ given commands $b_A$, $b_B$, and $b_E$. (The outcome $j$ can be seen as several fragments, for example one for Bob and one for Eve, allowing for analysis of mutual information between Eve and Bob, etc.) It is to be noticed that this procedure sets up a divide that runs through the CPC between state vectors as terms in models, on one side, and on the other side the commands to and results from instruments. A large part of the story told here amounts to noticing this divide.
Given a CPC-oriented model $\alpha$ of quantum key distribution that shows Alice and Bob to be secure against eavesdropping, one can envelop model $\alpha$ in a model $\beta$ that introduces a range of conditions; under some conditions model $\beta$ projects to model $\alpha$, agreeing with it, while under other conditions model $\beta$ leads to drastically different conclusions in conflict with those of model $\alpha$. Among these are conditions under which Eve can learn the key without leaving tracks that Alice and Bob can detect. This envelopment is possible because model $\beta$ can invoke states and their inner products that differ from those of model $\alpha$ while still agreeing with model $\alpha$ with respect to probabilities of outcomes for commands considered in model $\alpha$.
For example, we envelop model $\alpha$ with a model $\beta$ expressing conditions in which Alice's transmitter leaks light into a channel accessible to Eve, but that is unknown to Alice and Bob (and is not expressed in model $\alpha$ \cite{slutskyPRA}). There are two cases to consider, corresponding to two types of models. Deferring models of Eve's use of a probe, we start with the simpler case of a model that segments the transmission of signals from Alice to Bob into (1)
Alice's transmission to Eve, followed by (2) Eve's transmission to Bob. For such segmented transmission, suppose model $\alpha$ assumes that (1) Alice chooses commands from a set $A_\alpha = \{0,1\}$, with command $b_A$ generating a state vector $|v_\alpha(b_A) \rangle \in {\cal H}_\alpha$, and (2) Eve commands her listening instruments with a command $b_E \in E_\alpha$ to make a measurement expressed by a POVM $M_\alpha(b_E)$ which has a detection operator $M_\alpha(b_E;j_E)$ acting on ${\cal H}_\alpha$, associated with outcome $j_E$. Model $\alpha$ implies that the conditional probability of Eve obtaining the outcome $j_E$ given her command $b_E$ and Alice's command $b_A$ is
\begin{equation} \mbox{$\Pr_\alpha(j_E|b_A,b_E)$} =
\langle v_\alpha(b_A) | M_\alpha(b_E;j_E)| v_\alpha(b_A) \rangle. \label{eq:p_alpha}\end{equation}
\begin{prop} Given any such (segmented) model $\alpha$ with inner pro\-duct $\langle v_\alpha(0)|v_\alpha(1) \rangle$ and given any $0 \leq r < 1$, there is a model $\beta$ that gives the same conditional probabilities of Eve's outcomes for all her commands belonging to $E_\alpha$, so
\begin{equation} (\forall b_A \in A_\alpha, b_E \in E_\alpha) \mbox{$\Pr_\beta(j_E|b_A,b_E)$} =
\mbox{$\Pr_\alpha(j_E|b_A,b_E)$} \label{eq:same} \end{equation} while
\begin{equation} |\langle v_\beta(0)|v_\beta(1)\rangle | = r
|\langle v_\alpha(0)|v_\alpha(1)\rangle | . \label{eq:inner} \end{equation} \end{prop}
\noindent {\em Proof}: Motivated by the idea that, unknown to Alice, her transmitter signal might generate an additional ``leakage'' into an unintended spurious channel that Eve reads, we construct the following enveloping model $\beta$ which assumes: \begin{enumerate} \item the same set of commands for Alice, so $A_\beta = A_\alpha$, \item a larger Hilbert space ${\cal H}_\beta =
{\cal H}_{\rm leak} \otimes {\cal H}_\alpha$ in which Alice produces vectors $|v_\beta(b_A) \rangle = |w_\beta(b_A) \rangle \otimes
|v_\alpha(b_A) \rangle$, with $|w_\beta(b_A) \rangle \in {\cal H}_{\rm leak}$; \item a larger set of commands for Eve, $E_\beta = E_\alpha \sqcup E_{\rm extra}$ (disjoint union); \item a POVM-valued function of Eve's commands to her measuring instruments, with detection operators \begin{equation} M_\beta(b_E;j_E) = \left\{ \begin{array}{l} {\bf 1}_{\rm leak} \otimes M_\alpha(b_E;j_E) \mbox{ for all } b_E \in E_\alpha,\\
\mbox{Eve's choice of POVM to distinguish } |v_\beta(0) \rangle\\
\mbox{from } |v_\beta(1) \rangle \mbox{ if } b_E \in E_{\rm extra}. \end{array} \right. \end{equation} \end{enumerate} According to model $\beta$, if Eve chooses any measurement command of
$E_\alpha$, Eq.\ (\ref{eq:p_alpha}) holds. But model $\beta$ speaks not of the vectors $|v_\alpha(b_A) \rangle$ but of other vectors having an inner product of magnitude
\begin{equation} |\langle v_\beta(0)|v_\beta(1)
\rangle | = |\langle w(0)|w(1) \rangle |
|\langle v_\alpha(0)|v_\alpha(1) \rangle|. \label{eq:winner} \end{equation}
The unit vectors $|w(0) \rangle$ and $|w(1) \rangle$ can be specified at will, so that the factor $r
\stackrel{\rm def}{=} |\langle w(0) | w(1) \rangle |$ can be chosen to be as small as one pleases. $\Box$
If she can find and gain access to a channel carrying leakage states, Eve implements a model $\beta$ with a value of $r < 1$, in which case she uses an optimal POVM to distinguish Alice's 1's and 0's, with fewer `inconclusives' than Alice and Bob think possible, and hence with less impact on Bob's error rate. If Eve can do this, she has more information about the key for a given rate of Bob's errors than Alice and Bob found possible when they bet on model $\alpha$, thus vitiating Alice and Bob's attempt to distribute a key secure against undetected eavesdropping.
Whether Eve can implement a measurement of leakage as called for in model $\beta$ with $r < 1$ is unanswerable by modeling; it is a question that requires work on ``the other side of the divide.'' The point to be stressed is that the agreement between model $\alpha$ and a set of measured results, no matter what results, is no logical guarantee against Eve implementing model $\beta$ with a value of $r$ less than 1, or even a value of 0 which would give her the whole key while causing no errors for Alice and Bob to detect.
\subsection{Models involving a defense function}
When noise in communications channels is recognized, privacy amplification is necessary to distill a secure key \cite{bennett95}. Arguments for the security of quantum key distribution with noisy channels, summarized and refined in Refs.\ \cite{slutskyPRA,slutskyAO,brandt,brandt2}, center on a {\em defense function}. The existence of a defense function depends on a proof (within some model) of a relation between Eve's maximum Renyi information on whatever bits she directly or indirectly interrogates and a positive contribution to Bob's error rate in receiving bits.
Defense functions have been analyzed for models of Eve's use of a probe \cite{fuchs} and without restricting Alice's transmission to a choice of only two state vectors. In such a model $\alpha$, Alice chooses one of several state vectors in one Hilbert space ${\cal H}_{{\rm sig},\alpha}$ while Eve generates a fixed vector in a different Hilbert space ${\cal H}_{{\rm probe},\alpha}$, and the tensor product of Alice's choice of state vector and Eve's fixed probe vector evolves unitarily in an interaction, after which Eve and Bob make measurements, Eve confined to the probe sector and Bob to the signal sector. Like segmented models, probe models relate Eve's information to Bob's error rate in such a way that Bob's error rate depends on inner products ascribed to the state vectors among which Alice chooses; in particular if the inner products for distinct signal vectors are all zero, Eve can learn everything without causing any effect that Alice and Bob can detect.
The Appendix displays consequences of leakage of Alice's transmission for models involving Eve's use of a probe: just as for models that segment the transmission, the state vectors used to model Alice's transmission are model-dependent, and so are their inner products. To see the consequence for defense functions, suppose that Alice and Bob use model $\alpha$ which assumes that Alice chooses between state vectors $|v_\alpha(0) \rangle$ and $|v_\alpha(1) \rangle$ with inner product having a magnitude $S_\alpha = |\langle v_\alpha(1)|v_\alpha(0) \rangle |$. Assuming model $\alpha$, Alice and Bob determine a defense function $t(n,e_T)$, as discussed in \cite{slutskyAO}; in order to mark its dependence on model $\alpha$ and especially its dependence on the inner product of $S_\alpha$, we write this as $t_\alpha(n,e_T,S_\alpha)$. For any such model $\alpha$ and whatever the measured results with which it accords, we can show a model $\beta$ that agrees with model $\alpha$ insofar as these results are concerned, but disagrees with it about predictions of the detectability of Eve's eavesdropping, because in place of the inner product(s) of model $\alpha$, model $\beta$ has inner product(s) smaller by our choice of $r$, for any $0 \leq r < 1$. \begin{prop} If a model $\alpha$ asserts that Alice and Bob can distill a key that is secure against measurements commanded by Eve from a set of commands $E_\alpha$, then there exists another model $\beta$ that matches the predictions of model $\alpha$ for the commands in $E_\alpha$ but that makes additional commands available to Eve that make the key insecure. \end{prop}
To prove this, one uses Proposition 3 of the Appendix that envelops any model $\alpha$ with a model $\beta$ in which $S_\beta = r S_\alpha$ with $r$ as small as one pleases. The effect of making $S_\beta$ smaller than $S_\alpha$ is visible for the case of B92 models \cite{B92} in Figure 4 of \cite{slutskyAO}, where $S_\beta$ is denoted (in notation with which our notation regrettably clashes) by $\sin 2\alpha$. One sees that as $S_\beta$ gets smaller, $t_\beta$ gets bigger, so that at any fixed error rate, one can determine an $r$ for which model $\beta$ allows no distilled secure key. For the BB84 model as discussed in \cite{slutskyAO}, the effect of $r \ll 1$ in an enveloping model $\beta$ is to conflict with the BB84 model in such a way as to increase $t$ and allow undetected eavesdropping.
Thus, just as for segmented eavesdropping discussed above, Eve can try to implement a model $\beta$ which drastically increases what she can learn for a given error rate. Again, whether she can succeed in implementing such a model is another question, on the other side of the divide that runs through the CPC's between models and instruments. No matter what measured results they stand on, Alice and Bob always face a choice between a model $\alpha$ and an enveloping model $\beta$ that challenges the security asserted by model $\alpha$. Because both models make identical predictions about probabilities that connect with the measured data, Alice and Bob face a choice that no combination of logic and their fixed set of measured results can decide. They must make a judgment, or, to put it baldly, they must make a {\em guess} and act on it \cite{ams}.
\section{Modulation of clock rate to improve security}
While Alice and Bob may view their need for guesswork and judgment as bad news, they can put this need to good use if their system designer Diana recognizes that Eve is in the same boat: she too must act on guesswork. Recognizing this, Diana can design a key distribution system with features that make it harder for Eve to snoop.
As discussed in Section III, to accord with model $\alpha$, any receiver whether Bob's or Eve's, must maintain close synchrony with Alice's transmission in order to function. In both the segmented and the probe cases discussed above, the models $\alpha$ and $\beta$ can accord with measured results only if Bob's and Eve's receivers work in accord with enveloping models similar to model $\delta$ that expresses clock skew contained within an allowed leeway. Recall that model $\delta$ describes a receiver as parsing its results for each of Alice's bits into two parts, one indicating `0', `1', or `inconclusive', the other indicating skew to be contained by adjusting the faster-slower lever of Bob's clock, and, like Bob's receiver, Eve's must do this to make eavesdropping measurements at times that work.\footnote{In the segmented case, Bob, unaware, synchronizes his receiver to Eve's re-transmission, even though he supposes he is synchronizing with Alice's transmission.}
We suggest that Diana try to design Alice's transmitter and Bob's receiver to make the parsing by Eve's receiver impossible without use of prior information that Alice has also encoded, and that Bob has better access to than does Eve. The idea is for Alice's transmitter to be timed by a clock whose rate is intentionally randomly varied rapidly and over a wide range, and for Alice to encrypt indications of coming rate variations in her transmission to Bob. The eavesdropping problem is different (and harder) for these rate variations than for the key because they are more perishable. Quantum-mechanical models assert that the operation of the faster-slower lever on Eve's receiver cannot be corrected {\em ex post}; that is, if she intercepts Alice's signal and records it using a receiver clock unsynchronized to Alice's transmission, there is no way to reconstruct from her record what she would have received with a synchronized clock.
\section{Generalization}
Extending the proof in \cite{ams} that guesswork is necessary to the linking of quantum models to results of instruments, we have introduced the concept of enveloping models to prove that for any quantum-mechanical model $\alpha$ of key distribution there exists an enveloping model
$\beta$ that agrees with $\alpha$ for commands dealt with by $\alpha$, but encompasses other possibilities, and leads to conclusions about security that conflict with those implied by model $\alpha$. By drawing on the quantum-mechanical separation of states and outcomes, this proof used more than was really necessary. All that is necessary is the separation made in quantum mechanics between what happens at an occurrence of a measure, {\em an} outcome, and what might have happened, expressed as the {\em set of possible outcomes}. This separation is found not only in quantum mechanics, but in any statistical theory, and in particular in the usual electrical engineering of ``non-quantum'' systems, based on Maxwell's electromagnetics to which one adjoins ideas of noise or the generation of random signals. In this non-quantum framework, the statistical outlook alone allows one to introduce CPC's as a medium in which to see a divide between models and instruments within the CPC's that manage both. Doing this, one puts the statements of a model in the form $\Pr_\alpha(j|b_A,b_E)$. Then Propositions 1 and 2 can be proved without resort to quantum mechanics, so again the issues considered above arise: (a) what else might Eve measure that a model used by Alice and Bob has failed to account for, and (b) how might clock pumping help Alice and Bob? Thus the uncloseable possibility of enveloping any model $\alpha$ by another model $\beta$ that expresses extra conditions of the use of instruments is no peculiarity of using quantum rather than non-quantum models; it is endemic to any cryptographic modeling that invokes probabilities.
\acknowledgments We thank Howard E. Brandt for reading an early draft and giving us an astute critique, indispensable to this paper. The situations described in which Diana, Alice, Bob, and Eve take part are what Wittgenstein called {\em language games} \cite{witt1}, with the language being the quantum mechanics of CPC-oriented models.
\vfil \eject \appendix \section{Leakage channel in case Eve uses a probe}
This appendix proves for the case of Eve using a probe the analog of Proposition 1: if one model with its inner products predicts a set of probabilities for outcomes, so does another model having smaller inner products. (Thus, as in the segmented case, inner products depend on a choice of model undetermined by measured data.) The proof here makes no requirement that Alice choose among only two state vectors; she can choose from a set of any size. Transposing to the CPC-context a model expressing Eve's use of a probe \cite{slutskyPRA}, one obtains a model $\alpha$ that assumes: \begin{enumerate} \item a set of Alice's commands $A_\alpha$; \item a Hilbert space ${\cal H}_{{\rm sig},\alpha}$ for Alice's signals and a disjoint Hilbert space ${\cal H}_{{\rm probe},\alpha}$ for Eve's probe;
\item a function assigning Alice's states to commands (here only Alice's commands), $|v_\alpha \rangle : A_\alpha \rightarrow {\cal H}_{{\rm sig},\alpha}$;
\item a fixed starting state for Eve's probe of $|e_\alpha \rangle \in {\cal H}_{{\rm probe},\alpha}$; \item a set $E_\alpha$ of Eve's possible commands to her measuring instruments; \item unitary operators $U_\alpha(b_E)$ for $b_E \in E_\alpha$ acting on the product Hilbert space ${\cal H}_{{\rm probe},\alpha} \otimes {\cal H}_{{\rm sig},\alpha}$ for the interaction of Eve's probe with Alice's signal; \item a set $O_E$ of possible outcomes of Eve's measurements, indexed by $j_E$; \item for each of Eve's commands $b_E \in E_\alpha$, a POVM $M_{E,\alpha}(b_E)$ with detection operators $M_{E,\alpha}(b_E;j_E)$ that act on ${\cal H}_{{\rm probe},\alpha}$; \item a POVM for Bob's receiver acting on ${\cal H}_{{\rm sig},\alpha}$, with possible outcomes indexed by $j_B$ and detection operators $M_B(j_B)$.
\end{enumerate} This produces a quantum mechanical model $\alpha$ in which
\begin{eqnarray} \lefteqn{\mbox{$\Pr_\alpha$}(j_B,j_E|b_A,b_E) }\quad\nonumber \\ &=&
\langle v_\alpha (b_A)|\langle e_\alpha | U_\alpha^{\dagger}(b_E) [M_B(j_B)\otimes M_E(b_E;j_E)] U_\alpha(b_E)
|e_\alpha |\rangle |v_\alpha(b_A) \rangle. \end{eqnarray}
\begin{prop} Given any such (probe) model $\alpha$ with inner products
$\langle v_\alpha(b_A)|v_\alpha(b'_A)\rangle$ and any $0 \leq r < 1$, there is a model $\beta$ that gives the same conditional probabilities of Eve's and Bob's outcomes for each command $b_E \in E_\alpha$ and for all of Bob's commands, so
\begin{equation} (\forall b_A \in A_\alpha, b_E \in E_\alpha) \mbox{$\Pr_\beta(j_E|b_A,b_E)$} =
\mbox{$\Pr_\alpha(j_E|b_A,b_E)$} \label{eq:same2} \end{equation} while
\begin{equation} (\forall b_A \neq b'_A) |\langle v_\beta(b_A)|v_\beta(b'_A) \rangle | = r
|\langle v_\alpha(b_A)|v_\alpha(b'_A)\rangle | . \label{eq:p_alpha2} \end{equation} \end{prop}
\noindent{\em Proof}: The proof extends the construction used in the proof of Proposition 1, with a model $\beta$ defined by \begin{enumerate} \item the same command set for Alice: $A_\beta = A_\alpha$; \item signals expressed by a vector intended by Alice, as in model
$\alpha$, tensored in to an unintended vector in an additional Hilbert space ${\cal H}_{\rm leak}$, so Alice produces vectors $|v_\beta(b_A) \rangle = |w_\beta(b_A) \rangle
\otimes |v_\alpha(b_A) \rangle$, with $|w_\beta(b_A) \rangle \in {\cal H}_{\rm leak}$;
\item a fixed starting state for Eve's probe of $|e_\alpha \rangle \in {\cal H}_{{\rm probe},\alpha};$ \item a larger set of commands for Eve, $E_\beta = E_\alpha \sqcup E_{\rm extra}$ (disjoint union); \item unitary operators $U_\beta(b_E)$ acting on ${\cal H}_{\rm leak} \otimes {\cal H}_{{\rm probe},\alpha} \otimes {\cal H}_{{\rm sig},\alpha}$ for the interaction of Eve's probe with Alice's signal, defined so that \begin{equation} U_\beta(b_E) = \left\{ \begin{array}{l} {\bf 1}_{\rm leak} \otimes U_\alpha(b_E) \mbox{ for all } b_E \in E_\alpha, \\[10pt] \mbox{Eve's choice of unitary } U_\beta(b_E) \mbox{ if } b_E \in E_{\rm extra}; \end{array} \right. \end{equation} \item a POVM-valued function of Eve's commands to her measuring instruments, with detection operators defined so \begin{equation} M_\beta(b_E;j_E) = \left\{ \begin{array}{l} {\bf 1} \otimes M_\alpha(b_E;j_E) \mbox{ for all } b_E \in E_\alpha, \\ \mbox{Eve's choice of POVM on } \\ {\cal H}_{\rm leak} \otimes {\cal H}_{{\rm probe},\alpha} \mbox{ if } b_E \in E_{\rm extra}. \end{array} \right. \end{equation} \end{enumerate} According to model $\beta$, if Eve chooses any measurement command of
$E_\alpha$, Eq.\ (\ref{eq:same2}) holds. But model $\beta$ speaks not of the vectors $|v_\alpha(b_A) \rangle$ but of other vectors having an inner product (relevant to the security of quantum key distribution)
of \begin{equation} |\langle v_\beta(b_A)|v_\beta(b'_A) \rangle | =
|\langle w(b_A)| w(b'_A) \rangle | |\langle v_\alpha(b_A)
|v_\alpha(b'_A) \rangle|. \label{eq:winner2} \end{equation} The unit vectors $|w(b_A) \rangle$ can be specified so that $|w(b_A) \rangle =
r^{1/2} |u(b_A) \rangle + (1-r) |u_0 \rangle$, with $\langle u(b_A)
|u(b'_A) \rangle = 0$ for all $b_A \neq b'_A$ and $\langle u(b_A)|u_0 \rangle = 0$ for all $b_A \in A_\beta$. With this specification Eq.\ (\ref{eq:p_alpha}) holds, and furthermore $r$ can be chosen as small as one wishes. $\Box$
\goodbreak
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AXON: An In-situ TEM Software Platform Streamlines Image Acquisition, Metadata Synchronization and Data Analysis, Enabling Deeper Understanding, and Improved Reproducibility of In-situ Experimental Results
Madeline Dressel Dukes, Kate Marusak, Yaofeng Guo, Jennifer McConnell, Stamp Walden, John Damiano, David Nackashi
Published online by Cambridge University Press: 22 July 2022, pp. 108-109
AXON Dose: A Machine Vision Solution for Accurate, Quantifiable Dose Management in the Transmission Electron Microscope
Stamp Walden, Madeline Dressel Dukes, Kate Marusak, Yaofeng Guo, Jennifer McConnell, John Damiano, David Nackashi
The ASKAP Variables and Slow Transients (VAST) Pilot Survey
Australian SKA Pathfinder
Tara Murphy, David L. Kaplan, Adam J. Stewart, Andrew O'Brien, Emil Lenc, Sergio Pintaldi, Joshua Pritchard, Dougal Dobie, Archibald Fox, James K. Leung, Tao An, Martin E. Bell, Jess W. Broderick, Shami Chatterjee, Shi Dai, Daniele d'Antonio, Gerry Doyle, B. M. Gaensler, George Heald, Assaf Horesh, Megan L. Jones, David McConnell, Vanessa A. Moss, Wasim Raja, Gavin Ramsay, Stuart Ryder, Elaine M. Sadler, Gregory R. Sivakoff, Yuanming Wang, Ziteng Wang, Michael S. Wheatland, Matthew Whiting, James R. Allison, C. S. Anderson, Lewis Ball, K. Bannister, D. C.-J. Bock, R. Bolton, J. D. Bunton, R. Chekkala, A. P Chippendale, F. R. Cooray, N. Gupta, D. B. Hayman, K. Jeganathan, B. Koribalski, K. Lee-Waddell, Elizabeth K. Mahony, J. Marvil, N. M. McClure-Griffiths, P. Mirtschin, A. Ng, S. Pearce, C. Phillips, M. A. Voronkov
Journal: Publications of the Astronomical Society of Australia / Volume 38 / 2021
Published online by Cambridge University Press: 12 October 2021, e054
The Variables and Slow Transients Survey (VAST) on the Australian Square Kilometre Array Pathfinder (ASKAP) is designed to detect highly variable and transient radio sources on timescales from 5 s to $\sim\!5$ yr. In this paper, we present the survey description, observation strategy and initial results from the VAST Phase I Pilot Survey. This pilot survey consists of $\sim\!162$ h of observations conducted at a central frequency of 888 MHz between 2019 August and 2020 August, with a typical rms sensitivity of $0.24\ \mathrm{mJy\ beam}^{-1}$ and angular resolution of $12-20$ arcseconds. There are 113 fields, each of which was observed for 12 min integration time, with between 5 and 13 repeats, with cadences between 1 day and 8 months. The total area of the pilot survey footprint is 5 131 square degrees, covering six distinct regions of the sky. An initial search of two of these regions, totalling 1 646 square degrees, revealed 28 highly variable and/or transient sources. Seven of these are known pulsars, including the millisecond pulsar J2039–5617. Another seven are stars, four of which have no previously reported radio detection (SCR J0533–4257, LEHPM 2-783, UCAC3 89–412162 and 2MASS J22414436–6119311). Of the remaining 14 sources, two are active galactic nuclei, six are associated with galaxies and the other six have no multi-wavelength counterparts and are yet to be identified.
2510: QIPR: Creating a Quality Improvement Project Registry
Amber L. Allen, Christopher Barnes, Kevin S. Hanson, David Nelson, Randy Harmatz, Eric Rosenberg, Linda Allen, Lilliana Bell, Lynne Meyer, Debbie Lynn, Jeanette Green, Peter Iafrate, Matthew McConnell, Patrick White, Samantha Davuluri, Tarun Gupta Akirala
Journal: Journal of Clinical and Translational Science / Volume 1 / Issue S1 / September 2017
Published online by Cambridge University Press: 10 May 2018, pp. 20-21
OBJECTIVES/SPECIFIC AIMS: To create a searchable public registry of all Quality Improvement (QI) projects. To incentivize the medical professionals at UF Health to initiate quality improvement projects by reducing startup burden and providing a path to publishing results. To reduce the review effort performed by the internal review board on projects that are quality improvement Versus research. To foster publication of completed quality improvement projects. To assist the UF Health Sebastian Ferrero Office of Clinical Quality & Patient Safety in managing quality improvement across the hospital system. METHODS/STUDY POPULATION: This project used a variant of the spiral software development model and principles from the ADDIE instructional design process for the creation of a registry that is web based. To understand the current registration process and management of quality projects in the UF Health system a needs assessment was performed with the UF Health Sebastian Ferrero Office of Clinical Quality & Patient Safety to gather project requirements. Biweekly meetings were held between the Quality Improvement office and the Clinical and Translational Science – Informatics and Technology teams during the entire project. Our primary goal was to collect just enough information to answer the basic questions of who is doing which QI project, what department are they from, what are the most basic details about the type of project and who is involved. We also wanted to create incentive in the user group to try to find an existing project to join or to commit the details of their proposed new project to a data registry for others to find to reduce the amount of duplicate QI projects. We created a series of design templates for further customization and feature discovery. We then proceed with the development of the registry using a Python web development framework called Django, which is a technology that powers Pinterest and the Washington Post Web sites. The application is broken down into 2 main components (i) data input, where information is collected from clinical staff, Nurses, Pharmacists, Residents, and Doctors on what quality improvement projects they intend to complete and (ii) project registry, where completed or "registered" projects can be viewed and searched publicly. The registry consists of a quality investigator profile that lists contact information, expertise, and areas of interest. A dashboard allows for the creation and review of quality improvement projects. A search function enables certain quality project details to be publicly accessible to encourage collaboration. We developed the Registry Matching Algorithm which is based on the Jaccard similarity coefficient that uses quality project features to find similar quality projects. The algorithm allows for quality investigators to find existing or previous quality improvement projects to encourage collaboration and to reduce repeat projects. We also developed the QIPR Approver Algorithm that guides the investigator through a series of questions that allows an appropriate quality project to get approved to start without the need for human intervention. RESULTS/ANTICIPATED RESULTS: A product of this project is an open source software package that is freely available on GitHub for distribution to other health systems under the Apache 2.0 open source license. Adoption of the Quality Improvement Project Registry and promotion of it to the intended audience are important factors for the success of this registry. Thanks goes to the UW-Madison and their QI/Program Evaluation Self-Certification Tool (https://uwmadison.co1.qualtrics.com/SE/?SID=SV_3lVeNuKe8FhKc73) used as example and inspiration for this project. DISCUSSION/SIGNIFICANCE OF IMPACT: This registry was created to help understand the impact of improved management of quality projects in a hospital system. The ultimate result will be to reduce time to approve quality improvement projects, increase collaboration across the UF Health Hospital system, reduce redundancy of quality improvement projects and translate more projects into publications.
Geomorphology and glacial history of Rauer Group, East Antarctica
Duanne A. White, Ole Bennike, Sonja Berg, Simon L. Harley, David Fink, Kevin Kiernan, Anne McConnell, Bernd Wagner
Journal: Quaternary Research / Volume 72 / Issue 1 / July 2009
Published online by Cambridge University Press: 20 January 2017, pp. 80-90
The presence of glacial sediments across the Rauer Group indicates that the East Antarctic ice sheet formerly covered the entire archipelago and has since retreated at least 15 km from its maximum extent. The degree of weathering of these glacial sediments suggests that ice retreat from this maximum position occurred sometime during the latter half of the last glacial cycle. Following this phase of retreat, the ice sheet margin has not expanded more than ∼ 1 km seaward of its present position. This pattern of ice sheet change matches that recorded in Vestfold Hills, providing further evidence that the diminutive Marine Isotope Stage 2 ice sheet advance in the nearby Larsemann Hills may have been influenced by local factors rather than a regional ice-sheet response to climate and sea-level change.
Two-Color, Two-Photon Imaging at Long Excitation Wavelengths Using a Diamond Raman Laser
Johanna Trägårdh, Michelle Murtagh, Gillian Robb, Maddy Parsons, Jipeng Lin, David J. Spence, Gail McConnell
Journal: Microscopy and Microanalysis / Volume 22 / Issue 4 / August 2016
We demonstrate that the second-Stokes output from a diamond Raman laser, pumped by a femtosecond Ti:Sapphire laser, can be used to efficiently excite red-emitting dyes by two-photon excitation at 1,080 nm and beyond. We image HeLa cells expressing red fluorescent protein, as well as dyes such as Texas Red and Mitotracker Red. We demonstrate the potential for simultaneous two-color, two-photon imaging with this laser by using the residual pump beam for excitation of a green-emitting dye. We demonstrate this for the combination of Alexa Fluor 488 and Alexa Fluor 568. Because the Raman laser extends the wavelength range of the Ti:Sapphire laser, resulting in a laser system tunable to 680–1,200 nm, it can be used for two-photon excitation of a large variety and combination of dyes.
Identifications from the Parkes-MIT-NRAO Surveys
Niven J. Tasker, Alan E. Wright, David McConnell, Ann Savage, Michael J. Kesteven, Euan Troup, Mark Griffith
Journal: Publications of the Astronomical Society of Australia / Volume 10 / Issue 4 / 1993
The Parkes-MIT-NRAO (PMN) 4.85 GHz continuum radio survey of the southern sky was undertaken in 1990 June and November. This survey was performed on the Parkes 64 m telescope using the NRAO seven-beam receiver. A point-source catalogue of 36,640 radio sources has been produced for the Southern Survey zone −87.5° ≤δ ≤ −37° and for the Tropical Survey zone −29° ≤δ ≤ −9.5°. The flux limit of this survey varies with declination and is typically about 30 mJy.
We have begun to cross-correlate the PMN data with sources contained in catalogues compiled at radio, and other, wavelengths. We have found associations for 96% of the PKSCAT90 2700 MHz database sources, and 95% of the Molonglo 408 MHz Catalogue sources within in the PMN Southern and Tropical Survey zones.
A program to identify the optical counterparts of PMN Southern Survey point sources, S4.85 GHz ≥ 70 mJy, using the COSMOS database, is under way. To facilitate this programme we are improving the positional accuracy of PMN sources with observations made at the Australia Telescope National Facility compact array. We have developed a new "snapshot" mode of observing to process the large number of sources (~ 8000) in our sample. It is possible to obtain accurate positions from three snapshots efficiently with a total integration of < 3 minutes.
Evaluating the Economic Impact of Farmers' Markets Using an Opportunity Cost Framework
David W. Hughes, Cheryl Brown, Stacy Miller, Tom McConnell
Journal: Journal of Agricultural and Applied Economics / Volume 40 / Issue 1 / April 2007
Farmers' markets presumably benefit local economies through enhanced retention of local dollars. Unlike other studies, the net impact of farmers' markets on the West Virginia economy is examined. Producer survey results are used in estimating annual direct sales ($1,725 million). Using an IMPLAN-based input-output model, gross impacts are 119 jobs (69 full-time equivalent jobs) and $2,389 million in output including $1.48 million in gross state product (GSP). When the effect of direct revenue losses are included (primarily for grocery stores), the impact is reduced to 82 jobs (43 full-time equivalent jobs), $1,075 million in output, and $0,653 million in GSP.
Edited by Andrew Shifflett, Associate Professor of English at the University of South Carolina, Columbia, Edward Gieskes, Associate Professor of English at the University of South Carolina, Columbia
Book: Renaissance Papers 2012
Published by: Boydell & Brewer
Print publication: 01 November 2013, pp vii-viii
Cosmetic Blackness: East Indies Trade, Gender, and The Devil's Law-Case
Print publication: 01 November 2013, pp 83-96
With the recent shift in literary studies towards what is often described as a "global Renaissance," it is hardly surprising that figures of merchants and travelers both in early modern travelogues and plays have come under greater scrutiny as sites for understanding the formation of a fluid English identity, transnational commerce, emergent colonialism, and nation building. What still remains largely unexplored, however, particularly in the context of the East Indies trade, is the impact of this emergent globalization on the bodies of the European women who were closely related to the merchants or factors. While scholarship on plays such as Fletcher's The Island Princess or Dryden's Amboyna emphasizes the roles of both European men and their beloved native women, the white woman still remains a shadowy presence at the fringes of our current academic interest in the early modern spice trade.
This essay seeks to address this gap by turning to the public stage, particularly to a play that explores how the emergent trade with the East Indies appeared to affect the physical and moral complexion of one such European woman. In the trial scene of John Webster's play The Devil's Law-Case (1623), Jolenta, the sister of Romelio, an East Indies merchant enters with "her face colour'd like that of a Moore," accompanied by two Surgeons, "one of them like a Jew." Although the assembled people quickly recognize her they still comment on her changed complexion. Ariosto the advocate exclaims, "Shee's a blacke one indeed" (5.5.40) while Ercole, one of her suitors, wails "to what purpose / Are you thus ecclipst?" (5.5.57–58). Of course, Jolenta's transformation is temporary and apparently superficial; yet her blackening appears to gesture towards deeper concerns regarding the impact of the East Indies trade, particularly on a woman who has never left her home or sailed the high seas to profit from pepper, cinnamon, cardamom and mace.
Print publication: 01 November 2013, pp i-iv
From One Marvell to Another: Puritan Logic in "To His Coy Mistress"
Print publication: 01 November 2013, pp 97-104
Gods apprentice is a jorneyman: he must allwayes learne the mystery of his profession, & walking forward aime hard to the marke for the price of his high calling. as the teacher in Gods schoole must give Line upon Line, precept upon precept: so to the scholler likewise nulla dies sine linea, no day must pass without a new lesson, as Cato said, so Gods child must grow old every day learning many things. And so in practise also. he must adde to his faith vertue, | & to plowing, sowing. Like Charles the fifth, plus ultra must be his motto: he must go from strength to strength untill he appeare be=fore the Lord in Sion. And that because, he is leaving his abode in this world but an im=perfect pilgrime. he is not what, he is not where he should be … [There are those who] 'looke behind them, that turne their face in the day of battell, & quite give over Gods husbandry', [those] 'that forget their first love who though they forsake not the plough yet are they idle companions that do the worke of the Lord negligently … [For them] it had beene better never to have knowne the way of righ=teousness then that they should bee like a dog to his vomit & a sow to wallowing in the mire.
"Bred Now of Your Mud": Land, Generation, and Maternity in Antony and Cleopatra
Analyses of Antony and Cleopatra have long noted the dialectical opposition between Rome and Egypt, an opposition that sets up a concomitant correspondence between geography and gender. Although recent scholarship has destabilized the categories, Rome has traditionally represented the masculine—solid, controlled, bounded—while Egypt is feminine—fluid, unchecked, limitless, and thus constantly generating. Egypt in the play evokes an elemental fecundity that is spontaneous and natural at the same time that it is corrupting and degenerate, "dungy," in Antony's words. Further, the connection between Cleopatra and Egypt is inextricable in the play; she exists in metonymic relation to her country, the word "Egypt" used no less than seven times to refer to her directly. Picking up on Janet Adelman's argument that the play constructs Cleopatra as "one with her feminized kingdom as though it were her body," this essay examines the complex idea of Egyptian earthiness in connection with Cleopatra and her fertile/infertile body by reading it in conjunction with various theories of reproduction—what the Renaissance called generation. Specifically, I seek to show how the trope of spontaneous generation allows Shakespeare to expand his interrogation of procreation in the play, blurring gender boundaries as he does so.
Ovid, Lucretius, and the Grounded Goddess in Shakespeare's Venus and Adonis
In the sticky, sweet, and sweaty world in which Shakespeare situates his Venus and Adonis, something has gone awry. According to Venus, "Nature" is "at strife" with herself for having made Adonis. By "Nature" Venus is, of course, referring to herself. Compared to the Venus of book 10 of Ovid's Metamorphoses—a goddess who makes men and women fall in love, who brings stone to life, and whose magical doves transport her anywhere she wishes to go—Shakespeare's Venus is, by comparison, a much more natural being. A creature of the senses, most especially smell, Shakespeare's Venus does not so much manipulate the natural world as bond with it. She experiences heightened, animal-like sensibilities that allow her to commune with Adonis's horse and to imagine herself as the earthbound and hunted Wat the Hare.
But why would Shakespeare strip Ovid's goddess of her supernatural powers and drive her so literally down to earth? The answer, I would suggest, is that Venus and Adonis traces its ancestry not only to Ovid's Metamorphoses but also to Lucretius's De Rerum Natura. Consider the powerful invocation to Venus with which Lucretius begins his great philosophical poem on "the nature of things": "Venus, power of life, it is you who beneath the sky's sliding stars inspirit the ship-bearing sea, inspirit the productive land.
Getting Past the Ellipsis: The Spirit and Urania in Paradise Lost
Print publication: 01 November 2013, pp 117-125
In John Shawcross's book The Development of Milton's Thought: Law, Religion, and Government, he quotes that famous phrase from Milton, "fit audience, though few." I was brought up short while reading because this quotation does not include an ellipsis. Can even Shawcross nod? I was reassured when I realized that he had not cited book and line numbers for the quotation; he was simply quoting an oft-used phrase rather than Paradise Lost itself. I thus felt better about John, but continued to be troubled by the broader implications of "fit audience … though few," with or without the ellipsis. Here I shall argue that the ellipsis eliminates a central element, in the line and the poetic sentence and in terms of Milton's own concerns about the fate of his text. And what scholars so often omit by typifying Milton's audience using this phrase is the place of the ineffable Spirit of God in the communion or community of believers.
I shall dispense with the simple part first: how often is the ellipsis used, and what does it skate over? The phrase appears in the invocation to Book 7 of Paradise Lost:
Standing on Earth, not rapt above the Pole,
More safe I Sing with mortal voice, unchang'd
To hoarse or mute, though fall'n on evil days,
On evil days though fall'n, and evil tongues;
In darkness, and with dangers compast round,
And solitude; yet not alone, while thou
Visit'st my slumbers Nightly, or when Morn
Purples the East: still govern thou my Song,
Urania, and fit audience find, though few.
(7.23–31)
Renaissance Papers 2012
Edited by Andrew Shifflett, Edward Gieskes
Print publication: 01 November 2013
Renaissance Papers collects the best scholarly essays submitted each year to the Southeastern Renaissance Conference. The 2012 volume opens with two essays on sexuality in Elizabethan narrative poetry: on homoeroticism in Spenser's Faerie Queene and on Shakespeare's "swerve" into Lucretian imagery in Venus and Adonis. The volume then turns to Renaissance drama and its links to the wider culture: the commodification of spirit in Marlowe's Doctor Faustus, Shakespeare's evocation of the Acts of the Apostles in The Comedy of Errors, "summoning" in Hamlet and King Lear, discourses of procreation and generation in Antony and Cleopatra/, trade and gender in John Webster's Devil's Law-Case, and an examination of street scenes in Romeo and Juliet in relation to Paul's Cross Churchyard, the hub of the London bookselling market in the early modern period. The volume closes with essays on seventeenth-century literature and literary culture: on the "puritan logic" of the elder Andrew Marvell in his famous son's poem "To His Coy Mistress," on the "sociable lexicography" of a Royalist polymath attempting to reconcile with the English Commonwealth, and on the underestimated roles of Urania in Milton's Paradise Lost. Contributors: David Ainsworth, Thomas W. Dabbs, Sonya Freeman Loftis, Russell Hugh McConnell, Robert L. Reid, Amrita Sen, Susan C. Staub, Emily Stockard, Nathan Stogdill, Christina A. Taormina, Emma Annette Wilson. Andrew Shifflett and Edward Gieskes are Associate Professors of English at the University of South Carolina, Columbia.
Antipholus and the Exorcists: The Acts of the Apostles in Shakespeare's The Comedy of Errors
In act 4 of The Comedy of Errors, Adriana, in response to her husband's wildly erratic behaviour, recruits one Doctor Pinch to cure his apparent madness. Antipholus of Ephesus is of course not really mad at all, but understandably frustrated and confused with the events of the day, which have seen him locked out of his own house and accused of failing to pay for valuables that he actually never received, thanks to a series of misunderstandings involving his identical twin, Antipholus of Syracuse. When Antipholus of Ephesus refuses to cooperate with his wife's well-intentioned intervention, Doctor Pinch attempts an exorcism to drive away the demons which he supposes to possess him:
I charge thee, Satan, housed within this man,
To yield possession to my holy prayers
And to thy state of darkness hie thee straight:
I conjure thee by all the saints in heaven!
Of course, Pinch's attempted exorcism fails, because there is no Satan to exorcise, and Antipholus becomes so angry that Adriana has him forcibly bound and taken away in Pinch's custody. In a later scene a messenger reports that Antipholus and his servant Dromio have gnawed through their bonds and escaped and then captured and tormented the hapless Doctor Pinch, burning off his beard "with brands of fire," putting it out again with "Great pails of puddled mire," and then preaching patience to him while cutting him with scissors and concludes that "unless you send some present help, / Between them they will kill the conjurer" (5.1.172, 74–76, 177–78).
"An Heap Is Form'd into an Alphabet": Thomas Blount's Sociable Lexicography
In the criticism of the last thirty years, the consensus has been that the English Civil War was fought on the battlefield and on the page. Titles such as Writing the English Republic, Literature and Revolution, Literature and Dissent, Literature and Politics, and Poetry and Allegiance urge that politics and literature were inextricable in mid-century England. And, when talking about the English Civil War, when we say "political" we mean "partisan." Lucasta, we have learned, is a royalist rallying cry and Paradise Lost a "Republican epic." In these contentious times, the act of taking up the pen was a partisan one. But while much attention has been given to the expressions of partisanship, much less has been given to the concepts of sociability that marked its borders. While the violent disruptions of the English Civil War certainly created a culture of divisiveness, they also created an alternative culture increasingly attentive to the advantages of adaptability. Recent work on literary form in the period suggests that, far from sites of stable partisan expression, literary genres were inherently flexible spaces that lent themselves to experimentation not conflict. Lyric's association with "verses" or "turnings," romance's many reversals and transformations, and the essay's function as a genre of "attempts" provided authors opportunities to explore and enact strategies of "flexibleness." In this essay, I would like to consider this "flexibleness" as it appears in an unlikely place: the English vernacular dictionary.
Reconstructing the Bower of Bliss: Homoerotic Myth-Making in The Faerie Queene
Print publication: 01 November 2013, pp 1-12
Erotic encounters between women appear with surprising frequency in the middle books of The Faerie Queene. Both Guyon's adventure in the Bower of Bliss and Britomart's encounters with Malecasta and Amoret reflect Renaissance beliefs about female-female desire. Spenser consistently associates heterosexual intercourse with sexual maturity: his treatment of homoerotic desire as an "adolescent" stage of development adheres to early modern myths regarding same-sex attraction. Indeed, one of Spenser's goals in the middle books of The Faerie Queene seems to be discouraging women's non-teleological, non-reproductive eroticism, and encouraging them to "move forward" into a reproductive sexual marriage. The homoeroticism that fills Spenser's garden of sexual delight motivates Guyon's wrathful destruction of Acrasia's Bower—a destruction that points toward the potential threat of feminine stasis in non-reproductive sexual activity. In addition, Britomart's sexual encounters with other women become both a primary challenge to the lady Knight's quest and a central element of her sexual education as she learns to embrace her role as a mother and wife.
Reconstructing the Bower of Bliss
Spenser's Bower of Bliss is a space of feminine sexual autonomy. Although critics once viewed the sexuality displayed in the Bower as a show that exists only for Guyon, describing it as a realm of lust devoid of sexual fulfillment, a world of pornographic show as opposed to one of sexual satisfaction, more recent readings acknowledge the rampant female sexuality of the Bower. Only Guyon sees a show without sexual satisfaction. Indeed, the women in the fountain who "wrestle wantonly" are enjoying lust in action (2.12.63). Acrasia repeatedly "bedew[s] Verdant's lips . . . with kisses light" and "sucke[s] his spright,"—considering the early modern connection between "spright" and "semen," Acrasia is obviously engaged in sexual activity and satisfying her lust (2.12.72). For the women fondling each other in the fountain, and for Acrasia and Verdant, the Bower contains sex: the reader only sees a show if he or she looks through the eyes of the voyeuristic Guyon. Indeed, Spenser invites such a perspective: the Bower is full of references to eyes and sight.
The Summoning of Hamlet and Lear
Summoning provokes the psyche's most momentous unfoldings. The mental and visceral impact of a legal summons is obvious to all who receive one, unleashing a flood of piteous self-justification and sharp questioning of the Rule of Law. No one likes being called to judgment. To the discomfort of social courts Shakespeare's summonings add a spiritual burden. Because King Hamlet was killed as he slept, when his sinful soul was unready, his anxious ghost vanishes at cock-crow "like a guilty thing / Upon a fearful summons" (1.1.154–55). King Lear, distraught at lost power and public humiliation, allies his voice with heavenly thundering ("Close pent-up guilts, / Rive your concealing continents and cry / These dreadful summoners grace"), but he denies that the thunder summons him: "I am a man / More sinned against than sinning" (3.2.57–60).
Each play begins with a king, or a kingly wraith, summoning children for judgment, and each king's spectacular enactment of sovereignty goes terribly wrong. The "fearful summons" and "dreadful summoners" carry deep irony in that each king—one secretly slain, one publicly shamed—clings to an illusory summoner-power after having been quite defanged. Old Hamlet's ghost is "majestic" in complete armor. | CommonCrawl |
Home » Statistics » Kelly's Coefficient of Skewness for Ungrouped data | Formula | Examples
Kelly's Coefficient of Skewness for Ungrouped data | Formula | Examples
Kelly's Coefficient of Skewness for Ungrouped data
1 Kelly's Coefficient of Skewness for Ungrouped data
2 Kelly's Coefficient of Skewness Calculator for ungrouped data
3 How to calculate Kelly's Coefficient of Skewness for ungrouped data?
4 Range for Kelly's coefficient of Skewness
5 Kelly's coefficient of skewness Example 1
Kelly's Coefficient of Skewness for Ungrouped data
Kelly's coefficient of skewness is based on deciles or percentiles of the data. The Bowley's coefficient of skewness is based on the middle 50 percent of the observations of data set. It means the Bowley's coefficient of skewness leaves the 25 percent observations in each tail of the data set.
Kelly suggested a measure of skewness which is based on middle 80 percent of the observations of data set.
For a symmetric distribution, the first decile namely $D_1$ and ninth decile $D_9$ are equidistant from the median i.e. $D_5$. Thus, $D_9 – D_5 = D_5 -D_1$.
The Kelley's coefficient of skewness based is defined as
$$ \begin{aligned} S_k &= \frac{D_9+D_1 - 2D_5}{D_9 -D_1}\\ & OR \\ S_k &=\frac{P_{90}+P_{10} - 2P_{50}}{P_{90} -P_{10}} \end{aligned} $$
$D1=P{10}$ is the first decile or tenth percentile,
$D5=P{50}$ is the fifth decile or fiftieth percentile,
$D9=P{90}$ is the ninth decile or nineteenth percentile.
If $S_k < 0$, the data is negatively skewed.
If $S_k = 0$, the data is symmetric(i.e., not skewed).
If $S_k > 0$, the data is positively skewed.
Kelly's Coefficient of Skewness Calculator for ungrouped data
Use this calculator to find the Kelly's coefficient of skewness for ungrouped (raw) data.
Kelly's Coeff. of Skewness Calculator
Enter the X Values (Separated by comma,)
Number of Obs. (n):
Ascending order of X values :
First Decile : ($D_1$)
Sample Median : ($D_5$)
Ninth Decile : ($D_9$)
Kelly's Coeff. of Skewness :
How to calculate Kelly's Coefficient of Skewness for ungrouped data?
Step 1 – Enter the $x$ values separated by commas
Step 2 – Click on "Calculate" button to get Decile for ungrouped data
Step 3 – Gives the output as number of observations $n$
Step 4 – Gives the output as ascending order data
Step 5 – Gives the Deciles $D_1$,$D_5$ and $D_9$.
Step 6 – Gives output as Kelly's Coefficient of Skewness
Range for Kelly's coefficient of Skewness
Kelly's coefficient of skewness ranges from -1 to +1.
We know that, if $a>0$ and $b>0$, then $|a-b|\leq |a+b|$,
$$ \begin{aligned} & \text{i.e., } \bigg|\dfrac{a-b}{a+b} \bigg| \leq 1 \end{aligned} $$
Now, taking $a= D_9 – D_5$ and $b= D_5-D_1$ in \eqref{sb} we get
$$ \begin{aligned} & \bigg|\dfrac{(D_9 - D_5)-(D_5-D_1)}{(D_9 - D_5)+(D_5-D_1)}\bigg| \leq 1\\ &\Rightarrow \bigg|\dfrac{D_9 + D_1-2D_5}{D_9 -D_1}\bigg| \leq 1\\ & \Rightarrow |S_k|\leq 1\\ & \Rightarrow -1\leq S_k\leq 1. \end{aligned} $$
Thus, Kelly's coefficient of skewness ranges from -1 and +1.
Kelly's coefficient of skewness Example 1
The marks obtained by a sample of 20 students in a class test are as follows:
16, 15, 19, 22, 13, 17, 14, 18, 12, 9.
Find Kelly's Coefficient of Skewness.
Kelly's coefficient of skewness is
$$ \begin{aligned} S_k &= \frac{D_9+D_1 - 2D_5}{D_9 -D_1}\\ \end{aligned} $$
The formula for $i^{th}$ decile is
$D_i =$ Value of $\bigg(\dfrac{i(n+1)}{10}\bigg)^{th}$ observation, $i=1,2,3,\cdots, 9$
where $n$ is the total number of observations.
Arrange the data in ascending order
9, 10, 12, 13, 14, 15, 15, 16, 17, 17
First Decile $D_1$
The first decile $D_1$ can be computed as follows:
$$ \begin{aligned} D_1 &=\text{Value of }\bigg(\dfrac{1(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{1(20+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(2.1\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(2\big)^{th} \text{ obs.}\\ & +0.1 \big(\text{Value of } \big(3\big)^{th}\text{ obs.}-\text{Value of }\big(2\big)^{th} \text{ obs.}\big)\\ &=10+0.1\big(12 -10\big)\\ &=10.2 \end{aligned} $$
Fifth Decile $D_5$
The fifth decile $D_5$ can be computed as follows:
$$ \begin{aligned} D_5 &=\text{Value of }\bigg(\dfrac{5(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{5(20+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(10.5\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(10\big)^{th} \text{ obs.}\\ & +0.5 \big(\text{Value of } \big(11\big)^{th}\text{ obs.}-\text{Value of }\big(10\big)^{th} \text{ obs.}\big)\\ &=17+0.5\big(18 -17\big)\\ &=17.5 \end{aligned} $$
Ninth Decile $D_9$
The ninth decile $D_9$ can be computed as follows:
$$ \begin{aligned} D_9 &=\text{Value of }\bigg(\dfrac{9(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{9(20+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(18.9\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(18\big)^{th} \text{ obs.}+0.9 \big(\text{Value of } \big(19\big)^{th}\text{ obs.}-\text{Value of }\big(18\big)^{th} \text{ obs.}\big)\\ &=27\\ & +0.9\big(29 -27\big)\\ &=28.8 \end{aligned} $$
Kelly's coefficient of skewness
$$ \begin{aligned} S_k &= \frac{D_9+D_1 - 2D_5}{D_9 -D_1}\\ &=\frac{28.8+10.2 - 2* 17.5}{28.8 - 10.2}\\ &=0.2151 \end{aligned} $$
As the coefficient of skewness $S_k$ is $\text{greater than zero}$ (i.e., $S_k > 0$), the distribution is $\text{positively skewed}$.
The following data gives the hourly wage rates (in dollars) of 25 employees of a company.
22, 26, 25, 28, 31.
Find the Kelly's coefficient of skewness.
The sample size is $n = 25$.
$$ \begin{aligned} D_1 &=\text{Value of }\bigg(\dfrac{1(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{1(25+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(2.6\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(2\big)^{th} \text{ obs.}\\ & +0.6 \big(\text{Value of } \big(3\big)^{th}\text{ obs.}-\text{Value of }\big(2\big)^{th} \text{ obs.}\big)\\ &=18+0.6\big(19 -18\big)\\ &=18.6\text{ dollars} \end{aligned} $$
$$ \begin{aligned} D_5 &=\text{Value of }\bigg(\dfrac{5(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{5(25+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(13\big)^{th} \text{ obs.}\\ &=24\text{ dollars} \end{aligned} $$
$$ \begin{aligned} D_9 &=\text{Value of }\bigg(\dfrac{9(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{9(25+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(23.4\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(23\big)^{th} \text{ obs.}\\ & +0.4 \big(\text{Value of } \big(24\big)^{th}\text{ obs.}-\text{Value of }\big(23\big)^{th} \text{ obs.}\big)\\ &=31+0.4\big(32 -31\big)\\ &=31.4\text{ dollars} \end{aligned} $$
$$ \begin{aligned} S_k &= \frac{D_9+D_1 - 2D_5}{D_9 -D_1}\\ &=\frac{31.4+18.6 - 2* 24}{31.4 - 18.6}\\ &=0.1562 \end{aligned} $$
Blood sugar level (in mg/dl) of a sample of 20 patients admitted to the hospitals are as follows:
75,89,72,78,87, 85, 73, 75, 97, 87, 84, 76,73,79,99,86,83,76,78,73.
$$ \begin{aligned} D_1 &=\text{Value of }\bigg(\dfrac{1(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{1(20+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(2.1\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(2\big)^{th} \text{ obs.}\\ & +0.1 \big(\text{Value of } \big(3\big)^{th}\text{ obs.}-\text{Value of }\big(2\big)^{th} \text{ obs.}\big)\\ &=73+0.1\big(73 -73\big)\\ &=73\text{ mg/dl} \end{aligned} $$
$$ \begin{aligned} D_5 &=\text{Value of }\bigg(\dfrac{5(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{5(20+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(10.5\big)^{th} \text{ obs.}\\ &=78.5\text{ mg/dl} \end{aligned} $$
$$ \begin{aligned} D_9 &=\text{Value of }\bigg(\dfrac{9(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{9(20+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(18.9\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(18\big)^{th} \text{ obs.}\\ & +0.9 \big(\text{Value of } \big(19\big)^{th}\text{ obs.}-\text{Value of }\big(18\big)^{th} \text{ obs.}\big)\\ &=89+0.9\big(97 -89\big)\\ &=96.2\text{ mg/dl} \end{aligned} $$
$$ \begin{aligned} S_k &= \frac{D_9+D_1 - 2D_5}{D_9 -D_1}\\ &=\frac{96.2+73 - 2* 78.5}{96.2 - 73}\\ &=0.5259 \end{aligned} $$
Diastolic blood pressure (in mmHg) of a sample of 18 patients admitted to the hospitals are as follows:
65,76,64,73,74,80, 71, 68,66, 81, 79, 75, 70, 62, 83,63, 77, 78.
$$ \begin{aligned} D_1 &=\text{Value of }\bigg(\dfrac{1(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{1(18+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(1.9\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(1\big)^{th} \text{ obs.}\\ & +0.9 \big(\text{Value of } \big(2\big)^{th}\text{ obs.}-\text{Value of }\big(1\big)^{th} \text{ obs.}\big)\\ &=62+0.9\big(63 -62\big)\\ &=62.9\text{ mmHg} \end{aligned} $$
$$ \begin{aligned} D_5 &=\text{Value of }\bigg(\dfrac{5(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{5(18+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(9.5\big)^{th} \text{ obs.}\\ &=73.5\text{ mmHg} \end{aligned} $$
$$ \begin{aligned} D_9 &=\text{Value of }\bigg(\dfrac{9(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{9(18+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(17.1\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(17\big)^{th} \text{ obs.}\\ & +0.1 \big(\text{Value of } \big(18\big)^{th}\text{ obs.}-\text{Value of }\big(17\big)^{th} \text{ obs.}\big)\\ &=81+0.1\big(83 -81\big)\\ &=81.2\text{ mmHg} \end{aligned} $$
$$ \begin{aligned} S_k &= \frac{D_9+D_1 - 2D_5}{D_9 -D_1}\\ &=\frac{81.2+62.9 - 2* 73.5}{81.2 - 62.9}\\ &=-0.1585 \end{aligned} $$
As the coefficient of skewness $S_k$ is $\text{less than zero}$ (i.e., $S_k < 0$), the distribution is $\text{negatively skewed}$.
The following data are the heights, correct to the nearest centimeters, for a group of children:
137, 138, 141, 143, 144, 146, 147, 152, 154, 161
$$ \begin{aligned} D_1 &=\text{Value of }\bigg(\dfrac{1(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{1(20+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(2.1\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(2\big)^{th} \text{ obs.}\\ & +0.1 \big(\text{Value of } \big(3\big)^{th}\text{ obs.}-\text{Value of }\big(2\big)^{th} \text{ obs.}\big)\\ &=129+0.1\big(129 -129\big)\\ &=129\text{ cm} \end{aligned} $$
$$ \begin{aligned} D_5 &=\text{Value of }\bigg(\dfrac{5(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{5(20+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(10.5\big)^{th} \text{ obs.}\\ &=137\text{ cm} \end{aligned} $$
$$ \begin{aligned} D_9 &=\text{Value of }\bigg(\dfrac{9(n+1)}{10}\bigg)^{th} \text{ obs.}\\ &=\text{Value of }\bigg(\dfrac{9(20+1)}{10}\bigg)^{th} \text{ obs.}\\ &= \text{ Value of }\big(18.9\big)^{th} \text{ obs.}\\ &= \text{Value of }\big(18\big)^{th} \text{ obs.}\\ & +0.9 \big(\text{Value of } \big(19\big)^{th}\text{ obs.}-\text{Value of }\big(18\big)^{th} \text{ obs.}\big)\\ &=152+0.9\big(154 -152\big)\\ &=153.8\text{ cm} \end{aligned} $$
$$ \begin{aligned} S_k &= \frac{D_9+D_1 - 2D_5}{D_9 -D_1}\\ &=\frac{153.8+129 - 2* 137}{153.8 - 129}\\ &=0.3548 \end{aligned} $$
In this tutorial, you learned about formula for Kelly's coefficient of skewness for ungrouped data and how to calculate Kelly's coefficient of skewness for ungrouped data. You also learned about how to solve numerical problems based on Kelly's coefficient of skewness for ungrouped data.
To learn more about other descriptive statistics measures, please refer to the following tutorials:
Let me know in the comments if you have any questions on Kelly's coefficient of skewness calculator for ungrouped data with examples and your thought on this article.
Categories All Calculators, Descriptive Statistics, Statistics, Statistics-Calc Tags coefficient of skewness, descriptive statistics, Kelly's coefficient of skewness, skewness, Skewness calculator
Kelly's coefficient of skewness for grouped data | Formula | Examples
Confidence Interval for Mean Calculator | CommonCrawl |
In pentagon $ABCDE$, $BC=CD=DE=2$ units, $\angle E$ is a right angle and $m \angle B = m \angle C = m \angle D = 135^\circ$. The length of segment $AE$ can be expressed in simplest radical form as $a+2\sqrt{b}$ units. What is the value of $a+b$?
We draw the pentagon as follows, and draw altitude $\overline{BG}$ from $B$ to $\overline{AE}$. Since $\angle BAG = 45^\circ$, $AG=GB$.
[asy]
import olympiad;
draw((0,0)--(1,0)--(1+1/sqrt(2),1/sqrt(2))--(1+1/sqrt(2),1+1/sqrt(2))--(-1-1/sqrt(2),1+1/sqrt(2))--cycle);
draw((0,1+1/sqrt(2))--(0,0));
draw(rightanglemark((0,0),(0,1+1/sqrt(2)),(-1-1/sqrt(2),1+1/sqrt(2))));
label("$B$",(0,0),SW);
label("$G$",(0,1+1/sqrt(2)),N);
label("$C$",(1,0),SE);
label("$D$",(1+1/sqrt(2),1/sqrt(2)),E);
label("$E$",(1+1/sqrt(2),1+1/sqrt(2)),NE); label("$A$",(-1-1/sqrt(2),1+1/sqrt(2)),NW);
label("2",(.5,0),S); label("2",(1.7,1.2),E); label("2",(1.3,.5));
draw((1,0)--(1+1/sqrt(2),0)--(1+1/sqrt(2),1/sqrt(2)),dashed);
label("$F$",(1+1/sqrt(2),0),SE);
[/asy] We extend lines $BC$ and $ED$ past points $C$ and $D$ respectively until they intersect at $F$. $\triangle CFD$ is a 45-45-90 triangle with $CF=FD=\frac{2}{\sqrt{2}}=\sqrt{2}$. So $GBFE$ is a square with side length $2+\sqrt{2}$, and $AG = BG = 2+\sqrt{2}$. It follows that $AE = AG + GE = 2(2+\sqrt{2}) = 4+2\sqrt{2}$, and finally $a+b = \boxed{6}$. | Math Dataset |
Direct Interaction between Ras Homolog Enriched in Brain and FK506 Binding Protein 38 in Cashmere Goat Fetal Fibroblast Cells
Wang, Xiaojing;Wang, Yanfeng;Zheng, Xu;Hao, Xiyan;Liang, Yan;Wu, Manlin;Wang, Xiao;Wang, Zhigang 1671
https://doi.org/10.5713/ajas.2014.14145 PDF KSCI
Ras homolog enriched in brain (Rheb) and FK506 binding protein 38 (FKBP38) are two important regulatory proteins in the mammalian target of rapamycin (mTOR) pathway. There are contradictory data on the interaction between Rheb and FKBP38 in human cells, but this association has not been examined in cashmere goat cells. To investigate the interaction between Rheb and FKBP38, we overexpressed goat Rheb and FKBP38 in goat fetal fibroblasts, extracted whole proteins, and performed coimmunoprecipitation to detect them by western blot. We found Rheb binds directly to FKBP38. Then, we constructed bait vectors (pGBKT7-Rheb/FKBP38) and prey vectors (pGADT7-Rheb/FKBP38), and examined their interaction by yeast two-hybrid assay. Their direct interaction was observed, regardless of which plasmid served as the prey or bait vector. These results indicate that the 2 proteins interact directly in vivo. Novel evidence is presented on the mTOR signal pathway in Cashmere goat cells.
Genomic Selection for Adjacent Genetic Markers of Yorkshire Pigs Using Regularized Regression Approaches
Park, Minsu;Kim, Tae-Hun;Cho, Eun-Seok;Kim, Heebal;Oh, Hee-Seok 1678
This study considers a problem of genomic selection (GS) for adjacent genetic markers of Yorkshire pigs which are typically correlated. The GS has been widely used to efficiently estimate target variables such as molecular breeding values using markers across the entire genome. Recently, GS has been applied to animals as well as plants, especially to pigs. For efficient selection of variables with specific traits in pig breeding, it is required that any such variable selection retains some properties: i) it produces a simple model by identifying insignificant variables; ii) it improves the accuracy of the prediction of future data; and iii) it is feasible to handle high-dimensional data in which the number of variables is larger than the number of observations. In this paper, we applied several variable selection methods including least absolute shrinkage and selection operator (LASSO), fused LASSO and elastic net to data with 47K single nucleotide polymorphisms and litter size for 519 observed sows. Based on experiments, we observed that the fused LASSO outperforms other approaches.
Molecular Characterization and Tissue Distribution of Estrogen Receptor Genes in Domestic Yak
Fu, Mei;Xiong, Xian-Rong;Lan, Dao-Liang;Li, Jian 1684
Estrogen and its receptors are essential hormones for normal reproductive function in males and females during developmental stage. To better understand the effect of estrogen receptor (ER) gene in yak (Bos grunniens), reverse transcription-polymerase chain reaction (PCR) was carried out to clone $ER{\alpha}$ and $ER{\beta}$ genes. Bioinformatics methods were used to analyze the evolutionary relationship between yaks and other species, and real-time PCR was performed to identify the mRNA expression of $ER{\alpha}$ and $ER{\beta}$. Sequence analysis showed that the ER open reading frames (ORFs) encoded 596 and 527 amino acid proteins. The yak $ER{\alpha}$ and $ER{\beta}$ shared 45.3% to 99.5% and 53.9% to 99.1% protein sequence identities with other species homologs, respectively. Real-time PCR analysis revealed that $ER{\alpha}$ and $ER{\beta}$ were expressed in a variety of tissues, but the expression level of $ER{\alpha}$ was higher than that of $ER{\beta}$ in all tissues, except testis. The mRNA expression of $ER{\alpha}$ was highest in the mammary gland, followed by uterus, oviduct, and ovary, and lowest in the liver, kidney, lung, testis, spleen, and heart. The $ER{\beta}$ mRNA level was highest in the ovary; intermediary in the uterus and oviduct; and lowest in the heart, liver, spleen, lung, kidney, mammary gland, and testis. The identification and tissue distribution of ER genes in yaks provides a foundation for the further study on their biological functions.
VCS: Tool for Visualizing Copy Number Variation and Single Nucleotide Polymorphism
Kim, HyoYoung;Sung, Samsun;Cho, Seoae;Kim, Tae-Hun;Seo, Kangseok;Kim, Heebal 1691
Copy number variation (CNV) or single nucleotide phlyorphism (SNP) is useful genetic resource to aid in understanding complex phenotypes or deseases susceptibility. Although thousands of CNVs and SNPs are currently avaliable in the public databases, they are somewhat difficult to use for analyses without visualization tools. We developed a web-based tool called the VCS (visualization of CNV or SNP) to visualize the CNV or SNP detected. The VCS tool can assist to easily interpret a biological meaning from the numerical value of CNV and SNP. The VCS provides six visualization tools: i) the enrichment of genome contents in CNV; ii) the physical distribution of CNV or SNP on chromosomes; iii) the distribution of log2 ratio of CNVs with criteria of interested; iv) the number of CNV or SNP per binning unit; v) the distribution of homozygosity of SNP genotype; and vi) cytomap of genes within CNV or SNP region.
Maternal Low-protein Diet Alters Ovarian Expression of Folliculogenic and Steroidogenic Genes and Their Regulatory MicroRNAs in Neonatal Piglets
Sui, Shiyan;Jia, Yimin;He, Bin;Li, Runsheng;Li, Xian;Cai, Demin;Song, Haogang;Zhang, Rongkui;Zhao, Ruqian 1695
Maternal malnutrition during pregnancy may give rise to female offspring with disrupted ovary functions in adult age. Neonatal ovary development predisposes adult ovary function, yet the effect of maternal nutrition on the neonatal ovary has not been described. Therefore, here we show the impact of maternal protein restriction on the expression of folliculogenic and steroidogenic genes, their regulatory microRNAs and promoter DNA methylation in the ovary of neonatal piglets. Sows were fed either standard-protein (SP, 15% crude protein) or low-protein (LP, 7.5% crude protein) diets throughout gestation. Female piglets born to LP sows showed significantly decreased ovary weight relative to body weight (p<0.05) at birth, which was accompanied with an increased serum estradiol level (p<0.05). The LP piglets demonstrated higher ratio of bcl-2 associated X protein/B cell lymphoma/leukemia-2 mRNA (p<0.01), which was associated with up-regulated mRNA expression of bone morphogenic protein 4 (BMP4) (p<0.05) and proliferating cell nuclear antigen (PCNA) (p<0.05). The steroidogenic gene, cytochrome P450 aromatase (CYP19A1) was significantly down-regulated (p<0.05) in LP piglets. The alterations in ovarian gene expression were associated with a significant down-regulation of follicle-stimulating hormone receptor mRNA expression (p<0.05) in LP piglets. Moreover, three microRNAs, including miR-423-5p targeting both CYP19A1 and PCNA, miR-378 targeting CYP19A1 and miR-210 targeting BMP4, were significantly down-regulated (p<0.05) in the ovary of LP piglets. These results suggest that microRNAs are involved in mediating the effect of maternal protein restriction on ovarian function through regulating the expression of folliculogenic and steroidogenic genes in newborn piglets.
In vitro and Lactation Responses in Mid-lactating Dairy Cows Fed Protected Amino Acids and Fat
Nam, I.S.;Choi, J.H.;Seo, K.M.;Ahn, J.H. 1705
The objective of this study was to evaluate the effect of ruminally protected amino acids (RPAAs) and ruminally protected fat (RPF) supplementation on ruminal fermentation characteristics (in vitro) and milk yield and milk composition (in vivo). Fourteen mid-lactating Holstein dairy cows (mean weight $653{\pm}62.59kg$) were divided into two groups according to mean milk yield and number of days of postpartum. The cows were then fed a basal diet during adaptation (2 wk) and experimental diets during the treatment period (6 wk). Dietary treatments were i) a basal diet (control) and ii) basal diet containing 50 g of RPAAs (lysine and methionine, 3:1 ratio) and 50 g of RPF. In rumen fermentation trail (in vitro), RPAAs and RPF supplementation had no influence on the ruminal pH, dry matter digestibility, total volatile fatty acid production and ammonia-N concentration. In feeding trial (in vivo), milk yield (p<0.001), 4% fat corrected milk (p<0.05), milk fat (p<0.05), milk protein (p<0.001), and milk urea nitrogen (p<0.05) were greater in cows fed RPAAs and RPF than the corresponding values in the control group. With an index against as 0%, the rates of decrease in milk yield and milk protein were lower in RPAAs and RPF treated diet than those of basal diet group (p<0.05). In conclusion, diet supplemented with RPAAs and RPF can improve milk yield and milk composition without negatively affecting ruminal functions in Holstein dairy cows at mid-lactating.
Energy Requirements in Early Life Are Similar for Male and Female Goat Kids
Bompadre, T.F.V.;Neto, O. Boaventura;Mendonca, A.N.;Souza, S.F.;Oliveira, D.;Fernandes, M.H.M.R.;Harter, C.J.;Almeida, A.K.;Resende, K.T.;Teixeira, I.A.M.A. 1712
Little is known about the gender differences in energetic requirements of goats in early life. In this study, we determined the energy requirements for maintenance and gain in intact male, castrated male and female Saanen goat kids using the comparative slaughter technique and provide new data on their body composition and energy efficiency. To determine the energy requirements for maintenance, we studied 21 intact males, 15 castrated males and 18 females ($5.0{\pm}0.1kg$ initial body weight (BW) and $23{\pm}5d$ of age) using a split-plot design with the following main factors: three genders (intact males, castrated males, and females) and three dry matter intake levels (ad libitum, 75% and 50% of ad libitum intake). A slaughter group included three kids, one for each nutritional plane, of each gender, and all three animals within a group were slaughtered when the ad libitum kid reached 15 kg in BW. Net energy requirements for gain were obtained for 17 intact males, eight castrated males and 15 females ($5.1{\pm}0.4kg$ BW and $23{\pm}13d$ of age). Animals were fed ad libitum and slaughtered when they reached 5, 10, and 15 kg in BW. A digestion trial was performed with nine kids of each gender to determine digestible energy, metabolizable energy and energy metabolizability of the diet. Our results show no effect of gender on the energy requirements for maintenance and gain, and overall net energy for maintenance was $205.6kJ/kg^{0.75}$ empty body weight gain (EBW) ($170.3kJ/kg^{0.75}$ BW) from 5 to 15 kg BW. Metabolizable energy for maintenance was calculated by iteration, assuming heat production equal to metabolizable energy intake at maintenance, and the result was $294.34kJ/kg^{0.75}$ EBW and $k_m$ of 0.70. As BW increased from 5 to 15 kg for all genders, the net energy required for gain increased from 9.5 to 12.0 kJ/g EBW gain (EWG), and assuming $k_g=0.47$, metabolizable energy for gain ranged from 20.2 to 25.5 kJ/g EWG. Our results indicate that it is not necessary to formulate diets with different energetic content for intact male, castrated male and female Saanen goat kids weighing from 5 to 15 kg.
Effects of Coconut Materials on In vitro Ruminal Methanogenesis and Fermentation Characteristics
Kim, E.T.;Park, C.G.;Lim, D.H.;Kwon, E.G.;Ki, K.S.;Kim, S.B.;Moon, Y.H.;Shin, N.H.;Lee, S.S. 1721
The objective of this study was to evaluate the in vitro effects of coconut materials on ruminal methanogenesis and fermentation characteristics, in particular their effectiveness for mitigating ruminal methanogenesis. Fistulated Holstein cows were used as the donor of rumen fluid. Coconut materials were added to an in vitro fermentation incubated with rumen fluid-buffer mixture and timothy substrate for 24 h incubation. Total gas production, gas profiles, total volatile fatty acids (tVFAs) and the ruminal methanogens diversity were measured. Although gas profiles in added coconut oil and coconut powder were not significantly different, in vitro ruminal methane production was decreased with the level of reduction between 15% and 19% as compared to control, respectively. Coconut oil and coconut powder also inhibited gas production. The tVFAs concentration was increased by coconut materials, but was not affected significantly as compared to control. Acetate concentration was significantly lower (p<0.05), while propionate was significantly higher (p<0.05) by addition of the coconut materials than that of the control. The acetate:propionate ratio was significantly lowered with addition of coconut oil and coconut powder (p<0.05). The methanogens and ciliate-associated methanogens in all added coconut materials were shown to decrease as compared with control. This study showed that ciliate-associated methanogens diversity was reduced by more than 50% in both coconut oil and coconut powder treatments. In conclusion, these results indicate that coconut powder is a potential agent for decreasing in vitro ruminal methane production and as effective as coconut oil.
Effects of Acarbose Addition on Ruminal Bacterial Microbiota, Lipopolysaccharide Levels and Fermentation Characteristics In vitro
Yin, Yu-Yang;Liu, Yu-Jie;Zhu, Wei-Yun;Mao, Sheng-Yong 1726
This study investigated the effects of acarbose addition on changes in ruminal fermentation characteristics and the composition of the ruminal bacterial community in vitro using batch cultures. Rumen fluid was collected from the rumens of three cannulated Holstein cattle fed forage ad libitum that was supplemented with 6 kg of concentrate. The batch cultures consisted of 8 mL of strained rumen fluid in 40 mL of an anaerobic buffer containing 0.49 g of corn grain, 0.21 g of soybean meal, 0.15 g of alfalfa and 0.15g of Leymus chinensis. Acarbose was added to incubation bottles to achieve final concentrations of 0.1, 0.2, and 0.4 mg/mL. After incubation for 24 h, the addition of acarbose linearly decreased (p<0.05) the total gas production and the concentrations of acetate, propionate, butyrate, total volatile fatty acids, lactate and lipopolysaccharide (LPS). It also linearly increased (p<0.05) the ratio of acetate to propionate, the concentrations of isovalerate, valerate and ammonia-nitrogen and the pH value compared with the control. Pyrosequencing of the 16S rRNA gene showed that the addition of acarbose decreased (p<0.05) the proportion of Firmicutes and Proteobacteria and increased (p<0.05) the percentage of Bacteroidetes, Fibrobacteres, and Synergistetes compared with the control. A principal coordinates analysis plot based on unweighted UniFrac values and molecular variance analysis revealed that the structure of the ruminal bacterial communities in the control was different to that of the ruminal microbiota in the acarbose group. In conclusion, acarbose addition can affect the composition of the ruminal microbial community and may be potentially useful for preventing the occurrence of ruminal acidosis and the accumulation of LPS in the rumen.
Rice Distillers Dried Grain Is a Promising Ingredient as a Partial Replacement of Plant Origin Sources in the Diet for Juvenile Red Seabream (Pagrus major)
Choi, Jin;Rahman, Md. Mostafizur;Lee, Sang-Min 1736
This study was designed to test the effects of dietary distillers dried grain (DDG) level on the growth performance, feed utilization, body composition and antioxidant activity of juvenile red seabream (Pagrus major). Six isonitrogenous and isocaloric diets were formulated to contain 0%, 5%, 10%, 15%, 20%, and 25% DDG from rice (designated as DDG0, DDG5, DDG10, DDG15, DDG20, and DDG25), respectively. Juvenile red seabream averaging $10.1{\pm}0.05g$ were randomly distributed into 400-L tanks in a flow through systems. Three replicate groups of fish were fed one of the experimental diets to visual satiation two times a day for 10 weeks. Survival, weight gain, feed efficiency, protein efficiency ratio and hepatosomatic index of fish were not affected by dietary DDG levels (p>0.05). Proximate and amino acid composition of whole body in juvenile red seabream were not affected by dietary DDG levels (p>0.05). Plasma content of total protein, glucose, cholesterol, glutamic-pyruvic transaminase, phospholipid and triglyceride were not affected by dietary DDG levels (p>0.05). 1, 1-Diphenyl-2-picryl-hydrazyl radical and alkyl radical scavenging activities in plasma and liver of fish were not affected by dietary DDG levels (p>0.05). The results of this experiment suggest that DDG has the potential to replace plant origin ingredients such as wheat flour and corn gluten meal and could be used up to 25% in diet without incurring negative effects on the growth performance of juvenile red seabream.
Effects of Pyridoxine on Growth Performance and Plasma Aminotransferases and Homocysteine of White Pekin Ducks
Xie, Ming;Tang, Jing;Wen, Zhiguo;Huang, Wei;Hou, Shuisheng 1744
A dose-response experiment with seven supplemental pyridoxine levels (0, 0.66, 1.32, 1.98, 2.64, 3.30, and 3.96 mg/kg) was conducted to investigate the effects of pyridoxine on growth performance and plasma aminotransferases and homocysteine of White Pekin ducks and to estimate pyridoxine requirement for these birds. A total of 336 one-day-old male White Pekin ducks were divided to 7 experimental treatments and each treatment contained 8 replicate pens with 6 birds per pen. Ducks were reared in raised wire-floor pens from hatch to 28 d of age. At 28 d of age, the weight gain, feed intake, feed/gain, and the aspartate aminotransferase, alanine aminotransferase, and homocysteine in plasma of ducks from each pen were all measured. In our study, the pyridoxine deficiency of ducks was characterized by growth depression, decreasing plasma aspartate aminotransferase activity and increasing plasma homocysteine. The ducks fed vitamin $B_6$-deficient basal diets had the worst weight gain and feed/gain among all birds and this growth depression was alleviated (p<0.05) when pyridoxine was supplemented to basal diets. On the other hand, plasma aspartate aminotransferase and homocysteine may be the sensitive indicators for vitamin $B_6$ status of ducks. The ducks fed basal diets had much lower aspartate aminotransferase activity and higher homocysteine level in plasma compared with other birds fed pyridoxine-supplemented diets (p<0.05). According to quadratic regression, the supplemental pyridoxine requirements of Pekin ducks from hatch to 28 days of age was 2.44 mg/kg for feed/gain and 2.08 mg/kg for plasma aspartate aminotransferase and the corresponding total requirements of this vitamin for these two criteria were 4.37 and 4.01 mg/kg when the pyridoxine concentration of basal diets was included, respectively. All data suggested that pyridoxine deficiency could cause growth retardation in ducks and the deficiency of this vitamin could be indicated by decreasing plasma aspartate aminotransferase activity and increasing plasma homocysteine.
Evaluation of Dietary Multiple Enzyme Preparation (Natuzyme) in Laying Hens
Lee, K.W.;Choi, Y.I.;Moon, E.J.;Oh, S.T.;Lee, H.H.;Kang, C.W.;An, B.K. 1749
The current experiment was designed to evaluate the efficacy of adding the multi-enzyme mixture (Natuzyme) into layers' diets with different levels of energy and available phosphorus in relation to laying performance, egg qualities, blood cholesterol level, microflora and intestinal viscosity. Two hundred and fifty 43-wk-old Hy-Line commercial layers were divided into five groups with five replicates per group (10 birds per replicate) and fed one of five experimental diets. A corn and soybean meal-based control diet was formulated and used as a control diet. Two experimental control diets were formulated to reduce energy and crude protein contents (rE) or energy, crude protein and phosphorus contents (rEP). In addition, Natuzyme was added into either rE (rE-Natu500) or rEP (rEP-Natu500) diet to reach a concentration of 500 mg per kg of diet. The experiment lasted 8 weeks. There were no significant differences in feed intake, egg production, egg weight, egg qualities such as eggshell color or Haugh unit, total cholesterol, relative organ weights and cecal microflora profiles between any dietary treatments. Natu500 supplementation into the rE diet, but not rEP diet significantly increased egg mass and eggshell qualities such as strength and thickness, but it decreased cecal ammonia concentration and intestinal viscosity in laying hens. In conclusion, the present study shows that adding multiple enzyme preparation could improve performance of laying hens fed energy and protein restricted diets.
Effects of Maize Source and Complex Enzymes on Performance and Nutrient Utilization of Broilers
Tang, Defu;Hao, Shengyan;Liu, Guohua;Nian, Fang;Ru, Yingjun 1755
The objective of this study was to investigate the effect of maize source and complex enzymes containing amylase, xylanase and protease on performance and nutrient utilization of broilers. The experiment was a $4{\times}3$ factorial design with diets containing four source maize samples (M1, M2, M3, and M4) and without or with two kinds of complex enzyme A (Axtra XAP) and B (Avizyme 1502). Nine hundred and sixty day old Arbor Acres broiler chicks were used in the trial (12 treatments with 8 replicate pens of 10 chicks). Birds fed M1 diet had better body weight gain (BWG) and lower feed/gain ratio compared with those fed M3 diet and M4 diet (p<0.05). Apparent ileal crude protein digestibility coefficient of M2 was higher than that of M3 (p<0.05). Apparent metabolisable energy (AME) and nitrogen corrected AME (AMEn) of M1 were significant higher than those of M4 (p<0.05). Supplementation of the basal diets with enzyme A or B improved the BWG by 8.6% (p<0.05) and 4.1% (p>0.05), respectively. The fresh feces output was significantly decreased by the addition of enzyme B (p<0.05). Maize source affects the nutrients digestibility and performance of broilers, and a combination of amylase, xylanase and protease is effective in improving the growth profiles of broilers fed maize-soybean-rapeseed-cotton mixed diets.
The Expression of Carnosine and Its Effect on the Antioxidant Capacity of Longissimus dorsi Muscle in Finishing Pigs Exposed to Constant Heat Stress
Yang, Peige;Hao, Yue;Feng, Jinghai;Lin, Hai;Feng, Yuejin;Wu, Xin;Yang, Xin;Gu, Xianhong 1763
The objective of this study was to assess the effects of constant high ambient temperatures on meat quality, antioxidant capacity, and carnosine expression in longissimus dorsi muscle of finishing pigs. Castrated 24 male DLY (crossbreeds between Landrace${\times}$Yorkshire sows and Duroc boars) pigs were allocated to one of three treatments: constant ambient temperature at $22^{\circ}C$ and ad libitum feeding (CON, n = 8); constant high ambient temperature at $30^{\circ}C$ and ad libitum feeding (H30, n = 8); and constant ambient temperature at $22^{\circ}C$ and pair-fed with H30 (PF, n = 8). Meat quality, malondialdehyde (MDA) content, antioxidant capacity, carnosine content, and carnosine synthetase (CARNS1) mRNA expression in longissimus dorsi muscle were measured after three weeks. The results revealed that H30 had lower $pH_{24h}$, redness at 45 min, and yellowness at 24 h post-mortem (p<0.05), and higher drip loss at 48 h and lightness at 24 h post-mortem (p<0.01). Constant heat stress disrupted the pro-oxidant/antioxidant balance in longissimus dorsi muscle with higher MDA content (p<0.01) and lower antioxidant capacity (p<0.01). Carnosine content and CARNS1 mRNA expression in longissimus dorsi muscle of H30 pigs were significantly decreased (p<0.01) after three weeks at $30^{\circ}C$. In conclusion, constant high ambient temperatures affect meat quality and antioxidant capacity negatively, and the reduction of muscle carnosine content is one of the probable reasons.
Meat Quality Characteristics of Small East African Goats and Norwegian Crosses Finished under Small Scale Farming Conditions
Hozza, W.A.;Mtenga, L.A.;Kifaro, G.C.;Shija, D.S.N.;Mushi, D.E.;Safari, J.G.;Shirima, E.J.M. 1773
The aim of the experiment was to study the effect of feeding system on meat quality characteristics of Small East African (SEA) goats and their crosses with Norwegian ($SEA{\times}N$) goats finished under small scale farming conditions. Twenty four castrated goats at the age of 18 months with live body weight of $16.7{\pm}0.54kg$ from each breed (SEA and $SEA{\times}N$) were distributed in a completely randomized design in a $2{\times}3$ factorial arrangement (two breed, and three dietary treatments). The dietary treatments were; no access to concentrate (T0), 66% access to ad libitum concentrate allowance (T66) and 100% access to ad libitum concentrate allowance with 20% refusal (T100) and the experimental period was for 84 days. In addition, all goats were allowed to graze for 2 hours daily and later fed grass hay on ad libitum basis. Daily feed intakes were recorded for all 84-days of experiment after which the animals were slaughtered. Feed intake of T100 animals was 536 g/d, which was 183 g/d higher than that of T66 group. Supplemented goats had significantly (p<0.05) better feed conversion efficiency. The SEA had higher (p<0.05) hot carcass weight (8.2 vs 7.9 kg), true dressing percentage (54.5 vs 53.3) and commercial dressing percentage (43.3 vs 41.6) compared to $SEA{\times}N$. There was no significant difference (p>0.05) for dressing percentage and carcass conformation among supplemented goats except fatness score, total fat depots and carcass fat which increased (p<0.05) with increasing concentrate levels in the diet. Increasing level of concentrate on offer increased meat dry matter with subsequent increase of fat in the meat. Muscle pH of goats fed concentrate declined rapidly and reached below 6 at 6 h post-mortem but temperature remained at $28^{\circ}C$. Cooking loss and meat tenderness improved (p<0.05) and thawing loss increased (p<0.05) with ageing period. Similarly, meat tenderness improved (p<0.05) with concentrate supplementation. Shear force of muscles varied from 36 to 66, the high values been associated with Semimembranosus and Gluteobiceps muscles. The present study demonstrates that there are differences in meat quality characteristics of meat from SEA goats and their crosses with Norwegian breeds finished under small scale farming conditions in rural areas. Therefore, concentrate supplementation of goats of both breeds improves meat quality attributes.
Effects of Capsaicin on Adipogenic Differentiation in Bovine Bone Marrow Mesenchymal Stem Cell
Jeong, Jin Young;Suresh, Sekar;Park, Mi Na;Jang, Mi;Park, Sungkwon;Gobianand, Kuppannan;You, Seungkwon;Yeon, Sung-Heom;Lee, Hyun-Jeong 1783
Capsaicin is a major constituent of hot chili peppers that influences lipid metabolism in animals. In this study, we explored the effects of capsaicin on adipogenic differentiation of bovine bone marrow mesenchymal stem cells (BMSCs) in a dose- and time-dependent manner. The BMSCs were treated with various concentrations of capsaicin (0, 0.1, 1, 5, and $10{\mu}M$) for 2, 4, and 6 days. Capsaicin suppressed fat deposition significantly during adipogenic differentiation. Peroxisome proliferator-activated receptor gamma, cytosine-cytosine-adenosine-adenosine-thymidine/enhancer binding protein alpha, fatty acid binding protein 4, and stearoyl-CoA desaturase expression decreased after capsaicin treatment. We showed that the number of apoptotic cells increased in dose- and time-dependent manners. Furthermore, we found that capsaicin increased the expression levels of apoptotic genes, such as B-cell lymphoma 2-associated X protein and caspase 3. Overall, capsaicin inhibits fat deposition by triggering apoptosis. | CommonCrawl |
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MSP101 is an ongoing series of informal talks by visiting academics or members of the MSP group. The talks are usually Thursday 2pm in room LT1310 in Livingstone Tower. They are announced on the msp-interest mailing-list. The list of talks is also available as a RSS feed and as a calendar file. Due to COVID-19, the MSP 101 seminar is now held online. The Zoom link can be found on the msp-interest mailing list and the SPLS Zulipchat.
2022-01-27 (Online): Ways of Staring at Free Monoidal Categories (Conor McBride, MSP)
What's a free monoidal category? Imagine planar circuits connecting a sequence of inputs to a sequence of outputs, built from components similarly taking a sequence of inputs to a sequence of outputs. They are categorical: you can wire the outputs of one circuit to the inputs of another if they match. They are monoidal: you can place them in parallel, concatenating input and output sequences. They have an equational theory which amounts to allowing arbitrary jiggling around in the plane, just as long as you keep everything connected in the same way and treat wires and components as deformable but impassable boundaries. Can we decide this equational theory? Yes, in principle, but how might we do it in practice? What would it take to give unique normal forms for such circuits as inductively defined data? Fred, Jules and I were staring at this puzzle one evening, looking over a notebook for a clue to plant upon the page, in a pub not so many minutes away known for the central stairway up to its mezzanine. We were staring at circuits and staring at the stairway, until we began to see the stairways in the circuits. And it made us wonder!
2022-02-03 (Online): TBA (Matteo Capucci, MSP)
2022-02-10 (Online): TBA (Jérémy Ledent, MSP)
2022-02-17 (Online): TBA (Bruno Gavranović, MSP)
2022-02-24 (Online): TBA (Bob Atkey, MSP)
2022-03-03 (Online): TBA (Fredrik Nordvall Forsberg, MSP)
2022-03-09: SPLS (University of Edinburgh / Online)
2022-03-17 (Online): TBA (Andre Videla, MSP)
2022-03-24 (Online): TBA (William Waites, MSP)
List of previous talks
2022-01-12: MSP101 Planning meeting (Online)
2021-12-15, 15:00 (Online): Data types with negative information (Bob Atkey, MSP)
Inductive data types are a foundational tool for representing data and knowledge in dependently typed programming languages. The user provides a collection of rules that determine positive evidence for membership in the type. Elimination of an inductive type corresponds to structural induction on its members.
But what if our data modelling requires negative information as well as positive? For example, representing the result of a backtracking parser requires evidence that a certain parsing attempt *didn't* work. Standard formulations of inductive types do not allow negative information mixed with positive.
Mixing positive and negative information has been studied in logic programming, resulting in concepts like Negation as Failure, stable models, and answer set programming. Incorporating negative information into systems of rules leads us into the realm of non-monotonic logics, where simply adding knowledge does not necessarily preserve existing conclusions.
In this talk, I'll describe a way to understand data types with negative information in type theory by combining ideas from 3-valued stable models in logic programming and constructive negation (or refutation) from linear logic. The hope is that by adding negation to data types we will be able to represent some reasoning tasks more easily in dependent types.
2021-12-08, 15:00 (Online): Constructing disjunctive formulas in the modal mu-calculus (Clemens Kupke, MSP)
The modal mu-calculus is a general modal fixpoint logic that allows to reason about the ongoing, possibly infinite behaviour of a transition system. For example, the temporal logics from Sean's 101 talk a few weeks ago can be seen as fragments of the mu-calculus.
In this talk I am planning to give a brief introduction to the mu-calculus and then focus on a key result about the logic: every formula is equivalent to a so-called disjunctive formula. Constructing a disjunctive formula of "small" size is important e.g. for satisfiability checking. I am going to sketch a construction that yields a disjunctive formula that is single-exponential in the closure size of the input formula. The construction will highlight a few subtleties that arise when thinking about the size of a mu-calculus formula.
Based on joint work with Johannes Marti and Yde Venema.
2021-12-01, 15:00 (Online): When completeness is not enough: introduction to algebraisable logics (Georgi Nakov, MSP)
The relation between logic and algebra is often expressed by existence of equational completeness theorems stating that a certain logic L is complete with respect to a certain class of algebras K. Perhaps the most famous example is given by the class of Boolean algebras and classical propositional logic. However, CPC is also complete with respect to the class of Heyting algebras by a negative translation. In order to recover a univocal connection, we need to dig deeper into the theory of algebraisability [1] - an algebraisable logic admits a unique algebraic semantics.
In this talk we will introduce some notions and basic results from the theory of algebraisable logics. We will further propose an extension to the theory to accommodate for logics with weaker forms of substitutions (e.g. logics of dependence and independence, various epistemic logics).
This is joint work with Davide Quadrellaro.
[1] W. J. Blok and D. Pigozzi. Algebraizable logics, https://bookstore.ams.org/memo-77-396/
2021-11-24, 15:00 (Online): Homotopic and Compositional Aspects of (Hyper)graph Rewriting and Fundamental Physics (Jonathan Gorard, University of Cambridge)
Graph grammars and double-pushout rewriting are pretty old news at this point. But the abstract rewriting systems produced by (hyper)graph grammars actually carry an extremely rich compositional and algebraic structure, many aspects of which have remained woefully under-investigated. We will show how causality relations in non-deterministic hypergraph rewriting systems can effectively be captured by means of a strict monoidal 2-category (a "multiway system"), and how higher homotopy types may consequently be constructed by adjoining appropriate sets of additional rewriting rules, with the n → ∞ limit of the resulting n-fold category yielding an ∞-groupoid (the classifying space of which forms an (∞,1)-topos). We will discuss how this general framework has been applied pragmatically in the design of efficient automated theorem-proving and diagrammatic reasoning software, before concluding with a more speculative discussion of the potential applicability of the overall formalism to the foundations of physics (especially categorical quantum mechanics and discrete quantum gravity).
This is joint with Xerxes Arsiwalla.
2021-11-17, 15:00 (Online): Optics in three acts (Matteo Capucci, MSP)
Optics can be constructed in at least three different ways: from Tambara theory (profunctor encoding), as open diagrams (existential optics) and as free 'categories with a counit' (free teleological categories). Each of these three constructions highlights a different point of view on optics, and their interplay is an elegant mathematical story and a(n arguably) successful story for abstract nonsense. In this play talk, we look at all three constructions and contemplate their equivalence.
2021-11-10, 15:00 (Online): A Tour of Temporal Logic (Sean Watters, MSP)
Modal logic can be applied to a diverse range of use-cases by fixing different notions of model and modality. Our motivation today will be the formal verification technique of model checking, and with this in mind, I will focus on temporal logic. In temporal logics, the models represent systems which change over time, and the modalities allow us to reason about exactly how they proceed in time.
I will first introduce basic relational modal logic, and show its limitations in this domain. We will then go on a tour of LTL, CTL, CTL*, and the modal 𝜇-calculus. Key destinations on the tour will include the respective expressive power and model-checking complexity of each, particularly in comparison to the 𝜇-calculus. I will end the tour with a closer look at the model-checking problem of the 𝜇-calculus, and its relationship to parity games.
2021-11-03, 15:00 (Online): An Introduction to Finite Model Theory (Dylan Braithwaite, MSP)
Model theory in classical logic provides a number of tools useful for classifying structures described by sets of axioms. In computer science we are often interested in cases where these structures are finite, but many results in classical model theory fail to specialise to cases where we require all models to be finite. In this talk I will give an introduction to some of the alternative techniques used for studying expressibility of finite models and discuss some interesting connections with complexity theory.
2021-10-27, 15:00 (Online): Categories for persistent homology (Riu Rodríguez Sakamoto, MSP)
Data analysis has dealt traditionally with regression, clustering and dimensionality reduction. Persistent homology, a tool in Topological Data Analysis, can be thought of as the counterpart of clustering: Instead of looking at clusters of data, it classifies holes of data. And we mean `holes' in the most visual way: given some point cloud in a metric space, it addresses how do we characterize regions of the space where there's no data. This talk will introduce persistence modules, which are a central concept of the theory. We will overview their construction from data, how we can compare between, and stability properties.
2021-10-20: SPLS (Online)
2021-10-13, 15:00 (Online): The changing shapes of cybercats (Toby Smithe, Topos Institute and University of Oxford)
The ménagerie of categorical models of dynamical systems is becoming a veritable zoo, but what makes all these animals tick? In this talk, I will introduce a new specimen: a symmetric monoidal category of continuous-time open Markov processes with general state spaces. I will explain how this category is obtained from a category of ``continuous-time coalgebras'' opindexed by polynomials, and describe how this recipe also gives categories of nondeterministic systems in arbitrary (continuous) time. These new specimens are motivated by the cybernetic question of how to model systems that are continuously performing approximate Bayesian inference. I will therefore sketch why their better-known cousins weren't quite up to the job, and show that our new SMC admits Bayesian inversion. Finally, I will attempt to make contact with the MSP branch of categorical cybernetics, asking what makes the shapes of our structures seem so similar-but-different, and how we might begin to understand systems nested within systems.
2021-10-06, 15:00 (Online): A very abridged introduction to open games (Jules Hedges, MSP)
Open games allow you to talk about some amount of game theory (and thus, some amount of economics) in terms of a lot of familiar ideas from category theory and compositionality. It isn't possible to introduce open games adequately to anyone in this amount of time, but this talk will be the best I can manage in the circumstances. I'll explain approximately how open games work, and how they relate to some other interests of MSP. I'll also do a quick demo of the open game engine, a Haskell implementation that's robust enough to do some Real Economics ™.
2021-09-29, 15:00 (Online): Dependently typed programming: do you know what you are doing? (Fredrik Nordvall Forsberg, MSP)
Wouldn't it be nice if you could tell the computer what you mean, so that it could help you catch not only syntax errors, but also semantics errors? Even better, so it could help you and guide you towards the program that you want to write? So that knowing what you are doing could become a responsibility shared between human and machine, instead of being only your problem? Using dependent types, and some care, we can achieve this by encoding the precise meaning of the program in its type -- any implementation will consequently be correct *by construction*. I will give a demonstration of such dependently typed programming, based in part on a nice recent draft functional pearl by Wouter Swierstra: "A correct-by-construction conversion to combinators" [1]. I will not assume that you know much about correct-by-construction programming, or combinatory logic.
[1] https://webspace.science.uu.nl/~swier004/publications/2021-jfp-submission-2.pdf
Agda file
2021-09-23, 15:00 (Online): Building the behavior of graphs compositionally (Jade Master, MSP)
The free category on a graph may be understood as an operational semantics. The objects of this free category represent states and the morphisms represent possible sequences of events which may occur. How can this operational semantics be built from smaller components? We will see how gluing graphs together is the composition of category and how the operational semantics of graphs extends to a functor on this category. Blackboxing is a process which takes a system and focuses only the relationship it induces between its inputs and outputs. In this talk we will explain how blackboxing is almost a functor, and how the categorical framework developed so far gives insight into computation.
2021-09-16, 15:00 (Online): Introduction to universal coalgebra (Ezra Schoen, MSP)
The aim of this presentation is to give an overview of various concepts within universal coalgebra. We will explore such topics as corecursion, bisimulation and coalgebraic (modal) logic. No prior knowledge is required, save for some elementary category theory.
2021-09-09, 15:00 (Online): All kinds of types (André Videla, MSP)
What are types? What types of types exist? Why do we do bother and what does this buy us? This introduction to type theory aims to motivate the study of type systems and demonstrate their power. No knowledge is required except for basic programming concepts, like functions and data structures. Our journey will take us from the Set theory slums, through the curry-howard plains, into the type-theory ivory tower and end looking at the stars where modal logic and QTT live.
2021-07-15, 15:00 (Online): Expressivity of BCI algebras (Samuel Arsac, MSP)
I will have a look into the expressivity of BCI-algebras, by implementing them in Coq and by looking at them from a categorical point of view. The first step will be to implement the λ* operator as shown in [1], which will allow for the use of linear lambda-calculus. From this point it will be much easier to define useful terms with BCI combinators, and I will use the encoding of booleans in linear lambda-calculus defined in [2] to create a type of duplicable and discardable booleans. In parallel, I will talk about the categorical aspects with the notion of realisability and category of assemblies [1], with the aim of showing that we can obtain a linear-non-linear model.
[1] https://www.kurims.kyoto-u.ac.jp/~naophiko/paper/realizability.pdf
[2] https://www.cs.brandeis.edu/~mairson/Papers/jfp02.pdf
2021-07-08, 15:00 (Online): Bite me! (Rational Fixpoints of Containers) (Conor McBride, MSP)
Least fixpoints, often written mu X. Blah(X), give us inductive types (for strictly positive Blah), e.g. finite lists of A given as mu X. 1 + A*X: all the lists of finite length. Greatest fixpoints, often written nu X. Blah(X), give us possibly infinite type (for strictly positive Blah), e.g. finite lists, but also wild infinite lists like [0,1,2,3,...] which never repeat.
There is no Greek letter between mu and nu. But there is at least one interesting fixpoint between the least and the greatest: things which can loop back on themselves, but not diverge into the wide blue yonder. That is the rational fixpoint.
What are we to do with it? I don't know but I intend to find out.
I shall give a deeply unsatisfactory talk on this topic which leaves more questions than answers, for the purpose of infecting people with this problem. There will be at least one solid clue.
Haskell file
2021-07-01, 15:00 (Online): Translating Extensive Form Games to Open Games with Agency (Matteo Capucci, MSP)
We show open games cover extensive form games with both perfect and imperfect information. Doing so forces us to address two current weaknesses in open games: the lack of a notion of player and their agency within open games, and the lack of choice operators. Using the former we construct the latter, and these choice operators subsume previous proposed operators for open games, thereby making progress towards a core, canonical and ergonomic calculus of game operators. Collectively these innovations increase the level of compositionality of open games, and demonstrate their expressiveness.
This is a practice talk for ACT'21.
2021-06-24 (Online): Optics for generic declarative server APIs (André Videla, MSP)
Martin-Löf type theory has provided us with a new programming paradigm: One where types and terms have shed their differences in order to live harmoniously in the same universe. Despite this successful reunion we have yet to communicate this story to the people building today's software, for whom it is still a fairy-tale, rather than reality.
For this, I will demonstrate how to use dependent types for a purpose that is extremely common in commercial software: Web servers. Web servers are a great example because of how ubiquitous they are. Every service, company or product probably has a web server running behind it in some way (if only to serve web pages). Dependent types help the implementation of servers in small and big ways and the experience can be further enhanced by combining it with lenses in order to reach powerful new levels of abstraction.
2021-06-17, 15:00 (Online): Executable Storytelling with Rule-Based Models (William Waites, LSHTM Centre for the Mathematical Modelling of Infectious Diseases)
I will tell a (very simplified) story about how the adaptive immune system works through the medium of rules. Rules are compositional creatures that can be looked at from the perspective of structured cospans, algebras or double pushout graph rewriting systems. I won't belabour the abstract interpretation, rather I will concentrate on how they can be used to good effect to both explain (normal mode) immune response to a pathogen like SARS-CoV-2 and generate a model that can be simulated and reproduces some interesting heterogeneity that is observed in the world. Furthermore, compositionality means that this set of rules can be freely combined with sets of rules for transmission and diagnostic testing and will show this in action. Finally, I'll speculate about some other kinds of models that it might be interesting to incorporate.
2021-06-10 (Online): Real Numbers in Agda (Bob Atkey, MSP)
In a constructive logic real numbers are even more interesting than they are in the classical world. To demonstrate the differences, I'll talk about how we can construct a type of real numbers in Agda in terms of the Cauchy completion of the metric space of rational numbers. This yields an implementation of real numbers that is reasonably efficient and that we can do proofs about. The basic construction closely follows Russell O'Connor's "A monadic, functional implementation of real numbers" [1]. I'll also talk about using the completion of a metric space to implement quantitative equational theories over complete separable metric spaces in Agda. I'll try to work from the assumption that the audience knows nothing about metric spaces, completion, or construction of real numbers.
[1] https://arxiv.org/abs/cs/0605058
2021-06-03, 15:00 (Online): A relationally parametric model of Quantitative Type Theory (Georgi Nakov, MSP)
Polymorphism allows a single function to be instantiated with multiple types. It is parametric if all of the instances behave uniformly. Reynolds managed to give rigorous formalization of this notion in his abstraction theorem for polymorphic lambda calculus. The key insight is that types may also be interpreted as relations. In this talk, I will give an overview of relational parametricity and some of its consequences, focus on an approach of extending parametricity to dependent type theories using reflexive graphs (due to Atkey, Ghani and Johann) and finish by presenting a relationally parametric model of Quantitative Type Theory.
2021-05-20, 15:00 (Online): A Generic Framework for Analyzing (Featherweight) Java Programs (Chuangjie Xu, fortiss GmbH)
As a notion to represent disjoint sets of memory locations, regions serve as the basis of various techniques for e.g. memory management and pointer analysis. They are closely interrelated with effects, and have been illustrated to be useful for improving the precision of analysis. We generalize the notion of region to represent properties of values, introduce a region type system for Featherweight Java (FJ) that is parametrized with a monad-like structure, and prove a uniform soundness theorem. Its instances include some type systems studied by Martin Hofmann et al. as well as a new one that performs more precise analysis of trace-based program properties. Our region type system is separate from the FJ type system, making it simpler and also easier to move to larger fragments of Java. The uniform framework helps to avoid redundant work on the meta-theory when extending the system to cover other language features such as exception handling. This is joint work with Ulrich Schöpp.
2021-05-13, 15:00 (Online): Formalising the (Sub-)Structural Aspects of SystemVerilog, Again... (Jan de Muijnck-Hughes, Glasgow)
Hardware design is commoditised and it might be the case that several components of your design use encrypted bitstreams bought from third-parties. We must have faith that the encrypted bitstreams do what they are supposed to. In the Border Patrol Project we are interested in being able to reason about the structure & behaviour of designs as a whole, regardless of if we can inspect each module down to the individual gates.
Following on from my SPLS Nov '20 I want to update everyone on my journey in capturing the physical structure of hardware design using lambda-calculi. Specifically, I will re-introduce System-V, a typed lambda calculus that is based upon the well-known hardware description and verification language SystemVerilog. I will show how a System-V design can capture physical hardware design in the Verilog style, and its type-system enforce correct wiring. I will then show how we can formally look beyond System-V itself (and hardware design by extension) by leveraging known-things from programming-language theory, and posit on where to go from here.
2021-05-06, 15:00 (Online): Categorical semantics of the Simply typed lambda calculus (André Videla, MSP)
The Simply typed lambda calculus is a tried and true tool for experimentation and teaching, today we're going to use it as our guide through an introduction to category theory. Category theory without proper motivation or context can be a bit puzzling to get into. But using the simply typed lambda calculus as our framework we will see what it takes in order to interpret it as a Cartesian Closed Category. This walkthough should provide you with the tools to understand Cartesian Closed Category as well as some hands-on experience with proving your first results in Category theory!
2021-04-29, 15:00 (Online): Games with players (Matteo Capucci, MSP)
The algebra of open games is a compelling language for modelling large and medium scale games, with many interacting players. Its most striking features are compositionality and an intuitive and expressive graphical calculus. In this talk I'll go through some recent developments in this area to sketch a general way to build open games, starting from an informal specification, a classical game, or a bunch of given other games. The most important novelty is the presence of a well-defined and correct notion of player, which was missing until now. The guiding principle will be that 'games and players live in orthogonal planes', as suggested by the graphical language of the para construction.
2021-04-01, 15:00 (Online): All about convex sets (Jules Hedges, MSP)
This talk is about the algebras of the finite support probability monad on Set. They consist of a set equipped with a "mixture" operator satisfying some axioms, and are variously known as (abstract) convex sets, convex algebras, barycentric algebras, etc. Every actually-convex subset of a real vector space is a convex set, but there are less expected examples too: every join-semilattice can be seen as a convex set, and there are examples that combine aspects of both vector spaces and semilattices.
I will spend a lot of time on the basic theory and examples, which is mostly due to Marshall Stone. I'll then transition into talking about work in progress with Paolo Perrone and Sharwin Rezagholi, in which we aim to prove a version of Brouwer's fixpoint theorem for convex sets. Specifically, we aim to construct a "topological realisation" functor F: Conv -> Top and find sufficient convex properties on X such that F(X) satisfies the fixpoint property for continuous maps. Along the way we prove a classification theorem: every convex set can be decomposed as a family of convex subsets of vector spaces "fibred over" a semilattice (slightly improving a similar unpublished result by Tobias Fritz).
2021-03-25, 15:00 (Online): Type Preserving Crossover Operations for Genetic Programs (Donovan Crichton, Australian National University)
This talk serves as my formal introduction to the MSP101 Group, and covers work from my Bachelor thesis. We will look at how dependent types can be used to enable a type-safe crossover operation to allow the generation of program examples that are guaranteed to be well typed. We'll also cover alternate approaches I could've taken, as well as further work in the area.
2021-03-18, 15:00 (Online): Propositional Dynamic Logic(s) (Clemens Kupke, MSP)
I will provide an introduction to dynamic modal logics such as Propositional Dynamic Logic (PDL) and Game Logic (GL) and will then describe a (co)algebraic framework for these logics. This framework relates program/game constructs of these logics to monad structure. The axioms of these logics express compatibility requirements between the modal operators and the monad structure.
2021-03-11, 15:00 (Online): Quantitative containers (Fredrik Nordvall Forsberg, MSP)
Quantitative Type Theory combines linear types (where we keep track of how many times a variable is used) and dependent types (where terms can appear in types). This gives a logical system which is both expressive and precise with respect to the resource usage of programs and proofs, with a rich model theory. I will talk about work in progress investigating data types in this setting, in the form of "resource-aware" quantitative containers, and their initial algebra semantics.
This is joint work with Georgi Nakov.
2021-03-04, 15:00 (Online): Quantitative Iteration Theories (Radu Mardare, MSP)
We develop a fixed-point extension of quantitative equational logic and give semantics in one-bounded complete quantitative algebras. Unlike previous related work about fixed-points in metric spaces, we are working with the notion of approximate equality rather than exact equality. The result is a novel theory of fixed points which can not only provide solutions to the traditional fixed-point equations but we can also define the rate of convergence to the fixed point. We show that such a theory is the quantitative analogue of a Conway theory and also of an iteration theory; and it reflects the metric coinduction principle. We study the Bellman equation for a Markov decision process as an illustrative example.
This is a recent joint work with Gordon Plotkin and Prakash Panangaden.
2021-02-25, 15:00 (Online): Some Thoughts on a Datatype for Higher Genus Graphs (Malin Altenmüller, MSP)
Circuit diagrams are commonly modelled by graphs embedded into some oriented surface (maps). When the circuit's topology is non-trivial (e.g. for quantum circuits), the maps live on higher genus surfaces. I will give an introduction to the relevant graph theory and discuss some ideas on how we might be able to program with these higher genus structures. Starting from the plane case, a multi-stack approach seems promising for approaching these more complex maps. Strategies for modelling structures with multiple stacks exist in various different contexts and any of your experiences are most welcome in the discussion!
2021-02-18, 16:00 (Online): Internal ∞-Categories with Families (Nicolai Kraus, University of Nottingham)
It is natural to formalise the notion of a model of type theory (especially the syntax = intended initial model) inside type theory itself. This is often done by writing the definition of a category with families (CwF) as a generalised algebraic theory. What could the success criterion from a HoTT point of view be? The typical first goal is that the initial model is an h-set or even has decidable equality. Our second goal is to make the "standard model" work, i.e. the universe U should be a CwF in a straightforward way (cf. Mike Shulman's 2014 question whether the n-th universe in HoTT models HoTT with n-1 universes). Unfortunately, it is hard to combine these two goals. If we include set-truncatedness explicitly in the definition of a CwF, then the "standard model" is not a CwF. If we don't, then the initial model is not an h-set. The root of the problem is that 1-categories are not well-behaved concepts in an untruncated setting. The natural approach are higher categories, which corresponds to "equipping the syntax with all coherences" instead of truncating. In this talk, I will explain one approach to this based on a type-theoretic formulation of Segal spaces, expressed in HTS/2LTT. I will discuss what works and what is still open. The talk will be based on arXiv:2009.01883.
2021-02-11, 15:00 (Online): Categorical Foundation of Gradient-Based Learning (Bruno Gavranović, MSP)
I will give an introduction to the categorical foundation of gradient-based learning algorithms. I'll define three abstract constructions and show how they can be put together to form general neural networks. The Para construction is used to compose neural networks while keeping track of their weights. Lenses/Optics which are used to take care of the forward-backward data flow and lastly, reverse derivative categories are used to functorially construct the backward wires from the forward ones. In addition, we'll see that gradient descent, Momentum, and a number of optimizers are lenses too, and that this framework includes learning on boolean circuits, in addition to standard Euclidean spaces.
2021-02-04, 15:00 (Online): Reviewing (Bob Atkey, MSP)
I'll talk about how I go about writing, reading, and responding to reviews. Reviews are an essential part of academic publishing, but why do we do them (for free!), and why do we care what they say? I discuss how reviews are used by programme committees and chairs to decide what papers are selected for conferences (and journals), how I think you should go about doing reviews, and why you should write reviews. I'll also talk about reading reviews written about your own work, and how to go about the author response period effectively.
2021-01-28, 15:00 (Online): Combinatorics, Topology and Game theory: a proof of Nash's theorem (Jérémy Ledent, MSP)
The aim of this talk is to state and prove Nash's theorem, which says that every finite game with mixed strategies has a Nash equilibrium. The proof makes use of two other famous results: Sperner's lemma, a combinatorial result about coloring the vertices of a triangulation, and Brouwer's fixed-point theorem. Thus, my talk will be organised into three largely independent parts: first I will sketch the proof of Sperner's lemma; then I will use Sperner's lemma to prove Brouwer's theorem; and finally I will use Brouwer's theorem to prove the existence of Nash equilibria.
2021-01-07, 13:00 (Online): Generic metatheory of linear programming languages (James Wood, MSP)
I will give an introduction to generic metatheory in the style of Allais, Atkey, Chapman, McBride, and McKinna, leading into new work from me and Bob. Generic metatheory is about deriving proofs and operations for a whole class of programming languages, rather than the usual 1 language. The class we consider combines variable binding with usage-sensitivity, and contains many variants of linear natural deduction calculi (and, as a special case, intuitionistic and classical calculi). We build on the framework of Allais et al. and our recent use of linear algebra in linear metatheory.
2020-12-17, 13:00 (Online): Algebraic effects and effect handlers (Sam Lindley, University of Edinburgh)
I'll give an introduction to algebraic effects and effect handlers as a general approach to programming and reasoning about effectful computation. I'll present the notion of a computation over an algebraic effect as a command-response tree over an effect signature quotiented by some equational theory. I'll consider how to interpret command-response trees and motivate effect handlers as the reification of such interpretations as an object language feature that provides a generic implementation strategy for algebraic effects. I'll give examples to show that it can be useful to interpret the same command-response tree using different interpretations which may not respect the same equational theory. Thus effect handlers can provide an expressive programming feature independently of any non-trivial algebraic theory.
2020-12-10, 13:00 (Online): Session Types (Uma Zalakain, University of Glasgow)
This talk is a brief introduction to session types, a type formalism for structured communication between concurrent programs. Instead of typing programs, we will session type channel endpoints, to then ensure that the programs that make use of these endpoints do so in a principled manner, according to their session types. I will go over some of the key ideas that enable session-typed programming, and comment on the properties that session-typed programs exhibit. As an example, I will introduce (and comment on the oddities of) a type system that uses session types to type the pi calculus. I will close mentioning some of the extensions to session types and more advanced type systems.
2020-12-03, 13:00 (Online): Metrics in probabilistic context (Radu Mardare, MSP)
This talk will (probably) be a tutorial on how metrics meet probability theory and how this provides us with the context for innovative paradigms in computer science. The novel challenges that machine learning, cyber-physical systems and statistical computational methods rise to computer science require a fundamental change of the semantics of computation. While in the past we were happy to know whether two programs/algorithms/machines behave the same or not, today it is obvious that this is not enough for our purposes. We need, more and more, to be able to reason about and measure the similarity of non-identical computational behaviours. Moreover, we need to speak about randomness in computational phenomena, especially when we formalize learning or behaviours in unknown contexts. Last but not least, we need to eventually replace the classic metatheory of computation with a probabilistically-based one, where questions can be answered probabilistically within controlled confidence boundaries.
I will not have the time to approach all these during the talk, but I will try to give an overview of the mathematical instruments one needs to approach such a complex challenge.
2020-11-26, 13:00 (Online): Event Structures and Games (Glynn Winskel, MSP)
I'll introduce event structures, a model of computation in which behaviour is captured through partial orders of causal dependency between events. As an application I'll show their role in a theory of concurrent/distributed games and strategies, which has been useful in the semantics of computation, also for probabilistic and quantum programs. Though originally motivated by the limitations of traditional semantics, through determinacy and value theorems, and the preservation of winning/optimal strategies under composition, a form of structural game theory is emerging. In this talk I'll concentrate on what I see as connection points with work here at Strathclyde: as a lead in, relations with the stable domain theory of Berry and Girard; then, the view of strategies as profunctors, whence how they connect to the traditional domain theory of Scott; concluding with how distributed strategies support certain dialectica categories and lenses, and through them open games. My hope is that these connections will be fruitful. On the one side, distributed games can offer a rich metalanguage which extends to probabilistic and quantum computation.
2020-11-19, 13:00 (Online): Containers (Fredrik Nordvall Forsberg, MSP)
How do we represent and reason about data types, in general? The theory of containers is one answer, which is locally quite popular in the MSP group (and beyond). A container is given by a collection of shapes, and for each shape, a collection of positions, where one is meant to plug in data. Containers can be used to analyse generic constructions on data types, without resorting to a messy induction over syntax.
Each container induces a functor from Set to Set, which maps an "element type" X to the set of containers with data drawn from X. Furthermore, there is a natural notion of morphism of containers, with the remarkable property that it completely captures the natural transformations between the functors the containers represent: the interpretation functor from containers to the functor category from Set to Set is full and faithful. In general, the category of containers is extremely nice, eg containers are closed under almost all operations you can imagine (composition, products, coproducts, exponentials, initial algebras, final coalgebras, ...). I hope to give an introductory talk that will make the above more precise.
2020-11-12, 13:00 (Online): Lenses 101 (Jules Hedges, MSP)
I'll talk about the zoo of closely-related structures (with a lot of clashing terminology) known as "lenses" and "optics". Lenses and optics are a way of managing inherently state-heavy applications (for example videogames) in purely functional languages, which turn out to have a lot of surprisingly deep category theory behind them. I will also talk about some of the places where lenses have recently appeared far outside their original domain, for example in game theory, machine learning and systems theory.
2020-11-05, 13:00 (Online): Probability theory with string diagrams (Eigil Rischel, MSP)
Going back to Lawvere, people have tried to study probability theory in terms of monads - the idea being that for each "space" of some type, X, there should be a space PX of probability distributions on X, with point-distributions and integration of distributions giving the monad structure. The Kleisli category of this monad is of particular interest: it corresponds to "stochastic maps" between spaces.
It turns out that one can actually often do without the monad - the Kleisli category itself carries enough structure that one can do probability theory "internally", rederiving the notions of determinism, independence of measures, and many others. I will present this picture of probability, and give some examples of results that can be proven in this framework.
2020-10-29, 13:00 (Online): Realisability (Bob Atkey, MSP)
Sometimes, we think of the category of sets and functions as a simple model of programs in a functional programming language. But sets and functions are much too expressive -- there are plenty of set theoretic functions that aren't expressible in any implementable programming language. Realisability is a way to bring functions down to earth by requiring that they are computable in some model of computation. I'll cover the construction of categories of realizable functions, usually called the category of Assemblies, and its interesting sub-category, the category of Modest Sets or PERs. I'll also sketch how to interpret type theory in the category of assemblies. If there is time, I'll also cover linear realisability.
2020-10-22, 13:00 (Online): Fantastic sheaves and where to find them (Matteo Capucci, MSP)
Sheaves are among the oddest creatures roaming the world of mathematics. They come from the far land of algebraic geometry, they speak a tricky language, and they organize in unfathomably big herds ('topoi'). There's no reason to be afraid, though: sheaves can be the tamest beasts and provide many useful services to the mathematician who's willing to learn their ways. In this talk, I'll try to demystify sheaves by giving an elementary exposition of their basic features, trying to convey useful intuition about their behaviour. I will also try to give a taste some of their most astounding tricks, such as sheaf cohomology and topos-theoretic forcing.
2020-10-15, 13:00 (Online): Quantitative Type Theory and its application (André Videla, MSP)
To build upon last week's talk on Dependent types, we will develop the intuition around Quantitative types and their usage in modern programming languages. What does it mean to program when the type system is responsible for tracking resources rather than leave this task to the programmer? What new patterns do we see? And what are the limitations we encounter and how do we fix them? This talk aims to answers those questions in an approachable and interactive way.
2020-10-08, 13:00 (Online): Dependent Types (Conor McBride, MSP)
I'll give an introduction to the setup for dependent type theories, in the bidirectional style, again sketching why fundamental metatheoretic properties hold by not asking the wrong questions. In particular I'll develop the theory of dependent function types, then add lists. Let's conspire to make append associative, and if time permits, map a functorial action.
2020-10-01, 13:00 (Online): Introduction to Monoidal Categories (Joe Collins, MSP)
A nice, chill introduction to the basic definitions and theorems on monoidal categories. We will be defining monoidal categories and symmetric monoidal categories, and talking about the coherence theorem and string diagrams.
2020-09-24, 13:00 (Online): Bidirectional STLC (Conor McBride, MSP)
By way of kicking off this semester of MSP101, I'll give an introduction to our local dialect of bidirectional type systems, defined as mutually inductive systems of moded judgments. That is, we not only assign a syntactic category to each place in a judgment form, but also designate its mode as being "input", "subject", or "output". The syntactic categories of introduction and elimination forms are distinguished: the former have types as inputs, the latter types as outputs. I'll illustrate this using the Simply Typed Lambda-Calculus, and I'll rattle through as much metatheory thereof as I can before you bottle me off.
2020-09-10: MSP101 Planning meeting (https://meet.jit.si/MSP101/)
2020-07-03 (Online): Compositional Game Theory, compositionally (Jérémy Ledent, MSP)
The recent notion of Open Games allows to study game theory in a compositional way: complex games can be obtained from smaller ones using various operators such as sequential and parallel composition. Thus, Open Games form a symmetric monoidal category. There are many flavors of open games, ranging from small variations on how equilibria are treated, to more radical changes such as introducing probabilistic behavior. Proving from scratch that each variant still forms a monoidal category is tedious.
I will present a compositional construction of the category of Open Games, which consists of three main steps. In each step, one can swap a component for a similarly-behaved one, without disturbing the rest of the structure. Thus, one can define many variants of Open Games with minimal effort. This compositional approach is based on the notion of Arrows, a concept first introduced in functional programming.
This is a practise talk for ACT20 next week, and joint work with Bob, Bruno, Neil, Clemens and Fred.
2020-06-19 (Online): Building Resource-Dependent EDSLs in a Dependently-Typed Language (Jan de Muijnck-Hughes, Glasgow)
While many people use dependent types for theorem proving some of us like to use dependent types for building, and running, safer programs.
Idris' Effects library demonstrates how to embed resource dependent algebraic effect handlers into a dependently typed host language, providing run-time and compile-time based reasoning on type-level resources. Building upon this work, Resources is a framework for realising EDSLs with type systems that contain domain specific substructural properties. Differing from Effects, Resources allows a language's substructural properties to be encoded within type-level resources that are associated with language variables. Such an association allows for multiple effect instances to be reasoned about autonomically and without explicit type-level declaration. We use type-level predicates as proof that a language's substructural properties hold.
Using Resources we have shown how to provide correctness-by-construction guarantees that substructural properties of written programs hold when working with =Files=, specifying Domain Specific =BiGraphs=, and Global Session Descriptions---=Sessions=. With this talk I want to discuss the how and why of Resources, and show how we can use the framework to build EDSLs with interesting type-systems.
This talk is part practise talk for ECOOP 2020, part tutorial on an important idiom for practical dependently-typed programming, and part chance to highlight more how dependent-types are useful when programming in the real world.
2020-06-12 (Online): Interpreting Dependent Type Theory in Containers (Bob Atkey, MSP)
In her PhD thesis, Tamara von Glehn showed that the Category of Containers (a.k.a. Category of Polynomials, Category of Dependent Lenses, Dialectica Category) supports an interpretation of dependent types. I'll present the basic constructions used, and, if I get to it, show that the model refutes the principle of functional extensionality.
2020-06-05: Verification of Session Types Workshop (VEST) (Online)
2020-05-29, 11:00 (Online): The 'graded types' paradigm: past, present, and a possible future (Dominic Orchard, University of Kent)
In the 1970s, the notion of 'grading' appeared in analytic philosophy and logic as a way of making reasoning more fine-grained, capturing 'degrees' of necessity or possibility. Somewhat independently, grading has become a topic of interest in type theory and programming language semantics, with graded structures again providing a means of more fine-grained reasoning. In this talk, I will give an overview of this idea (both its past and present) and explore two uses: (1) using graded modal types for program reasoning in the context of an experimental functional language with linear and indexed types, called Granule; (2) specialising program semantics to give correct-by-construction arguments about program analyses and transformations. I will also give a mathematical characterisation of grading which suggests a broad paradigm, and will briefly mention various ongoing works.
2020-05-22 (Online): A quantitative model for λ-calculus (Jérémy Ledent, MSP)
Resource monoids and length spaces are a semantic framework inspired from realizability, which was introduced by Dal Lago and Hofmann in the context of implicit complexity. It has been used to define quantitative models for various programming languages (such as Elementary Affine Logic, LFPL), and deduce soundness properties of the form: "Every definable function lies in a given complexity class". In this talk, I will show how this framework can be used to measure different quantitative properties of a language than time complexity. Namely, I will present a model of simply-typed λ-calculus such that the interpretation of a λ-term contains an upper bound on the size of its normal form.
2020-05-15 (Online): A Linear Algebra Approach to Linear Metatheory (James Wood, MSP)
Since its introduction, linear logic has been the cause of and solution to many problems in theoretical computer science. There are many examples in the literature of calculi which restrict usage of variables, allowing them to capture linearity, monotonicity, sensitivity analysis, privacy constraints, and other coeffects. However, the syntactic metatheory of these calculi is often difficult, and rarely in a form amenable to formalisation in a proof assistant.
In this talk, I will introduce the notion of kits and environments yielding generic traversals – a method developed by Conor for proving renaming and substitution lemmas. Then, I will discuss recent work by myself and Bob on adapting this method to semiring-graded calculi. The result is a pleasingly and surprisingly minor variation on the original, comprising the introduction of a linear map to mediate usage annotations. This solution gives us confidence that we can tackle all of quantitative metatheory in the familiar intuitionistic style.
2020-05-01 (Online): Stone duality for Markov processes (second attempt) (Radu Mardare, MSP)
The talk will focus on the category of Markov processes, on Markovian logics and Aumann algebras. Markovian logics are modal logics designed to specify (approximated) properties of Markov processes and characterize their bisimilarity. Aumann algebras are the algebraic counterpart of Markovian logics, i.e., they are Boolean algebras with operators that encode probabilistic information. Markovian logics are not compact and for this reason the "classic" Stone duality fails. I will present a different version of Stone duality constructed on top of Rasiowa-Sikorski lemma, which is a result bringing topological results into Model Theory.
This talk summarizes results obtained in collaboration with Dexter Kozen and Prakash Panangaden.
2020-04-24 (Online): Something Else About Mary (Conor McBride, MSP)
This is very much work in progress, jointly with Guillaume Allais (now of St Andrews) and Fredrik Nordvall Forsberg, with occasional involvement from others of the usual MSP suspects.
Mary is our attempt to build a virtual learning environment which offers effective support to our own classes, building from the experience (but thankfully not the codebase) of the tools I built for teaching and assessing my first year hardware class. Last time I talked about Mary, she was just a twinkle in my eye. Now, she is at least a toddler.
Mary is a variant on pandoc markdown which allows code fragments to be embedded in documents and document fragments to be embedded in code. This talk will focus mainly on the programming language embedded in the system which is currently called "Shonkier", as it is a variant on the "Shonky" language to which Frank compiles. It is untyped, for the time being, but that's an opportunity.
The key design choice in Shonkier is that the C-style function application notation, f(e1,..,en), is used for all sorts of contextualization, and there are many such sorts. "Functions" (which already generalize to effect handlers) are but one form of context. We have also added first class environments (computed by pattern matching expressions) and guarding contexts (computed by Boolean expressions). One recent tweak which pays dividends is that effect handlers can now reply to requests with effects as well as with values. We are now free to negotiate various forms of contingency in a refreshingly direct style.
In short, we have abandoned the naive delights of being context-free in favour of being context-negotiable.
2020-04-17: MSP101 Planning meeting (Online at https://meet.jit.si/MSP101)
2020-04-03 (https://meet.jit.si/MSP101): Completeness of Modal Logic via Stone Duality (Clemens Kupke, MSP)
This talk is intended to be introductory, building on Radu's earlier intro to Stone duality. In my presentation I will show how to use duality for proving completeness of basic (propositional) modal logic. To this aim I will first recall what completeness means and then discuss two different kinds of semantics for modal logic: its algebraic semantics, wrt which completeness of the logic is relatively straightforward, and Kripke semantics, which is the semantics one usually is interested in. Stone Duality allows to establish a connection between the two types of semantics such that completeness of modal logic wrt Kripke semantics follows.
2020-03-27 (https://meet.jit.si/MSP101): Multidimensionally-correct by construction programming (Fredrik Nordvall Forsberg, MSP)
Physical dimensions such as length, mass and time can be used to ensure that equations are physically meaningful -- it makes no sense to add 7 metres to 12 seconds, and in an equation such as PV = nRT, it better be the case that both sides have the same physical dimension (in this case pressure times length^3). Even better, dimensions can not only be used to tell scientists off when making mistakes, but also to deduce non-trivial formulas (the famous example being that of the period of a pendulum as a function of its mass and length, and the acceleration of gravity).
From an MSP perspective, there is a strong similarity between physical dimensions and types in programming languages, and indeed, this connection was explored in the 1990's by various researchers. A concrete outcome is Andrew Kennedy's implementation of units of measure in Microsoft's F# language, which makes it possible to write F# programs that are dimensionally-correct by construction.
However, a lot of numerical software is written manipulating not single numbers, but vectors and matrices, using the methods of linear algebra. I will report on work in progress with Conor on extending dimensional analysis to multiple dimensions in this second sense.
2020-03-06: Classic puns (James Wood, MSP)
There are several principles that are taken to be characteristic of non-constructive reasoning. These include the principles of excluded middle, double negation elimination, and choice.
On the other side, there are several formal systems that are well known to capture only constructive reasoning. These include dependent type theory (à la Martin-Löf) and linear logic.
However, each of the systems I just listed is able to derive at least one theorem that looks an awful lot like one of the non-constructive principles I listed earlier. What's going on here?
2020-02-28: The Dialectica Categories (Georgi Nakov, MSP)
The Dialectica Categories were introduced in de Paiva's eponymous work as an internalized version of Gödel's functional interpretation. The interpretation translates Heyting Arithmetic (HA) into System T (intended as an axiomatization of primitive recursive functionals of finite type) and was originally developed as a tool to prove the relative consistency of HA. Translating the contraction rule poses certain problems and as a solution, Gödel requires decidability of atomic formulae. Several variants exist that lift this restriction.
In this talk, I will present the categorical constructions from de Paiva's paper. We will investigate their structure and see how the different versions of the interpretation are accommodated in this setting. Finally, we will conclude that in the process we have obtained a model of Intuitionistic Linear Logic.
2020-02-14: Towards Compositional Structures in Neural Networks (Bruno Gavranović, MSP)
Neural networks have become an increasingly popular tool for solving many real-world problems. They are a general framework for differentiable optimization which includes many other machine learning approaches as special cases. However, at the moment there is no comprehensive mathematical account of their behavior. I'm exploring the hypothesis that the language of category theory could be well suited to describe these systems in a precise manner. I will give a short tour of recent developments in this area, mostly based around the notion of lenses.
2020-02-07 (LT1415): Stone Duality in Stochastic Context (Radu Mardare, MSP)
We define Aumann algebras, an algebraic analog of probabilistic modal logic. An Aumann algebra consists of a Boolean algebra with operators modeling probabilistic transitions. We prove that countable Aumann algebras and countably-generated continuous-space Markov processes are dual in the sense of Stone. Our results subsume existing results on completeness of probabilistic modal logics for Markov processes.
This summarizes results obtained in collaboration with Dexter Kozen and Prakash Panangaden.
2020-02-06 (LT1415): Rust, Servo and Mixed Reality Research at Mozilla (Alan Jeffrey, Mozilla)
Mozilla is a non-profit whose mission is to expand access to, and protect privacy on, the Internet. Mozilla Research investigates emerging technologies, such as programming languages and new web platforms. This talk is an overview of research efforts around the Rust programming language, the Servo web engine, and Mixed Reality (VR and AR) on the web.
2020-01-31: Resource Constrained Programming with Full Dependent Types (Bob Atkey, MSP)
I will talk about a system that combines Dependent Types and Linear Types. As an application of this system, I will show how to transport Martin Hofmann's LFPL and Amortised Resource analysis systems for resource constrained computing to full dependent types. This results in a theory where unconstrained computations are allowed at the type level, but only polynomial time computations at the term level. The combined system now allows one to explore the world of propositions whose proofs are not only constructive, but also of restricted complexity.
2020-01-24: Simplicial Models for Multi-Agent Epistemic Logic (Jérémy Ledent, MSP)
Epistemic Logic is the modal logic of knowledge. It allows one to reason about a finite set of agents who can know facts about the world, and about what the other agents know. The traditional way to interpret epistemic logic formulas is by using Kripke models: that is, graphs whose vertices represent the possible worlds, and whose edges indicate the agents that cannot distinguish between two worlds. I will present an alternative kind of model for epistemic logic based on chromatic simplicial complexes. Simplicial models are equivalent to Kripke models; thus, this connection uncovers the higher-dimensional geometric nature of knowledge. Finally, I will show how to adapt these geometric models in order to interpret other epistemic notions, such as belief, distributed knowledge, and more.
2020-01-17: Something About Mary (Conor McBride, MSP)
Quite a few colleagues in the MSP group have experienced both the pleasure and the pain of working with my Marx system in the course of delivering our classes. (I never meant to write a virtual learning environment, but somehow, I sort of did.) I propose to re-engineer it from scratch and do a rather better job (suitable for wider deployment) deliberately, and am keen to solicit assistance in this mission, lest it be yet another engineering project distinguished by my madness and dissolution that you are, even so, obliged to put up with. Inevitably, I will struggle and rapidly fail to exclude interesting computer science from the design of the system. There will be fun.
Mary will be a content management system with pages written in Markdown and stored in git repositories on the department's GitLab server. However, every such page is also a form, supporting interactive content. Pages will therefore need to embed code for processing the data in the form, certainly on the server, but preferably (unlike in Marx) on the client. So Mary will embed a programming language that we might have fun designing and implementing. I propose to base this language on Frank, an effects and handlers language that I cooked up a while back. Access to form fields can be seen as an effect: by remember the resumption for each such access, we can model what to recompute in the client when fields change, after the fashion of spreadsheets, programmed in apparently direct style.
Mary will also need to maintain a database to achieve cross-session persistence of student work and staff configuration data. The Marx approach to analytics over this database amounted to "grep". In the meantime, however, Fred and I spent quite a while thinking about how to give an account of data models and analytics at a higher level of abstraction using carefully undermarketed ideas from dependent type theory. We should consider how to adapt these ideas to manage our own data.
Let's get excited and make things!
(Comrades who are not Strathclyders but who are interested in effects, types, or just having better tools to survive this business we call higher education, should very much feel invited to engage. I will happily export this project.)
2020-01-16: MSP101 Planning meeting (LT1310)
2019-12-12, 13:30: Potato Powered Proofs (Conor McBride, MSP)
Stuart and I have been working on interpreters for predicative calculi which are "potato-powered" in that they work by structural recursion on inductive data which may look suspiciously like types, but are not statically checked to be the actual types of the programs in question. This works because the calculi are presented in a bidirectional style which, by design, causes every redex to carry some "potatoes", which hopefully contain enough inductive carbohydrate to keep you going for the whole of the computation to be done.
Naturally, it would be lovely if we could prove that the interpreter's output is genuinely a reduct of its input, and that well typed input yields normal form output (i.e., that an honest type gives the potatoes required to do the whole computation). How is it done? Potato-powered logical relations, of course! I'll give a crash course on cooking programs and proofs with potatoes.
2019-12-05, 13:30: Boolean-valued semantics (Radu Mardare, MSP)
This talk is based on the paper with the same title that I have presented at LICS 2018. It is a joint work with Dana Scott, Dexter Kozen, Prakash Panangaden, Robert Furber and Giorgio Bacci. We extend Dana's Boolean-valued set theory (introduced in 50's to demonstrate the independence of the continuum hypothesis) to get, initially, a denotational model of untyped lambda calculus, and eventually extend it to a model of stochastic lambda calculus. The model is constructed over a space of random variables, which inherit a natural continuous Boolean algebra structure.
2019-11-28, 13:30 (off-campus): Selective Applicative Functors (not a seminar, due to the strike) (Bob Atkey, MSP)
At this year's ICFP, Mohkov, Lukyanov, Marlow, and Dimino introduced "Selective Applicative Functors", a programming interface that is an intermediate stage between monads and applicative functors. I'll motivate what they're for, and describe what they are, and I'll talk about a more abstract way of thinking about them in terms of "right-skew monoidal categories".
2019-11-21, 13:30: Expressive Logics for Coinductive Predicates (Clemens Kupke, MSP)
The classical Hennessy-Milner theorem says that two states of an image-finite transition system are bisimilar if and only if they satisfy the same formulas of a certain modal logic. I will place this type of result in a general context, moving from transition systems to coalgebras and from bisimilarity to coinductive predicates. Furthermore I will present a sufficient condition for a coalgebraic logic to be expressive, i.e., to fully characterise a coinductive predicate on coalgebras. Our approach will be illustrated with logics characterising similarity, divergence and a behavioural metric on automata.
This is joint work with Jurriaan Rot to be presented at CSL 2020.
2019-11-14, 13:30: Programs and Proofs in Linear Algebra (James Wood, MSP)
I try to understand the ways of linear algebraists from a programming languages perspective. In particular, I investigate the practice of specifying a linear map by giving only its action on basis elements. We can turn this practice into a nice programming language for linear maps. Furthermore, our reasoning about these maps is significantly simplified by only considering the basis elements. Finally, keeping track of basis elements amounts to a typing discipline, and we can expect to get some nice properties just by observation of the types.
Meanwhile, I will try to elucidate the similarity between, on one side, categories of spaces and linear maps and, on the other, Rel, the category of sets and relations. This forms an important part of our programming languages view.
2019-11-07, 13:30: MSP 101: Coends and Kan extensions (Neil Ghani, MSP)
2019-10-31, 13:30: Directed algebraic topology for concurrency (Jérémy Ledent, MSP)
I will give an introduction to the geometric semantics for concurrent programs that have been developed by Goubault, Fajstrup, Raussen et al. since the 1990's. A concurrent program can be viewed as a topological space equipped with a notion of direction, modeling the passage of time. Thus, in such a space, the existence of a path from A to B does not guarantee the existence of a path from B to A. A path in a directed space corresponds to an execution of the program; and two such executions are equivalent when the corresponding paths are homotopic, that is, when they can be deformed continuously into one another. This idea motivated the development of directed algebraic topology, the analogue of algebraic topology for directed spaces.
2019-10-30: SPLS (University of Glasgow)
2019-10-24, 13:30: Hopf Monads and Formally Adding in Morphisms (Joe Collins, MSP)
It should be noted that this is ongoing work. The talk that you will see is going to be very unpolished; this is stuff that I have been working on for the past while, and the material will not be fully formed yet. The things I say may not end up matching with reality. So with that minor caveat, let us continue.
A Hopf algebra is a monoid, a comonoid, and an extra endomorphism such that various commutation rules are obeyed, and Hopf monads are a strange generalisation of Hopf algebras in the direction of monads. I have previously talked about when a Hopf algebra is a Hopf-Frobenius algebra, and my goal now is to generalise this to the case of Hopf Monads. However, to keep things interesting, I am doing this in a weird way.
Let T be a Hopf monad in monoidal category C. T is isomorphic to a Hopf algebra if there exists a natural transformation e: T -> 1 that respects the Hopf monad structure. There are many examples of Hopf monads which are not isomorphic to any Hopf algebras, so this morphism does not exist in C in general. However, what happens if I formally add in this morphism, creating a new category C_e? In this category, T would presumably be isomorphic to a Hopf algebra, and this begs the question: what can T in C_e tell us about T in C? By looking at the natural functor C -> C_e, can I use this to transfer my theorem about Hopf algebras to Hopf monads? I certainly believe so, but let's find out together!
2019-10-17, 13:30: Ordinals below epsilon-zero in cubical Agda (Fredrik Nordvall Forsberg, MSP)
Ordinals and ordinal notation systems play an important role in program verification, since they can be used to prove termination of programs. We present three ordinal notation systems representing ordinals below epsilon_0 in type theory, using recent type-theoretical innovations such as mutual inductive-inductive definitions and higher inductive types. As case studies, we show how basic ordinal arithmetic can be developed for these systems, and how they admit a transfinite induction principle. We prove that all three notation systems are equivalent, so that we can transport results between them using the univalence principle. All our constructions have been implemented in cubical Agda.
2019-10-10, 13:30: Quantitative algebras: towards a quantitative theory of effects (Radu Mardare, MSP)
We develop a quantitative analogue of equational reasoning meant to provide metric semantics for stochastic/probabilistic/quantitative systems and programming languages. Quantitative algebras are algebras over metric spaces defined by these quantitative equational theories, and they implicitly define monads over the category of (extended) metric spaces. We have a few relevant examples of such algebras, where the induced free algebras correspond to well-known structures; among these are Hausdorff metrics from quantitive semilattices, p-Wasserstein metrics (hence also the Kantorovich metric) from barycentric algebras, the total variation metrics from a variant of barycentric algebras, and more.
The talk is based on a series of joint works with Prakash Panangaden and Gordon Plotkin. The main results have been presented at LICS'16, LICS'17 and LICS'18.
2019-10-03, 13:30: Summer trip reports (James Wood, Joe Collins, Bob Atkey, Guillaume Allais, Fredrik Nordvall Forsberg, MSP)
2019-08-30, 11:00 (LT1415): A combinatorial representation of the operad of plane graphs (Malin Altenmüller, MSP)
I will present how we can formalise non-symmetric monoidal categories (i.e. string diagrams) in a combinatorial way. These kind of diagrams will be represented by plane graphs with a distinguished boundary vertex. For encoding plane graphs it is sufficient to store the order of neighbours for each vertex, called a rotation system. I'll show how to define these sorts of graphs, how to compose them and — together with the right notion of rewriting — how they form an operad.
This is practice for a talk I will give at STRINGS in Birmingham next week.
2019-08-16, 16:00 (LT1415): Polynomial certificates for nondeterministic automata (Rob Myers)
Given a deterministic finite automaton accepting L, another accepting the reverse language, and an integer n, does there exist a nondeterministic acceptor with n or fewer states? We've proved that this problem is in NP i.e. polynomial certificates exist. All previous algorithms constructing small nondeterministic automata for arbitrary regular languages are at best PSPACE-complete. We achieve this by explaining and substantially improving the Kameda-Weiner algorithm using coalgebraic methods. At the underlying level we use a categorical equivalence between finite join-semilattices and bipartitioned graphs.
2019-07-05, 16:00: The Thing with Thinnings: CodeBruijn is (Free;Stuff) (Conor McBride, MSP)
I've been hacking away on tools to do metatheory using "codebruijn" representation of syntax. The method, as with scoped de Bruijn representation, is to index terms by their scope, but the codebruijn method goes further, by systematically treating the support of terms, i.e., the particular subscope of variables which are actually relevant, paired with the thinning that embeds support into scope. The key codebruijn type constructor is "pair a thing with a thinning from some support", and it turns any old sort of thing into something which uniformly admits further thinning just by postcomposition, without affecting either the thing or its support.
Stuff-which-uniformly-admits-thinning is stuff with structure, so there is a forgetful functor back to plain old Stuff. The Free functor which adds uniform thinning, is its left adjoint. The codebruijn type constructor is the monad (Free;Stuff). The corresponding comonad, (Stuff;Free), is rather more familiar to semanticists as the abstraction over larger scopes found in Kripke models.
I've always been a bit shaky on adjunctions, presheaves and what have you, so this talk will not assume categorical confidence, but rather seek to build intuition, guided by a concrete example grounded in the practicalities of manipulating syntax.
2019-06-27, 16:00: Isbell Duality, Stone Duality, and Physical Theories (Kevin Dunne)
Isbell duality is an adjunction between a category of "generalised algebras" and a category of "generalised spaces", and is an abstraction of Stone duality. Broadly speaking, a Stone duality is an equivalence between some category of commutative algebras, and some category of spaces, for example: Boolean algebras and Stone spaces; or, commutative C*-algebras and Hausdorff spaces.
There is a Stone duality which applies in the setting of Newtonian mechanics, between the category of smooth manifolds and a category of commutative algebras, and this equivalence of categories admits an extremely elegant and intuitive interpretation from the perspective of physics.
I am going to discuss how to build an analogous interpretation for quantum theory using the machinery of Isbell duality, in such a way that directly generalises Newtonian mechanics, with the ultimate aim being to resolve some of the conceptual problems of quantum theory.
2019-06-07, 16:00 (LT1415): TYPES 2019 practice talks (Malin Altenmüller and James Wood, MSP)
Malin and James will practice giving their talks "Containers of Applications and Applications of Containers" and "Linear metatheory via linear algebra" respectively.
Containers of Applications and Applications of Containers (abstract)
Linear metatheory via linear algebra (abstract)
2019-05-29, 13:00 Departmental seminar (LT1415): The next 700 abstract machines (Dan Ghica, Birmingham)
We propose a new formalism for representing programming languages, based on a universal graph-rewriting abstract machine. The calculus itself only deals with the basic structural aspects of a programming languages, namely variables, names (e.g. memory locations) and thunks (i.e. fragments with delayed execution). Everything else needs to be supplied as extrinsic operations, with associated rewrite rules. This basic calculus allows us to represent all known sequential programming language features in a uniform framework, reason abstractly about their cost, and also reason about challenging equivalences.
This is joint work with Koko Muroya and Todd Waugh Ambridge.
2019-05-24, 16:00: Bigraphs with sharing and their algebra (Michele Sevegnani, Glasgow)
Bigraphical Reactive Systems (BRS) are a universal model of computation introduced by Robin Milner for the representation of interacting systems that evolve in both time and space. Bigraphs have been shown forming a category called symmetric partial monoidal category and their dynamic theory is defined in terms of rewriting and transition. A limitation of bigraphs is that the underlying model of location is a forest, which means there is no straightforward representation of locations that can overlap or intersect. In this talk, I will present bigraphs with sharing, an extension of bigraphs which solves this problem by defining the model of location as a directed acyclic graph, thus allowing a natural representation of overlapping or intersecting locations. I will give a complete presentation of the extended theory, including a categorical semantics, algebraic properties, a normal form and several essential procedures for computation.
2019-05-15, 16:00: Getting the unifier on your side (Guillaume Allais, MSP)
I will explain how to write practical generic n-ary functions and combinators for n-ary relations.
Agda Standard Library pull request
2019-05-10, 13:00 (LT1415): A Purely Functional Array Language and its Optimising GPU Compiler (Troels Henriksen, University of Copenhagen)
Futhark is a small programming language designed to be compiled to efficient parallel code. It is a statically typed, data-parallel, and purely functional array language, and comes with a optimising ahead-of-time compiler that generates GPU code via OpenCL and CUDA. Futhark is not designed for graphics programming, but instead uses the compute power of the GPU to accelerate data-parallel array computations ("GPGPU").
This talk presents the design of the Futhark language, as well as outlines several of the key compiler optimisations that have enabled performance comparable to hand-written GPU code. Through the use of a functional source language, we obtain strong invariants that simplify and empower the application of traditional compiler optimisation techniques. In particular, I will discuss (i) aggressively restructuring transformations such as fusion, (ii) loop interchange and distribution to extract flat parallel kernels from nested parallel source programs, (iii) multi-versioned code generation that exploits as much parallelism as necessary and efficiently sequentialises the rest, and (iv) data layout transformations to ensure coalesced memory access on GPUs.
2019-05-03, 16:00: MSP 101: Fixed points of indexed containers (Conor McBride, MSP)
In an attempt to con Agda into allowing alternation of least and greatest fixed points, I try to remember how to construct these things for indexed containers.
2019-04-12, 16:00: A Type-System of Sorts for Bigraphs (Jan de Muijnck-Hughes, Glasgow)
Bigraphs are a mathematical model (hypergraph) for representing the spatial and communication structures of networked entities [fn:1]. Bigraph Reactive Systems (BRS) build upon bigraphs by incorporating temporal changes to a model as specified with reaction rules. There is a wealth of theory (category) that supports reasoning about bigraphs, and an emergent tool (BigraphER [fn:2]) for their modelling.
Bigraphs have an elegant algebraic structure that is general purpose and simply typed. However, it is common to place restrictions on the shape of a bigraph by a system of sorts that are applied as side-conditions. The details, and application, of these sorts are presented as an aside from the bigraph definition itself. This makes transformation of Bigraph models harder to enforce during manipulation as part of a BRS. We must provide runtime checks to provide relevant guarantees over model correctness.
In this talk, I want to introduce you to the interesting world of Bigraph specification, and my current and ongoing work on developing a dependent type-system to embed a system-of-sorts directly into the algebraic definition of bigraphs. This allows one to provide correct-by-construction guarantees that a given model is correct w.r.t. a provided system-of-sorts.
[fn:1] Robin Milner. 2009. The Space and Motion of Communicating Agents. Cambridge University Press.
[fn:2] http://dcs.gla.ac.uk/~michele/bigrapher.html
2019-04-05, 16:00: Shonan "Programming and Reasoning with Algebraic Effects and Effect Handlers" trip report (Bob Atkey and Conor McBride, MSP)
2019-04-01, 16:00: Determinacy and the red-green chase (Johannes Marti, Universität Bremen)
The determinacy problem is to decide for queries v1,...,vn and q whether for every database D knowing the answers to v1,...,vn in D suffices to deduce the answer to q in D. It has been shown that this problem is undecidable if v1,...,vn and q are arbitrary conjunctive queries.
In this talk I give a gentle introduction to some of the formal tools that are used to study the determinacy problem. Especially, I focus on the red-green chase, which is a neat construction that links the problem of determinacy with query answering relative to ontologies. Lastly, I might present some of our own, very modest, results about classes of conjunctive queries for which the determinacy problem is decidable.
2019-03-20, 16:00: Probabilistic open games (Alasdair Lambert, MSP)
We extend the Open Games framework for compositional game theory to encompass also mixed strategies, making essential use of the discrete probability distribution monad on Set. We show how to compose the resulting probabilistic games in parallel and sequentially, and illustrate the framework on the well-known Matching Pennies game.
2019-03-15, 16:00: Fropf algebras (Joe Collins, MSP)
Last time I talked, you heard about Interacting Frobenius algebras — this time, get ready for "Hopf-Frobenius algebras", as Ross likes to call them, or "Fropf algebras", as Ross doesn't like to call them. I am in the process of writing about them, so this is going to be quite similar to the last talk except we're coming at them from a new angle, and with new results.
2019-03-08, 16:00 (LT1415): Coalgebra Learning via Duality (Clemens Kupke, MSP)
A key result in computational learning theory is Dana Angluin's L* algorithm that describes how to learn a deterministic finite automaton (DFA) using membership and equivalence queries. I will present a generalisation of this algorithm to the level of coalgebras. The approach relies on the use of logical formulas as tests, based on a dual adjunction between states and logical theories. This allows us to learn, e.g., labelled transition systems, using modal formulas as tests.
Joint work with Simone Barlocco and Jurriaan Rot.
2019-02-22, 16:00: MSP 101: Proof-irrelevance for Dependent Type Theory (Fredrik Nordvall Forsberg, MSP)
Proof-irrelevance can mean two similar but different things: on the one hand, irrelevant proofs can be discarded when extracting programs to execute (run-time erasability), and on the other hand, one might want to treat proofs as irrelevant during type checking (type-checking time erasability). I will give an overview of the subject, focusing on type-checking time erasability. I suspect that at least three people in the audience knows more than me, so my main function will be to keep things accessible. I plan to follow the recent paper "Degrees of Relatedness: A Unified Framework for Parametricity, Irrelevance, Ad Hoc Polymorphism, Intersections, Unions and Algebra in Dependent Type Theory" by Andreas Nuyts and Dominique Devriese (LICS 2018).
"Degrees of Relatedness" by Andreas Nuyts and Dominique Devriese (LICS 2018) [PDF]
2019-02-15, 16:00: Finitary indexed containers (Malin Altenmüller, MSP)
I will talk about two different representatives of indexed containers which — all together in the end — will construct an application that manages windows on a screen.
The first instance of containers are rectangular windows. Defining these in terms of finite indexed containers (finite in the number of positions where elements can be stored) lets us interpret indices as outer boundaries. I will show some constructions on these space partitioning structures, e.g. product operations and overlaying of windows (this will hopefully include some pretty pictures!).
Secondly I will explain how we can encode applications as indexed containers (not necessarily finite, despite the title), being defined in terms of commands and responses. To sum up I present the window managing application where all the above things will occur and get combined.
I am working on these topics together with Conor and we are currently collocating them in a paper.
2019-02-08, 16:00: Evaluators for bidirectional type systems (Stuart Gale, MSP)
In this talk I'll introduce syntax and semantics of a generic bidirectionally type checked lambda calculus. I'll also show that lumps of syntax that act as a type annotation are in fact the fuel which powers structural evaluation.
2019-02-01, 16:00: Linear logic: how hard can it be? (James Wood, MSP)
In the linear world, we are no longer allowed to freely discard or duplicate hypotheses and conclusions. This should mean that there are fewer possible strategies for proving a given proposition, making the task of proving things easier. However, we find that linear logic proof search is a surprisingly powerful model of computation. In this talk, I'll give an introduction to linear logic sequent calculus. I'll then go through a neat construction of [0] embedding quantified boolean formulae into linear logic without bangs, and sketch the proof from the same paper that linear logic with bangs is Turing-complete. Some content from CS106 may come in handy.
[0]: Lincoln, Mitchell, Scedrov, Shankar, 1992, "Decision problems for propositional linear logic"
2019-01-25, 16:00: MSP 101: Chu spaces (Bob Atkey, MSP)
I'll talk about Chu Spaces, a general construction of *-autonomous categories (a.k.a. models of classical linear logic). Chu spaces are interesting because they come with an inbuilt notion of duality, which has been interpreted as a kind of player/opponent duality.
2019-01-18, 16:00: Metametatheory of Bidirectional Type Systems (Conor McBride, MSP)
In previous talks, I've written down bidirectional versions of particular type systems, with type checking for introduction forms and type synthesis for elimination forms. But what are the metarules for writing down the rules? How do we show that following the metarules ensures good properties of the rules? I'll report some progress towards capturing the syntactic properties of "good" rules which might push us closer to the goal of metatheory by construction.
NB This talk has nothing to do with bidirectional transformations.
2018-12-17: SYCO 2 (Strathclyde)
2018-12-14, 16:15 (LT1415): Abstract differential geometry matters! (Robin Cockett, University of Calgary )
The last few years has seen the development — largely in Canada and Australia — of an axiomatic approach to differential geometry based on tangent categories. Tangent categories incorporate the previous leading settings for differential geometry: finite dimensional manifolds, synthetic differential geometry, convenient manifolds, etc. In addition they widen the scope significantly as they also include combinatorial species, Goodwillie Functor calculi, and examples from computer science. The talk will give a survey of tangent categories and some of the developments so far.
2018-12-07, 16:00: Interacting Frobenius Algebras (Joe Collins, MSP)
What is a Frobenius algebra? What is a Hopf algebra? And why are they such good friends? In this talk, I will be talking about PROPs, what an interacting Frobenius algebra is, and some weird stuff that appears with them, and I shall be drawing lots of pretty pictures as well.
2018-11-30, 16:15: Dual adjunctions (Simone Barlocco, MSP)
Coalgebras provide an abstract framework to represent state-based transition systems. Modal logics provide a formal language to specify such systems. In our recently submitted work (joint work with Clemens Kupke and Jurriaan Rot) we devise a general algorithm to learn coalgebras. Modal formulas are used as tests to probe the behaviour of states.
In my introductory talk, I will discuss how to set up a general framework that connects coalgebras and their modal logics via a dual adjunction. Moreover, I will show a known result which guarantees that indistinguishable states wrt to modal formulas are behavioural equivalent, a key fact that entails that — whenever possible — our algorithm learns a minimal representation of a coalgebra.
2018-11-23, 16:15: The reachable part of a coalgebra (Clemens Kupke, MSP)
State-based transition systems are often studied relative to a specified initial state. System behavior then only depends on those states that are "reachable" from the initial state. This has both consequences for the theory (e.g. by allowing to prove non-definability results in modal logic) and practice (by making seemingly large systems more manageable) of such systems.
Coalgebras provide a general model for transition systems. In this introductory talk I will discuss how to define the reachable part of a coalgebra via the notion of T-base for an endofunctor T from [1]. We will first discuss this notion and then provide a sufficient categorical condition for the existence of the T-base. We will then show how to characterise the reachable part of a coalgebra as least fixpoint of a monotone operator.
This is based on joint work with Simone Barlocco and Jurriaan Rot.
[1] Alwin Blok. Interaction, observation and denotation. Master's thesis, ILLC Amsterdam, 2012.
2018-11-16, 16:15: MSP 101: Polynomial time programming languages (Bob Atkey, MSP)
Implicit Computational Complexity is the study of programming languages or logical systems that capture complexity classes. Roughly, every program that can be written in the language is in some complexity class. Many of the languages that have been proposed for capturing useful classes like PTIME are not much fun to program in. However, the work of the late Martin Hofmann included work on languages like LFPL, which only allows polynomial time computation, but is also reasonably usable. I'll talk about LFPL, and how the proof of polynomial time bounds works.
2018-11-09, 16:00: The semantics of worldly type systems (Ben Price, MSP)
I will present a type theory whose judgements are indexed by a preorder of "worlds", representing for example nodes in a distributed computation, or a security level. This means a term may only typecheck at particular worlds, and will be mobile upwards along the preorder (for instance every low security value is also a high security one). To enable talking about the world structure without compromising mobility, the terms can talk about "shifts", which describe relative worlds.
I then give a semantic model based on the usual presheaf model for STLC where worlds form the base category, and shifts are endofunctors on the worlds. This semantics will show our programs are indeed oblivious to data they cannot "see". Examples will be given to demonstrate this framework in some concrete cases, and to motivate future work.
2018-11-02, 16:00: The Next 700 Program Transformers (Geoff Hamilton, Dublin City University)
In this talk, I will describe a hierarchy of program transformers in which the transformer at each level builds on top of the transformers at lower levels. The program transformer at the bottom of the hierarchy corresponds to positive supercompilation, and that at the next level corresponds to distillation. I will then try to characterise the improvements that can be made at each level, and show how the transformers can be used for program verification and theorem proving.
2018-10-26, 16:00: Syrup: circuits with memory (Conor McBride, MSP)
By way of giving CS106 students better tools to tackle the concept of memory in circuits, I implemented a small programming language called (for reasons which are unlikely to become clear) Syrup. Syrup is suspiciously like a dialect of Haskell, except that the blessed monad allows bits of state. Marking homework done in Syrup necessitates checking whether two circuits have the same externally observable behaviour, which makes it a matter of bisimulation. I'll talk a bit about how much fun it was implementing the decision procedure to find either a bisimulation or a countermodel (for purposes of decent feedback).
2018-10-19, 16:00: SPLS Post-mortem (Stuart Gale, MSP)
2018-10-17: SPLS (Strathclyde)
2018-10-12, 16:00: Why quantum computers suck, and what we might do about it (Ross Duncan, Cambridge Quantum Computing)
Near-term quantum computers have many limitations which make them difficult to use for stuff. I will outline some of the difficulties and handwave at some new ideas from compositional mathematics that might help us address these problems.
2018-10-05, 16:00: ICFP trip report (James Wood, MSP)
Last week, I attended the ICFP conference and various associated workshops in St Louis. In this trip report, I will talk about selected talks and the people I met there. If time allows, I may also cover the adventures of Ioan, one of our summer interns and current undergraduate.
2018-09-21, 16:00: First-order unification (Conor McBride, MSP)
I'll do a 101 today, filed under "stuff everybody should know" about first-order unification (the algorithm underlying Hindley-Milner type inference, Prolog, etc). But then I'll throw in the twist of considering syntax with binding. The way I cook it, this makes essential use of the structure of the category of thinnings.
2018-08-31, 15:00 (LT1415): Telescopic [Constraint] Trees, or: Information-Aware Type Systems In Context (Philippa Cowderoy)
The minimalist tradition in type systems makes for easy mathematics, but often leaves their mechanisms needlessly obscured.
I build a structure for Hindley-Milner checking problems in the tradition of Type Inference in Context. This structure is derived from typing rules in the style of my first talk and mirrors data structures used for elaboration problems in dependent type systems — offering a notation that can be used among designers and implementors of type systems and even in explaining their behaviour to users.
2018-08-29, 15:00 (LT1415): Information-Aware Type Systems (Philippa Cowderoy)
One possible remedy is to track the behaviour of information in a system — its creation, its destruction and how it flows between constraints and source locations. I illustrate this with the Simply-Typed Lambda Calculus.
2018-07-18, 12:00: Finite and Infinite Traces, Inductively and Coinductively (Jurriaan Rot, Radboud University Nijmegen)
It is a well-known fact (used e.g. in model checking) that, on finitely branching transition systems, finite trace equivalence coincides with infinite trace equivalence. I will show how to prove this coinductively, which is arguably nicer than the standard inductive proof.
2018-06-13, 11:00: Gillette Fusion: Kits for Hutton's Razor or Type Unsafe and Scope Unsafe Programs and Their Proofs (James Chapman, MSP)
The paper Type-and-Scope Safe Programs and Their Proofs abstracts the common type-and-scope safe structure from computations on lambda-terms that deliver, e.g., renaming, substitution, evaluation, CPS-transformation, and printing with a name supply. By exposing this structure, we can prove generic simulation and fusion lemmas relating operations built this way. In this talk I will present this approach but for simpler setting of Hutton's Razor. This reduces the mathematical structures involved from relative structures to the ordinary counterparts.
2018-06-05: SPLS (Cairn Auditorium, PG G.01, Postgraduate Centre, Heriot-Watt University)
2018-05-30, 11:00: Multi-Dimensional Arrays and their Types (Artjoms Šinkarovs and Peter Hancock, Heriot-Watt)
Though Multi-Dimensional Arrays (MDAs) seem conceptually straightforward, it's not easy to come up with a mathematical theory of arrays that can be used within optimising compilers. We'd like to treat arrays as functions from indices to values with some domain restrictions. It is desirable that these domain restrictions are specified in a compact form, and are equipped with closed algebraic operations like intersection, union, etc. We are going to consider a few typical models like hyperrectangulars, grids and polyhedrons.
When typing array operations, ideally we'd like to find a balance between tracking all the shapes of arrays and allowing for generic array operations. This proves to be tricky, for reasons we'll explain.
We will propose, tentatively, an analysis of MDAs in terms of container functors. The aim is to supply concepts helpful when thinking about MDAs, e.g. when designing notations for coding with arrays. Some intriguing gadgetry shows up.
2018-05-29, 13:00: A Coinductive Proof of Policy Iteration Correctness (Helle Hvid Hansen, TU Delft)
This is the second half of Helle's talk on a (co)algebraic treatment of Markov Decision Processes. It focuses on a coinductive explanation of policy improvement using a new proof principle, based on Banach's Fixpoint Theorem, that we call contraction coinduction.
2018-05-23, 16:00: How Do We Model a Problem Like Mutable State? (Bob Atkey, MSP)
2018-05-17, 13:00: HoTT for sets (Thorsten Altenkirch, Nottingham)
Before getting lost in the realms of higher dimensions we should see wether we can interpret set-level HoTT. We know how to deal with functional extensionality and a static universe of propositions (see Observational Type Theory) but what about a dynamic universe of propositions, i.e. one reflecting HProps that also validates propositional extensionality. I will discuss the problems modelling this and a possible solution using globular setoids.
The dynamic prop corresponds to a subobject classifier in a topos (in particular we get unique choice) while the static universe corresponds to a quasitopos I am told.
2018-05-16, 11:00: Long-Term Values in Markov Decision Processes, (Co)Algebraically (Helle Hvid Hansen, TU Delft)
In this talk, we study Markov decision processes (MDPs) with the discounted sums criterion from the perspective of coalgebra and algebra. Probabilistic systems, similar to MDPs but without rewards, have been extensively studied, also coalgebraically, from the perspective of program semantics. Here, we focus on the role of MDPs as models in optimal planning, where the reward structure is central. Our main contributions are: (i) a coinductive explanation of policy improvement using a new proof principle, based on Banach's Fixpoint Theorem, that we call contraction coinduction, and (ii) showing that the long-term value function of a policy can be obtained via a generalized notion of corecursive algebra, which takes boundedness into account.
This is joint work with Frank Feys (Delft) and Larry Moss (Indiana).
2018-05-04, 11:00: MSP 101: The Dialectica Interpretation (Fredrik Nordvall Forsberg, MSP)
A proof interpretation translates proofs of one logical system into proofs of another (example: the double-negation interpretation of classical logic into constructive logic). This often reveals some information about the original system (e.g. classical logic is equiconsistent with constructive logic). Gödel's Dialectica interpretation (named after the journal it was published in) translates Heyting arithmetic (the constructive theory of the natural numbers, including induction) into System T (the quantifier-free theory of the simply typed lambda calculus with natural numbers) — quantifier complexity is traded for higher type complexity. Combining this translation with (a refined) double negation translation, one can extract System T programs from a proof of a forall-exists statement, even if this proof is using non-constructive priciples such as Markov's Principle, Excluded Middle, or the Quantifier-Free Axiom of Choice. I've always found the Dialectica translation mystifying, so I'll try to explain the intuition behind it.
2018-04-19, 11:00: Midlands Graduate School Trip Report (James Wood, MSP)
2018-04-12, 11:00 (LT1415): Quotient Inductive-Inductive Types (Fredrik Nordvall Forsberg, MSP)
Higher inductive types (HITs) in Homotopy Type Theory (HoTT) allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types and allow to define types which are not sets in the sense of HoTT (i.e. do not satisfy uniqueness of equality proofs) such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define sets, such as the Cauchy reals, the partiality monad, and the internal, total syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductive-inductive definitions. We call those HITs quotient inductive-inductive types (QIITs).
Although there has been recent progress on the general theory of HITs, there isn't yet a theoretical foundation of the combination of equality constructors and induction-induction, despite having many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initial-algebra semantics and show that this is equivalent to the section induction principle, which justifies the intuitively expected elimination rules.
2018-04-06, 11:00: Setoids and Quotients (James Chapman, MSP)
In preparation for Fred's talk about "Quotient Inductive-Inductive Types" next week I will introduce quotients and setoids in type theory and some of the issues surrounding them. The use of setoids is discouraged by many doctors and can lead to a contagious and incurable condition: relation preservation. Quotients on the other hand are dangerous if not correctly handled and can lead to unsightly things appearing where they shouldn't such as inhabitants of the excluded middle.
2018-03-28, 12:30: What is a category, actually? (Thorsten Altenkirch, Nottingham)
Let us revisit the definition of a category and define it in a way which has the advantage that we can generalize it to higher dimensions. Why am I interested in higher categories (or specifically (\infty,1)-categories)? I have a few problems in Homotopy Type Theory which I think can be solved using these beasts: the coherence problem for type theory in type theory (in the moment I cannot even define the standard model) and generalizing the Hungarian approach to Quotient Inductive Inductive Types (QIITs) to Higher Inductive Inductive Types (HIITs).
2018-03-21, 15:00 (LT1415): Equivariant ZFA and the foundations of nominal techniques (Jamie Gabbay, Heriot-Watt)
The sets foundations of nominal techniques are usually taken to be Fraenkel-Mostowski set theory (which is ZFA + a finite support property). I will argue that in many situations, a new foundation which I call Equivariant ZFA (EZFA) may be a better choice, because you can do everything in EZFA that you can do in FM and furthermore EZFA with Choice (EZFAC) is consistent whereas FM with Choice is not.
I will define EZFA and how it interacts with Choice.
I will prove that EZFA is equivalent to ZFA.
I will then prove that EZFA is not equivalent to ZFA.
I will explain why I think EZFA(C) may be useful, why my last three papers were actually written in EZFAC, and finally I will discuss the implications this may have for mathematical foundations in general.
This talk will be based on the following paper: http://www.gabbay.org.uk/papers.html#equzfn
2018-03-14, 15:00: Quantum Computing with ZX Calculus (Joe Collins, MSP)
Quantum mechanics is dope, so it makes sense that making a computer using the principles of quantum mechanics would also be pretty sick. However, the formalism that is used by physicists, called a Hilbert space, is not specialised for this purpose. In particular, it is
1) Difficult to prove properties about programs for quantum computers
2) Difficult to see what is these programs are actually doing
Thankfully, category theory is very cool! Using ZX calculus, we can talk about quantum computing in a much clearer manner. I will be introducing some fundamental quantum mechanics and ZX calculus, and then using ZX calculus I will talk about Shor's algorithm.
2018-03-09, 16:00: MSP 101: Kan extensions (Alasdair Lambert and Ben Price, MSP)
2018-03-07: SPLS (Level 5, School of Computing Science, University of Glasgow)
2018-02-21, 15:00: A Type-System for describing the Structural Topology of System-on-a-Chip Architectures (Jan de Muijnck-Hughes, Glasgow)
The protocols that describe the interactions between IP Cores on System-on-a-Chip architectures are well-documented. These protocols described not only the structural properties of the physical interfaces but also the behaviour of the emanating signals. However, there is a disconnect between the design of SoC architectures, their formal description, and the verification of their implementation in known hardware description languages.
Within the Border Patrol project we are investigating how to capture and reason about the structural and behavioural properties of SoC architectures using state-of-the-art advances in programming language research. Namely, we are investigating using dependent types and session types for the description of hardware communication.
In this talk I will discuss my work in designing a linked family of languages that captures and reasons about the topological structure of a System-on-a-Chip. These languages provide correct-by-construction guarantees over the interaction protocols themselves; the adherence of a component that connects using said protocols; and the validity of the specified connections. These guarantees are provided through abuse of indexed monads to reason about resource usage; and general (ab)use of dependent types as presented in Idris.
I will not cover all aspects of the languages but will concentrate my talk detailing the underlying theories that facilitate the correct-by-construction guarantees.
2018-02-14, 15:00: MSP 101: Domain Theory (Fredrik Nordvall Forsberg, MSP)
I will give an introduction to Domain Theory, focusing on motivation. I hope to cover recursive definitions, and solving domain equations using retraction pairs.
2018-02-08, 15:00 (LT1212): Introduction to coalgebra (Alexander Kurz, Leicester)
2018-02-07, 15:00 Departmental seminar (LT1415): From Couplings to Probabilistic Relational Program Logics (Justin Hsu, UCL)
Many program properties are relational, comparing the behavior of a program (or even two different programs) on two different inputs. While researchers have developed various techniques for verifying such properties for standard, deterministic programs, relational properties for probabilistic programs have been more challenging. In this talk, I will survey recent developments targeting a range of probabilistic relational properties, with motivations from privacy, cryptography, machine learning. The key idea is to meld relational program logics with an idea from probability theory, called a probabilistic coupling. The logics allow a highly compositional and surprisingly general style of analysis, supporting clean proofs for a broad array of probabilistic relational properties.
2018-02-07, 15:00: MSP 101: The adjoint functor theorem (Fredrik Nordvall Forsberg, MSP)
Adjoint functors arise everywhere, but how do we find them? It is a fun exercise to prove that right adjoints preserve limits, and, dually, that left adjoints preserve colimits. An adjoint functor theorem is a statement that (under certain conditions) the converse holds: a functor which preserves limits is a right adjoint. I will discuss the General Adjoint Functor Theorem, and why Peter Johnstone considers it fundamentally useless.
2018-01-31, 15:00: Reflections on the PhD process (Kevin Dunne, MSP)
2018-01-24, 15:00: Coinduction and the Companion (Jurriaan Rot, Radboud University Nijmegen)
The coinductive proof method can be enhanced by several techniques, often referred to as up-to-techniques. I will talk about the basic theory of coinduction-up-to, and a little about the more recent notion of companion. This companion is the largest valid up-to technique for a given predicate, and gives a nice way of working with coinduction up-to.
2018-01-16: MSP101 Planning session (LT1310)
2017-12-08, 16:00: Dijkstra Monads (Bob Atkey, MSP)
One monad to the tune of another.
2017-12-06, 13:00 (LT1415): The Power of Parameterization in Coinductive Proof (Chung-Kil Hur, Seoul National University)
In this talk, I will present a simple yet powerful principle for coinductive reasoning, which we call "parameterized coinduction". More specifically, it is as simple as the Knaster-Tarski theorem without requiring any syntactic checking, yet as powerful as Coq's syntactic guarded coinduction supporting incremental reasoning. As an important consequence, parameterized coinduction can easily support complex nested induction-coinduction.
We also implemented the parameterized coinduction as the Coq library called "paco", which can be found at:
http://plv.mpi-sws.org/paco
This is joint work with Georg Neis, Derek Dreyer and Viktor Vafeiadis, and was presented at POPL'13.
2017-12-01, 16:00: MSP 101: Hereditary substitution (Conor McBride, MSP)
2017-11-17, 16:00: Deep Learning (Ross Duncan, MSP)
I'll explain the first concepts of Deep Learning. This is an advert for a reading group on the topic which will run over the next few weeks.
2017-11-10, 16:00: Infinite Iteration of Games (Alasdair Lambert, MSP)
My talk will be based on our recent paper "A Compositional Treatment of Iterated Open Games". In this paper we introduce a new operator on open games to capture the concept of subgame perfect equilibrium as well as providing final coalgebra semantics of infinite games.
2017-11-03, 16:00: Measuring the sizes of trees (Bob Atkey, MSP)
I'll talk about a way of measuring the sizes of trees using weighted tree automata, in a compositional way that works well with pattern matching. This is based on some work by Georg Moser and Martin Hofmann.
2017-10-27, 16:00: Size-Matters in the Modal mu-Calculus (Johannes Marti, Universität Bremen)
I talk about three methods for measuring the size of formulas in the modal mu-calculus and explore how the choice between them influences the complexity of computations on formulas. Especially, I focus on the guarded transformation, which is a simple syntactic transformation on formulas that is commonly assumed to be polynomial but has recently been shown to require exponential time.
I will complain about the mess in the literature and present two of our (Clemens, Yde and me) own preliminary results:
1) There is a polynomial guarded transformation if we measure the input formula in the number of its subformulas and measure the output formula in the size of its closure.
2) If there is a polynomial guarded transformation where we measure the input formula in the size of its closure then there is a polynomial algorithm for solving parity games. Hence finding such a transformation is at least as hard as solving parity games, which is commonly believed to be quite hard.
We employ an automata-theoretic approach that relates the different measures for the size of a formula to different constraints on the transition structure of an automaton corresponding to the formula.
This is a very technical talk but there will be many pictures!
2017-10-13, 16:00: Coalgebraic Learning (Simone Barlocco, MSP)
Automata learning is a well known technique to infer a finite state machine from a set of observations. One important algorithm for automata learning is the L* algorithm by Dana Angluin. In this 101 I will present a new perspective on L* using ideas from coalgebra and modal logic. After a brief recap of how L* works, I will describe a generalisation of the L* algorithm to the coalgebraic level. I will conclude my talk with two concrete instances of the general framework: the learning of Mealy machines and of bisimulation quotients of probabilistic transition systems. Joint work with Clemens Kupke.
2017-10-11: SPLS (Informatics Forum, Edinburgh)
2017-10-06, 16:00: Tetration (Peter Hancock)
In ghci, you can write something like
let two f = f . f in two two two two (+1) 0
and it will print "65536". If you write something like
let two f = f . f ; four = two . two in four ($ two) id (+1) 0
it will print "Occurs check: cannot construct the infinite type: a ~ a -> a".
Can haskell not count to four?(*)
The topic is the compact representation of unfeasibly large numbers using a pure (Glaswegian?) universe. Some excursions into the hinterland of the topic are likely.
By the way, tetration is the function m n |-> n ^ .. ^ n with m n's.
(*) No doubt mad haskell mambo or pragma exists that will trick it into counting up to four. I will reward the most amusing/revolting demonstration with a large (reasonably priced) malt whisky.
2017-09-29, 16:00: Worlds (Ben Price, MSP)
Building on last week's introduction to Martin-Löf 1971, we describe a toy type theory which not only accounts for what type terms have, but also where they live. This extra information can be interpreted as where (physically) data lives, at what phase (typechecking vs runtime) it exists, when it exists, or who has access to the data.
In return for caring about these "worlds" describing where data lives, we get applications to distributed computing and erasure for efficient code generation, with future work to consider productivity and security.
2017-09-22, 16:00: Bidirectional Martin-Löf 1971 (Conor McBride, MSP)
Per Martin-Löf's 1971 Theory of Types is the ancestor of the type systems used today in Agda, Coq, Idris, NuPRL, and many other variations on the theme of dependent types. Its principal virtue is its simplicity: it has very few moving parts (but they move quite a lot). Its well known principal vice is its inconsistency: you can write a looping program inhabiting any type (thus 'proving' any proposition). I'll be talking about the design principles for constructing dependent type systems which are bidirectional — clearly split into a type checking part and a type synthesis part. By following these principles, it gets easier to establish good safety properties of these systems. In particular, I'll sketch how to keep type safety ("well typed programs don't go wrong") separate from normalization ("all computations terminate"). Martin-Löf's 1971, reformulated bidirectionally, makes a good example, because it's small and type-safe, but not normalizing.
2017-09-11 (LT1101 (Board room)): Adding Cubes to Agda (Andrea Vezzosi, Chalmers)
Cubical Type Theory (CTT) provides an extension of Martin-Löf Type Theory (MLTT) where we can interpret the univalence axiom while preserving the canonicity property, i.e. every closed term actually computes to a value. The typing and equality rules of CTT come as a fairly well-behaved extension of the ones of MLTT and the denotational model and prototype implementation help clarifying the system further.
Given the above it felt reasonable to introduce the features of CTT into a more mature proof assistant like Agda, and this talk reports the status of this endeavour. In short:
The univalence axiom is proven as a theorem and we successfully tested its computational behavior on small examples.
comp computes for any parametrized data or record types, including coinductive ones, but it is stuck for inductive families.
The interaction of the path type and copatterns gives extensionality principles for coinductive records.
The interval I is an actual type, we also have restriction types A[φ ↦ u] and types for partial elements Partial φ A. Their sort makes sure comp does not apply to them.
Examples are collected at https://github.com/Saizan/cubical-demo.
2017-09-06: Binding and Substitution in String Diagrams (Ross Duncan, MSP)
In the categorical semantics of (e.g.) the simply typed lambda calculus substitution of a variable by a term is achieved by composing morphisms. What is the equivalent notion in diagrammatic languages? What even is a "variable" in this context? I'll sketch some (pretty) rough ideas for the beginnings of a "functional language" of diagrams including substitution, binding, and pattern matching. It turns out to all be about operads and co-operads.
2017-07-27 (LT1415): The Rise and Fall of Cooperative Communities (Matteo Cavaliere, Edinburgh)
Social, biological and economic networks grow and decline with recurrent fragmentation and re-formation, often explained in terms of external perturbations. I will present a model of dynamical networks and evolutionary game theory that explains these phenomena as consequence of imitation and endogenous conflicts between "cooperators" and "cheaters". Cooperators promote well-connected prosperous (but fragile) networks and cheaters cause the network to fragment and lose its prosperity. Once the network is fragmented it can be reconstructed by a new invasion of cooperators, leading to recurrent cycles of formation and fragmentation observed, for instance, in bacterial communities and socio-economic networks. In the last part of the talk, I will briefly introduce my current works on the role of individual decision-making in cooperative communities and the possibility of synthetic biology to address these ideas in microbial communities.
M. Cavaliere, S. Sedwards, C.E. Tarnita, M.A. Nowak, A. Csikasz-Nagy.
Prosperity is Associated with Instability in Dynamical Networks.
Journal of Theoretical Biology, 299, 2012.
Plasticity Facilitates Sustainable Growth in the Commons.
M. Cavaliere, J.F. Poyatos.
Journal of the Royal Society Interface, 10, 2013.
Eco-Evolutionary Feedbacks can Rescue Cooperation in Microbial Populations.
C. Moreno-Fenoll, M. Cavaliere, E. Martinez-Garcia, J.F. Poyatos.
Scientific Reports, 7, 2017.
2017-07-10: Open Games Workshop (LT1415)
2017-07-05 (LT1415): Some enumerative, topological, and algebraic aspects of linear lambda calculus (Noam Zeilberger, Birmingham)
Enumeration of graphs on surfaces (or "maps") is an active topic of research in combinatorics, with links to wide-ranging domains such as algebraic geometry, knot theory, and mathematical physics. In the last few years, it has also been found that map enumeration is related to the combinatorics of lambda calculus, with various well-known families of maps in 1-to-1 correspondence with various natural families of linear lambda terms. In the talk I will begin by giving a brief survey of these enumerative connections, then use those to motivate a closer look at the surprisingly rich topological and algebraic properties of linear lambda calculus.
2017-06-29: Syntax and semantics of Quantitative Type Theory (Bob Atkey, MSP)
At last year's WadlerFest celebration, Conor presented a dependent type theory where variables are tagged with information about how they are used. Variable usage tagging has been developed in the non dependent setting, starting with Girard's Linear Logic, and culminating with recent work in contextual effects, coeffects, and quantitative type theories. The subtlety with dependent types lies in how to account for the difference between usage in types and terms. Conor's system handles this by treating usage in types as a "zero" usage, so that it doesn't affect the usage in terms. This is a departure from previous "linear" type theories that maintains a strict separation between usage controlled information, which types cannot depend on, and unrestricted information, which types can depend on.
Conor presented a syntax and typing rules for the system, as well as an erasure property that exploits the difference between "not used" and "used", but doesn't do anything with the finer usage information available. I'll present a collection of models for the system that fully exploit the usage information. This will give interpretations of type theory in resource constrained computational models, Geometry of Interaction models, and imperative models. To maintain order, I will gather all these notions of model under a new concept of "Quantitative Category with Families", a generalisation of the standard "Category with Families" class of models of dependent types.
2017-06-15: Informal introduction to knot theory and the unknotting problem (Stuart Gale, MSP)
This is an informal talk on the interesting properties I've found when playing with the unknotting problem (knot simplification moves that help to establish whether any given knot is a loop in complicated disguise, or something really knotted).
I'll discuss the syntax that I've used for annotating knots that leads to a(n almost) syntax based method for unknotting, but that hints further at unknotting in a more interesting way by using an unintentional property of the syntax.
I'll present some examples of the problems with representing knots and how the syntax and reduction rules help, in my opinion, to make unknotting more tangible.
2017-06-08: TYPES 2017 Trip report (Fredrik Nordvall Forsberg, MSP)
I'll tell you about the most interesting talks, ideas and gossip that came out of the TYPES conference in BudaPest last week.
2017-06-01: Optimal nondeterminism: explaining the Kameda-Weiner algorithm (Rob Myers)
The Kameda-Weiner algorithm takes a machine (nondeterministic finite automaton) as input, and provides an optimal machine (state-minimal nondeterministic finite automaton) as output. In this talk I will discuss work which provides a clear explanation of it, by translating the various syntactic constructs into more meaningful order-theoretic ones, and then composing them together to prove correctness.
2017-05-25, 11:00: Learning via Coalgebraic Logic (Clemens Kupke, MSP)
A key result in computational learning theory is Dana Angluin's L* algorithm that describes how to learn a regular language, or a deterministic finite automaton (DFA), using membership and equivalence queries. In my talk I will present a generalisation of this algorithm using ideas from coalgebra and modal logic — please note, however, that prior knowledge of these topics will not be required.
In the first part of my talk I will recall how the L* algorithm works and establish a link to the notion of a filtration from modal logic. Furthermore I will provide a brief introduction to coalgebraic modal logic. In the second part of my talk I will present a generalisation of Angluin's original algorithm from DFAs to coalgebras for an arbitrary finitary set functor T in the following sense: given a (possibly infinite) pointed T-coalgebra that we assume to be regular (i.e. having an equivalent finite representation) we can learn its finite representation by (i) asking "logical queries" (corresponding to membership queries) and (ii) making conjectures to which a teacher has to reply with a counterexample (equivalence queries). This covers (known variants of) the original L* algorithm and algorithms for learning Mealy and Moore machines. Other examples are infinite streams, trees and bisimulation quotients of various types of transition systems.
Joint work with Simone Barlocco.
2017-05-18, 15:00: The joy of QIITs (Thorsten Altenkirch, Nottingham)
Quotient inductive inductive types (QIITS) are set-truncated mutually defined higher inductive types. I am going to discuss two applications of QIITs:
1. define an internal syntax of Type Theory without reference to untyped preterms;
2. define a version of the partiality monad that doesn't require countable choice.
On the one hand I think that these applications are interesting because they represent applications of HoTT which have nothing to do with homotopy theory; on the other hand they are clearly not very higher order (in the sense of truncation levels) but can be defined in the set-truncated fragment of HoTT. Hence my question: what are interesting applications of higher types which are not directly related to synthetic homotopy theory?
This talk is based on joint work with Paolo Capriotti, Nils Anders Danielsoon, Gabe Dijkstra, Ambrus Kaposi and Nicolai Kraus.
2017-05-04: MSP 101: Differential Operators (Conor McBride, MSP)
A traditional source of complaint from CS undergraduates (especially in the USA, but in other places, too) is that they are made to learn too much standard issue mathematics with little apparent relevance to computation. Differential calculus (with its usual presentational focus on physical systems) is often picked upon as the archetype. What we see in action is the fragile male ego: they are not so quick to complain about the unimportance of things they do not find difficult. All of which makes more delicious the irony that differential operators have a key role to play in understanding discrete structures, such as automata, datatypes, execution stacks, and plenty more.
The basic idea is as follows: to put your finger over any single K in the pair of words
BREKEKEKEX KOAXKOAX
you must choose either to put your finger over a single K in BREKEKEKEX and pair with KOAXKOAX intact, or to leave BREKEKEKEX intact and cover a K in KOAXKOAX. You have just followed Leibniz's rule for differentiating a product (with respect to K), and computed a one-hole context for a K in a data structure.
Newton, of course, would point out that such derivatives arise as the limit of a divided difference, a concept worthy of study in more generality. I would point out that divided differences are often definable, even in situtations when neither division nor difference makes much apparent sense. Notably, Brzozowski's derivative for regular languages is a divided difference (even though it is not Leibniz's derivative).
I'll work mainly with containers (which look a lot like power series) but make sure there are plenty of concrete examples. In practice, it becomes rather useful to compute derivatives by pattern matching on types, which is especially funny as symbolic differentiation is the first example in the literature of computing anything by pattern matching at all.
2017-04-27: What I, Dunne, done in my PhD (Kevin Dunne, MSP)
2017-04-20: Modular Datalog (Bob Atkey, MSP)
2017-04-10: ALCOP2017 (Strathclyde, Room MC301 (McCance Building))
2017-04-05: CLAP (Strathclyde, room LT1415)
2017-03-30: Cubical adventures in Connecticut (Fredrik Nordvall Forsberg, MSP)
I'll report on my attempts to design a cubical type theory together with Dan Licata and Ed Morehouse during my visit to Wesleyan University, Middletown, Connecticut. We had something which seemed quite promising, but that falls apart just short of the finish line; I'll tell you about it in the hope of miraculous rescue from the audience. However, I'll start from basics so that everyone has a chance to join in in the fun. Mentions of Donald Trump will be kept to a minimum.
2017-03-23: Blockchain discussion (Bob Atkey, MSP)
2017-03-16: MSP 101: Operational and Denotational Semantics for PCF (Bob Atkey, MSP)
PCF is the prototypical functional programming language, with two data types (naturals and booleans), lambda-abstraction and recursion. PCF was introduced by Gordon Plotkin in his seminal "LCF Considered as a Programming Language" paper from 1977. Despite PCF's simplicity, its semantics is theoretically interesting. I will introduce PCF, its operational semantics, the "standard" domain-theoretic denotational semantics and show that the two agree on closed programs. Finally, I will discuss observational equivalence for PCF and show that the denotational semantics fails to be "fully abstract".
2017-03-09: MSP 101: Indexed Containers/Interaction Structures (Conor McBride, MSP)
2017-03-02: MSP 101: Separation Logic and Hoare Logic (Bob Atkey, MSP)
Hoare Logic is a logic for proving properties of programs of the form: if the initial state satisfies a precondition, then the final state satisfies a postcondition. Hoare logic proofs are structured around the structure of the program itself, making the system a compositional one for reasoning about pieces of programs. I'll introduce Hoare Logic for a little imperative language with WHILE loops. I'll then motivate Separation Logic, which enriches Hoare Logic with a Frame Rule for local reasoning.
2017-02-23 (LT1415): MSP 101: Blockchain (James Chapman, MSP)
I will try to follow on from yesterday's introduction by getting to nitty gritty of bitcoin/blockchain. I won't assume attendance of the seminar but will try not to repeat it!
2017-02-22, 15:00 Departmental seminar (LT1415): An introduction to Blockchains (Andrea Bracciali, Stirling)
Blockchains, i.e. decentralised, distributed data structures which can also carry executable code for a non-standard execution environment, introduce new models of computation. Decentralised, here, means, informally speaking, "without central control", e.g. a currency without a (central) bank, but much more. Blockchains support the recently introduced virtual currencies, a la Bitcoin, and a new class of decentralised applications, including smart contracts. In this talk we will introduce the main aspects of a blockchain, with particular reference to the Bitcoin blockchain as a paradigmatic case of such a new model of computation, and also touching on smart contracts. No previous knowledge of bitcoin/blockchain required for this introductory talk.
2017-02-16: MSP 101: Automata on infinite words (Clemens Kupke, MSP)
In this 101 I plan to discuss omega-automata, i.e., finite automata that operate on infinite words/streams. These automata form an important tool for the specification and verification of the ongoing, possibly infinite behaviour of a system. In my talk I will provide the standard definition(s) of omega-automata and highlight what makes omega-automata difficult from a coalgebraic perspective. Finally, I am going to discuss the work by Ciancia & Venema that provides a first coalgebraic representation of a particular type of omega-automata, so-called Muller automata.
2017-02-15, 13:00 Departmental seminar (LT1415): Semantics for probabilistic programming (Chris Heunen, Edinburgh)
Statistical models in e.g. machine learning are traditionally expressed in some sort of flow charts. Writing sophisticated models succinctly is much easier in a fully fledged programming language. The programmer can then rely on generic inference algorithms instead of having to craft one for each model. Several such higher-order functional probabilistic programming languages exist, but their semantics, and hence correctness, are not clear. The problem is that the standard semantics of probability theory, given by measurable spaces, does not support function types. I will describe how to get around this.
2017-02-09: MSP 101: Automata learning (Simone Barlocco, MSP)
Automata learning is a well known technique to infer a finite state machine from a set of observations. One important algorithm for automata learning is the L* algorithm by Dana Angluin. In this 101, I will explain how the L* algorithm works via an example. Afterwards, I will discuss the ingredients of the algorithm both in the standard framework by Angluin and in a recently developed categorical/coalgebraic framework by Jacobs & Silva. Lastly, I plan to outline the proof of the minimality of the automaton that is built by the learning algorithm.
2017-02-06 (LT1415): On the expressive power of user-defined effects: effect handlers, monadic reflection, delimited control without answer-type-modification (Ohad Kammar, Oxford)
We compare the expressive power of three programming abstractions for user-defined computational effects: Bauer and Pretnar's effect handlers, Filinski's monadic reflection, and delimited control. This comparison allows a precise discussion about the relative merits of each programming abstraction.
We present three calculi, one per abstraction, extending Levy's call-by-push-value. These comprise syntax, operational semantics, a natural type-and-effect system, and, for handlers and reflection, a set-theoretic denotational semantics. We establish their basic meta-theoretic properties: adequacy, soundness, and strong normalisation. Using Felleisen's notion of a macro translation, we show that these abstractions can macro-express each other, and show which translations preserve typeability. We use the adequate finitary set-theoretic denotational semantics for the monadic calculus to show that effect handlers cannot be macro-expressed while preserving typeability either by monadic reflection or by delimited control. We supplement our development with a mechanised Abella formalisation.
Joint work with Yannick Forster, Sam Lindley, and Matija Pretnar.
2017-02-02: MSP 101: First-order logic (Johannes Marti, MSP)
In this 101 I outline the syntax and semantics of classical first order predicate logic. I try to also mention some of the characteristic properties of first order logic such as compactness, the Löwenheim-Skolem theorem or locality properties in finite model theory.
2017-01-26: La Vie Parisienne: POPL trip report (Conor McBride, James Chapman, and Bob Atkey, MSP)
Our POPL attendees will tell us about their favourite talks, the latest research gossip and show us their most scenic photos from POPL in Paris.
2017-01-19: MSP 101: Automata and games for fixpoint logics (Johannes Marti, MSP)
I explain how we can use automata and games to understand the behaviour of modal fixpoint logics.
2016-12-14, 16:00 Departmental seminar (LT1415): Do be do be do (Sam Lindley, Edinburgh)
We explore the design and implementation of Frank, a strict functional programming language with a bidirectional effect type system designed from the ground up around a novel variant of Plotkin and Pretnar's effect handler abstraction.
Effect handlers provide an abstraction for modular effectful programming: a handler acts as an interpreter for a collection of commands whose interfaces are statically tracked by the type system. However, Frank eliminates the need for an additional effect handling construct by generalising the basic mechanism of functional abstraction itself. A function is simply the special case of a Frank operator that interprets no commands.
Moreover, Frank's operators can be multihandlers which simultaneously interpret commands from several sources at once, without disturbing the direct style of functional programming with values.
Effect typing in Frank employs a novel form of effect polymorphism which avoid mentioning effect variables in source code. This is achieved by propagating an ambient ability inwards, rather than accumulating unions of potential effects outwards.
I'll give a tour of Frank through a selection of concrete examples.
(Joint work with Conor McBride and Craig McLaughlin)
2016-12-14, 11:00: Compositional Game Theory (Alasdair Lambert, MSP)
I will be discussing composition in a model of economic game theory and methods for representing the impact of choice on subsequent games. Time permitting I will also work through some games using this model.
2016-12-07, 11:00: Nominal filters and semantics of predicate logic (Jamie Gabbay, Heriot-Watt)
A filter P is a consistent deductively closed set of predicates. A filter is prime when
(φ ∨ ψ) ∈ P ⇒ (φ ∈ P ∨ ψ ∈ P)
In words: if phi-or-psi is in P then phi is in P or psi is in P. Primeness gives soundness for disjunction.
Using this it is not hard to construct a semantics to propositional logic in which a predicate φ "means" the set of prime filters containing it. This is a standard "trick" for building semantics and is an extremely useful proof-method.
I have developed a semantics for predicate logic and also for the lambda-calculus based on similar notions of filter, but in a nominal context — meaning that filters are developed using Fraenkel-Mostowski (FM) set theory instead of Zermelo-Fraenkel (ZF) set theory. What matters here is that FM sets have additional name structure over ZF sets, and this additional structure can be exploited to give semantics to the extra structure that predicates have over propositions, and in particular the additional name structure lets us write down primeness conditions for soundness for universal quantification.
The resulting semantics is rich and interesting. In a sentence: nominal techniques help us to extend the notion of Stone representation and duality from propositional logic to full first-order logic (also with equality, if we wish, and also to other logics and calculi with variables and quantifiers).
I will give a detailed description of the filter-style conditions involved, and discuss some of what I think they tell us about predicates and quantification in logic and computation. More information can also be found in two papers here:
http://www.gabbay.org.uk/papers.html#semooc
http://www.gabbay.org.uk/papers.html#repdul
2016-11-30: CLAP (Strathclyde)
2016-11-23, 11:00: Some model theory for the modal mu-calculus (Yde Venema, ILLC, Amsterdam)
We discuss a number of semantic properties pertaining to formulas of the modal mu-calculus. For each of these properties we provide a corresponding syntactic fragment, in the sense that a mu-calculus formula \phi has the given property iff it is equivalent to a formula \phi' in the corresponding fragment. Since this formula \phi' will always be effectively obtainable from \phi, as a corollary, for each of the properties under discussion, we prove that it is decidable in elementary time whether a given mu-calculus formula has the property or not.
The properties that we study have in common that they all concern the dependency of the truth of the formula at stake, on a single proposition letter p. In each case the semantic condition on \phi will be that \phi, if true at a certain state in a certain model, will remain true if we restrict the set of states where p holds, to a special subset of the state space. Important examples include the properties of complete additivity and (Scott) continuity, where the special subsets are the singletons and the finite sets, respectively.
Our proofs for these characterisation results will be automata-theoretic in nature; we will see that the effectively defined maps on formulas are in fact induced by rather simple transformations on modal automata.
2016-11-16, 11:00: MSP 101: Modal logic (Johannes Marti, MSP)
Modal logic provides a simple, yet surprisingly powerful, language for specifying properties of coalgebras. In this talk I introduce the basic modal logic that is interpreted on relational structures. My aim is to provide an idea how modal logic relates to other logics, such as first-order and intuitionistic logic, and to the duality between algebraic and coalgebraic structures.
If time permits, I might also give a very informal warm-up for the modal mu-calculus which is the topic of next week's talk.
2016-11-09: SPLS (Room 301, McCance building, Strathclyde)
2016-11-09, 10:30: MSP 101: Meta-theory of lambda-calculi (Stuart Gale, MSP)
I'll give a standard overview of Simply Typed Lambda Calculus (STLC) (syntax, typing and computation rules) in a well-typed setting, and then modify it to show STLC in a bidirectional setting.
Afterwards I'll show Strong Confluence (Church-Rosser theorem) in the bidirectional setting.
2016-11-02 (LT1415): An introduction to many-valued logics and effect algebras (Phil Scott, University of Ottawa)
The algebras of many-valued Lukasiewicz logics (MV algebras) as well as the theory of quantum measurement (Effect algebras) have undergone considerable development in the 1980s and 1990s; they now constitute important research fields, with connections to several contemporary areas of mathematics, logic, and theoretical computer science.
Both subjects have recently attracted considerable interest among groups of researchers in categorical logic and foundations of quantum computing. I will give a leisurely introduction to MV algebras (and their associated logics), as well as the more general world of effect algebras. If time permits, we will also illustrate some new results (with Mark Lawson, Heriot-Watt) on coordinatization of some concrete MV-algebras using inverse semigroup theory.
2016-11-02, 11:00: System F and proof-relevant parametricity (Ben Price, MSP)
I shall give a brief introduction to System F.
I will then explain how to capture our intuition about polymorphic functions behaving uniformly by relational parametricity, and talk about ongoing work to find a notion of proof-relevant parametricity.
2016-10-26, 11:00: MSP 101: Data types and initial-algebra semantics (Fredrik Nordvall Forsberg, MSP)
I will give a basic introduction to data types and initial-algebra semantics. The meaning of a data type is given as the initial object in a category of types with the corresponding constructors. Initiality immediately allows the modelling of a non-dependent recursion principle. I'll show how this can be upgraded to full dependent elimination, also known as induction, by using the uniqueness of the mediating arrow; in fact, induction is equivalent to recursion plus uniqueness. All possibly unfamiliar terms in this abstract will also be explained.
2016-10-19, 11:00: MSP 101: A first introduction to coalgebra (Clemens Kupke, MSP)
The core subject of Computer Science is "generated behaviour" (quiz: who said this?). Coalgebra provides the categorical formalisation of generated behaviour. I am planning to provide a first, very basic introduction to coalgebra. This will consist of two parts: i) coinduction & corecursion as means to define & reason about the (possibly) infinite behaviour of things; ii)modal logics for coalgebras.
2016-10-12, 11:00: MSP 101: Category Theory (Neil Ghani, MSP)
2016-10-05, 11:00: MSP 101: Rewriting, and operads (Ross Duncan, MSP)
2016-09-28, 15:00: Towards a Generic Treatment of Syntaxes with Binding (Guillaume Allais, Radboud University Nijmegen)
The techniques used by the generic programming community have taught us that we can greatly benefit from exposing the common internal structure of a family of objects. One can for instance derive once and for all a wealth of iterators from an abstract characterisation of recursive datatypes as fixpoints of functors.
Our previous work on type and scope preserving semantics and their properties has made us realise that numerous semantics of the lambda calculus can be presented as instances of the fundamental lemma associated to an abstract notion of 'Model'. This made it possible to avoid code duplication as well as prove these semantics' properties generically.
Putting these two ideas together, we give an abstract description of syntaxes with binding making both their recursive and scoping structure explicit. The fundamental lemma associated to these syntaxes can be instantiated to provide the user with proofs that its language is stable under renaming and substitution as well as provide a way to easily define various evaluators.
2016-09-28, 11:00: ICFP trip report (James Chapman, MSP)
2016-09-21, 11:00: Eventual image functors (Kevin Dunne, MSP)
For a category C we consider the endomorphism category End(C) and the subcategory of automorphisms Aut(C) -> End(C). It has been observed that for C the category of finite sets, finite dimensional vector spaces, or compact metric spaces this inclusion functor admits a simultaneous left and right adjoint.
We give general criteria for the existence of such adjunctions for a broad class of categories which includes FinSet, FinVect and CompMet as special cases. This is done using the language of factorisation systems and by introducing a notion of eventual image functors which provide a general method for constructing adjunctions of this kind.
2016-09-15, 15:00: Morphisms of open games (Jules Hedges, Oxford)
2016-08-24, 16:00 (LT1415): Information Effects for Understanding Type Systems (Philippa Cowderoy)
Or: how someone else found the maths to justify my dogma
Slides (Philippa says: beware, slightly wordy slides!)
Haskell implementation for simply-typed lambda calculus
2016-07-01 (LT1415): A New Perspective On Observables in the Category of Relations (Kevin Dunne, MSP)
Practice talk for Quantum Interactions.
2016-07-01 (LT1415): Interacting Frobenius Algebras are Hopf (Ross Duncan, MSP)
Practice talk for LICS.
Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi have shown that, given a suitable distribution law, a pair of Hopf algebras forms two Frobenius algebras. Coming from the perspective of quantum theory, we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise \cite{Bonchi2014a} by including non-trivial dynamics of the underlying object – the so-called phase group – and investigate the effects of finite dimensionality of the underlying model, and recover the system of Bonchi et al as a subtheory in the prime power dimensional case. We show that the presence of a non-trivial phase group means that the theory cannot be formalised as a distributive law.
2016-06-06: Quantum Physics and Logic 2016 (McCance building, Strathclyde)
2016-06-01, 12:00: Interacting Frobenius algebras (Kevin Dunne, MSP)
Practice talk for QPL.
Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise (BSZ 2014) by including non-trivial dynamics of the underlying object – the so-called phase group – and investigate the effects of finite dimensionality of the underlying model. We recover the system of Bonchi et al as a subtheory in the prime power dimensional case, but the more general theory does not arise from a distributive law.
2016-05-25, 15:30 Departmental seminar (LT908): A Compositional Approach to Game Theory (Neil Ghani, MSP)
I will sketch an alternative approach to economic game theory based upon the computer science idea of compositionality: concretely we i) give a number of operators for building up complex and irregular games from smaller and simpler games; and ii) show how the Nash equilibrium of these complex games can be defined recursively from their simpler components. We apply compositional reasoning to sophisticated games where agents must reason about how their actions affect future games and how those future games effect the utility they receive. This forces us into a second innovation — we augment the usual lexicon of games with a dual notion to utility because, in order for games to accept utility, this utility must be generated by other games. Our third innovation is to represent our games as string diagrams so as to give a clear visual interface to manipulate them. Our fourth, and final, innovation is a categorical formalisation of these intuitive diagrams which ensures our reasoning about them is fully rigorous.
The talk will be general so as appeal to as wide an audience as possible. In particular, no knowledge of category theory will be assumed!
2016-05-04, 11:00: Factorisation Systems and Algebra (Kevin Dunne, MSP)
I'll show how to generalise some results from algebra (think groups, rings, R-modules etc.) to a categorical setting using factorisation systems and an appropriate notion of finiteness on the objects of a category.
2016-04-27, 11:00: Typing with Leftovers (Guillaume Allais, Radboud University Nijmegen)
2016-04-19, 16:00 Departmental seminar (LT1415): Formal languages, coinductively formalized (Andreas Abel, Chalmers and Gothenburg University)
Formal languages and automata are taught to every computer science student. However, the student will most likely not see the beautiful coalgebraic foundations.
In this talk, I recapitulate how infinite trees can represent formal languages (sets of strings). I explain Agda's coinduction mechanism based on copatterns and demonstrate that it allows an elegant representation of the usual language constructions like union, concatenation, and Kleene star, with the help of Brzozowski derivatives. We will also investigate how to reason about equality of languages using bisimulation and coinductive proofs.
2016-04-14: Categories, Logic, and Physics, Scotland ( School of Informatics, Edinburgh)
2016-03-31, 10:30 (LT1415): Presentations by candidates for the 'Coalgebraic Foundations of Semi-Structured Data' RA position
2016-03-23, 16:00 Departmental seminar (LT1415): A static analyser for concurrent Java (Bob Atkey, MSP)
ThreadSafe is a static analysis tool for finding bugs in concurrent Java code that has been used by companies across the world to analyse and find bugs in large mission industrial applications. I will talk about how ThreadSafe works, and our experiences in applying static analysis technology to the "real world".
ThreadSafe is available from http://www.contemplateltd.com/
2016-03-17, 11:00: On the rule algebraic reformulation of graph rewriting (Nicolas Behr, Edinburgh)
Motivated by the desire to understand the combinatorics of graph rewriting systems, it proved necessary to invent a formulation of graph rewriting itself that is not based on category theoretic structures as in the traditional framework, but on the concept of diagrammatic combinatorial Hopf algebras and reductions thereof. In this talk, I will present how the classical example of the Heisenberg-Weyl algebra of creation and annihilation of indistinguishable particles, which can alternatively be interpreted as the algebra of discrete graph rewriting, gave the initial clues for this novel framework. In hindsight, to pass from the special case of discrete graph rewriting to the case of general graph rewriting required every aspect of the framework of diagrammatic combinatorial Hopf algebras as a guideline for the construction, yet none of the traditional category theoretic ideas, whence one might indeed consider this reformulation as an independent formulation of graph rewriting.
The new framework results in a number of surprising results even directly from the formulation itself: besides the two main variants of graph rewriting known in the literature (DPO and SPO rewriting), there exist two more natural variants in the new framework. For all four variants, graph rewriting rules are encoded in so-called rule diagrams, with their composition captured in the form of diagrammatic compositions followed by one of four variants of reduction operations. Besides the general structure theory of the resulting algebras aka the rule algebras, one of the most important results to date of this framework in view of applications is the possibility to formulate stochastic graph rewriting systems in terms of the canonical representations of the rule algebras. Notably, this is closely analogous to the formulation of chemical reaction systems in terms of the canonical representation of the Heisenberg-Weyl algebra aka the bosonic Fock space. The presentation will not assume any prior knowledge of the audience on the particular mathematics required for this construction, and will be given on the whiteboard. The work presented is the result of a collaboration with Vincent Danos and Ilias Garnier (ENS Paris/LFCS University of Edinburgh), and (in an earlier phase) with Tobias Heindel (University of Copenhagen).
2016-03-03, 11:00: Comprehensive Parametric Polymorphism (Fredrik Nordvall Forsberg, MSP)
In this talk, we explore the fundamental category-theoretic structure needed to model relational parametricity (i.e., the fact that polymorphic programs preserve all relations) for the polymorphic lambda calculus (a.k.a. System F). Taken separately, the notions of categorical model of impredicative polymorphism and relational parametricity are well-known (lambda2-fibrations and reflexive graph categories, respectively). Perhaps surprisingly, simply combining these two structures results in a notion that only enjoys the expected properties in case the underlying category is well-pointed. This rules out many categories of interest (e.g. functor categories) in the semantics of programming languages.
To circumvent this restriction, we modify the definition of fibrational model of impredicative polymorphism by adding one further ingredient to the structure: comprehension in the sense of Lawvere. Our main result is that such comprehensive models, once further endowed with reflexive-graph-category structure, enjoy the expected consequences of parametricity. This is proved using a type-theoretic presentation of the category-theoretic structure, within which the desired consequences of parametricity are derived. Working in this type theory requires new techniques, since equality relations are not available, so that standard arguments that exploit equality need to be reworked.
This is joint work with Neil Ghani and Alex Simpson, and a dry run for a talk in Cambridge the week after.
2016-02-25, 11:00: Excuse My Extrusion (Conor McBride, MSP)
I have recently begun to learn about the Cubical Type Theory of Coquand et al., as an effective computational basis for Voevodsky's Univalent Foundations, inspired by a model of type theory in cubical sets. It is in some ways compelling in its simplicity, but in other ways intimidating in its complexity. In order to get to grips with it, I have begun to develop my own much less subtle variation on the theme. If I am lucky, I shall get away with it. If I am unlucky, I shall have learned more about why Cubical Type Theory has to be as subtle as it is.
My design separates Coquand's all-powerful "compose" operator into smaller pieces, dedicated to more specific tasks, such as transitivity of paths. Each type path Q : S = T, induces a notion of value path s {Q} t, where either s : S, or s is •, "blob", and similarly, t : T or t = •. A "blob" at one end indicates that the value at that end of the path is not mandated by the type. This liberalisation in the formation of "equality" types allows us to specify the key computational use of paths between types, extrusion:
if Q : S = T and s : S, then s ⌢• Q : s {Q} •
That is, whenever we have a value s at one end of a type path Q : S = T, we can extrude that value across the type path, getting a value path which is s at the S end, but whose value at the T end is not specified in advance of explaining how to compute it. Extrusion gives us a notion of coercion-by-equality which is coherent by construction. It is defined by recursion on the structure of type paths. Univalence can be added to the system by allowing the formation of types interpolating two equivalent types, with extrusion demanding the existence of the corresponding interpolant values, computed on demand by means of the equivalence.
So far, there are disconcerting grounds for optimism, but the whole of the picture has not yet emerged: I may just have pushed the essential complexity into one corner, or the whole thing may be holed below the waterline. But if it does turn out to be nonsense, it will be nonsense for an interesting reason.
2016-02-18: Coalgebraic Dynamic Logics (Clemens Kupke, MSP)
I will present work in progress on a (co)algebraic framework that allows to uniformly study dynamic modal logics such as Propositional Dynamic Logic (PDL) and Game Logic (GL). Underlying our framework is the basic observation that the program/game constructs of PDL/GL arise from monad structure, and that the axioms of these logics correspond to compatibility requirements between the modalities and this monad structure. So far we have a general soundness and completeness result for PDL-like logics wrt T-coalgebras for a monad T. I will discuss our completeness theorem, its limitations and plans for extending our results. [For the latter we might require the help of koalas, wallabies and wombats.]
2016-02-11, 11:00: Introduction to (infinity, 1)-categories (Fredrik Nordvall Forsberg, MSP)
Infinity-categories simultaneously generalise topological spaces and categories. Intuitively, a (weak) infinity-category should have objects, morphisms, 2-morphisms, 3-morphisms, ... and identity morphisms and composition which is suitably unital and associative up to a higher (invertible) morphism (the number 1 in (infinity, 1)-category means that k-morphisms for k > 1 are invertible) . The trouble begins when one naively tries to make these coherence conditions precise; already 4-categories famously requires 51 pages to define explicitly. Instead, one typically turns to certain "models" of infinity-categories that encode all this data implicitly, usually as some kind of simplicial object with additional properties. I will introduce two such models: quasicategories and complete Segal spaces. If time allows, I will also discuss hopes and dreams about internalising these notions in Type Theory, which should give a satisfactory treatment of category theory in Type Theory without assuming Uniqueness of Identity Proofs.
2016-02-04, 11:00: Two Constructions on Games (Neil Ghani, MSP)
I've been working with Jules Hedges on a compositional model of game theory. After briefly reminding you of the model, I'll discuss where we are at – namely the definition of morphisms between games, and the treatment of choice and iteration of games. I'm hoping you will be able to shed some light on this murky area. There is a draft paper if anyone is interested.
2016-01-28, 11:00: Introduction to sheaves (Kevin Dunne, MSP)
2016-01-26, 15:00: Rewriting modulo symmetric monoidal structure (Aleks Kissinger, Nijmegen)
String diagrams give a powerful graphical syntax for morphisms in symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and control theory.
An important role in many such approaches is played by equational theories of diagrams, which can be oriented and used as rewrite systems. In this talk, I'll lay the foundations for this form of rewriting by interpreting diagrams combinatorially as typed hypergraphs and establish the precise correspondence between diagram rewriting modulo the laws of SMCs on the one hand and double pushout (DPO) rewriting of hypergraphs, subject to a soundness condition called convexity, on the other. This result rests on a more general characterisation theorem in which we show that typed hypergraph DPO rewriting amounts to diagram rewriting modulo the laws of SMCs with a chosen special Frobenius structure.
If there's time, I'll also discuss some of the results we have in developing the rewrite theory of hypergraphs for SMCs, namely termination proofs via graph metrics and strongly convex critical pair analysis.
2016-01-26, 10:30: Enriched Lawvere Theories (John Power, Bath)
In this talk, we consider extending Lawvere theories to allow enrichment in a base category such as CMonoid, Poset or Cat. In doing so, we see that we need to alter the formulation in a fundamental way, using the notion of cotensor, a kind of limit that is hidden in the usual accounts of ordinary category theory but is fundamental to enriched category theory. If time permits, we will briefly consider the specific issues that arise when one has two-dimensional structure in the enriching category, as exemplified by Poset and Cat.
2016-01-25, 10:30: Lawvere Theories (John Power, Bath)
In 1963, Bill Lawvere characterised universal algebra in category theoretic terms. His formulation being category theoretic was not its central contribution: more fundamental was its presentation independence. Three years later, monads were proposed as another category theoretic formulation of universal algebra. Overall, the former are technically better but the relationship is particularly fruitful and the latter are more prominent, cf Betamax vs VHS. So we study Lawvere theories carefully in the setting of universal algebra and in relation to monads.
2016-01-21, 11:00: Overview: A Type Theory for Probabilistic and Bayesian Reasoning (Guillaume Allais, MSP)
The probabilistic calculus introduced in the eponymous paper by Robin Adams and Bart Jacobs is inspired by quantum theory by considering that conditional probabilities can be seen as side-effect-free measurements. A type-theoretic treatment of this semantic observation leads, once equipped with suitable computation rules, to the ability to do exact conditional inference.
I will present the type theory and the accompanying computation rules proposed in the paper and discuss some of the interesting open questions I will be working on in the near future.
2015-12-02, 11:00: Introduction to coherence spaces, and how to use them for dependent session types (Bob Atkey, MSP)
Coherence spaces are a simplification of Scott domains, introduced by Girard to give a semantics to the polymorphic lambda-calculus. While investigating the structure of coherence spaces, Girard noticed that the denotation of the function type in coherence spaces can be decomposed into two independent constructions: a linear ("use-once") function space, and a many-uses-into-one-use modality. And so Linear Logic was discovered.
Coherence spaces are interesting because they model computation at a low level in terms of interchange of atomic 'tokens' of information. This makes them a useful tool for understanding several different computational phenomena.
In this talk, I'll show how coherence spaces naturally model session types, via Wadler's interpretation of Classical Linear Logic as a session-typed pi-calculus, and how that interpretation extends to an interpretation of a dependently typed version of session types.
2015-11-18, 11:00: The categorical structure of von Neumann algebras (Bram Westerbaan, Nijmegen)
I would like to speak about the categorical structure of the category of von Neumann algebras, with as morphisms normal, completely positive, unital linear maps. For some years my colleagues and I have worked on identifying basic structures in this category, and while surprisingly many things do not exist or do not work in this category (it's not a topos or even an extensive category, there's no epi-mono factorisation system, there is no dagger, colimits — if they exist at all — are horrendous...), we did find some structure (the products behave reasonable in some sense, there is a 'quotient', and 'comprehension', and we have a universal property for the minimal Stinespring dilation, and a universal property for M_2—the qubit). There is no deep category theory involved by any standards, and I promise I will spare you the functional analysis, so it should be a light talk.
2015-11-11, 11:00: The Power of Coalitions (Clemens Kupke, MSP)
Due to popular demand I am going to give a brief introduction to Marc Pauly's Coalition Logic, a propositional modal logic that allows to reason about the power of coalitions in strategic games. I will provide motivation and basic definitions. Furthermore I am planning to discuss how the logic can be naturally viewed as a coalgebraic logic and what we gain from the coalgebraic perspective. Finally — if (preparation) time permits — I am going to say how the logic can be applied to the area of mechanism design.
2015-10-28, 11:00: Interacting Frobenius Algebras Are Hopf (Ross Duncan, MSP)
Commutative Frobenius algebras play an important role in both Topological Quantum Field Theory and Categorical Quantum Mechanics; in the first case they correspond to 2D TQFTs, while in the second they are non-degenerate observables. I will consider the case of "special" Frobenius algebras, and their associated group of phases. This gives rise to a free construction from the category of abelian groups to the PROP generated by this Frobenius algebra. Of course a theory with only one observable is not very interesting. I will consider how two such PROPs should be combined, and show that if the two algebras (i) jointly form a bialgebra and (ii) their units are "mutually real"; then they jointly form a Hopf algebra. This gives a "free" model of a pair of strongly complementary observables. I will also consider which unitary maps must exist in such models.
Slides are here if you want a preview.
2015-10-22, 15:00 (LT1415): Semantics for Social Systems where Theory about the System Changes the System (Viktor Winschel, ETH Zurich)
In societies the notion of a law is not given by nature. Instead social dynamics are driven by the theories the citizens have about the dynamics of the social system. Obviously self-referential mathematical structures, developed in computer science, are candidates to be applied in social sciences for this foundational issue. We will see a prototypical game theoretical problem where several computer scientific tools can help to discuss these structures. It is a long standing problem in economics and of human kind and their scarce recourses: "should we go to a bar that is always so overcrowded"?
2015-10-21: SPLS (Edinburgh)
2015-10-14, 11:00: Type and Scope Preserving Semantics (Guillaume Allais, MSP)
We introduce a notion of type and scope preserving semantics generalising Goguen and McKinna's "Candidates for Substitution" approach to defining one traversal generic enough to be instantiated to renaming first and then substitution. Its careful distinction of environment and model values as well as its variation on a structure typical of a Kripke semantics make it capable of expressing renaming and substitution but also various forms of Normalisation by Evaluation as well as, perhaps more surprisingly, monadic computations such as a printing function.
We then demonstrate that expressing these algorithms in a common framework yields immediate benefits: we can deploy some logical relations generically over these instances and obtain for instance the fusion lemmas for renaming, substitution and normalisation by evaluation as simple corollaries of the appropriate fundamental lemma. All of this work has been formalised in Agda.
Github repository with paper and sources
2015-10-07: STP (Dundee)
2015-10-06, 15:00 (LT1415): Proof-theoretic semantics for dynamic logics (Alessandra Palmigiano, TU Delft)
Research in the proof theory of dynamic logics has recently gained momentum. However, features which are essential to these logics prevent standard proof-theoretic methodologies to apply straightforwardly. In this talk, I will discuss the main properties proof systems should enjoy in order to serve as suitable environments for an inferential theory of meaning (proof-theoretic semantics). Then, I'll identify the main challenges to the inferential semantics research agenda posed by the very features of dynamic logics which make them so appealing and useful to applications. Finally, I'll illustrate a methodology generating multi-type display calculi, which has been successful on interesting case studies (dynamic epistemic logic, propositional dynamic logic, monotone modal logic).
1. S. Frittella, G. Greco, A. Kurz, A. Palmigiano, V. Sikimić, A Proof-Theoretic Semantic Analysis of Dynamic Epistemic Logic, Journal of Logic and Computation, Special issue on Substructural logic and information dynamics (2014), DOI:10.1093/logcom/exu063.
2. S. Frittella, G. Greco, A. Kurz, A. Palmigiano, V. Sikimić, Multi-type Display Calculus for Dynamic Epistemic Logic, Journal of Logic and Computation, Special issue on Substructural logic and information dynamics (2014), DOI:10.1093/logcom/exu068.
3. S. Frittella, G. Greco, A. Kurz, A. Palmigiano, Multi-type Display Calculus for Propositional Dynamic Logic, Special issue on Substructural logic and information dynamics (2014), DOI:10.1093/logcom/exu064.
4. S. Frittella, G. Greco, A. Kurz, A. Palmigiano, V. Sikimić, Multi-type Sequent Calculi, Proc. Trends in Logic XIII, A. Indrzejczak, J. Kaczmarek, M. Zawidski eds, p 81-93, 2014.
5. G. Greco, A. Kurz, A. Palmigiano, Dynamic Epistemic Logic Displayed, Proc. Fourth International Workshop on Logic, Rationality and Interaction (LORI 2013) Eds: Huaxin Huang, Davide Grossi, Olivier Roy eds, 2013.
2015-09-30, 11:00: ICFP trip report (Bob Atkey, MSP)
2015-09-25, 16:00 (LT1415): Coinduction, Equilibrium and Rationality of Escalation (Pierre Lescanne, ENS Lyon)
Escalation is the behavior of players who play forever in the same game. Such a situation is a typical field for application of coinduction which is the tool conceived for reasoning in infinite mathematical structures. In particular, we use coinduction to study formally the game called dollar auction, which is considered as the paradigm of escalation. Unlike what is admitted since 1971, we show that, provided one assumes that the other agent will always stop, bidding is rational, because it results in a subgame perfect equilibrium. We show that this is not the only rational strategy profile (the only subgame perfect equilibrium). Indeed if an agent stops and will stop at every step, whereas the other agent keeps bidding, we claim that he is rational as well because this corresponds to another subgame perfect equilibrium. In the infinite dollar auction game the behavior in which both agents stop at each step is not a Nash equilibrium, hence is not a subgame perfect equilibrium, hence is not rational. Fortunately, the notion of rationality based on coinduction fits with common sense and experience. Finally the possibility of a rational escalation in an arbitrary game can be expressed as a predicate on games and the rationality of escalation in the dollar auction game can be proved as a theorem which we have verified in the proof assistant COQ. In this talk we will recall the principles of infinite extensive games and use them to introduce coinduction and equilibria (Nash equilibrium, subgame perfect equilibrium). We will show how one can prove that the two strategy profiles presented above are equilibria and how this leads to a "rational" escalation in the dollar auction. We will show that escalation may even happen in much simpler game named 0,1-game.
2015-09-16, 11:00: Contextuality, path logic and a modal logic for Social Choice Theory (Giovanni Ciná, Amsterdam)
Social Choice functions are procedures used to aggregate the preferences of individuals into a collective decision. We outline two recent abstract approaches to SCFs: a recent sheaf treatment of Arrow's Theorem by Abramsky and a modal logic studied by Ulle Endriss and myself. We show how to relate the categorical modelling of Social Choice problems to said work in Modal Logic. This insight prompts a number of research questions, from the relevance of sheaf-like condition to the modelling of properties of SCFs on varying electorates.
2015-09-02, 11:00: Concurrent games (Ross Duncan, MSP)
Whiteboard photos
2015-08-26, 11:00: Comonadic Cellular Automata II (Kevin Dunne, MSP)
This is a sequel to my last 101 where I spoke about describing cellular automata as algebras of a comonad on Set. I'll describe how we can make sense of "generalised cellular automata" (probabilistic/non-deterministic/quantum, for example) as comonads on other categories via distributive laws of monads and comonads.
2015-08-13, 15:00 (LT1415): The Polymorphic Blame Calculus and Parametricity (Jeremy Siek, Indiana University)
The Polymorphic Blame Calculus (PBC) of Ahmed et al. (2011) combines polymorphism, as in System F, with type dynamic and runtime casts, as in the Blame Calculus. The PBC is carefully designed to ensure relational parametricity, that is, to ensure that type abstractions do not reveal their abstracted types. The operational semantics of PBC uses runtime sealing and syntactic barriers to enforce parametricity. However, it is an open question as to whether these mechanisms actually guarantee parametricity for the PBC. Furthermore, there is some question regarding what parametricity means in the context of the PBC, as we have examples that are morally parametric but not technically so. This talk will review the design of the PBC with respect to ensuring parametricity and hopefully start a discussion regarding what parametricity should mean for the PBC.
2015-07-23, 11:00: Free interacting observables (Ross Duncan, MSP)
2015-05-26, 16:00 (LT1415): Final coalgebras from corecursive algebras (Paul Levy, Birmingham)
We give a technique to construct a final coalgebra out of modal logic. An element of the final coalgebra is a set of modal formulas. The technique works for both the finite and the countable powerset functors. Starting with a corecursive algebra, we coinductively obtain a suitable subalgebra. We see - first with an example, and then in the general setting of modal logic on a dual adjunction - that modal theories form a corecursive algebra, so that this construction may be applied.
We generalize the framework to categories other than Set, and look at examples in Poset and in the opposite category of Set.
2015-05-13, 11:00: Two-dimensional proof-relevant parametricity (Federico Orsanigo, MSP)
Relational parametricity is a fundamental concept within theoretical computer science and the foundations of programming languages, introduced by John Reynolds. His fundamental insight was that types can be interpreted not just as functors on the category of sets, but also as equality preserving functors on the category of relations. This gives rise to a model where polymorphic functions are uniform in a suitable sense; this can be used to establish e.g. representation independence, equivalences between programs, or deriving useful theorems about programs from their type alone.
The relations Reynolds considered were proof-irrelevant, which from a type theoretic perspective is a little limited. As a result, one might like to extend his work to deal with proof-relevant, i.e. set-valued relations. However naive attempts to do this fail: the fundamental property of equality preservation cannot be established. Our insight is that just as one uses parametricity to restrict definable elements of a type, one can use parametricity of proofs to ensure equality preservation. The idea of parametricity for proofs can be formalised using the idea of 2-dimensional logical relations. Interestingly, these 2-dimensional relations have clear higher dimensional analogues where (n+1)-relations are fibered over a n-cube of n-relations. Thus the story of proof relevant logical relations quickly expands into one of higher dimensional structures similar to the cubical sets which arises in Homotopy Type Theory. Of course, there are also connections to Bernardy and Moulin's work on internal parametricity.
2015-05-08, 15:00 Departmental seminar (LT1415): The probability of the Alabama paradox (Svante Linusson, KTH, Stockholm)
There exists various possible methods to distribute seats proportionally between states (or parties) in a parliament. Hamilton's method (also known as the method of largest reminder) was abandoned in the USA because of some drawbacks, in particular the possibility of the Alabama paradox, but it is still in use in many other countries.
In recent work (joint with Svante Janson) we give, under certain assumptions, a closed formula for the probability that the Alabama paradox occurs given the vector p_1,...,p_m of relative sizes of the states.
From the theorem we deduce a number of consequences. For example it is shown that the expected number of states that will suffer from the Alabama paradox is asymptotically bounded above by 1/e. For random (uniformly distributed) relative sizes p_1,...,p_m the expected number of states to suffer from the Alabama paradox converges to slightly more than a third of this, or approximately 0.335/e=0.123, as m -> infinity.
I will assume no prior knowledge of electoral mathematics, but begin by giving a brief background to various methods suggested and used for the distribution of seats proportionally in a parliament (it's all in the rounding).
2015-05-06, 11:00: Game theory in string diagrams (Jules Hedges, Queen Mary University of London)
We define a category whose morphisms are 'games relative to a continuation', designed to allow games to be built recursively from simple components. The resulting category has interesting structure similar to (but weaker than) compact closed, and comes with an associated string diagram language.
2015-04-29, 15:00 Departmental seminar (LT1415): Cyclic homology from mixed distributive laws (Uli Kraehmer, University of Glasgow)
In pure mathematics, cyclic homology is an invariant of associative algebras that is motivated by algebra, topology and even mathematicial physics. However, when studied from an abstract point of view it turns out to be an invariant of a pair of a monad and a comonad that are related by a mixed distributive law, and I speculate that this could lead to some potential applications in computer science.
(based on joint work with Niels Kowalzig and Paul Slevin)
2015-04-29, 11:00: An Introduction to Differential Privacy (Bob Atkey, MSP)
Let's say you have a database of people's private information. For SCIENCE, or some other reason, you want to let third parties query your data to learn aggregate information about the people described in the database. However, you have a duty to the people whose information your database contains not to reveal any of their individual personal information. How do you determine which queries you will let third parties execute, and those you will not?
"Differential Privacy" defines a semantic condition on probabilistic queries that identifies queries that are safe to execute, up to some "privacy budget".
I'll present the definition of differential privacy, talk a bit about why it is better than some 'naive' alternatives (e.g., anonymisation), and also describe how the definition can be seen as an instance of relational parametricity.
A good place to read about the definition of differential privacy is the book "The Algorithmic Foundations of Differential Privacy" by Cynthia Dwork and Aaron Roth.
The Algorithmic Foundations of Differential Privacy by Cynthia Dwork and Aaron Roth
2015-04-22: Patterns to avoid: (dependent) stringly-typed programming (Guillaume Allais, MSP)
Type : Set
Type = String -> Bool
2015-03-20, 15:00 Departmental seminar (LT1415): Approximating transition systems (Chris Heunen, Oxford)
Classical computation, invertible computation, probabilistic computation, and quantum computation, form increasingly more sophisticated labelled transition systems. How can we approximate a transition system by less sophisticated ones? Considering all ways to get probabilistic information out of a quantum system leads to domain-theoretic ideas, that also apply in the accompanying Boolean logic. I will survey to what extent these domains characterise the system, leading with examples from quantum theory, in a way that is accessible to a broad audience of computer scientists, mathematicians, and logicians.
2015-03-11, 11:00: Collapsing (Peter Hancock)
The topic comes from theory of infinitary proofs, and cut-elimination. In essence it is about nicely-behaved maps from higher "infinities" to lower ones, as the infinitary proofs are er, infinite, and can be thought of as glorified transfinite iterators. What might nice behaviour mean?
You can think of it as how to fit an uncountable amount of beer into a bladder whose capacity is merely countable. (Or maybe even finite.)
The most ubiquitous form of infinity is the regular cardinal, iepassing from a container F to F + (mu F -> _), where mu is the W-type operation. I'll "explain" regular collapsing as being all about diagonalisation.
2015-03-04: A HoTT-Date with Thorsten Altenkirch (LT1415)
2015-02-18: SPLS @ Strathclyde (Royal College Building, room RC512)
2015-02-11, 11:00 (LT1415): Totality versus Turing Completeness? (Conor McBride, MSP)
I gave an SPLS talk, which was mostly propaganda, about why people should stop claiming that totality loses Turing completeness. There was some technical stuff, too, about representing a recursive definition as a construction in the free monad whose effect is calling out to an oracle for recursive calls: that tells you what it is to be recursive without prejudicing how to run it. I'm trying to write this up double-quick as a paper for the miraculously rubbery MPC deadline, with more explicit attention to the monad morphisms involved. So I'd be grateful if you would slap down the shtick and make me more morphic. The punchline is that the Bove-Capretta domain predicate construction is a (relative) monad morphism from the free monad with a recursion oracle to the (relative) monad of Dybjer-Setzer Induction-Recursion codes. But it's worth looking at other monad morphisms (especially to the Delay monad) along the way.
2015-01-28, 16:30 (LT1415): Runners for your computations (Tarmo Uustalu, Institute of Cybernetics, Tallinn)
What structure is required of a set so that computations in a given notion of computation can be run statefully this with set as the state space? Some answers: To be able to serve stateful computations, a set must carry the structure of a lens; for running interactive I/O computations statefully, a "responder-listener" structure is necessary etc. I will observe that, in general, to be a runner of computations for an algebraic theory (defined as a set equipped with a monad morphism between the corresponding monad and the state monad for this set) is the same as to be a comodel of this theory, ie a coalgebra of the corresponding comonad. I will work out a number of instances. I will also compare runners to handlers.
2015-01-22, 11:00: Termination, later (James Chapman, Institute of Cybernetics, Tallinn)
It would be a great shame if dependently-typed programming (DTP) restricted us to only writing very clever programs that were a priori structurally recursive and hence obviously terminating. Put another way, it is a lot to ask of the programmer to provide the program and its termination proof in one go, programmers should also be supported in working step-by-step. I will show a technique that lowers the barrier of entry, from showing termination to only showing productivity up front, and then later providing the opportunity to show termination (convergence). I will show an example of a normaliser for STLC represented in Agda as a potentially non-terminating but nonetheless productive corecursive function targeting the coinductive delay monad.
(Joint work with Andreas Abel)
2014-12-17, 11:00: Worlds, Types and Quantification (Conor McBride, MSP)
I've managed to prove a theorem that I've been chasing for a while. The trouble, of course, was stating it. I'll revisit the motivation for extending type systems with an analysis of not just what things are but where, when, whose, etc. The idea is that typed constructions occur in one of a preordered set of worlds, with scoping restricted so that information flows only "upwards" from one world to another. Worlds might correspond to "at run time" and "during typechecking", or to computation in distinct cores, or in different stages, etc. What does the dependent function space mean in this setting? For a long time, I thought that each world had its own universal quantifier, for abstracting over stuff from that world. Failure to question this presumption is what led to failure to state a theorem I could prove. By separating quantifiers from worlds, I have fixed the problem. I'll show how to prove the key fact: if I can build something in one world and then move it to another, then it will also be a valid construction once it has arrived at its destination.
2014-12-10, 11:00: ZX and PROPs (Aleks Kissinger, Oxford)
2014-12-03: HoTT reading group @ Strathclyde (LT1310)
We will read the paper A Model of Type Theory in Cubical Sets by Marc Bezem, Thierry Coquand and Simon Huber. Thierry's Variation on cubical sets might also be useful. Administrative details: meet for lunch at 12am for those who want to, reading group starts at 2pm.
2014-11-26, 11:00: Diagrammatic languages for monoidal categories (Ross Duncan, MSP)
Monoidal categories are essentially 2-dimensional things, so why on earth would we represent them using a linear string of symbols? I'll talk about how to use string diagrams for monoidal categories, graph rewriting for reasoning within them, and how the syntax can be extended to handle certain kinds of infinitary expressions with the infamous !-box. If there's time I'll finish with some half-baked (eh... basically still looking for the on switch of the oven...) ideas of how to generalise them.
2014-11-19, 12:00: Arrow's Theorem and Escalation (Neil Ghani and Clemens Kupke, MSP)
Neil and Clemens will report back from the Lorentz Center Workshop on Logics for Social Behaviour.
2014-11-12, 11:00: Lambda-abstraction and other diabolical contrivances (Peter Hancock)
The topic is the unholy trinity of eta, zeta, and xi. I'll indicate how Curry managed to give a finite combinatorial axiomatisation of this nastiness, by anticipating (almost-but-not-quite) McBride et al's applicative functors.
2014-11-06, 09:00: Comonadic Cellular Automata (Kevin Dunne, MSP)
Kevin will be giving an informal talk about some of the stuff he has been learning about. He'll give the definition of a cellular automaton and then talk about how this definition can be phrased in terms of a comonad.
2014-11-05, 11:00 (Boardroom (LT1101d)): Logical Relations for Monads by Categorical TT-Lifting (Shin-ya Katsumata, Kyoto University)
Logical relations are widely used to study various properties of typed lambda calculi. By extending them to the lambda calculus with monadic types, we can gain understanding of the properties on functional programming languages with computational effects. Among various constructions of logical relations for monads, I will talk about a categorical TT-lifting, which is a semantic analogue of Lindley and Stark's leapfrog method.
After reviewing some fundamental properties of the categorical TT-lifting, we apply it to the problem of relating two monadic semantics of a call-by-value functional programming language with computational effects. This kind of problem has been considered in various forms: for example, the relationship between monadic style and continuation passing style representations of call-by-value programs was studied around '90s. We give a set of sufficient conditions to solve the problem of relating two monadic semantics affirmatively. These conditions are applicable to a wide range of such problems.
2014-10-29: SICSA CSE Meeting on Effects and Coeffects Systems and their use for resource control (Dundee)
2014-10-27 (Boardroom (LT1101d)): Constructing analysis-directed semantics (Dominic Orchard, Imperial College London)
All kinds of semantics are syntax directed: the semantics follows from the syntax. Some varieties of semantics are syntax and type directed. In this talk, I'll discuss syntax, type, and analysis directed semantics (analysis-directed semantics for short!), for analyses other than types. An analysis-directed semantics maps from terms coupled with derivations of a static program analysis into some semantic domain. For example, the simply-typed lambda calculus with an effect system maps to the category generated by a strong parametric effect monad (due to Katsumata) and a bounded-linear logic-like analysis (described as a coeffect systems) maps to a category generated by various structures related to monoidal comonads. I'll describe a general technique for building analysis-directed semantics where semantic objects and analysis objects have the same structure and are coupled by lax homomorphisms between them. This aids proving semantic properties: the proof tree of an equality for two program analyses implies the rules needed to prove equality of the programs' denotations.
2014-10-22, 11:00: Higher dimensional parametricity and its cubical structure (Neil Ghani, MSP)
Neil will talk about partial progress made during the summer on higher dimensional parametricity and the cubical structures that seem to arise.
Details will be kept to a minimum and, of course, concepts stressed.
2014-10-15: SPLS (Heriot Watt)
2014-10-08, 11:00: "Real-world data" and dependent types (Conor McBride, MSP)
Conor has offered to talk to us about what he has been thinking about recently. He says this includes models, views, and dependent types.
2014-10-01, 11:00: Initial algebras via strong dinaturality, internally (Fredrik Nordvall Forsberg, MSP)
Or: My summer with Steve
Or: How Christine and Frank were right, after all
Or: Inductive types for the price of function extensionality and impredicative Set
Christine Paulin-Mohring and Frank Pfenning suggested to use impredicative encodings of inductive types in the Calculus of Constructions, but this was later abandoned, since it is "well-known" that induction principles, i.e. dependent elimination, can not be derived for this encoding. It now seems like it is possible to give a variation of this encoding for which the induction principle is derivable after all. The trick is to use identity types to cut down the transformations of type (Pi X : Set) . (F(X) -> X) -> X to the ones that are internally strongly dinatural, making use of a formula for a "generalised Yoneda Lemma" by Uustalu and Vene.
2014-08-15, 15:00: Internal parametricity (Fredrik Nordvall Forsberg, MSP)
2014-05-28, 11:00: Graphical algebraic foundations for monad stacks (Ohad Kammar, Cambridge)
Ohad gave an informal overview of his current draft, with the following abstract:
Haskell incorporates computational effects modularly using sequences of monad transformers, termed monad stacks. The current practice is to find the appropriate stack for a given task using intractable brute force and heuristics. By restricting attention to algebraic stack combinations, we provide a linear-time algorithm for generating all the appropriate monad stacks, or decide no such stacks exist. Our approach is based on Hyland, Plotkin, and Power's algebraic analysis of monad transformers, who propose a graph-theoretical solution to this problem. We extend their analysis with a straightforward connection to the modular decomposition of a graph and to cographs, a.k.a. series-parallel graphs.
We present an accessible and self-contained account of this monad-stack generation problem, and, more generally, of the decomposition of a combined algebraic theory into sums and tensors, and its algorithmic solution. We provide a web-tool implementing this algorithm intended for semantic investigations of effect combinations and for monad stack generation.
2014-05-21, 11:00: Coalgebraic Foundations of Databases (Clemens Kupke, MSP)
This 101 is intended to be a brainstorming session on possible links between the theory of coalgebras and the theory of databases. I will outline some ideas in this direction and I am looking forward to your feedback.
2014-05-14, 11:00: Resource aware contexts and proof search for IMLL (Guillaume Allais, MSP)
In Intuitionistic Multiplicative Linear Logic, the right introduction rule for tensors implies picking a 2-partition of the set of assumptions and use each component to inhabit the corresponding tensor's subformulas. This makes a naive proof search algorithm intractable. Building a notion of resource availability in the context and massaging the calculus into a more general one handling both resource consumption and a notion of "leftovers" of a subproof allows for a well-structured well-typed by construction proof search mechanism.
Here is an Agda file implementing the proof search algorithm.
2014-04-07, 11:00: Nominal sets (Jamie Gabbay, Heriot-Watt)
2014-04-02, 11:00: Towards cubical type theory (Thorsten Altenkirch, Nottingham)
2014-03-19, 11:00: The selection monad transformer (Guillaume Allais, MSP)
Guillaume presented parts of Hedges' paper Monad transformers for backtracking search (accepted to MSFP 2014). The paper extends Escardo and Oliva's work on the selection and continuation monads to the corresponding monad transformers, with applications to backtracking search and game theory.
2014-03-05, 15:00: Lagrange inversion (Stuart Hannah, Strathclyde Combinatorics)
Stuart spoke about Lagrange inversion, a species-theoretic attempt to discuss the existence of solutions to equations defining species.
2014-02-28: Species (Neil Ghani, MSP)
Neil spoke about how adding structured quotients to containers gives rise to a larger class of data types.
2014-02-19, 11:00: Synthetic Differential Geometry (Tim Revell, MSP)
Tim gave a brief introduction to Synthetic Differential Geometry. This is an attempt to treat smooth spaces categorically so we can extend the categorical methods used in the discrete world of computer science to the continuous work of physics.
2014-02-12, 11:00: Worlds (Conor McBride, MSP)
Conor talked about worlds (aka phases, aka times, ...): why one might bother, and how we might go about equipping type theory with a generic notion of permitted information flow.
2014-02-05, 11:00: Operads (Miles Gould, Edinburgh)
Miles has kindly agreed to come through and tell us about Operads, thus revisiting the topic of his PhD and the city in which he did it.
2014-01-22, 11:00: Overview of (extensions of) inductive-recursive definitions (Lorenzo Malatesta, MSP)
2014-01-08, 11:00: On the recently found inconsistency of the univalence axiom in current Agda and Coq (Conor McBride, MSP)
2013-12-18, 11:00: Quantum Mechanics (Ross Duncan, MSP)
2013-11-20, 11:00: Classical Type Theories (Robin Adams, MSP visitor)
In 1987, Felleisen showed how to add control operators (for things like exceptions and unconditional jumps) to the untyped lambda-calculus. In 1990, Griffin idly wondered what would happen if one did the same in a typed lambda calculus. The answer came out: the inhabited types become the theorems of classical logic.
I will present the lambda mu-calculus, one of the cleanest attempts to add control operators to a type theory. We'll cover the good news: the inhabited types are the tautologies of minimal classical logic, and Godel's Double Negation translation from classical to intuitionistic logic turns into the CPS translation.
And the bad news: control operators don't play well with other types. Add natural numbers (or some other inductive type), and you get inconsistency. Add Sigma-types, and you get degeneracy (any two objects of the same type are definitionally equal). It gets worse: add plus-types, and you break Subject Reduction.
2013-11-13, 11:00: Continuation Passing Style (Guillaume Allais, MSP)
I chose to go through (parts of) Hatcliff and Danvy's paper "A Generic Account of Continuation-Passing Styles" (POPL 94) which gives a nice factorization of various CPS transforms in terms of:
embeddings from STLC to Moggi's computational meta-language (either call-by-value, call-by-name, or whatever you can come up with)
followed by a generic CPS transform transporting terms from ML back to STLC
Here is an Agda file containing what we had the time to see.
Generated by a Haskell program. | CommonCrawl |
Abstract: The purpose of this paper is to provide a brief review of some recent developments in quantum feedback networks and control. A quantum feedback network (QFN) is an interconnected system consisting of open quantum systems linked by free fields and/or direct physical couplings. Basic network constructs, including series connections as well as feedback loops, are discussed. The quantum feedback network theory provides a natural framework for analysis and design. Basic properties such as dissipation, stability, passivity and gain of open quantum systems are discussed. Control system design is also discussed, primarily in the context of open linear quantum stochastic systems. The issue of physical realizability is discussed, and explicit criteria for stability, positive real lemma, and bounded real lemma are presented. Finally for linear quantum systems, coherent $H^\infty$ and LQG control are described. | CommonCrawl |
Final Review (All Quizzes)
emmalynn799Plus
Terms in this set (105)
In Congress, a bill must be introduced in
a. the Office of the President.
b. the House of Representatives or the Senate.
c. any federal court.
d. the House of Representatives.
e. the Senate.
The House of Representatives or the Senate
Gina spray painted graffiti on Monica's building without Monica's permission. They both live in New Mexico. The damage to Monica's building totaled $12,000. This case will be heard in:
a. state court, because state courts are courts of limited jurisdiction.
b. federal court, because there is complete diversity.
c. federal court, because federal courts are courts of general jurisdiction.
d. federal court, because there is a federal question.
e. state court, because there is no basis for federal jurisdiction.
State court, because there is no basis for federal jurisdiction
The Supremacy Clause
a. permits the United States to intervene in the affairs of foreign nations for humanitarian crises.
b. permits the United States to lead at the United Nations.
c. prohibits federal law from conflicting with local ordinances.
d. prohibits state law from conflicting with federal law.
e. prohibits federal law from conflicting with state law.
Prohibits state law from conflicting with federal law
The following are courts of limited jurisdiction, except:
a. U.S. bankruptcy court
b. family law court
c. probate court
d. U.S. Tax court
e. Oregon circuit courts
Oregon circuit courts
This state power allows the states to regulate for the health, welfare, and safety of its citizens:
a. police power
b. preemption
c. precedent
d. preponderance of the evidence
e. plain meaning rule
Police power
The doctrine of stare decisis indicates that judges will generally follow established:
a. precedents.
b. public opinion.
c. petitions.
d. preponderance of the evidence.
e. pleadings.
Precedents
Randy owned a company. He asked Stan to dump the company's pollution into a local river, rather than paying for proper disposal. Stan dumped the company's pollution into the local river, as directed. This action violated the Clean Water Act. Vincent saw Stan dump the pollution into the river. Vincent brought a civil suit against Randy, Stan, and the company. Vincent's civil suit will:
a. not be permitted to move forward, because Vincent does not have standing.
b. be permitted to move forward, because of carpe diem.
c. not be permitted to move forward, because Stan was simply following orders.
d. be permitted to move forward, because the Clean Water Act contains a citizen suit provision.
e. not be permitted to move forward, because the river would have to bring its own claim for damages to itself.
Be permitted to move forward, because the Clean Water Act contains a citizen suit provision.
Claims arising under administrative rules and regulations will not be heard by a court without the plaintiff having first:
a. obtained a favorable judgment at the state court level or the U.S. District Court level.
b. been granted a pardon by the president or by the governor of a state.
c. exhausted all administrative remedies.
d. made a motion for the production of real evidence.
e. made a motion for summary judgment.
Exhausted all administrative remedies
When a violation of the federal Clean Water Act occurs within a specific state, the case may be brought in:
a. federal court, because there will be no damages to a legal person.
b. state court, because there is no federal question.
e. state court, because damages are greater than $75,000.
Federal court, because there is a federal question
The plain meaning rule:
a. used by defendants in civil cases as an affirmative defense.
b. the guiding principle of extraterritorial application of U.S. laws.
c. is used by judges to determine statutory meaning.
d. has no place in legal interpretation.
e. is constitutional law.
Is used by judges to determine statutory meaning
Enabling acts create:
a. Congress.
b. federal courts.
c. administrative agencies.
d. Redistricting.
e. Ordinances.
Administrative agencies
People who have specific roles in a civil lawsuit may include all of the following except:
a. defendant
b. judge
c. plaintiff
d. bail bondsman
e. juror
Any powers not given to the federal government are reserved to the states or to the people by:
a. the Fifth Amendment.
b. the Tenth Amendment.
c. the First Amendment.
d. the Fourth Amendment.
e. the Fourteenth Amendment.
The Tenth Amendment
The U.S. Circuit Courts of Appeals are:
a. trial courts.
c. courts of general jurisdiction.
d. inferior courts.
e. state courts.
U.S. statutes are codified in the:
a. Federal Register.
b. U.S. Code.
c. Code and Register of Federal Actions.
d. Federal Reporter.
e. Code of Federal Regulations.
Stan jokingly gave Robert the deed to 5 acres that Stan owned. However, Stan did not really intend to give Robert 5 acres. Before handing Robert the deed, Stan said, "This is just a joke, OK?" When Robert was handed the deed, he said, "Thanks! I love this gift!" If Robert sues Stan to enforce this gift, Robert's suit will probably fail because:
a. there was no actual delivery.
b. there was no actual acceptance.
c. there was no intent to transfer title.
d. Robert does not have capacity.
e. Stan does not have capacity.
There was no intent to transfer title
Real property may be acquired by all of the following methods, except:
a. purchase.
b. adverse possession.
c. gift.
d. capture.
e. bequest.
The following are all required elements of adverse possession, except:
a. actual possession.
b. exclusivity.
c. open and notorious possession.
d. statutory length of time.
e. breaking the sod.
Breaking the sod
Ann and Marc have equal right to the possession and use of an entire property, even if they own unequal shares of the property. However, upon the death of either party, ownership immediately vests in the living party. Which type of ownership interests do Ann and Marc have?
a. Periodic tenancy
b. Tenancy in common
c. Tenancy at will
d. Common law property
e. Joint tenancy
Tracy and Rashad have equal right to the possession and use of an entire property, even if they own unequal shares of the property. They have separate individual interests, including the right to sell those interests without consent of the other party. Which type of ownership interest do Stuart and Bub have?
a. Joint tenancy
b. Tenancy at will
c. Periodic tenancy
d. Tenancy in common
e. Community property
Which of the following real property is likely to be subject to public trust doctrine restrictions on title transfer?
a. A pier
b. A boathouse
c. A shoreline
d. A boat
e. A beach house
A shoreline
The ownership of land was granted to Tehva by the following language: "To Tehva for as long as she lives. Then to Abigail." Abigail's interest is best described as that of a:
a. tenancy by the entirety.
b. tenancy at sufferance.
c. life tenant.
d. remainderman.
e. life estate.
Remainderman
All of the following are personal property, except:
a. the grain in a silo.
b. a barrel of hazardous waste sitting on the land.
c. a dog.
d. a forest.
e. lumber that will be used to build a house.
A forest
Ayesha is the sole owner of her land. She has the greatest possible ownership estate in her land. She has a:
b. defeasible fee.
c. fee simple defeasible.
d. fee simple absolute.
Fee simple absolute
A tenancy at will is characterized by:
a. a prohibition on tenant vacancy.
b. an indefinite duration.
c. a prohibition on renewal.
d. a definite beginning and a definite end.
e. the Statute of Frauds.
An indefinite duration
Dwight lives west of the Mississippi River. Dwight is the sole owner of land that abuts the Rogue River. Dwight is:
a. a riparian land owner.
b. tenant in common.
c. an adverse possessor.
d. in violation of the public trust doctrine.
e. a remainderman.
A riparian land owner
This type of restriction on real property runs with the land.
a. Regulatory takings
b. An easement
c. A tenancy
d. The rule against perpetuities
e. A covenant
A covenant
An association of two or more persons formed to carry on a business for profit is a:
a. committee.
b. partnership.
c. commune.
d. tenancy.
e. legislature.
Ann and Marc have equal right to the possession and use of an entire property, even if they own unequal shares of the property. However, upon the death of either party, ownership immediately vests in the living party. That concept where ownership immediately vests in the remaining living party is known as:
a. the right of survivorship.
b. public trust right.
c. right to quiet title.
d. right to open and notorious use of property.
e. riparian right.
The right of survivorship
The following are examples of real property or fixtures, except:
a. an RV.
b. a county road.
c. a permanently affixed road sign.
d. timber, before it is felled.
e. an office building.
An RV
A civil wrong or injury is:
a. federalism.
b. a tort.
c. criminal.
d. forfeiture.
e. community property.
A tort
This tort requires proof that the harm that a statute was designed to prevent occurred to a person whom the statute was intended to protect.
a. Negligence per se
b. Non sequitur
c. Nuisance
d. Novation
e. Negligence
Negligence per se
Assumption of the risk is a defense to:
a. a violation of the public trust doctrine.
b. breach of contract.
c. trespass to land.
d. strict liability.
e. negligence.
Shawn loves to grill hot peppers outside on her grill. She grills hundreds of pounds of hot peppers during the spring, summer, and fall. The smoke from the hot pepper grilling burns her neighbors' eyes and, consequently, her neighbors cannot go outside in their own yards during Shawn's grilling season. Shawn is committing the following tort:
a. Trespass to land
b. Strict liability
c. Negligence
d. Nuisance
e. Negligence per se
A plaintiff in a tort case may:
a. bring more than one tort claim at a time.
b. bring more than one statutory claim at a time.
c. bring only one tort claim.
d. bring only one statutory claim.
e. all of the above
This type of injury must exist for a negligence claim to succeed:
a. physical.
b. punitive.
c. speculative.
d. permanent.
e. actual.
The following is a possible remedy in a tort case:
a. Liquidated damages
b. Specific performance
c. Punitive damages
d. Civil disabilities
e. Incarceration
The following are different classifications of tort, except:
a. strict liability.
b. negligence.
c. intentional.
d. juvenile.
e. All of these are different classifications of tort.
Cynthia decided to dump toxic waste down her kitchen sink. If her actions demonstrated a conscious disregard for a known risk of probable harm to others, she would be described as:
a. a bystander.
b. res ipsa loquitur.
c. distinguished on the facts.
d. reckless.
e. a scapegoat.
The following are defenses to various torts, except:
a. assumption of the risk.
b. entrapment.
c. consent.
d. comparative fault.
e. coming to the nuisance.
Entrapment
A monetary award designed to deter a defendant from similar future actions is:
a. right to open and notorious use of property.
b. the rule against perpetuities.
c. punitive damages.
d. the right of survivorship.
e. the public trust doctrine.
To prove negligence, an plaintiff must prove:
a. entrapment.
b. detrimental reliance.
c. recklessness.
d. duty of care.
e. intent.
Proximate cause can be described as all of the following, except:
a. no unforeseen intervening events.
b. foreseeable injury.
c. a required element of negligence.
d. distinct from actual cause.
e. breach of the duty of care.
Breach of the duty of care
The court ordered Xavier to stop dumping industrial waste into the city sewer system. This order is called:
a. an invective.
b. an irreconcilable difference.
c. an inconvenient truth.
d. an injunction.
e. an intentional tort.
An injunction
This type of liability may be imposed on persons engaged in abnormally dangerous or ultrahazardous activity:
a. reckless
b. depraved
c. strict
d. permanent
e. negligent
The process of evaluating a property's environmental conditions and assessing potential liability for any contamination is known as:
a. all available inquiries.
b. warrantying the product.
c. construing the terms against the drafter.
d. disclaiming liability.
e. indemnification.
All available inquiries
Modern law expresses ______________, as evidenced by the many warranty obligations implied in law.
a. a duty based worldview
b. caveat emptor
c. a violation of the public trust doctrine
d. liquidated damages
e. caveat venditor
Caveat venditor
When a contract's terms are ambiguous, a judge may:
a. indemnify the state.
b. assign an error of law.
c. issue an eviction order.
d. award punitive damages.
e. construe the terms against the drafter of the contract.
Construe the terms against the drafter of the contract
Leased residential real property is subject to the implied warranty of:
a. habitability.
b. due care.
c. survivorship.
d. indemnification.
e. express.
The following are theories of product liability, except:
c. failure to warn.
d. breach of warranty.
e. misuse.
Georgina advertised "ocean front" property" for sale. Sebastian agreed to buy the land. Unfortunately, after the transaction was complete, Sebastian learned that the property was not within sight of the ocean. If Georgina lied to sell the property, she committed:
a. fraud.
b. a mistake.
c. caveat venditor.
d. nuisance.
e. trespass.
This is a statement in a contract that one party will not be liable for damages to the other for breach of contract under certain circumstances.
a. Disclaimer
b. Express warranty
c. Injunction
d. Internment
e. Indemnification agreement
Damages may be recovered for a product that is sold by a merchant but that is not fit for common, ordinary usage under this warranty:
a. implied warranty of purpose
b. implied warranty of merchantability
c. implied warranty of habitability
d. implied warranty of fitness
e. implied warranty of title
Implied warranty of merchantibility
One of the strongest incentives for conducting an environmental audit prior to purchasing real property is the availability of certain defenses to liability under this statute:
a. NEPA
b. TSCA
c. ESA
d. CAA
e. CERCLA
Persons who own commercial land that is environmentally contaminated, but who did not themselves contaminate the land:
a. may be a potentially responsible party under CERCLA.
b. may be liable for cleanup of the land.
c. may be able to assert the innocent landowner defense.
d. could have engaged in all available inquiries.
Which type of warranty is written?
a. Implied-in-fact
b. Fitness for a particular purpose
c. Implied-in-law
d. Express
e. Expected
For contract formation to be valid, a bargained for exchange must have occurred. A bargained for exchange is known as:
a. contestability.
b. caveat venditor.
d. consideration.
e. caveat emptor.
This type of warranty is not stated in actual words but it exists by fact or by law:
a. express
b. interest
c. implied
d. expected
e. indemnity
This agreement is one where one party promises to reimburse another party, or hold that person harmless, for loss or for damage.
a. Disclaimer of the implied warranties
b. Information system
c. Express warranty
d. Indemnification agreement
e. Identity protection
Indemnification agreement
Engaging in ultrahazardous or abnormally dangerous activity can result in this type of liability:
a. intentional
b. negligent
c. punitive
d. speculative
e. strict
A parcel of land that has been divided into two or more units is a:
a. waste.
b. subdivision.
c. regulatory taking.
d. variance.
e. nuisance.
In the following case, the state held that when the government required an easement as a condition of granting a building permit, this constituted a taking. This is because the condition placed on the building permit was unrelated to the state's purposes of reducing obstacles to public viewing and use of beaches.
a. Township of Blair v. Lamar OCI North Corp
b. Nollan v. Coastal California Commission
c. Tahoe-Sierra Preservation Council v. Tahoe Regional Planning Agency
d. Lucas v. South Carolina Coastal Council
e. Kelo v. City of New London
Nollan v. Coastal California Commission
Gina believed that construction of a hazardous waste incinerator was very important to ensure that hazardous waste was disposed of properly. However, she did not want it located in her neighborhood. This is an example of:
a. NIMBY.
b. GMHB.
c. GMA.
d. CZMA.
If a state creates a statute that blatantly discriminates against out of state sellers, the state has created a facially discriminatory law. This law violates:
a. the public trust doctrine.
c. the dormant commerce clause.
d. the implied warranty of purpose.
e. the Property Clause.
The dormant commerce clause
In this case, the court held that an economic development plan that is designed to rejuvenate the community may be a proper purpose for the exercise of the government's power of eminent domain.
a. Tahoe-Sierra Preservation Council v. Tahoe Regional Planning Agency
c. Lucas v. South Carolina Coastal Council
d. Township of Blair v. Lamar OCI North Corp
Big Company, Inc. placed a billboard along a well-travelled road. After the billboard had been in place for five years, a zoning ordinance was imposed on that section of the road, which prohibited billboards. The billboard is an example of:
a. an indemnification.
b. a regulatory taking.
c. an exaction.
d. a nonconforming use.
e. a violation of the public trust doctrine.
Nonconforming use
Rezoning of a single parcel is known as ________________, and it is generally not permitted.
a. nonconforming use
b. spot zoning
c. a nuisance
d. a variance
e. a mistake
Spot zoning
An exception to or waiver of the zoning code that is permitted is:
a. spot zoning
b. a nuisance
c. an indemnification
d. a regulatory taking
e. a variance
A variance
Congress manages wild horses on federal lands through powers derived from:
a. the dormant commerce clause.
b. the Property Clause.
c. Article III
d. the Copyright Clause.
e. the Takings Clause.
The property clause
Priyanka did not like the zoning ordinance enacted by her city council, because she feared that her property value would decrease. Priyanka:
a. does not have a good claim for a taking.
b. has violated the public trust doctrine with her concerns.
c. has a good claim for a traditional taking.
d. has a good claim for a regulatory taking.
e. should ask for a variance.
should ask for a variance
The states' power to legislate for the safety of its citizens is known as:
a. power of attorney.
b. police power.
c. consideration.
d. exaction.
The legal process by which government exercises its right of eminent domain and acquires private land for public use is known as a:
a. field preemption.
b. variance.
c. dormant commerce transaction.
d. hearing on the merits.
e. condemnation proceeding.
Condemnation proceeding
Congress enacted a statute that required interstate trucking companies to display signs on trucks carrying certain hazardous materials. Congress has the power to enact such a statute because of its powers drawn from:
a. the Supremacy Clause.
b. the U.S. President.
c. the Property Clause.
d. the National Highway Traffic Safety Commission
e. the Commerce Clause.
The commerce clause
The power of eminent domain allows government to:
a. violate the dormant commerce clause.
b. take property and use it for a public purpose.
c. preempt the field.
d. violate the public trust doctrine.
e. avoid paying just compensation.
Take property and use it for a public purpose
The courts have interpreted "public use" in the takings clause to mean:
a. public vision.
b. public purpose.
c. public party.
d. public surplus.
e. public-private partnership.
An agency rule for governing the agency's organization is an example of:
a. a nuisance.
b. a delegation of duties.
c. an interpretive rule.
d. a substantive rule.
e. a procedural rule
A procedural rule
An agency rule that prescribes law or policy is:
a. an interpretive rule.
b. a procedural rule.
c. a nuisance.
d. a delegation of duties.
e. a substantive rule.
A substantive rule
Notice of proposed notice-and-comment rulemaking appears in:
a. the U.S. Code.
b. the Code of Federal Regulations.
c. the U.S. Reporter.
d. the Federal Register.
e. the Federal Reporter.
The Federal Register
This is the standard of review typically applied by the courts in reviewing agency action:
a. strict scrutiny
b. arbitrary and capricious
c. abuse of discretion
d. clear and convincing
e. de novo
Arbitrary and capricious
This case stands for the proposition that courts must defer to an agency's interpretation of the language in the statute that it is authorized to administer.
a. Chevron v. NRDC
b. Citizens to Preserve Overton Park v. Volpe
c. Lujan v. Defenders of Wildlife
d. Massachusetts v. EPA
e. In re Quechee Lakes Corp
Chevron v. NRDC
An administrative agency may lawfully engage in action permitted by its:
a. FOIA.
b. enabling legislation.
c. implied warranty.
d. indemnification authority.
e. exaction.
Threshold issues to litigation for judicial review of an administrative agency decision include all of the following except:
a. standing.
b. live case or controversy.
c. finality.
d. ripeness.
e. arbitration.
Jane had the right to challenge an action in court. Jane had:
b. the public trust.
c. conflict preemption.
d. a variance.
e. delegated authority.
Administrative agency powers typically include all of the following, except:
a. executive.
b. legislative.
c. judicial.
d. rulemaking.
e. delegation.
The following are examples of executive powers that may be exercised by administrative agencies, except:
a. seizing property.
b. conducting searches.
c. enforcement of laws.
d. issuing subpoenas.
e. rulemaking.
Administrative agencies create the following forms of law:
a. rules and regulations
b. statutes
c. common law
d. ordinances
e. executive orders
The following limit administrative agency power, except:
a. executive restraints.
b. Congressional restraints.
c. agency restraints.
d. international restraints.
e. judicial review.
International restraints
An agency statement that presents the agency's understanding of the meaning of the language in its regulations or in the statutes it administers is:
b. an interpretive rule.
c. a substantive rule.
d. a procedural rule.
e. a delegation of duties.
An interpretive rule
Most agencies use this type of rulemaking procedure:
a. informal
b. hybrid
c. preemptive
d. implied
e. formal
This is a legal doctrine that addresses the question of whether a branch of government may constitutionally assign some of its powers or delegate some of its duties to an administrative agency.
a. The doctrine of laches
b. The doctrine of res judicata
c. The assignment doctrine
d. The public trust doctrine
e. The delegation doctrine
The delegation doctrine
Roads are generally not permitted in these federally designated areas:
a. desert
b. forests
c. riparian
d. coastal
e. wilderness
This federal statute is a good example of federalism, because it requires states to develop plans that meet the requirements of federal law:
a. National Environmental Policy Act
b. Atomic Energy Act
c. Coastal Zone Management Act
d. National Parks Service Organic Act
e. Nuclear Waste Policy Act
Coastal Zone Management Act
The following information may be relied upon to determine whether a species should be listed as endangered or threatened under the Endangered Species Act:
a. the "cute" factor
b. scientific data
c. economic impact
d. political favors
e. lobbyist efforts
This statute establishes a health-based safety standard for pesticide residues in all foods:
a. SDWA
b. CERCLA
c. RCRA
d. FQPA
e. FDCA
FQPA
If a state fails to achieve compliance within its borders to meet the EPA standards for air quality, the state may be subject to a:
a. NAAQS.
b. FONSI.
c. NPDES.
d. FIP.
e. SIP.
One way that federal agencies show that they have considered consequences of their proposed actions is through the preparation of an:
a. EIS.
b. EPA.
c. ESA.
d. EOP.
e. EHS.
This statute establishes a national hazardous waste management program, provides for state or regional solid waste plans, and regulates underground storage tanks:
a. RCRA
b. SDWA
c. CERCLA
d. FIFRA
e. TSCA
RCRA
The Premanufacture Notification Program was established by:
a. CERCLA.
b. TSCA.
c. SDWA.
d. FIFRA.
e. RCRA.
TSCA
In designating critical habitat for an endangered species, the following must be taken into consideration:
a. political favors
b. economic impact
c. lobbyist efforts
d. whether the species should be listed as endangered
e. the "cute" factor
The EPA has the authority to regulate certain pesticides, including conducting inspections and requiring labeling of pesticides, under this statute:
b. NEPA
FIFRA
A species that is likely to become endangered in the future is:
a. a charismatic species.
b. an invasive species.
c. a keystone species.
d. a threatened species.
e. a nuisance species.
A threatened species
Under the Endangered Species Act, ________________ is defined as "harass, harm, pursue, hunt, shoot, wound, kill, trap, capture, or collect, or attempt to engage in any such conduct."
a. a recovery
b. an extinction
c. a taking
d. a listing
e. a threat
A taking
This Clean Air Act requires the EPA to establish standards for ________ criteria pollutants.
a. five
b. four
c. seven
d. six
e. three
This agency has the authority manage national forests:
a. NOAA
b. USFS
c. FDA
d. BLM
e. EPA
USFS
This statute requires that the EPA establish tolerances for pesticide residues in food or animal feed:
b. RCRA
c. FDCA
d. CERCLA
e. FQPA
FDCA
Soil Test 1
Botany Final
Ecology - Exam 2; Organisms Adaptations: Temperatu…
Ecology - Exam 2; Natural Selection, Evolution, an…
Verified questions
Graph the equation using the slope and $y$-intercept, as in previous examples. \ $$ y = x + 3 $$
The average rate of return on investments in large stocks has outpaced that on investments in Treasury bills by about 7% since 1926. Why, then, does anyone invest in Treasury bills?
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The following technology matrix for a simple economy describes the relationship of certain industries to each other in the production of $1$ unit of product. $$ \overset{\textbf{A}\quad\quad \textbf{M}\quad \quad\ \textbf{F}\quad \quad\textbf{U}}{\left[\begin{array}{cr} 0.36& 0.03& 0.10& 0.04\\ 0.06& 0.42& 0.25 &0.33\\ 0.18& 0.15 &0.10& 0.41\\ 0.10 &0.20 &0.31 &0.15 \end{array}\right]}\begin{array}{c}\text{Agriculture}\\\text{Manufacturing}\\\text{Fuel}\\\text{Utiilies}\end{array} $$ (a) For each $100$ units of manufactured products produced, how many units of fuels are required? (b) How many units of utilities are required to produce $40$ units of agricultural products?
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ϵ\epsilon-δ\delta proof that lim\lim\limits_{x \to 1} \frac{1}{x} = 1.
I'm starting Spivak's Calculus and finally decided to learn how to write epsilon-delta proofs.
I have been working on chapter 5, number 3(ii). The problem, in essence, asks to prove that
\lim\limits_{x \to 1} \frac{1}{x} = 1.
Here's how I started my proof,
\left| f(x)-l \right|=\left| \frac{1}{x} – 1 \right| =\left| \frac{1}{x} \right| \left| x – 1\right| < \epsilon \implies \left| x-1 \right| < \epsilon |x|
I haven't made any further progress past this point. Is it possible to salvage this proof? Should I try an alternate approach?
Update 2/19/2018: It appears that this answer has received a lot of attention, which I'm very glad to know about. When you're reading through this answer and you're trying to learn about \delta-\epsilon proofs for the first time, I would recommend skipping the sections labeled Addendum. on your first read. Please let me know of any other clarifications that you would like with this answer.
Whenever I am doing a \delta-\epsilon proof, I do some scratch work (note, this is NOT part of the proof) to figure out what to choose for \delta. I always tell students to think about the following:
What are you given?
What do you want to show?
In the definition of the limit, you are given an arbitrary \epsilon > 0 and you want to find \delta such that 0 < |x - 1| < \delta implies \left|\dfrac{1}{x} - 1 \right| < \epsilon\text{.} You have control over what to choose for your \delta in this case. The idea of this \delta-\epsilon proof is to work with the expression |x - 1| < \delta and get \left|\dfrac{1}{x} - 1 \right| < \epsilon at the end.
Let's do some scratch work (again, NOT part of the proof).
Scratch Work
Let's start with what we want to show for our scratch work (starting with what you want to show is bad to do 100\% of the time when you're doing proofs - again, this is scratch work and not actually part of the proof).
We want to show that \left|\dfrac{1}{x} - 1 \right| < \epsilon. Let's work backwards and try to turn the expression \left|\dfrac{1}{x} - 1 \right| into some form of |x-1|.
So, note that \left|\dfrac{1}{x} - 1 \right| =\left|\dfrac{1-x}{x} \right| = \left|\dfrac{-(x-1)}{x} \right| = \left|\dfrac{x-1}{x} \right| since |y|=|-y| for all y in \mathbb{R}.
The last expression can be rewritten as \dfrac{\left|x-1 \right|}{\left| x \right|}. Looking at this expression, we do have |x-1| in the numerator, which is good. But we have that pesky |x| in the denominator.
Since we do have control of what |x-1| is less than (this is our \delta), let's choose a really convenient, small number to work with that is greater than 0. Let's say \delta = \dfrac{1}{2}.
Well, if |x - 1| < \dfrac{1}{2}, then -\dfrac{1}{2} < x-1 < \dfrac{1}{2} \implies \dfrac{1}{2} < x < \dfrac{3}{2} \implies \dfrac{1}{2} < |x| < \dfrac{3}{2}\text{.} So if we choose \delta = \dfrac{1}{2}, \dfrac{1}{2} <|x| < \dfrac{3}{2}.
Addendum. In many examples, \delta is usually chosen to be 1. Why did we elect not to do that in this case?
It's because it wouldn't work.
Intuitively, here's why it doesn't: when you consider the neighborhood of radius 1 centered around x = 1, you get the interval (0, 2). f(x) = \dfrac{1}{x} doesn't have a finite limit at x = 0, so this makes \delta = 1 a bad choice.
This isn't the case if \delta = 1/2. The neighborhood of radius 1/2 around x = 1 is (1/2, 3/2). f has limits at every x-value in the interval (1/2, 3/2), including the endpoints.
In terms of the algebra, if we had chosen \delta = 1, the algebra wouldn't have worked out. We would've gotten 0 < x < 1 and would not have been able to obtain a finite upper bound for \dfrac{1}{x}. That is,
0 < x < 1 \implies 1 < \dfrac{1}{x} < \infty\text{.}
We do not have a finite upper bound for \dfrac{1}{x} in this case, and hence why \delta = 1 will not work for this purpose.
If \dfrac{1}{2} <|x| < \dfrac{3}{2}, then \dfrac{2}{3} <\dfrac{1}{|x|} < 2 and \dfrac{1}{|x|} < 2 \implies \dfrac{\left|x-1 \right|}{\left| x \right|} < 2\left| x-1 \right|\text{.} Now we have control over what |x-1| is less than. So to get \epsilon, we choose \delta = \dfrac{\epsilon}{2}.
But, wait - didn't I say that we chose \delta = \dfrac{1}{2} earlier? A simple solution would be to minimize \delta, i.e., make \delta = \min\left(\dfrac{\epsilon}{2} , \dfrac{1}{2} \right).
Addendum. To see why \delta = \min\left(\dfrac{\epsilon}{2} , \dfrac{1}{2} \right) works, suppose \dfrac{\epsilon}{2} > \dfrac{1}{2}, so that \delta = \dfrac{1}{2}. Then \epsilon > 1. Then \dfrac{\left|x-1 \right|}{\left| x \right|} < 2|x-1| < 2 \cdot \dfrac{1}{2} = 1 < \epsilon\text{.}
Now suppose \dfrac{\epsilon}{2} \leq \dfrac{1}{2}, so that \delta
= \dfrac{\epsilon}{2}.
Then \dfrac{\left|x-1 \right|}{\left| x \right|} < 2|x-1| < 2 \cdot \dfrac{\epsilon}{2} = \epsilon\text{.}
In both cases, we have \dfrac{\left|x-1 \right|}{\left| x \right|} < \epsilon, as desired.
See also Why do we need min to choose \delta?.
So now we've found our \delta and can use this to write out the proof.
The Proof
Proof. Let \epsilon > 0 be given. Choose \delta := \min\left(\dfrac{\epsilon}{2} , \dfrac{1}{2} \right). Then
\left|\dfrac{1}{x} - 1 \right| = \left|\dfrac{x-1}{x} \right| =
\dfrac{\left|x-1 \right|}{\left| x \right|} < 2\left| x-1 \right|
(since if |x - 1| < \dfrac{1}{2}, \dfrac{1}{|x|} < 2) and 2\left| x-1 \right| < 2\delta \leq 2\left(\dfrac{\epsilon}{2}\right) = \epsilon\text{. }\square
Success at last.
Addendum. Note that the end goal above was achieved, namely to show that \left|\dfrac{1}{x}-1\right| < \epsilon\text{.}
In the step 2\left| x-1 \right| < 2\delta \leq
2\left(\dfrac{\epsilon}{2}\right) = \epsilon\text{,} textbooks
usually omit the step with the \delta and just write 2\left| x-1
\right| < 2\left(\dfrac{\epsilon}{2}\right) =
\epsilon\text{.}
Addendum. It may seem that the note "(since if |x - 1| < \dfrac{1}{2}, \dfrac{1}{|x|} < 2)" may be an additional assumption added to the problem - i.e., that \delta has to be \dfrac{1}{2}. This is not the case for the following reason: given |x-1| < \delta, we have
|x-1| < \min\left(\dfrac{\epsilon}{2}, \dfrac{1}{2}\right)\text{.}
Obviously, if \epsilon \geq 1, we end up with |x - 1| < \dfrac{1}{2}, as stated above. But let's suppose that \epsilon < 1. Then
|x - 1| < \dfrac{\epsilon}{2} < \dfrac{1}{2}
and you end up with |x - 1| < \dfrac{1}{2}, so the \dfrac{1}{|x|} < 2 implication holds in either case.
Source : Link , Question Author : Gamma Function , Answer Author : Clarinetist
Categories epsilon-delta, graph-limits, real-analysis Tags epsilon-delta, graph-limits, real-analysis Post navigation
Is √64\sqrt{64} considered 88? or is it 8,−88,-8?
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\begin{definition}[Definition:Surface Area]
'''Surface area''' is the measure of the total area of the surface of a body.
It has two dimensions and is specified in units of length squared.
Hence the term '''square''' as a general vague term for a quantity in two dimensions.
It is a scalar quantity.
\end{definition} | ProofWiki |
Praxis II Practice Tests
Praxis Elementary Education
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Early Childhood Education Praxis®: Practice Test & Study Guide
Physical Education Praxis® 5091 Guide and Test
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Praxis Elementary Education: Multiple Subjects Practice Test | Praxis 5001 Study Guide
Praxis® Elementary Education: Multiple Subjects Practice Test | Praxis® 5001 Study Guide
What is the Praxis Elementary Education: Multiple Subjects (5001) Exam?
Taking the Praxis (5001) Exam
Preparing for the Praxis 5001: Practice Tests and Study Guides
Start Practice Test
The Praxis 5001, also called the Praxis 2 Elementary Education: Multiple Subjects tests, is used throughout the United States as part of many states' teacher certification processes. This Praxis test for elementary teachers evaluates whether entry-level teachers have the content knowledge necessary for the multiple subjects they will be teaching in elementary school. The Praxis Elementary Education: Multiple Subjects exam therefore is meant to support a generalist elementary school license.
The Praxis 5001 can be taken as one exam or as four subtests. These subtests consist of Reading and Language Arts (5002), Mathematics (5003), Social Studies (5004), and Science (5005). When taken as one exam, the Praxis test cost is $180. When taken as individual subtests, each subtest costs $64.
Practice tests give you a better idea of the topics you have mastered and those you should keep studying.
The Praxis 2 Elementary Education: Multiple Subjects (5001) exam is a computer-based test that comprises four subtests. They may be taken as a combination of all the subjects or as individual subtests. Overall, the time taken for the entire test is 4 hours 35 minutes.
The questions in each test are selected-response questions, constructed-response questions, or numeric entry. Selected-response questions involve picking from a list of choices or otherwise selecting something on the screen. Constructed-response questions involve creating an answer of one's own. Numeric entry asks the test-taker to enter a numeric value in the answer field.
The Praxis (5002) Reading and Language Arts test has 80 questions a test-taker must answer in 90 minutes, 50 questions in 65 minutes in the Praxis (5003) Mathematics test, 60 questions in 60 minutes in the Praxis (5004) Social Studies test, and 55 questions in 60 minutes in the Praxis (5005) Science test.
Elementary Education: Reading and Language Arts Subtest (5002)
The Reading and Language Arts subtest (5002) takes 1.5 hours to complete. It focuses on Reading, Writing, Speaking, and Listening. The test will consist of 80 questions overall, broken down into two categories, Reading and Writing, Speaking, and Listening. The 80 questions include single-selection multiple choice items, as well as question types involving multiple selection, order matching, or grids.
There are 38 questions for Reading, with a score weight worth 47% of the Reading and Language Arts subtest score. This category evaluates foundational skills, such as the roles of phonological awareness, phonics and word analysis, and fluency in literacy development, as well as comprehending literature and informational texts. The Writing, Speaking and Listening test has 42 questions, with a score weight worth 53% of the Reading and Language Arts subtest. This category evaluates writing, such as knowledge of the different developmental stages of writing; language, including grammar, usage, and mechanics; and speaking and listening, including characteristics of effective oral presentations.
Most states set 157 as the minimum score for the Reading and Language Arts subtest.
Elementary Education: Mathematics Subtest (5003)
The Mathematics Subtest (5003) takes 65 minutes to complete. It focuses on 50 selected-response questions as well as numeric questions. An online scientific calculator is available to assist test-takers as needed. The subtest is divided into three categories, Numbers and Operations, Algebraic Thinking, and Geometry and Measurement, Data, Statistics, and Probability. Questions on this subtest include selected-response, including both single- and multiple-selection multiple choice, and numeric entry questions.
Numbers and Operations has a total of 20 questions and a score weight of 40% of the Mathematics subtest and involves testing one's understanding of things like rational numbers, operations, percentages, and basic concepts of number theory. Algebraic Thinking has a total of 15 questions and a weight score of 30% of the Mathematics subtest and includes questions on evaluating and manipulating algebraic expressions and understanding the solutions to linear equations and inequalities, among other related topics. Geometry and Measurement, Data, Statistics, and Probability has a total of 15 questions and a score weight of 30% of the Mathematics subtest and asks test-takers about figures, perimeter and area, coordinate planes, and basic statistics.
Most states also set 157 as their minimum passing score for this test.
Elementary Education: Social Studies Subtest (5004)
The Social Studies subtest (5004) takes 1 hour to complete. There are 60 questions in this test, divided into three categories, on United States History, Government, and Citizenship; Geographic, Anthropology, and Sociology; and World History and Economics. All questions in this subtest are selected-response.
There are 27 questions in the first section on United States History, Government, and Citizenship, which comes with a score weight of 45% of the Social studies subtest and includes questions on major events in U.S. history; the nature, purpose, and forms of government; and the rights and responsibilities of citizenship. The second section is Geography, Anthropology and Sociology, with a total of 18 questions and a score weight of 30%, which evaluates one's understanding of world and regional geography, the interaction of human and physical systems, and how people of different cultural backgrounds interact with different elements of their environment. The final section is World History and Economics with 15 questions and a score weight worth 25% of the Social Studies subtest score; this section asks about classical civilizations (Egypt, Greece, Rome), cross-cultural comparisons in world history instruction, key terms and basic concepts of economics, and the role of government in economics.
Most states set the passing Praxis score for this subtest at 155.
Elementary Education: Science Subtest (5005)
The Science Subtest (5005) takes 1 hour to complete. There are 55 selected-response questions with an on-screen scientific calculator to assist test-takers. The test is placed into three separate categories, Earth Science, Life Science, and Physical Science. The questions in this subtest are selected-response.
Earth Science has between 17-18 questions, its score weight covers 33% of the Science subtest, and it involves an understanding of Earth structures and processes, Earth history and paleontology, and Earth and the universe. The second subsection is Life Science with a total of between 18-19 questions and a score weight worth 33% of the Science subtest, which asks about the structure and function of living systems, reproduction and heredity, life cycles, the interdependence of organisms, and personal health. The final section is Physical Science with a total number of between 18-19 questions and a score weight worth 33% of the Science subtest, which includes questions on physical and chemical structure and properties of matter, forces and motions, and energy. All categories of this subtest may further ask science as inquiry, as a human endeavor, science's unifying processes, and how to use resources and research materials in science.
Most states set 159 as the minimum score for this subtest.
The best way to prepare for the Praxis 5001 is for test-takers to familiarize themselves with the Educational Testing Service (ETS), which administers the Praxis test, and the resources it offers. The ETS page on the Praxis 5001 includes links to study resources and practice tests for each of the subtests, as well as a comprehensive Praxis study guide for Elementary Education. ETS also provides videos on topics such as what to expect on the Praxis test day and how to actually navigate a Praxis exam. There are other study resources available both online and as physical books that test-takers might find helpful, but ultimately, the most authoritative source for what is likely to be on the actual exam is ETS, so it should always be the priority.
Praxis test preparation should also include research about state-specific requirements. Praxis passing scores are set by each individual state, so aspiring teachers will need to make sure they understand what their own state requires. ETS further provides pages for each state so that test-takers may more easily find all the information they need.
The Praxis 5001 Study Guide and Test Preparation Resources
ETS provides a number of preparation materials for this exam. Test-takers will find:
A Praxis 5001 study guide with a comprehensive overview of the test, the topics it covers, and tips and strategies for studying
Information on calculator use, including practice using the on-screen calculator that will be provided for the Mathematics and Science subtests
A Microsoft Word document of resources for each subtest that includes links to materials on every topic that may appear on the Praxis exam
Microsoft Word documents with study plans for each subtest
Videos about the test and how to prepare for it
Praxis 5001 practice questions for each subtest that simulate the test itself
Test takers will be able to use these resources to make a schedule and organize their time around studying for the Praxis 5001, learn about the actual administration of the test, and understand how this exam is scored, as well as review the content knowledge being evaluated.
Individuals may also find non-ETS resources to be helpful study materials. There are many webinars, video lessons, and practice tests available both online and in physical book form that test-takers can use to prepare for the Praxis 5001.
Praxis 5001 Practice Tests
As part of its preparation materials, ETS sells interactive Praxis 5001 practice tests. Because they come directly from the test administrator, these practice tests are the ones most likely to simulate the actual test in both administration and content. When possible, these practice tests should be a test taker's priority.
However, both the internet and retail bookstores also carry Praxis study materials, including practice tests. These materials can include:
Images and video information
Praxis 5001 quizzes
Short mini-tests
Multiple choice practice questions
Video tutorials and webinars
Interactive practice tests that can provide a result of one's score immediately after completing it
Full-length practice tests
Many of the online Praxis multiple subjects 5001 practice tests are also free, which may make them more accessible than the ones provided by ETS.
Jenifer Woodson
Jenifer Woodson has been a teacher for over 6 years, and an artist and art gallery owner for over 11 years. Currently, she teaches art to grades 6-12. She has passed the Praxis exam. She is currently earning a Ph.D. in Leadership with a specialization in Education from the University of the Cumberlands. Jenifer holds an M.A.T. from the University of Louisville and a B.A. in Studio Arts from Kentucky State University.
How much does it cost to take the Praxis 5001?
It cost $180 to take the Praxis 5001 Elementary Education: Multiple Subjects exam if taken as one test. If the four subtests are taken separately, each costs $64. There may also be special surcharges depending on the state or for additional services.
How do I prepare for Praxis 5001?
Preparing for the Praxis can be difficult, so it is important to spend a fair amount of time on each individual topic. Using a study plan will help you know what you need to focus on. There are complete practice tests available that will initially enable test-takers to know where their weaknesses and strengths are. They will then be able to approach their studying with a more targeted focus.
What are the passing scores for Praxis 5001?
Individual states set the passing scores for each Praxis exam, which is further individualized for each of the four subtests. Aspiring teachers should therefore research their own state's requirements. The ETS provides individual pages for each state where test-takers can find this information.
Is there an essay on the Praxis 5001?
The Praxis 5001 Multiple Subjects exam does not include questions that require an essay. They are mainly mulltiple choice answers, with a few math problems included.
Take a Praxis Elementary Education Practice Test Online
Exam Instructions:
Complete the practice test below to test your knowledge of Praxis Elementary Education.
Choose your answers below. Complete the 15 questions then click "See Results."
You have answered 0 out of 15 correctly.
The correct answers are highlighted with green below.
Solve the following algebraic equation.
{eq}6x + 10 - 8x = 3 {/eq}
Students are asked to solve and simplify the following: 1/8 * 12/5. What would be the correct answer?
19 and 1/5
2 and 2/5
What is the best type of graph to use when you want to display percentages?
Bar Graph
Circle Graph
Which is NOT a benefit of the prewriting stage of the writing process?
Prewriting allows you to establish the audience for which you are writing.
Prewriting allows you to let your writing flow as a stream of consciousness.
Prewriting allows you to plan out the message of what you want to write.
Prewriting allows you to determine your purpose for writing.
Which of the following is the smallest unit of sound?
Phoneme
Vowel
What is not an example of 3D shape?
The purpose of narrative writing is to
convince someone to get something
give background on a person's life
explain something factual
Which quadrant can have a negative {eq}x {/eq} coordinate and a positive {eq}y {/eq} coordinate?
Fourth quadrant
Second quadrant
First quadrant
Third quadrant
Patricia walked a distance of {eq}m, {/eq} in miles. Jennifer walked 2 times as many miles as Patricia. Which expression shows the distance Jennifer walked?
m + 2
m - 2
{eq}\frac{m}{2} {/eq}
Which explorer first charted North America for the English government?
What is the rule?
Output = Input + 4
Output = Input - 4
Output = Input * -2
All of these statements describe a hypothesis EXCEPT
It is often written as an if/then statement.
It is an educated guess.
It can be tested.
It is comprised of several sentences.
Which figure best represents a ray?
To what does the obliquity of the Earth refer?
The angle of the Earth's tilt on its rotational axis
The shape of the Earth's orbit around the Sun.
The wobbling of the Earth as it spins on its axis.
The distance between the Earth and the Sun.
The document that announced America's independence from Britain is called: | CommonCrawl |
Research | Open | Published: 23 May 2015
A distributed approach to the OPF problem
Tomaso Erseghe1
This paper presents a distributed approach to optimal power flow (OPF) in an electrical network, suitable for application in a future smart grid scenario where access to resource and control is decentralized. The non-convex OPF problem is solved by an augmented Lagrangian method, similar to the widely known ADMM algorithm, with the key distinction that penalty parameters are constantly increased. A (weak) assumption on local solver reliability is required to always ensure convergence. A certificate of convergence to a local optimum is available in the case of bounded penalty parameters. For moderate sized networks (up to 300 nodes, and even in the presence of a severe partition of the network), the approach guarantees a performance very close to the optimum, with an appreciably fast convergence speed. The generality of the approach makes it applicable to any (convex or non-convex) distributed optimization problem in networked form. In the comparison with the literature, mostly focused on convex SDP approximations, the chosen approach guarantees adherence to the reference problem, and it also requires a smaller local computational complexity effort.
One of the key aspects of the current research trends for the future smart grid is the possibility of devising distributed algorithms for solving a global problem. This corresponds to the idea of a decentralized access to generation/storage resources, as well as to the much more challenging task of decentralized control.
The typical smart grid problem taken into consideration for distributed optimization is that of optimal power flow (OPF), that is, the optimal management of electrical power throughout the grid under a number of (electrical) constraints (e.g., the satisfaction of a power request from a load, the presence of a dispatchable/non dispatchable renewable generator or of a storage system). The OPF problem, being non-convex in nature in both the target function and the constraints, is very difficult to solve. For this reason, a widely used approach is to map it into a (somehow) close convex problem, and then solve the convex counterpart by means of distributed methods, e.g., the alternating direction method of multipliers. In this context, semi definite programming (SDP) relaxations have emerged as a common option, e.g., see Lavaei and Sojoudi et al. [1-4], Lam, Tse, and Zhang et al. [5,6], Dall'Anese and Giannakis et al. [7,8], Gayme and Topcu [9], and Erseghe and Tomasin [10]. One of the limits of this approach lies in the lack of adherence to the original problem, and in fact, optimality of the solution can only be ensured for very specific networks. But complexity is also an issue, since the number of variables involved in the local processing is squared with respect to its natural size. A few other worth mentioning approaches are available from the literature. S̆ulc et al. [11] exploit the (convex) LinDistFlow approximation as a lower complexity alternative to SDP relaxation. Magnusson et al. [12] avoid SDP relaxation and propose a sequential convex approximation approach, which, however, is known to imply slow convergence speeds. Instead, the consensus and innovation approach has been applied to the (convex) DC-OPF problem by Hug and Kar et al. [13,14], but the chosen distributed algorithm only provides approximate solutions even in the considered convex scenario.
The kind of approach we follow is alternative to the main trend in the literature, in the sense that we do not consider any convex relaxation and work directly on the non-convex OPF problem. In this way, we can guarantee adherence to the original problem and develop an algorithm which is capable of identifying local minima. This idea was originally exploited in [15] where a distributed algorithm based upon ADMM was proposed. This algorithm provided undeniable evidence of the goodness of the intuition but had two major drawbacks. First, optimization for speed was cumbersome and required centralized coordination. Second, no guarantee on convergence was available, and in fact the algorithm often failed to converge. Although the convergence failure did not practically prevent the algorithm output for being usable, convergence is an issue that practically limits the algorithm speed.
In this paper, we wish to solve the above cited issues. To simplify system parameters and improve convergence speed, we remap the distributed problem in such a way to reveal the network power flow. In the ADMM formulation, the power flow variables are adequately weighted in order to force the algorithm to solve an approximate linear problem in the power flow variables in the first iterations (similarly to what happens with DC-OPF). The approximation is progressively abandoned in later iterations. This corresponds to the practical intuition that a linear power flow exchange problem provides a solution which is close to the optimum (some preliminary results on this aspect were recently presented at an international conference [16]). We also modify the plain ADMM algorithm and reinterpret it as a non-convex augmented Lagrangian method (see the work of Martinez and Birgin et al. [17,18]) where penalty parameters are constantly updated (increased) to always guarantee convergence. More specifically, a global convergence guarantee is available under the assumption that local solvers are efficient, in the sense that they can guarantee the identification of a (feasible) local minimum. This might not be an easy task in general, but it is a reasonable assumption when the number of local variables is controlled. Furthermore, a certificate of convergence to a local optimum is available when penalty parameters are bounded. The kind of coordination involved in this process is only local and therefore defines a fully distributed algorithm.
The rest of this paper is organized as follows. First, the reference OPF problem is presented and put in a networked form readily usable for obtaining a distributed algorithm. Then the distributed approach is discussed in abstract form and its convergence properties proved. Application to the specific OPF problem is then detailed, and the proposed distributed algorithm is finally tested in meaningful scenarios.
The OPF problem
We first introduce the OPF problem in its natural (centralized) formulation.
Standard formulation
Consider an electrical network of N nodes at steady state, where V i , P i , and Q i represent, respectively, the local complex voltage, and the node's active and reactive powers. Assume that, at node i, a local cost is associated to active power production through a cost function f i (P i ). Assume that the electrical neighbors of node i are identified through the neighbors set $\mathcal {N}_{i}$ , and that the line admittance Y i,j , $j\in \mathcal {N}_{i}$ , is known for each physical connection. Then the standard OPF problem has the form
$$ \begin{aligned} &\text{min} \sum\limits_{i\in\mathcal{N}} f_{i}(P_{i})\\ &\text{w.r.t.}\,\, V_{i}\in {\mathbb{C}}, P_{i}, Q_{i}{\in{\mathbb{R}}}, i\in\mathcal{N}\\ &\text{s.t.}\,\,\, P_{i}+{jQ}_{i} = V_{i} \sum\limits_{j\in {\mathcal{N}}_{i}} Y_{i,j}^{*} V_{j}^{*}\\ &\quad\underline{V}_{i}\le |V_{i}|\le \overline{V}_{i}\\ &\quad\underline{P}_{i}\le P_{i}\le \overline{P}_{i},\\ &\quad\underline{Q}_{i}\le Q_{i}\le \overline{Q}_{i}\\ \end{aligned} $$
((1))
where $\mathcal {N}=\{1,\ldots,N\}$ is the nodes set. The first constraint in (1) refers to power flow equations (i.e., Kirchoff's laws). The remaining constraints are voltage and power constraint limitations, with $\underline {V}_{i}$ , $\overline {V}_{i}$ , $\underline {P}_{i}$ , $\overline {P}_{i}$ , $\underline {Q}_{i}$ , $\overline {Q}_{i}$ local upper and lower bounds.
For the ease of simplicity here we refer to a basic OPF problem, but additional constraints can be easily added to (1), e.g., power flow constraints on specific lines. Constraints referred to resources such as storage systems and renewable generators (dispatchable or not dispatchable) can be included by suitably selecting the cost factor f i , by introducing proper corrections to the cost function, or by inserting a time variable. Discrete variables can be also included in the problem formulation (e.g., the tap changing of the transformers, or the cost to turn on/off a generator), in which case a mixed-integer programming solver will be needed. The results that follow are valid for all the above generalizations.
Region-based formulation
We now wish to fully capture the network relations in (1), in such a way to be used in a distributed implementation. The idea is to partition the network in R regions, where the sets $\mathcal {R}_{k}$ , k=1,…,R, identify nodes belonging to region k. We have
$$ \mathcal{N} = \bigcup_{k=1}^{R} \mathcal{R}_{k}\;,\qquad \mathcal{R}_{k}\cap \mathcal{R}_{h} = \emptyset, \forall k\neq h\;. $$
Because of power flow equations in (1), the voltages of interest in region k are those belonging to set
$$ \mathcal{V}_{k} = \bigcup_{i\in \mathcal{R}_{k}} \mathcal{N}_{i} $$
where $\mathcal {N}_{i}$ identify the neighbors of node i. Note that set $\mathcal {V}_{k}$ includes set $\mathcal {R}_{k}$ as a subset, as well as all those nodes which belong to neighbor regions and which have a direct connection (edge) with one of the nodes of $\mathcal {R}_{k}$ . Accordingly, we identify the local voltage vectors x k with entries x k,ℓ by
$$ \boldsymbol{x}_{k} = [x_{k,\ell} ]_{\ell \in \mathcal{V}_{k}}\;,\qquad x_{k,\ell}=V_{\ell} $$
and the corresponding constraint region
$$ \begin{aligned} \mathcal{X}_{k}=& \left\{\, \underline{V}_{\ell}\le |x_{k,\ell}|\le \overline{V}_{\ell}, \forall \ell\in\mathcal{V}_{k},\right.\\ & \quad \underline{P}_{i}\le P_{i}\le \overline{P}_{i},\;\\ &\quad \underline{Q}_{i}\le Q_{i}\le \overline{Q}_{i},\;\\ &\left.\quad P_{i}+{jQ}_{i} = x_{k,i} \sum_{j\in\mathcal{N}_{i}} Y_{i,j}^{*} x_{k,j}^{*}, \forall i\in\mathcal{R}_{k}\right\} \end{aligned} $$
collecting voltage constraints, active and reactive power constraints, and power flow constraints, and to which we may add any additional constraint of interest. Regions $\mathcal {X}_{k}$ are deliberately chosen to be compact (closed and bounded) in order to strengthen later derivations and results.
Hence, a region-based equivalent formalization for (1) corresponds to the non-convex problem
$$ \begin{aligned} & \text{min} \sum\limits_{k \in \mathcal{R}} F_{k}(\boldsymbol{x}_{k})\\ & \text{w.r.t.}\,\, \boldsymbol{x}_{k} \in \mathcal{X}_{k}, k \in \mathcal{R}\\ & \text{s.t.}\,\, x_{k,\ell} = x_{h, \ell},\; \forall \ell \in \mathcal{V}_{k} \cap \mathcal{V}_{h}, k,h\in\mathcal{R} \end{aligned} $$
where $\mathcal {R}=\{1,\ldots,R\}$ , function
$$ F_{k}(\boldsymbol{x}_{k}) = \sum\limits_{\ell\in\mathcal{R}_{k}} f_{\ell}(P_{\ell}) $$
collects local cost functions, and where the constraint is forcing equivalence between duplicated (voltage) variables in vectors x k .
Capturing the power flow
The formalization given in (6), although correct, is somehow unsatisfactory in terms of the slow convergence speed involved with its distributed implementation, and in terms of the difficulty in optimizing its system parameters (see [15]). The key point is that we are not using any electrical intuition that could help the distributed processing. The intuition we use is illustrated in Figure 1.
A way to capture the power flow on edge (i,j) with $i\in \mathcal {R}_{k}$ and $j\in \mathcal {R}_{h}$ .
The idea with Figure 1 is the following. Consider two neighboring regions k, and h, and edge (i,j) connecting the two regions, i.e., with $i\in \mathcal {R}_{k}$ and $j\in \mathcal {R}_{h}$ . It also is $\{i,j\}\subset \mathcal {V}_{k}$ and $\subset \mathcal {V}_{h}$ . Then, equivalence between the local variables can be written in the form
$$ \begin{aligned} x_{k,i}& = x_{h,i}\\ x_{k,j}& = x_{h,j} \end{aligned} $$
which is equivalent to the constraint in (6). However, equivalence can be also written in the form
$$ \begin{aligned} x_{k,i} - x_{k,j}& = x_{h,i} - x_{h,j}\\ x_{k,i} + x_{k,j}& = x_{h,i} + x_{h,j} \end{aligned} $$
where the first equivalence captures the power flow, since the power flowing through line (i,j) is of the form Z i,j |V i −V j |2, i.e., it only depends on voltage differences as from the first of (9).
The corresponding formulation for the OPF problem can then be compactly written by using sets
$$ \mathcal{O}_{k} = \left\{(i,j)\Big| i\in\mathcal{R}_{k}, j\in\mathcal{N}_{i}\cap(\mathcal{V}_{k}\backslash\mathcal{R}_{k})\right\} $$
((10))
collecting in region k those edges connecting a node of $\mathcal {R}_{k}$ to a node in a neighbor region. By further introducing two auxiliary variables z − and z + belonging to the linear spaces
$$ \begin{aligned} \mathcal{Z}^{-}& =\{\boldsymbol{z}^{-}|z^{-}_{i,j} = - z^{-}_{j,i}, \,\forall(i,j)\in\mathcal{O}_{k}, k\in\mathcal{R}\}\\ \mathcal{Z}^{+} &=\{\boldsymbol {z}^{+}|z^{+}_{i,j} = z^{+}_{j,i}, \,\forall(i,j)\in\mathcal{O}_{k}, k\in\mathcal{R}\} \end{aligned} $$
the OPF problem becomes
$$ \begin{aligned} &\text{min} \sum\limits_{k\in\mathcal{R}} F_{k}(\boldsymbol{x}_{k})\\ & \text{w.r.t.}\,\, \boldsymbol{x}_{k}\in\mathcal{X}_{k}, k\in \mathcal{R}\\ &\quad\boldsymbol{z}^{-}\in\mathcal{Z}^{-},\boldsymbol{z}^{+}\in\mathcal{Z}^{+}\\ & \text{s.t.}\,\, \rho\,(x_{k,i} - x_{k,j}) = z^{-}_{i,j},\\ &\quad\zeta\,(x_{k,i} + x_{k,j}) = z^{+}_{i,j}, \forall (i,j)\in\mathcal{O}_{k}, k\in\mathcal{R} \end{aligned} $$
where two positive constants ρ, ζ are used to differently weigh the power flow constraint on z − (providing convergence on an approximate linear problem on power flow variables) from the full equivalence constraint on z +. The linear constraints in (12) can be also expressed in the compact matrix notation
$$ \boldsymbol{z}_{k} =\left[ \begin{array}{l} z_{i,j}^{-}\\ z_{i,j}^{+} \end{array} \right]_{(i,j)\in\mathcal{O}_{k}} = \boldsymbol{A}_{k}\boldsymbol{x}_{k} $$
where A k is a sparse matrix of size $2|\mathcal {O}_{k}|\times |\mathcal {V}_{k}|$ . In the typical case of large regions having a few connections with neighbors it is $|\mathcal {O}_{k}|\ll |\mathcal {V}_{k}|$ .
The distributed approach
We now introduce the distributed algorithm in a general and abstract form, in order to assess its properties and capture its structure with a compact notation.
Reference optimization problem
The kind of problem we wish to solve in (12) is a non-convex problem of the form
$$ \begin{aligned} &\text{min}\,\, F(\boldsymbol{x})\\ &\text{w.r.t.}\,\,\boldsymbol{x} \in \mathcal{X}, \boldsymbol{z} \in \mathcal{Z}\\ & \text{s.t.}\,\,\boldsymbol{A} \boldsymbol{x}=\boldsymbol{z} \end{aligned} $$
where $\boldsymbol {x}=\;[\boldsymbol {x}_{k}]_{k\in \mathcal {R}}$ collects all variables, $\boldsymbol {z}=\;[\boldsymbol {z}_{k}]_{k\in \mathcal {R}}$ collects all auxiliary variables, $F(\boldsymbol {x})=\sum _{k\in \mathcal {R}}F_{k}(\boldsymbol {x}_{k})$ is separable, $\mathcal {X} = \mathcal {X}_{1}\times \ldots \times \mathcal {X}_{R}$ is a Cartesian product, set $\mathcal {Z}=\mathcal {Z}^{-}\times \mathcal {Z}^{+}$ is a linear space with associated projector $\boldsymbol {L}_{\mathcal {Z}}$ , and A=diag(A 1,…,A R ) has a block diagonal form. The results given in the following further consider bounded (as we already assumed), and F(x) continuous. We finally assume that (14) has a solution.
The smoothness of functions involved with the OPF problem ensure that a one-to-one relation exists between local minima of problem (14) and the corresponding Karush Kuhn Tucker (KKT) conditions. We have (e.g., see [19])
Theorem 1.
(KKT stationary points) The KKT stationary point conditions associated with the primal problem ( 14 ) are given by
$$ \begin{aligned} \boldsymbol{0} &\in \partial F(\boldsymbol{x}) + \partial \eta_{\mathcal{X}}(\boldsymbol{x})+ \boldsymbol{A}^{T}\boldsymbol{\lambda}\\ \boldsymbol{A} \boldsymbol{x} &= \boldsymbol{z}\\ \boldsymbol{x}& \in \mathcal{X}\;,\;\boldsymbol{z} \in\mathcal{Z}\;,\;\boldsymbol{\lambda} \perp \mathcal{Z} \end{aligned} $$
where ∂is the proximal sub-gradient operator, and where $\eta _{\mathcal {A}}$ is the indicator function of set , with $\eta _{\mathcal {A}}(\boldsymbol {a})=0$ if $\boldsymbol {a} \in \mathcal {A}$ and +∞ if $\boldsymbol {a}\not \in \mathcal {A}$ . Conditions (15) identify the local minima of (14). □
Augmented Lagrangian formalization
No global minimum ensurance is given in the present context, since the Lagrangian associated with problem (14) may suffer of a primal-dual gap. A remedy in this respect is to use a Powell Hestenes Rockafellar (PHR) augmented Lagrangian formulation. The augmented Lagrangian associated with problem (14) can be written in the form
$$ \begin{aligned} L(\boldsymbol{x},\boldsymbol{z},\boldsymbol{\lambda}, \boldsymbol{\epsilon}) & = F(\boldsymbol{x}) +\eta_{\mathcal{X}}(\boldsymbol{x})+ \eta_{\mathcal{Z}}(\boldsymbol{z}) \\ & \qquad + \boldsymbol{\lambda}^{T}(\boldsymbol{A} \boldsymbol{x}-\boldsymbol{z}) +\frac{1}{2}\|\boldsymbol{A} \boldsymbol{x}-\boldsymbol{z}\|_{\boldsymbol{\epsilon}}^{2} \end{aligned} $$
where $\|\boldsymbol {x}\|^{2}_{\boldsymbol {\epsilon }}=\boldsymbol {x}^{T}\text {diag}(\boldsymbol {\epsilon })\boldsymbol {x}$ is a scaled norm, and where the entries of ε are strictly positive. In (16), the couple (x,z) plays the role of primal variables, while (λ,ε) play the role of dual variables (Lagrange multipliers). The dual function associated with (16) is
$$ D(\boldsymbol{\lambda},\boldsymbol{\epsilon}) = \min\limits_{\boldsymbol{x},\boldsymbol{z}} L(\boldsymbol{x},\boldsymbol{z}, \boldsymbol{\lambda},\boldsymbol{\epsilon})\;. $$
The PHR augmented Lagrangian of (16) is well defined, in the sense that it ensures the typical properties of ordinary Lagrangians of convex functions, i.e., the zero duality gap property and the applicability of a saddle point theorem. The result is given in ([20], Theorem 11.59). Incidentally, we are using a vector of weighting factors ε instead of a unique multiplication by scalar factor ε. This, however, does not modify derivation nor the final result.
(Rockafellar-Wets) 1. Zero duality gap Let (x ∗,z ∗)be a solution to the primal problem (14), and let (λ ∗,ε ∗) be any maximizer of the dual function (17). The corresponding duality gap is zero, that is, we have
$$ F(\boldsymbol{x}^{*})=D(\boldsymbol{\lambda}^{*},\boldsymbol{\epsilon}^{*})\;. $$
2. Saddle point The solutions in 1 identify a saddle point of PHR augmented Lagrangian (16), that is
$$ \begin{aligned} (\boldsymbol{x}^{*},\boldsymbol{z}^{*}) & \in {\underset{\textbf{x},\textbf{z}}{\text{argmin}}}\; L(\boldsymbol{x},\boldsymbol{z},\boldsymbol{\lambda}^{*},\boldsymbol{\epsilon}^{*})\\ (\boldsymbol{\lambda}^{*},\boldsymbol{\epsilon}^{*}) & \in {\underset{\boldsymbol{\lambda}, \boldsymbol{\epsilon}\ge \boldsymbol{0}}{\text{argmax}}}\; L(\boldsymbol{x}^{*},\boldsymbol{z}^{*},\boldsymbol{\lambda}, \boldsymbol{\epsilon})\;. \end{aligned} $$
Conversely, any saddle point (19) identifies a primal and dual solution, as from 1. □
In this context, the search for an optimum point can be turned into the search for a saddle point of the PHR augmented Lagrangian, which is in general more effective in terms of efficiency and speed. However, since only a local optimization point may be available for the first of (19) (because of non-convexity), then only local saddle points can be practically identified. It is then interesting to observe the following result, which is a straightforward consequence of the fact that local minima/maxima conditions of (19) correspond to KKT stationary point conditions (15), as the reader can easily verify.
There exists a one-to-one correspondence between local minima of the original problem (14), KKT stationary points (15), and local saddle points of the PHR augmented Lagrangian in (19). □
As a consequence, the search for local minima can be mapped into a search for local saddle points of the augmented Lagrangian.
Alternating direction search for a local saddle point
The search for a local saddle point can be dealt with by using the method of [17] (see also [18]). In our context, the method can be mapped into an alternating direction algorithm of the form
$$ \begin{aligned} \boldsymbol{x}_{t+1} & \in \arg\min_{\boldsymbol{x} \in \mathcal{X}} L(\boldsymbol{x},\boldsymbol {z}_{t},\boldsymbol{\lambda}_{t},\boldsymbol{\epsilon}_{t})\\ \boldsymbol{z}_{t+1} & \in \arg\min_{\boldsymbol{z} \in \mathcal{Z}} L(\boldsymbol{x}_{t+1},\boldsymbol {z},\boldsymbol{\lambda}_{t},\boldsymbol{\epsilon}_{t})\\ \boldsymbol{\lambda}_{t+1} & = \boldsymbol{\lambda}_{t} + \boldsymbol{E}_{t} (\boldsymbol{A} \boldsymbol{x}_{t+1}-\boldsymbol{z}_{t+1}) \end{aligned} $$
where E t =diag(ε t ), and where ε t is suitably updated at each cycle by guaranteeing ε t+1≥ε t . Note that, differently from [17], and similarly to what we have in ADMM, an independent update is used for x t and z t . In turn, differently from ADMM, the weighting parameters ε t are updated in order to ensure convergence of the process in a non-convex scenario.
Throughout the process, we assume that the commutation property
$$ \boldsymbol{L}_{\mathcal{Z}} \boldsymbol{E}_{t} = \boldsymbol{E}_{t} \boldsymbol{L}_{\mathcal{Z}} $$
holds, which corresponds to the request
$$ \epsilon_{k,i,j} = \epsilon_{h,j,i}\;,\qquad (i,j)\in\mathcal{O}_{k}, j\in\mathcal{R}_{h}, k,h\in\mathcal{R}\;. $$
We also assume that
$$ \boldsymbol{\lambda}_{0}\perp \mathcal{Z}\;. $$
These are light hypotheses guaranteeing that (20) simplifies to updates
$$ \begin{aligned} \boldsymbol{x}_{t+1} & \in \arg\min_{\boldsymbol{x} \in \mathcal{X}} F(\boldsymbol{x}) +\frac{1}{2}\|\boldsymbol{A} \boldsymbol{x} - (\boldsymbol{z}_{t}-\boldsymbol{E}_{t}^{-1}\boldsymbol{\lambda}_{t}) \|_{\boldsymbol{\epsilon}_{t}}^{2}\\ \boldsymbol{z}_{t+1} & = \boldsymbol{L}_{\mathcal{Z}}\boldsymbol{A} \boldsymbol{x}_{t+1}\\ \boldsymbol{\lambda}_{t+1} & = \boldsymbol{\lambda}_{t} + \boldsymbol{E}_{t} (\boldsymbol{A} \boldsymbol{x}_{t+1}-\boldsymbol{z}_{t+1}) \end{aligned} $$
and we also have
$$ \boldsymbol{z}_{t+1} \in \mathcal{Z}\;,\qquad \boldsymbol{\lambda}_{t+1}\perp \mathcal{Z} $$
so that the third line in KKT conditions (15) is satisfied throughout the iterative process. Note that the update of x t in the first of (24) corresponds to the parallel of a number of local updates because F is separable, and is a Cartesian product. In addition, since the full minimum for the first of (24) may be not available, we relax the result by assuming that a local minimum is achieved and that the target function in this local minimum x t+1 is smaller than or equal to the function value in x t . Therefore, a reliability assumption on the local solver is required. Although this might be in general a strong request (e.g., see [21]), especially when the local constraints identify a very small feasibility region, we expect it to be reasonably met when the number of local variables is not too large (i.e., for small regions).
Interestingly, given the fact that is bounded, then both sequences {x t } and {z t } are bounded. This may not be the case for {λ t }, but it is convenient to force this property by assuming
$$ \boldsymbol{\lambda}_{t+1} = \mathcal{P}[\boldsymbol{\lambda}_{t} + \boldsymbol{E}_{t} (\boldsymbol{A} \boldsymbol{x}_{t+1}-\boldsymbol{z}_{t+1})] $$
with $\mathcal {P}[\boldsymbol {\lambda }]= \max (\boldsymbol {\lambda }_{\text {min}}, \min (\boldsymbol {\lambda },\boldsymbol {\lambda }_{\text {max}}))$ a projection onto a compact box. The reason for this action will become clearer later on in the proof of Theorem 5.
Concerning penalty parameters ε t , in the centralized fashion of [17] the update criterion on ε t is of the form
$$ \boldsymbol{\epsilon}_{t+1} =\left\{ \begin{array}{ll} \boldsymbol{\epsilon}_{t} & \text{if}\, \Gamma_{t+1}\le\theta\,\Gamma_{t}\\ \tau \boldsymbol{\epsilon}_{t} & \text{otherwise} \end{array} \right. $$
with constants 0<θ<1 and τ>1, and with
$$ \Gamma_{t}=\| \boldsymbol{A} \boldsymbol{x}_{t}-\boldsymbol{z}_{t}\|_{\infty} $$
a measure of the primal gap (in infinity norm), in such a way to increase the penalty only if the primal gap is not decreasing sufficiently. The criterion can be also made local. The approach we propose is the following. We first check the primal gap decrease in region k via
$$ \check{\boldsymbol{\epsilon}}_{k,t+1} =\left\{ \begin{array}{ll} \|\boldsymbol{\epsilon}_{k,t}\|_{\infty} \boldsymbol{1} & \text{if}\, \Gamma_{k,t+1}\le\theta\,\Gamma_{k,t}\\ \tau \|\boldsymbol{\epsilon}_{k,t}\|_{\infty} \boldsymbol{1} & \text{otherwise} \end{array} \right. $$
with 1 the all-ones vector, and with
$$ \Gamma_{k,t}=\| \boldsymbol{A}_{k}\boldsymbol{x}_{k,t}-\boldsymbol{z}_{k,t}\|_{\infty} $$
the local gap. We then select the smallest $\boldsymbol {\epsilon }_{t+1}\ge \check {\boldsymbol {\epsilon }}_{t+1}$ satisfying (29), which in our context implies
$$ \epsilon_{k,i,j,t+1} = \max\left(\check{\epsilon}_{k,i,j,t+1},\check{\epsilon}_{h,j,i,t+1}\right) $$
where $(i,j)\in \mathcal {O}_{k}, j\in \mathcal {R}_{h}, k,h\in \mathcal {R}$ . This approach only requires local message exchanges. With this definition, the update is such that if one value of ε k,t grows to ∞, then all the values in the network do so, as it is for the centralized counterpart (27).
The proposed solution is summarized in Algorithm 1.
Convergence guarantees
The important characteristic of Algorithm 1 is that, in the given scenario, it provides a distributed solution. The main difference with the inspiring technique of [17] lays in the use of an alternating search with respect to x and z (versus the joint minimum search on (x,z)), this being the key point for obtaining a distributed algorithm. Nevertheless, the algorithm always converges (despite the non-convex scenario), and convergence guarantees essentially equivalent to those of [17] can be derived.
We separately treat the case where the penalty constant parameters are bounded and the case where they are unbounded. For bounded parameters we have the following result.
(Bounded penalties)Consider Algorithm 1, and assume that the sequence of penalty parameters {ε t } is bounded. We have:
Sequences {z t } and {λ t } converge to finite values, z ∗ and λ ∗, respectively.
There exists a finite limit point (accumulation point) for the sequence {x t }, and if A T A is invertible then sequence {x t } is further guaranteed to converge to a finite value x ∗.
The triplets (x ∗,z ∗,λ ∗), with x ∗ any limit point of {x t }, satisfy the KKT conditions of (15), hence all limit points x ∗ identify a local minimum to the original problem. Even more, in the limit t→∞ any triplet (x t ,z t ,λ t ) satisfies the KKT stationarity conditions, i.e., identifies a local minimum and satisfies the constraint A x t =z t . □
Proof of Theorem 4.
Consider that the sequence of penalty parameters {ε t } is bounded, to have ε t =ε ∞ for t≥t 0. For both (27) and (29), we have that Γ t+1≤θ Γ t for t>t 0, and therefore λ t is bounded and converges to a finite value λ ∞ (also in case the projection (26) is limiting the value to its maximum).
Now, by exploiting equivalence $\boldsymbol {z}_{t}=\boldsymbol {L}_{\mathcal {Z}} \boldsymbol {A} \boldsymbol {x}_{t}$ , we rewrite the update of x t in (24) in the form
$$ \begin{aligned} \boldsymbol{x}_{t+1} & \in {\underset{\boldsymbol{x}\in\mathcal{X}}{\text{argmin}}}\; F(\boldsymbol{x}) +\frac{1}{2}\|(\boldsymbol{I}- \boldsymbol{L}_{\mathcal{Z}})\boldsymbol{A} \boldsymbol{x} \|_{\boldsymbol{\epsilon}_{t}}^{2}\\ & +\frac{1}{2}\|\boldsymbol{L}_{\mathcal{Z}} \boldsymbol{A}(\boldsymbol{x}- \boldsymbol{x}_{t}) \|_{\boldsymbol{\epsilon}_{t}}^{2} + {\boldsymbol{\lambda}_{t}^{T}} \boldsymbol{A} \boldsymbol{x}\;. \end{aligned} $$
By then using the shorthand notation
$$\begin{array}{@{}rcl@{}} g_{t} & =& F(\boldsymbol{x}_{t}) + \eta_{\mathcal{X}}(\boldsymbol{x}_{t}) + \frac{1}{2}\|\boldsymbol{A} \boldsymbol{x}_{t}- \boldsymbol{z}_{t} \|_{\boldsymbol{\epsilon}_{\infty}}^{2} + \boldsymbol{\lambda}_{\infty}^{T}\boldsymbol{A} \boldsymbol{x}_{t}\\ \zeta_{t} &=& (\boldsymbol{\lambda}_{t}-\boldsymbol{\lambda}_{\infty})^{T} \boldsymbol{A}(\boldsymbol{x}_{t}-\boldsymbol{x}_{t+1})\;, \end{array} $$
and Δ g t =g t+1−g t , from (32) we have
$$ \Delta g_{t} +\frac{1}{2}\|\boldsymbol{z}_{t+1} - \boldsymbol{z}_{t} \|_{\boldsymbol{\epsilon}_{\infty}}^{2} \le \zeta_{t} \le |\zeta_{t}| \;,\quad t>t_{0} $$
which implies Δ g t ≤|ζ t | for t>t 0. By noting that ∥A(x t − x t+1)∥ is bounded because is assumed bounded, and by recalling that ${\lim }_{\textit {t}\rightarrow \infty } \boldsymbol {\lambda }_{t}= \boldsymbol {\lambda }_{\infty }$ , then it also is ${\lim }_{\textit {t}\rightarrow \infty }|\zeta _{t}|=0$ . This is sufficient to guarantee that Δ g t converges to 0 for t→∞, which can be proved by contradiction. Specifically, if Δ g t does not converge to 0 then there exists an infinite sequence for which |Δ g t |≥ε>0. Moreover, since Δ g t ≤|ζ t |, where the right value can be made arbitrarily small for large t, there also exists an infinite sequence for which Δ g t ≤−ε. By denoting the sequence as $\mathcal {S}_{\epsilon }\subset (t_{0},\infty)$ , this would imply
$$g_{\infty} -g_{t_{0}} = \sum_{t\not\in\mathcal{S}_{\epsilon}} \Delta g_{t} + \sum_{t\in\mathcal{S}_{\epsilon}} \Delta g_{t} \le \sum_{t\not\in\mathcal{S}_{\epsilon}} |\zeta_{t}| - \sum_{t\in\mathcal{S}_{\epsilon}} \epsilon \;. $$
Since |ζ t | is guaranteed to be exponentially decreasing because of the assumption Γ t+1≤θ Γ t , the above implies g ∞ =−∞, hence a contradiction. Therefore, g t converges to a finite value, and, as a consequence of (33), the weighted norm ${\| \boldsymbol {z}_{t+1} - \boldsymbol {z}_{t} \|}_{\epsilon _{\infty }}^{2}$ converges to 0, i.e., z t converges to a finite value too. These results justify points 1 and 2.
To conclude with point 3, since x t+1 is assumed a local minimum, from (32) we also have, for t>t 0,
$$\begin{array}{*{20}l} {\boldsymbol{0}} & \in \partial F(\boldsymbol{x}_{t+1}) + \partial\eta_{\text{\c{X}}}(\boldsymbol{x}_{t+1})+ \boldsymbol{A}^{T}\boldsymbol{\lambda}_{\infty} \cr & + \boldsymbol{A}^{T} \boldsymbol{E}_{\infty}(\boldsymbol{I}- \boldsymbol{L}_{\mathcal{Z}}) \boldsymbol{A} \boldsymbol{x}_{t+1}\cr & + \boldsymbol{A}^{T} \boldsymbol{E}_{\infty}(\boldsymbol{z}_{t+1}- \boldsymbol{z}_{t}) + \boldsymbol{A}^{T}(\boldsymbol{\lambda}_{t}-\boldsymbol{\lambda}_{\infty}) \end{array} $$
and since the values on the second and third lines tend to 0 in the limit, then in the limit, the KKT stationary point conditions (15) are satisfied.
As a consequence, bounded penalty parameters guarantee a convergence of the algorithm to a KKT stationary point, i.e., they imply the identification of a local minimum. Note that the result is sufficiently strong also in the case where A T A is not invertible (see second part of point 3). This is an important property since the invertibility of A T A is only ensured for a single-node regions choice $\mathcal {R}_{k}=\{k\}$ .
The result for unbounded parameters assumes that the ill conditioning associated with very large/infinite values is adequately solved, e.g., by locally normalizing the minimization in (32) by the maximum penalty value ∥ε k,t ∥ ∞ . We have
Theorem 5 (Unbounded penalties).
Consider Algorithm 1, and assume that the sequence of penalty parameters {ε t } is unbounded. We have:
Sequence {z t } converges to a finite value, z ∗.
There exists a finite limit point for the sequence {x t }, and if A T A is invertible then sequence {x t } is ensured to converge to a finite value x ∗. □
The results in the proof of Theorem 4 can be applied by suitably (locally) normalizing parameters. The kind of replacements we use are
$$\begin{array}{*{20}l} \boldsymbol{\epsilon}_{t} &\quad\Longrightarrow\quad \tilde{\boldsymbol{\epsilon}}_{t} = \left[\frac{ \boldsymbol{\epsilon}_{k,t}}{\|\boldsymbol{\epsilon}_{k,t}\|_{\infty}}\right]_{k=1,\ldots,R}\cr F(\boldsymbol{x}) &\quad\Longrightarrow\quad \tilde{F}(\boldsymbol{x}) = \sum_{k=1}^{R} \frac{F_{k}(\boldsymbol{x}_{k})}{\|\boldsymbol{\epsilon}_{k,t}\|_{\infty}} \\ &\boldsymbol{\lambda}_{t}\Longrightarrow\quad \tilde{\boldsymbol{\lambda}}_{t} = \left[\frac{\boldsymbol{\lambda}_{k,t}}{\|\boldsymbol{\epsilon}_{k,t}\|_{\infty}}\right]_{k=1,\ldots,R} \end{array} $$
which have the characteristic of providing bounded quantities. For both (27) and (29), all entries ε k,t are diverging by construction, hence $\tilde {\boldsymbol {\lambda }}_{t}$ is ensured to converge to 0 in the limit. Convergence is also guaranteed to be exponential, because of the presence of parameter τ>1 in the update of penalty parameters. These properties are fundamental and are ensured by use of projection (26). Furthermore, $\tilde {\boldsymbol {\epsilon }}_{t} $ is guaranteed to converge to the all ones vector 1. By then investigating the counterparts to g t and ζ t , namely,
$$\begin{array}{*{20}l}\tilde{g}_{t} & = \tilde{F}(\boldsymbol{x}_{t}) + \eta_{\text{\c{X}}}(\boldsymbol{x}_{t}) + \frac{1}{2}\|\boldsymbol{A} \boldsymbol{x}_{t}- \boldsymbol{z}_{t} \|_{\tilde{\epsilon}_{t}}^{2} \\\tilde{\zeta}_{t} & = \tilde{\boldsymbol{\lambda}}_{t}^{T} \boldsymbol{A}(\boldsymbol{x}_{t}- \boldsymbol{x}_{t+1}) \end{array} $$
we still verify that properties ${\lim }_{\textit {t}\rightarrow \infty }|\tilde {\zeta }_{t}|=0$ and
$$ \Delta\tilde{g}_{t} +\frac{1}{2}\| \boldsymbol{z}_{t+1} - \boldsymbol{z}_{t} \|_{\tilde{\epsilon}_{t}}^{2} \le \tilde{\zeta}_{t}\le |\tilde{\zeta}_{t}| $$
hold, and we also have that $\Delta \tilde {g}_{t}$ converges to 0. Hence $\tilde {g}_{t}$ converges to a finite value, so that there exist limit points for the sequence {x t }. From (34) we also find that z t converges to a finite value. This proves the theorem.
Note that Theorem 5, although being able to prove convergence of both sequences {x t } and {z t }, cannot guarantee that the limit solution is feasible, i.e., it satisfies A x t =z t . As a matter of fact, in the limit, the minimization in (32) assumes the (approximate) form
$$ \boldsymbol{x}_{t+1}\in\underset{x\in\mathcal{X}}{\text{argmax}}\ \|(\boldsymbol{I}- \boldsymbol{L}_{\mathcal{Z}}) \boldsymbol{A} \boldsymbol{x} \|^{2} + \| \boldsymbol{L}_{\mathcal{Z}} \boldsymbol{A}(\boldsymbol{x}- \boldsymbol{x}_{t}) \|^{2} $$
which corresponds to an iterative algorithm for performing a projection of x onto the feasible space $\mathcal {X}\cap \{\boldsymbol { x}| \boldsymbol {A} \boldsymbol {x}= \boldsymbol {L}_{\mathcal {Z}} \boldsymbol { A}\boldsymbol { x}\}$ , and in this context, the contribution $ \| \boldsymbol {L}_{\mathcal {Z}}\boldsymbol { A}(\boldsymbol { x}-\boldsymbol { x}_{t}) \|^{2}$ plays the role of a proximity operator, forcing vicinity to the solution available from the previous step. Therefore, if the algorithm used to solve the local problem (32) is sufficiently powerful, then convergence to a feasible point is also ensured in the limit. This is the case, in practice, only for moderately non-convex scenarios.
The distributed OPF algorithm
The distributed OPF algorithm that we obtain by applying Algorithm 1 to problem (12) is summarized in Algorithm 2. The local penalty parameters update (29)-(31) is used.
Note that two local message exchanges (denoted with arrows) are required in lines 10 to 11 and lines 22 to 23 to exchange, respectively, the updated values x k,t (in order to update auxiliary variables) and the temptative penalty parameters updates $\check {\boldsymbol {\epsilon }}_{k,t}$ (in order to make sure that the final update satisfies (21)). In principle, a single message exchange could be obtained by postponing the penalty parameters correction of line 24 after the auxiliary variable update in line 13, at the cost of some sub optimality in performance.
Overall, the local processing effort of Algorithm 2 is light. The algorithm complexity is determined by the update of x t in line 5, which corresponds to a region-based optimization problem, and which can be efficiently solved by state-of-the-art methods, e.g., interior point methods (IPMs). The remaining actions require a limited effort, especially in the standard case where a few connections are active with neighboring regions and auxiliary vectors are short (i.e., $|\mathcal {O}_{k}|\ll |\mathcal {V}_{k}|$ ).
We finally underline that five key parameters are used in Algorithm 2, and these need to be accurately set for good performance. We have:
Weighting constants ρ and ζ (they define matrices A k , see (12)-(13)). They should be chosen in such a way that ρ≫ζ>0, in order to force the algorithm towards an approximate linear solution on power flow variables.
Initialization value for penalty parameters ξ. It should be set to a small value to guarantee a good algorithm outcome even when starting from a point very far from the optimum.
Penalty parameters update constants 0<θ<1 and τ>1. In order to avoid a rapid increasing behavior on penalty parameters, the constants should be set to values close to 1.
The algorithm performance is tested using three different scenarios, namely: 1) the wide area network IEEE Power System Test Case Archive [22]; 2) the IEEE PES Distribution Test Feeder [23,24]; 3) a microgrid topology generated according to the model proposed in [25]. The networks in Scenarios 2) and 3) have a tree topology, while Scenario 1) involves networks with many loops where algorithm convergence may be an issue. All chosen scenarios are moderate sized networks, with moderate non-convexities, which constitute the applicability field of the proposed algorithm. Applicability to more complex networks with more severe non-convexities and a high number of loops (e.g., the Polish system models) requires use of some additional (quasi centralized) coordination between entities, and will be the subject of future investigation.
Description of the scenarios
A power losses minimization problem under voltage and power constraints is considered (i.e., f i (P i )=P i ), and the following settings are used in the various scenarios:
Networks sizes N=30, 57, 118, and 300 are used. Constraints and load requests are set as from the MATPOWER distribution [26].
The N=123 nodes network is used in single-phase fashion. The chosen settings are inspired by [6]. Load requests are set as given in the dataset, and generating capabilities ranges are added in the form |Q G,i |≤1.2|Q L,i |, and 0≤P G,i ≤30 kW, where the subscript L stands for load and G for generation. Voltage regulation is applied with 0.94≤|V i |≤1.06.
A unique network is selected with N=120. The network is generated as four joint small-world graphs with 30 nodes (to limit the depth of the graph) and rewiring probability p=0.4 (see also details in [16]). Lines lengths have an exponential distribution with parameter μ=65.86 m and a minimum distance set to 10 m. The impedance value is chosen 2.9400+j0.0861 Ω/km (class 1, 10 mm 2 cables). Load requests are randomly generated with an uniform distribution in [0,3] kW, and with a uniform cosϕ with $\phi \in [-\frac \pi 8,\frac \pi 8]$ . 20% of the nodes are given generation capabilities, randomly distributed in [0,10] kW for active power and [−20,20] kVAr for reactive power. Voltage regulation is applied in the range 0.9≤|V i |≤1.1.
Region partitioning
Region partitioning is a fundamental aspect for ensuring a good performance. Ideally, compact regions with very few outer connections guarantee limited complexity, accuracy of the solution, and controlled computational time. In the considered scenarios, region partitioning is chosen in such a way that a unique generator is available in each region, and the region further includes those loads which are electrically closer (in terms of line impedance) to the generator. Since this corresponds to an excessively fine partitioning in Scenario 2), for the IEEE feeder, the region choice is made in such a way that a local controller is placed at each network bifurcation point, and the associated region corresponds to all those nodes which are electrically closer to it (in terms of line impedance).
Simulation tools
The local optimization problem (see line 5 of Algorithm 2, or see the first of (24)) is solved by using IPOPT [27], an efficient IPM solver which allows a MatLab interface. Although a true optimality guarantee is not available, IPM methods are known to perform very well for OPF kind of problems. MUMPS linear solver is used within IPOPT, and the warm start option is used in such a way to start the local minimization process using the solution available from the previous iteration (this reduces computational times). The code is run on a MacBook Air and is written in MatLab [28].
Convergence test in the considered scenarios
A test on the behavior of Algorithm 2 in the three different scenarios using the parameters of Table 1 is illustrated in Figure 2. The starting point is chosen to be the all-ones vector x k,0=1, and Lagrange multipliers are initially set to zero, λ k,0=0. This corresponds to the unavailability of any a priori information on both position and Lagrange multipliers and is therefore a worst case scenario. Iterations are stopped (and convergence is declared) when the primal gap ∥A x t −z t ∥ ∞ (infinity norm) reaches 10−4. The maximum values for Lagrange multipliers are set to λ max=103·1, λ min=−103·1.
Performance of distributed OPF with IEEE and microgrid networks.
Table 1 Performance starting from a remote point
For the three scenarios considered, Figure 2 shows in the first column the voltages V i (amplitude and phase diagram) at convergence, together with the active voltage constraints. Observe that all voltage limitations are met.
The second column of Figure 2 shows the behavior of the primal gap in norm 2 and norm ∞ as a function of the iteration number t. Although the curves are not strictly decreasing, they are clearly diminishing to zero-gap value. The penalty parameters update, illustrated in the third column of Figure 2, shows the ability of (29)-(31) of keeping a small gap between maximum and minimum values of ε t . The fact that the parameters are always increasing is due to the sub optimality of the distributed criterion with respect to the centralized criterion (27) which would be more effective in limiting the increase of penalty parameters. Nevertheless, the algorithm converges to points very close to the optimum (see Table 1) despite the very badly chosen initial point. In this respect, the local IPM solvers are fully capable of resolving the limit problem (35) and hence guarantee convergence to a feasible point. Note that the slower convergence is experienced with Scenario 2), i.e., the IEEE feeder with N=123. This is due to the fact that this is the network with highest depth due to its radial structure. This makes the distributed process particularly challenging since agreement must be obtained between regions that are very far one from the other.
Finally, in the fourth column of Figure 2, we provide the locally determined reactive power regulation (Q G,i stands for reactive power at generators), which show a converging behavior in accordance with the fact that the primal gap is vanishing. A perfectly equivalent behavior is found for active powers (but this is not shown in figure).
A more in-depth performance measure for the tests of Figure 2 is given in Table 1, where the distributed approach of Algorithm 2 is compared with the performance of a centralized IPOPT solver.
Note that the performance gap with respect to a central solver is always below a 1% error, which is an impressive performance considering that we are dealing with a worst case situation, and that we are approaching the problem in distributed form with a severe network partitioning. As a matter of fact, the outstanding performance of IPMs is mainly due to their central coordination capabilities (e.g., see [15]). Incidentally, we observed that the performance of Algorithm 2 is almost independent of the chosen settings. As a consequence, the performance gap in Table 1 coincides with the ultimate accuracy that could be achieved after thousands of iterations for every studied case.
By inspecting the references, the reader can further appreciate the substantial improvement with respect to the performance of the ADMM-based algorithm of [15], and the sensibly improved network size and partitioning performance with respect to the preliminary algorithm version of [16].
Some information on the processing times involved with Algorithm 2 is given in both Table 1 and Figure 3.
Local and aggregate processing times per iteration.
Figure 3 shows, for the six networks under consideration, the maximum local processing time and the aggregate processing time per iteration. These are almost constant throughout the iterative process, evidencing the fact that the processing time is approximately linear in the number of iterations. From Table 1, we can further extract some information on the time needed per region (the max processing time per region), which is in a range between 2 and 13 s, the value being in agreement with the literature on distributed OPF (e.g., see [6]).
Observe that communication delays were not taken into account in Figure 3 and Table 1, and in fact these can be made negligible by choosing a suitable communication technique. High data rate communication standards with associated short packet lengths are to be preferred. This is the case, for example, of broadband power line communication techniques which can guarantee packet lengths of less than a millisecond [29] and which can be deployed in small area applications (e.g., in micro grids). WiMax is a wireless alternative in these scenarios. For wide area applications, instead, optical fiber communications (e.g., gigabit Ethernet) are an appropriate solution.
In this paper, we proposed a distributed algorithm for OPF regulation based upon a non-convex formulation. By suitably controlling penalty parameters, the algorithm was proven to always converge under a proper assumption on local solver reliability. A certificate of convergence to a local minimum is also available under the request that penalty factors are bounded. The algorithm was shown to provide a reliable performance also in a worst case situation where the search for the optimum is initialized on a point very far from its final destination. The algorithm was proven to be efficient and fast and to be also robust with respect to a severe network partitioning. Its required computational effort was found to be of the order of state-of-the-art methods (using convex problem approximations to ease the convergence issue), with the added value of allowing for a full adherence to the original problem since no (convex) approximation is used.
On the applicability side, the distributed algorithm is readily applicable on moderate time scales (tens of seconds) and on moderate sized networks (up to 300 nodes) for system optimization purposes, not concerning fast regulation (e.g., fault or protection issues require much faster time scales). In this scenario, the algorithm is also expected to be robust to packet losses, because of its alternating direction structure.
Applicability to larger network sizes, with many loops, and more severe non-convexities is instead out of the scope of the present work. As a matter of fact, the proposed alternating direction search allows distributing the processing burden, but might not find an agreement (or it might take too long) in harsh situations. To overcome these difficulties, two strategies can be jointly employed. On the one side, some criteria to determine the optimal region partition strategy should be identified. On the other side, some additional coordination between agents should be used, e.g., a proper distributed generalization of the techniques used in the work of Martinez and Birgin et al. [18] which could also be capable of closing the performance gap with respect to a centralized solver. Use of recent advances on ADMM accelerated methods and scaling techniques (e.g., see [30]) is also an interesting option but need to be suitably adapted to a non-convex context. These aspects are left for future investigations.
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Università di Padova, Dipartimento di Ingegneria dell'Informazione, via G. Gradenigo 6/b, Padova, Italy
Tomaso Erseghe
Search for Tomaso Erseghe in:
Correspondence to Tomaso Erseghe.
The author declares that he has no competing interests.
Tomaso Erseghe was born in 1972. He received the Laurea (M.Sc degree) and the Ph.D. in Telecommunication Engineering from the University of Padova, Italy in 1996 and 2002, respectively. Since 2003, he is Assistant Professor (Ricercatore) at the Department of Information Engineering, University of Padova. His current research interest is in the fields of distributed algorithms for telecommunications, and smart grids optimization. His research activity also covered the design of ultra-wideband transmission systems, properties and applications of the fractional Fourier transform, and spectral analysis of complex modulation formats.
https://doi.org/10.1186/s13634-015-0226-x
Alternating direction method of multipliers
Augmented Lagrangian methods
Convergence guarantee
Distributed processing
Optimal power flow
Advanced signal processing techniques and telecommunications network infrastructures for Smart Grid analysis, monitoring and management | CommonCrawl |
\begin{definition}[Definition:Ellipse/Eccentricity]
Let $K$ be an ellipse specified in terms of:
:a given straight line $D$
:a given point $F$
:a given constant $e$ such that $0 < e < 1$
where $K$ is the locus of points $P$ such that the distance $p$ from $P$ to $D$ and the distance $q$ from $P$ to $F$ are related by the condition:
:$q = e p$
The constant $e$ is known as the '''eccentricity''' of the ellipse.
\end{definition} | ProofWiki |
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Volume 950 - 10 November 2022
Page/Article number: low to high Page/Article number: high to low Title Type Online publication date
Cover image:
Graphical abstract from Toghraei, I. & Billant, P. 2022 Dynamics of a stratified vortex under the complete Coriolis force: two-dimensional three-components evolution. J. Fluid Mech.950, A29. doi:10.1017/jfm.2022.812.
Functional renormalisation group for turbulence
Léonie Canet
Published online by Cambridge University Press: 24 October 2022, P1
Turbulence is a complex nonlinear and multi-scale phenomenon. Although the fundamental underlying equations, the Navier–Stokes equations, have been known for two centuries, it remains extremely challenging to extract from them the statistical properties of turbulence. Therefore, for practical purposes, a sustained effort has been devoted to obtaining an effective description of turbulence, that we may call turbulence modelling, or statistical theory of turbulence. In this respect, the renormalisation group (RG) appears as a tool of choice, since it is precisely designed to provide effective theories from fundamental equations by performing in a systematic way the average over fluctuations. However, for Navier–Stokes turbulence, a suitable framework for the RG, allowing in particular for non-perturbative approximations, has been missing, which has thwarted RG applications for a long time. This framework is provided by the modern formulation of the RG called the functional renormalisation group (FRG). The use of the FRG has enabled important progress in the theoretical understanding of homogeneous and isotropic turbulence. The major one is the rigorous derivation, from the Navier–Stokes equations, of an analytical expression for any Eulerian multi-point multi-time correlation function, which is exact in the limit of large wavenumbers. We propose in this JFM Perspectives article a survey of the FRG method for turbulence. We provide a basic introduction to the FRG and emphasise how the field-theoretical framework allows one to systematically and profoundly exploit the symmetries. We stress that the FRG enables one to describe fully developed turbulence forced at large scales, which was not accessible by perturbative means. We show that it yields the energy spectrum and second-order structure function with accurate estimates of the related constants, and also the behaviour of the spectrum in the near-dissipative range. Finally, we expound the derivation of the spatio-temporal behaviour of $n$-point correlation functions, and largely illustrate these results through the analysis of data from experiments and direct numerical simulations.
Vortex patterns in rapidly rotating Rayleigh–Bénard convection under spatial periodic forcing
Shan-Shan Ding, Hong-Lin Zhang, Dong-Tian Chen, Jin-Qiang Zhong
Published online by Cambridge University Press: 13 October 2022, R1
Pattern-forming with externally imposed symmetry is ubiquitous in nature but little studied. We present experimental studies of pattern formation and selection by spatial periodic forcing in rapidly rotating convection. When periodic topographic structures are constructed on the heated boundary, they modulate the local temperature and velocity fields. Symmetric convection patterns in the form of regular vortex lattices are observed near the onset of convection, when the periodicity of the external forcing is set close to the intrinsic vortex spacing. We show that the new patterns arise as a dynamical process of imperfect bifurcation which is well described by a Ginzburg–Landau-like model. We explore the phase diagram of buoyancy strength and periodicity of external forcing to find the optimal experimental settings for which the vortex patterns best match that of the external forcing.
The effect of particle anisotropy on the modulation of turbulent flows
Stefano Olivieri, Ianto Cannon, Marco E. Rosti
You have access Access
We investigate the modulation of turbulence caused by the presence of finite-size dispersed particles. Bluff (isotropic) spheres versus slender (anisotropic) fibres are considered to understand the influence of the shape of the objects on altering the carrier flow. While at a fixed mass fraction – but different Stokes number – both objects provide a similar bulk effect characterized by a large-scale energy depletion, a scale-by-scale analysis of the energy transfer reveals that the alteration of the whole spectrum is intrinsically different. For bluff objects, the classical energy cascade shrinks in its extension but is unaltered in the energy content and its typical features, while for slender ones we find an alternative energy flux which is essentially mediated by the fluid–solid coupling.
Size-dependent transient nature of localized turbulence in transitional channel flow
Duo Xu, Baofang Song
It has been reported that a fully localized turbulent band in channel flow becomes sustained when the Reynolds number is above a threshold. Here we show evidence that turbulent bands are of a transient nature instead. When the band length is controlled to be fixed, the lifetime of turbulent bands appears to be stochastic and exponentially distributed, a sign of a memoryless transient nature. Besides increasing with the Reynolds number, the mean lifetime also strongly increases with the band length. Given that the band length always changes over time in real channel flow, this size dependence may translate into a time dependence, which needs to be taken into account when clarifying the relationship between channel flow transition and the directed percolation universality class.
Shape of sessile drops at small contact angles
Ehud Yariv
The shape of a sessile drop on a horizontal substrate depends upon the Bond number $Bo$ and the contact angle $\alpha$. Inspired by puddle approximations at large $Bo$ (Quéré, Rep. Prog. Phys., vol. 68, 2005, p. 2495), we address here the limit of small contact angles at fixed drop volume and arbitrary $Bo$. It readily leads to a pancake shape approximation, where the drop height and radius scale as $\alpha$ and $\alpha ^{-1/2}$, respectively, with capillary forces being appreciable only near the edge. The pancake approximation breaks down for $Bo=\textrm {ord}(\alpha ^{2/3})$. In that distinguished limit, capillary and gravitational forces are comparable throughout, and the drop height and radius scale as $\alpha ^{2/3}$ and $\alpha ^{-1/3}$, respectively. For $Bo\ll \alpha ^{2/3}$ these scalings remain, with the drop shape turning into a spherical cap. The asymptotic results are compared with a numerical solution of the exact problem.
When do shape changers swim upstream?
R.N. Bearon
Published online by Cambridge University Press: 13 October 2022, F1
Using a multiple-scale analysis, Walker et al. (J. Fluid Mech., vol. 944, 2022, R2) obtain the long-time behaviour of a shape-changing swimmer in a Poiseuille flow. They show that the behaviour falls into one of three categories: endless tumbling at increasing distance from the midline of the flow; preserved initial behaviour of the swimmer; or convergence to upstream rheotaxis, where the swimmer is situated at the midline of the flow. Furthermore, a single swimmer-dependent constant is identified that determines which of the three behaviours is realised.
JFM Papers
Ascending non-Newtonian long drops in vertical tubes
S. Longo, L. Chiapponi, D. Petrolo, S. Bosa, V. Di Federico
Published online by Cambridge University Press: 13 October 2022, A1
We report on theoretical and experimental studies describing the buoyancy-driven ascent of a Taylor long drop in a circular vertical pipe where the descending fluid is Newtonian, and the ascending fluid is non-Newtonian yield shear thinning and described by the three-parameter Herschel–Bulkley model, including the Ostwald–de Waele model as a special case for zero yield. Results for the Ellis model are included to provide a more realistic description of purely shear-thinning behaviour. In all cases, lubrication theory allows us to obtain the velocity profiles and the corresponding integral variables in closed form, for lock-exchange flow with a zero net flow rate. The energy balance allows us to derive the asymptotic radius of the inner current, corresponding to a stable node of the differential equation describing the time evolution of the core radius. We carried out a series of experiments measuring the rheological properties of the fluids, the speed and the radius of the ascending long drop. For some tests, we measured the velocity profile with the ultrasound velocimetry technique. The measured radius of the ascending current compares fairly well with the asymptotic radius as derived through the energy balance, and the measured ascent speed shows a good agreement with the theoretical model. The measured velocity profiles also agree with their theoretical counterparts. We have also developed dynamic similarity conditions to establish whether laboratory physical models, limited by the availability of real fluids with defined rheological characteristics, can be representative of real phenomena on a large scale, such as exchanges in volcanic conduits. Appendix B contains scaling rules for the approximated dynamic similarity of the physical process analysed; these rules serve as a guide for the design of experiments reproducing real phenomena.
A critical analysis of turbulence dissipation in near-wall flows, based on stereo particle image velocimetry and direct numerical simulation data
William K. George, Michel Stanislas, Jean Marc Foucaut, Christophe Cuvier, Jean Philippe Laval
An experiment was performed using stereo particle image velocimetry (SPIV) in the Laboratoire de Mécanique des Fluides de Lille boundary layer facility to determine all the derivative moments needed to estimate the average dissipation rate of the turbulence kinetic energy $\epsilon = 2 \nu \langle {\mathsf{s}}_{ij}{\mathsf{s}}_{ij} \rangle$, where ${\mathsf{s}}_{ij}$ is the fluctuating strain rate and $\langle ~\rangle$ denotes ensemble averages. Also measured were all the moments of the full average deformation rate tensor, as well as all of the first, second and third fluctuating velocity moments except those involving pressure. The Reynolds number was $Re_\theta = 7634$ or $Re_\tau = 2598$. The present paper gives the measured average dissipation, $\epsilon$ and the derivative moments comprising it. The results are compared with the earlier measurements of Balint, Wallace & Vukolavcevic (J. Fluid Mech., vol. 228, 1991, pp. 53–86) and Honkan & Andreopoulos (J. Fluid Mech., vol. 350, 1997, pp. 29–96) at lower Reynolds numbers and to new results from a plane channel flow DNS at comparable Reynolds number. Of special interest is the prediction by George & Castillo (Appl. Mech. Rev., vol. 50, 1997, pp. 689–729) and Wosnik, Castillo & George (J. Fluid Mech., vol. 421, 2000, pp. 115–145) that $\epsilon ^+ \propto {x_2^+}^{-1}$ for streamwise homogeneous flows and a nearly indistinguishable power law, $\epsilon \propto {x_2^+}^{\gamma -1}$, for boundary layers. In spite of the modest Reynolds number, the predictions seem to be correct. Then the statistical character of the velocity derivatives is examined in detail, and a particular problem is identified with the breakdown of local homogeneity inside $x_2^+ = 100$. A more general alternative for partially homogeneous turbulence flows is offered which is consistent with the observations. With the help of DNS, the spatial characteristics of the dissipation very near the wall are also examined in detail.
Understanding of turbulence modulation and particle response in a particle-laden jet from direct numerical simulations
Hua Zhou, Evatt R. Hawkes, Timothy C.W. Lau, Rey Chin, Graham J. Nathan, Haiou Wang
Point-particle direct numerical simulations have been employed to quantify the turbulence modulation and particle responses in a turbulent particle-laden jet in the two-way coupled regime with an inlet Reynolds number based on bulk velocity and jet diameter $({D_j})$ of ~10 000. The investigation focuses on three cases with inlet bulk Stokes numbers of 0.3, 1.4 and 11.2. Special care is taken to account for the particle–gas slip velocity and non-uniform particle concentrations at the nozzle outlet, enabling a reasonable prediction of particle velocity and concentration fields. Turbulence modulation is quantified by the variation of the gas-phase turbulent kinetic energy (TKE). The presence of the particle phase is found to damp the gas-phase TKE in the near-field region within $5{D_j}$ from the inlet but subsequently increases the TKE in the intermediate region of (5–20)Dj. An analysis of the gas-phase TKE transport equation reveals that the direct impact of the particle phase is to dissipate TKE via the particle-induced source term. However, the finite inertia of the particle phase affects the gas-phase velocity gradients, which indirectly affects the TKE production and dissipation, leading to the observed TKE attenuation and enhancement. Particle response to the gas-phase flow is quantified. Particles are found to exhibit notably stronger response to the gas-phase axial velocity than to the radial velocity. A new dimensionless figure is presented that collapses both the axial and radial components of the particle response as a function of the local Stokes number based on their respective integral length scales.
Low-Reynolds-number oscillating boundary layers on adiabatic slopes
Bryan E. Kaiser, Lawrence J. Pratt, Jörn Callies
We investigate the instabilities and transition mechanisms of Boussinesq stratified boundary layers on sloping boundaries when subjected to oscillatory body forcing parallel to the slope. We examine idealized forms of boundary layers on hydraulically smooth abyssal slopes in tranquil mid- to low-latitude regions, where low-wavenumber internal tides gently heave isopycnals up and down adiabatic slopes in the absence of mean flows, high-wavenumber internal tides, shelf breaks, resonant tide–bathymetry interactions (critical slopes) and other phenomena associated with turbulence 'hot spots'. In non-rotating low-Reynolds-number flow, increased stratification on the downslope phase has a relaminarizing effect, while on the upslope phase we find transition-to-turbulence pathways arise from shear production triggered by gravitational instabilities. When rotation is significant (low slope Burger numbers) we find that boundary layer turbulence is sustained throughout the oscillation period, resembling stratified Stokes–Ekman layer turbulence. Simulation results suggest that oscillating boundary layers on smooth slopes at low Reynolds number ( $\textit {Re}\leqslant 840$), unity Prandtl number and slope Burger numbers greater than unity do not cause significant irreversible turbulent buoyancy flux (mixing), and that flat-bottom dissipation rate models derived from the tide amplitude are accurate within an order of magnitude.
Slowing down convective instabilities in corrugated Couette–Poiseuille flow
N. Yadav, S.W. Gepner
Couette–Poiseuille (CP) flow in the presence of longitudinal grooves is studied by means of numerical analysis. The flow is actuated by movement of the flat wall and pressure imposed in the opposite direction. The stationary wall features longitudinal grooves that modify the flow, change hydrodynamic drag on the driving wall and cause onset of hydrodynamic instability in the form of travelling waves with a consequent supercritical bifurcation, already at moderate ranges of the Reynolds number. We show that by manipulating this system it is possible to significantly decrease phase speed of the unstable wave and to effectively decouple time scales of wave propagation and amplification with a potential to significantly reduce the distance required for the onset of nonlinear effects. Current analysis begins with concise characterization of stationary, laminar CP flow and the effects of applying a selected corrugation pattern, followed by determination of conditions leading to the onset of instabilities. In the second part we illustrate selected nonlinear solutions obtained for low, supercritical values of the Reynolds numbers and due to the amplification of unstable travelling waves of possibly low phase velocities. This work is concluded with a short discussion of a linear evolution of a wave packet consisting of a superposition of a number of unstable waves and initiated by a localized pulse. This part illustrates that in addition to the reduction of the phase velocity of a single, unstable mode, imposition of the Couette component also reduces group velocity of a wave packet.
Exponential asymptotics for elastic and elastic-gravity waves on flow past submerged obstacles
Christopher J. Lustri
Linearized flow past a submerged obstacle with an elastic sheet resting on the flow surface are studied in the limit that the bending length is small compared to the obstacle depth, in two and three dimensions. Gravitational effects are included in the two-dimensional geometry, but absent in the three-dimensional geometry; the Froude number is chosen so that gravitational and elastic restoring forces are comparable in size. In each of these problems, the waves are exponentially small in the asymptotic limit, and can be computed using exponential asymptotic methods. In the two-dimensional problem, flow past a submerged step is considered. It is found that the relative strengths of the gravitational and elastic restoring forces produce two distinct classes of elastic sheet behaviour. In one parameter regime, constant-amplitude elastic waves and gravity waves extend indefinitely upstream and downstream from the obstacle. In the other parameter regime, all waves decay exponentially away from the obstacle. The equivalent nonlinear two-dimensional geometry is then studied; this asymptotic analysis predicts the existence of a third intermediate regime in which waves persist indefinitely in only one direction, depending on whether the submerged step rises or falls. In the three-dimensional geometry, it is predicted that the elastic waves extend ahead of the submerged source, decaying algebraically in space. The form of these elastic waves is computed, and validated by comparison with numerical computations of the elastic sheet behaviour.
Effects of a single spanwise surface wire on a free-ended circular cylinder undergoing vortex-induced vibration in the lower synchronization range
E. Vaziri, A. Ekmekci
This experimental study investigated the control induced by a spanwise surface wire on a rigid circular cylinder undergoing vortex-induced vibration (VIV) under the conditions of low mass damping in the lower synchronization branch. Being motivated by the idea of VIV-based energy harvesting from ocean and river flows, this elastically mounted cylinder was immersed in a water channel, leaving a free end at its bottom spanwise end, while the free water surface bounded its top. The cylinder was constrained to vibrate in the cross-stream direction. The wire diameter was 6.25 % of the cylinder diameter. Experimental research was conducted by attaching this large-scale wire along the span of the cylinder at various angular positions ranging from 0° to 180° (with respect to the most upstream point of the cylinder) at a fixed Reynolds number of 104 (based on the cylinder diameter). Simultaneous to measuring the trajectory of the cylinder motion via a laser distance sensor, the instantaneous velocity field in the near wake of the cylinder was obtained using particle image velocimetry. Several VIV response categories were identified depending on the angular position of the wire, which led to the classification of distinct angular ranges for the wire application. Associated with the structural vibrations in these categories, different vortex-formation modes induced by the wire were revealed. For specific wire positions, decreases of up to 98 % and increases of up to 102 % were identified in the oscillation amplitude of the cylinder compared with the amplitude of the clean cylinder under similar conditions.
The critical layer in quadratic flow boundary layers over acoustic linings
Matthew J. King, Edward J. Brambley, Renan Liupekevicius, Miren Radia, Paul Lafourcade, Tauqeer H. Shah
A straight cylindrical duct is considered containing an axial mean flow that is uniform everywhere except within a boundary layer near the wall, which need not be thin. Within this boundary layer the mean flow varies parabolically. The linearized Euler equations are Fourier transformed to give the Pridmore-Brown equation, for which the Green's function is constructed using Frobenius series. The critical layer gives a non-modal contribution from the continuous spectrum branch cut, and dominates the downstream pressure perturbation in certain cases, particularly for thicker boundary layers. The continuous spectrum branch cut is also found to stabilize what are otherwise convectively unstable modes by hiding them behind the branch cut. Overall, the contribution from the critical layer is found to give a neutrally stable non-modal wave when the source is located within the sheared flow region, and to decay algebraically along the duct as $O(x^{-{5}/{2}})$ for a source located with the uniform flow region. The Frobenius expansion, in addition to being numerically accurate close to the critical layer where other numerical methods lose accuracy, is also able to locate modal poles hidden behind the branch cut, which other methods are unable to find; this includes the stabilized hydrodynamic instability. Matlab code is provided to compute the Green's function.
Improved convergence of the spectral proper orthogonal decomposition through time shifting
Diego C.P. Blanco, Eduardo Martini, Kenzo Sasaki, André V.G. Cavalieri
Spectral proper orthogonal decomposition (SPOD) is an increasingly popular modal analysis method in the field of fluid dynamics due to its specific properties: a linear system forced with white noise should have SPOD modes identical to response modes from resolvent analysis. The SPOD, coupled with the Welch method for spectral estimation, may require long time-resolved datasets. In this work, a linearised Ginzburg–Landau model is considered in order to study the method's convergence. Spectral proper orthogonal decomposition modes of the white-noise forced equation are computed and compared with corresponding response resolvent modes. The quantified error is shown to be related to the time length of Welch blocks (spectral window size) normalised by a convective time. Subsequently, an algorithm based on a temporal data shift is devised to further improve SPOD convergence and is applied to the Ginzburg–Landau system. Next, its efficacy is demonstrated in a numerical database of a boundary layer subject to bypass transition. The proposed approach achieves substantial improvement in mode convergence with smaller spectral window sizes with respect to the standard method. Furthermore, SPOD modes display growing wall-normal and spanwise velocity components along the streamwise direction, a feature which had not yet been observed and is also predicted by a global resolvent calculation. The shifting algorithm for the SPOD opens the possibility for using the method on datasets with time series of moderate duration, often produced by large simulations.
Experimental investigation of three-dimensional modes in the wake of a rotationally oscillating cylinder
Soumarup Bhattacharyya, Izhar Hussain Khan, Shivam Verma, Sanjay Kumar, Kamal Poddar
Published online by Cambridge University Press: 17 October 2022, A10
Three-dimensionalities in the wake of flow past a circular cylinder executing sinusoidal rotary oscillations about its axis is studied experimentally. The results of water tunnel experiments on a rotationally oscillating cylinder for Reynolds number of 250 with varying amplitude and forcing frequency are discussed. Qualitative studies using hydrogen bubble and laser-induced fluorescence flow visualisation techniques are performed. Observation made for oscillating amplitude, $\theta _{0} = {\rm \pi}/4$ and $\theta _{0}=3{\rm \pi} /4$, and a low normalised forcing frequency, $FR$, of 0.75 and 0.5, respectively, confirmed a mode having a spanwise non-dimensional wavelength of $\sim$1.8 which is also observed for a rotating cylinder. On increasing forcing frequency this mode disappears and a new mode with a bean-shaped structure and a much smaller spanwise normalised wavelength of $\sim$0.8 appears at an $FR$ of 1 and an oscillation amplitude of ${\rm \pi} /2$. This mode remains almost stable until a forcing frequency of $FR=1.4$. At higher forcing frequency, $FR=2.75$, and oscillation amplitude of $3{\rm \pi} /4$, a mode with cellular structure and a normalised spanwise wavelength of $\sim$1.6 is identified. The cells in this mode flatten up with increasing downstream distance and are shed alternately with respect to the adjacent cell. Certain combinations of forcing parameters resulted in a forced two-dimensionality of the wake. Quantitative studies using hot-wire measurements and particle image velocimetry confirm the presence of these modes and wake characteristics. Wake mode map in the forcing frequency and amplitude plane is presented showing regions of newly discovered modes and wake lock-on boundaries.
Patterned convection in inclined slots
J.M. Floryan, A. Baayoun, S. Panday, Andrew P. Bassom
An analysis of laminar natural convection in inclined slots subjected to patterned heating has been performed. The imposed heating takes a simple form characterized by a single Fourier mode combined with uniform heating. It is shown that periodic heating applied at the lower plate produces no net flow when the slot is either horizontal or vertical, but a net upward flow is generated when the slot is tilted. Periodic heating applied at the upper plate produces net downward flow in the inclined situation. The addition of uniform heating promotes the upward flow while cooling has the opposite effect. There is a critical inclination angle at which the maximum net flow rate is greatest. Dynamic and thermal boundary layers are present when the wavenumber of the imposed heating is large. The use of heating at both plates, with the same wavenumber, leads to a flow dominated by the plate exposed to a more intense heating; when the two plates are heated equally no net flow is observed irrespective of the inclination angle. Changes of the relative positions of the two patterns can change the net flow rate by up to 50 %. The intensity of the flow increases with reduction of the Prandtl number. If the heating applied to the plates is of different wavelength, but of the same intensity, a wide range of behaviours of the flow system is possible. The details of this response are sensitive to the ratio of the two wavenumbers.
Three-dimensional buoyant hydraulic fractures: constant release from a point source
Andreas Möri, Brice Lecampion
Hydraulic fractures propagating at depth are subjected to buoyant forces caused by the density contrast between fluid and solid. This paper is concerned with the analysis of the transition from an initially radial fracture towards an elongated buoyant growth – a critical topic for understanding the extent of vertical hydraulic fractures in the upper Earth crust. Using fully coupled numerical simulations and scaling arguments, we show that a single dimensionless number governs buoyant hydraulic fracture growth, namely the dimensionless viscosity of a radial hydraulic fracture at the time when buoyancy becomes of order 1. It quantifies whether the transition to buoyancy occurs when the growth of the radial hydraulic fracture is either still in the regime dominated by viscous flow dissipation or already in the regime where fracture energy dissipation dominates. A family of fracture shapes emerge at late time from finger-like (toughness regime) to inverted elongated cudgel-like (viscous regime). Three-dimensional toughness-dominated buoyant fractures exhibit a finger-like shape with a constant-volume toughness-dominated head and a viscous tail having a constant uniform horizontal breadth: there is no further horizontal growth past the onset of buoyancy. However, if the transition to buoyancy occurs while in the viscosity-dominated regime, both vertical and horizontal growths continue to match scaling arguments. As soon as the fracture toughness is not strictly zero, horizontal growth stops when the dimensionless horizontal toughness becomes of order 1. The horizontal breadth follows the predicted scaling.
Extended kinetic theory for granular flow in a vertical chute
Mudasir Ul Islam, J. T. Jenkins, S. L. Das
We consider steady, fully-developed flows of deformable, inelastic grains driven by gravity between identical bumpy walls. Using constitutive relations from extended kinetic theory (EKT) for the erodible bed near the centreline and the collisional flow between the surfaces of the bed and the walls, we calculate the fields of mean velocity, fluctuation velocity and solid volume fraction across the chute. We consider both situations in which the solid volume fraction at and near the centreline is high enough to form a bed and when it is not. We compare results predicted by EKT with recent discrete element simulations results, and obtain very good agreement.
The Mott-Smith solution to the regular shock reflection problem
M.Yu. Timokhin, A.N. Kudryavtsev, Ye.A. Bondar
The classical Mott-Smith solution for one-dimensional normal shock wave structure is extended to the two-dimensional regular shock reflection problem. The solution for the non-equilibrium molecular velocity distribution function along the symmetry-plane streamline is obtained as a weighted sum of four Maxwellians. An analysis of applicability of the solution has been performed using the results of direct simulation Monte Carlo calculations for a range of incident shock wave intensities. Accuracy of the solution improves with increasing $Ma_n$, the Mach number normal to the shock front, so that the solution becomes rather accurate for strong shocks with $Ma_n>8$. | CommonCrawl |
Math 3001, Spring 2020
Analysis I
Instructor: Yuhao Hu
Email: [email protected]
Office: Math 225
Office Hours: WF 4:00-5:45pm
Lectures: MWF 9:00-9:50am at ECCR 110
In some sense, mathematical analysis was introduced as a new language of mathematics in order to bring rigor to the subject of calculus (e.g., to understand the meaning of "limit"). The development of this 'language', like the invention of a telescope, allows one to make more accurate observations, put forward precise questions and propositions, and construct useful theories and techniques. These new questions, theorems, etc., in turn, enrich not only the language but also the body of mathematics that's tied to it. Almost 200 years after Cauchy's 1821 book Cours d'analyse, analysis is now a 'universe' in itself.
This course is a beginning step towards understanding analysis. We start with a bit of set theory, followed by an introduction to the structure of real numbers. We develop language/tools that allow us to make sense of limit, which further allows us to take a more precise view of functions and their continuity. This course ends with basic theories of differentiation and integration.
T1. Understanding Analysis, 2nd ed. by Stephen Abbott
T2. (Optional) Analysis I, 3rd ed. by Terence Tao
(weekly, only the first meeting of the week is dated; "S" stands for "Section" in T1)
01/13 Number systems, sets (S1.1); functions, logic (S1.2)
01/22 Characterization of $\mathbb{R}$, axiom of completeness (S1.3); Consequences of completeness (S1.4)
01/27 Cardinality (S1.5); Cantor's theorem (S1.6); puzzles arising from infinite sums (S2.1)
02/03 Limit of a sequence (S2.2);
02/17 (Midterm I on Friday)
03/23-27 (Spring break; no class)
04/06 (Midterm II on Friday)
(Due on each Friday, unless specified otherwise.)
Homework 1 (Lectures 1-2) due Jan. 24
[Lecture 1]
Reading: pp. 1-7 (note: The default textbook is 'Understanding Analysis'.)
Exercises: [p. 11, Sec. 1.2: 1, 2, 3(a,c)]
Reading: pp. 7-11
Exercises: [p. 12, Sec. 1.2: 6(a,b), 8, 10, 11, 12]
Midterm I: 2/21 Friday
Midterm II: 4/10 Friday
Final: TBD
Two lowest Homework grades will be dropped.
Homework: 30% Midterm I: 20% Midterm II: 20% Final Exam: 30% | CommonCrawl |
The crustal stress field of Germany: a refined prediction
Steffen Ahlers ORCID: orcid.org/0000-0001-9094-01161,
Luisa Röckel2,
Tobias Hergert1,
Karsten Reiter1,
Oliver Heidbach3,4,
Andreas Henk1,
Birgit Müller2,
Sophia Morawietz3,4,
Magdalena Scheck-Wenderoth5,6 &
Denis Anikiev5
Geothermal Energy volume 10, Article number: 10 (2022) Cite this article
Information about the absolute stress state in the upper crust plays a crucial role in the planning and execution of, e.g., directional drilling, stimulation and exploitation of geothermal and hydrocarbon reservoirs. Since many of these applications are related to sediments, we present a refined geomechanical–numerical model for Germany with focus on sedimentary basins, able to predict the complete 3D stress tensor. The lateral resolution of the model is 2.5 km, the vertical resolution about 250 m. Our model contains 22 units with focus on the sedimentary layers parameterized with individual rock properties. The model results show an overall good fit with magnitude data of the minimum (Shmin) and maximum horizontal stress (SHmax) that are used for the model calibration. The mean of the absolute stress differences between these calibration data and the model results is 4.6 MPa for Shmin and 6.4 MPa for SHmax. In addition, our predicted stress field shows good agreement to several supplementary in-situ data from the North German Basin, the Upper Rhine Graben and the Molasse Basin.
The prediction of the recent crustal stress field is important for many applications regarding the exploitation or use of the subsurface, particularly for directional drilling, stimulation and exploitation of geothermal or hydrocarbon reservoirs. Another currently important application is the search for and long-term safety assessment of a high-level nuclear waste deposit. However, up to now the knowledge of the crustal stress field for Germany is limited. It is essentially based on two major databases regarding stress tensor orientations and stress magnitudes (Heidbach et al. 2016; Morawietz et al. 2020), several 2D numerical models (Grünthal and Stromeyer 1994; Marotta et al. 2002; Kaiser et al. 2005; Jarosiński et al. 2006; Cacace et al. 2008) and some regional scale 3D geomechanical–numerical models (Buchmann and Connolly 2007; Heidbach et al. 2014; Hergert et al. 2015; Ziegler et al. 2016). The only large scale 3D model that covers entire Germany has been presented by Ahlers et al. (2021a). However, this model focuses on the large-scale stress pattern in the entire crust with low resolution in sediments that are represented with homogeneous mean rock properties.
To take a further step towards a robust prediction of the recent crustal stress state, we developed a geomechanical–numerical model of Germany based on Ahlers et al. (2021a). It provides a continuous prediction of the crustal stress in 3D with focus on the sedimentary basins. The work of Ahlers et al. (2021a) also provided a continuous description of the stress state of Germany but focused on basement structures and included a homogenous sedimentary layer without mechanical stratification in a coarse resolution. The model presented here has been significantly improved with a differentiated sedimentary layer, consisting of 15 units with specific material properties (density, Poisson's ratio and Young's modulus) and an 18-time higher resolution in the upper part of the crust. An improved differentiation of the sedimentary layer is essential, since the majority of applications focuses on sedimentary basins particularly for geothermal and hydrocarbon exploitation. At the same time, stress conditions within sedimentary units can be particularly challenging due to structural, lithological and mechanical variability. Mechanical properties varying with depth—mainly stiffness contrasts—can lead to differing stress magnitudes, differential stresses and perturbations in the orientation of the maximum horizontal stress (SHmax) (Cornet and Röckel 2012; Heidbach et al. 2014; Hergert et al. 2015). Extreme cases include very weak layers of salt or clay, leading to a nearly lithostatic stress state, which can mechanical decouple the overburden from the underburden layers (Roth and Fleckenstein 2001; Röckel and Lempp 2003; Heidbach et al. 2007; Ahlers et al. 2019). Furthermore, our new model is calibrated with minimum horizontal stress (Shmin) and SHmax magnitudes which significantly improve the reliability of the predicted stress state compared to the model Ahlers et al. (2021a), which could only be calibrated with Shmin values.
We assume linear elasticity and use the finite element method to solve the partial differential equations of the equilibrium of forces. First, an appropriate initial stress state is defined representing an undisturbed state of stress governed by gravity. In a second step, the stress state in the model is calibrated with magnitude data by varying displacement boundary conditions defined at the model edges. This modeling approach has been used for different tectonic settings and scales and is described in detail in Buchmann and Connolly (2007), Hergert and Heidbach (2011), Heidbach et al. (2014), Reiter and Heidbach (2014), Hergert et al. (2015) and Ahlers et al. (2021a). For the construction and discretization of the 3D model geometry and the assignment of rock properties to individual finite elements the software packages GOCAD™, HyperMesh™ and ApplePY (Ziegler et al. 2019) are used. As solver, we use the commercial finite element software package Abaqus™ v2019. For post-processing Tecplot 360™ enhanced with the GeoStress add-on (Stromeyer and Heidbach 2017) is used.
Geology of the study area
The diverse history of the model area lead to the complex geological structure observed today (Fig. 1c, d). The upper crust can be subdivided into four parts: the East European Craton (EEC) in the northeast amalgamated with Avalonia further south during the Caledonian orogeny, the Armorican Terrane Assemblage (ATA) added during the Variscan orogeny and finally the Alp–Carpathian–Pannonian (ALCAPA) part as a result of the Alpine orogeny (Ziegler and Dèzes 2006; McCann 2008; Linnemann and Romer 2010). Since these units are identical to Ahlers et al. (2021a) we refer to this publication for further information. The basement of the model area is mainly covered by late Paleozoic to Cenozoic sediments, with the exception of the Rhenish and Bohemian massifs commonly interpreted as long-lived highs (e.g., Eynatten et al. 2021), the Alpine mountain chain and parts of the Mid German Crystalline High (MGCH) and Moldanubian Zone (MDZ). During the final phase of the Variscan orogeny from the late Carboniferous to Permian time the model area was affected by extension leading to the origin of several basins filled up with debris of the eroding Variscan orogeny and contemporaneous volcanic activity mainly located in NE Germany (McCann et al. 2008; Scheck-Wenderoth et al. 2008). Due to the large amount of mostly reddish clastic and volcanic rocks deposited, this time period is called Rotliegend. The largest basin developed was the Southern Permian Basin with a maximum extent of ~ 1700 km covering large areas of the northern model area (Stollhofen et al. 2008). During the late Permian, this basin was flooded from the north leading to the deposition of the so-called 'Zechstein' evaporites (McCann et al. 2008). The following Triassic development of the model area was controlled by the breakup of Pangea and the westward opening of the Tethys leading to an E–W dominated extensional tectonic regime and the development of N–S-oriented graben systems mainly in the north, e.g., the Central Graben or the Glückstadt Graben (Fig. 1c) (Scheck-Wenderoth and Lamarche 2005; Kley et al. 2008). During this time period the southern 'Alpine' part of the model area was characterized by open marine conditions of the Tethys shelf, whereas continental to shallow marine conditions by repeated incursions of the Tethys dominated the northern 'Germanic' domain (Feist-Burkhardt et al. 2008). The Jurassic was dominated by the progressive breakup of Pangea and mostly marine conditions (Pienkowski et al. 2008). Central Europe was still affected by extensional tectonics but the extension direction changed to NW–SE during the Late Jurassic (Kley et al. 2008). During the Early Cretaceous this development continued with depocenters in the northern part of the model area (Voigt et al. 2008). However, the deposition of sediments was restricted to these depocenters that evolved as en echelon subbasins along the southern margin of the Permian Basin in a transtensional regime (Scheck-Wenderoth et al. 2008).The largest part of the former Permian Basin area in the northern domain of the model area was uplifted during Late Jurassic to Early Cretaceous times. In the Late Cretaceous the tectonic setting and depositional conditions changed. Due to an eustatic sea-level rise large parts of Central Europe were flooded and predominantly carbonates and sandstones were deposited (Scheck-Wenderoth et al. 2008). The rotation of Iberia reversed the tectonic regime leading to the inversion of former depocenters, the formation of thrust faults and basement uplifts, e.g., the Harz mountains (Kley et al. 2008). Additional processes for the Late Cretaceous to Paleogene exhumation are still discussed (Eynatten et al. 2021). The Cenozoic development of the model area was mainly influenced by the collision of Africa and Eurasia leading to the rise of the Alpine mountain chains and the evolution of the Molasse Basin (MB). In addition, the Cenozoic Rift System developed, e.g., the Upper Rhine Graben (URG) and the Lower Rhine Graben (LRG) (Ziegler and Dèzes 2006) and the uplift of the Rhenish Massif began (Reicherter et al. 2008). Except the sedimentary basins of the MB, the URG, the LRG and the North German Basin (NGB) large parts of the model area were affected by erosion (Rasser et al. 2008). The Cenozoic tectonics north of the Alps were accompanied by various volcanic activities, e.g., the Vogelsberg Complex, the Eifel, Ohře Graben or in the vicinity of the URG (Litt et al. 2008; Reicherter et al. 2008).
Overview of the geology of the model region and stress data used for comparison and calibration. The black frame shows the key area of the model with increased stratigraphic resolution based on Anikiev et al. (2019). a Distribution of the stress magnitude data of Morawietz and Reiter (2020). The color-coded dots indicate the quality assigned by Morawietz et al. (2020). Displayed are all data from a true vertical depth (TVD) > 200 m. b Data of the World Stress Map (WSM) of Heidbach et al. (2016) and Levi et al. (2019). Color-coded lines indicate the stress regime and the orientation of SHmax. Grey lines show the mean orientation of SHmax on a regular grid (Details see chapter 3.1). c Geological framework of the model area based on Asch (2005) and Kley and Voigt (2008). BF—Black Forest, CG—Central Graben, EI—Eifel, HG—Horn Graben, HZ—Harz Mountains, GG—Glückstadt Graben, LRG—Lower Rhine Graben, OD—Odenwald, OG—Ohře Graben, SNB—Saar–Nahe Basin, SP—Spessart, TW—Tauern Window, URG—Upper Rhine Graben, VB—Vogelsberg Complex, VG—Vosges. d Overview of the basement structures based on Kroner et al. (2010) and Brückl et al. (2010). The red frame shows the entire model area. ALCAPA—Alps–Carpathian–Pannonian, ATA—Armorican Terrane Assemblage, EEC—East European Craton
Model geometry
All basic information about the model area and geometry is described in Ahlers et al. (2021a). However, to refine the resolution w.r.t. to the former model we subdivide the sediment layer. The subdivision is based on the 3D-Deutschland (3DD) model of Anikiev et al. (2019) and is, therefore, only resolved in the key area of our model (Fig. 2). We did not extend the higher stratigraphic resolution to the whole model area, since the data availability is poor and our focus area is covered by the 3DD data. Thus, the sedimentary units outside of the 3DD area are combined to a single unit. Furthermore, the stress data used for calibration are mainly located within the 3DD model area. The geometry data include a gap between the base of the Rotliegend—the deepest layer almost completely contained in the 3DD model—and the surface of the crystalline basement of Ahlers et al. (2021a). All lithological units between are represented by one unit named Pre-Permian unit though we aware that this is a very heterogeneous unit comprising early- to mid-Paleozoic low-grade metamorphic sediments and late-Paleozoic sediments.
Overview of the discretized model showing the internal structure including 22 units. ATA—Armorican Terrane Assemblage, ALCAPA—Alpine–Carpathian–Pannonian, EEC—East European Craton, NGB—North German Basin, URG—Upper Rhine Graben
Model discretization
The average lateral resolution of the model is 2.5 km and constant over the entire depth range of the model, the vertical resolution varies with depth. The mesh is divided into three vertical zones with a decreasing resolution with increasing depth. The deepest and most coarsely resolved unit is the lithospheric mantle limited by the bottom of the model and the Mohorovičić discontinuity with five element layers leading to a vertical resolution of 10 to 15 km. The mesh of the crust is subdivided at 10 km depth. Below 10 km depth the mesh contains ten element layers with a vertical resolution between 2.5 and 4 km. Above 10 km depth the mesh contains 43 element layers with a vertical resolution of about 240 m. Overall, the mesh contains about 11.1 million hexahedral elements.
Due to the complex geometry of the individual layers, especially of the sedimentary units, we did not create an individual mesh for each of them. We use ApplePY (Ziegler et al. 2019) to assign the individual mechanical properties of each unit to the finite elements of the mesh. Therefore, apart from the lithospheric mantle, which is the deepest meshed zone, the geometry of the individual units are not directly represented by the mesh. The final model consists of 22 units (Table 1): the lithospheric mantle, the lower crust, four units of the upper crust (EEC, Avalonia, ATA, ALCAPA), the Pre-Permian unit, Rotliegend volcanics and sediments, Zechstein carbonates and salt, Mesozoic units of the Triassic, Jurassic and Cretaceous, thrusted units of the Alps, folded units of the MB, three lateral subdivided units of Cenozoic sediments (NGB, URG and MB), Cenozoic volcanics and the sediments outside the 3DD area. In addition, there are some relict elements (Table 1, Layer ID: 2) which arise from the element assignment with ApplePY if the surface of the mesh does not fit perfectly to the surface defined by the geometry used. This occurs, because the geometry of the 3DD model has a higher resolution (1 × 1 km) than the mesh of our model (~ 2.5 × 2.5 km).
Table 1 Overview of all units defined in the model and parameters used
Individual material properties are assigned to all 22 units of the model (Table 1). The density and the elastic properties (Young's modulus and Poisson's ratio) are defined as material properties for the numerical calculation, i.e., linear-elastic material behavior is assumed. Friction angle, cohesion and tensile strength are only used for post-processing analysis later on. Although our model contains 22 units, the Triassic and Jurassic units still contain mechanically very different subunits. For example, the Triassic unit comprises the sandstone-dominated Buntsandstein, the carbonate- and evaporate-dominated Muschelkalk and the claystone-dominated Keuper (Feist-Burkhardt et al. 2008). Therefore, we calculated the arithmetic mean of the subunits for the mechanical properties of these two units. Another challenge was the parametrization of the thrusted units of the Alps, the sediments outside the 3DD area and the relict elements unit. Since the relict elements can only occur at the top of the model, we used an average value of the three Cenozoic sedimentary units (NGB, URG and MB). For the sediments outside of the 3DD model area, we have chosen roughly estimated values. For the thrusted units of the Alps, we decided to use the mean of the Mesozoic units with the exception of the density. As for almost all other units, we use the density values from the 3DD model of Anikiev et al. (2019). The elastic properties are mainly based on the P3 database (Bär et al. 2020) and Hergert et al. (2015). In addition, the friction angle and cohesion are mainly based on the latter. The tensile strength is assumed to be 5 MPa for almost all units, since available data are limited. In general, all values in Table 1 without a reference are roughly estimated. The friction angle, cohesion and tensile strength of the Zechstein salt are not defined due to the visco-elastic properties of salt (e.g., Urai et al. 2008). In addition, we defined a Young's modulus gradient for all units with the exception of the sediments outside 3DD as it led to convergence problems at the model edges. The Young's modulus gradient should mimic the tendency of the rock mass to strengthen with increasing depth due to compaction and increasing confining pressure. An effect that can be seen, for example, in the decrease of porosity and permeability with increasing depth (Ingebritsen and Manning 1999). Our Young's modulus gradient reaches from 1.5 km depth to the surface, describing a reduction of the Young's modulus from 100% (of the values in Table 1) to 10% at the surface. The reduction to 10% is derived from Hudson and Harrison (1997) and the depth of the gradient was iterated by preliminary tests.
Initial stress state
An initial stress state is established, describing an only gravity-driven undisturbed, non-tectonic stress field within the upper crust before the displacement boundary conditions are applied. To achieve such an initial stress state, we extend our model with a sideburden (dark blue), an underburden (light blue) and a stiff shell (green) (Fig. 3a, b). The shell has a conic shape with a theoretical intersection point at the center of the earth emulating the naturally increasing confining pressure with depth (Zang and Stephansson, 2010). The elastic properties (Young's modulus and Poisson's ratio) of the sideburden and underburden are the adjusting screws to set a best-fit initial stress state. For calibration we use a semi-empirical function of Sheorey (1994) describing the undisturbed stress state of the earth as stress ratio (k):
Top (a) and side view (b) of the extended model box used to achieve an initial stress state. c Best-fit of 29 virtual wells (blue curves) and blue dots in a in comparison to calculated curves (red) for a Young's modulus of 30 and 70 GPa based on Sheorey (1994). There is a change of the scale of the depth axis below 7000 m TVD
$$k=\frac{{S}_{\mathrm{Hmean}}}{{S}_{V}}=\frac{{S}_{\mathrm{Hmax}}+{S}_{\mathrm{hmin}}}{{2S}_{V}}$$
depending on depth (z) and Young's modulus (E):
$$k=0.25+7\text{E}\left(0.001+\frac{1}{z}\right)$$
we compare the theoretical k values with k values of our model using 29 virtual wells up to 25 km true vertical depth (TVD) (blue dots in Fig. 2a). Subsequently, we vary the gravity-driven settlement of the model by varying the Young's modulus of the underburden and increase or decrease the influence of SV on the horizontal stresses by varying the Poisson's ratio within the model and the sideburden. This workflow has been used and described in detail by several authors before, e.g., Buchmann and Connolly (2007), Hergert (2009), Hergert and Heidbach (2011), Reiter and Heidbach (2014) or Ahlers et al. (2021a).
Our best-fit regarding the theoretical stress state of Sheorey (1994) is displayed in Fig. 3c. There is a change of scale along the depth axis below 7000 m TVD, since a uniform stress state and an almost perfect fit to the theoretical curve (red curve) is achieved at greater depths when all wells reach the upper crustal units (Layer ID: 18–21) with a homogeneous Young's modulus of 70 GPa. About half of the calibration wells (blue curves) show an almost vertical progression within the upper 1500 m, while the other half follows the progression of the theoretical curve of Sheorey (1994) to higher k values with decreasing depth. This effect occurs due to the Young's modulus gradient defined within the main area of our model up to 1500 m TVD. Figure 3c includes the function of Eq. (2) for a Young's modulus of 30 GPa typical for the sediments and a Young's modulus of 70 GPa representing crystalline basement units.
Displacement boundary conditions and calibration
After the initial stress state is reached, the model is calibrated with measured in-situ stress data using variable displacement boundary conditions. The bottom of the model is fixed vertically, horizontal displacements are allowed and the model surface is free. Displacement boundary conditions are defined at the five vertical model edges (Fig. 4). The directions of displacements are predefined by the mean SHmax orientation derived from the World Stress Map (WSM, Heidbach et al. 2016) data (Ahlers et al. 2021a). At the northern and southern edges, where SHmax is oriented perpendicular to the model boundary, compression is applied. Accordingly, extension is applied at the western and eastern edges, where SHmax is parallel to the model boundary. The magnitudes of the displacement boundary conditions are derived through a calibration process using in-situ stress magnitudes from the database of Morawietz and Reiter (2020). Since SV is based almost entirely on density and is not influenced by displacement boundary conditions applied, only the SHmax and Shmin magnitudes are used for calibration. We use only data qualities from A to C and values from TVDs > 200 m to minimize possible topographical effects. Thus, in total 73 Shmin and 56 SHmax magnitudes from 200 to ~ 4700 m TVD are available from the database of Morawietz and Reiter (2020). Since the calibration data are unevenly distributed with depth (Figs. 6 and 8) a depth-weighted median for depth intervals of 500 m is used as decisive calibration value. The best-fit is achieved with a total shortening of 442 m in N–S direction and an extension of 560 m in E–W direction (Fig. 4).
Displacement boundary conditions applied at the discretized model. Arrows indicate direction of the displacements applied, numbers the magnitudes of displacements defined for the best-fit
Orientation of S Hmax
The orientations of SHmax in comparison to the mean orientation of SHmax derived from the WSM (Heidbach et al. 2016) and additional data by Levi et al. (2019) are displayed in Fig. 5. Figure 5a shows the modelled SHmax orientation at 5000 m TVD (red lines) and the mean SHmax orientation (black lines) with standard deviation (grey wedges) based on the WSM data. The angular deviation between these two are displayed as a color plot (Fig. 5b) and a histogram (Fig. 5c). The mean orientation of SHmax and standard deviation based on the WSM data is calculated using the stress2grid script of Ziegler and Heidbach (2019b) that uses the statistics for circular data (Mardia, 1972). Input data within a 200 km search radius are weighted by data quality and distance to a point of the 0.5° × 0.5° grid (Ziegler and Heidbach, 2019a). Furthermore, at least 10 data records—with a quality of A to C—within the search radius must be available to return a mean SHmax orientation. In addition, data from the NGB within or above the Zechstein salt are sorted out to avoid effects due to salt decoupling (e.g., Roth and Fleckenstein 2001; Röckel and Lempp 2003 and Heidbach et al. 2007, since visco-elastic properties are not included in our model.
Orientation of SHmax predicted by the model in comparison to the mean orientation of the WSM (Heidbach et al., 2016) and Levi et al. (2019). Details are described in the text. a Model results at 5 km TVD (red lines), the mean orientation derived from the WSM (black lines) with standard deviation (grey wedges). b Color-coded deviation between the results and the mean orientation of the WSM. Blue indicates an anti-clockwise rotation of the model results and orange a clockwise rotation of the model results w.r.t the mean orientation of the WSM. c Histogram showing the deviation displayed in a and b
As shown in Fig. 5a, our model predicts an almost homogeneous NNE–SSW orientated stress pattern. The median deviation of 0.3° indicates an overall good fit to the mean WSM data. Furthermore, almost all results are within the standard deviation of the mean WSM data. However, Fig. 5a, b indicates several model regions with significant deviations. For example, the local perturbations in the southern part of the model area where NW–SE orientations of SHmax dominate in the west and N–S orientations in the east or the NW–SE orientations in central Germany. In addition, the histogram (Fig. 5c) shows an uneven distribution of the deviation.
Stress magnitudes and absolute stress state
Magnitudes of S hmin
The magnitudes of Shmin predicted by the model are displayed in comparison to calibration data by Morawietz and Reiter (2020). The differences are calculated as model results minus calibration data. Thus, positive differences indicate too high values predicted by the model and negative differences too low values. The differences in Fig. 6a, c are color-coded regarding their qualities. We use 73 values, from TVDs > 200 m, with a quality of A to C from twelve localities mainly located in south Germany with the exception of a data record from Peckensen (Röckel and Lempp 2003). The localities are displayed in Fig. 6b with numbers indicating multiple magnitudes from different depths at one location. In general, the fit regarding the Shmin magnitudes of Morawietz and Reiter (2020) is good with differences in the range of − 20 to 7.5 MPa and a mean of the absolute differences of 4.6 MPa. Figure 6a shows a depth trend from negative differences in the upper 1500 m indicating too low magnitudes predicted by the model to positive differences in greater depths indicating slightly too high magnitudes predicted. Due to the large amount of data in the upper 1000 m this leads to the unweighted median of − 3.7 MPa and skewed histogram with a peak at − 2.5 to − 5 MPa. Therefore, we decided to use a depth-weighted median during the model calibration (chapter 2.7). A dependency on data qualities is not recognizable.
Shmin magnitudes of the model in comparison to data of Morawietz and Reiter (2020) used for calibration. a Color-coded differences versus depth. Differences are calculated as model results minus calibration data. b Spatial distribution of stress magnitude data, used for calibration. Numbers indicate localities with multiple data records. c Color-coded histogram of the differences displayed in a
The depth sections displayed in Fig. 7 show the Shmin magnitudes at 500, 1500, 2500 and 5000 m TVD. The Shmin at 500 m TVD shows a homogeneous stress distribution with values mainly between 0 and 10 MPa. Only the northern and southern model edges show some larger magnitudes with the exception of the Tauern Window showing lower stresses equal to the main part of the model section. The section at 1500 m shows a less homogeneous stress distribution with dominant values between 10 and 30 MPa. The highest values, up to 70 MPa in the southeast, are again, related to the model boundaries. The lowest values (0 to 10 MPa) occur along the border between the Rhenoherzynian zone (RHZ) and the MGCH (Fig. 1d) and in the western part of the Tauern Window. Striking are the lower values (10 to 20 MPa) within the area of the MDZ and the MGCH. Sine, at 2500 m TVD these regions show the same magnitudes as the adjacent regions, in general, a homogeneous distribution with values between 30 and 40 MPa. The lowest values with 20 to 30 MPa again occur in the western part of the Tauern Window and at the border between RHZ and MGCH. The highest values are associated with the model edges. At 5000 m TVD, the stress distribution within the area of the MDZ and the MGCH seems to be inverted in comparison to the distribution at 1500 m TVD, since the stress magnitudes are larger than in the adjacent areas, e.g., the Saxothuringian zone (SXZ). In general, the four depth sections of the model show a quite homogeneous distribution of Shmin indicated by a maximum range of 20 to 30 MPa for each depth section, except for the model edges.
Depth sections showing the lateral distribution of the Shmin magnitudes predicted by the model at four depth sections. TVD—true vertical depth. Black titles (details in Fig. 1d): MDZ—Moldanubian Zone, MGCH—Mid German Crystalline High, RHZ—Rhenoherzynian Zone, SXZ—Saxothuringian Zone. White title (details in Fig. 1c): TW—Tauern Window
Magnitudes of S Hmax
Figure 8 shows the SHmax magnitudes of the model in comparison to the calibration data of Morawietz and Reiter (2020). Differences displayed in Fig. 8a and c are calculated as model results minus calibration data. Thus, positive differences indicate too high values of SHmax predicted by the model and negative differences too low SHmax magnitudes predicted. The differences are color-coded depending on their qualities defined in the magnitude database by Morawietz et al. (2020). The localities of the data used are displayed in Fig. 8b. All data used, with a quality of A to C and from a TVD > 200 m are located at eight localities in southern Germany. The differences versus depth (Fig. 8a) and the histogram (Fig. 8c) show a uniform distribution with a scattering of ± 20 MPa. A dependency of depth or quality is not visible. The homogeneous fit between the model results and the calibration data is also indicated by the (unweighted) median of − 0.9 MPa, which is almost equal to the depth-weighted median of 0.0 MPa. However, we also use the weighted median for the calibration with the SHmax magnitudes as for the Shmin magnitudes to use a constant calibration value.
SHmax magnitudes of the model in comparison to data of Morawietz and Reiter (2020) used for calibration. a Color-coded differences versus depth. Differences are calculated as model results minus calibration data. b Spatial distribution of stress magnitude data, used for calibration. Numbers indicate localities with multiple data records. c Color-coded histogram of the differences displayed in a
To show the lateral distribution of the SHmax magnitudes four horizontal sections at 500, 1500, 2500 and 5000 m TVD are displayed in Fig. 9. At 500 m TVD, the magnitudes range from 0 to 30 MPa. The lowest magnitudes with 0 to 10 MPa are located within the MGCH in the vicinity to the RHZ and in the Tauern Window. The highest values of > 20 MPa are associated with the model edges in the north and south and the basement outcrops of the southwestern Rhenish Massif, the Bohemian Massif, the Vosges, the Black Forest, the Odenwald and the Spessart. The section at 1500 m TVD shows a more differentiated distribution of SHmax with values ranging from 20 to 60 MPa, with some exception along the model edges. The higher values are again associated with outcropping basement structures, such as the Bohemian Massif. Regions with lower SHmax magnitudes between 20 and 40 MPa are located in the MB, the URG, the Saar–Nahe Basin (SNB), the NGB and the northern part of the MGCH in the vicinity of the RHZ. The results at 2500 and 5000 m TVD confirm the trend of higher magnitudes within the areas belonging to basement structures and lower magnitudes belonging to sedimentary units. An exception is the southwestern part of the Rhenish Massif, where the magnitudes do not increase as much as within the areas of crystalline basements. Finally, at 5000 m TVD the higher SHmax magnitudes correspond to areas with outcropping or shallow lying crystalline basement structures and lower SHmax magnitudes to sedimentary or low-grade metamorphic units.
Depth sections showing the lateral distribution of the SHmax magnitudes predicted by the model at four depth sections. TVD—true vertical depth. Black titles (details in Fig. 1d): MDZ—Moldanubian Zone, MGCH—Mid German Crystalline High, RHZ—Rhenoherzynian Zone, SXZ—Saxothuringian Zone. White titles (details in Fig. 1c): BF—Black Forest, M—Massif, OD—Odenwald, SNB—Saar–Nahe Basin, SP—Spessart, TW—Tauern Window, URG—Upper Rhine Graben, VG—Vosges
Magnitudes of S V
The differences between SV magnitudes and the data by Morawietz and Reiter (2020) are displayed in Fig. 10. They are calculated as model results minus data of the magnitude database. Thus, too high model results lead to positive differences, too low model results to negative differences. Since the SV magnitudes depend almost entirely on the density and are not influenced by displacement boundary conditions applied, we did not use these data for calibration but for validation to check if the densities chosen are reasonable. We use 71 values of Morawietz and Reiter (2020) from twelve localities (Fig. 10b). As the results show, the data used from the 3DD model (Anikiev et al. 2019) are very appropriate. With the exception of two values at 750 and 3700 m TVD all differences are in the range of − 2.5 to 2.5 MPa, resulting in a mean of the absolute differences of 1.1 MPa and a median of 0.0 MPa.
Fig. 10
SV magnitudes of the model in comparison to data of Morawietz and Reiter (2020). a Color-coded differences versus depth. Differences are calculated as model results minus data of Morawietz and Reiter (2020). b Spatial distribution of stress magnitude data, used for comparison. Numbers indicate localities with multiple data. c Color-coded histogram of the differences displayed in a
Regime Stress Ratio
To indicate the stress regime predicted by the model, the Regime Stress Ratio (RSR) for four model sections at 500, 1500, 2500 and 5000 m TVD is shown in Fig. 11. The RSR is a unitless value between 0 and 3 describing seven stress states defined by Simpson (1997): radial extension (0), pure normal faulting (0.5), transtension (1), pure strike-slip (1.5), transpression (2), pure reverse faulting (2.5) and constriction (3). The RSR (Eq. 3) is derived from the regime index n (Eq. 4, Anderson 1905) and the ratio of stress differences ϕ (Eq. 5, Angelier 1979):
Four depth sections showing the lateral distribution of the Regime Stress Ratio (RSR) predicted by the model. TVD—true vertical depth. Black titles (details in Fig. 1d): MDZ—Moldanubian Zone, MGCH—Mid German Crystalline High, RHZ—Rhenoherzynian Zone, SXZ—Saxothuringian Zone. White titles (details in Fig. 1c): GG—Glückstadt Graben, M—Massif, SNB—Saar–Nahe Basin, URG—Upper Rhine Graben
$$\mathrm{RSR}=\left(n+0.5\right)+{\left(-1\right)}^{n}\left(\phi -0.5\right)$$
$$n=\left\{\begin{array}{c}0 \quad { S}_{\text{hmin}}<{S}_{\text{Hmax}}<{S}_{V}\\ 1 \quad {S}_{\text{hmin}}<{S}_{V}<{S}_{\text{Hmax}}\\ 2 \quad { S}_{V}<{S}_{\text{hmin}}<{S}_{\text{Hmax}}\end{array}\right.$$
$$\phi =\frac{\left({\sigma }_{2}-{\sigma }_{3}\right)}{\left({\sigma }_{1}-{\sigma }_{3}\right)}$$
The calculated RSR values at 500 m TVD show a very inhomogeneous distribution, displaying the whole range from thrust to normal faulting. Very high values—indicating a thrust faulting regime—can be found, e.g., at the model edges. Low values, indicating a normal faulting regime, occur, e.g., in the SNB, the URG, the NGB and in the MB. The model section at 1500 m TVD shows a more homogeneous pattern than at 500 m TVD. RSR values larger than 2 occur still at the model edges and within the area of the Glückstadt Graben. The lowest values still occur in the SNB, the URG, the NGB and some areas within the MB. This trend continues at 2500 m TVD. Here, almost the entire SNB show values < 0.5 indicating a pure normal faulting regime. As at 1500 m TVD, also the NGB and the southern part of the MB show a normal faulting regime and additionally also the SXZ. In contrast to the 1500 m TVD section, several regions show lower values at 2500 m TVD. The deepest model section shown at 5000 m TVD shows a much more homogeneous distribution than at 500 m with large areas indicating a normal faulting regime. The trends described for 2500 m TVD continues, except within the URG. Up to a TVD of 2500 m the RSR in this area is lower in comparison to the surrounding areas, but at 5000 m TVD the RSR is higher.
Stress gradients
Figure 12 shows modeling results from three major sedimentary basins in Germany, the NGB, the URG and the MB, in comparison to available data partially also used for calibration. Results and data of the upper 200 m TVD are not displayed to avoid showing data influenced by topography or free surface effects. The model results are displayed as magnitude sets (SV, Shmin and SHmax) from virtual wells up to 8 km TVD located at the locations from which measurement data are shown. This results in a compilation of 12 magnitude sets for the MB, five magnitude sets for the URG and 45 magnitude sets for the NGB.
Results from multiple virtual wells in the corresponding sedimentary basin of the model in comparison to measured and calculated magnitudes of SV, Shmin, and SHmax. The uncertainties of the magnitudes if specified are displayed as error bars. North German Basin (NGB): Röckel and Lempp (2003), Fleckenstein et al. (2004), Stöckhert et al. (2013). Upper Rhine Graben (URG): Cornet and Burlet (1992), Klee and Rummel (1993), Valley and Evans (2007), Häring et al. (2008), Meixner et al. (2014), Azzola et al. (2019). Molasse Basin (MB): nagra (2001), Seithel et al. (2015), Backers et al. (2017), Budach et al. (2017), Drews et al. (2019), Garrard et al. (2021)
The results for the NGB show quite uniform magnitudes for all 45 magnitude sets displayed. The modeled Shmin magnitudes show the smallest range of values with a maximum range of ~ 10 MPa except for two outliers below ~ 5000 m TVD. The range of the SHmax magnitudes are larger with an average range of ~ 15 MPa, a maximum of ~ 25 MPa at ~ 5000 m TVD and again two outliers. The SV magnitudes show a trend of increasing scattering with depth, starting with a range of < 5 MPa at 200 m TVD to a range of ~ 20 MPa at 8 km TVD.
The five data sets from the URG show similar gradients for Shmin and SV but significant differences for the SHmax magnitudes. The SHmax magnitudes are also the only ones that show significant changes of the gradients with depth. While the Shmin and SV magnitudes show an almost linear increase with depth, some SHmax magnitudes between 2000 and 3500 m increase sharply and some even show decreasing or constant values. The compilation of the MB shows the most inhomogeneous results. At about 1000 m TVD, the magnitudes of SHmax split in two groups. The magnitudes of three virtual wells show an increase to ~ 40 MPa at 2500 m TVD in contrast to nine virtual wells, which show continuous SHmax magnitudes. The SV and Shmin values of these three virtual wells also increase but less obviously to 5 to 10 MPa higher values. With increasing depth at about 2500, 3200, 3700, 4200, 5000 and 6000 m TVD, other virtual wells also show such increasing magnitudes and converge to the higher magnitude trend. In general, there seems to be a lower and a higher magnitude gradient with a transition zone between 1000 and 6500 m TVD changing from one dominating gradient (0–1000 m TVD) to another (> 6500 m TVD). At 8 km TVD, all magnitudes are roughly homogenous with Shmin magnitudes of 130 to 140 MPa, SHmax magnitudes of 180 to 200 MPa and SV magnitudes of 200 to 215 MPa.
Fracture Potential
As an additional result the Fracture Potential (FP) of four depth sections is displayed in Fig. 13. The FP based on Connolly and Cosgrove (1999) and Eckert and Connolly (2004) is a dimensionless value indicating how close to failure the stress state is. The calculation is described in detail by Heidbach et al. (2020). The FP is defined as actual maximum shear stress divided by the acceptable shear stress (Eq. 6):
Depth sections showing the lateral distribution of the Fracture Potential calculated for the model results. TVD—true vertical depth. Regions, where Zechstein salt occurs are left white. Black titles (details in Fig. 1d): MDZ—Moldanubian Zone, MGCH—Mid German Crystalline High, RHZ—Rhenoherzynian Zone, SXZ—Saxothuringian Zone. White titles (details in Fig. 1c): BF—Black Forest, M—Massif, SNB—Saar–Nahe Basin, TW—Tauern Window, VG—Vosges
$$\mathrm{FP}=\frac{actual \,maximum \,shear \,stress}{acceptable \,shear \,stress}$$
Therefore, an FP of > 1 indicates failure and lower FPs represent a stable state of stress. The actual maximum shear stress is calculated as mean of the maximum (σ1) and minimum (σ3) principal stress (Eq. 7):
$$actual\, maximum \,shear \,stress=\frac{1}{2}\left({\sigma }_{1}-{\sigma }_{3}\right)$$
The acceptable maximum shear stress is calculated as (Eq. 8):
$$acceptable \,shear \,stress=C cos\phi +\frac{1}{2}\left({\sigma }_{1}+{\sigma }_{3}\right) sin \phi$$
The FP is calculated with individual cohesion (C) and friction angle (φ) for each model unit (Table 1), except for the Zechstein salt unit, since salt behaves visco-elastic. Therefore, the regions where Zechstein salt occur are left white in Fig. 13. The results in Fig. 13 show a stable stress state with an FP between 0 and 0.6. The highest values of 0.5 to 0.6 occur at 1500 m depth. In general, there is an increase of the FP up to 1500 m TVD. With further increasing depth the FP decreases. At 500 m TVD relative low values are associated with sedimentary units as within the NGB, the SNB or the Mesozoic units in southern Germany. Higher values are mainly associated with outcropping basement units, for example within the Bohemian and Rhenish Massif, the Vosges and the Black Forest. This trend is also visible at 1500 m TVD with a generally higher FP. High values are associated with crystalline basement units, e.g., the MDZ, the MGCH or the Tauern Window, low values are predicted for the NGB, the SNB or the MB. At 2500 m TVD such a clear trend is not visible anymore, but a low FP is still located within in the SNB. An interpretation of the results from the NGB is difficult for this depth section, since the Zechstein salt unit is dominant here. At 5000 m TVD the NGB is still a region with a relative low FP. However, in contrast to the depth sections at 500 or 1500 m TVD some regions show opposite trends. For example, the Bohemian Massif with former relatively high values or the SNB with former relatively low values.
Although our results lie almost entirely within the standard deviation of the SHmax orientation derived from the WSM and some additional data of Levi et al. (2019) and a median of deviation of 0.3° suggest a very good fit, a closer look indicates some discrepancies (Fig. 5). The orientations of SHmax predicted by our model (Fig. 5a) show a homogenous NNW–SSE pattern with some small deviation to N–S but the mean SHmax orientation of the WSM show several regions with divergent patterns, e.g., within the eastern part of the NGB showing N–S to NNE–SSW orientations or the central part of Germany with dominant NW–SE orientations. These results indicate that our model does probably not include some relevant factors or our displacement boundary conditions applied are too simple to reproduce the pattern of the orientation of SHmax. The median deviation of 0.3° and the distribution of the histogram support the former since differing displacements, e.g., at the eastern and western edges would probably only shift the distribution as a whole. Implemented lateral stiffness contrasts do not seem to have a significant impact on the regional stress field, e.g., predicted by Grünthal and Stromeyer (1994), Marotta et al. (2002) or Reiter (2021), despite Young's modulus contrasts of > 50 GPa, e.g., at the southern edge of the SNB between weak Rotliegend sediments (15 GPa) and the stiff upper crust (70 GPa). However, laterally there are no contrasts of the Young's modulus in the upper crystalline crust. Thus, the softer units lie on a homogeneous and stiff block. This could be an explanation of the quite homogeneous orientations of SHmax of our model within in the sedimentary units but also for parts of the crust, since our model assumption of linear elasticity does not represents the ductile behavior of lower parts of the crust and the mantle (e.g., Stüwe 2007).
Processes not included in our model, which may also affect the stress field within the model region, are isostatic buoyancy effects in Scandinavia (Kaiser et al. 2005) or in the south due to erosion, deglaciation and potential slab break off below the Alpine chain (Przybycin et al. 2015; Sternai et al. 2019). In addition, the lithosphere–asthenosphere-boundary (Cacace 2008) or density contrasts in the lower crust, e.g., the Pritzwalk anomaly (Krawczyk et al. 2008) could explain the misfit in the northeast. Effects due to salt decoupling leading to a regional stress field below and a more local stress field above the Zechstein unit (Roth and Fleckenstein 2001; Röckel and Lempp 2003; Heidbach et al. 2007) should have no influence, since we did not use the data from the units above.
Absolute stress and stress regime
In general, the predicted Shmin magnitudes fit the values of Morawietz and Reiter (2020) quite well with differences between − 20 and + 7.5 MPa and a mean of the absolute differences of 4.6 MPa (Fig. 6) and an overall good fit to additional data in Fig. 12. However, the depth dependent differences (Fig. 6a) and the compilation of the URG and the MB (Fig. 12) indicate too low magnitudes of Shmin within the upper ~ 1500 m of the model. Since the shallow calibration data (Fig. 6b) and additional data of the MB (Fig. 12) are partly located in young sedimentary units, the Young's modulus in the model could possibly be too high for these units, despite the implemented Young's modulus gradient. Furthermore, unconsolidated sediments can behave in a visco-elastic manner (e.g., Chang and Zoback 1998; Zoback 2007), which the linear elastic properties of our model cannot represent. Both, a lower Young's modulus and visco-elastic properties would lead to higher Shmin magnitudes, since the Shmin magnitudes would approach the SV magnitudes. Missing visco-elastic properties can also explain the too low values at ~ 800 m TVD, data from Wittelsheim, in the southwestern URG (Fig. 12) measured in an evaporitic layer (Cornet and Burlet 1992). Another clue to explain the low values could be the geographic distribution of the data. Almost all calibration data used and the data of the MB indicating this trend are located in the southern part of Germany. Available high quality data from the northern part of Germany are sparse, but the data used for comparison do not confirm this trend (Fig. 12). A possible trend to slightly too high values in the lower part of the model indicated by the three deepest values in the general comparison (Fig. 6) cannot be confirmed by the results in Fig. 12.
The predicted SHmax magnitudes also show a good fit to the values of the magnitude database by Morawietz and Reiter (2020) (Fig. 8) despite a wider range of differences between -20 and + 20 MPa and a higher mean of the absolute differences of 6.4 MPa in comparison to the Shmin magnitudes. However, available SHmax magnitudes are usually calculated and not measured and, therefore, have larger uncertainties (Morawietz et al. 2020). Furthermore, calibration data from units which are parametrized with mean mechanical properties, e.g., the Triassic unit or from thin units which are numerically not sufficiently represented by our model resolution can explain the wider range. Since, differing stiffnesses have a significantly higher influence on SHmax magnitudes than on Shmin magnitudes (Fig. 12). A quantification of the influence of mean properties on the results is possible, e.g., using the HIPSTER tool of Ziegler (2021) but not feasible for such a large-scale model, since thousands of calculations would be necessary for such an estimation.
Although the SHmax magnitudes of our model do not show a general depth trend as described for the Shmin magnitudes, the differences in the upper 1000 m TVD show a distribution that is very similar to the one of the Shmin magnitudes within this depth range (Figs. 6 and 8). This does not fit to the assumption that the Young's modulus might be too high, because too low SHmax values rather indicate too low Young's modulus values. Incorporation of visco-elastic properties could increase both magnitudes if the SHmax magnitudes are lower than SV, which is partly the case indicated by a normal faulting regime in Figs. 11 and 12. Another possibility to increase both horizontal magnitudes at the same time is an increase of the vertical stress, a higher Poisson's ratio, a higher stress input due to the boundary conditions or a higher k ratio of the initial stress. Increasing stresses by increasing the density does not seem to make sense, since the fit with the SV magnitudes (Fig. 10) is almost perfect. A higher Poisson's ratio for different units within this specific depth interval only is difficult to explain. The third and fourth possibility, an increased shortening and reduced extension by the displacement boundary conditions or an increased k-ratio within the upper part of the model only, are difficult to implement and are difficult to explain from a tectonic point of view.
Another view inside the model is given by the compilation of virtual wells in Fig. 12. The magnitudes used for comparison from the NGB are mainly based on two data sets. A compilation of Röckel and Lempp (2003) from the eastern and central part of the NGB and a data set by Fleckenstein et al. (2004) from the western part of the NGB. The compilation of Röckel and Lempp (2003) only contains SV and Shmin magnitudes, the compilation of Fleckenstein et al. (2004) all three principal stresses. However, the data sets show some general differences. The data from Fleckenstein et al. (2004) show a near isotropic stress state, particularly well visible between 4750 and 5250 m where the stress magnitudes of Shmin, SHmax and SV are very close to each other. The smaller magnitudes of Shmin and also the larger SV magnitudes visible within this depth range are from Röckel and Lempp (2003). In general, the model results of all virtual wells displayed show a better fit to the data of Röckel and Lempp (2003). However, it is remarkable that the SHmax magnitudes contrary to the Shmin and SV magnitudes fit the data by Fleckenstein et al. (2004) quite well. The misfit between these data sets could be explained due to the different location within the NGB. Another possible reason for this discrepancy could be the different measurement methods, since the data by Fleckenstein et al. (2004) are based on core samples, while the data of Röckel and Lempp (2003) are in-situ measurements. However, despite these inconsistencies the overall fit of the model results to the displayed data within the NGB is good.
The fit of the second virtual well compilation for the URG in comparison to the predicted magnitudes is also good. Regarding the Shmin magnitudes, the results show only significant differences to the measured magnitudes at 800 m TVD and between 3500 and 4500 m TVD. The differences at ~ 800 m TVD, indicating ~ 10 MPa too low magnitudes, are probably a result of inappropriate material properties in the model for this area. As mentioned before, these data set is measured in an evaporitic layer at Wittelsheim (Cornet and Burlet 1992). However, our model does not include visco-elastic properties. Between 3500 and 4500 m TVD the model results indicate a trend to too high Shmin. However, the deepest data record at 5000 m TVD fits to the model results. The SHmax data in general show a wider range of values as the Shmin data. At ~ 800 and ~ 5000 m TVD our results indicate too low values. However, the value at 5000 m TVD from Basel (Häring et al. 2008) must be interpreted with care due to the very high uncertainty indicated by the error bar. The differences between 2000 and 3500 m TVD show significant deviations up to ~ 30 MPa but all data records are within the range of our predicted values. At ~ 2200 m TVD the lowest differences occur for all locations, except the results from Soultz-sous-Forets. Although both measured SHmax magnitudes shown at 2200 m TVD are from Soultz-sous-Forets (Klee and Rummel 1993) the model results from this region indicate too high values.. A possible explanation are altered granites in the upper part of the crystalline basement as described by Aichholzer et al. (2016) which are not well represented by our high Young's modulus of 70 GPa. A lower Young's modulus would possibly lead to lower SHmax and a better fit. However, the altered granites are only about 150 m thick and, therefore, not able to explain the full discrepancy. Remarkable are the SHmax magnitudes between 2500 and 3500 m TVD predicted for Bruchsal, showing a very low gradient leading to ~ 30 to 40 MPa lower values as predicted for the other locations. This trend can be explained by a thick Rotliegend layer within this region (GeORG-Projektteam 2013; Meixner et al., 2014), with a low Young's modulus of 15 GPa. In addition, four of the five virtual wells show a significant change of the SHmax gradients with depth. For example, Rittershoffen and Soultz-sous-Forets—located only ~ 10 km apart from each other—show almost similar SHmax values below 3000 m TVD, but the gradient change occur below 1500 m TVD at Soultz-sous-Forets and below 2250 m TVD at Rittershoffen. The depth at which the 'jump' of the SHmax magnitudes occurs fits quite perfectly to the boundary between the crystalline basement and the sedimentary units of the URG. Rittershoffen is a little east to Soultz-sous-Forets and further away from the western graben shoulder of the URG, and therefore, the top of the crystalline basement in Rittershoffen is located at ~ 2250 m TVD and at ~ 1550 m TVD in Soultz-sous-Forets (Aichholzer et al. 2016).
The Shmin magnitudes of our model within the MB show a good fit to the comparison data displayed below 2000 m TVD. However, as already mentioned, our model results indicate partially too low values in the upper 1500 m. The SHmax magnitudes again show larger variations. Similar to the results of the URG, in all virtual wells the SHmax magnitudes increase if the crystalline basement is reached. Thus, since the depth of the crystalline basement differs in almost every virtual well due to the southward increase of the sediment thickness in the MB, the resulting stresses are inhomogeneous. A remarkable outlier is the data record at ~ 3600 m from Mauerstetten (Backers et al. 2017).
In general, the predicted SHmax magnitudes of the MB and URG show the impact of the vertical model resolution. The transition zone between sediments and crystalline basement is ~ 750 m, which corresponds to about three element rows. A higher model resolution could decrease this transition zone significantly.
Since the calibration data (Figs. 6 and 8) and the compilations of Fig. 12 only show pointwise data the Shmin and SHmax magnitudes are additionally displayed by several depth sections to focus on the lateral distribution. In general, the depth sections of Fig. 7 displaying the lateral distribution of the Shmin magnitudes show a more homogeneous pattern than the SHmax magnitudes in Fig. 9. However, in both figures high magnitudes of Shmin or SHmax are associated with crystalline basement units, e.g., the Bohemian Massif in the southeast or the EEC in the northeast and low magnitudes are often related to sedimentary basins, e.g., the NGB or the MB or regions with outcropping low-grade metamorphic sediments, e.g., the RHZ or SXZ. This correlation can be explained by the combination of stiff units with relatively high densities and softer units, which often have lower densities. However, a high stiffness alone is not sufficient for high horizontal stress magnitudes. This is because high stiffness leads to an increase in SHmax magnitudes as a result of applied shortening but at the same time to a decrease in the Shmin magnitude as a result of applied extension. Accordingly, a higher Young's modulus leads to higher SHmax magnitudes and lower Shmin magnitudes. This effect can be reduced or increased by the influence of SV. SV reduces or increases the Shmin and SHmax magnitudes equally, and since it is defined by density, a high density leads to higher horizontal stresses and a lower density to lower horizontal stresses. An example for this combined effect can be seen within the MDZ and MGCH, where an inversion of the Shmin magnitudes relative to the adjacent regions occurs between 1500 and 5000 m TVD (Fig. 7). The MDZ and MGCH are regions with a thin sedimentary cover. Therefore, the low Shmin magnitudes at 1500 m occur due to the high Young's modulus of 70 GPa and a lower magnitude of SV due to the thin sedimentary cover. The adjacent regions show higher Shmin magnitudes, e.g., the Bohemian Massif with a similar Young's modulus but a higher SV due to the missing sedimentary cover or the SXZ due to a lower Young's modulus and a higher SV. At 5000 m TVD the MDZ and MGCH show higher Shmin values than the SXZ but more similar to the ones in the Bohemian Massif. This shows that the effect of the thin sedimentary cover vanishes with depth. Furthermore, the relation between the Shmin magnitudes in the Bohemian Massif and the RHZ and NGB shows that the influence of a higher density (Bohemian Massif > RHZ & NGB) exceeds the effect of a lower Young's modulus (RHZ & NGB < Bohemian Massif). The lateral distribution of the SHmax magnitudes is easier to explain, since a higher density and a higher Young's modulus result in high stresses and a lower density and a lower Young's modulus in low stresses. Some boundary effects are visible at the northern and southern model edge, where the compressional boundary conditions are defined, showing the highest values for Shmin and SHmax. An exception is the stiff Tauern window (Young's modulus: 70 GPa), showing lower magnitudes of Shmin and SHmax up to 1500 m TVD than the weaker surrounding thrusted units of the Alps (Young's modulus: 23 GPa) do. This is probably due to the fact, that the Tauern window is pushed into the softer units and the soft units absorb the stress. This is also indicated by the high RSR values within the surrounding units (Fig. 11, 500 to 2500 m TVD), indicating high horizontal stresses. Furthermore, remarkable low magnitudes of Shmin and SHmax occur at the boundary between the RHZ and the MGCH up to 2500 m TVD. This might be an effect due to the vertical boundary between these units (Ahlers et al. 2021a).
The prediction of the stress regime and possible stress regime changes are important for the stimulation of geothermal reservoirs (Azzola et al. 2019), borehole stability and for directional drilling (Rajabi et al. 2016). In particular, if the stress regime change leads to an increase of the differential stress, e.g., due to a change from a normal faulting to a strike-slip faulting regime. For this, the RSR indicating lateral and vertical stress regime changes is a useful parameter. For example, the RSR values in the Glückstadt Graben up to 2500 m TVD, which are higher as in the adjacent areas of the NGB. Probably, the predicted strike-slip regime is related to major salt walls within this region (Maystrenko et al., 2005). The low density of salt (2100 kg/m3) leads to a relative low SV magnitude and, as a result, the horizontal stresses exceed SV and even a thrust faulting regime at 1500 m TVD is partially established. Very low RSR values at 2500 m TVD in the SNB show an opposite effect. The weak Rotliegend sediments with a Young's modulus of 15 GPa lead to low horizontal stresses, and therefore, the RSR value decreases, leading to a normal faulting regime. This is also well visible in the depth section at 5000 m TVD in the NGB. The trend towards relatively higher values in the URG than in the surrounding areas at 5 km TVD is related to a lower SV in comparison to the graben shoulders because of the sedimentary fill of the URG but similar horizontal stresses. Above 5 km TVD, the horizontal stresses are lower in the URG due to the lower Young's modulus of the sediments. However, at 5 km TVD the basement of the URG is reached with a similar Young's modulus as for the graben shoulders. In general, a trend towards a normal faulting stress regime due to increasing dominance of SV with depth is visible. Thus, the depth section at 5000 m TVD shows mainly RSR values smaller than 1.0.
This trend is also clearly visible in Fig. 12. All results displayed show a stress regime change from a dominant strike-slip regime in the uppermost part of the model to a normal faulting regime at 8000 m TVD. However, within the depth range of 200 to 8000 m TVD the results of the individual virtual wells differ sometimes significantly. The results from the NGB show an almost continuous change from a strike-slip regime in the upper 500 m to a normal faulting regime at TVDs > 3000 m with a transition zone between 500 and 3000 m TVD, where the stress regime is not the same for all virtual wells. The virtual wells located in the URG show a more complex stress regime change with depth, with differences up to 7500 m TVD. In the upper 1500 m, the magnitudes of SV and SHmax are almost equal resulting in a strike-slip to reverse faulting regime or transpressional regime, respectively. Between 1500 and 2500 m TVD almost all SHmax magnitudes get higher than SV resulting in a strike-slip regime. Between ~ 5500 and 7500 m TVD, the stress regime changes to a normal faulting regime. An exception are the results at Bruchsal, showing several changes of the stress regime up to 5500 m TVD between strike-slip and normal faulting regimes. The results of the MB show almost similar results as the results from the URG. Starting with a strike-slip to reverse faulting regime within the upper parts of the model, followed by a strike-slip regime and finally a normal faulting regime.
The model presented is a further step towards a robust prediction of the crustal stress state of Germany with focus on sedimentary basins. It is based on Ahlers et al. (2021a), but significantly improved. An 18-time higher mesh resolution resulting in a lateral resolution of 2.5 km and a vertical resolution of up to 240 m, 15 additional units within the sedimentary layer and an additional model calibration with SHmax magnitudes, provide a refined prediction of the crustal stress field of Germany. The 3D geomechanical–numerical model provides the complete 3D stress tensor for the entire model volume. Overall, the results show a good fit to all three principal stress magnitudes SV, Shmin and SHmax indicated by absolute differences of 0.0 MPa for SV, 4.6 MPa for Shmin and 6.4 MPa for SHmax. The differences to the calibration data are mainly within in a range of ± 10 MPa for the Shmin magnitudes and within a range of ± 20 MPa for the SHmax magnitudes. Despite the overall good fit, some data indicate too low Shmin values in the upper 1500 m TVD of our model. However, additional data from the NGB does not confirm a general trend. Apart from the magnitudes we compare our results also with a mean orientation of SHmax derived from the WSM (Heidbach et al. 2016) and additional data of Levi et al. (2019). Our predicted orientations of SHmax show an overall good fit with a median of 0.3° and our results lie almost entirely within the standard deviation of the derived WSM data. However, our model does not resolve perturbations of SHmax on smaller local scales as indicated by the data of the WSM.
Some limitations result from the size of our model. Due to the size, it is not possible to define visco-elastic properties, e.g., for the Zechstein salt unit. Furthermore, the vertical resolution is still too low to numerically represent all units sufficiently and at the same time some inhomogeneous units such as the Triassic are still combined. Accordingly, sub modeling is a useful method to enable a higher resolution and stratigraphic refinement (e.g., Ziegler et al. 2016). Such smaller models on local or reservoir scale also enable consideration of varying rock properties (e.g., Ziegler 2021), a quantification of model uncertainties and the implementation of structures that influence the stress field on a local scale, e.g., faults. In addition, more high quality data records, especially magnitude data from north Germany are necessary for a more reliable calibration.
The model geometry and the results of our model have been published and are publicly available at: https://doi.org/10.48328/tudatalib-437.5 (see Ahlers et al. 2022).
3DD:
3D Deutschland
ALCAPA:
Alps–Carpathian–Pannonian
Armorican Terrane Assemblage
CG:
Central Graben
EEC:
East European Craton
EI:
FP:
HG:
Horn Graben
HZ:
Harz Mountains
GG:
Glückstadt Graben
LRG:
Lower Rhine Graben
MDZ:
Moldanubian Zone
MGCH:
Mid German Crystalline High
OG:
Ohře Graben
RHZ:
Rhenoherzynian Zone
RSR:
S Hmax :
Maximum horizontal stress
S hmin :
Minimum horizontal stress
SNB:
Saar–Nahe Basin
S V :
Vertical stress
SXZ:
Saxothuringian Zone
TVD:
True vertical depth
TW:
Tauern Window
URG:
Upper Rhine Graben
VB:
Vogelsberg Complex
VG:
WSM:
World Stress Map
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This study is part of the SpannEnD Project (http://www.SpannEnD-Projekt.de, last access: 2022/06/17), which is supported by Federal Ministry for Economic Affairs and Energy (BMWI) and managed by Projektträger Karlsruhe (PTKA) (project code: 02E11637A). Calculations for this research were conducted on the Lichtenberg high performance computer of the Technische Universität Darmstadt. Coastlines and borders used in the figures are based on the Global Self-consistent Hierarchical High-resolution Geography (GSHHG) of Wessel and Smith (1996). We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG—German Research Foundation) and the Open Access Publishing Fund of Technische Universität Darmstadt.
Open Access funding enabled and organized by Projekt DEAL. This research has been supported by the Bundesministerium für Wirtschaft und Energie (Grant No. 02E11637A).
Engineering Geology, Institute of Applied Geosciences, TU Darmstadt, 64287, Darmstadt, Germany
Steffen Ahlers, Tobias Hergert, Karsten Reiter & Andreas Henk
Technical Petrophysics, Institute of Applied Geosciences, KIT, 76131, Karlsruhe, Germany
Luisa Röckel & Birgit Müller
Seismic Hazard and Risk Dynamics, GFZ German Research Centre for Geosciences, 14473, Potsdam, Germany
Oliver Heidbach & Sophia Morawietz
Institute for Applied Geosciences, TU Berlin, 10587, Berlin, Germany
Basin Modelling, GFZ German Research Centre for Geosciences, 14473, Potsdam, Germany
Magdalena Scheck-Wenderoth & Denis Anikiev
Department of Geology, Geochemistry of Petroleum and Coal, Faculty of Georesources and Material Engineering, RWTH Aachen University, Aachen, Germany
Magdalena Scheck-Wenderoth
Steffen Ahlers
Luisa Röckel
Tobias Hergert
Karsten Reiter
Oliver Heidbach
Andreas Henk
Birgit Müller
Sophia Morawietz
Denis Anikiev
Conceptualization of the project was done by AH, TH, KR, OH and BM. Construction, discretization and calibration of the model were done by SA. Data for the model and its calibration were collected and provided by SA, LR, SM, MSW and DA. Evaluation of the model results and their interpretation were performed by SA with the support of AH, TH, KR, BM, LR, OH and SM. SA wrote the paper with help from all coauthors. All authors read and approved the final manuscript.
Correspondence to Steffen Ahlers.
Ahlers, S., Röckel, L., Hergert, T. et al. The crustal stress field of Germany: a refined prediction. Geotherm Energy 10, 10 (2022). https://doi.org/10.1186/s40517-022-00222-6
3D geomechanical–numerical model
Stress tensor
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Stress state
North German Basin
Molasse Basin | CommonCrawl |
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