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Lost in my Life (twist ties), 2009
Pigmented ink print, edition of 3
90 x 60 inches
Courtesy of the Artist, Barbara Krakow Gallery (Boston), Gallery Joe (Philadelphia), and Yancey Richardson Gallery (New York)
Rachel Perry Welty has been creating obsessive, repetitive, and process-based works about aspects of her own life for over a decade. 24/7 is her first large-scale solo museum show and features Welty's major works in drawing, sculpture, collage, installation, video, photography, and social media–some of which has been created especially for this exhibition.
As a conceptual artist, Welty utilizes whatever methods or materials best communicate the concept or idea for the artwork. Her disparate practices are unified by an interest in investigating language and systems as well as a visual aesthetic that combines the spare, precision of Minimalism with the vivid color and irony of Pop Art. Welty's beautifully crafted work addresses a wide variety of issues including consumerism, suburbia, narcissism, information overload, language, the fleeting nature of experience, the passage of time, humor, and ultimately, life and death.
Welty takes daily life as her subject, incorporating the mundane and the extraordinary in equal measure. She appropriates the material that annoyingly, and sometimes mistakenly, inundates our lives as her subject, including spam emails, wrong number voice messages, receipts, twist ties, fruit stickers, Facebook updates, and even Muzak. The disposable minutiae of life is collected, organized, and transformed in poignant and visually surprising ways that uncover the poetic in the everyday. Whether painstakingly coding and copying her son's hospital receipts and records, creating brightly colored wallpaper from fruit stickers, or meticulously reporting her every action on Facebook, Welty's surprising transformations continually comment on and record what Welty calls, "the business of living."
Major funding provided by James and Audrey Foster, The Goldhirsh Foundation, a grant from the Artists' Resource Trust, Katherine Kirk and Malcolm Gefter, Barbara and Jonathan Lee, and an anonymous donor. Additional support provided by AT&T and John and Deborah French.
Watch videos of Rachel Perry Welty in her studio and installing at deCordova.
Follow Rachel Perry Welty on Twitter!
Rachel Perry Welty lives in Gloucester, Massachusetts and New York, New York. She is a graduate of the Fifth Year Certificate program of the School of the Museum of Fine Arts, Boston, and participated in the Traveling Scholar's exhibition at the Museum of Fine Arts in both 2002 and 2006. In 2006 she was also a finalist for the Foster Prize competition at the Institute of Contemporary Art. She is represented by Barbara Krakow Gallery in Boston, Gallery Joe in Philadelphia, and Yancey Richardson Gallery in New York.
Artist & Curator Talk: Rachel Perry Welty and Deputy Director for Curatorial Affairs Nick Capasso
Saturday, April 2, 3 pm
Eye Wonder Family Program: Sunday, April 3, 1–3 pm
Afternoon Lecture and Tour: Understanding Conceptual Art: Deputy Director for Curatorial Affairs Nick Capasso and Associate Curator Dina Deitsch Saturday, April 9, 3 pm
Process Gallery
Explore 24/7 in a behind-the-scenes way through hands-on interactive activities, touchable materials samples, and a prop from one of Rachel's Lost in my Life sets! Find out more about the artist's creative process in this interactive gallery.
Cell Phone Audio Tour
Bring your cell phone to 24/7 and let the artist lead you through his exhibition. Listen as Rachel Perry Welty discusses her creative process.
Family Activity Kit
This free kit includes art making materials and activities that focus on the Rachel Perry Welty exhibition. It is designed for ages 6–12, and is available at the Front Desk.
Family Gallery Guides
Gallery Guides are available throughout the museum and provide family-friendly information about current exhibitions.
| 392,610
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Kelly, Stephen W. and Dominguez, R.
(2002)
Speech Rehabilitation using the SNORS+ System.
In: CIE 2002 (Octava Conferencia de Ingenieria Electrica), 2002 4-6th September, Mexico.
(The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)
| 253,780
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\begin{document}
\title{Complexity of multivariate polynomial evaluation}
\author{E. Ballico, M. Elia, and M. Sala}
\date{}
\maketitle
\begin{abstract}
We describe a method to evaluate multivariate polynomials over a finite field
and discuss its multiplicative complexity.
\end{abstract}
{\bf Keywords}: multiplicative complexity, complexity, multivariate polynomials, finite field, computational algebra\\
{\bf MSC}: 12Y05, 94B27\\
\section{Introduction}
Many applications require the evaluation of multivariate polynomials over finite fields. For instance, the so called affine codes (also called evaluation or functional or algebraic geometry codes) are obtained evaluating a finite-dimensional linear subspace
of $\mathbb {F}_q[x_1,\dots ,x_r]$ at a finite set $S\subseteq \mathbb {F}_q^r$ (\cite{g}, \cite{h}, \cite{l} and several other papers).
When the degree $n$ of the polynomials is small, and/or the number $r$ of variables is also small the direct computation is
efficient, however as $n$, or $r$, or both become large, evaluation becomes an issue. The case of univariate polynomials was considered by several authors, see e.g \cite{patterson}, \cite{pan} and some recent papers (\cite{ers1}, \cite{ERS}).
In this paper we propose an evaluation method for multivariate polynomials
which reduces significantly the multiplicative complexity and hence
the computational burden.
\noindent
Set $M_r(n)=\binom{n+r}{r}$, and let $p(x_1, \ldots,x_r)$ be a polynomial of degree $n$ in $r$
variables with coefficients in a finite field $\mathbb{F}_{p^s}$; the number of monomials occurring in
$p(x_1, \ldots,x_r)$ is $M_r(n)$.
We will consider the evaluation of $p(x_1, \ldots,x_r)$ at a point
$\mathbf a=(\alpha_1, \ldots, \alpha_r) \in F_{p^m}^{r}$, where $m$ is divisible by $s$.
A direct evaluation of $p(\alpha_1, \ldots, \alpha_r)$
is obtained from the evaluation of the $M_r(n)$ distinct monomials, a task requiring $M_r(n)-r-1$
multiplications. Therefore we perform $M_r(n)-1$ multiplications, and a total number
$A_r(n)=M_r(n)-1$ of additions. The total number of required multiplications is
$$ P_r(n)= 2M_r(n)-r-2 = 2\frac{n^r}{r!}+\frac{r+1}{(r-1)!}n^{r-1}+ \cdots -1-r \,, $$
however different computing strategies can require a significant smaller number of multiplications.
To the aim of developing some of these strategies, the polynomial $p(x_1, \ldots,x_r)$ is written as a sum
\begin{equation}
\label{eqint}
p(x_1, \ldots,x_r) = \sum_{i=0}^{s-1} \beta^i q_i(x_1, \ldots,x_r)
\end{equation}
of $s$ polynomials, where each $q_i(x_1, \ldots,x_r)$ is a polynomial of degree $n$ in $r$ variables
with coefficients in the prime field $\mathbb {F}_p$, the value $p(\alpha_1, \ldots, \alpha_r)$ can be obtained from the $s$ values
$q_i(\alpha_1, \ldots, \alpha_r)$, $2s-2$ multiplications, and $s$ additions in $\mathbb {F}_{p^m}$. In these
computations $\beta$ and its powers constitute a basis of $\mathbb {F}_{p^m}$.
Therefore, we may restrict our attention to the evaluation at a point $\mathbf a \in \mathbb{F}_{p^m}^{r}$
of a polynomial $q(x_1, \ldots,x_r)$, in $r$ variables, of degree $n$ over $\mathbb{F}_p$.
\vspace{3mm}
\noindent
As pointed out in \cite{ERS}, \S 2.1, the prime $2$ is particularly interesting because
of its occurrence in many practical applications, for example in error correction coding. Furthermore, in
$\mathbb{F}_2$ multiplications are trivial.
Therefore, we give first a description of our method in the easiest
case, that is, over $\FF_2$ and with two variables.
Later, we generalize to any setting.
\section{Our computational model}
There are two kinds of multiplications that are involved in our computations:
field multiplications in the coefficient field $\FF_{p^s}$ and in extension
field $\FF_{p^m}$.
We assign cost $1$ to any of these, except for the multiplications
by $0$ or $1$, that cost $0$ in our model.
\begin{remark}
There can be multiplications that cost much less, such as squares
in characteristic $2$, but we still treat them as cost $1$.
\end{remark}
As customary, we assign cost $0$ to any data reading.
We could count separately field sums, but our aim is to minimize
the number of field multiplications, and so we use as implicit upper
bound for the number of sums the value $2M_r(n)$, that is, twice the number of
all monomials.
We will not discuss of the number of sums any further.
We assume that an ordering on monomials is chosen once and for all,
e.g. the degree lexicographical ordering
(see \cite{CGC-cd-inbook-D1moratech}), so that
our input data can be modeled as an $\FF_{p^s}$ string, any entry
corresponding to a polynomial coefficient.
\begin{remark} \label{qq}
A well-established method to evaluate all monomials up to degree $n$
at a given point is to start from degree-$1$ monomials and then
iterate from degree-$r$ monomials to degree-$r+1$ monomials,
since the computations of any degree-$r+1$ monomial requires only
one multiplication, once you have in memory all degree-$r$ monomials.
\end{remark}
We remark here that our algorithm accepts as input {\emph any} polynomial
of a given total degree and so our estimates are worst-case complexity,
which translates in considering dense polynomials.
Clearly, other faster methods could be derived for special classes of polynomials,
such as sparse polynomials or polynomials with a predetermined algebraic structure.
We will not discuss the memory requirement of our methods, but one can
see easily by inspecting the following algorithms that it is negligible
compared to their computational effort.
\section{The case $r=2$, $p=2$}
A polynomial $P(x, y)$ of degree $n$ in $2$ variables over the binary field may be decomposed
into a sum of $4$ polynomials as
\begin{eqnarray}
\label{eq0}
P(x, y) &=& P_{0,0} (x^2, y^2)+ xP_{1,0} (x^2, y^2) + yP_{0,1} (x^2, y^2) + x yP_{1,1} (x^2, y^2) \nonumber \\
&=& P_{0,0} (x, y)^2+ xP_{1,0} (x, y)^2 + yP_{0,1} (x, y)^2 + x yP_{1,1} (x, y)^2.
\end{eqnarray}
where $P_{i, j}(x, y)$ are polynomials of degree $\lfloor \frac{n-i-j}{2}\rfloor$.
Therefore the value of $P(x, y)$ in the point $\mathbf a = (\alpha_1, \alpha_2) \in \mathbb{F}_{2^m}^2$
can be obtained by computing the $4$ numbers $P_{i, j} (\alpha_1, \alpha_2)$, the monomial
$\alpha_1 \alpha_2$, performing $3$ products $\alpha_1 P_{1, 0} (\alpha_1, \alpha _2)$,
$\alpha_2 P_{0, 1} (\alpha_1, \alpha_2)$, and $\alpha_1 \alpha_2 P_{1, 1} (\alpha_1, \alpha_2)$,
and finally performing $3$ additions.
We observe that all $P_{i,j}$'s have the same possible monomials, i.e. all
monomials of degree up to $\lfloor \frac{n}{2}\rfloor$.
There is no need to store separately $P_{0,0},P_{0,1},P_{1,0},P_{1,1}$,
because the selection of any of these is obtained by a trivial indexing
rule.
The polynomials $P_{i, j}(\alpha_1, \alpha_2)$ can be evaluated as sums of such monomials,
which can be evaluated once for all.
Therefore, $P(\alpha_1, \alpha_2)$ is obtained performing (see Remark \ref{qq})
a total number of
$$P_r(n)=4+3+ \frac{(\lfloor \frac{n}{2} \rfloor +2)(\lfloor \frac{n}{2} \rfloor +1) }{2} -3 \approx
\frac{n^2}{8} $$
multiplications, a figure considerably less than $\frac{n^2}{2}$ as required by the direct computation.
However, the mechanism can be iterated, and the point is to find the number of steps yielding
the maximum gain, that is to find the most convenient degree of the polynomials that should be
directly evaluated. We have the following:
\begin{Theorem}
\label{theo1}
Let $P(x,y)$ be a polynomial of degree $n$ over $\mathbb{F}_2$, its evaluation at a point
$(\alpha_1, \alpha_2) \in \mathbb{F}_{2^m}^2$ performed by applying repeatedly the
decomposition (\ref{eq0}), requires a number $G_2(n,2,L_{opt})$ of products which asymptotically is
$$
G_2(n,2,L_{opt}) \approx c \sqrt{\frac{7}{6} } n~~~~c < 5 ~~~~~~~~~~.
$$
where $L_{opt}$, the number of iterations yielding the minimum of $G_2(n,2,L)$,
is an integer included into the interval
$$ - \frac{1}{2} + \log_4 (\sqrt{\frac{6}{7}} n) +\epsilon < {L_{op}} <
\log_4 ( \sqrt{\frac{6}{7}}) n +\epsilon' ~~, $$
where $\epsilon$ and $\epsilon'$ are less than $1$ and $O(\frac{1}{\sqrt n})$.
\end{Theorem}
\noindent
{\sc Proof}. The polynomial $P(x,y)$ is decomposed into the sum of $4$ polynomials that are
perfect squares over $\mathbb{F}_2$, each of which is the similarly decomposed.
Let $P_{i, j}^{(L,h)} (x, y)$ denote the polynomials at the $L$-step of this iterative process,
with $h$ varying from $1$ to $4^{L-1}$. The number of polynomials after $L$ steps is $4^L$,
while their degrees are not greater than $\lfloor \frac{n}{2^L}\rfloor$.
The value $P(\alpha_1, \alpha_2)$ is obtained performing backward the reconstruction process
obtaining at each step the values $P_{i,j}^{(\ell-1,h)} (\alpha_1, \alpha_2)$ from the values
$P_{i,j}^{(\ell,h)} (\alpha_1, \alpha_2)$, whereas the $4^L$ numbers $P_{i,j}^{(L,h)} (\alpha_1, \alpha_2)$, $i,j \in \{0,1 \}$ and $h=0, \ldots , 4^{L-1}$, are computed from the direct
evaluation of $M_2(\lfloor \frac{n}{2^L}\rfloor )$ monomials using $M_2(\lfloor \frac{n}{2^L}\rfloor )-3$ multiplications.
Therefore the total number of multiplications necessary to obtain $P(\alpha_1, \alpha_2)$ is a sum of
$M_2(\lfloor \frac{n}{2^L}\rfloor)-3$ with
\begin{itemize}
\item[-] the number of squares
$$ \frac{4}{3}(4^{L}-1) = [4^{L}+ 4^{L-1}+\cdots +4^{L-L+1} ] $$
\item[-] the number of multiplications of kind $x^iy^j P_{i,j}(\alpha_1, \alpha_2)$
$$ 4^{L}-1= 3 [4^{L-1}+ 4^{L-2}+\cdots +4^{L-L} ] $$
\end{itemize}
that is the total number is:
$$ G_2(n,2,L) = \frac{7}{3}(4^{L}-1)+\frac{(\lfloor \frac{n}{2^L} \rfloor +1)(\lfloor \frac{n}{2^L} \rfloor +2)}{2}-3 $$
The number of products required to evaluate $P(\alpha_1, \alpha_2)$ in this way is a function
of $L$, and the values of $L$ that correspond to local minima are specified by the conditions
$$ G_2(n,2,L) \leq G_2(n,2, L-1) ~~\mbox{and}~~ G_2(n,2,L) \leq G_2(n,2,L + 1) ~~, $$
from which, it is straightforward to obtain the conditions
$$ \begin{array}{l}
4^L - \frac{3}{56} \frac{n^2}{4^L} > \frac{n}{2^L14}( \frac{3}{2}-\{\frac{n}{2^L}\})+ (\{\frac{n}{2^L}\}- \{\frac{n}{2^L 2}\}) \left(\frac{-3}{14}+ \frac{1}{14} (\{\frac{n}{2^L}\}+\{\frac{n}{2^L 2}\})+ \frac{1}{2} \right) \\
\\
4^L - \frac{6}{7} \frac{n^2}{4^L} <
\frac{4n}{2^L7} (\frac{3}{2}+\{\frac{n}{2^L}\}-2 \{\frac{2n}{2^L}\})- \frac{2}{7} (\{\frac{n}{2^L}\}^2-
\{\frac{2n}{2^L}\}^2) +\frac{6}{7} (\{\frac{n}{2^L}\}- \{\frac{2n}{2^L}\}) \\
\end{array} $$
where $\{x\}$ denotes the fractional part of $x$.
These inequalities show that there is only one minimum that corresponds to a value of $L$ such that
$$ - \frac{1}{2} + \log_4 (\sqrt{\frac{6}{7}} n) +\epsilon < {L_{op}} <
\log_4 ( \sqrt{\frac{6}{7}}) n +\epsilon' ~~, $$
where $\epsilon$ and $\epsilon'$ are $O(\frac{1}{\sqrt n})$.
Therefore, the minimum value of $G_2(n,2,L)$ is asymptotically
$$ G_2(n,2,L_{op}) \approx c \sqrt{\frac{7}{6}} n $$
where $c$ is a constant less than $5$. \QED
\begin{remark}\label{b1}
In the computations of our bounds we essentially compute separately each monomial. Hence our approach seems to be very efficient for the computation of several polynomials at the same point. This fact is exploited
in the computation of the required number of multiplications when the polynomial coefficients are in $\mathbb{F}_{2^s}$. An application of equation (\ref{eqint}) and Theorem \ref{theo1} would give
the asymptotic estimate
$$ G_{2^s}(n,2,L_{op}) \approx c \sqrt{\frac{7}{6}} n s~~. $$
since the evaluation of any $q_i$ would cost $c \sqrt{\frac{7}{6}}n$.
However, the polynomials $q_i(x,y)$ can be evaluated contemporarily. Therefore, computing the power necessary
to evaluate the polynomial at step $L$ only once, this lead to a total number or required multiplications
$$ G_{2^s}(n,2,L) = s\frac{7}{3}(4^{L}-1)+\frac{(\lfloor \frac{n}{2^L} \rfloor+1)(\lfloor \frac{n}{2^L} \rfloor+2)}{2}-3 $$
because only the reconstruction operations need to be repeated $s$ times. By repeating the argument outlined in the proof of Theorem \ref{theo1}, the conclusion is that the optimal value of $L$ depends also on $s$ and asymptotically the required value of multiplications is
$$ G_{2^s}(n,2,L_{op}) \approx c' \sqrt{\frac{7}{6}} n \sqrt s ~~. $$
\end{remark}
\section{The case $r>2$, $p=2$}
The evaluation of a polynomial $P(x_1, \ldots, x_r)$ in $r$ variables can be done writing $P$, similarly to equation (\ref{eq0}),
this polynomial as a sum of $2^r$ polynomials
\begin{eqnarray}
\label{eq1}
P(x_1, \ldots, x_r) &=& \sum_{i_1, \ldots, i_r \in \{0,1 \}}
x_1^{i_1} \ldots x_r^{i_r} P_{i_1, \ldots, i_r} (x_1^{2}, \ldots, x_r^{2}) \nonumber \\
&=& \sum_{i_1, \ldots, i_r \in \{0,1\}} x_1^{i_1} \ldots x_r^{i_r}
\left( P_{i_1, \ldots, i_r} (x_1, \ldots, x_r) \right)^2\,,
\end{eqnarray}
where $P_{i_1, \ldots, i_r} (x_1, \ldots, x_r)$ is a polynomial of degree
$\frac{n-\sum i_j}{2}$.
The argument of Theorem \ref{theo1} still applies, and the minimum number of steps
is obtained in the following theorem.
\begin{Theorem}
\label{theo2}
Let $L_{opt}$ be the number of steps of this method yielding the minimum number of products, $G_2(n,r,L_{op})$, required to evaluate a polynomial of degree $n$ in $r$ variables,
with coefficients in $\mathbb{F}_2$. Then $L_{opt}$ is an integer that asymptotically is
included into the interval
$$ -\frac{1}{2} + \frac{1}{2r} \log_2 \frac{2^r-1}{r!(2^{r+1}-1)} +\frac{\log_2 n}{2} \leq L_{op} \leq
\frac{1}{2} + \frac{1}{2r} \log_2 \frac{2^r-1}{r!(2^{r+1}-1)} +\frac{\log_2 n}{2} $$
that is $L_{op}$ is the integer closest to $\frac{1}{2r} \log_2 \frac{2^r-1}{r!(2^{r+1}-1)} +\frac{\log_2 n}{2}$.
Asymptotically the minimum $G_2(n,r,L_{op})$ is included into the interval:
$$ \frac{1}{\sqrt{2^r}} \sqrt{4\frac{2^{r+1}-1}{2^r-1} \frac{1}{r!}} n^{r/2} < G_2(n,r,L_{op}) <
\sqrt{2^r} \sqrt{4\frac{2^{r+1}-1}{2^r-1} \frac{1}{r!}} n^{r/2}~~. $$
\end{Theorem}
\noindent
{\sc Proof}. Using equation (\ref{eq1}) the polynomial $P(x_1, \ldots, x_r)$ evaluated at the point
$\mathbf a = (\alpha_1, \ldots, \alpha_r) \in \mathbb{F}_{2^m}^r$
can be obtained from the evaluation of all $P_{i_1, \ldots, i_r} (x_1, \ldots, x_r)$ at $\mathbf a$,
by evaluating $2^r$ monomials $\alpha_1^{i_1} \ldots \alpha_r^{i_r}$ (which
require $2^r-r-1$ multiplications), performing $2^r$ squaring,
combining these factors with $2^r-1$ multiplications, and finally
adding the results. \\
We can iterate this procedure: at each step the number of polynomials
$P_{i,j}$'s is multiplied by $2^r$ and their
degrees are divided at least by $2$. Therefore, after $L$ steps the number of polynomials is $2^{Lr}$ and their degrees are not greater than $\lfloor \frac{n}{2^L} \rfloor$. Once the $2^r$ numbers $P_{i_1, \ldots, i_r} (\alpha_1, \ldots, \alpha_r)$ are known, the total number of squaring is
$$ 2^{rL}+ 2^{r(L-1)}+\cdots +2^{r(L-L+1))} = \frac{2^r}{2^r-1} (2^{rL}-1) $$
and the number of products necessary to obtain $P(\alpha)$ is
$$ (2^r-1) [2^{r(L-1)}+ 2^{r(L-2)}+\cdots +2^{r(L-L))} ]=2^{rL}-1 \,, $$
hence the total number of required multiplications is
$$ \frac{2^{r+1}-1}{2^r-1} (2^{rL}-1)\,. $$
Since the total number of monomials in $r$ variables in a generic polynomial of degree
$\lfloor \frac{n}{2^L} \rfloor$ is $M_r(\lfloor \frac{n}{2^L} \rfloor)$,
then $M_r(\lfloor \frac{n}{2^L} \rfloor)-r-1$ is the number of products necessary to evaluate
all independent monomials.
Therefore, the total number of multiplications for evaluating $P(\mathbf a)$ is
$$ G_2(n,r,L) = \frac{2^{r+1}-1}{2^r-1} (2^{rL}-1)+M_r(n)-r-1 ~~. $$
We look for the optimal value $L_{op}$ giving the minimum $ G_2(n,r,L_{op})$.
Since
$$ M_r(\lfloor \frac{n}{2^L} \rfloor)=\frac{1}{r!} \left( \frac{n}{2^L}-\{ \frac{n}{2^L} \} \right)^r \prod_{j=1}^{r} \left( 1+ \frac{j}{\frac{n}{2^L}-\{ \frac{n}{2^L} \} } \right) ~~, $$
then $M_r(\lfloor \frac{n}{2^L} \rfloor)$ is an expression that is $\frac{1}{r!}(\frac{n}{2^L})^r+ O(\frac{2^L}{n})$ asymptotically in $n$. \\
The local optima are given by the values of $L$ such the
$$ G_2(n,r,L) \leq G_2(n,r, L-1) ~~\mbox{and}~~ G_2(n,r,L) \leq G_2(n,r,L + 1) \ . $$
Then, considering the asymptotic expression
$$ G_2(n,r,L) = \frac{2^{r+1}-1}{2^r-1} 2^{rL} + \frac{1}{r!} (\frac{n}{2^L})^r ~~, $$
it is immediate
to obtain the conditions
$$ \begin{array}{l}
\displaystyle 2^{2rL} > \frac{1}{r!} \frac{n^r}{2^r} \frac{2^r-1}{2^{r+1}-1} \\
\\
\displaystyle 2^{2rL} < 2^r n^r \frac{1}{r!} \frac{2^r-1}{2^{r+1}-1} ~~, \\
\end{array} $$
showing that asymptotically $L_{op}$ must satisfy the inequalities
$$ -\frac{1}{2} + \frac{1}{2r} \log_2 \frac{2^r-1}{r!(2^{r+1}-1)} +\frac{\log_2 n}{2} < L_{op} <
\frac{1}{2} + \frac{1}{2r} \log_2 \frac{2^r-1}{r!(2^{r+1}-1)} +\frac{\log_2 n}{2} ~~. $$
Therefore $L_{op}$ is the closest integer to
$\frac{1}{2r} \log_2 \frac{2^r-1}{r!(2^{r+1}-1)} +\frac{1}{2} \log_2 n $,
and the total number of products asymptotically is included into the interval:
$$ \frac{1}{\sqrt{2^r}} \sqrt{4\frac{2^{r+1}-1}{2^r-1} \frac{1}{r!}} n^{r/2} < G_2(n,r,L_{op}) < \sqrt{2^r} \sqrt{4\frac{2^{r+1}-1}{2^r-1} \frac{1}{r!}} n^{r/2}~~. $$
\QED
\section{The case $r\geq 2$, $p>2$}
A polynomial $P(x_1, \ldots, x_r)$ of degree $n$, in $r$ variables over the field $\mathbb{F}_p$,
is simply decomposed into a sum of $p^r$ polynomials as
\begin{eqnarray}
\label{eq2}
P(x_1, \ldots, x_r) &=& \sum_{i_1, \ldots, i_r \in \{0,1, \ldots, p-1 \}}
x_1^{i_1} \ldots x_r^{i_r} P_{i_1, \ldots, i_r} (x_1^{p}, \ldots, x_r^{p}) \nonumber \\
&=& \sum_{i_1, \ldots, i_r \in \{0,1, \ldots, p-1\}} x_1^{i_1} \ldots x_r^{i_r}
\left( P_{i_1, \ldots, i_r} (x_1, \ldots, x_r) \right)^p
\end{eqnarray}
where $P_{i_1, \ldots, i_r} (x_1, \ldots, x_r)$ is a polynomial of degree
$\lfloor \frac{n-\sum i_j}{p}\rfloor$.
Therefore the polynomial $P(x_1, \ldots, x_r)$ evaluated at the point
$\mathbf a = (\alpha_1, \ldots, \alpha_r) \in \mathbb{F}_{p^m}^r$
can be obtained from the evaluation of all polynomials $P_{i_1, \ldots, i_r} (x_1, \ldots, x_r)$
at $\mathbf a$, by evaluating the $p^r$ monomials $\alpha_1^{i_1} \ldots \alpha_r^{i_r}$ (which require $p^r-r-1$ multiplications), performing $p^r$ computations of $p$-powers,
combining these factors with $p^r$ multiplications, and finally adding all results.\\
The argument of Theorem \ref{theo1} and \ref{theo2} still applies, and the minimum number
of steps is obtained in the following theorem.
\begin{Theorem}
\label{theo3}
Let $L_{opt}$ be the number of steps of this method yielding the minimum number of products, $G_p(n,r,L_{op})$, required to evaluate a polynomial of degree $n$ in $r$ variables,
with coefficients in $\mathbb{F}_p$. Then $L_{opt}$ is an integer that asymptotically is
included into the interval
$$ -\frac{1}{2} +B +\frac{\log_p n}{2} \leq L_{op}
\leq \frac{1}{2} + B +\frac{\log_p n}{2} $$
where $B=\frac{1}{2r} \log_p \frac{(p-1)(p^r-1)}{r!(2~ p^{r}-1)}$,
that is, $L_{op}$ is the integer closest to $B +\frac{\log_p n}{2}$.
Asymptotically the minimum $G_p(n,r,L_{op})$ is included into the interval:
$$ \frac{2}{\sqrt{p^r}} \sqrt{(p-1)\frac{2p^{r}-1}{p^r-1} \frac{1}{r!}} n^{r/2} < G_p(n,r,L_{op}) <
2\sqrt{p^r} \sqrt{(p-1)\frac{2p^{r}-1}{p^r-1} \frac{1}{r!}} n^{r/2}~~. $$
\end{Theorem}
\noindent
{\sc Proof}. Using equation (\ref{eq2}) the polynomial $P(x_1, \ldots, x_r)$ evaluated at the point
$\mathbf a = (\alpha_1, \ldots, \alpha_r) \in \mathbb{F}_{p^m}^r$
can be obtained from the evaluation of all $P_{i_1, \ldots, i_r} (x_1, \ldots, x_r)$ at $\mathbf a$,
by evaluating $p^r$ monomials $\alpha_1^{i_1} \ldots \alpha_r^{i_r}$ (which
require $p^r-r-1$ multiplications), computing $p^r$ $p$-powers,
combining these factors with $p^r-1$ multiplications, and finally performing
the required additions. \\
We can iterate this procedure: at each step the number of polynomials is multiplied by $p^r$ and their
degrees are at least divided by $p$. Therefore, after $L$ steps the number of polynomials is $p^{rL}$ and their degrees are not greater than $\lfloor \frac{n}{p^L} \rfloor$. Once the $p^r$ numbers $P_{i_1, \ldots, i_r} (\alpha_1, \ldots, \alpha_r)$ are known, the total number of $p$-powers is
$$ p^{rL}+ p^{r(L-1)}+\cdots +p^{r(L-L+1))} =\frac{p^r}{p^r-1} (p^{rL}-1) $$
and the number of products necessary to obtain $P(\alpha)$ is
$$ (p^r-1) [p^{r(L-1)}+ p^{r(L-2)}+\cdots +p^{r(L-L))} ]=p^{rL}-1 ~~, $$
hence the total number of required multiplications is
$$ \frac{2p^{r}-1}{p^r-1} (p^{rL}-1)~~. $$
The total number of multiplications for computing all the monomials of all the polynomials arising at step $L$ is $M_r(\lfloor \frac{n}{2^L} \rfloor)-r-1$, and further $(p-2) (M_r(\lfloor \frac{n}{2^L} \rfloor)-r-1) $ products are necessary to provide every possible term
occurring in the polynomials at step $L$. As a consequence the total number of multiplications necessary to evaluate
$P(\mathbf a)$ is
$$ \frac{2p^{r}-1}{p^r-1} (p^{rL}-1) + (p-1) (M_r(\lfloor \frac{n}{2^L} \rfloor)-r-1) \,.$$
The same passages used in Theorem \ref{theo2} allow us to conclude that $L_{op}$ is asymptotically
identified by the chain of inequalities
$$ \sqrt{\frac{1}{p^r}} \sqrt{\frac{(p-1)(p^r-1)n^r}{r!(2p^r-1)}} \leq p^{rL_{op}} \leq \sqrt{\frac{(p-1)(p^r-1)n^r}{r!(2p^r-1)}} \sqrt{p^r} $$
which written in the form
$$ -\frac{1}{2} +B +\frac{\log_p n}{2} \leq L_{op}
\leq \frac{1}{2} + B +\frac{\log_p n}{2} $$
shows that the unique optimal value is the integer closest to $ B +\frac{\log_p n}{2} $, where $B=\frac{1}{2r} \log_p \frac{(p-1)(p^r-1)}{r!(2~ p^{r}-1)}$.
The minimum number of multiplications is asymptotically included into the interval
$$ \frac{2}{\sqrt{p^r}} \sqrt{(p-1)\frac{2p^{r}-1}{p^r-1} \frac{1}{r!}} n^{r/2} < G_p(n,r,L_{op}) <
2\sqrt{p^r} \sqrt{(p-1)\frac{2p^{r}-1}{p^r-1} \frac{1}{r!}} n^{r/2}~~. $$
\begin{remark}\label{b2}
Our proofs start with the evaluations of certain monomials. Hence they may be extended verbatim to other finite-dimensional linear subspaces
of $\mathbb {F}_{p}[x_1,\dots ,x_r]$, just taking their dimension $\alpha$ as vector spaces instead of the integer $\binom{n+r}{r}$. For a suitable
linear space $V$ in Theorem 3 we could get a bound of order
$c_3\sqrt{\alpha}$ with $c_3\sim 2\sqrt{2p^{r+1}}$. For instance, call
$V(r,n)$ the linear subspace of $\mathbb {F}_{p}[x_1,\dots ,x_r]$ formed by all polynomials whose degree in each variable is at most $n$.
We have $\dim (V(n,r)) = (n+1)^r$. In this case iterating this procedure we arrive at each step at a vector
space $V(\lceil n/p^L\rceil ,r)$. Taking $L$ such that $2p^{rL} \sim p\dim (V(\lceil n/p^L\rceil,r))$, i.e. taking $L\sim \log _p n/2 + B$
with $B\sim \frac{1}{2r}\log _p(p)/2 \sim \frac{1}{2r} \sim - \frac{1}{2r}\log _p 2$, we get an upper bound
of order $2\sqrt{2p^{r+1}}n^{r/2}$.
\end{remark}
\section{Further remarks}
The complexity of polynomial evaluation is crucial in the determination
of the complexity of several computational algebra methods, such as
the Buchberger-Moeller algorithm
(\cite{CGC-alg-art-buchmoeller,CGC-cd-inbook-D1morafglm}),
other commutative algebra methods (\cite{CGC-cd-inbook-D1moratech}),
the Berlekamp-Massey-Sakata algorithm (\cite{CGC-cd-art-sakata88,CGC-cd-inbook-D1sakata1}).
In turn, these algorithms are the main tools used in algebraic
coding theory (and in cryptography).
This justifies our special interest in the finite field case.
For example, the previous algorithms can be adapted naturally
to achieve iterative decoding of algebraic codes and algebraic-geometry
codes, see e.g. \cite{CGC-cd-inbook-D1sakata2,CGC-cd-inbook-D1eleanna}.
Other versions can decode and construct more general geometric
codes, see e.g. \cite{g}.
\section{Acknowledgments}
The authors would like to thank C. Fontanari for valuable discussions.
The first author would like to thank MIUR and GNSAGA of INDAM.
The third author would like to thank MIUR for the programme
``Rientro dei cervelli''.
This work has been partially done while the second author was Visiting Professor
with the University of Trento, funded by CIRM.
| 158,315
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\begin{document}
\date{}
\maketitle
\begin{abstract}
We show that many standard results of Lorentzian causality theory remain valid if the
regularity of the metric is reduced to $C^{1,1}$. Our approach is based on regularisations of
the metric adapted to the causal structure.
\vskip 1em
\noindent
{\em Keywords:} Causality theory, low regularity.
\end{abstract}
\section{Introduction}
Traditionally, general relativity as a geometric theory has been formulated
for smooth space-time metrics. However, over the decades the PDE point of
view has become more and more prevailing. After all, general relativity as a
physical theory is governed by field equations and
questions of regularity are essential in the context of solving the initial
value problem. Already the classical local existence theorem for the vacuum
Einstein equations (\cite{CG69}) deals with space-time metrics in
$H^s_{\mbox{\small loc}}$ with $s>5/2$ (which merely guarantees the metric on
the spatial slices to be $C^1$) and more recent studies have significantly
lowered the regularity (\cite{KR05,M06,KRS12}).
Also from the physical point of view non-smooth solutions are of vital interest.
For example, one would like to study systems where different regions of
space-time have different matter contents, e.g. inside and outside a star, or in
the case of shock waves. On matching these regions the matter variables become
discontinuous, which via the field equations forces the differentiability of
the metric to be below $C^2$. E.g.\ a metric of regularity $C^{1,1}$
(continuously differentiable with locally Lipschitz first order
derivatives, often also denoted by $C^{2-}$) corresponds to finite jumps of the matter variables.
In the standard approach (\cite{L55}) one deals with metrics
which are piecewise $C^3$ but globally are only $C^1$. Even more extreme situations
are exemplified by impulsive waves (e.g.\
\cite[Ch.\ 20]{GP09}) where the metric is still $C^3$ off the impulse but
globally is merely $C^0$.
On the other hand, in the bulk of the literature in general relativity it seems
to be assumed (sometimes implicitly) that the differentiability of the
space-time metric is at least $C^2$, especially so in the standard references on
causality theory. More precisely, the presentations
in \cite{P72,ON83,BEE96,Kriele,MS} generally (seem to) assume smoothness,
while \cite{HE,Seno1,Seno2,Chrusciel_causality} assume $C^2$-differentiability.
This mismatch in regularity---the quest for low regularity from physics
and analysis versus the need for higher regularity to maintain standard
results from geometry---has of course been widely noted, see
e.g.\ \cite{HE,MS,Clarke,Seno1,Seno2,Chrusciel_causality,SW} for a review of
various approaches to causal structures and discussions of
regularity assumptions. The background of this ``annoying problem''
(\cite[\S2]{Seno2}) is that for $C^2$-metrics the existence of totally normal (convex)
neighbourhoods is guaranteed. Furthermore, as emphasised by Senovilla,
$C^2$-differentiability of the metric is one of the fundamental assumptions of
the singularity theorems (see \cite[\S6.1]{Seno1} for a discussion of regularity
issues in this context). Finally, in
\cite{Chrusciel_causality} it has recently been explicitly demonstrated that
assuming the metric to be $C^2$ allows one to retain many of the
standard causality properties of smooth metrics.
However, if one attempts to lower the differentiability of the metric below
$C^2$ one encounters serious problems. It \emph{is} possible to develop some of
the elements of causality theory in low regularity: E.g.,
smooth time functions exist on domains of dependence even for continuous metrics (\cite{FS11,CG}) and the
space of causal curves is still compact in this case (\cite{SW}).
On the other hand it is well-known that some essential building
blocks of the theory break down for general $C^1$-metrics. Explicit
counterexamples by Hartman and Wintner, \cite{Hartman,HW} (in
the Riemannian case) show that for
connections of H\"older regularity $C^{0,\alpha}$ with $0<\alpha<1$ convexity
properties in small neighbourhoods may fail to hold. For example, radial
geodesics may fail to be minimising between any two points they contain.
Also recently a study of the causality of continuous metrics in \cite{CG} has
revealed a dramatic failure of fundamental results of smooth causality:
e.g., light cones no longer need to be topological hypersurfaces of codimension
one. In fact, for any $0<\alpha<1$ there are metrics of regularity $C^{0,\alpha}$, called
`bubbling metrics', whose light-cones have nonempty
interior, and for whom the push-up principle ceases to hold (there exist
causal curves that are not everywhere null but for which there
is no fixed-endpoint deformation into a timelike curve).
For these reasons there has for some time been considerable interest in
determining the minimal degree of regularity of the metric for which standard
results of Lorentzian causality remain valid. A reasonable candidate is the
regularity class $C^{1,1}$ since it marks the
threshold where one still has unique solvability of the geodesic
equation, and the above remarks show that lower regularity will
in general prevent reasonable convexity properties.
However, the main ingredient for studying local causality, the
exponential map, is now only locally Lipschitz and while
it was well-known (\cite{Whitehead}) that it is a local homeomorphism, only
recently in \cite{KSS} it was shown to be in fact bi-Lipschitz. More precisely,
using approximation techniques and employing new methods of Lorentzian
comparison geometry (\cite{CleF}) it was shown in \cite[Th.\ 2.1]{KSS} that the
exponential map retains maximal regularity in the following sense:
\begin{Theorem}\label{mainpseudo}
Let $M$ be a smooth manifold with a $C^{1,1}$-pseudo-Riemannian
metric $g$ and let $p\in M$. Then there exist open neighbourhoods
$\tilde U$ of $0\in T_{p}M$ and $U$ of $p$ in M such that
\begin{equation*}
\exp_{p}:\tilde U\rightarrow\ U
\end{equation*}
is a bi-Lipschitz homeomorphism.
\end{Theorem}
\medskip\noindent
It then follows from Rademacher's theorem that both $\exp_p$ and
$\exp_p^{-1}$ are differentiable almost everywhere. If $\exp_p: \tilde
U \to U$ is a bi-Lipschitz homeomorphism and $\tilde U$ is star-shaped
around $0$ we call $U$ a normal neighbourhood of $p$. If $U$ is a
normal neighbourhood of each of its elements then it is called totally
normal. In the literature (e.g., \cite{ON83}), totally normal sets are
also called convex sets. Any totally normal set $U$ is geodesically
convex in the sense that for any two points in $U$ there is a unique
geodesic contained in $U$ that connects them. Totally normal sets play
an important role in local causality theory, see Section
\ref{mainsec} below. The following result, proved in \cite[Th.\ 4.1]{KSS}
ensures that locally there always exist such neighbourhoods:
\begin{Theorem} \label{totally} Let $M$ be a smooth manifold with a
$C^{1,1}$-pseudo-Riemannian metric $g$. Then each point $p\in M$
possesses a basis of totally normal neighbourhoods.
\end{Theorem}
\noindent
The aim of this paper is to develop the key elements of causality theory for
$C^{1,1}$-Lorentzian metrics based on the above results as well as on refined
regularisation techniques, extending the approach of \cite{CG},
thereby demonstrating that indeed $C^{1,1}$ is the
minimal degree of regularity where a substantial part of
smooth causality theory remains valid.
\medskip\noindent While we were in the final stages of preparing the
present paper we learned that an alternative approach to causality
theory for $C^{1,1}$-Lorentzian metrics by E.\ Minguzzi had recently
appeared in \cite{M}. This paper also establishes the fact
$\exp_p$ is a bi-Lipschitz homeomorphism, and in addition shows that
$\exp$ is a bi-Lipschitz homeomorphism on a neighbourhood of the zero-section
in $TM$ and is strongly differentiable over this zero section
\cite[Th.\ 1.11]{M}. In this work, the required properties of the
exponential map are derived from a careful analysis of the
corresponding ODE problem based on Picard-Lindel\"of approximations,
as well as from an inverse function theorem for Lipschitz maps. In
\cite{M} the author also goes on to establish the Gauss Lemma and to develop
the essential elements of $C^{1,1}$-causality, thereby obtaining many
of the results that are also contained in the present work, some even in
greater generality.
\medskip\noindent
Nevertheless, we believe that our approach is of interest,
and that in fact the approach in \cite{M} and ours nicely complement
each other, for the following reasons: Our methods are a direct
continuation
of the regularisation approach of P.\ Chrusciel and J.\ Grant
(\cite{CG}) and are completely independent from those employed in \cite{M}.
The basic idea is to approximate a given metric of low regularity
(which may be as low as $C^0$) by two
nets of smooth metrics ${\check g}_\epsilon$ and ${\hat g}_\epsilon$
whose light cones sandwich those of $g$. We then continue the line of
argument of \cite{CG,KSS} to establish the key results of causality theory
for a $C^{1,1}$-metric (thereby answering a corresponding question
in \cite{CG} which mainly motivated this work, namely whether the results
of \cite{Chrusciel_causality} remain true for $C^{1,1}$-metrics).
The advantage of these methods is that they quite easily adapt to
regularity below $C^{1,1}$, which as far as we can see is the natural
lower bound for the applicability of those employed in \cite{M}.
As an example, we note that the push-up lemmas from \cite{CG},
cf.\ Prop.\ \ref{push-up1} and \ref{nonnull} below, in fact even hold for
$C^{0,1}$-metrics (or, more generally, for causally plain $C^0$-metrics),
whereas the corresponding results in \cite[Sec.\ 1.4]{M} require the
metric to be $C^{1,1}$.
\medskip\noindent
Furthermore, although considerable work still
needs to be done, we believe that the regularisation approach adopted
here, together with methods from Lorentzian comparison geometry as used
in \cite{CleF} and \cite{KSS}, will allow us to address some of the
other results required (such as curvature estimates, variational properties
of curves, and existence of focal points) in order to establish singularity
theorems for $C^{1,1}$-metrics, where so far only limited results are
available (\cite{Seno1}). Indeed, we note that the relevance of the
kind of approximation techniques advocated in \cite{CG,KSS} for such
questions was already pointed out in \cite[Sec.\ 8.4]{HE}.
\medskip\noindent The plan of the paper is as follows. In section 2 we
introduce the regularisation techniques
and show how they may be applied to establish the Gauss Lemma
(Theorem \ref{Gauss}) for a $C^{1,1}$-pseudo-Riemannian metric. Section 3 deals
with the key elements of $C^{1,1}$-causality theory and in Theorem \ref{lcb}
we again use regularisation methods to show that the local causal
structure is given by the image of the null cone under the exponential
map. This is then used to show
that if a causal curve from $p$ ends at a point in $\partial J^+(p)$
then it is a null geodesic. We then go on to deduce the basic
elements of causality theory using standard methods. Finally in
section 4 we refer to the results of \cite{CG} to show that all the
major building blocks are in place to follow the $C^2$-proofs as given
in \cite{Chrusciel_causality} to establish those elements of causality
theory that do not rely on continuity of the curvature.
\section{Regularisation techniques}\label{mainsec}
Throughout this paper we assume $M$ to be a $C^\infty$-manifold and only lower
the regularity of the metric. This is no loss of generality since any
$C^k$-manifold with $k\ge 1$ possesses a unique $C^\infty$-structure that is
$C^k$-compatible with the given $C^k$-structure on $M$ (see \cite[Th.\
2.9]{Hirsch}).
As already mentioned in the introduction a fundamental tool in our approach
is approximating a given metric of regularity $C^{1,1}$ by a net
$g_\eps$ of $C^\infty$-metrics, in the following sense:
\begin{remark}\label{approxrem}
We cover $M$ by a countable and locally finite collection of
relatively compact chart neighbourhoods and
denote the corresponding charts by $(U_i,\psi_i)$ ($i\in \N$). Let $(\zeta_i)_i$ be a
subordinate partition of unity with $\mathrm{supp}(\zeta_i)\Subset U_i$ (i.e.,
$\mathrm{supp}(\zeta_i)$ is a compact subset of $U_i$) for all $i$
and choose a family of cut-off functions $(\chi_i)_i\in\mathscr{D}(U_i)$ with
$\chi_i\equiv 1$ on a
neighbourhood of $\mathrm{supp}(\zeta_i)$. Finally, let $\rho\in
\mathscr{D}(\R^{n})$ be a test function with unit integral and define the
standard mollifier $\rho_{\eps}(x):=\eps^{-n}\rho\left (\frac{x}{\eps}\right)$
($\eps>0$). Then denoting by $f_*$ (resp.\ $f^*$) push-forward
(resp.\ pullback) under a map $f$, the following formula defines a family
$(g_\eps)_\eps$ of smooth sections of $T^0_2(M)$
\[
g_\eps:=\sum\limits_i\chi_i\,
g_\eps^i:=\sum\limits_i\chi_i\,\psi_i^*\Big(\big(\psi_{i\,*} (\zeta_i\,
g)\big)*\rho_\eps\Big)
\]
which satisfies
\begin{itemize}
\item[(i)] $g_\eps$ converges to $g$ in the $C^1$-topology as $\eps\to 0$, and
\item[(ii)] the second derivatives of $g_\eps$ are bounded, uniformly in $\eps$, on compact sets.
\end{itemize}
On any compact subset of $M$, therefore, for $\eps$ sufficiently
small the $g_\eps$ form a family of pseudo-Riemannian metrics of the
same signature as $g$ whose Riemannian curvature tensors $R_\eps$ are
bounded uniformly in $\eps$. Indeed, properties (i) and
(ii) were the only ones required to derive all results given in \cite{KSS}.
\medskip\noindent
Also observe that the above procedure can be applied even to distributional
sections of any vector bundle $E\to M$ (using the corresponding vector bundle
charts) and that the usual convergence properties of smoothings via convolution
are preserved.
\end{remark}
\medskip\noindent
To distinguish exponential maps stemming from metrics $g_\eps$, etc.,
we will write $\exp_p^{g_\eps}$, etc.. For brevity we will drop this
superscript for the $C^{1,1}$-metric $g$ itself, though.
We shall need the following properties of the exponential maps
corresponding to an approximating net as above:
\begin{Lemma}\label{uniform}
Let $g$ be a $C^{1,1}$-pseudo-Riemannian metric on $M$ and let
$g_\eps$ be a net of smooth pseudo-Riemannian metrics that satisfy
conditions (i) and (ii) of Remark \ref{approxrem}. Then any $p\in M$ has a
basis of normal neighbourhoods $U$ such that, with $\exp_p: \tilde U
\to U$, all $\exp_p^{g_\eps}$ are diffeomorphisms with domain
$\tilde U$ for $\eps$ sufficiently small. Moreover, the inverse maps
$(\exp_p^{g_\eps})^{-1}$ also are defined on a common neighbourhood
of $p$ for $\eps$ small, and converge locally uniformly to
$\exp_p^{-1}$.
\end{Lemma}
\begin{proof}
The claims about the common domains of $\exp_p^{g_\eps}$, resp.\ of
$(\exp_p^{g_\eps})^{-1}$ follow from \cite[Lemma 2.3 and
2.8]{KSS}. To obtain the convergence result, we first note that
without loss, given a common domain $V$ of the
$(\exp_p^{g_\eps})^{-1}$ for $\eps<\eps_0$, we may assume that
$\bigcup_{\eps<\eps_0} (\exp_p^{g_\eps})^{-1}(V)$ is relatively
compact in $\tilde U$: this follows from the fact that the maps
$(\exp_p^{g_\eps})^{-1}$ are Lipschitz, uniformly in $\eps$ (see
\cite{KSS}, the argument following Lemma 2.10).
\medskip\noindent
Now if $(\exp_p^{g_\eps})^{-1}$ did not converge uniformly to
$\exp_p^{-1}$ on some compact subset of $V$ then by our compactness
assumptions we could find a sequence $q_k$ in $V$ converging to some
$q\in V$ and a sequence $\eps_k\searrow 0$ such that
$w_k:=(\exp_p^{g_{\eps_k}})^{-1}(q_k)\to w \not =
\exp_p^{-1}(q)$. But since $(\exp_p^{g_\eps}) \to \exp_p$ locally
uniformly (by \cite[Lemma 2.3]{KSS}), we arrive at $q_k = \exp_p^{g_{\eps_k}}(w_k) \to
\exp_p(w)\not=q$, a contradiction.
\end{proof}
\medskip\noindent
In the particular case of $g$ being Lorentzian, a more sophisticated
approximation procedure, adapted to the causal structure of $g$, was
given in \cite[Prop.\ 1.2]{CG}.
\medskip\noindent
To formulate this result, we first recall that a space-time is a time-oriented
Lorentzian manifold (of signature $(-+\dots+)$), with time-orientation determined by some continuous timelike
vector field. In what follows, all Lorentzian manifolds will be supposed to be
time-oriented. Also we recall from \cite{CG} that for two Lorentzian metrics $g$,
$h$, we say that $h$ has strictly larger light cones than $g$, denoted by $g\prec h$, if for any tangent
vector $X\not=0$, $g(X,X)\le 0$ implies that $h(X,X)<0$.
\medskip\noindent
We will also need the following technical tools:
\begin{Lemma}\label{sbe} Let $(K_m)$ be an exhaustive sequence of compact subsets of a manifold $M$
($K_m\subseteq K_{m+1}^\circ$, $M=\bigcup_m K_m$), and let $\eps_1\ge \eps_2 \ge \dots >0$ be given.
Then there exists some $\psi\in C^\infty(M)$ such that $0<\psi(p)\le \eps_m$ for $p\in K_m\setminus K_{m-1}^\circ$
(where $K_{-1}:=\emptyset$).
\end{Lemma}
\begin{proof} See, e.g., \cite[Lemma 2.7.3]{GKOS}.
\end{proof}
For what follows, recall that $K\Subset M$ denotes that $K$ is a compact subset of $M$.
\begin{Lemma}\label{globallem} Let $M$, $N$ be manifolds, and set $I:=(0,\infty)$.
Let $u: I\times M \to N$ be a smooth map
and let (P) be a property attributable to values $u(\eps,p)$, satisfying:
\begin{itemize}
\item[(i)] For any $K\Subset M$ there exists some $\eps_K>0$ such that (P) holds for
all $p\in K$ and $\eps<\eps_K$.
\item[(ii)] (P) is stable with respect to decreasing $K$ and $\eps$:
if $u(\eps,p)$ satisfies (P) for all $p\in K\Subset M$ and
all $\eps$ less than some $\eps_K>0$ then for any compact set $K'\subseteq K$
and any $\eps_{K'} \le \eps_K$, $u$ satisfies (P) on $K'$ for all
$\eps\le \eps_{K'}$.
\end{itemize}
Then there exists a smooth map $\tilde u: I\times M \to N$
such that (P) holds for all $\tilde u(\eps,p)$ ($\eps\in I$, $p\in M$) and
for each $K\Subset M$ there exists some $\eps_K\in I$ such that
$\tilde u(\eps,p) = u(\eps,p)$ for all $(\eps,p)\in (0,\eps_K] \times K$.
\end{Lemma}
\begin{proof} See \cite[Lemma 4.3]{HKS}.
\end{proof}
\medskip\noindent
Based on these auxiliary results, we can prove the following refined version
of \cite[Prop.\ 1.2]{CG}:
\begin{Proposition}\label{CGapprox} Let $(M,g)$ be a space-time with a
continuous Lorentzian metric, and $h$ some smooth
background Riemannian metric on $M$. Then for any $\eps>0$, there exist smooth
Lorentzian metrics $\check g_\eps$ and $\hat g_\eps$ on $M$ such that $\check g_\eps
\prec g \prec \hat g_\eps$ and $d_h(\check g_\eps,g) + d_h(\hat g_\eps,g)<\eps$,
where
$$
d_h(g_1,g_2) := \sup_{0\not=X,Y\in TM} \frac{|g_1(X,Y)-g_2(X,Y)|}{\|X\|_h \|Y\|_h}.
$$
Moreover, $\hat g_\eps$ and $\check g_\eps$ depend smoothly on $\eps$, and if
$g\in C^{1,1}$ then $\check g_\eps$ and $\hat g_\eps$ additionally satisfy
(i) and (ii) from Rem.\ \ref{approxrem}.
\end{Proposition}
\begin{proof} First we use time-orientation to obtain a $C^{1,1}$
timelike one-form $\tilde\omega$ (the $g$-metric equivalent of a smooth timelike vector field).
Using the smoothing procedure of Rem.\ \ref{approxrem}, on each $U_i$ we can pick $\eps_i>0$ so small
that $\tilde\omega_{\eps_i}$ is timelike on $U_i$. Then $\omega := \sum_i \zeta_i \tilde \omega_{\eps_i}$ is a
smooth timelike one-form on $M$. By compactness we obtain on every $U_i$ a constant $c_i>0$ such that
\begin{equation}\label{ci}
|\omega(X)|\geq c_i\quad \text{for all $g$-causal vector fields $X$
with $\|X\|_h=1$}.
\end{equation}
\medskip\noindent
Next we set on each $U_i$ and for $\eta>0$ and $\lambda<0$
\begin{equation}\label{gel}
{\hat g}^i_{\eta,\lambda}=g_\eta^i+\lambda\, \omega\otimes\omega,
\end{equation}
where $g^i_\eta$ is as in Remark \ref{approxrem} (set $\eps:=\eta$ there and $g^i_\eta:=g_\eta|_{U_i}$).
Let $\Lambda_k$ ($k\in\N$) be a compact exhaustion of $(-\infty,0)$. For each $k$, there exists
some $\eta_k>0$ such that $\eta_k< \min_{\lambda\in \Lambda_k} |\lambda|$, $\eta_k>\eta_{k+1}$ for all $k$, and
\begin{equation}\label{geps-g}
|g^i_\eta(X,X)-g(X,X)|\leq |\lambda|\,\frac{c_i^2}{2}
\end{equation}
for all $g$-causal vector fields $X$ on $U_i$ with $\|X\|_h=1$, all $\lambda \in \Lambda_k$, and all $0< \eta\le \eta_k$.
Thus by Lemma \ref{sbe} there exists a smooth function $\lambda\mapsto \eta(\lambda,i)$ on $(-\infty,0)$
with $0<\eta(\lambda,i)\le |\lambda|$ and such that \eqref{geps-g} holds
for all $g$-causal vector fields $X$ on $U_i$ with $\|X\|_h=1$, all $\lambda$, and all $0< \eta\le \eta(\lambda,i)$.
\medskip\noindent
Combining
\eqref{ci} with \eqref{geps-g} we obtain
\[
{\hat g}^i_{\eta,\lambda}(X,X)
=g(X,X)+(g^i_\eta-g)(X,X)+\lambda\,\omega(X)^2
\leq0+\Big(|\lambda|\frac{c_i^2}{2}+\lambda c_i^2\Big)\|X\|_h^2<0,
\]
for all $g$-causal $X$ and hence $g \prec {\hat g}^i_{\eta,\lambda}$
for all $\lambda<0$ and $0<\eta\leq\eta(\lambda,i)$.
\medskip\noindent
Given a compact exhaustion $E_k$ ($k\in \N$) of $(0,\infty)$, for each $k$ there exists
some $\lambda_k<0$ such that $|\lambda_k|<\min_{\eps\in E_k} \eps$,
$\lambda_k<\lambda_{k+1}$ for all $k$, and
\[
d_{U_i}(\hat g^i_{\eta(\lambda,i),\lambda},g)
:=\sup_{0\not=X,Y\in TU_i}\,\frac{|\hat g^i_{\eta(\lambda,i),\lambda}(X,Y)-g(X,Y)|}{
\|X\|_h\|Y\|_h}
<\,\frac{\eps}{2^{i+1}}.
\]
for all $\eps\in E_k$ and all $\lambda_k\le \lambda <0$. Again by Lemma \ref{sbe} we
obtain a smooth map $(0,\infty) \to (-\infty,0)$, $\eps\to \lambda_i(\eps)$ such that $|\lambda_i(\eps)|< \eps$
for all $\eps$, and
$d_{U_i}(\hat g^i_{\eta(\lambda_i(\eps),i),\lambda_i(\eps)},g)<\frac{\eps}{2^{i+1}}$
for all $\eps>0$.
We now consider the smooth symmetric $(0,2)$-tensor field on $M$,
\[
g_\eps:=\sum_i\chi_i \hat g^i_{\eta(\lambda_i(\eps),i),\lambda_i(\eps)}.
\]
By construction, $(\eps,p)\mapsto g_\eps(p)$ is smooth, and $g_\eps$ converges to $g$
locally uniformly as $\eps\to 0$. Therefore, for any $K\Subset M$
there exists some $\eps_K$ such that for all $0<\eps<\eps_K$, $g_\eps$ is of the same
signature as $g$, hence a Lorentzian metric on $K$,
with strictly larger lightcones than $g$. We are thus in a position to apply Lemma \ref{globallem}
to obtain a smooth map $(\eps,p)\mapsto \hat g_\eps(p)$ such that for each fixed $\eps$, $\hat g_\eps$ is a globally
defined Lorentzian metric which on any given $K\Subset M$ coincides with $g_\eps$ for
sufficiently small $\eps$.
\medskip\noindent
Then $d_h(\hat g_\eps,g)< \eps/2$, and $\eps\to 0$ implies $ \lambda_i(\eps) \to 0$ and a fortiori $\eta(\lambda_i(\eps),i)\to 0$
for each $i\in \N$.
\medskip\noindent
From this, by virtue of \eqref{gel}, (i) and (ii) of Remark
\ref{approxrem} hold for $\hat g_\eps$ if $g\in C^{1,1}$.
\medskip\noindent
The approximation $\check g_\eps$ is constructed analogously choosing $\lambda>0$.
\end{proof}
\begin{remark}
\begin{itemize}
\item[(i)] From Rem.\ \ref{approxrem} and the above proof it follows that, given a
Lorentzian metric of some prescribed regularity (e.g., Sobolev, H\"older, etc.),
the inner and outer regularisations $\check g_\eps$ and $\hat g_\eps$ converge to $g$
as good as regularisations by convolution do locally.
\item[(ii)] If $g$ is a metric of general pseudo-Riemannian signature, then
since $g_\eps$ in Rem.\ \ref{approxrem} depends smoothly on $\eps$, also in this case an application
of Lemma \ref{globallem} allows
to produce regularisations $\tilde g_\eps$ that are pseudo-Riemannian metrics on all of $M$
of the same signature as $g$ and satisfy (i) and (ii) from that remark.
\end{itemize}
\end{remark}
To conclude this section we derive the Gauss lemma for $C^{1,1}$-metrics.
This result has first appeared (in a more general form) in \cite{M}. In the spirit of
our approach, we include an independent proof using regularisation methods.
\begin{Theorem}\label{Gauss} (The Gauss Lemma)
Let $g$ be a $C^{1,1}$-pseudo-Riemannian metric on $M$, and let
$p\in M$. Then $p$ possesses a basis of normal neighbourhoods $U$ with the following
properties: $\exp_p: \tilde U \to U$ is a bi-Lipschitz
homeomorphism, where $\tilde U$ is an open star-shaped neighbourhood
of $0$ in $T_pM$. Moreover,
for almost all $x\in \tilde U$, if $v_x$, $w_x\in T_x(T_pM)$ and
$v_x$ is radial, then
$$
\langle T_x\exp_p(v_x), T_x\exp_p(w_x)\rangle = \langle v_x, w_x \rangle.
$$
\end{Theorem}
\begin{proof} Let $g_{\eps}$ be approximating smooth metrics as in Rem.\
\ref{approxrem}. Take $U$, $\tilde U$ as in Lemma \ref{uniform} and let
$x\in \tilde U$ be such that $T_x\exp_p$ exists. By bilinearity,
we may assume that $x=v_x=v$ and $w_x=w$. Let
$f(t,s):=\exp_p(t(v+sw))$. Then $(t,s)\mapsto f(t,s)$ is $C^{0,1}$
hence $(t,s)\mapsto f_{t}(t,s)\in L^{\infty}_{\text{loc}}$ and
$(t,s)\mapsto f_{s}(t,s)\in L^{\infty}_{\text{loc}}$. For any fixed
$s$, however, $t\mapsto f(t,s)$ is $C^{2}$, as is $s\mapsto f(t,s)$
for any $t$ fixed (both curves being geodesics).
\medskip\noindent Let
$f^{\eps}(t,s):=\exp_{p}^{g_{\eps}}(t(v+sw))$. Then by the smooth
Gauss lemma, for all $\eps$ we have:
\begin{equation*}
\langle T_{v}\exp_{p}^{g_{\eps}}(v),T_{v}\exp_{p}^{g_{\eps}}(w)\rangle
=\langle f^{\eps}_{s}(1,0),f^{\eps}_{t}(1,0)\rangle
=\langle v,w \rangle.
\end{equation*}
By standard ODE estimates (see \cite[Lemma 2.3]{KSS} and the
discussion following it) it follows that $\forall v$:
$\exp_{p}^{g_{\eps}}(tv)\rightarrow \exp_{p}(tv)$ in $C^{1}(\R)$ for
$\eps\to 0$. Thus, we have:
\begin{align*}
f_{t}^{\eps}(1,0)&=\partial_{t}|_{1}\exp_{p}^{g_{\eps}}(tv)\rightarrow \partial_{t}|_{1}\exp_{p}(tv)\ (\eps\rightarrow 0)\\
f_{s}^{\eps}(1,0)&=\partial_{s}|_{0}\exp_{p}^{g_{\eps}}(v+sw)\rightarrow \partial_{s}|_{0}\exp_{p}(v+sw)\ (\eps\rightarrow 0)
\end{align*}
and therefore, whenever $T_{v}\exp_{p}$ exists,
\begin{equation*}
\langle T_{v}\exp_{p}(v), T_{v}\exp_{p}(w)\rangle =\langle v,w \rangle.
\end{equation*}
\end{proof}
\section{Causality theory}
As in \cite{Chrusciel_causality} we will base our approach to
causality theory on locally Lipschitz curves.
We note that this definition differs from that in \cite{M}, where
the corresponding curves are required to be $C^1$ (see, however, Cor.\ \ref{lipisc1} below or \cite[Th.\ 1.27]{M}).
Any locally Lipschitz curve $c$ is
differentiable almost everywhere (Rademacher's theorem) and we call
$c$ timelike, causal, spacelike or null, if $c'(t)$ has the
corresponding property whenever it exists. If the time-orientation of
$M$ is determined by a continuous timelike vector field $X$ then a causal curve
$c$ is called future- resp.\ past-directed if $\langle
X(c(t)),c'(t)\rangle < 0$ resp.\ $>0$ wherever $c'(t)$ exists. With
these notions we have:
\begin{definition}\label{causalitydef}
Let $g$ be a $C^0$-Lorentzian metric on $M$. For $p\in A \subseteq M$
we define the relative chronological, respectively causal future of $p$ in $A$ by
(cf.\ \cite[2.4]{Chrusciel_causality}):
\begin{equation*}
\begin{split}
I^{+}(p,A) &:=\{q\in A |\ \text{there exists a future directed timelike curve in $A$ from $p$ to $q$ }\} \\
J^{+}(p,A) &:=\{q\in A |\ \text{there exists a future directed causal curve in $A$ from $p$ to $q$ }\}\cup A.
\end{split}
\end{equation*}
For $B\subseteq A$ we set $I^+(B,A) := \bigcup_{p\in B} I^+(p,A)$ and analogously for $J^+(B,A)$.
We set $I^+(p):=I^+(p,M)$. Replacing `future directed' by `past-directed' we obtain the corresponding
definitions of the chronological respectively causal pasts $I^-$, $J^-$.
\end{definition}
\medskip\noindent
Below we will formulate all results for $I^+$, $J^+$. By symmetry, the corresponding claims for
chronological or causal pasts follow in the same way.
\medskip\noindent
As usual, for $p$, $q\in M$ we write $p < q$, respectively $p\ll q$, if
there is a future directed causal, respectively timelike, curve from $p$
to $q$. By $p\le q$ we mean $p=q$ or $p<q$.
\medskip\noindent
We now recall some definitions that were introduced in \cite{CG} and
results there obtained which will be of use in this paper.
\begin{definition}
A locally Lipschitz curve $\alpha: [0,1]\rightarrow M$ is said to be
locally uniformly timelike (l.u.-timelike) with respect to the $C^0$-metric $g$
if there exists a smooth
Lorentzian metric $\check g\prec g$ such that $\check g(\alpha',\alpha')<0$
almost everywhere. Then for $p\in A\subseteq M$
\begin{equation*}
\check I^+_g(p,A):=\{q\in A|\ \text{there exists a future directed
l.u.-timelike curve in $A$ from $p$ to $q$} \}.
\end{equation*}
\end{definition}
Thus $\check I^+_g(A)=\bigcup_{\check g\prec g}I^+_{\check g}(A)$, hence it is open (\cite[Prop.\ 1.4]{CG}).
The following definition (\cite[Def.\ 1.8]{CG}) introduces a highly useful substitute for
normal coordinates in the context of metrics of low regularity
\begin{definition}\label{def:cylindrical chart}
Let $(M,g)$ be a smooth Lorentzian manifold with continuous metric $g$
and let $p\in M$. A relatively compact open subset $U$ of $M$ is called
a cylindrical neighbourhood of $p\in U$ if there exists a smooth chart
$(\vphi,U)$, $\vphi= (x^0,...,x^{n-1})$ with $\vphi(U)=I\times V$,
$I$ an interval around $0$ in $\R$ and $V$ open in $\R^{n-1}$, such that:
\begin{enumerate}
\item $\frac{\partial}{\partial x^0}$ is timelike and $\frac{\partial}{\partial x^i}$,
$i=1,...,n-1$, are spacelike,
\item For $q\in U,\ v\in T_qM$, if $g_q(v,v)=0$ then $\frac{|v^0|}{\|\vec{v}\|}\in (\frac{1}{2},2)$
(where $T_q\vphi(v)=(v^0,\vec{v})$, and $\|\,\|$ is the Euclidean norm on $\R^{n-1}$),
\item $(\varphi_\ast g)_{\vphi(p)}=\eta$ (the Minkowski metric).
\end{enumerate}
\end{definition}
By \cite[Prop.\ 1.10]{CG}, every point in a spacetime with continuous metric possesses a basis
of cylindrical neighbourhoods.
According to \cite[Def.\ 1.16]{CG}, a Lorentzian manifold $M$ with $C^0$-metric $g$ is called causally plain if for
every $p\in M$ there exists a cylindrical neighbourhood $U$ of $p$ such
that $\partial \check I^\pm (p,U)=\partial J^\pm (p,U)$. This condition excludes
causally `pathological' behaviour (bubbling metrics).
By \cite[Cor.\ 1.17]{CG}, we have:
\begin{Proposition}\label{lipplain}
Let $g$ be a $C^{0,1}$-Lorentzian metric on M. Then $(M,g)$ is
causally plain.
\end{Proposition}
The most important property of causally plain Lorentzian manifolds for our purposes is
given in the following result (\cite[Prop.\ 1.21]{CG}).
\begin{Proposition} \label{ichecki}
Let $g$ be a continuous, causally plain Lorentzian metric and let $A\subseteq M$. Then
\begin{equation}
I^\pm(A)=\check I^\pm(A).
\end{equation}
\end{Proposition}
Furthermore, we will make use of the following `push-up' results (\cite[Lemma 1.22]{CG},
\cite[Prop.\ 1.23]{CG}):
\begin{Proposition}\label{push-up1} Let $g$ be a causally plain $C^0$-Lorentzian metric on $M$ and
let $p,\, q,\, r\in M$ with $p\le q$ and $q\ll r$ or $p\ll q$ and $q\le r$. Then $p\ll r$.
\end{Proposition}
\begin{Proposition}\label{nonnull}
Let $M$ be a spacetime with a
continuous causally plain metric $g$. Consider a causal future-directed curve $\alpha:[0,1]\to M$
from $p$ to $q$. If there exist $s_1$, $s_2\in [0, 1]$, $s_1 < s_2$, such that $\alpha|_{[s_1,s_2]}$
is timelike, then in any neighbourhood of $\alpha([0,1])$ there exists a timelike future-directed curve from $p$ to $q$.
\end{Proposition}
\medskip\noindent
Returning now to our main object of study, for the remainder of the paper
$g$ will denote a $C^{1,1}$-Lorentzian metric. Then in particular, $g$ is
causally plain by Prop.\ \ref{lipplain}.
To analyse the local causality for $g$ in terms of
the exponential map we first introduce some terminology. Let $\tilde
U$ be a star-shaped neighbourhood of $0\in T_pM$ such that $\exp_p:
\tilde U \to U$ is a bi-Lipschitz homeomorphism (Th.\
\ref{mainpseudo}). On $T_pM$ we define the position vector field
$\tilde P: v \mapsto v_v$ and the quadratic form $\tilde Q: T_pM \to
\R$, $v\mapsto g_p(v,v)$. By $P$, $Q$ we denote the push-forwards of
these maps via $\exp_p$, i.e.,
\begin{equation*}
\begin{split}
P(q) &:= T_{\exp_p^{-1}(q)}\exp_p (\tilde P (\exp_p^{-1}(q)) )\\
Q(q) &:= \tilde Q (\exp_p^{-1}(q)).
\end{split}
\end{equation*}
As $\exp_p$ is locally Lipschitz, $P$ is an $L^\infty_{\text{loc}}$-vector field on $U$, while $Q$ is
locally Lipschitz (see, however, Rem.\ \ref{regrem} below).
\medskip\noindent
Let $X$ be some smooth vector field on $U$ and denote by $\tilde X$ its pullback $\exp_p^*X$ (note that
$T_v\exp_p$ is invertible for almost every $v\in\tilde U$).
Then by Th.\ \ref{Gauss}, for almost every $q\in U$ we have, setting $\tilde q := \exp_p^{-1}(q)$:
\begin{align*}
\langle \grad Q(q), X(q) \rangle &=X(Q)(q)=\tilde{X}(\tilde Q)(\tilde q)=\langle \grad \tilde{Q},\tilde{X}\rangle|_{\tilde q}=
2\langle \tilde{P},\tilde{X} \rangle|_{\tilde q}=2\langle P,X \rangle|_{q}.
\end{align*}
It follows that $\grad Q = 2P$.
\medskip\noindent
\begin{remark}\label{regrem}
It is proved in \cite{M} that the regularity of both $P$ and $Q$ is better than would
be expected from the above definitions. Indeed, \cite[Prop.\ 2.3]{M} even shows that $P$, as
a function of $(p,q)$ is strongly differentiable on a neighbourhood of the diagonal in
$M\times M$, and by \cite[Th.\ 1.18]{M}, $Q$ is in fact $C^{1,1}$ as a
function of $(p,q)$. We will however not
make use of these results in what follows and only remark that slightly weaker
regularity properties of $P$ and $Q$ (as functions of $q$ only) can also be obtained
directly from standard ODE-theory. In fact, setting $\alpha_v(t):=\exp_p(tv)$ for $v\in T_pM$,
it follows that $P(q)=\alpha_{v_q}'(1)$, where $v_q:=\exp_p^{-1}(q)$.
Since $t\mapsto (\alpha_v(t),\alpha_v'(t))$ is the solution
of the first-order system corresponding to the geodesic equation with initial value $(p,v)$,
and since the right-hand side of this system is Lipschitz-continuous, \cite[Th.\ 8.4]{Amann}
shows that $v\mapsto \alpha_v'(1)$ is Lipschitz-continuous. Since also $q\mapsto v_q$
is Lipschitz, we conclude that $P$ is Lipschitz-continuous.
From this, by the above calculation, it follows that $Q$ is $C^{1,1}$.
\end{remark}
\medskip\noindent
As in the smooth case, we may use $\exp_p$ to introduce normal
coordinates. To this end, let $e_0$, \dots, $e_n$ be an orthonormal
basis of $T_pM$ and for $q\in U$ set $x^i(q)e_i:=\exp_p^{-1}(q)$. The
coordinates $x^i$ then are of the same regularity as $\exp_p^{-1}$,
i.e., locally Lipschitz. The coordinate vector fields
$\left.\frac{\partial}{\partial x^i}\right|_q =
T_{\exp_p^{-1}(q)}\exp_p(e^i)$ themselves are in
$L^\infty_{\text{loc}}$. Note, however, that in the $C^{1,1}$-setting
we can no longer use the relation $g_p = \eta$ (the Minkowski-metric
in the $x^i$-coordinates), since it is not clear a priori that
$\exp_p$ is differentiable at $0$ with
$T_0\exp_p=\text{id}_{T_pM}$\footnote{See, however, \cite{M} where it
is shown that indeed $\exp_p$ is even strongly differentiable at $0$
with derivative $\text{id}_{T_pM}$.}. Due to the additional loss in
regularity it is also usually not advisable to write the metric in
terms of the exponential chart (the metric coefficients in these
coordinates would only be $L^\infty_{\text{loc}}$).
The following is the main result on the local causality in normal neighbourhoods.
\begin{Theorem}\label{lcb}
Let $g$ be a $C^{1,1}$-Lorentzian metric, and let $p\in M$. Then $p$ has a basis of normal neighbourhoods $U$,
$\exp_p: \tilde U\to U$ a bi-Lipschitz homeomorphism, such that:
\begin{equation*}
\begin{split}
I^{+}(p,U)=\exp_{p}(I^{+}(0)\cap \tilde{U})\\
J^{+}(p,U)=\exp_{p}(J^{+}(0)\cap \tilde{U}) \\
\partial I^{+}(p,U) = \partial J^{+}(p,U) =\exp_{p}(\partial I^{+}(0)\cap \tilde{U})
\end{split}
\end{equation*}
Here, $I^+(0)= \{v\in T_pM \mid \tilde Q(v)<0 \}$, and $J^+(0)= \{v\in T_pM \mid \tilde Q(v)\le 0 \}$.
In particular, $I^+(p,U)$ (respectively $J^+(p,U)$) is open (respectively closed) in $U$.
\end{Theorem}
\begin{proof} We first note that the third claim follows from the first two and the fact that
$\exp_p$ is a homeomorphism on $U$. For the proof of the first two claims we take a normal neighbourhood $U$ that is contained in a cylindrical neighbourhood of $p$.
In addition, we pick a regularising net $\hat g_\eps$ as
in Prop.\ \ref{CGapprox} and let $U$, $\tilde U$ as in Lemma
\ref{uniform} (fixing a suitable $\eps_0>0$).
\noindent $(\supseteq)$
Let $v\in \tilde U$ and let $\alpha:= t\mapsto \exp_p(tv)$, $t\in [0,1]$.
Set $\alpha_\eps(t):=\exp_p^{\hat g_\eps}(tv)$. Then by continuous dependence on initial data
we have that $\alpha_\eps\rightarrow \alpha$ in $C^{1}$ (cf.\ \cite[Lemma 2.3]{KSS}).
Hence applying the smooth Gauss lemma for each $\eps$ it follows that for each $t\in [0,1]$
we have
$$
g(\alpha'(t),\alpha'(t)) = \lim_{\eps\to 0} \hat g_\eps(\alpha_\eps'(t),\alpha_\eps'(t))
= \lim_{\eps\to 0} (\hat g_\eps)_p(v,v) = g_p(v,v).
$$
Also, time-orientation is respected by $\exp_p$ since both $I(0)\cap \tilde U$ and
$I(p,U)$ (by \cite[Prop.\ 1.10]{CG}) have two connected components,
and the positive $x^0$-axis in $\tilde U$ is mapped to $I^+(p,U)$.
\medskip\noindent
$(\subseteq)$: We denote the
position vector fields and quadratic forms corresponding to
$\hat g_\eps$ by $\tilde P_\eps$, $P_\eps$ and
$\tilde Q_\eps$, $Q_\eps$, respectively.
\medskip\noindent
If $\alpha:[0,1]\to U$ is a future-directed causal curve in $U$
emanating from $p$ then $\alpha$ is timelike with respect to each
$\hat g_\eps$. Set $\beta:=(\exp_p)^{-1}\circ\alpha$ and $\beta_\eps:=
(\exp^{\hat g_\eps}_p)^{-1}\circ \alpha$. By \cite[Prop.\
2.4.5]{Chrusciel_causality}, $\beta_\eps([0,1]) \subseteq I^+_{\hat
g_\eps(p)}(0)$ for all $\eps<\eps_0$. Then by Lemma \ref{uniform}
we have that $\beta_\eps \to \beta$ uniformly, and that
$\tilde Q_\eps\to \tilde Q$ locally uniformly, so $\tilde Q(\beta(t)) = \lim
\tilde Q_\eps(\beta_\eps(t)) \le 0$ for all $t\in [0,1]$, and therefore
$\beta((0,1])\subseteq J^+(0)\cap \tilde U$. Together with the first
part of the proof it follows that $\exp_p(J^+(0)\cap \tilde U) = J^+(p,U)$.
Now assume that $\alpha$ is timelike. Then by Prop.\ \ref{ichecki},
$\alpha((0,1])\subseteq \check{I}^{+}(p,U)$.
This means that there exists a smooth metric $\check{g}\prec g$ such
that $\alpha$ is $\check{g}$-timelike. Let $f_{\check{g}}, f_{g}$
denote the graphing functions of $\partial I^{+}_{\check{g}}(p,U)$
and $\partial J^{+}(p,U)$, respectively (in a cylindrical chart, see \cite[Prop.\ 1.10]{CG}).
Then by \cite[Prop.\ 1.10]{CG}, since $\alpha$ lies in $I^{+}_{\check{g}}(p,U)$,
it has to lie strictly above $f_{\check{g}}$, hence also strictly above $f_{g}$,
and so $\alpha((0,1])\cap \partial J^{+}(p,U)=\emptyset$.
But then, since $\exp_p$ is a homeomorphism on $U$, we have that
$$ \beta((0,1])\cap (\partial J^+(0)\cap \tilde U) =
\beta((0,1])\cap \exp_{p}^{-1}(\partial J^{+}(p,U))=
\exp_p^{-1}(\alpha((0,1])\cap \partial J^{+}(p,U))=\emptyset$$
Hence $\beta$ lies entirely in $I^+(0)\cap \tilde U$, as claimed.
\end{proof}
\begin{Corollary}\label{lipisc1}
Let $U\subseteq M$ be open, $p\in U$. Then the sets $I^+(p,U)$,
$J^+(p,U)$ remain unchanged if, in Def.\ \ref{causalitydef},
Lipschitz curves are replaced by piecewise $C^1$ curves, or in fact
by broken geodesics.
\end{Corollary}
\begin{proof}
Let $\alpha: [0,1] \to U$ be a, say, future directed timelike
Lipschitz curve in $U$. By Th.\ \ref{totally} and Th.\ \ref{lcb} we
may cover $\alpha([0,1])$ by finitely many totally normal open sets
$U_i\subseteq U$, such that there exist $0=t_0 < \dots <t_N = 1$
with $\alpha([t_i,t_{i+1}])\subseteq U_{i+1}$ and
$I^{+}(\alpha(t_i),U_i)=\exp_{\alpha(t_i)}(I^{+}(0)\cap
\tilde{U_i})$ for $0\le i < N$. Then the concatenation of the radial
geodesics in $U_i$ connecting $\alpha(t_i)$ with $\alpha(t_{i+1})$
gives a timelike broken geodesic from $\alpha(0)$ to $\alpha(1)$ in
$U$.
\end{proof}
\medskip\noindent
The following analogue of \cite[Cor.\ 2.4.10]{Chrusciel_causality} provides more
information about causal curves intersecting the boundary of
$J^+(p,U)$:
\begin{Corollary}\label{boundary}
Under the assumptions of Th.\ \ref{lcb}, suppose that $\alpha: [0,1]
\to U$ is causal and $\alpha(1)\in \partial J^+(p,U)$. Then $\alpha$
lies entirely in $\partial J^+(p,U)$ and there exists a
reparametrisation of $\alpha$ as a null-geodesic segment.
\end{Corollary}
\begin{proof}
Suppose to the contrary that there exists $t_0\in (0,1)$ such that
$\alpha(t_0)\in I^+(p,U)$. Then there exists a future directed
timelike curve $\gamma$ from $p$ to $\alpha(t_0)$.
Applying Prop. \ref{nonnull} to the concatenation
$\gamma\cup \alpha|_{[t_0,1]}$ it follows that there exists a future
directed timelike curve from $p$ to $\alpha(1)$. But then
$\alpha(1)\in I^+(p,U)$, a contradiction. Thus $\alpha(t)\in \partial
J^+(p,U),\ \forall t\in[0,1]$,
implying that $\beta(t)=\exp_p^{-1}\circ\alpha(t)\in \partial J^+(0),\ \forall t\in[0,1]$, so
$\tilde{Q}(\beta(t))=0,\ \forall t\in [0,1]$ and for almost all $t$ we have
\begin{equation*}
0=\frac{d}{dt}\tilde Q(\beta(t))=g_p(\grad \tilde Q(\beta(t)),\beta'(t))
=g_p(2\tilde P(\beta(t)),\beta'(t)).
\end{equation*}
Hence $\beta(t)$ is collinear with $\beta'(t)$
almost everywhere, and it is easily seen that this implies
the existence of some $v\neq 0, v\in \partial J^+(0)$, and of some
$h:\R\rightarrow \R$ such that $\beta(t)=h(t)v$. The function
$h$ is locally Lipschitz since $\beta$ is, and injective since $\alpha$
is (on every cylindrical neighbourhood there is a natural time function).
Thus $h$ is strictly monotonous, and in fact strictly
increasing since otherwise $\beta$ would enter $J^-(0)$. Thus
$\beta'(t)=f(t)\beta(t)$ where $f(t):= \frac{h'(t)}{h(t)}\in
L^\infty_{\text{loc}}$. From here we may argue exactly as in
\cite[Cor.\ 2.4.10]{Chrusciel_causality}: the function $r(s):=\int_0^s f(\tau)\,
d\tau$ is locally Lipschitz and strictly increasing, hence a
bijection from $[0,1]$ to some interval $[0,r_0]$. Thus so is its
inverse $r\to s(r)$, and we obtain $\beta(s(r))' =
\beta'(s(r))/f(s(r)) = \beta(s(r))$ a.e., where the right hand side
is even continuous. It follows that in this parametrisation, $\beta$
is $C^1$ and in fact is a straight line in the null cone, hence
$\alpha$ can be parametrised as a null-geodesic segment, as claimed.
\end{proof}
\begin{Corollary}
The relation $\ll$ is open: if $p\ll q$ then there exist
neighbourhoods $V$ of $p$ and $W$ of $q$ such that $p'\ll q'$ for all
$p'\in V$ and $q'\in W$. In particular, for any $p\in M$, $I^+(p)$
is open in $M$.
\end{Corollary}
\begin{proof}
Let $\alpha$ be a future-directed timelike curve from $p$ to $q$ and
pick totally normal neighbourhoods $N_p$, $N_q$ of $p$, $q$ as in
Th.\ \ref{lcb}. Now let $p'\in N_p$ and $q'\in N_q$ be points on
$\alpha$. Then $V:=I^-(p',N_p)$ and $W:=I^+(q',N_q)$ have the
required property.
\end{proof}
\medskip\noindent
From this we immediately conclude:
\begin{Corollary} Let $A\subseteq U \subseteq M$, where $U$ is open. Then
$$
I^+(A,U) = I^+(I^+(A,U)) = I^+(J^+(A,U)) = J^+(I^+(A,U)) \subseteq J^+(J^+(A,U)) = J^+(A,U)
$$
\end{Corollary}
\medskip\noindent
A consequence of Prop.\ \ref{nonnull} is that the causal future
of any $A\subseteq M$ consists (at most) of $A$, $I^+(A)$ and of
null-geodesics emanating from $A$:
\begin{Corollary}\label{on}
Let $A\subseteq M$ and let $\alpha$ be a causal curve from some
$p\in A$ to some $q\in J^+(A)\setminus I^+(A)$. Then $\alpha$ is a
null-geodesic that does not meet $I^+(A)$.
\end{Corollary}
\begin{proof}
By Prop.\ \ref{nonnull}, $\alpha$ has to be a null curve. Moreover,
if $\alpha(t)\in I^+(A)$ for some $t$ then for some $a\in A$ we
would have $a\ll \alpha(t) \le q$, so $q\in I^+(A)$ by Prop.\ \ref{push-up1}, a
contradiction. Covering $\alpha$ by totally normal neighbourhoods as in Cor.\ \ref{lipisc1}
and applying Cor.\ \ref{boundary} gives the claim.
\end{proof}
\medskip\noindent
Following \cite[Lemma 14.2]{ON83} we next give a more refined
description of causality for totally normal sets. For this, recall
from the proof of \cite[Th.\ 4.1]{KSS} that the map $E: v\mapsto
(\pi(v),\exp_{\pi(v)}(v))$ is a homeomorphism from some open neighbourhood $S$
of the zero section in $TM$ onto an open neighbourhood $W$ of the
diagonal in $M\times M$. If $U$ is totally normal as in Th.\ \ref{lcb} and such that $U\times
U \subseteq W$ then the map $U\times U \to TM$, $(p,q)\mapsto
\overrightarrow{pq}:=\exp_p^{-1}(q) = E^{-1}(p,q)$ is continuous.
\begin{Proposition} Let $U\subseteq M$ be totally normal as in Th.\ \ref{lcb}.
\begin{itemize}
\item[(i)] Let $p$, $q\in U$. Then $q\in I^+(p,U)$ (resp.\ $\in J^+(p,U)$) if and only
if $\overrightarrow{pq}$ is future-directed timelike (resp.\ causal).
\item[(ii)] $J^+(p,U)$ is the closure of $I^+(p,U)$ relative to $U$.
\item[(iii)] The relation $\le$ is closed in $U\times U$.
\item[(iv)] If $K$ is a compact subset of $U$ and $\alpha: [0,b)\to K$ is causal, then $\alpha$
can be continuously extended to $[0,b]$.
\end{itemize}
\end{Proposition}
\begin{proof}
(i) and (ii) are immediate from Th.\ \ref{lcb}.
\medskip\noindent
(iii) Let $p_n \le q_n$, $p_n\to p$, $q_n\to q$. By (i),
$\overrightarrow{p_nq_n}$ is future-directed causal for all $n$. By
continuity (\cite[Th.\ 4.1]{KSS}), therefore,
$\langle\overrightarrow{pq},\overrightarrow{pq}\rangle \le 0$, so
$\overrightarrow{pq}$ is future-directed causal as well.
\medskip\noindent
(iv) Let $0< t_1 < t_2 < \dots \to b$. Since $K$ is compact,
$\alpha(t_i)$ has an accumulation point $p$ and it remains to show
that $p$ is the only accumulation point. Suppose that $q\not=p$ is
also an accumulation point. Choose a subsequence $t_{i_k}$ such that
$\alpha(t_{i_{2k}}) \to p$ and $\alpha(t_{i_{2k+1}}) \to q$. Then
since $\alpha(t_{i_{2k}})\le \alpha(t_{i_{2k+1}}) \le
\alpha(t_{i_{2k+2}})$, (iii) implies that $p\le q\le p$. By (i),
then, $\overrightarrow{pq}$ would be both future- and past-directed,
which is impossible.
\end{proof}
\medskip\noindent
From this, with the same proof as in \cite[Lemma 14.6]{ON83} we
obtain:
\begin{Corollary} Let $A\subseteq M$. Then
\begin{itemize}
\item[(i)] $J^+(A)^\circ=I^+(A)$.
\item[(ii)] $J^+(A)\subseteq \overline{I^+(A)}$.
\item[(iii)] $J^+(A) = \overline{I^+(A)}$ if and only if $J^+(A)$ is closed.
\end{itemize}
\end{Corollary}
\medskip\noindent
Finally, as in the smooth case, one may introduce a notion of
causality also for general continuous curves (cf.\ \cite[p.\ 184]{HE},
\cite[Def.\ 8.2.1]{Kriele}):
\begin{definition}\label{contcaus}
A continuous curve $\alpha: I \to M$ is called future-directed
causal (resp.\ timelike) if for every $t\in I$ there exists a
totally normal neighbourhood $U$ of $\alpha(t)$ such that for any
$s\in I$ with $\alpha(s)\in U$ and $s>t$,
$\alpha(s)\in J^+(\alpha(t))\setminus \{\alpha(t)\}$ (resp.\
$\alpha(s)\in I^+(\alpha(t))\setminus \{\alpha(t)\}$), and analogously
for $s<t$ with $J^-$ resp.\ $I^-$.
\end{definition}
\medskip\noindent
Then the proof of \cite[Lemma\ 8.2.1]{Kriele}) carries over to the
$C^{1,1}$-setting, showing that any continuous causal (resp.\
timelike) curve is locally Lipschitz.
\begin{remark}
While a continuous causal curve $\alpha$ need not
be a causal Lipschitz curve in the sense of our definition (cf.\ \cite[Rem.\ 1.28]{M}),
it still follows that $\langle \alpha'(t),\alpha'(t)\rangle \le 0$ wherever $\alpha'(t)$ exists
(however, $\alpha'(t)$ might be $0$).
\medskip\noindent
To see this, consider first the case where $g$ is smooth. Set
$p:=\alpha(t)$, pick a normal neighbourhood $U$ around $p$ and set
$\beta:=\exp_p^{-1}\circ \alpha$. Then $\beta'(t)=\alpha'(t)$ and by
Def.\ \ref{contcaus} and Th.\ \ref{lcb}, $\beta(s)\in J^+(0)$
for $s>t$ small. Therefore, $\beta'(t)\in J^+(0)$, so $\langle
\alpha'(t),\alpha'(t)\rangle \le 0$. In the general case, where $g$ is
only supposed to be $C^{1,1}$, pick a regularisation $\hat g_\eps$ as
in Prop.\ \ref{CGapprox}. Then $\hat g_\eps(\alpha'(t),\alpha'(t)) \le
0$ for all $\eps$ by the above and letting $\eps\to 0$ gives the
claim.
\end{remark}
\section{Further aspects of causality theory}\label{further}
In the previous section we have shown that the fundamental
constructions of causality theory remain valid for
$C^{1,1}$-metrics. It was demonstrated by P.\ Chru\'sciel in
\cite{Chrusciel_causality} that to obtain a consistent causality
theory for $C^2$-metrics one needs two main ingredients: on the one
hand, a push-up Lemma, as given by Prop.\ \ref{push-up1},
\ref{nonnull}. The second pillar in the development of the theory is
the fact that accumulation curves of causal curves are causal again.
Here, if $\alpha_n:I\to M$ is a sequence of paths (parametrised
curves) then a path $\alpha:I\to M$ is called an accumulation curve of
the sequence $(\alpha_n)$ if there exists a subsequence
$(\alpha_{n_k})$ that converges to $\alpha$ uniformly on compact
subsets of $I$. It was shown in \cite[Th.\ 1.6]{CG} that limit
stability of causal curves holds in fact even for continuous metrics:
\begin{Theorem}\label{accu}
Let $g$ be a $C^0$-Lorentzian metric on $M$, and let $\alpha_n: I\to
M$ be a sequence of causal curves that accumulate at some $p\in M$
($\alpha_n(0)\to p$). Then there exists a causal curve $\alpha$ that
is an accumulation curve of $\alpha_n$.
\end{Theorem}
\medskip\noindent
With these key tools at hand, and the results obtained so far,
causality theory for $C^{1,1}$-metrics can be further developed by
following the proofs given in \cite{Chrusciel_causality} for
$C^2$-metrics. In the remainder of this section we list some
main results that can be derived in this way.
Extendability of geodesics is characterised as follows (cf.
\cite[Prop.\ 2.5.6]{Chrusciel_causality}):
\begin{Proposition}
Let $(M,g)$ be a spacetime with a $C^{1,1}$-Lorentzian metric $g$.
A geodesic $\alpha: I\to M$ is maximally extended as a geodesic if
and only if it is inextendible as a causal curve.
\end{Proposition}
Furthermore, it is already shown in \cite[Th.\ 2.5.7]{Chrusciel_causality})
that even if the metric is merely supposed to be continuous, every
future directed causal (resp.\ timelike) curve possesses an
inextendible causal (resp.\ timelike) extension of $\alpha$.
As a direct consequence of Cor.\ \ref{boundary} we obtain (cf.\ \cite[Prop.\ 2.6.9]{Chrusciel_causality}):
\begin{Proposition}
Let $g$ be a $C^{1,1}$-Lorentzian metric on $M$. If $\alpha$ is an achronal
causal curve, then $\alpha$ is a null geodesic.
\end{Proposition}
For sequences of curves, \cite[Prop.\ 2.6.8, Th.\ 2.6.10]{Chrusciel_causality}) give:
\begin{Proposition}
Let $g$ be a $C^{1,1}$-Lorentzian metric on $M$. If $\alpha_n: I\to M$
is a sequence of maximally extended geodesics accumulating at $\alpha$,
then $\alpha$ is a maximally extended geodesic.
\end{Proposition}
\begin{Theorem}
Let $(M,g)$ be a spacetime with a $C^{1,1}$-Lorentzian metric $g$ and
let $\alpha_n:I\to M$ be a sequence of achronal causal curves accumulating
at $\alpha$. Then $\alpha$ is achronal.
\end{Theorem}
\medskip\noindent
Causality conditions and notions such as domains of dependence and Cauchy
horizons can be defined independently of the regularity of
the metric. As an example of the interrelation of causality conditions
for metrics of low regularity, we mention
\cite[Prop.\ 2.7.4]{Chrusciel_causality}, which shows that if a spacetime with
continuous metric is stably causal then it is strongly causal.
Turning now to globally hyperbolic spacetimes,
\cite[Prop.\ 2.8.1, Cor.\ 2.8.4, Th.\ 2.8.5]{Chrusciel_causality} give:
\begin{Proposition}
Let $(M,g)$ be a globally hyperbolic spacetime with $g$ a $C^{1,1}$-Lorentzian
metric and let $\alpha_n$ be a sequence of causal curves accumulating at
both $p$ and $q$. Then there exists a causal curve $\alpha$ which is an
accumulation curve of the $\alpha_n$'s and passes through $p$ and $q$.
\end{Proposition}
\begin{Corollary}
If $M$ is a spacetime with a $C^{1,1}$-Lorentzian metric $g$ that is
globally hyperbolic, then
\begin{equation*}
\overline{I^{\pm}(p)}=J^{\pm}(p).
\end{equation*}
\end{Corollary}
\begin{Theorem}
For a globally hyperbolic spacetime $M$ with a $C^{1,1}$-metric $g$, if
$q\in I^+(p)$, resp. $q\in J^+(p)$, there exists a timelike, resp. causal,
future directed geodesic from $p$ to $q$.
\end{Theorem}
Moreover, the proof of \cite[Th.\ 2.9.9]{Chrusciel_causality} can be
adapted to show:
\begin{Theorem}
Let $M$ be a spacetime with a $C^{1,1}$-Lorentzian metric $g$ and let $S$
be an achronal hypersurface in $(M,g)$. Suppose that the interior
$D^\circ_I(S)$ of the domain of dependence $D_I(S)$ is nonempty. Then
$D^\circ_I(S)$ equipped with the metric obtained by restricting $g$ is
globally hyperbolic.
\end{Theorem}
Note that here the definition of domains of dependence is based on timelike
curves, as is the definition of Cauchy horizons. Finally, the analogue
of \cite[Prop.\ 2.10.6]{Chrusciel_causality} establishes the existence of
generators of Cauchy horizons :
\begin{Proposition}
Let $g$ be a $C^{1,1}$-Lorentzian metric on $M$ and let $S$ be a
spacelike $C^1$-hypersurface in $M$. For any point $p$
in the Cauchy horizon $H^+_I(S)$ there exists a past directed null geodesic
$\alpha_p\subset H^+_I(S)$ starting at $p$ which either does not have an
endpoint in $M$, or has an endpoint in $\bar{S}\setminus S$.
\end{Proposition}
Based on these foundations, a deeper study of causality theory, in particular
in the direction of singularity theorems for metrics of low regularity can
be undertaken. As detailed in \cite[Sec.\ 6.1]{Seno1}, this will require
to solve a whole range of analytical problems that go beyond the results of this
paper, in particular concerning variational properties of curves, control of curvature
quantities, and a study of focal points. As already mentioned in the introduction,
we hope that the techniques developed in \cite{CleF,CG,KSS,M} as well as in this paper
can contribute to this task.
\medskip\noindent
{\bf Acknowledgements.} We would like to thank James D.\ E. Grant for
helpful discussions. The authors acknowledge the support of FWF
projects P23714 and P25326, as well as OeAD project WTZ CZ 15/2013.
We are indebted to the referees of this paper for several comments that
have led to substantial improvements in the presentation.
| 190,412
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Yesterday there were gusts of wind of up to 40 knots so we stayed in Almerimar.
Yesterday there were gusts of wind of up to 40 knots so we stayed in Almerimar.:
That’s it, we have called today “day off”, but we all know that it doesn’t mean that we are not going to be working.
It is impressive how, after almost 50 dives made by Oceana with the underwater robot (ROV) in this area, we can still find new habitats. This time, the surprise has come by the hand of a mollusk, the giant oyster Neopycnodonte zibrowii. More than 400 metres in depth, over some rocks, we have found several individuals of this long-lived animal, who lives up to 500 years and is considered a living fossil.
Last year, when we were working in the same area –the western half of the area designated to be studied within the framework of the LIFE+INDEMARES Project–, we also found groups of common bottlenose dolphins, swimming, resting or feeding. This time, we saw two groups of at least 15 individuals, something similar to what we were able to observe in 2010 while we were working in the same area.
Seco de los Olivos (aka Chella Bank) has shown us why it is today an area exploited by fishermen looking for two of the most valued treasures of the Mediterranean: the grouper and the red coral, the latter one harvested here until a few decades ago.
Because of the wind yesterday, we have a strong swell when we set sail to the seamount, but it won’t be a problem because the weather forecast for today is good, so we count on the conditions improving throughout the day.
ROV dives.
| 48,221
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432 Owen
734.487.1215
sock.chung@emich.edu
Ph.D. Auburn University
M.B.A Auburn University
B.Sc. Seoul National University
Dr. Chung teaches undergraduate and graduate courses in Database Management, ERP/Supply Chain Management and IT Project Management. His research interests include IS Strategy, Enterprise Systems and Supply Chain Management. Dr. Chung is actively involved in the Association for Information Systems and Decision Sciences Institute (DSI).
COB 200 Introduction to Business
IS 215 End-User Computing
IS 350 Enterprise Resource Planning
IS 380 Introduction to Databases
IS 421 Advanced Database Design
IS 499 3 Credit Independent Study
IS 632 Global Dimensions of Information Technology
IS 645 Database Management Systems
IS 650 Enterprise Resource Planning
IS 697 1 Credit Independent Study
IS 699 3 Credit Independent Study
IT Strategy; Database Management; Enterprise Resource Planning (ERP)/SAP
Chung, S. (2005). Technological Factors Relevant to Continuity on ERP for e-Business Platform: Integration, Modularity, and Flexibility. Journal of Internet Commerce, 4 (4).
Chung, S., H., B., T.A., L., & Ford, F.N. (in press, 2005). An Empirical study of Information Technology Infrastructure Flexibility Construct for Mass Customization. Database for Advances in Information Technology.
Lewis, B.R., Chung, S., & Rainer, R.K. (in press, 2005). The Current Status of the IT Application Portfolio. The Communications of Association for Information Systems.
Chung, Sock. H., Tummala, R., & Tang, H., (2004). Technological Factors Linking ERP to Supply Chain Management: Modularity and Integration. 2004 National Conference of the Academy of Business Administration.
Chung, S., Lewis, B. R. , & Rainer, R. K. (in press, 2003). The Current Status of the IT Application Portfolio. The Communications of Association for Information Systems.
Eastern Michigan University
Ypsilanti, MI, USA 48197
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\begin{document}
\title{A Characterization of Easily Testable Induced Digraphs and $k$-Colored Graphs}
\author{Lior Gishboliner\thanks{ETH Zurich.
Email: lior.gishboliner@math.ethz.ch.}
}
\maketitle
\begin{abstract}
We complete the characterization of the digraphs $D$ for which the induced $D$-removal lemma has polynomial bounds, answering a question of Alon and Shapira. We also study the analogous problem for $k$-colored complete graphs. In particular, we prove a removal lemma with polynomial bounds for Gallai colorings.
\end{abstract}
\section{Introduction}
In this paper we are concerned with binary combinatorial objects, such as graphs, digraphs and $k$-colored graphs. A removal lemma is a statement of the following form: Suppose that $G,F$ are binary combinatorial objects of the same type, where we think of $G$ as large and of $F$ as small and fixed. For every $\varepsilon > 0$ there is $\delta = \delta(\varepsilon) > 0$, such that if $G$ contains at most $\delta n^{v(F)}$ (induced) copies of $F$, then $G$ can be made (induced) $F$-free by changing at most $\varepsilon n^2$ entries in its adjacency matrix, where $n = v(G)$.
The first result of this type was the famous {\em triangle removal lemma} of Ruzsa and Szemer\'edi \cite{RuzsaSz76}, which states that if an $n$-vertex graph contains at most $\delta(\varepsilon)n^3$ triangles, then it can be made triangle-free by deleting at most $\varepsilon n^2$ edges. This result played a key role in the development of extremal combinatorics, and its proof is one of the first applications of the celebrated Szemer\'edi regularity lemma \cite{Szemeredi}. The original proof generalizes from triangles to arbitrary graphs, giving the {\em graph removal lemma}. Later, Alon, Fischer, Krivelevich and Szegedy \cite{AFKS} proved an analogous result for induced subgraphs, the so-called {\em induced removal lemma}. This result states that if a graph contains at most $\delta(\varepsilon)n^{v(F)}$ induced copies of $F$, then it can be made induced $F$-free by adding/deleting at most $\varepsilon n^2$ edges. Analogous results have later been proved for other combinatorial structures, such as digraphs \cite{AS_digraphs} and ordered graphs \cite{ABF}. In another direction, the induced removal lemma was generalized to arbitrary hereditary graph properties \cite{AS_hereditary}.
A common feature of all of the above results is that their proof uses Szemer\'edi's regularity lemma or a generalization thereof. Consequently, these proofs give quite weak, tower-type (or worse) bounds on $\delta(\varepsilon)$. For example, in the case of the graph removal lemma, the best known bound \cite{Fox} is
$1/\delta \leq \text{tower}(O(\log 1/\varepsilon))$, where $\text{tower}(x)$ is a tower of $x$ exponents. For the induced removal lemma and for other structures (e.g. ordered graphs), the best known general bounds are even worse, see e.g. \cite{CF} for the state of the art. However, for particular graphs $F$, better bounds are known. This has raised the natural question of characterizing the cases where one can prove a removal lemma with polynomial bounds, i.e. when $1/\delta$ can be taken to be polynomial in $1/\varepsilon$.
The first result of this type was obtained by Alon \cite{Alon}, who proved that for a graph $F$, the $F$-removal lemma has polynomial bounds if and only if $F$ is bipartite. Later, Alon and Shapira \cite{AS_induced} obtained a nearly complete characterization for the induced case, showing that the induced $F$-removal lemma has polynomial bounds if $F \in \{P_2,P_3,\overline{P_2},\overline{P_3}\}$, and that it does not have polynomial bounds if $F \notin \{P_2,P_3,\overline{P_2},\overline{P_3},P_4,C_4,\overline{C_4}\}$, where $P_k$ and $C_k$ are the path and cycle with $k$ vertices, respectively, and $\overline{F}$ denotes graph complement. The case of $P_4$ was later settled by Alon and Fox \cite{AF}. The author and Shapira \cite{GS_C4} proved an exponential bound for the case of $C_4$. For similar results for certain families of graph properties, see \cite{GS_poly}.
Similar characterizations of polynomial removal lemmas were also obtained for other combinatorial structures, e.g. for tournaments \cite{FGSY} and for digraphs \cite{AS_digraphs}. In particular, Alon and Shapira \cite{AS_digraphs} characterized the digraphs $D$ for which the $D$-removal lemma has polynomial bounds, and asked for a characterization in the induced case. They showed \cite{AS_induced} that the induced $D$-removal lemma does not have polynomial bounds whenever $v(D) \geq 5$. Here we answer their question by completing the characterization. Before stating our results, let us introduce the following commonly used terminology:
for a graph/digraph $F$, we say that (induced) $F$-freeness is {\em easily testable} if the (induced) $F$-removal lemma has polynomial bounds, and otherwise we say that it is {\em hard to test} (or just {\em hard}). This terminology comes from the field of property testing, where the goal is to design fast algorithms which distinguish between graphs satisfying a certain property and graphs which are $\varepsilon$-far from the property. The efficiency of such testers is measured by the number of queries they make to the input graph, and for many properties one can design testers whose query complexity is independent of the size of the input, i.e. depends only on $\varepsilon$. For hereditary graph properties, the query complexity of the best (one-sided error) tester is essentially given by the function $\delta(\varepsilon)$ in the corresponding removal lemma. We refer the reader to the book \cite{Goldreich} for an introduction to property testing.
The following theorem gives the characterization of digraphs $D$ for which the induced $D$-removal lemma has polynomial bounds.
\begin{theorem}\label{thm:digraphs}
For a digraph $D$, induced $D$-freeness is easily testable if and only if $v(D) = 2$.
\end{theorem}
\noindent
Observe that the ``if'' part of Theorem \ref{thm:digraphs} is trivial.
Induced digraphs can encode $3$-colored complete graphs, where the color of a pair $\{i,j\}$ is the number of directed edges between $i$ and $j$, namely $0$, $1$ or $2$ (as in \cite{AS_digraphs}, we allow anti-parallel edges, but not parallel edges).
By {\em $k$-colored complete graph} we mean a coloring of the edges of a complete graph with $k$ colors. So in particular, a graph can be thought of as a $2$-colored complete graph.
The removal lemma generalizes in a straightforward manner to $k$-colored complete graphs (where instead of edge addition/deletion, one speaks of edge color changes).
Now, given a digraph $D$, let $C(D)$ denote the corresponding $3$-colored complete graph; namely, $C(D)$ has the same vertex-set as $D$, and the color of a pair $\{i,j\}$ is the number of edges in $D$ between $i$ and $j$ (either $0$, $1$ or $2$). Note that the map $D \mapsto C(D)$ is not one-to-one. Indeed, if $D'$ is obtained from $D$ by reversing the direction of some single edges (i.e. edges $(i,j) \in E(D)$ for which $(j,i) \notin E(D)$), then $C(D') = C(D)$.
Two subgraphs of a graph/digraph/$k$-colored graph are called {\em pair-disjoint} if they share at most one vertex. Throughout the paper, we will use the obvious fact that if a graph/digraph/$k$-colored graph $G$ contains $\varepsilon n^2$ pair-disjoint (induced) copies of $F$, then one must add/delete/change the color of at least $\varepsilon n^2$ edges in order to make $G$ (induced) $F$-free. By a
{\em hardness construction} for (induced) $F$-freeness, we mean a graph $G$ which contains a collection of $\varepsilon n^2$ (induced) pair-disjoint copies of $F$, but only $\delta n^{v(F)}$ (induced) copies of $F$ overall, where $\delta \ll \poly(\varepsilon)$ (namely, $\delta$ goes to $0$ faster than any polynomial in $\varepsilon$).
So a hardness construction (for every $\varepsilon$ and $n$) shows that (induced) $F$-freeness is hard to test.
The following (almost immediate) proposition shows that for a digraph $D$, a hardness construction for $C(D)$-freeness implies a hardness construction for induced $D$-freeness.
\begin{proposition}\label{prop:coloring}
Let $D$ be a digraph. For $\varepsilon,\delta > 0$ and $n \geq 1$, suppose that there is a $3$-colored complete graph $G$ on $n$ vertices which contains $\varepsilon n^2$ pair-disjoint copies of $C(D)$, but only $\delta n^{v(D)}$ copies of $C(D)$ overall. Then there is a digraph $G'$ on $n$ vertices which contains $\varepsilon n^2$ induced pair-disjoint copies of $D$, but only $\delta n^{v(D)}$ induced copies of $D$ overall.
\end{proposition}
\begin{proof}
Choose $G'$ such that $C(G') = G$, and such that each of the $\varepsilon n^2$ pair-disjoint copies of $C(D)$ in $G$ makes an induced copy of $D$ in $G'$.
\end{proof}
Proposition \ref{prop:coloring} suggests the problem of characterizing the easily testable $3$-colored complete graphs. It turns out that here the situation is somewhat different from that of induced digraphs: while all induced digraphs on at least $3$ vertices are hard (by Theorem \ref{thm:digraphs}), there is a $3$-colored complete graph on $3$ vertices which is easily testable, namely the rainbow triangle.
This assertion is the main part of our next result, which characterizes the easily testable $3$-colored complete graphs:
\begin{theorem}\label{thm:3-colored_graphs}
Let $F$ be a $3$-colored complete graph. Then $F$-freeness is easily testable if and only if $v(F) = 2$ or $F$ is the rainbow triangle.
\end{theorem}
The main part in the proof of Theorem \ref{thm:3-colored_graphs} is to show that the property of having no rainbow triangles is easily testable. This is done in Section \ref{sec:3-colored}. The structure of $3$-colored complete graphs with no rainbow triangles (also called Gallai colorings) was described by a fundamental result of Gallai \cite{Gallai} (see also \cite{GS}). This result states that if $G$ has no rainbow triangles, then $G$ is obtained from a 2-colored complete graph by replacing each vertex with a 3-colored complete graph without rainbow triangles (and replacing edges with complete bipartite graphs of the same color). Moreover, every 3-colored complete graph obtained in this way has no rainbow triangles.
This structure result (stated below as Lemma \ref{lem:Gallai}) will play a key role in the proof.
There are two digraphs $D$ for which $C(D)$ is the rainbow triangle. Let us denote them by $D_1,D_2$. Even though the rainbow triangle is easily testable, it turns out that induced $D_i$-freeness is hard to test for each $i = 1,2$.
Theorem \ref{thm:3-colored_graphs} does imply however that the property of avoiding {\em both} $D_1,D_2$ as induced subdigraphs is easily testable. These digraphs $D_1,D_2$ are the only cases of Theorem \ref{thm:digraphs} which are not covered by using Theorem \ref{thm:3-colored_graphs} and Proposition \ref{prop:coloring}.
To complement Theorem \ref{thm:3-colored_graphs}, we show that for $k \geq 4$, there are no non-trivial easily testable $k$-colored complete graphs.
\begin{proposition}\label{prop:k-colored}
Let $k \geq 4$ and let $F$ be a $k$-colored complete graph. Then $F$-freeness is easily testable if and only if $v(F) = 2$.
\end{proposition}
\noindent
The proof of the hardness direction of Theorems \ref{thm:digraphs}-\ref{thm:3-colored_graphs} and Proposition \ref{prop:k-colored} appears in Section \ref{sec:hard}.
\section{Testing for Gallai colorings}\label{sec:3-colored}
In this section we prove that the property of having no rainbow triangles is easily testable. We restate this result as follows.
\begin{theorem}\label{thm:Gallai_coloring}
Let $\varepsilon > 0$ be small enough, and let $G$ be an $n$-vertex $3$-colored complete graph with at most $\varepsilon^{36} n^3$ rainbow triangles. Then $G$ can be made rainbow-triangle-free by changing the color of at most $\varepsilon n^2$ edges.
\end{theorem}
The proof is similar in spirit to the argument used by Alon and Fox \cite{AF} to show that the property of being a cograph (or, equivalently, of having no induced path on four vertices) is easily testable.
We now introduce the necessary definitions.
Let $\mathcal{P} = (V_1,\dots,V_m)$ be a vertex-partition of a $3$-colored complete graph. For colors $a,b \in [3]$, we say that $\mathcal{P}$ is {\em $(a,b)$-monochromatic} if each of the bipartite graphs $(V_i,V_j)$ is monochromatic in color $a$ or in color $b$. Denote by $E(\mathcal{P})$ the set of all edges which go between the sets $V_1,\dots,V_m$, and put $e(\mathcal{P}) := |E(\mathcal{P})| = \sum_{1 \leq i < j \leq m}{|V_i||V_j|}$. We say that $\mathcal{P}$ is {\em $\varepsilon$-close to being $(a,b)$-monochromatic} if one can turn $\mathcal{P}$ into an $(a,b)$-monochromatic partition by changing the color of at most $\varepsilon \cdot e(\mathcal{P})$ of the edges in $E(\mathcal{P})$.
Gallai \cite{Gallai,GS} proved the following fundamental fact about colorings with no rainbow triangles.
\begin{lemma}[\cite{Gallai,GS}]\label{lem:Gallai}
If $G$ is a $3$-colored complete graph with no rainbow triangles, then there exist two colors $a,b \in [3]$ such that $G$ admits an $(a,b)$-monochromatic partition (with at least two parts).
Conversely, if $\mathcal{P}$ is an $(a,b)$-monochromatic partition of $G$ (for some two colors $a,b$), and $G[X]$ has no rainbow triangles for every $X \in \mathcal{P}$, then $G$ has no rainbow triangles.
\end{lemma}
\noindent
Before proceeding, let us prove the following very simple lemma:
\begin{lemma}\label{lem:balanced}
Let $m,d,a_1,\dots,a_p \geq 0$ such that $a_1 + \dots + a_p = m$ and $a_i \leq m - d$ for every $1 \leq i \leq p$. Then $\sum_{1 \leq i < j \leq p}{a_ia_j} > d \cdot \frac{m-d}{2}$.
\end{lemma}
\begin{proof}
Without loss of generality, assume that $a_1 \leq \dots \leq a_p$. Let $1 \leq i \leq p$ be minimal with $a_1 + \dots + a_i \geq d$. We have $i \leq p-1$, because otherwise we would have $a_p > m - d$, a contradiction.
Note that $a_{i+1} + \dots + a_p \geq a_{i+1} \geq a_i$ and $a_{i+1} + \dots + a_p = m - (a_1 + \dots + a_i) = m - (a_1 + \dots + a_{i-1}) - a_i >
m - d - a_i$. Summing these two inequalities and dividing by $2$, we obtain that $a_{i+1} + \dots + a_p > \frac{m-d}{2}$. Now,
$
\sum_{1 \leq i < j \leq p}{a_ia_j} \geq (a_1 + \dots + a_i) \cdot (a_{i+1} + \dots + a_p) > d \cdot \frac{m - d}{2},
$
as required.
\end{proof}
The main step in the proof of Theorem \ref{thm:Gallai_coloring} is the following approximate version of Lemma \ref{lem:Gallai}. It states that if a $3$-colored complete graph $G$ has few rainbow triangles, then for some two colors $a,b$, $G$ has a partition which is close to being $(a,b)$-monochromatic.
\begin{lemma}\label{lem:approximate partition}
Let $\varepsilon > 0$ be small enough, and let $G$ be an $n$-vertex $3$-colored complete graph with at most $\varepsilon^{33} n^3$ rainbow triangles. Then there exist two colors $a,b \in [3]$ and a partition $\mathcal{P}$ of $V(G)$ which is $\varepsilon$-close to being $(a,b)$-monochromatic.
\end{lemma}
\begin{proof}
We will denote the color of an edge $\{x,y\}$ by $c(x,y)$.
Let $d_i(x)$ be the degree of $x$ in color $i$ (for $i \in [3]$).
If there is a vertex $x$ and a color $i \in [3]$ such that $d_i(x) \geq (1-\varepsilon)(n-1)$, then the partition $\{x\},V(G) \setminus \{x\}$ satisfies the requirement in the lemma. So from now on, suppose that $d_i(x) < (1-\varepsilon)(n-1)$ for every vertex $x$ and color $i \in [3]$.
If there are less than $\varepsilon \binom{n}{2}$ edges of some color $i \in [3]$, then we can take $\mathcal{P}$ to be the partition of $V(G)$ into singletons (taking $a,b$ to be the two colors which are not $i$). So we may assume that for each color $i \in [3]$, there are at least $\varepsilon \binom{n}{2}$ edges of color $i$.
For a color $i \in [3]$, let $\text{SMALL}_i$ be the set of all vertices $v$ with $d_i(v) \leq \frac{\varepsilon^2}{128}n$.
Note that $|\text{SMALL}_i| \leq (1-\frac{\varepsilon}{2})n$ because otherwise there would be less than $\varepsilon \binom{n}{2}$ edges in color $i$.
Set
$$
s := \frac{128\log(2000/\varepsilon^2)}{\varepsilon^2},
$$
$$
\delta := \frac{\varepsilon^2}{64s^2},
$$
$$
k := \frac{128}{\varepsilon^2},
$$
and
$$
t := \frac{2(k+s\log s)}{\delta}.
$$
Note that $s = \tilde{O}(\frac{1}{\varepsilon^2})$, $\delta = \tilde{\Omega}(\varepsilon^6)$ and $t = \tilde{O}(\frac{1}{\varepsilon^8})$.
We will later use the fact that
\begin{equation}\label{eq:s,t}
2^k e^{-\delta t} \ll s^{-s},
\end{equation}
which easily follows from our choice of $t$. Here, the $\ll$ means that the left-hand side is smaller than $C$ times the right hand side for a fixed constant $C$, provided that $\varepsilon$ is small enough.
Sample $s + kt$ vertices of $G$ uniformly at random and independently. Let $S$ be the set of the first $s$ vertices, $T_1$ be the set of the next $t$ vertices, $T_2$ the set of the next $t$, and so on.
Put $T := T_1 \cup \dots \cup T_k$ and $R := S \cup T$.
Let $E_0$ be the event that $G[R]$ contains no rainbow triangles.
We have $|R| = s + kt = \tilde{O}(\frac{1}{\varepsilon^{10}}) \leq \frac{1}{2\varepsilon^{11}}$, say, where the last inequality holds if $\varepsilon$ is small enough.
Since $G$ contains at most $\Delta := \varepsilon^{33} n^3$ rainbow triangles, the probability that $G[R]$ contains a rainbow triangle is at most
$\binom{|R|}{3} \cdot \Delta \cdot 6 \cdot \frac{1}{n^3} \leq
\frac{|R|^3\Delta}{n^3} \leq \frac{1}{8}$. Namely, $\mathbb{P}[E_0] \geq \frac{7}{8}$.
Say that a vertex $v \in V(G)$ is {\em bad} if there is a color $i \in [3]$ such that $v \notin \text{SMALL}_i$, and yet $v$ has no neighbour of color $i$ in $S$. Observe that if $v \in V(G) \setminus \text{SMALL}_i$, then the probability that $v$ has no color-$i$ neighbour in $S$ is at most
$(1 - \varepsilon^2/128)^s \leq e^{-s\varepsilon^2/128} = \frac{\varepsilon^2}{2000}$.
Hence, the expected number of bad vertices is at most $3 \cdot \frac{\varepsilon^2}{2000} \cdot n \leq \frac{\varepsilon^2}{512}n$. Let $E_1$ be the event that there are at most $\frac{\varepsilon^2}{128}n$ bad vertices. By Markov's inequality, $\mathbb{P}[E_1] \geq \frac{3}{4}$.
Let $Z$ be the set of all vertices $v$ such that all edges between $v$ and $S$ have the same color.
For a vertex $v$, recall that $d_i(v) < (1 - \varepsilon)(n-1)$ for every color $i$.
Hence, $\mathbb{P}[v \in Z] \leq 3 \cdot (1-\varepsilon)^s \leq
3 e^{-\varepsilon s}$. It follows that $\mathbb{E}[|Z|] \leq 3 e^{-\varepsilon s}n$, and hence $\mathbb{P}[|Z| \geq 24 e^{-\varepsilon s}n] \leq \frac{1}{8}$ by Markov's inequality.
Let $E_2$ be the event that $T \cap Z = \emptyset$.
By the union bound, we have $\mathbb{P}[T \cap Z \neq \emptyset] \leq |T| \cdot |Z|/n$.
So if
$|Z| \leq 24 e^{-\varepsilon s}n$, then
$\mathbb{P}[Z \cap T \neq \emptyset] \leq |T| \cdot 24 e^{-\varepsilon s} \leq kt \cdot 24 e^{-\varepsilon s} \leq \frac{1}{8}$, where the last inequality holds for $\varepsilon$ small enough, as $e^{\varepsilon s}$ is (at least) exponential in $1/\varepsilon$, while $kt$ is polynomial in $1/\varepsilon$. So overall,
$\mathbb{P}[E_2] \geq \frac{3}{4}$.
For a partition $S = U_1 \cup \dots \cup U_p$ and for a set $X \supseteq S$, we say that a partition $X = W_1 \cup \dots \cup W_q$ (where $q \geq p$) {\em extends} $(U_1,\dots,U_p)$ if $W_i \cap S = U_i$ for every $1 \leq i \leq p$.
Suppose that $E_0$ and $E_2$ happened.
Since $E_0$ happened, by Lemma \ref{lem:Gallai} there exists an $(a,b)$-monochromatic partition $R = W_1 \cup \dots \cup W_q$, $q \geq 2$, for some two colors $a,b$.
Let $U_i := W_i \cap S$, and suppose without loss of generality that $U_1,\dots,U_{p}$ are the nonempty sets among $U_1,\dots,U_q$. We claim that
$p \geq 2$. Indeed, if $p = 1$ then $S \subseteq W_1$. But then, for any $w \in W_2$, we have that all edges between $w$ and $S$ have the same color. This however implies that $w \in Z$, contradicting that $E_2$ happened.
So we see that if $E_0$ and $E_2$ happened, then there exist two colors $a,b$, a partition
$S = U_1 \cup \dots \cup U_p$ with $p \geq 2$, and an $(a,b)$-monochromatic partition $R = W_1 \cup \dots \cup W_q$ which extends $(U_1,\dots,U_p)$.
The main step in the proof is to establish the following:
\begin{claim*}\label{statement:approximate_partition main}
Fix any choice of $S$ and suppose that $E_1$ happened. Then, either there is a partition $\mathcal{P}$ as in the statement of the lemma, or with probability larger than $\frac{3}{4}$ the following holds: for every two colors $a,b \in [3]$ and for every $(a,b)$-monochromatic partition $S = U_1 \cup \dots \cup U_p$ with $p \geq 2$, there is no $(a,b)$-monochromatic partition of $R$ which extends $(U_1,\dots,U_p)$.
\end{claim*}
\begin{proof}
Fix two colors $(a,b)$ and an $(a,b)$-monochromatic partition $S = U_1 \cup \dots \cup U_p$ with $p \geq 2$. Without loss of generality, suppose that $a = 1, b = 2$. For $1 \leq i < j \leq p$, let $c_{i,j} \in \{1,2\}$ be the color of the monochromatic bipartite graph $(U_i,U_j)$.
Let $\mathcal{A}$ be the event that there is a $(1,2)$-monochromatic partition of $R$ which extends $(U_1,\dots,U_p)$.
We will show that either there is a partition $\mathcal{P}$ as in the statement of the lemma, or $\mathbb{P}[\mathcal{A}] \ll s^{-s}$.
We then take the union bound over all at most $3 \cdot s^s$ choices of $a,b$ and $(U_1,\dots,U_p)$, to get the required result.
Let us define sets $V_i^{(\ell)}$, $1 \leq \ell \leq k$ and $1 \leq i \leq p$, as follows. The definition is by induction on $\ell$.
For $1 \leq i \leq p$, define $V^{(1)}_i$ as the set of vertices $x \in V(G) \setminus S$ such that there is an edge of color $3$ between $x$ and $U_i$.
Since $E_1$ happened,
all but $\frac{\varepsilon^2}{128} n$ of the vertices in $V(G) \setminus \text{SMALL}_3$ belong to $V_1^{(1)} \cup \dots \cup V_p^{(1)}$.
Hence, $|V_1^{(1)} \cup \dots \cup V_p^{(1)}| \geq n - |\text{SMALL}_3| - \frac{\varepsilon^2}{128} n \geq
\frac{\varepsilon}{2}n - \frac{\varepsilon^2}{128} n \geq \frac{\varepsilon}{4}n$.
Now let $2 \leq \ell \leq k$, and suppose we already defined $V^{(\ell-1)}_1,\dots,V^{(\ell-1)}_p$. For $1 \leq i \leq p$, let $V^{(\ell)}_i$ be the set of all $x \in V(G) \setminus S$ such that either $x \in V^{(\ell-1)}_i$, or
there are at least $\delta n$ edges of color $3$ between $x$ and $V^{(\ell-1)}_i$, or
for each color $c \in \{1,2\}$, there are at least $\delta n$ edges of color $c$ between $x$ and $V^{(\ell-1)}_i$.
\begin{claim}\label{claim:forcing}
Let $1 \leq \ell \leq k$, $1 \leq i \leq p$ and $x \in V^{(\ell)}_i$. Then with probability at least $1 - (2^{\ell} - 2)e^{-\delta t}$ over the choice of $T_1,\dots,T_{\ell-1}$, the following holds:
if $(W_1,\dots,W_q)$ is a $(1,2)$-monochromatic partition of $S \cup T_1 \cup \dots \cup T_{\ell-1} \cup \{x\}$ which extends $(U_1,\dots,U_p)$, then $x \in W_i$.
\end{claim}
\begin{proof}
We prove the statement by induction on $\ell$. For $\ell = 1$, it follows immediately from the definition of the set $V^{(1)}_i$ that if $(W_1,\dots,W_q)$ is a $(1,2)$-monochromatic partition of $S \cup \{x\}$ which extends $(U_1,\dots,U_p)$, then $x \in W_i$ (with probability $1$). Let now $\ell \geq 2$, and let $x \in V^{(\ell)}_i$. If $x \in V^{(\ell-1)}_i$, then the assertion follows from the induction hypothesis.
Otherwise, either there are at least $\delta n$ edges of color $3$ between $x$ and $V^{(\ell-1)}_i$, or for each color $c \in \{1,2\}$, there are at least $\delta n$ edges of color $c$ between $x$ and $V^{(\ell-1)}_i$. We will assume that the latter case holds; the former case can be handled similarly (and more easily).
So for each $c = 1,2$, let $Y_c \subseteq V^{(\ell-1)}_i$ be a set of at least $\delta n$ vertices which are connected to $x$ in color $c$.
The probability that $T_{\ell-1} \cap Y_1 = \emptyset$ or $T_{\ell-1} \cap Y_2 = \emptyset$ is at most $2(1 - \delta)^{t} \leq 2e^{-\delta t}$. Suppose that $T_{\ell-1} \cap Y_1 \neq \emptyset$ and $T_{\ell-1} \cap Y_2 \neq \emptyset$, and fix vertices $y_c \in T_{\ell-1} \cap Y_c$, $c = 1,2$.
By the induction hypothesis, with probability at least $1 - 2 \cdot (2^{\ell-1} - 2)e^{-\delta t} = 1 - (2^{\ell} - 4)e^{-\delta t}$ over the choice of $T_1,\dots,T_{\ell-2}$, the following holds: for every $(1,2)$-monochromatic partition $(W_1,\dots,W_q)$ of $S \cup T_1 \cup \dots \cup T_{\ell-2} \cup \{y_1,y_2\}$ which extends $(U_1,\dots,U_p)$, it holds that $y_1,y_2 \in W_i$.
Assume that this event holds.
Let $(W_1,\dots,W_q)$ be a $(1,2)$-monochromatic partition of $S \cup T_1 \cup \dots \cup T_{\ell-1} \cup \{x\}$ which extends $(U_1,\dots,U_p)$. We have that $y_1,y_2 \in W_i$ and $\{x,y_1\},\{x,y_2\}$ have distinct colors. Hence, $x$ must be in $W_i$ as well. The probability that this fails is at most
$2e^{-\delta t} + (2^{\ell} - 4)e^{-\delta t} = (2^{\ell} - 2)e^{-\delta t}$, as required.
\end{proof}
\begin{claim}\label{claim:balanced}
Let $1 \leq \ell \leq k$ and $1 \leq i \leq p$ and suppose that $|V^{(\ell)}_i| \geq (1 - \frac{\varepsilon}{2})n$. Then $\mathbb{P}[\mathcal{A}] \ll s^{-s}$.
\end{claim}
\begin{proof}
For convenience, let us assume that $i = 1$. Fix a vertex $u \in U_2$. By our assumption, there are at least $\varepsilon (n-1) - |S| \geq \frac{3\varepsilon}{4}n$ vertices $x \in V(G) \setminus S$ such that the color of $\{x,u\}$ is not $c_{1,2}$.
Since $|V^{(\ell)}_1| \geq (1 - \frac{\varepsilon}{2})n$, at least $\frac{\varepsilon}{4}n$ of these vertices are in $V^{(\ell)}_1$.
The probability that $T_k$ contains no such vertex $x \in V^{(\ell)}_1$ is at most $(1 - \frac{\varepsilon}{4})^t \leq e^{-\varepsilon t/4}$. Suppose that $T_k$ contains such a vertex $x$. By Claim \ref{claim:forcing}, with probability at least $1 - (2^k - 2) e^{-\delta t}$ over the choice of $T_1,\dots,T_{k-1}$, it holds that if $(W_1,\dots,W_q)$ is a $(1,2)$-monochromatic partition of $S \cup T_1 \cup \dots \cup T_{k-1} \cup \{x\}$ extending $(U_1,\dots,U_p)$, then $x \in W_1$. Assume that this event happens; we show that then $\mathcal{A}$ fails. Indeed, suppose by contradiction that $(W_1,\dots,W_q)$ is a $(1,2)$-monochromatic partition of $R = S \cup T_1 \cup \dots \cup T_k$ extending $(U_1,\dots,U_p)$. We have $x \in T_k$, so $x \in W_1$. However, the color of $\{x,u\}$ is not $c_{1,2}$, contradicting the fact that the bipartite graph $(W_1,W_2)$ is monochromatic. It follows that
$
\mathbb{P}[\mathcal{A}] \leq e^{-\varepsilon t/4} + (2^k - 2) e^{-\delta t} \leq 2^k e^{-\delta t} \ll s^{-s}$, where the last inequality holds by \eqref{eq:s,t}. This proves the claim.
\end{proof}
\begin{claim}\label{claim:approximate_monochromatic}
Let $1 \leq \ell \leq k$ and $1 \leq i < j \leq p$, and suppose that there are at least $\delta n^2$ edges between $V^{(\ell)}_i$ and $V^{(\ell)}_j$ whose color is not $c_{i,j}$. Then $\mathbb{P}[\mathcal{A}] \ll s^{-s}$.
\end{claim}
\begin{proof}
The probability that $T_k$ contains no edge $\{v_i,v_j\} \in E(V^{(\ell)}_i,V^{(\ell)}_j)$ with $c(v_i,v_j) \neq c_{i,j}$, is at most $(1 - 2\delta)^{t/2} \leq e^{-\delta t}$. Suppose that $T_k$ contains such an edge $\{v_i,v_j\}$. By applying Claim \ref{claim:forcing} to $v_i$ and $v_j$, we get the following: with probability at least $1 - 2 \cdot (2^k - 2) e^{-\delta t}$ over the choice of $T_1,\dots,T_{k-1}$, it holds that if $(W_1,\dots,W_q)$ is a $(1,2)$-monochromatic partition of $S \cup T_1 \cup \dots \cup T_{k-1} \cup \{v_i,v_j\}$ extending $(U_1,\dots,U_p)$, then $v_i \in W_i$ and $v_j \in W_j$. But $c(v_i,v_j) \neq c_{i,j}$, contradicting the fact that the bipartite graph $(W_i,W_j)$ should be monochromatic in color $c_{i,j}$. The probability of failure is at most
$e^{-\delta t} + 2 \cdot (2^k - 2) e^{-\delta t} \leq 2 \cdot 2^k e^{-\delta t} \ll s^{-s}$, where the last inequality holds by \eqref{eq:s,t}.
\end{proof}
Put $V^{(\ell)} := V^{(\ell)}_1 \cup \dots \cup V^{(\ell)}_p$. By construction, we have $V^{(\ell)} \subseteq V^{(\ell+1)}$ for every $\ell \geq 1$. By our choice of $k$, there must be some $1 \leq \ell \leq k-1$ such that $|V^{(\ell+1)}| \leq |V^{(\ell)}| + \frac{\varepsilon^2}{128}n$. From now on we fix such an $\ell$.
By Claims \ref{claim:balanced} and \ref{claim:approximate_monochromatic}, we may assume that:
\begin{enumerate}
\item[(a)] $|V^{(\ell)}_i| \leq (1 - \frac{\varepsilon}{2})n$ for every $1 \leq i \leq p$.
\item[(b)] For every pair $1 \leq i < j \leq p$,
all but at most $\delta n^2$ of the edges between $V^{(\ell)}_i$ and $V^{(\ell)}_j$ \nolinebreak have \nolinebreak color \nolinebreak $c_{i,j}$.
\end{enumerate}
We would like the sets $V^{(\ell)}_1,\dots,V^{(\ell)}_p$ to be pairwise disjoint; hence, if an element belongs to several of these sets, then we place it in one of them arbitrarily, removing it from all others. Items (a)-(b) continue to hold.
Recall that
$|V^{(\ell)}| \geq |V^{(1)}| \geq \frac{\varepsilon}{4}n$.
Put $V' := V^{(\ell+1)} \setminus V^{(\ell)}$
and note that $|V'| \leq \frac{\varepsilon^2}{128}n$ by our choice of $\ell$. Also, set $X := V(G) \setminus (S \cup V^{(\ell)} \cup V') = V(G) \setminus (S \cup V^{(\ell+1)})$. Observe that by definition, if
$x \in X$
then for every $1 \leq i \leq p$, all but at most $2\delta n$ of the edges between $x$ and $V^{(\ell)}_i$ have the same color $c_{x,i} \in \{1,2\}$ (because $x \notin V^{(\ell+1)}$).
Consider the partition $\mathcal{P}$ of $V(G)$ having the following parts: $V^{(\ell)}_1,\dots,V^{(\ell)}_p$; $V' \cup S$; and the vertices of
$X$ as singletons.
We claim that $e(\mathcal{P}) \geq \frac{\varepsilon}{16}n^2$. Indeed, first note that $|V^{(\ell)}| + |X| = n - |V' \cup S| \geq (1 - \frac{\varepsilon}{8})n$, say.
If
$|X| \geq \frac{\varepsilon}{8}n$
then $e(\mathcal{P}) \geq |V^{(\ell)}| \cdot |X| \geq
(n - \frac{\varepsilon}{8}n - |X|) \cdot |X| \geq
(n - \frac{\varepsilon}{4}n) \cdot \frac{\varepsilon}{8}n \geq \frac{\varepsilon}{16}n^2$.
On the other hand, if
$|X| \leq \frac{\varepsilon}{8}n$ then
$|V^{(\ell)}| \geq (1 - \frac{\varepsilon}{4})n$. Now, as $|V^{(\ell)}_i| \leq (1 - \frac{\varepsilon}{2})n$ for every $1 \leq i \leq p$, we get from Lemma \ref{lem:balanced} (with parameters $m = (1 - \frac{\varepsilon}{4})n$ and $d = \frac{\varepsilon}{4}n$) that
$e(\mathcal{P}) \geq
\sum_{1 \leq i < j \leq p}{|V^{(\ell)}_i| \cdot |V^{(\ell)}_j|} \geq
\frac{\varepsilon}{4}n \cdot \frac{(1 - \frac{\varepsilon}{2})n}{2} \geq
\frac{\varepsilon}{16}n^2$.
So indeed $e(\mathcal{P}) \geq\frac{\varepsilon}{16}n^2$ in both cases.
We now modify at most $\varepsilon \cdot e(\mathcal{P})$ of the edges in $E(\mathcal{P})$ to turn $\mathcal{P}$ into a $(1,2)$-monochromatic partition.
The changes we make are as follows:
\begin{itemize}
\item For every $1 \leq i < j \leq p$, make $(V^{(\ell)}_i,V^{(\ell)}_j)$ monochromatic in color $c_{i,j}$. This is a total of at most $\binom{p}{2} \cdot \delta n^2 \leq s^2 \delta n^2 \leq \frac{\varepsilon^2}{64}n^2$ edge changes altogether.
\item For each $x \in X$ and $1 \leq i \leq p$, make all edges between $x$ and $V^{(\ell)}_i$ have the same color $c_{x,i} \in \{1,2\}$; this can be done with at most $2\delta n$ edge changes. Thus, this step requires at most $n \cdot p \cdot 2\delta n \leq 2s\delta n^2 \leq \frac{\varepsilon^2}{64}n^2$ edge changes altogether.
\item Change the color of all color-$3$ edges inside $X$. Recall that all but at most $\frac{\varepsilon^2}{128} n$ of the vertices in $X$ are in $\text{SMALL}_3$, because $X \cap V^{(1)} = \emptyset$ and as $E_1$ happened.
Recall also that each vertex in $\text{SMALL}_3$ is incident to at most $\frac{\varepsilon^2}{128} n$ edges of color $3$. Hence, this step requires at most $\frac{\varepsilon^2}{128} n \cdot |X| + |X| \cdot \frac{\varepsilon^2}{128} n \leq \frac{\varepsilon^2}{64} n^2$ edge changes.
\item Color all edges between $V' \cup S$ and $V(G) \setminus (V' \cup S)$ with color $1$ (say). This step requires at most $|V' \cup S| \cdot n \leq \frac{\varepsilon^2}{64}n^2$ edge changes.
\end{itemize}
The total number of edge changes in the above four items is at most $\frac{\varepsilon^2}{16} n^2 \leq \varepsilon \cdot e(\mathcal{P})$.
After these changes, $\mathcal{P}$ is $(1,2)$-monochromatic.
This proves the main claim.
\end{proof}
Let us now complete the proof of Lemma \ref{lem:approximate partition} using the main claim. Suppose by contradiction that there is no partition $\mathcal{P}$ of $V(G)$ as in the statement of the lemma. Then by the main claim, and as $\mathbb{P}[E_1] \geq 3/4$, we have the following: with probability larger than $1/2$, there does not exist an $(a,b)$-monochromatic partition $S = U_1 \cup \dots \cup U_p$, where $p \geq 2$, and an $(a,b)$-monochromatic partition of $R$ which extends $(U_1,\dots,U_p)$. On the other hand, we saw that such partitions $S = U_1 \cup \dots \cup U_p$ and $R = W_1 \cup \dots \cup W_q$ do exist if $E_0$ and $E_2$ happen, which has probability at least $1/2$ as $\mathbb{P}[E_0],\mathbb{P}[E_2] \geq 3/4$. This contradiction completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:Gallai_coloring}]
We decompose $G$ by repeatedly applying Lemma \ref{lem:approximate partition}. It is convenient to describe the decomposition using a tree, where each node corresponds to a subset of $V(G)$. The root is $V(G)$. At each step, if there is a leaf $X$ with $|X| \geq \varepsilon n$, then apply Lemma \ref{lem:approximate partition} to $G[X]$. As $\varepsilon^{33}|X|^3 \geq \varepsilon^{36}n^3$, we know that $G[X]$ contains at most $\varepsilon^{33}|X|^3$ rainbow triangles. Thus, Lemma \ref{lem:approximate partition} gives a partition $\mathcal{P}_X$ of $X$ which is $\varepsilon$-close to being $(a,b)$-monochromatic, for some two colors $a,b \in [3]$. Now add all the sets $Y \in \mathcal{P}_X$ as children of $X$. When this process stops, every leaf is of size at most $\varepsilon n$. For each non-leaf $X$, turn $\mathcal{P}_X$ into an $(a,b)$-monochromatic partition (for the two suitable colors $a,b$) by changing the colors of at most $\varepsilon \cdot e(\mathcal{P})$ of the edges in $E(\mathcal{P})$. This requires in total at most $\varepsilon\binom{n}{2} \leq \varepsilon n^2/2$ edge changes altogether. Next, for each leaf $X$, make $G[X]$ rainbow-triangle-free. This requires at most
$$
\sum_{X \text{ leaf}}{\binom{|X|}{2}} \leq \frac{\varepsilon n - 1}{2} \cdot
\sum_{X \text{ leaf}}{|X|} \leq \varepsilon n^2/2
$$
additional edge changes. So the total number of edge-changes is at most $\varepsilon n^2$. After these edge-changes, the resulting $3$-colored complete graph has no rainbow triangles, by the ``conversely'' part of Lemma \ref{lem:Gallai}. This completes the proof.
\end{proof}
\section{Lower Bounds}\label{sec:hard}
In this section we prove the ``only if'' parts of Theorems \ref{thm:digraphs} and \ref{thm:3-colored_graphs} and of Proposition \ref{prop:k-colored}.
The proofs use Behrend-type constructions, similarly to \cite{Alon,AS_induced}. Due to this similarity, we will be somewhat brief.
We need the following simple lemma.
\begin{lemma}\label{lem:design}
For $d \geq 2$ and $r \geq 2d$, there is a collection $R \subseteq [r]^d$, $|R| \geq (r/2)^2$, such that any two $d$-tuples in $R$ agree on at most one coordinate.
\end{lemma}
\begin{proof}
Let $p$ be a prime such that $r/2< p\leq r$; such a prime exists by Bertrand's postulate. For $a,b \in \mathbb{F}_p$, let $x_{a,b} \in \mathbb{F}_p^d$ be the $d$-tuple $x_{a,b}(i) = a + (i-1)b$, $i = 1,\dots,d$.
Observe that for $(a_1,b_1) \neq (a_2,b_2)$, there is at most one $1 \leq i \leq d$ with $x_{a_1,b_1}(i) = x_{a_2,b_2}(i)$. Indeed, if there are two such $1 \leq i < j \leq d$, then $a_1 + (i-1)b_1 = a_2 + (i-1)b_2$ and $a_1 + (j-1)b_1 = a_2 + (j-1)b_2$. Solving this system of equations gives $a_1 = a_2$ and $b_1 = b_2$, a contradiction. Here we use the fact that $i \not\equiv j \pmod{p}$, which follows from $p > r/2 \geq d$.
\end{proof}
\begin{lemma}\label{lem:RS_triangle}
Let $k \geq 2$, let $F$ be a $k$-colored complete graph, and suppose that there is a triangle in $F$ whose edges avoid one of the colors. Then for every small enough $\varepsilon > 0$ and large enough $n$, there is an $n$-vertex $k$-colored complete graph $G$ which contains $\varepsilon n^2$ pair-disjoint copies of $F$, but only $\varepsilon^{\Omega(\log1/\varepsilon)}n^{v(F)}$ copies of $F$ altogether.
\end{lemma}
\begin{proof}
Put $f = v(F)$ and suppose that $V(F) = [f]$. Without loss of generality, suppose that $F$ has a triangle whose edges avoid the color $k$.
By \cite[Lemma 4.1]{AS_induced}, for every $m \geq 1$, there is a set $S \subseteq [m]$ of size at least $m/e^{C\sqrt{\log m}}$, such that for all
$1 \leq p,q \leq f-1$, there is no solution to $p s_1 + q s_2 = (p+q) s_3$ with distinct $s_1,s_2,s_3 \in S$.
Let $m$ be the maximal integer satisfying $e^{-C\sqrt{\log m}} \geq 4f^4\varepsilon$. It is easy to check that $m \geq (1/\varepsilon)^{\Omega(\log 1/\varepsilon)}$. Let $S \subseteq [m]$ be as above; so $|S| \geq 4f^4 \varepsilon m$.
Define a $k$-colored complete graph $H$ consisting of $f$ disjoint sets $V_1,\dots,V_f$, where $V_i = [i \cdot m]$. So $v(H) = \binom{f+1}{2}m$, and hence $f^2m/2 \leq v(H) \leq f^2m$.
For each $x \in [m]$ and $s \in S$, add a copy $F_{x,s}$ of $F$ in which $v_i := x + (i-1)s \in V_i$ plays the role of $i$ for every $i \in [f] = V(F)$. All edges in $H$ which do not belong to any of the copies $F_{x,s}$ (in particular, the edges inside the sets $V_1,\dots,V_f$) are colored with color $k$.
We claim that the copies $F_{x,s}$, $(x,s) \in [m] \times S$, are pair-disjoint.
Indeed, if $F_{x_1,s_1}$ and $F_{x_2,s_2}$ have the same vertex in $V_i$ and $V_j$, then $x_1 + (i-1)s_1 = x_2 + (i-1)s_2$ and $x_1 + (j-1)s_1 = x_2 + (j-1)s_2$. Solving this system of equations, we get that $x_1 = x_2$ and $s_1 = s_2$. So we conclude that the copies $F_{x,s}$ are indeed pair-disjoint. The number of these copies is $m|S| \geq 4f^4 \varepsilon m^2$.
Next, we bound the number of triangles in $H$ which avoid the color $k$.
Such a triangle cannot contain two vertices from the same $V_i$, since the edges inside $V_1,\dots,V_f$ are colored with color $k$. Let $1 \leq a < b < c \leq f$, and let $x \in V_a, y \in V_b, z \in V_c$ be a triangle avoiding the color $k$. By construction, there are $s_1,s_2,s_3 \in S$ such that
$y - x = (b - a)s_1$, $z - y = (c - b)s_2$ and $z - x = (c - a)s_3$. So, setting $p := b - a$ and $q := c - b$, we have $ps_1 + qs_2 = (p + q)s_3$. By our choice of $S$, we have $s_1 = s_2 = s_3 =: s$. So each such triangle is determined by the choice of $x \in V_a$ and $s$. There are $|V_a| = a \cdot m \leq f \cdot m$ choices for $x$ and $|S| \leq m$ choices for $s$. Hence, the total number of triangles
in $H$ avoiding the color $k$ is at most $\binom{f}{3} \cdot fm \cdot |S| \leq f^4m^2$.
Now let $G$ be the $\frac{n}{v(H)}$-blowup of $H$, where all edges inside the blowup of each $V_i$ are colored with $k$. Each copy of $F$ in $H$ gives rise to $(\frac{n}{2v(H)})^2$ pair-disjoint induced copies of $F$ in $G$, by Lemma \ref{lem:design} with parameters $r=\frac{n}{v(H)}$ and $d = f$. Hence, $G$ contains a collection of
$4f^4\varepsilon m^2 \cdot (\frac{n}{2v(H)})^2 \geq \varepsilon n^2$ pair-disjoint induced copies of $F$. To complete the proof, we bound the total number of copies of $F$ in $G$. Each copy of $F$ must contain a triangle which avoids the color $k$. The number of such triangles in $H$ is at most $f^4m^2$, and each of these triangles in $H$ gives rise to $(\frac{n}{v(H)})^3$ such triangles in $G$. Hence, the total number of triangles in $G$ avoiding the color $k$ is at most $f^4m^2 \cdot (\frac{n}{v(H)})^3 \leq f^4m^2 \cdot (\frac{2n}{f^2m})^3 \leq \frac{n^3}{m}$. It follows that the number of copies of $F$ in $G$ is at most $\frac{n^3}{m} \cdot n^{f-3} = \frac{n^f}{m} \leq \varepsilon^{\Omega(\log 1/\varepsilon)} \cdot n^f$, as required.
\end{proof}
Lemma \ref{lem:RS_triangle} immediately implies that for $k \geq 4$, every $k$-colored complete graph with at least $3$ vertices is hard. For $k = 3$, observe that if $F$ is a $3$-colored complete graph and two of the edges incident to some $v \in V(F)$ have the same color, then $F$ has a triangle avoiding one of the colors. If $v(F) \geq 5$ then such a vertex $v$ must exist. If $v(F) = 3$ then such a vertex exists unless $F$ is a rainbow triangle. And if $v(F) = 4$ then such a vertex exists unless each color spans a matching of size $2$. Let $F_4$ denote this $3$-colored complete graph; namely, $V(F_4) = \{a_1,a_2,a_3,a_4\}$; $\{a_1,a_2\},\{a_3,a_4\}$ have color $1$; $\{a_1,a_4\},\{a_2,a_3\}$ have color $2$; and $\{a_1,a_3\},\{a_2,a_4\}$ have color $3$.
To complete the proof of Theorem \ref{thm:3-colored_graphs}, we now describe a variant of the above construction suited for $F_4$.
\begin{lemma}\label{lem:RS_F4}
For every small enough $\varepsilon > 0$ and large enough $n$, there is an $n$-vertex $3$-colored complete graph $G$ which contains $\varepsilon n^2$ pair-disjoint copies of $F_4$, but only $\varepsilon^{\Omega(\log1/\varepsilon)}n^{4}$ copies of $F_4$ altogether.
\end{lemma}
\begin{proof}
By \cite[Lemma 3.1]{Alon}, for every $m \geq 1$, there is a set $S \subseteq [m]$ of size at least $m/e^{C\sqrt{\log m}}$ containing no solution to $s_1 + s_2 + s_3 = 3s_4$ with distinct $s_1,s_2,s_3,s_4$.
Let $m$ be the maximal integer satisfying $e^{-C\sqrt{\log m}} \geq 400\varepsilon$. It is easy to check that $m \geq (1/\varepsilon)^{\Omega(\log 1/\varepsilon)}$. Let $S \subseteq [m]$ be as above; so $|S| \geq 400\varepsilon m$.
Define a $3$-colored complete graph $H$ consisting of $4$ disjoint sets $V_1,V_2,V_3,V_4$, where $V_i = [i \cdot m]$; so $v(H) = 10m$.
For each $x \in [m]$ and $s \in S$, add a copy $F_{x,s}$ of $F_4$ on the vertices $v_i = x + (i-1)s \in V_i$, where $v_i$ plays the role of $a_i$ for each $1 \leq i \leq 4$.
All edges not participating in one of these copies are colored with color $3$.
Observe that all edges between $V_1$ and $V_3$ and between $V_2$ and $V_4$ have color $3$.
As before, the copies $F_{x,s}$ are pair-disjoint. Their number is $m|S| \geq 400 \varepsilon m^2$.
Observe that if $F$ is a copy of $F_4$ in $H$, then $F$ must contain one vertex from each of the sets $V_1,\dots,V_4$.
Indeed, note that for every pair $1 \leq i < j \leq 4$, the edges in $V_i \cup V_j$ use only two colors. So $|V(F) \cap (V_i \cup V_j)| \leq 2$ for all $i,j$ (since every triangle in $F_4$ is rainbow). Hence, $|V(F) \cap V_i| = 1$ for every $1 \leq i \leq 4$. It is now easy to see that every copy of $F_4$ in $H$ is of the form $v_1,\dots,v_4$, where $v_i \in V_i$ plays the role of $a_i$. Fix such a copy $v_1,\dots,v_4$. By construction, there must be $s_1,s_2,s_3,s_4$ such that
$v_2 - v_1 = s_1$, $v_3 - v_2 = s_2$, $v_4 - v_3 = s_3$ and $v_4 - v_1 = 3s_4$. So $s_1 + s_2 + s_3 = 3s_4$, and hence $s_1 = s_2 = s_3 = s_4$ by our choice of $S$. It follows that the number of copies of $F_4$ in $H$ is $m|S| \leq m^2$.
Let $G$ be the $\frac{n}{v(H)}$-blowup of $H$, where all edges inside the blowup of each $V_i$ are colored with color $3$. Each copy of $F_4$ in $H$ gives rise to $(\frac{n}{2v(H)})^2$ pair-disjoint copies of $F_4$ in $G$ by Lemma \ref{lem:design}. Hence, $G$ contains a collection of
$400 \varepsilon m^2 \cdot (\frac{n}{2v(H)})^2 = \varepsilon n^2$ pair-disjoint copies of $F_4$.
Let us now upper-bound the total number of copies of $F_4$ in $G$. By the same argument as above, every copy of $F_4$ in $G$ must be of the form $v_1,\dots,v_4$ with $v_i$ belonging to the blowup of $V_i$ and playing the role of $a_i$ in the copy. So
every copy of $F_4$ in $G$ corresponds to a copy of $F_4$ in $H$. On the other hand, every copy of $F_4$ in $H$ gives rise to $(\frac{n}{v(H)})^4$ copies of $F_4$ in $G$. So overall, there are at most $m^2 \cdot (\frac{n}{v(H)})^4 \leq \frac{n^4}{m} \leq \varepsilon^{\Omega(\log1/\varepsilon)}n^{4}$ copies of $F_4$ in $G$, as required.
\end{proof}
To complete the proof of Theorem \ref{thm:digraphs}, we need to handle the two digraphs $D$ whose corresponding $3$-colored complete graph $C(D)$ is the rainbow triangle. These digraphs are obtained from each other by reversing the direction of all edges. So by symmetry, it remains to handle just one of them. Let then $D_3$ be the digraph with vertices $a_1,a_2,a_3$ and edges $(a_1,a_3),(a_2,a_3),(a_3,a_2)$.
\begin{lemma}
For every small enough $\varepsilon > 0$ and large enough $n$, there is an $n$-vertex digraph $G$ which contains $\varepsilon n^2$ pair-disjoint induced copies of $D_3$, but only $\varepsilon^{\Omega(\log1/\varepsilon)}n^3$ induced copies of $D_3$ altogether.
\end{lemma}
\begin{proof}
By \cite[Lemma 3.1]{Alon}, for every $m \geq 1$, there is a set $S \subseteq [m]$ of size at least $m/e^{C\sqrt{\log m}}$ containing no solution to $s_1 + s_2 = 2s_3$ with distinct $s_1,s_2,s_3$.
Let $m$ be the maximal integer satisfying $e^{-C\sqrt{\log m}} \geq 144\varepsilon$. It is easy to check that $m \geq (1/\varepsilon)^{\Omega(\log 1/\varepsilon)}$. Let $S \subseteq [m]$ be as above; so $|S| \geq 144\varepsilon m$.
Define a digraph $H$ consisting of $3$ disjoint sets $V_1,V_2,V_3$, where $V_i = [i \cdot m]$; so $v(H) = 6m$.
For each $x \in [m]$ and $s \in S$, add a copy $D_{x,s}$ of $D_3$ on the vertices $v_1 = x \in V_1$, $v_2 = x + s \in V_2$, $v_3 = x + 2s \in V_3$, where $v_i$ plays the role of $a_i$ for each $1 \leq i \leq 3$.
For all pairs of vertices $\{x,y\}$ not participating in one of these copies, put exactly one edge between $x$ and $y$, and if $x \in V_1, y \in V_3$ then direct this edge from $y$ to $x$. This way, the only edges going from $V_1$ to $V_3$ are those participating in one of the copies $D_{x,s}$. Note that, in particular, each of the sets $V_1,V_2,V_3$ spans a tournament.
As before, the copies $D_{x,s}$ are pair-disjoint. Their number is $m|S| \geq 144 \varepsilon m^2$.
It is easy to see that every induced copy of $D_3$ in $H$ must be of the form $v_1,v_2,v_3$ with $v_i \in V_i$ playing the role of $a_i$. If $v_1,v_2,v_3$ is such a copy, then by construction there are $s_1,s_2,s_3 \in S$ with $v_2 - v_1 = s_1$, $v_3 - v_2 = s_2$ and $v_3 - v_1 = 2s_3$. So $s_1 + s_2 = 2s_3$, implying that $s_1 = s_2 = s_3$. It follows that $H$ contains at most $|S|m \leq m^2$ induced copies of $D_3$.
Let $G$ be the $\frac{n}{v(H)}$-blowup of $H$, where the blowup of each $V_i$ is a tournament. Every induced copy of $D_3$ in $H$ gives rise to $(\frac{n}{2v(H)})^3$ pair-disjoint induced copies of $D_3$ in $G$, by Lemma \ref{lem:design}. Hence, $G$ contains a collection of $144\varepsilon m^2 \cdot (\frac{n}{2v(H)})^2 = \varepsilon n^2$ pair-disjoint induced copies of $D_3$. On the other hand, it is easy to see that every induced copy of $D_3$ in $G$ corresponds to an induced copy of $D_3$ in $H$, so overall $G$ has at most $m^2 \cdot (\frac{n}{v(H)})^3 \leq \frac{n^3}{m} \leq \varepsilon^{\Omega(\log1/\varepsilon)}n^3$ induced copies of $D_3$.
\end{proof}
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\section{Conclusion}
In this paper, we propose \alg, a novel adaptive control algorithm that efficiently learns the truncated Markov parameters of the underlying dynamical system and deploys projected online gradient descent to design a controller.
We show that in the presence of convex set of persistently exciting linear controllers and strongly convex loss functions, \alg achieves a regret upper bound of polylogarithmic in number of agent-environment interactions.
The unique nature of \alg which combines occasional model estimation with continual online convex optimization allows the agent to achieve significantly improved regret in the challenging setting of adaptive control in partially observable linear dynamical systems.
In this work, we relaxed the requirement in a priori knowledge of the variance of the Gaussian process noise, and measurement noise to just their upper and lower bounds. For the future work, we plan to extend the study of \alg to more general sub-Gaussian noise, and potentially to adversarial perturbations~\citep{simchowitz2020improper}.
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BAYTOWN, Texas (KTRK) --If organizing and shipping out oddly named home furnishings sounds like a duty you could do full time, then IKEA has a job for you.
The Sweden-based retailer announced the operation of a distribution center in Baytown. The facility, which will focus on delivering items to online customers or those who purchased larger items at a store, will hold about 200 positions.
Today, IKEA is planning a hiring event to fill permanent full-time roles at the Baytown IKEA center.
The positions available include general warehouse co-worker, warehouse team lead, and safety & security coordinator.
If you're interested, head to the IKEA careers site first to apply.
Then, visit the IKEA Distribution Center at 4762 Borusan Rd. in Baytown during the hiring event from 2 p.m. to 8 p.m.
Report a typo to the ABC13 staff
| 40,247
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Posted by Anonymous on Friday, May 6, 2011 at 11:03pm.
Your information about MacDonnell is excellent. If you really want more, check out this information.
Red River Colony was a group of people. Red River Valley is the low-lying area around the Red River.
I used Wikipedia as my source to get the information above, there are some things in there which I don't get...have I covered everything? Because I tried to gather all the details about Mile Macdonnel.
| 89,288
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The Ultimate in ON-Site Repair, Restoration, Protextion & Dyeing of Leather,Vinyl, Plastic and Fabric.
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| 124,934
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I’m super excited to have been invited to take part in the Summer Swimwear Tour hosted by Ajaire, Jess and Saskia. I don’t own too many swimwear patterns so it was fun to be able to choose a new pattern to try out. I had already purchased the women’s version of the Swim and Surf shorts pattern by Gracious Threads and decided to make the girl’s version for Silly Bean.
Boardshort fabric was impossible to find locally and I wasn’t sure about buying online mainly because the shipping situation with Canada Post was up in the air for a little while there. So I bought a XXXL men’s boardshorts on deep discount from Old Navy and was able to get a pair of shorts out of them for Silly Bean. There’s enough left over I could make something for their dolls too.
The original Old Navy shorts:
The new Swim and Surf shorts:
Note: When I took the modelled shots it was before I fixed the elastic to have all three rows of elastic. My oldest hates modelling for me so getting her to try things on is like pulling teeth sometimes. So I had to take advantage of the fitting time to be also picture time.
Silly Bean lasted a few minutes, and then Miss V wanted to try them on. So Miss V ran around the yard wearing them. They had been playing in the sprinkler just before the photos were taken and already had swimsuits on.
I was hoping that the gold bias trim would make Silly Bean like them, but no go. She wants nothing to do with them. (I should have used pink trim instead!) I think that was part of why she wouldn’t model for me. Blue looks so pretty on her but she’s very into pink and girly stuff. Luckily they fit Miss V fine.
I blended sizes and sewed the size 4 width with the size 6 length for my skinny minny 6-year-old Silly Bean. 3-year-old Miss V’s waist measurement is just a tiny bit smaller and she fits into the size 3. The size 4 fits her fine though and she has growing room. I had cut the shorts at the swim length, and it makes a good run length for Miss V ha. I also omitted the drawstring for these. They look cute with the drawstring but honestly I like the look better with the grommets than with buttonholes but I didn’t have any on hand.
I love that there is an optional integrated swim bottoms included so that you could wear these alone for swimming and not need to wear with a separate bikini bottom. AND since they’re made for woven fabrics, you could use the pattern to sew up some regular shorts as well.
The pattern includes the printing by layers option which makes it easy to blend sizes. I really like all the patterns I’ve sewn up so far from Gracious Threads, and can’t wait to make myself a pair of swim shorts too. I know I’d feel a lot more comfortable wearing a pair of those in public.
Thanks so much for including me on the tour!
That’s 17 patterns + $30 in fabric!!
8 thoughts on “Swimwear Tour: Swim and Surf Shorts”
I love the running pics haha. There IS a running length 😉 The gold bias tape looks great! Thanks for being on the tour!
Thank you for having me! I have to figure out how to make a gif of all the running pics I got, it was funny. Her running in her running length.
What a great idea! Love the gold too!
Thanks!! 🙂
Love the gold bias tape! What a cute girl!
Thank you! The gold bias tape is so fun, I don’t even remember buying it, lol. I was happy I found it in my stash 🙂
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09 238 7758
info@pukekohebuilders.co.nz
St Andrews Centre Presbyterian Church, Queen Street, Waiuku
Designers: Jasmax Ltd
Project Architect: Jeremy Bennett
Construction period: 10 months
This project involved the construction of an extension and refurbishment of the existing church. The extension was constructed with a concrete floor, structural steel frame, full height glazing and a coloursteel roof.
Category: Seismic Tourism and Leisure Project
"One of the country's most awarded architectural firms has completed a $1.5 million refurbishment and enlargement of a church just south of Auckland.
Auckland firm Jasmax designed changes to the Waiuku Districts Combined Church, extending it by designing a new sanctuary, lounge, kitchen, office and meeting rooms.
Hamish Boyd, project director at Jasmax, said the original A-frame structure was retained and the new sanctuary was inserted on the south side to increase seating capacity and change the focus into a semicircle community model.
The church space was reorganised and Jasmax said the intention was to open up the facility more to the community.
The new additions intersect the existing dominant A-frame structure, with large amounts of glazing to each side.
"The notion of wrapping is taken from the original church and developed with the new works, with careful attention placed on transitions between old and new," Jasmax said of the project.
"The facility is intended to become more than a simple church, with the ability to operate as a theatre for performance, as a functions venue and youth and community centre."
As well as functioning as a church, the building is hired out for dinners, performances, conferences, meetings and special events.
Last month, Jasmax won one of four supreme awards from the Institute of Architects for its 10-level AUT Business School project.
Jasmax said the AUT building celebrated the university's relationship with the city.
"The building form, which is articulated as a woven basket, has been prised apart to reveal the community using the space created within."
(Above content from article in the New Zealand Herald by Anne Gibson.)
Auckland
Telephone; 09 365 9626
New Zealand Herald
Arch Daily:.
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Am still missing the Firm and would love to have it return. Would suggest better promotion for intelligent, riveting story line.
Loved the show, bring it back
The Firm’s first season was great. It should continue. Perhaps it was not promoted well.
So disappointed. My husband and I enjoyed The Firm. To have it end without closure makes us feel like we wasted our time on any of it. We enjoyed the characters, the story. Why cancel it. It’s like cutting off a great movie 2/3 of the way thru.
Great show, get the second season made!
My husband and I just loved every season . We aren’t generally binge watchers but we watched between 3 and 4 episodes every night until it was done. I don’t understand why it wasn’t a hit for everyone. possibly because they wasn’t enough sex for people. Plus not much swearing like a lot of shows. A little too clean for the average viewers.
Just finished binge watching season one “The Firm”. We searched the web to see what happened to season 2. Cancelled????? Can’t be, every episode was a cliff hanger. So many issues left that had to have closure. Please…. season 2 and beyond!!
I highly enjoyed the TV series “The Firm” and was very disappointed about the series being cancelled after season one on NBC. I never missed an episode.
I was looking forward to season two.
What a bummer.
Well what a disappointment, we have been left hanging after 22 episodes. Can’t believe that they are not making a season 2. It may not have been the best series on the television but it was a long long way off the worst.
I liked the original movie and was thrilled to see a series- wish it would continue! Binge watched it and was eagerly awaiting the next series. Sad it is no longer on.
This is dreadful news, there is no closure whatsoever. How can they lead us down 22 episodes and drop a cliff hanger like that.
Terrible, very disappointed.
Prime really should not have run it at all.
Thanks,
Dougal
Just watched on Tubi, upset that it just ended and left it all hanging. One of the best series I’ve ever watched. Fast paced, great episodes that were gripping, great writing and actors perfectly cast. I watched this practically non-stop it was so gripping. Better than anything that is called entertainment on now. Do something even if it is years later! I thought with so many episodes in one season there would be a big season finale that wrapped it all up. This will teach me not to watch one season shows any more.
We really enjoyed this series. Please don’t leave us hanging
I have really enjoyed it – watching on Amazon Prime video. One of the best series I have seen for many years
Has to be another season, can’t leave us here
Would love a series 2 of the firm. Its brilliantly produced, directed, acted, and the story lines are gripping. Much much better than lots of crap that’s shown on tv as “entertainment”
I want to see more.
I couldn’t agree more with everything you said! I just finished watching episode 22 and was shocked to discover that it didn’t continue into the 2nd season! I finally discovered something worthwhile to watch and then it’s gone!!
My husband and I just finished watching season 1 of the Firm and SO DISSAPOINTED that the show was cancelled. Excellent writing, great plot, loved the characters, bring it back on cable! Would love to see a season 2, 3 etc..
| 276,180
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ihateit to I have to take my brother to baseball practice.
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About our bed and breakfast
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\section{Problem Formulation} \label{sec:form2}
\subsection{Active Model Discrimination Problem}
Designing a separating input for model discrimination is equivalent to finding an admissible input for the system, such that if the system is excited with this input, any observed trajectory is consistent with only one model, regardless of any realization of uncertain parameters. In addition, the designed separating input must be optimal for a given cost function $J(u_T)$. The problem of input design for model discrimination can be defined formally as follows:
\begin{problem}[Exact Active Model Discrimination] \label{prob:1}
Given $N$ well-posed affine models $\mathcal{G}_i$, and state, input and noise constraints,
\eqref{eq:bmx0_polytope},\eqref{eq:y_polytope},\eqref{eq:d_polytope}-\eqref{eq:v_polytope},
find an optimal input sequence $u^*_T$ to minimize a given cost function $J(u_T)$ such that for all possible initial states $\bm{x}_0$, uncontrolled input ${d}_T$, process noise $w_T$ and measurement noise $v_T$, only one model is valid, i.e., the output trajectories of any pair of models have to differ in at least one time instance. The optimization problem can be formally stated as follows:
\begin{subequations} \label{eq:quant}
\begin{align}
\nonumber \min_{u_T,x_T} J(u_T) & \\
\text{s.t. }\hspace{0.2cm}
\forall k \in \mathbb{Z}_{T-1}^0:&\ \text{\eqref{eq:u_polytope} hold},\\
\begin{array}{l} \begin{rcases}
\forall i,j \in \mathbb{Z}_\mathrm{N}^{+}, i<j, \forall k \in \mathbb{Z}_T^{0},\\
\forall \bm{x}_0,y_T ,d_T,w_T, v_T:\\ \text{\eqref{eq:state_eq}-\eqref{eq:bmx0_polytope},\eqref{eq:x_polytope},\eqref{eq:y_polytope},\eqref{eq:d_polytope}-\eqref{eq:v_polytope} hold} \end{rcases}: \end{array}
&\ \begin{array}{l}\exists k\in \mathbb{Z}_T^{0}, \\
z_{i}(k) \neq z_{j}(k)\end{array} \label{eq:separability_logic1}
\end{align}
\end{subequations}
\end{problem}
The separation condition \eqref{eq:separability_logic1} ensures that for each pair of models and for all possible values of the uncertain variables, there must exist at least one time instance such that the output values of two models are different. This means that we are first dealing with the uncertainties, then considering the quantifier on $k$ (time instance) for each uncertainty. If we change the order of quantification as was done in \cite{harirchi2017active}, by first considering the existence quantifier and then dealing with all uncertainties, a conservative active model discrimination approach will be obtained.
\begin{problem}[Conservative Active Model Discrimination \cite{harirchi2017active}] \label{prob:2}
Given $N$ well-posed affine models $\mathcal{G}_i$, and state, input and noise constraints
\eqref{eq:bmx0_polytope},\eqref{eq:y_polytope},\eqref{eq:d_polytope}-\eqref{eq:v_polytope}, find an optimal input sequence $u^*_T$ to minimize a given cost function $J(u_T)$ such that there exists at least one time instance at which the output trajectories of each pair of models are different for all possible initial states $\bm{x}_0$, uncontrolled input ${d}_T$, process noise $w_T$ and measurement noise $v_T$. The optimization problem can be formally stated as follows:
\begin{subequations} \label{eq:quant2}
\begin{align}
\nonumber \min_{u_T,x_T} J(u_T) & \\
\text{s.t. }\hspace{0.2cm}
\forall k \in \mathbb{Z}_{T-1}^0:&\ \text{\eqref{eq:u_polytope} hold},\\
\ \begin{array}{l}\exists k\in \mathbb{Z}_T^{0}, \\
z_{i}(k) \neq z_{j}(k)\end{array} :
&
\begin{array}{l} \begin{cases}
\forall i,j \in \mathbb{Z}_\mathrm{N}^{+}, i<j, \forall k \in \mathbb{Z}_T^{0},\\
\forall \bm{x}_0,y_T ,d_T,w_T, v_T:\\ \text{\eqref{eq:state_eq}-\eqref{eq:bmx0_polytope},\eqref{eq:x_polytope},\eqref{eq:y_polytope},\eqref{eq:d_polytope}-\eqref{eq:v_polytope} hold.} \end{cases} \end{array} \label{eq:separability_logic2}
\end{align}
\end{subequations}
\end{problem} \vspace*{-0.15cm}
Note that the quantifier order matters. In the first optimal formulation in Problem \ref{prob:1}, the `for all' quantifier precedes the `there exists' quantifier, implying that the time instance at which separation is enforced can be dependent on the realization of the uncertain variables (similar to adjustable robust optimization \cite{ben2004adjustable}). In the latter formulation in Problem \ref{prob:2} \cite{harirchi2017active}, the `there exists' quantifier precedes the `for all' quantifier, thus separation is enforced at the same time instance for all realizations of the uncertain variables.
In other words, the exact formulation in Problem \ref{prob:1} need not consider the worst-case uncertainties at each time instance and can learn or adapt from previous time steps to consider increasingly smaller uncertainty sets. Meanwhile, the conservative formulation in Problem \ref{prob:2} is a robust solution that considers the worst-case uncertainties at each time instance. To illustrate this difference,
we consider a $N$ second-order system with only uncertainties in the initial conditions. The sets of uncertain variables $\bm{x}(0)$ at time instance $k$ for which we need to find a separating input are shown in Fig. \ref{fig:quantifier} in grey. From the top row, we can see that for the formulation of Problem \ref{prob:1}, the sets are shrunk because once we know all feasible inputs that are separating for some initial conditions at time instance $k$, we can consider the remaining initial conditions at the next time instance (the area between the solid line and dashed line). By contrast, for the formulation in Problem \ref{prob:2}, the sets of uncertain variables are the same at each time instance because we consider all worst-case
uncertain variables at each time instance (bottom row).
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.325,trim=0mm 12mm 0mm -5mm]{Figures/quantifier.pdf}
\caption{The set of uncertain variables for which we need to find a separating input (in grey). The top row depicts the case for the exact formulation in Problem \ref{prob:1} and the bottom row depicts the scenario for the conservative formulation in Problem \ref{prob:2}. \label{fig:quantifier}\vspace*{-0.3cm} }
\end{center}
\end{figure}
| 171,717
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\begin{document}
\title[Tilting Modules over APD's]{Tilting Modules over Almost Perfect Domains}
\author{Jawad Abuhlail}
\address{Department of Mathematics and Statistics\\
Box 5046, KFUPM, 31261 Dhahran, KSA}
\email{abuhlail@kfupm.edu.sa}
\urladdr{http://faculty.kfupm.edu.sa/math/abuhlail}
\thanks{The authors were supported by KFUPM under Research Grant \# MS/Rings/351.}
\author{Mohammad Jarrar}
\address{Department of Mathematics and Statistics\\
Box 5046, KFUPM, 31261 Dhahran, KSA}
\email{mojarrar@kfupm.edu.sa}
\urladdr{}
\date{}
\subjclass{Primary 13C05; Secondary 13D07, 13H99}
\keywords{tilting module, cotilting module, Fuchs-Salce tilting module, perfect ring,
almost perfect domain, coprimely packed ring, Dedekind domain, $1$
-Gorenstein domain, $h$-local domain, Matlis domain}
\maketitle
\begin{abstract}
We provide a complete classification of all \emph{tilting modules} and \emph{
tilting classes} over almost perfect domains, which generalizes the
classifications of tilting modules and tilting classes over Dedekind and $1$
-Gorenstein domains. Assuming the APD is Noetherian, a complete
classification of all \emph{cotilting modules} is obtained (as duals of the
tilting ones).
\end{abstract}
\section{Introduction}
Throughout, $R$ is a commutative ring with $1_{R}\neq 0_{R}$ and all $R$
-modules are unital. With $Z(R)$ we denote the set of zero-divisors of $R$
and set $R^{\times }:=R\backslash Z(R).$ With $Q=(R^{\times })^{-1}R$ we
denote the total ring of quotients of $R$ (the field of quotients, if $R$ is
an integral domain). With $R$\textrm{-}$\mathrm{Mod}$ we denoted the
category of $R$-modules.
Let $M$ be an $R$-module. The \emph{character module} of $M$ is $M^{c}:=
\mathrm{Hom}_{\mathbb{Z}}(M,\mathbb{Q}/\mathbb{Z}).$ With $\mathrm{Max}(M)$
we denote the (possibly empty) spectrum of maximal $R$-submodules and define
\begin{equation*}
\mathrm{rad}(_R M):=\bigcap\limits_{L\in \mathrm{Max}(M)}L\text{ }\;\text{(}
=M,\text{ if }\mathrm{Max}(M)=\varnothing \text{).}
\end{equation*}
In particular, $\mathrm{Max}(R)$ is the spectrum of maximal $R$-ideals and $
J(R):=\mathrm{rad}(_R R)\;$is the Jacobson radical of $R.$ We denote with $
\mathrm{p.d.}_{R}(M)$ (resp. $\mathrm{i.d.}_{R}(M),$ $\mathrm{w.d.}_{R}(M)$)
the projective (resp. injective, weak or flat) dimension of $_{R}M.$
Moreover, we set
\begin{equation*}
\begin{tabular}{lllllll}
$\mathcal{P}_{n}$ & $:=$ & $\{_{R}M\mid \mathrm{p.d.}_{R}(M)\leq n\};$ & & $
\mathcal{P}$ & $:=$ & $\dbigcup\limits_{n=0}^{\infty }\mathcal{P}_{n};$ \\
$\mathcal{I}_{n}$ & $:=$ & $\{_{R}M\mid \mathrm{i.d.}_{R}(M)\leq n\};$ & & $
\mathcal{I}$ & $:=$ & $\dbigcup\limits_{n=0}^{\infty }\mathcal{I}_{n};$ \\
$\mathcal{F}_{n}$ & $:=$ & $\{_{R}M\mid \mathrm{w.d.}_{R}(M)\leq n\};$ & & $
\mathcal{F}$ & $:=$ & $\dbigcup\limits_{n=0}^{\infty }\mathcal{F}_{n}.$
\end{tabular}
\end{equation*}
In particular, $\mathcal{PR}:=\mathcal{P}_{0}$ is the class of projective $R$
-modules, $\mathcal{IN}:=\mathcal{I}_{0}$ is the class of injective $R$
-modules, and $\mathcal{FL}:=\mathcal{F}_{0}$ is the class of flat\emph{\ }$
R $-modules. The class of torsion-free $R$-modules will be denoted with $
\mathcal{TF}.$ For a multiplicative subset $S\subseteq R^{\times },$ the
class of $S$\emph{-divisible} $R$-modules is
\begin{equation*}
\mathcal{D}_{S}:=\{_{R}M\mid sM=M\text{ for every }s\in S\}.
\end{equation*}
In particular, $\mathcal{DI}:=\mathcal{D}_{R^{\times }}$ is the class of
\emph{divisible} $R$-modules. For any unexplained definitions and
terminology on domains and their modules we refer to \cite{FS2001}.
It is well known that every module over any ring has an \emph{injective
envelope} as shown by B. Eckmann and A. Schopf \cite{ES1953} (see \cite[17.9]
{Wis1991}). The dual result does not hold for the categorical dual notion of
\emph{projective covers}. Rings over which every (finitely generated)$\;$module has
a projective cover were considered first by H. Bass \cite{Bas1960} and
called (\emph{semi}-)\emph{perfect rings}. At the beginning of the current
century, L. Bican, R. El Bashir, and E. Enochs \cite{BEE2002} solved the so-called \emph{flat cover conjecture} proving that every module has a flat
cover. Recalling that the class of strongly flat modules $\mathcal{SFL}$
lies strictly between $\mathcal{FL}$ and $\mathcal{PR},$ rings over which
every (finitely generated torsion)\ module has a \emph{strongly flat cover}
were studied by S. Bazzoni and L. Salce \cite{BS2002}; such rings were
characterized as being \emph{almost }(\emph{semi-})\emph{perfect,} in the
sense that every proper homomorphic image of such rings is (semi-)perfect (see also
\cite{BS2003}). Since almost perfect rings that are not domains are perfect,
and since perfect domains are fields, the interest is restricted to almost
perfect domains (\emph{APD}'s). Although local APD's were studied earlier by
R. Smith \cite{Smi1969} under the name ``\emph{local domains with
topologically }$T$-\emph{nilpotent radical}''\emph{\ }(\emph{local
TTN-domains}), the interest in them resurfaced only recently in connection
with the revival of theory of \emph{cotorsion pairs} introduced by L. Salce
\cite{Sal1979}. Our main reference on APD's and their modules is the survey by L. Salce \cite{Sal}
(see also \cite{BS2002}, \cite{Zan2002}, \cite{BS2003}, \cite{SZ2004}, \cite {Sal2005}, \cite{Sal2006}, \cite{Zan2008}, \cite{FL2009}).
\emph{Tilting modules}\ were introduced by S. Brenner and M. Butler \cite
{BB1980} and then generalized by several authors (e.g. \cite{HR1982}, \cite
{Miy1986}, \cite{CT1995}, \cite{Wis1998}, \cite{A-HC2001}). Cotilting
modules appeared as vector space duals of tilting modules over finite
dimensional (Artin) algebras (e.g. \cite[IV.7.8.]{Hap1988}) and then
generalized in a number of papers (e.g. \cite{CDT1997}, \cite{A-HC2001},
\cite{Wis2002}, \cite{Baz2004}). A classification of (co)tilting modules
over special classes of commutative rings and domains was initiated by R.
G\"{o}bel and Trlifaj \cite{GT2000}, who classified (co)tilting Abelian
groups (assuming G\"{o}del's axiom of constructibility; a condition removed
later in \cite{BET2005}). (Co)tilting modules were classified also over
Dedekind domains by S. Bazzoni et al. \cite{BET2005} (removing set
theoretical assumptions in \cite{TW2002}), over valuation and Pr\"{u}fer
domains by L. Salce in \cite{Sal2004} and \cite{Sal2005B}, and recently over
arbitrary $1$-Gorenstein rings by J. Trlifaj and D. Posp\'{i}\v{s}il \cite
{TP2009}.
An open problem in \cite[Page 254]{GT2006} is ``\emph{Characterize all
tilting modules and classes over Matlis domains}'' ($R$ is Matlis, iff $\mathrm{
p.d.}_{R}(Q)=1$). Recalling that APD's are Matlis domains by
\cite[Proposition 2.5]{Sal}, a natural question in this connection was
raised to the first author by L. Salce: ``\emph{Characterize all tilting
modules and classes over APD's}''. Our main result (Theorem \ref{MAIN})
provides a complete answer:
\vspace*{2mm} \textbf{MAIN THEOREM.} Let $R$ be an APD that is not a field.
\begin{enumerate}
\item All tilting $R$-modules are $1$-tilting and represented (up to
equivalence) by
\begin{equation*}
\{T(X):=\bigcap\limits_{\frak{m}\in X}R_{\frak{m}}\bigoplus \frac{
\bigcap\limits_{\frak{m}\in X}R_{\frak{m}}}{R}\mid X\subseteq \mathrm{Max}
(R)\}.
\end{equation*}
\item $\{X$\textrm{-}$\mathrm{Div}\mid X\subseteq \mathrm{Max}(R)\}$ is the
class of all tilting classes, where
\begin{equation*}
X\text{\textrm{-}}\mathrm{Div}:=\{_{R}M\mid \frak{m}M=M\text{ for every }
\frak{m}\in X\}.
\end{equation*}
\item If $R$ is \emph{coprimely packed}, then the set of \emph{Fuchs-Salce
tilting modules }
\begin{equation*}
\{\delta _{S}\mid S\subseteq R^{\times }\text{ is a multiplicative subset}\}
\end{equation*}
classifies all tilting $R$-modules (up to equivalence).
\end{enumerate}
This provides a partial solution to the above mentioned open problem on
Matlis domains and generalizes the classification of tilting modules over $1$
-Gorenstein domains (which are properly contained in the class of APD's) and
Dedekind domains.
The paper is organized as follows. After this introductory section, we
collect in Section 2 some preliminaries on (semi-)perfect rings and almost
(semi-)perfect domains. In Section 3, we characterize some classes of
modules over APD's:
\begin{equation*}
\mathcal{I}=\mathcal{I}_{1},\mathcal{F}=\mathcal{F}_{1}=\mathcal{P}_{1}=
\mathcal{P},\mathcal{IN}=\mathcal{DI}\cap \mathcal{I}_{1},\mathcal{FL}=
\mathcal{TF}\cap \mathcal{P}_{1},\mathcal{DI}=\{M\mid \mathrm{rad}(_R M)=M\}.
\end{equation*}
Although these results are meant to serve in proving the main result (Theorem \ref{MAIN}),
we include them in a separate section since we believe they are interesting for their own.
In Section 4, we present our main results. Since $\mathcal{I}=\mathcal{I}_{1}$ and $\mathcal{P}=\mathcal{P}_{1},$ we notice
first that all (co)tilting modules over APD's are $1$-(co)tilting. Moreover,
we conclude (analogous to the case of Pr\"{u}fer domains) that all
torsion-free tilting modules over APD's are projective. In the local case,
we prove that every tilting module over a local APD is either divisible or
projective (see Theorem \ref{T=D-P}). Finally, we present in Theorem \ref
{MAIN} a complete classification of all tilting modules over APD's that are
not fields. Assuming moreover that the APD $R$ is \emph{coprimely packed}
(e.g. $R$ is a semilocal), we show that any tilting module is equivalent to
a \emph{Fuchs-Salce tilting }$R$\emph{-module} $\delta _{S}$ for some
suitable multiplicative subset $S\subseteq R^{\times }.$ If $R$ is a
coherent (whence Noetherian) APD, then the cotilting $R$-modules are
precisely the (dual) character modules of the tilting ones (see Corollary
\ref{Cot-APD}).
\section{Preliminaries}
In this section, we collect some preliminaries on (semi-)perfect rings and
almost (semi-)perfect domains.
\begin{definition}
(\cite{Bas1960}) The ring $R$ is said to be (\textbf{semi-})\textbf{perfect}
, iff every (finitely generated) $R$-module has a projective cover.
\end{definition}
For the convention of the reader, we collect in the following lemma some of the characterizations of perfect commutative rings (e.g. \cite[Section 28]
{AF1993}, \cite[Section 43]{Wis1991}, \cite[Chapter 8]{Lam2001}, \cite[Theorem 1.1]{BS2003}):
\begin{lemma}
\emph{\label{perfect}}The following are equivalent:
\begin{enumerate}
\item $R$ is perfect;
\item every semisimple $R$-module has a projective cover;
\item every flat $R$-module is \emph{(}self-\emph{)}projective;
\item direct limits of projective $R$-modules are \emph{(}self-\emph{)}
projective;
\item $R$ is semilocal and every non-zero $R$-module has a maximal
submodule;
\item $R$ is semilocal and every non-zero $R$-module contains a simple
submodule;
\item $R$ contains no infinite set of orthogonal idempotents and every
non-zero $R$-module contains a simple submodule;
\item $R/J(R)$ is semisimple and $J(R)$ is $T$-nilpotent;
\item $R/J(R)$ is semisimple and $R$ is semiartinian;
\item $R$ satisfies the DCC for principal \emph{(}finitely generated\emph{)}
ideals;
\item Any $R$-module satisfies the DCC on its cyclic \emph{(}finitely
generated\emph{)} $R$-submodules;
\item Any $R$-module satisfies the ACC on its cyclic $R$-submodules;
\item $R$ is a finite direct product of local rings with $T$-nilpotent
maximal ideals;
\item $R$ is semilocal and $R_{\frak{m}}$ is a perfect ring for every $
\frak{m}\in \mathrm{Max}(R);$
\item $R$ is semilocal and semiartinian.
\end{enumerate}
\end{lemma}
\begin{definition}
(\cite{BS2002}, \cite{BS2003}) $R$ is an \textbf{almost (semi-)perfect ring}
, iff $R/I$ is (semi-)perfect for every non-zero ideal $0\neq
I\trianglelefteq R.$
\end{definition}
\begin{remark}
An almost perfect ring that is not a domain is necessarily perfect by
\cite[Proposition 1.3]{BS2003}. On the other hand, any perfect domain is a field (e.g. \cite[Corollary 1.3]{Sal}). This restricts the
interest to \emph{almost perfect domains} (\emph{APD}'s).
\end{remark}
\begin{lemma}
\label{ASPD}\emph{(\cite[Theorem 4.9]{BS2002}, \cite[Theorem IV.3.7]{FS2001})
} The following are equivalent for an integral domain $R:$
\begin{enumerate}
\item $R$ is almost semi-perfect;
\item every finitely generated torsion $R$-module has a strongly flat cover;
\item $Q/R\simeq \bigoplus\limits_{\frak{m}\in \mathrm{Max}(R)}(Q/R)_{\frak{
m}}$ canonically;
\item $R$ is $h$-local \emph{(}i.e. $R/I$ is semilocal for every non-zero
ideal $0\neq I\trianglelefteq R$ and $R/P$ is local for every non-zero prime
ideal $0\neq P\in \mathrm{Spec}(R)$\emph{)}.
\end{enumerate}
\end{lemma}
In the following lemma we collect several characterizations of APD's (see
\cite[Main Theorem]{Sal}, \cite{BS2002}, and \cite{BS2003}):
\begin{lemma}
\label{APD-Main}For an integral domain $R$ with $Q\neq R$ the following are
equivalent:
\begin{enumerate}
\item $R$ is an APD;
\item $R$ is almost semi-perfect and $R_{\frak{m}}$ is an APD for every $
\frak{m}\in \mathrm{Max}(R);$
\item $R$ is $h$-local and $R_{\frak{m}}$ is an APD for every $\frak{m}\in
\mathrm{Max}(R);$
\item $R$ is $h$-local and $Q/R$ is semiartinian;
\item $R$ is $h$-local and for every proper non-zero ideal $I\neq 0,R,$ the
$R$-module $R/I$ contains a simple $R$-submodule.
\item every flat $R$-module is strongly flat;
\item every $R$-module has a strongly flat cover;
\item every weakly cotorsion $R$-module is cotorsion;
\item every $R$-module with weak dimension at most $1$ has projective
dimension at most $1$ \emph{(}i.e. $\mathcal{F}_{1}=\mathcal{P}_{1}$\emph{)};
\item every divisible $R$-module is weak-injective.
\end{enumerate}
\end{lemma}
\begin{remarks}
\label{rem-APD}Let $R$ be an integral domain.
\begin{enumerate}
\item $R$ is a coherent APD if and only if $R$ is Noetherian and $1$
-dimensional (see \cite[Propositions 4.5, 4.6]{BS2002}). Whence, Dedekind
domains are precisely the Pr\"{u}fer APD's.
\item A valuation domain $R$ is an APD if and only if $R$ is a DVR (e.g.
\cite[Example 2.2]{Sal}).
\item We have the following implications (e.g. \cite{FS2001}, \cite{Sal}): $
R$ is Dedekind $\Rightarrow R$ is $1$-Gorenstein $\Rightarrow R$ is $1$
-dimensional and Noetherian $\Rightarrow R$ is an APD $\Rightarrow R$ is a $1
$-dimensional $h$-local $\Rightarrow $ $R$ is a Matlis domain.
\end{enumerate}
\end{remarks}
The following examples illustrate that the implications above are not reversible:
\begin{examples}
\begin{enumerate}
\item \label{ex-APD}Let $d$ be a square-free integer such that $d\equiv 1$
(mod $4$) and consider the commutative Noetherian subring
\begin{equation*}
R:=\{\frac{m}{2n+1}+\frac{m^{\prime }}{2n^{\prime }+1}\sqrt{d}\mid
m,m^{\prime },n,n^{\prime }\in \mathbb{Z}\}\subseteq \mathbb{Q}[\sqrt{d}].
\end{equation*}
By \cite[Corollary 4.5]{Smi2000}, $R$ is a $1$-Gorenstein domain that is not
Dedekind.
\item Let $K$ be a field. Then $R=K[|t^{3},t^{5},t^{7}|]$ is a Noetherian $1
$-dimensional domain which is not $1$-Gorenstein (e.g. \cite[Ex. 18.8]{Mat}).
\item Let $K$ be a field and $V=(K[[x]],M)$ the local domain of power
series in the indeterminate $x$ with coefficients in $K$ and with maximal
ideal $M:=xK[[x].$ Let $(D,\frak{m})$ be a local subring of $K$ and consider
the local integral domain $R:=(D+M,\frak{m}+M$). By \cite[Lemma 3.1]{BS2003}
, $R$ is an APD if and only if $D$ is a field. Moreover, by \cite[Example
3.3]{BS2003}, if $D=F$ is a field and $K=F(X),$ then $R$ is Noetherian if
and only if $[K:F]<\infty .$ So, if $[K:F]=\infty $ then $R$ is a \emph{
non-Noetherian} APD whence not $1$-Gorenstein.
\item Any rank-one non-discrete valuation domain is a $1$-dimensional local
Matlis domain that is not an APD (a concrete example is \cite[Example 1.3]{Zan2008}).
\item Any almost Dedekind domain which is not Dedekind is a $1$-dimensional Matlis domain that is not of finite character, whence not $h$-local (for a concrete example see \cite[Example III.5.5]{FS2001}).
\end{enumerate}
\end{examples}
Generalizing the so-called \emph{Prime Avoidance Theorem} (e.g. \cite[3.61]
{Sha2000}) by allowing \emph{infinite} unions of prime ideals led to the
following notions.
\begin{punto}
(\cite{RV1970}, \cite{Erd1988}) An ideal $I$ of a commutative ring $R$ is
said to be \emph{coprimely packed} (resp. \emph{compactly packed}), iff for
any set of maximal (resp. prime) $R$-ideals $\{P_{\lambda }\}_{\Lambda }$ we
have
\begin{equation}
I\subseteq \bigcup_{\lambda \in \Lambda }P_{\lambda }\Rightarrow I\subseteq
P_{\lambda _{0}}\text{ for some }\lambda _{0}\in \Lambda . \label{packed}
\end{equation}
A class of $R$-ideals $\mathcal{E}$ said to be \emph{coprimely packed}
(resp. \emph{compactly packed}), iff every ideal in $\mathcal{E}$ is so. The
ring $R$ is said to be \emph{coprimely packed} (resp. \emph{compactly packed}
), iff every ideal of $R$ is coprimely packed (resp. compactly packed).
\end{punto}
\begin{remark}
\label{rem-cp}By \cite[Lemma 2]{EM1994} (resp. \cite[Theorem 2.3]{BO1992}),
a ring $R$ is coprimely packed (resp. compactly packed) if and only if $
\mathrm{Spec}(R)$ is coprimely packed (resp. compactly packed). Indeed, $1$
-dimensional rings (e.g. APD's) are coprimely packed if and only if they are
compactly packed. By \cite{RV1970} a Dedekind domain is compactly packed
(equivalently coprimely packed) if and only if its ideal class group is
torsion (see also \cite[Theorem 1.4]{Erd1988}). Semilocal rings are
obviously coprimely packed (by the Prime Avoidance Theorem). A coprimely
packed domain $R$ is $h$-local if, for example, $R$ is $1$-dimensional by
\cite[Proposition 1.3]{Erd1988} and \cite[Theorem 3.22]{Mat1972} (see also
\cite[Theorem 3.7, EX. IV.3.3]{FS2001}) or if $Q/R$ is injective by
\cite[Theorem 9]{Bra1975}. While clearly all compactly packed rings are
coprimely packed, it had been shown in \cite{RV1970} that a Noetherian
compactly packed ring has Krull dimension at most one; thus any semilocal
Noetherian ring with Krull dimension at least $2$ is coprimely packed but
not compactly packed.
\end{remark}
\begin{example}
\label{ex-cp}Let $K$ be an algebraically closed field and $F$ a proper subfield such
that $[K:F]=\infty $ and $X$ an indeterminate. By \cite[Example 5.5]{Sal}, $
R:=F+XK[X]$ is a non-coherent APD with $\mathrm{Max}(R)=\{XK[X]\}\cup
\{(1-aX)R\mid a\in K^{\times }\}.$ Clearly, $R$ is a coprimely packed
(compactly packed) APD that is not semilocal.
\end{example}
\section{Modules over APD's}
In this section, we characterize the injective modules, the torsion-free
modules, and the divisible modules over almost perfect domains. Moreover, we
show that over such integral domains $\mathcal{I}=\mathcal{I}_{1},$ $
\mathcal{F}=\mathcal{F}_{1}=\mathcal{P}_{1}=\mathcal{P}.$ Throughout in this
section, $R$ is an almost perfect domain with $Q\neq R.$
Dedekind domains are characterized by the fact that every divisible module
is injective (e.g. \cite[Theorem 4.24]{Ro}, \cite[40.5]{Wis1991}). This
inspires:
\begin{proposition}
\label{inj-APD}An $R$-module $M$ is injective if and only if $M$ is
divisible and $\mathrm{i.d.}_{R}(M)\leq 1,$ i.e.
\begin{equation}
\mathcal{IN}=\mathcal{DI}\cap \mathcal{I}_{1}. \label{IN=}
\end{equation}
\end{proposition}
\begin{Beweis}
$(\Rightarrow )$ Injective modules over any ring are divisible (e.g.
\cite[16.6]{Wis1991}).
$(\Leftarrow )$ Assume that $_{R}M$ is divisible and $\mathrm{i.d.}
_{R}(M)\leq 1.$
\textbf{Case 1. }$(R,\frak{m})$ is \emph{local}. Let $0\neq r\in R$ be
arbitrary. By Lemma \ref{APD-Main} (5), the $R$-module $R/Rr$ contains a
simple $R$-submodule $J/Rr$ ($\simeq R/\frak{m},$ since $\mathrm{Max}(R)=\{
\frak{m}\}$). So, we have a short exact sequence of $R$-modules
\begin{equation*}
0\rightarrow J/Rr\rightarrow R/Rr\rightarrow R/J\rightarrow 0.
\end{equation*}
Applying the contravariant functor $\mathrm{Hom}_{R}(-,M),$ we get a long
exact sequence
\begin{equation*}
\cdots \rightarrow \mathrm{Ext}_{R}^{1}(R/Rr,M)\rightarrow \mathrm{Ext}
_{R}^{1}(J/Rr,M)\rightarrow \mathrm{Ext}_{R}^{2}(R/J,M)\rightarrow \cdots
\end{equation*}
Since $_{R}M$ is divisible, we have $\mathrm{Ext}_{R}^{1}(R/Rr,M)=0$ by
\cite[Lemma I.7.2]{FS2001}; and since $\mathrm{i.d.}_{R}(M)\leq 1,$ we have $
\mathrm{Ext}_{R}^{2}(R/J,M)=0.$ It follows that $\mathrm{Ext}_{R}^{1}(R/
\frak{m},M)\simeq \mathrm{Ext}_{R}^{1}(J/Rr,M)=0,$ whence $_{R}M$ is
injective by \cite[Proposition 8.1. (1)]{Sal}.
\textbf{Case 2.} $R$ is arbitrary. Let $\frak{m}\in \mathrm{Max}(R)$ be
arbitrary. Since $R$ is $h$-local, it follows by \cite[Theorem IX.7.6]
{FS2001} that localizing any injective coresolution of $R$-modules at $\frak{
m}$ yields an injective coresolution of $R_{\frak{m}}$-modules, hence $
\mathrm{i.d.}_{R_{\frak{m}}}(M_{\frak{m}})\leq 1.$ Since $_{R_{\frak{m}}}M_{
\frak{m}}$ is also divisible, we conclude that $_{R_{m}}M_{\frak{m}}$ is
injective by the proof of Case 1. Since $R$ is $h$-local, we have (e.g. \cite{Mat1972}, \cite[Theorem IX.7.6]{FS2001})
\begin{equation*}
\mathrm{i.d.}_{R}(M)=\sup \{\mathrm{i.d.}_{R_{\frak{m}}}(M_{\frak{m}})\mid
\frak{m}\in \mathrm{Max}(R)\} = 0. \blacksquare
\end{equation*}
\end{Beweis}
It is well-known that for $1$-Gorenstein domains (and general $1$-Gorenstein rings), we have $\mathcal{I}= \mathcal{I}_{1} = \mathcal{F}=\mathcal{F}_{1} = \mathcal{P}=\mathcal{P}_{1}$ (e.g. \cite[9.1.10]{EJ2000}, \cite[7.1.12]{GT2000}). For the strictly larger class of APD's (see Example \ref{ex-APD} (3)), these hold partially.
\begin{proposition}
\label{I-1-P}We have
\begin{equation}
\mathcal{I}=\mathcal{I}_{1},\text{ }\mathcal{F}=\mathcal{F}_{1}=\mathcal{P}
_{1}=\mathcal{P}. \label{I=, P=}
\end{equation}
\end{proposition}
\begin{Beweis}
Let $R$ be an APD.
\begin{itemize}
\item We prove, by induction, that any $R$-module $M$ with finite injective
dimension at most $n$ has injective dimension at most $1.$ If $n=0,$ we are
done. Let $n\geq 1$ and assume the statement is true for $n-1.$ Let
\begin{equation*}
0\rightarrow M\overset{f_{0}}{\longrightarrow }E_{0}\overset{f_{1}}{
\longrightarrow }E_{1}\rightarrow \cdots \longrightarrow E_{n-2}\overset{
f_{n-1}}{\longrightarrow }E_{n-1}\overset{f_{n}}{\longrightarrow }
E_{n}\longrightarrow 0
\end{equation*}
be an injective coresolution of $_{R}M$ and $L:=\mathrm{Im}(f_{n-1})=\mathrm{
Ker}(f_{n}).$ Being a homomorphic image of a divisible $R$-module, $L$ is
divisible and obviously $\mathrm{i.d.}_{R}(L)\leq 1$ whence $_{R}L$ is
injective by Proposition \ref{inj-APD}. It follows that $\mathrm{i.d.}
_{R}(M)\leq n-1,$ whence $\mathrm{i.d.}_{R}(M)\leq 1$ by the induction
hypothesis.
\item Let $M$ be with finite weak (flat)\ dimension at most $n.$ By
\cite[Proposition IX. 7.7]{FS2001} we have for any injective cogenerator $
_{R}\mathbf{E}:$
\begin{equation}
\mathrm{i.d.}_{R}(\mathrm{Hom}_{R}(M,\mathbf{E}))=\mathrm{w.d.}_{R}(M)
\label{w.d.}
\end{equation}
and we conclude that $\mathrm{w.d.}_{R}(M)\leq 1$ by the first part of the
proof.
\item Let $_{R}M$ be with finite projective dimension at most $n.$ Since $
\mathrm{w.d.}_{R}(M)\leq \mathrm{p.d.}_{R}(M)\leq n,$ we have $M\in \mathcal{
F}_{1}=\mathcal{P}_{1}$ by Lemma \ref{APD-Main} (9).$\blacksquare $
\end{itemize}
\end{Beweis}
Using Proposition \ref{I-1-P} we conclude that an APD is either Dedekind or
has (weak) global dimension $\infty $. This provides new characterizations
of Dedekind domains and recovers the fact that Dedekind domains are
precisely the Pr\"{u}fer APD's.
\begin{corollary}
\label{Dedekind}An arbitrary integral domain $R$ is Dedekind if and only if $
R$ is an APD with finite \emph{(}weak\emph{)} global dimension if and only
if $R$ is an APD with \emph{(}weak\emph{)} global dimension at most one if
and only if $R$ is a Pr\"{u}fer APD.
\end{corollary}
\begin{proposition}
\label{flat-APD}An $R$-module $M$ is flat if and only if $M$ is torsion-free
and $\mathrm{p.d.}_{R}(M)\leq 1,$ i.e.
\begin{equation}
\mathcal{FL}=\mathcal{TF}\cap \mathcal{P}_{1} = \mathcal{TF}\cap \mathcal{F}_{1}. \label{FL=}
\end{equation}
\end{proposition}
\begin{Beweis}
$(\Rightarrow )$ Follows by the well-known fact that flat modules over
domains are torsion-free (e.g. \cite[36.7]{Wis1991}). So, we are done by $
\mathcal{F}_{1}=\mathcal{P}_{1}$ (Lemma \ref{APD-Main} (9)).
$(\Leftarrow )$ Since $_{R}M$ is torsion-free, it embeds in a vector space
over $Q$ (e.g. \cite[Lemma 4.33]{Ro}). So, we have a short exact sequence of
$R$-modules
\begin{equation*}
0\rightarrow M\rightarrow Q^{(\Lambda )}\rightarrow Q^{(\Lambda
)}/M\rightarrow 0.
\end{equation*}
Since $_{R}Q^{(\Lambda )}$ is flat, $\mathrm{p.d.}_{R}(Q^{(\Lambda )})\leq 1$
by Lemma \ref{APD-Main} (9). It follows by \cite[Lemma VI.2.4]{FS2001} that $
\mathrm{p.d.}_{R}(Q^{(\Lambda )}/M)<\infty ,$ whence $Q^{(\Lambda )}/M\in
\mathcal{P}_{1}=\mathcal{F}_{1}$ by Proposition \ref{I-1-P}. Consequently, $
_{R}M$ is flat.$\blacksquare $
\end{Beweis}
\begin{punto}
(\cite{GT2006}) An $R$-module over an (arbitrary ring) $R$ is said to be
\emph{\ }\textbf{strongly finitely presented}, iff it possesses a projective
resolution consisting of finitely generated $R$-modules. With $R$-$\mathrm{
mod}$ we denote the class of such modules. In case $R$ is coherent, $R$-$
\mathrm{mod}$ coincides with the class of finitely presented $R$-modules.
\end{punto}
\begin{proposition}
\label{APD-div}The following are equivalent for an $R$-module $M:$
\begin{enumerate}
\item $_{R}M$ is divisible;
\item $\mathrm{rad}(_{R}M)=M$ \emph{(}i.e. $M$ has no maximal $R$-submodules
\emph{)};
\item $\frak{m}M=M$ for every $\frak{m}\in \mathrm{Max}(R).$
\end{enumerate}
\end{proposition}
\begin{Beweis}
The result is obvious for $M=0.$ So, assume $M\neq 0.$ The equivalence $
(1)\Leftrightarrow (3)$ is already known for APD's (e.g. L. Salce
\cite[Proposition 8.1]{Sal}).
$(1)\Rightarrow (2)$ Suppose that $M$ contains a maximal $R$-submodule $L.$ Then $M/L\simeq R/\frak{m}$
for some maximal ideal $\frak{m}\trianglelefteq R.$ Since $_{R}M$ is
divisible by assumption, it follows that $R/\frak{m}$ is also a divisible $R$-module (a contradiction).
$(2)\Rightarrow (1)$ Suppose $_{R}M$ is not divisible. Then there exists $
0\neq r\in R$ such that $rM\neq M.$ By Lemma \ref{perfect} (5), the non-zero
$R/rR$-module $M/rM$ contains a maximal submodule $N/rM.$ Then there exists $
\frak{m}\in \mathrm{Max}(R),$ such that
\begin{equation*}
R/\frak{m}\simeq (R/rR)/(\frak{m}/rR)\simeq (M/rM)/(N/rM)\simeq M/N.
\end{equation*}
This implies that $N\in \mathrm{Max}(_{R}M)$ (a contradiction).$\blacksquare $
\end{Beweis}
\begin{definition}
A non-empty set $\mathcal{L}$ of $R$-ideals is said to be a \textbf{
localizing system} (or a \textbf{Gabriel topology}), iff for any ideals $
I,J\trianglelefteq R$ we have:
(LS1) If $I\in \mathcal{L}$ and $I\subseteq J,$ then $J\in \mathcal{L};$
(LS2) If $I\in \mathcal{L}$ and $(J:_{R}r)\in \mathcal{L}$ for every $r\in I,
$ then $J\in \mathcal{L}.$
\end{definition}
\begin{definition}
Let $R$ be an integral domain and $\mathcal{E}$ be a class of $R$-ideals. We
say an $R$-module $M$ is $\mathcal{E}$-divisible, iff $IM=M$ for every $I\in
\mathcal{E}.$
\end{definition}
For any classes $\mathcal{M}$ of $R$-modules and $\mathcal{E}$ of $R$-ideals
we set
\begin{equation*}
\begin{tabular}{lll}
$\mathcal{D}(\mathcal{M})$ & $:=$ & $\{I\trianglelefteq R\mid IM=M$ for
every $M\in \mathcal{M}\};$ \\
$\mathcal{E}$\textrm{-}$\mathrm{Div}$ & $:=$ & $\{_{R}M\mid IM=M$ for every $
I\in \mathcal{E}\}.$
\end{tabular}
\end{equation*}
If $R$ is a domain, then $\mathcal{D}(_{R}M)$ is a localizing system by
\cite[Lemma 1.1]{Sal2005}.
\begin{lemma}
\label{F-Div}Let $R$ be an APD and $\frak{F}$ a localizing system. An $R$
-module $M$ is $\frak{F}$-divisible if and only if $\frak{m}M=M$ for all
maximal ideals $\frak{m}$ in $\frak{F},$ i.e.
\begin{equation}
\frak{F}\text{\textrm{-}}\mathrm{Div}=(\frak{F}\cap \mathrm{Max}(R))\text{
\textrm{-}}\mathrm{Div}. \label{D(F)=}
\end{equation}
\end{lemma}
\begin{Beweis}
Let $M\in (\frak{F}\cap \mathrm{Max}(R))$\textrm{-}$\mathrm{Div}.$ Let $I\in
\frak{F}$ be arbitrary and set $\mathcal{M}(I):=\{\frak{m}\in \mathrm{Max}
(R)\mid I\subseteq \frak{m}\}\subseteq \frak{F}$ by (LS1). Let $\frak{m}\in
\mathrm{Max}(R)$ be arbitrary. If $\frak{m}\in \mathcal{M}(I),$ then $\frak{m
}_{\frak{m}}M_{\frak{m}}=(\frak{m}M)_{\frak{m}}=M_{\frak{m}}$ whence the $R_{
\frak{m}}$-module $M_{\frak{m}}$ is divisible by Proposition \ref{APD-div},
and it follows that $(IM)_{\frak{m}}=I_{\frak{m}}M_{\frak{m}}=M_{\frak{m}}.$
On the other hand, if $\frak{m}\notin \mathcal{M}(I),$ then $I_{\frak{m}}=R_{
\frak{m}}$ and so $(IM)_{\frak{m}}=R_{\frak{m}}M_{\frak{m}}=M_{\frak{m}}.$
Since $(IM)_{\frak{m}}=M_{\frak{m}}$ for every $\frak{m}\in \mathrm{Max}(R),$
we conclude that $IM=M$ (i.e. $M\in \frak{F}$\textrm{-}$\mathrm{Div}$).$
\blacksquare $
\end{Beweis}
\section{Tilting and Cotilting Modules}
This section is devoted to the classification of (co)tilting modules over
APD's. For any unexplained definitions we refer to \cite{GT2006}.
For any class of $R$-modules $\mathcal{M}$ we set
\begin{equation*}
\begin{tabular}{lll}
$\mathcal{M}^{\perp _{\infty }}$ & $:=$ & $\{_{R}N\mid \mathrm{Ext}
_{R}^{i}(M,N)=0\text{ for all }i\geq 1$ and every $M\in \mathcal{M}\};$ \\
$^{\perp _{\infty }}\mathcal{M}$ & $:=$ & $\{_{R}N\mid \mathrm{Ext}
_{R}^{i}(N,M)=0\text{ for all }i\geq 1$ and every $M\in \mathcal{M}\};$
\end{tabular}
\end{equation*}
Moreover, we set
\begin{equation*}
\mathcal{M}^{\perp }:=\bigcap\limits_{M\in \mathcal{M}}\mathrm{Ker}(\mathrm{
Ext}_{R}^{1}(M,-))\text{ and }^{\perp }\mathcal{M}:=\bigcap\limits_{M\in
\mathcal{M}}\mathrm{Ker}(\mathrm{Ext}_{1}^{R}(-,M)).
\end{equation*}
\begin{punto}
For $_{R}X,$ let $\mathrm{Gen}_{n}(_{R}X)$ be the class of $R$-modules $M$
possessing an exact sequence of $R$-modules $X^{(\Lambda _{n})}\rightarrow
\cdots \rightarrow X^{(\Lambda _{1})}\rightarrow M\rightarrow 0$ (for index
sets $\Lambda _{1},\cdots ,\Lambda _{n}$). Dually, let $\mathrm{Cogen}
_{n}(_{R}X)$ be the class of $R$-modules $M$ possessing an exact sequence of
$R$-modules $0\rightarrow M\rightarrow X^{\Lambda _{1}}\rightarrow \cdots
\rightarrow X^{\Lambda _{n}}$ (for index sets $\Lambda _{1},\cdots ,\Lambda
_{n}$). In particular, $\mathrm{Gen}(_{R}X):=\mathrm{Gen}_{1}(_{R}X)$ is the
class of $X$\emph{-generated }$R$-modules and $\mathrm{Cogen}(_{R}X):=
\mathrm{Cogen}_{1}(_{R}X)$ is the class of $X$\emph{-cogenerated} $R$
-modules.
\end{punto}
\begin{punto}
Let $\mathcal{A}$ and $\mathcal{B}$ be two classes of $R$-modules. Then $(
\mathcal{A},\mathcal{B})$ is said to be a \textbf{cotorsion pair}, iff $
\mathcal{A}=$ $^{\perp }\mathcal{B}$ and\textbf{\ }$\mathcal{B}=\mathcal{A}
^{\perp }.$ If, moreover, $\mathrm{Ext}_{R}^{i}(A,B)=0$ for all $i\geq 1$
and $A\in \mathcal{A},$ $B\in \mathcal{B}$ we say $(\mathcal{A},\mathcal{B})$
is \textbf{hereditary}. Each class $\mathcal{M}$ of $R$-modules \emph{
generates} a cotorsion pair $(^{\perp }(\mathcal{M}^{\perp }),\mathcal{M}
^{\perp })$ and \emph{cogenerates} a cotorsion pair $(^{\perp }\mathcal{M}
,(^{\perp }\mathcal{M})^{\perp }).$ For two cotorsion pairs $(\mathcal{A},
\mathcal{B}),$ $(\mathcal{A}^{\prime },\mathcal{B}^{\prime }),$ we have $
\mathcal{A}=\mathcal{A}^{\prime }$ if and only if $\mathcal{B}=\mathcal{B}
^{\prime }.$
\end{punto}
\begin{punto}
An $R$-module $T$ is said to be $n$-\textbf{tilting}, iff $\mathrm{Gen}
_{n}(_{R}T)=T^{\perp _{\infty }};$ the \textbf{induced }$n$\textbf{-tilting
class} $T^{\perp _{\infty }}$ cogenerates a \emph{hereditary cotorsion pair }
$(^{\perp }(T^{\perp _{\infty }}),T^{\perp _{\infty }})$ with $\mathcal{A}:=$
$^{\perp }(T^{\perp _{\infty }})\subseteq \mathcal{P}_{n}$ by \cite[Lemma
5.1.8]{GT2006} (in particular, $\mathrm{p.d.}_{R}(T)\leq n$). By \cite[Lemma
6.1.2]{GT2006} (see also \cite[Theorem 3.11]{Baz2004}), $_{R}T$ is $1$
-tilting if and only if $\mathrm{Gen}(_{R}T)=T^{\perp }.$ An $R$-module $T$
is \textbf{tilting}, iff $T$ is $n$-tilting for some $n\geq 0.$ Two tilting $
R$-modules $T_{1},$ $T_{2}$ are said to be \textbf{equivalent} ($T_{1}\sim T_{2}$), iff
$T_{1}^{\perp _{\infty }}=T_{2}^{\perp _{\infty }}.$
\end{punto}
\begin{punto}
An $R$-module $C$ is said to be $n$-\textbf{cotilting}, iff $\mathrm{Cogen}
_{n}(_{R}C)=$ $^{\perp _{\infty }}C;$ the \textbf{induced} $n$\textbf{
-cotilting class }$^{\perp _{\infty }}C$ generates a \emph{hereditary
cotorsion pair} $(^{\perp _{\infty }}C,(^{\perp _{\infty }}C)^{\perp })$
with $\mathcal{B}:=(^{\perp _{\infty }}C)^{\perp }\subseteq \mathcal{I}_{n}$
by \cite[Lemma 8.1.4]{GT2006} (in particular, $\mathrm{i.d.}_{R}(C)\leq n$).
By \cite[Lemma 8.2.2]{GT2006} (see also \cite[Theorem 3.11]{Baz2004}), $_{R}C
$ is $1$-cotilting if and only if $\mathrm{Cogen}(_{R}C)=$ $^{\perp }C.$ An $
R$-module $C$ is said to be \textbf{cotilting}, iff $C$ is $n$-cotilting for
some $n\geq 0.$ Two cotilting $R$-modules $C_{1},$ $C_{2}$ are said to be \textbf{equivalent} ($C_{1}\sim C_{2}$), iff $^{\perp _{\infty }}C_{1}=$ $^{\perp
_{\infty }}C_{2}.$
\end{punto}
\begin{remark}
Obviously, the $0$-tilting modules are precisely the projective generators,
while the $0$-cotilting modules are precisely the injective cogenerators.
\end{remark}
\begin{example}
Let $R$ be an integral domain, $S\subseteq R^{\times }$ a multiplicative
subset, and $\omega =()$ be the empty sequence. Let $F$ be the \emph{free} $R
$-module with basis
\begin{equation*}
\beta :=\{(s_{0},\cdots ,s_{n})\mid n\geq 0\text{ and }s_{j}\in S\text{ for }
0\leq j\leq n\}\cup \{\omega \}
\end{equation*}
and $G$ the $R$-submodule of $F$ (which is in fact \emph{free}) generated by
\begin{equation*}
\{(s_{0},\cdots ,s_{n})s_{n}-(s_{0},\cdots ,s_{n-1})\mid n>0\text{ and }
s_{j}\in S\text{ for }0\leq j\leq n\}\cup \{(s)s-\omega \}.
\end{equation*}
The $R$-module $\delta _{S}:=F/G$ is a $1$-tilting $R$-module with $\delta
_{S}^{\perp }=\mathrm{Gen}(\delta _{S})=\mathcal{D}_{S}$ as shown in \cite
{FS1992} and we call it the \textbf{Fuchs-Salce module.} It generalizes the
\textbf{Fuchs module }$\delta :=\delta _{R^{\times }}$ (introduced in \cite
{Fuc1984}), which was studied and shown to be $1$-tilting with $\delta
^{\perp }=\mathrm{Gen}(_{R}\delta )=\mathcal{DI}$ by A. Facchini in \cite
{Fac1987} and \cite{Fac1988}.
\end{example}
\begin{definition}
(\cite{GT2006}) A \textbf{Matlis localization} of the commutative ring $R$
is $S^{-1}R,$ where $S\subseteq R^{\times }$ is a multiplicative subset and $
\mathrm{p.d.}_{R}(S^{-1}R)\leq 1.$
\end{definition}
\begin{lemma}
\label{eqv}\emph{(\cite[Proposition 5.2.24]{GT2006}, \cite[Theorem 1.1]
{A-HHT2005}) }Let $R$ be a commutative ring and $S\subseteq R^{\times }$ a
multiplicative subset.
\begin{enumerate}
\item Let $T$ be an $n$-tilting $R$-module, $\mathcal{T}:=T^{\perp _{\infty
}}$ the induced $n$-tilting class and
\begin{equation*}
\mathcal{T}_{S}:=\{_{S^{-1}R}N\mid N\simeq S^{-1}M\text{ for some }M\in
\mathcal{T}\}.
\end{equation*}
Then $S^{-1}T$ is an $n$-tilting $S^{-1}R$-module and its induced $n$-tilting
class is
\begin{equation*}
(S^{-1}T)^{\perp _{\infty }}:=\bigcap_{i\geq 1}\mathrm{Ker}(\mathrm{Ext}
_{S^{-1}R}^{i}(S^{-1}T,-))=\mathcal{T}_{S}=T^{\perp _{\infty }}\cap S^{-1}R
\text{\textrm{-}}\mathrm{Mod}\text{.}
\end{equation*}
Moreover, $_{R}M\in \mathcal{T}$ if and only if $M_{\frak{m}}\in \mathcal{T}
_{\frak{m}}$ for every $\frak{m}\in \mathrm{Max}(R).$ If $T^{\prime }$ is
another $n$-tilting $R$-module, then
\begin{equation}
T\sim T^{\prime }\Leftrightarrow T_{\frak{m}}\sim T_{\frak{m}}^{\prime }
\text{ for all maximal ideals }\frak{m}\in \mathrm{Max}(R). \label{TeqvT'}
\end{equation}
\item The following are equivalent:
\begin{enumerate}
\item $\mathrm{p.d.}_{R}(S^{-1}R)\leq 1$ \emph{(}i.e. $S^{-1}R$ is a Matlis
localization\emph{)};
\item $T(S):=S^{-1}R\oplus \frac{S^{-1}R}{R}$ is a $1$-tilting $R$-module;
\item $\mathrm{Gen}(_{R}S^{-1}R)=\mathcal{D}_{S}.$
\end{enumerate}
Moreover, in this case $T(S)^{\perp _{\infty }}=\mathrm{Gen}(T(S))=\mathcal{D
}_{S}.$
\end{enumerate}
\end{lemma}
We prove now some fundamental properties of (co)tilting modules over APD's,
some of which are analogous to the case of Pr\"{u}fer domains:
\begin{proposition}
\label{tilt-1-cotilt}Let $R$ be an APD with $R\neq Q.$
\begin{enumerate}
\item All tilting $R$-modules are $1$-tilting.
\item The torsion-free tilting $R$-modules are precisely the projective
generators \emph{(}i.e. the $0$-tilting $R$-modules\emph{) }and are all
equivalent to $R.$
\item Every divisible tilting $R$-modules generates $\mathcal{DI},$ whence
is equivalent to $\delta .$
\item All localizations of $R$ are Matlis localizations. For every
multiplicative subset $S\subseteq R^{\times }$ we have a tilting $R$-module $
T(S):=S^{-1}R\oplus S^{-1}R/R\sim \delta _{S}$ and a cotilting $R$-module $
T(S)^{c}\sim \delta _{S}^{c}.$
\item All cotilting $R$-modules are $1$-cotilting.
\item The divisible cotilting $R$-modules are precisely the injective
cogenerators \emph{(}i.e. the $0$-cotilting $R$-modules\emph{)} and are
equivalent to $R^{c}:=\mathrm{Hom}_{\mathbb{Z}}(R,\mathbb{Q}/\mathbb{Z}).$
\end{enumerate}
\end{proposition}
\begin{Beweis}
\begin{enumerate}
\item Follows directly from $\mathcal{P}=\mathcal{P}_{1}$ (\ref{I=, P=}).
\item If $_{R}T$ is a torsion-free tilting $R$-module, then by ``1'': $T\in
\mathcal{TF}\cap \mathcal{P}_{1}\overset{\text{(\ref{FL=})}}{=}\mathcal{FL},$
whence $_{R}T$ is projective (since flat $1$-tilting modules over arbitrary
rings are projective by \cite[Corollary 2.8]{BH}). In this case, $\mathrm{Gen
}(_{R}T)=T^{\perp }=R$\textrm{-}$\mathrm{Mod}=R^{\perp };$ consequently, $
_{R}T$ is a projective generator and $T\sim R.$
\item Recall that $\mathcal{F}_{1}$ generates a cotorsion pair $(\mathcal{F}
_{1},\mathcal{WI}),$ where (by definition) $\mathcal{WI}:=\mathcal{F}
_{1}^{\perp }$ is the class of \emph{weak-injective} $R$-modules. Notice that conditions (8) and (9) of Lemma \ref
{APD-Main} can be expressed as $(\mathcal{F}_{1},\mathcal{WI})=(\mathcal{P}
_{1},\mathcal{DI}).$ Let $T$ be a tilting $R$-module and consider the
induced cotorsion pair $(^{\perp }(T^{\perp }),T^{\perp }).$ If $_{R}T$ is
divisible, then $T^{\perp }=\mathrm{Gen}(_{R}T)\subseteq \mathcal{DI},$
whence $\mathcal{P}_{1}=$ $^{\perp }\mathcal{DI}\subseteq $ $^{\perp
}(T^{\perp })\subseteq \mathcal{P}_{1}.$ So, $\delta ^{\perp }=\mathcal{DI}=
\mathcal{P}_{1}^{\perp }=T^{\perp }=\mathrm{Gen}(_{R}T),$ i.e. $T$ generates
$\mathcal{DI}$ and $T\sim \delta .$
\item For every multiplicative subset $S\subseteq R^{\times },$ the
localization $S^{-1}R$ is a flat $R$-module whence $\mathrm{p.d.}
_{R}(S^{-1}R)\leq 1$ by Lemma \ref{APD-Main} (9). It follows by Lemma \ref
{eqv} (2)\ that $T(S):=S^{-1}R\oplus \frac{S^{-1}R}{R}$ is a tilting $R$
-module with $T(S)^{\perp }=\mathcal{D}_{S}=\delta _{S}^{\perp },$ whence $
T(S)\sim \delta _{S}.$ The character module of any tilting $R$-module is
cotilting by \cite[Theorem 8.1.2]{GT2006}, whence $T(S)^{c}$ is a cotilting $
R$-module which is equivalent to $\delta _{S}^{c}$ (e.g. \cite[Theorem 8.1.13]{GT2006}
).
\item Follows directly from $\mathcal{I}=\mathcal{I}_{1}$ (\ref{I=, P=}).
\item If $_{R}C$ is a divisible cotilting $R$-module, then by ``6'': $C\in
\mathcal{DI}\cap \mathcal{I}_{1}\overset{\text{(\ref{IN=})}}{=}\mathcal{IN}.$
In this case, $\mathrm{Cogen}(_{R}C)=$ $^{\perp }C=R$\textrm{-}$\mathrm{Mod}=
$ $^{\perp }R^{c};$ consequently, $_{R}C$ is an injective cogenerator and $
C\sim R^{c}.\blacksquare $
\end{enumerate}
\end{Beweis}
The following is a key-result that will be used frequently in the sequel.
\begin{theorem}
\label{T=D-P}Let $(R,\frak{m})$ be a local APD with $R\neq Q.$ Any tilting $R
$-module is either projective or divisible. Hence, $R$ has exactly two
tilting modules $\{R,\delta \}$ \emph{(}up to equivalence\emph{)} and
exactly two tilting classes $\{R$\textrm{-}$\mathrm{Mod},$ $\mathcal{DI}\}.$
\end{theorem}
\begin{Beweis}
Let $T$ be a tilting $R$-module and assume that $_R T$ is not divisible. Then $
T\neq 0$ and contains by Proposition \ref{APD-div} a maximal $R$-submodule $N
$ such that $T/N\simeq R/\frak{m}.$ By \cite{BS2007} all tilting modules
(over arbitrary rings) are of finite type. So, there exists $\mathcal{S}
\subseteq \mathcal{P}_{1}\cap R$-$\mathrm{mod}$ such that $R/\frak{m}\in
\mathrm{Gen}(_{R}T)=T^{\perp }=\mathcal{S}^{\perp }.$ Let $M\in \mathcal{S}$
be arbitrary, so that $\mathrm{Ext}_{R}^{1}(M,R/\frak{m})=0.$ Since the field $R/\frak{m}$
is indeed injective as a module over itself, it follows (e.g. \cite[Page 34 (6)]{FS2001}) that
\begin{equation*}
\begin{tabular}{lll}
$\mathrm{Tor}_{1}^{R}(R/\frak{m},M)$ & $\simeq $ & $\mathrm{Tor}_{1}^{R}(
\mathrm{Hom}_{R/\frak{m}}(R/\frak{m},R/\frak{m}),M)$ \\
& $\simeq $ & $\mathrm{Hom}_{R/\frak{m}}(\mathrm{Ext}_{R}^{1}(M,R/\frak{m}
),R/\frak{m})=0.$
\end{tabular}
\end{equation*}
By \cite[II.3.2.Corollary 2]{B}, $_{R}M$ is projective (being finitely
presented and flat). So, $\mathcal{S}\subseteq \mathcal{PR},$ whence $_{R}T$
is projective.$\blacksquare $
\end{Beweis}
Recall (from \cite{Ham1971}) that an $R$-submodule $M$ of an $R$-module $N$
is said to be a \textbf{restriction submodule}, iff $M_{\frak{m}}=N_{\frak{m}
}$ or $M_{\frak{m}}=0$ for every $\frak{m}\in \mathrm{Max}(R).$ For any
subset $X\subseteq \mathrm{Max}(R),$ we set
\begin{equation*}
R_{(X)}:=\bigcap\limits_{\frak{m}\in X}R_{\frak{m}}\text{ (}:=Q,\text{ if }
X=\varnothing \text{) .}
\end{equation*}
\begin{lemma}
\label{loc-int}Let $R\neq Q,$ $X\subseteq \mathrm{Max}(R),$ $X^{\prime }:=
\mathrm{Max}(R)\backslash X$ and consider
\begin{equation*}
M_{1}:=\frac{R_{(X)}}{R}\text{ and }M_{2}:=\frac{R_{(X^{\prime })}}{R}.
\end{equation*}
\begin{enumerate}
\item If $R$ is an $h$-local domain, then $M_{1},M_{2}\subseteq \frac{Q}{R}$
are restriction $R$-submodules and
\begin{equation}
\frac{Q}{R}=M_{1}\oplus M_{2}=\frac{R_{(X)}}{R}\oplus \frac{R_{(X^{\prime })}
}{R}. \label{Q/R}
\end{equation}
\item If $R$ is a $1$-dimensional $h$-local domain, then
\begin{equation*}
T(X):=R_{(X)}\bigoplus \frac{R_{(X)}}{R}\text{ \ \emph{(}}=Q\oplus \frac{Q}{R
},\text{ if }X=\varnothing \text{\emph{)}}
\end{equation*}
is a $1$-tilting $R$-module.
\end{enumerate}
\end{lemma}
\begin{Beweis}
Recall first that if $\frak{m},\frak{m}^{\prime }\in \mathrm{Max}(R)$ are
such that $\frak{m}\neq \frak{m}^{\prime },$ then we have by \cite[Theorem
3.19]{Mat1972} (see also \cite[IV.3.2]{FS2001}):
\begin{equation}
R_{\frak{m}}\otimes _{R}R_{\frak{m}^{\prime }}\simeq (R_{\frak{m}})_{\frak{m}
^{\prime }}=Q. \label{R_m_m'}
\end{equation}
Moreover, if $\{R_{\lambda }\}_{\Lambda }$ is a class of $R$-submodules of $Q
$ with $\bigcap_{\lambda \in \Lambda }R_{\lambda }\neq 0,$ then it follows
from \cite[IV.3.10]{FS2001} that
\begin{equation}
(\bigcap_{\lambda \in \Lambda }R_{\lambda })_{\frak{m}}=\bigcap_{\lambda \in
\Lambda }(R_{\lambda })_{\frak{m}}\text{ for every }\frak{m}\in \mathrm{Max}
(R). \label{int-loc}
\end{equation}
\begin{enumerate}
\item Clearly $M_{1}\cap M_{2}=0.$ Let $\frak{m}^{\prime }\in \mathrm{Max}
(R)$ be arbitrary. Then
\begin{equation*}
(M_{1})_{\frak{m}^{\prime }}=\frac{(R_{(X)})_{\frak{m}^{\prime }}}{R_{\frak{m
}^{\prime }}}\overset{\text{(\ref{int-loc})}}{=}\frac{\bigcap\limits_{\frak{m
}\in X}(R_{\frak{m}})_{\frak{m}^{\prime }}}{R_{\frak{m}^{\prime }}}\overset{
\text{(\ref{R_m_m'})}}{=}\left\{
\begin{tabular}{lll}
$0,$ & & $\frak{m}^{\prime }\in X$ \\
& & \\
$\frac{Q}{R_{\frak{m}^{\prime }}},$ & & $\frak{m}^{\prime }\notin X$
\end{tabular}
\right. .
\end{equation*}
\newline
Similarly,
\begin{equation*}
(M_{2})_{\frak{m}^{\prime }}=\left\{
\begin{tabular}{lll}
$\frac{Q}{R_{\frak{m}^{\prime }}},$ & & $\frak{m}^{\prime }\in X$ \\
& & \\
$0,$ & & $\frak{m}^{\prime }\notin X$
\end{tabular}
\right. .
\end{equation*}
So, $M_{1},M_{2}\subseteq \frac{Q}{R}$ are restriction $R$-submodules.
Moreover, we have $(M_{1}\oplus M_{2})_{\frak{m}^{\prime }}=(M_{1})_{\frak{m}
^{\prime }}\oplus (M_{2})_{\frak{m}^{\prime }}=\frac{Q}{R_{\frak{m}^{\prime
}}}=(\frac{Q}{R})_{\frak{m}^{\prime }}$ for all $\frak{m}^{\prime }\in
\mathrm{Max}(R),$ and so $\frac{Q}{R}=M_{1}\oplus M_{2}.$
\item Notice first that a $1$-dimensional $h$-local domain is a Matlis
domain (in fact $\mathrm{p.d.}_{R}(Q)=\mathrm{p.d.}_{R}(\frac{Q}{R})=1$ as
shown in \cite[Lemma 2.4]{Sal}). For any $X\subseteq \mathrm{Max}(R),$ we
have $\frac{Q}{R}\overset{\text{(\ref{Q/R})}}{=}\frac{R_{(X)}}{R}\oplus
\frac{R_{(X^{\prime })}}{R}$ and so $T(X)$ is a $1$-tilting $R$-module by
\cite[Theorem 8.2]{A-HHT2005}.$\blacksquare $
\end{enumerate}
\end{Beweis}
\begin{remark}
Although we proved (\ref{Q/R}) for general $h$-local domains, we point out
here that it can be obtained for an \emph{APD} $R$ by applying
\cite[Theorem 3.10]{A-HHT2005} to $M_{1}:=\frac{R_{(X)}}{R}.$ Then $X_{1}:=
\mathrm{Supp}(M_{1})=\mathrm{Max}(R)\backslash X$ and $X_{2}:=\mathrm{Supp}
(Q/R)\backslash X_{1}=X.$ Consider the embedding $\varphi :\frac{Q}{R}
\rightarrow \prod\limits_{\frak{m}\in \mathrm{Max}(R)}(\frac{Q}{R})_{\frak{m}
}.$ Since $R$ is $h$-local, it follows by \cite[Theorem IV.3.7]{FS2001} (3)
that $M_{1}\simeq \bigoplus\limits_{\frak{m}\notin \mathrm{Max}(R)}(M_{1})_{
\frak{m}}=\bigoplus\limits_{\frak{m}\in X}\frac{Q}{R_{\frak{m}}}.$ So, $
M_{2}:=\varphi ^{-1}(\prod\limits_{\frak{m}\in X}(\frac{Q}{R})_{\frak{m}})=
\frac{R_{(X^{\prime })}}{R}.$ Notice that $\mathrm{w.d.}_{R}(\frac{Q}{R_{(X)}
})\leq 1$ and so $\mathrm{p.d.}_{R}(\frac{Q}{R_{(X)}})\leq 1$ by Lemma \ref
{APD-Main} (9). The equality (\ref{Q/R}) follows now by \cite[Theorem 3.10]
{A-HHT2005}.
\end{remark}
\begin{lemma}
\label{1-dim-h-local}Let $R$ be an APD with $R\neq Q.$ If $T$ is a tilting $R
$-module, then
\begin{equation}
T^{\perp _{\infty }}=\mathrm{Gen}(_{R}T)=\mathcal{D}(_{R}T)\text{-}\mathrm{
Div}. \label{D(D(T))}
\end{equation}
\end{lemma}
\begin{Beweis}
Clearly $\mathrm{Gen}(_{R}T)\subseteq \mathcal{D}(T)$\textrm{-}$\mathrm{Div}.
$ Let $M\in \mathcal{D}(T)$\textrm{-}$\mathrm{Div},$ $\frak{m}\in \mathrm{Max
}(R)$ be arbitrary and consider the tilting $R_{\frak{m}}$-module $T_{\frak{m
}}.$ By Theorem \ref{T=D-P}, $_{R_{\frak{m}}}T_{\frak{m}}$ is either
divisible or projective. If $\frak{m}\in \mathcal{D}(T),$ then $T_{\frak{m}}$
is divisible and generates all divisible $R_{\frak{m}}$-modules by
Proposition \ref{tilt-1-cotilt} (3). Moreover, $\frak{m}_{\frak{m}}M_{\frak{m
}}=({\frak{m}}M)_{\frak{m}}=M_{\frak{m}}$ and it follows by Proposition \ref
{APD-div} that $M_{\frak{m}}$ is a divisible $R_{\frak{m}}$-module, whence $
M_{\frak{m}}\in \mathrm{Gen}(_{R_{\frak{m}}}T_{\frak{m}}).$ On the other
hand, if $\frak{m}\notin \mathcal{D}(T)$ then $T_{\frak{m}}$ is a projective
$R_{\frak{m}}$-module whence a generator in $R_{\frak{m}}$\textrm{-}$\mathrm{
Mod}$ by Proposition \ref{tilt-1-cotilt} (2). In either cases $M_{\frak{m}
}\in \mathrm{Gen}(_{R_{\frak{m}}}T_{\frak{m}})=T_{\frak{m}}^{\perp _{\infty
}}\;$for every $\frak{m}\in \mathrm{Max}(R),$ whence $M\in T^{\perp _{\infty
}}=\mathrm{Gen}(_{R}T)$ by Lemma \ref{eqv} (1).$\blacksquare $
\end{Beweis}
\begin{theorem}
\label{MAIN}Let $R$ be an APD with $R\neq Q.$
\begin{enumerate}
\item The set
\begin{equation*}
\{T(X)\mid X\subseteq \mathrm{Max}(R)\}
\end{equation*}
is a representative set \emph{(}up to equivalence\emph{) }of all tilting $R$
-modules.
\item There is a bijective correspondence between the set of all tilting
torsion classes of $R$-modules and the power set of the maximal spectrum $
\frak{B}(\mathrm{Max}(R)).$ The correspondence is given by the mutually
inverse assignments:
\begin{equation*}
\begin{tabular}{lllll}
& $\mathcal{T}\mapsto $ & $\mathcal{DM}(\mathcal{T})$ & $:=$ & $\{\frak{m}
\in \mathrm{Max}(R)\mid \frak{m}M=M$ for every $M\in \mathcal{T}\};$ \\
and & & & & \\
& $X\mapsto $ & $X$\textrm{-}$\mathrm{Div}$ & $:=$ & $\{_{R}M\mid \frak{m}M=M
$ for every $\frak{m}\in X\}.$
\end{tabular}
\end{equation*}
\item If $R$ is coprimely packed, then the class of Fuchs-Salce tilting
modules
\begin{equation*}
\{\delta _{S}\mid S\subseteq R^{\times }\text{ is a multiplicative subset}\}
\end{equation*}
classifies all tilting $R$-modules \emph{(}up to equivalence\emph{)}.
\end{enumerate}
\end{theorem}
\begin{Beweis}
\begin{enumerate}
\item Let $T$ be a tilting $R$-module and set
\begin{equation*}
\begin{tabular}{lll}
$\Omega _{1}$ & $:=$ & $\{\frak{m}\in \mathrm{M}\mathrm{ax}(R)\mid T_{\frak{m
}}$ is a divisible $R_{\frak{m}}$-module$\};$ \\
$\Omega _{2}$ & $:=$ & $\{\frak{m}\in \mathrm{M}\mathrm{ax}(R)\mid T_{\frak{m
}}$ is a projective $R_{\frak{m}}$-module$\}.$
\end{tabular}
\end{equation*}
Notice first that $\mathrm{M}\mathrm{ax}(R)=\Omega _{1}\cup \Omega _{2}$ by
Theorem \ref{T=D-P} (a disjoint union by applying Proposition \ref
{tilt-1-cotilt} (2) $\&\;$(3) to the ring $R_{\frak{m}}$).
\textbf{Claim}: $T\sim T(\Omega _{2}).$ One can show (as in the proof of
Lemma \ref{loc-int}), that if $\frak{m}\in \mathrm{Max}(R)$ then
\begin{equation*}
T(\Omega _{2})_{\frak{m}}=\left\{
\begin{tabular}{lll}
$Q\oplus \frac{Q}{R_{\frak{m}}},$ & & $\frak{m}\in \Omega _{1}$ \\
& & \\
$R_{\frak{m}},$ & & $\frak{m}\in \Omega _{2}$
\end{tabular}
\right. .
\end{equation*}
So, $T_{\frak{m}}\sim T(\Omega _{2})_{\frak{m}}$ for every $\frak{m}\in
\mathrm{Max}(R)$ whence $T\sim T(\Omega _{2})$ by (\ref{TeqvT'}).
\item Let $\mathcal{T}=T^{\perp _{\infty }}$ be a tilting torsion class for
some tilting $R$-module $T.$ Then
\begin{equation*}
\mathcal{DM}(\mathcal{T})\text{\textrm{-}}\mathrm{Div}=\mathcal{DM}(T)\text{
\textrm{-}}\mathrm{Div}\overset{\text{(\ref{D(F)=})}}{=}\mathcal{D}(T)\text{
\textrm{-}}\mathrm{Div}\overset{\text{(\ref{D(D(T))})}}{=}\mathrm{Gen}
(_{R}T)=T^{\perp _{\infty }}=\mathcal{T}.
\end{equation*}
On the other hand, let $X\subseteq \mathrm{Max}(R),$ $\overline{X}:=\mathrm{
Max}(R)\backslash X,$ and $T^{\prime }:=T(\overline{X}).$ Then clearly $
\mathcal{DM}(T^{\prime })=X$ and so
\begin{equation*}
\mathcal{DM}(X\text{\textrm{-}}\mathrm{Div})=\mathcal{DM}(\mathcal{DM}
(T^{\prime })\text{\textrm{-}}\mathrm{Div})=\mathcal{DM}(T^{\prime })=X.
\end{equation*}
\item Let $R$ be compactly packed. Let $\Omega _{1}$ and $\Omega _{2}$ be
as in ``1''.
\textbf{Case 1.} $\mathrm{Max}(R)=\Omega _{1}$ (i.e. $T_{\frak{m}}$ is a
divisible $R_{\frak{m}}$-module for all $\frak{m}\in \mathrm{Max}(R)$). In
this case, $_{R}T$ is divisible whence $T\sim Q\oplus Q/R$ and we can take $
S=R^{\times }.$
\textbf{Case 2.} $\mathrm{Max}(R)=\Omega _{2}$ (i.e. $T_{\frak{m}}$ is a
projective $R_{\frak{m}}$-module for all $\frak{m}\in \mathrm{Max}(R)$). In
this case, $_{R}T$ is projective whence $T\sim R$ and we can take $S=\{1\}.$
\textbf{Case 3.} $\mathrm{Max}(R)\neq \Omega _{1}$ and $\mathrm{Max}(R)\neq
\Omega _{2}.$ Let
\begin{equation*}
S:=R\backslash \bigcup\limits_{\frak{m}\in \Omega _{2}}\frak{m}\text{ and }
T(S):=S^{-1}R\oplus S^{-1}R/R.
\end{equation*}
Let $\frak{m}\in \Omega _{2},$ so that $T_{\frak{m}}$ is projective and $
S\subseteq R\backslash \frak{m}.$ Then $(S^{-1}R)_{\frak{m}}=R_{\frak{m}}.$
Therefore $(T(S))_{\frak{m}}=(S^{-1}R)_{\frak{m}}\oplus (S^{-1}R/R)_{\frak{m}
}=R_{\frak{m}}$ is equivalent to the projective $R_{\frak{m}}$-module $T_{
\frak{m}}.$ On the other hand, let $\frak{m}\in \Omega _{1}$ so that $T_{
\frak{m}}$ is a divisible $R_{\frak{m}}$-module. Then $\frak{m}\cap S\neq
\varnothing $ (otherwise $\frak{m}\subseteq \bigcup\limits_{\frak{m}\in
\Omega _{2}}\frak{m}$ and so $\frak{m}\in \Omega _{2}$ since $R$ is
coprimely packed; a contradiction since $\Omega _{1}\cap \Omega
_{2}=\varnothing $). Let $\widetilde{s}\in S\cap \frak{m}.$ Clearly $
\widetilde{s}(S^{-1}R)_{\frak{m}}=(S^{-1}R)_{\frak{m}},$ whence $(S^{-1}R)_{
\frak{m}}$ is a divisible $R_{\frak{m}}$-module by Proposition \ref{APD-div}
. It follows that $(T(S))_{\frak{m}}=(S^{-1}R)_{\frak{m}}\oplus (S^{-1}R)_{
\frak{m}}/R_{\frak{m}}$ is a divisible $R_{\frak{m}}$-module, whence $T(S)_{
\frak{m}}\sim T_{\frak{m}}$ as $R_{\frak{m}}$-modules by Proposition \ref
{tilt-1-cotilt} (3) (applied to the ring $R_{\frak{m}}$). Since $T_{\frak{m}
}\sim T(S)_{\frak{m}}$ for all $\frak{m}\in \mathrm{Max}(R),$ we conclude
that $T\sim T(S)$ by (\ref{TeqvT'}).$\blacksquare $
\end{enumerate}
\end{Beweis}
\begin{remark}
\label{Bass}Let $R$ be a $1$-Gorenstein ring and $_{R}T$ be a tilting $R$
-module. By \cite{TP2009} there exists $X\subseteq \mathbf{P}_{1}$ (the set
of prime ideals of height $1$) and some (\emph{unique}) $R$-module $R_{X},$
satisfying $R\subseteq R_{X}\subseteq Q$ and fitting in an exact sequence
\begin{equation*}
0\rightarrow R\rightarrow R_{X}\rightarrow \bigoplus_{\frak{m}\in X}\mathrm{E
}(R/\frak{m})\rightarrow 0,
\end{equation*}
such that $T$ is equivalent to the so-called \textbf{Bass tilting module }$
B(X):=R_{X}\oplus \bigoplus_{\frak{m}\in X}\mathrm{E}(R/\frak{m}).$\textbf{\
}Let $\frak{m}\in \mathrm{Max}(R)$ be arbitrary. By the proof of
\cite[Theorem 0.1]{TP2009}, the $R_{\frak{m}}$-module $B(X)_{\frak{m}}$ is
injective, whence divisible, if $\frak{m}\in X$ and projective if $\frak{m}
\notin X.$ If $R$ is a $1$-Gorenstein domain (whence an APD), the same holds
for the $R_{\frak{m}}$-module $T(X^{\prime })_{\frak{m}},$ where $X^{\prime
}:=\mathrm{Max}(R)\backslash X.$ It follows that, in this case, $B(X)\sim
T(X^{\prime })$ by (\ref{TeqvT'}) and so $T\sim T(X^{\prime }).\blacksquare $
\end{remark}
A direct application of Theorem \ref{MAIN},
and \cite[Theorem 8.2.8]{GT2006} yields
\begin{corollary}
\label{Cot-APD}Let $R$ be a coherent \emph{(}Noetherian\emph{)} APD.
\begin{enumerate}
\item All cotilting $R$-modules are of cofinite type and $\{T(X)^{c}\mid
X\subseteq \mathrm{Max}(R)\}$ is a representative set \emph{(}up to
equivalence\emph{) }of all cotilting $R$-modules.
\item If $R$ is coprimely packed, then $\{\delta _{S}^{c}\mid S\subseteq
R^{\times }$ is a multiplicative subset$\}$ classifies all cotilting $R$
-modules \emph{(}up to equivalence\emph{)}.
\end{enumerate}
\end{corollary}
| 27,457
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Mariners ranging from the operators of small boats to captains of large cargo carriers and cruise ships at Port Everglades will now be able to get information in real-time on tides, currents, water levels, wind speed and other metrological information online or by phone.
Leaders at the port celebrated the installation of the Port Everglades Physical Oceanographic Real-Time System (PORTS), an advanced sensor device developed and installed by the National Oceanic and Atmospheric Administration (NOAA).
Glenn Wiltshire, Port Everglades Deputy Port Director said the new system will make it easier and safer for large ships to navigate in and around the port because they will have better and more up-to-date information on weather, wind speed, and water level.
“Boaters and commercial mariners will be able to make safer choices based on the data from this device because it gives real-time information and forecasts,” said Wiltshire. “Before this equipment went into service, mariners had to rely on data for Lake Worth in Palm Beach County or from Virginia Key in Miami to estimate conditions here.”
Rich Edwing, director of NOAA’s Center for Operational Oceanographic Products and Services is pleased with the new device. “Port Everglades PORTS will give vessel operators the tide and weather information they need to help them navigate these waters more safely and efficiently. This is a win for the port, the surrounding communities, and every business and industry that depends on these cruise and cargo ships.”
Broward County Commissioner Chip LaMarca said, “Having real-time tide, water level, and other metrological information all online or via a toll-free phone number is truly an asset we can all enjoy.” He added, “This equipment is collecting data about our natural resources that we can apply to preparation for storms, climate changes, and sea level rise.”
Recent Comments
| 27,341
|
ALIMONY
The purpose of a "maintenance" (or spousal support) award is to help the receiving spouse become financially independent after a divorce. Maintenance ends on the death of either party, or the remarriage of the receiving party, or a date specified in an agreement between the parties, or a date determined by a judge.
Factors Considered
In determining the amount and duration of maintenance the court shall consider:
459-a of the social services law;
.
[Domestic Relations Law § 236B]
An attorney can help protect your post-marital rights. They can help either spouse to negotiate an alimony agreement that achieves the results sought. Or they can petition a court to order or enforce the terms sought.
| 341,803
|
\begin{document}\sloppy
\maketitle
\begin{abstract}
We study symmetric Killing 2-tensors on Riemannian manifolds and show that several additional conditions can be realised only for Sasakian manifolds and Euclidean spheres.
In particular we show that (three)-Sasakian manifolds can also be characterized by properties of the symmetric products of their characteristic 1-forms.
Moreover, we recover a result of S.~Gallot on the characterization of spheres by means of functions satisfying a certain differential equation of order three.
\\
\noindent
\noindent
2010 {\it Mathematics Subject Classification}: Primary 53C05, 53C25, 53C29.
\noindent{\it Keywords}:
Killing tensors, prolongation of the Killing equation, spaces of constant curvature, Sasakian manifolds, metric cone, Young symmetrizer
\end{abstract}
\section{Introduction}
Symmetric tensors on a Riemannian manifold $(M^n,g)$ are by definition invariant under an arbitrary interchange of indices. On the other hand, (alternating) forms reverse their sign whenever one interchanges two different indices. More generally, the symmetry type of a tensor of valence $d$ reflects its behaviour under the action of the symmetric group $\rmS_d$ on its indices. Another famous example are the algebraic identities satisfied by the Riemannian curvature tensor like the first Bianchi identity.
According to Schur-Weyl duality, symmetry types of tensors are in one-one correspondence with irreducible representations of the special linear group $\SL(n)$ (cf.~\cite[Theorem 6.3]{FH}). The Killing equation in its general form is then stated as follows. Every function is by definition Killing, i.e. the Killing condition is empty for sections of the trivial vector bundle $M\times\R$. If $V$ is a non-trivial irreducible representation of the (special) linear group and $\gamma$ is a section of the vector bundle $VM$ associated to the frame bundle of $M$, then $\nabla \gamma$ is a section of $TM^* \otimes VM$. Decomposing the latter into irreducible subbundles according to the branching rules of Littlewood-Richardson (cf.~\cite[App.~A]{FH}), the Killing equation demands the vanishing of the Cartan component of $\nabla \gamma$.
Because the Killing equation is of finite type, the existence and uniqueness of the prolongation of the Killing equation is a priori ensured from an abstract point of view (cf.~\cite{BCEG}). Thus there exists a vector bundle $\mathrm{Prol}(VM) \to M$ equipped with a linear connection and a linear assignment $\Gamma(VM)\to \Gamma(\mathrm{Prol}(VM))$ such that Killing tensors correspond with parallel sections of $\mathrm{Prol}(VM)$.
\subsection{Killing forms}
For example, the Killing equation for $p$-forms is $\d\omega = (p+1) \nabla\omega$ for all $p\geq 0$, where $\d$ and $\nabla$ denote the exterior differential and the Levi-Civita connection.
In other words, $\omega$ is Killing if and only if $\nabla\omega$ is alternating, again.
In~\cite{Se} U.~Semmelmann studied more generally conformal Killing forms on Riemannian manifolds and gave partial classifications.
In particular, he found the complete classification of so called {\em special} Killing forms on compact Riemannian manifolds.
A Killing form $\omega$ is called special if it satisfies $\nabla_X \d \omega = - c\, X^\sharp\wedge \omega$
for some constant $c > 0$, where $X^\sharp$ denotes the dual of a vector field $X$ (cf.~\cite[Chapter~3]{Se}).
To every pair $(\omega,\eta)$ of a $p$-form $\omega$ and a $p+1$-form $\eta$ on $M$ we associate the $p+1$-form $r^p \d r \wedge \omega + \frac{r^{p+1}}{p+1} \eta$
on the cone $\hat M := M\times \R_+$ (cf.~\cite[3.2.4]{Se}).
Then it is easy to see that $\omega$ is a special Killing form if and only if the $p+1$-form $\hat\omega := \frac{1}{p+1} \d ( r^{p+1} \omega)$ associated with the pair $(\omega,\d \omega)$ is Levi-Civita parallel with respect to the cone metric.
On the other hand, a parallel $p+1$-form on the cone yields a special Killing $p$-form on $M$ by inserting the
radial vector field at $r=1$. Thus there is a 1-1 correspondence between parallel $p+1$-forms on the cone and special Killing $p$-forms on $M$. Using a classical
theorem of S.~Gallot (cf.~\cite[Proposition~3.1]{Ga}) one obtains the classification of special Killing forms on compact Riemannian
manifolds via the holonomy principle (cf.~\cite[Theorem~3.2.6]{Se}). Besides the usually considered examples (euclidean spheres and Sasakian manifolds), this
includes also certain exceptional geometries in dimensions six and seven. Moreover, specifying $p = 0$, i.e. $\omega$ is a function $f$, we recover that a non-vanishing function $f$ satisfying the second order differential equation $\nabla^2 f = n\cdot g\cdot f$ (denoted by $(E_1)$ in~\cite{Ga}) exists only on the unit-sphere.
Every Killing $p$-form $\omega$ also satisfies the second order equation $\nabla_X\d\, \omega = \frac{p+1}{p}R^+(X)\omega$, the prolongation of the Killing equation. Here $R^+(X)$ denotes the natural action of degree one of the endomorphism valued 1-form $R_{X,\bullet}$ on forms (cf.~\cite[Chapter~4]{Se}). In particular, a form $\omega$ is Killing if and only if
the prolongation $(\omega,\d \omega)$ is parallel with respect to a natural linear connection on the corresponding vector bundle, the Killing
connection. Moreover, on the unit sphere we have $R^+(X)\omega = - p\,X^*\wedge \omega$. In other words, a Killing form is special if and only if the prolonged Killing
equation has the same form as generally for Killing forms on a round sphere.
\subsection{A brief overview on our results for symmetric tensors}
Since the Killing equation for 1-forms (i.e. Killing vector fields) is well understood, we focus on symmetric 2-tensors:
\bigskip
\begin{definition}
A symmetric 2-tensor $\kappa$ is Killing if the completely symmetric
part of $\nabla\kappa$ vanishes, i.e.
\begin{equation}\label{eq:def_Killing}
\forall p\in M,x\in T_pM:\;\nabla_x\kappa(x,x)\;\;=\;\;0.
\end{equation}
\end{definition}
The crucial difference when compared with forms comes from the form of the prolongation and the lift to the cone associated with it. As laid out before, we can see the pair $(\omega,\d\omega)$ associated with a form $\omega$ not only as a set of variables of the prolongation, but also as a form on the cone.
Quite differently, for symmetric 2-tensors the variables of the prolongation are triples $(\alpha,\beta,\gamma)\in \bbS_{T_0}V^*\oplus \bbS_{T_1}V^*\oplus\bbS_{T_2}V^*$ of tensors of different symmetry types described by Young tableaus $T_i$ of shapes $(2,i)$ for $i =0,1,2$,
see~\eqref{eq:kappa_prolong_1}-\eqref{eq:kappa_prolong_2} and~\eqref{eq:def_C_kappa}-\eqref{eq:def_S_kappa} below.
Such a triple can be naturally seen as the components of a (symmetrized) algebraic curvature tensor on the cone via $S := r^2\, \alpha\owedge \d r\bullet \d r + r^3\, \beta\owedge \dr + r^4\, \gamma$, where $\bullet$ is the usual symmetric product and $\owedge$ is a product which will be introduced in the next section. We will understand under which conditions $S$ is Levi-Civita parallel on the metric cone (see Theorem~\ref{th:main_1}) and relate these conditions to the geometry of $M$ (see Theorem~\ref{th:main_2}). Finally, given a function $f$ on $M$ which satisfies the partial differential equation $(E_2)$ of order three considered in~\cite{Ga} (see~\eqref{eq:Gallots_Gleichung} below),
we associate with $f$ a Killing tensor which fits into Theorem~\ref{th:main_1}. Thus we recover the main result of~\cite{Ga} on the characterization of the unit sphere by the existence of such functions
(see Corollary~\ref{co:Gallot}).
Even though already for valence two these constructions are most efficently formulated by means of
Young symmetrizers, explicit descriptions avoiding this formalism are possible,
see for example~\cite{HM} (for valence two) and~\cite{Th,W} (for arbitrary valence). For symmetric tensors of arbitrary valence, Y.~Houri and others~\cite{HTY} have given an explicit construction of the Killing prolongation via a recursive construction
using the technique of Young symmetrizers. Although their arguments can not immediately be generalised to other symmetry types,
nevertheless one would conjecture that similar explicit constructions are possible for all symmetry types of tensors.
\section{Weyls construction of irreducible representations of the general linear group}
\label{se:introduction_to_Young_symmetrizers}
We recall the notion of Young tableaus, the action of the associated symmetrizers and projectors on tensor spaces
and their relation to irreducible representations of the general linear group via the Schur functor.
The notation is in accordance with~\cite{Sch}, for details see Fulton-Harris~\cite{F,FH}.
A partition $\lambda_1\geq \cdots \geq \lambda_k >0$ of an integer $d$ can be depicted through a
Young frame, an arrangement of $d$ boxes aligned from the left in $k$-rows of length $\lambda_i$ counted from top to bottom.
For example the frame corresponding to $(5,3,2)$ is
$$
\begin{array}{|c|c|c|c|c|}
\cline{1-5}
& & & & \\
\cline{1-5}
& & & \multicolumn{2}{c}{\;\;}\\
\cline{1-3}
& & \multicolumn{3}{c}{\;\;}\\
\cline{1-2}
\end{array}
$$
Filling the diagram with $d$ different numbers $\{i_1,\ldots,i_d\}$ we
obtain a {\em Young tableau} of {\em shape} $\lambda$ (cf.~\cite{F}). For example,
\begin{equation}\label{eq:normal_standard_Young_tableau}
T= \begin{array}{|c|c|c|c|c|}
\cline{1-5}
1 & 10 & 9 & 2 & 5\\
\cline{1-5}
8 & 7 & 4 & \multicolumn{1}{c}{\;\;} \\
\cline{1-3}
3 & 6 & \multicolumn{2}{c}{\;\;}\\
\cline{1-2}
\end{array}
\end{equation}
is a Young tableau of shape $(5,3,2)$. For simplicity we assume
in the following that $\{i_1,\ldots,i_d\} = \{1,\ldots,d\}$. When these
numbers are in order, left to right and top to bottom, the tableau is called
{\em normal.} Since there is only one normal diagram of a given shape $\lambda$, we will denote this by
$T_\lambda$.
Let $V^n$ be some vector space with dual space $V^*$.
The tensor product $\bigotimes^d V^*$ can be seen as the space of multilinear forms of degree $d$ of $V$.
Here we have the natural right action of the symmetric group $\rmS_d$ given by
\begin{equation*}
\lambda \cdot \sigma(v_1, \cdots, v_d) := \lambda(v_{\sigma^{-1}(1)}, \cdots,v_{\sigma^{-1}(d)}).
\end{equation*}
Let $\rmS_c$ and $\rmS_r$ denote the
subgroup of $\rmS_d$ preserving columns and rows, respectively, of some fixed Young tableau $T$ of shape $\lambda$. The row symmetrizer
and the column anti-symmetrizer are
\begin{align}\label{eq:def_r_lambda}
&r_T \colon \bigotimes^dV^* \to \bigotimes^dV^*,\ \lambda \mapsto
\sum_{\sigma\in \rmS_r} \lambda\cdot \sigma,\\
\label{eq:def_c_lambda}
&c_T \colon \bigotimes^dV^*\to \bigotimes^dV^*,\ \lambda \mapsto \sum_{\sigma\in \rmS_c} (-1)^{|\sigma|} \lambda\cdot\sigma.
\end{align}
The Young symmetrizer and its adjoint associated to a tableau $T$ are defined by
\begin{align}
\label{eq:def_S_*_lambda}
&\rmS_T \;\;:=\;\; r_T\circ c_T,\\
\label{eq:def_S_lambda}
&\rmS^\star_T \;\;:=\;\; c_T\circ r_T.
\end{align}
The endomorphisms $r_T$ and $c_T$ on
$\bigotimes^dV^*$ are called intertwining maps.
The images $\bbS_TV^* := \rmS_T(\bigotimes^dV^*)$ and $\bbS^\star_TV^* :=
\rmS^\star_T(\bigotimes^dV^*)$ both are irreducible representations of the
general linear group $\GL(n)$, dual to the one of highest weight $\lambda$. The intertwining maps yield by construction explicit invariant isomorphisms between $\bbS_TV^*$ and $\bbS^\star_TV^*$.
The assignment $V \mapsto \bbS_TV^*$ (or $V \mapsto \bbS^\star_TV^*$) is called the Schur functor for covariant tensors associated with $T$.
In particular, by means of Schurs lemma there exists a constant $h$ such that $\rmP_T := \frac{1}{h_\lambda} \rmS_T$ and $\rmP^\star_T := \frac{1}{h_\lambda} \rmS^\star_T$ both are projectors, the Young projectors associated with $T$.
This constant is actually an integer which depends only on
the underlying Young frame and hence we can write $h = h_\lambda$. It is given as follows:
A {\em hook} of length $d$ is a Young frame with $d$ boxes but only one row and one column of length larger than
one. The box in the first row on the uttermost left is called its center. For example, a hook of length $4$ is given by
\begin{equation}\label{eq:hook}
\begin{array}{|c|c|c|}
\cline{1-3}
& & \\
\cline{1-3}
& \multicolumn{2}{c}{\;\;}\\
\cline{1-1}
\end{array}
\end{equation}
For every box of $T$ there is a unique maximal hook inscribed into the
diagram centered at the given box. Its length is called the hook length of the
box. Then $h_\lambda$ is the product of all hook numbers taken over all boxes of the frame.
\paragraph{}
Following~\cite[Ch.~15.5]{FH}, there is another characterization of the representation space $\bbS^\star_TV^*$. Let $\mu_1\geq \cdots \geq \mu_\ell$ denote the conjugate
partition, i.e. column lengths. Then $\bbS^\star_TV^*$ is by construction a subspace of $\Lambda^{\mu_1}V^*\otimes\cdots \otimes \Lambda^{\mu_\ell}V^*$. Further, as a consequence of the branching rules due
to Littlewood-Richardson, it is immediately clear that $\bbS^\star_TV^*$ is a subspace of the kernel of the bilinear map
\begin{align}
&\ell_{ij}^\star \colon \Lambda^{\mu_i}V^*\times \Lambda^{\mu_j}V^*\to
\Lambda^{\mu_i+1}V^*\otimes \Lambda^{\mu_j-1}V^*,\quad (\omega_i,\omega_j)
\mapsto \ell_{ij}^\star(\omega_i,\omega_j):\\
& \ell_{ij}^\star(\omega_i,\omega_j)(v_1,\ldots,v_{\mu_i+\mu_j})\;\;:=\;\; \sum_{a=1}^{\mu_i+1}(-1)^{a+\mu_i+1}\omega_i(v_1,\cdots,\hat v_a,\ldots,v_{\mu_i+1})\omega_j(v_a,v_{\mu_i+2},\ldots,v_{\mu_i+\mu_j}),
\end{align}
which anti-symmetrizes all the indizes of $\omega_i$ with one further
index of $\omega_j$ for $i<j$. Moreover, it is also easy to see that $\bbS^\star_TV^*$ is in fact equal to the intersection of the
Kernels of all $\ell_{ij}^\star$ taken over all $1\leq i < j \leq \ell$.
For $\gamma\in \bigotimes_{a=1}^\ell \Lambda^{\mu_a}V^*$ let
$\gamma_{ij} \colon \Lambda^{\mu_i}V\times \Lambda^{\mu_j}V \to \bigotimes_{a\neq i,j} \Lambda^{\mu_a}V^*$ be the natural map. Then the previous is equivalent to
\begin{equation}\label{eq:exchange_rule_column}
\gamma_{ij}(v_1,\cdots,v_{\mu_i},v_{\mu_i+1},\cdots,v_{\mu_i+\mu_j}) = \sum_{a=1}^{\mu_i}\gamma(v_1,\cdots,v_{a-1},v_{\mu_i+1},v_{a+1},\ldots,v_{\mu_i},v_a,v_{\mu_i+2},\ldots,v_{\mu_i+\mu_j}),
\end{equation}
i.e. the elements of $\bbS^\star_TV^*$ respect the Plücker relations described in~\cite[15.53]{FH}. In~\cite[Ch.~8]{F} the above equation is seen
as an ``exchange rule'' of length one between the $i$-th and $j$-th column of $T$. Exchange rules of greater lengths (like pair symmetry for algebraic curvature tensors) then follow
automatically, cf. also~\cite[Exercise 15.54]{FH}.
\bigskip
\begin{example}
Consider the tableau $T := \scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 3 \\
\cline{1-2}
2 & 4 \\
\cline{1-2}
\end{array}}{16pt}$. The tensor products $\Lambda^2V^*\otimes \Lambda^2V^*$ and $\Lambda^3V^*\otimes V^*$ decompose according to the Littlewood-Richardson rules as follows
\begin{align*}
\Lambda^2V^*\otimes \Lambda^2V^* & = \Lambda^4V^*\oplus \bbS^\star_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 3\\
\cline{1-2}
2 & \multicolumn{1}{c}{} \\
\cline{1-1}
4 & \multicolumn{1}{c}{}\\
\cline{1-1}
\end{array}}{12pt}} V^*\oplus \bbS^\star_TV^*,\\
\Lambda^3V^*\otimes V^* &= \Lambda^4V^*\oplus \bbS^\star_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 3\\
\cline{1-2}
2 & \multicolumn{1}{c}{} \\
\cline{1-1}
4 & \multicolumn{1}{c}{}\\
\cline{1-1}
\end{array}}{12pt}} V^*.
\end{align*}
The first two factors are already skew-symmetric in the first three indices and hence $\ell_{12}^\star$ yields isomorphisms between the first and second summands in each decomposition, respectively. Therefore, it
necessarily vanishes on $\bbS^\star_TV^*$. From this we see that $\bbS^\star_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 3 \\
\cline{1-2}
2 & 4 \\
\cline{1-2}
\end{array}}{12pt}}V^*$ describes the subspace of $\Lambda^2V^*\otimes \Lambda^2V^*$ given by those tensors which satisfy the first Bianchi identity, i.e algebraic prototypes of the
curvature tensor $\tilde R(x_1,x_2,x_3,x_4)$ of some Riemannian manifold
$\tilde M$ at some point $p\in \tilde M$ (cf. also~\cite[p.12]{Sch}). The
only exchange rule of order two leads to pair symmetry, which is a well
known algebraic consequence of the first Bianchi identity in $\Lambda^2V^*\otimes \Lambda^2V^*$.
\end{example}
Dividing through the action of the determinant (which acts as a scalar
according to Schurs lemma), we obtain an irreducible representation of the special linear
group $\SL(n)$ and, conversely, up to isomorphy every irreducible representation of $\SL(n)$ is obtained in this way (cf.~\cite[Ch.~15]{FH}).
The representation space $\bbS_TV^*$ has a description completely analogous to $\bbS^\star_TV^*$ (although this fact seems to be less popular).
By definition, $\bbS_TV^*$ is a subspace of $\Sym^{\lambda_1}V^*\otimes\cdots \otimes \Sym^{\lambda_k}V^*$ (where as before $\lambda_i$ are the lengths of the rows of $T$).
Then it follows by similar arguments as above that $\bbS_TV^*$ is given by the
intersection of the kernels of the linear maps
\begin{align}
& \ell_{ij} \colon \Sym^{\lambda_i}V^*\otimes \Sym^{\lambda_j}V^*\to \Sym^{\lambda_i+1}V^*\otimes
\Sym^{\lambda_j-1}V^*,\quad (\alpha_i,\alpha_j)\mapsto \ell_{ij}(\alpha_i,\alpha_j):\\
&\ell_{ij}(\alpha_i,\alpha_j)(v_1,\ldots,v_{\lambda_i+\lambda_j})\;\;:=\;\;
\sum_{a=1}^{\lambda_i+1}\alpha_i(v_1,\cdots,\hat v_a,\ldots,v_{\lambda_i+1})\alpha_j(v_a,v_{\lambda_i+2},\ldots,v_{\lambda_i+\lambda_j})
\end{align}
with $i< j$.
As before, exchange rules between rows of greater length follow automatically.
\bigskip
\begin{example}\label{ex:Young_symmetrizer}
\begin{enumerate}
\item For every covariant 2-tensor $\alpha$ \begin{equation}\label{eq:Young_(2,0)}
\rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
\end{array}}{6pt}}\alpha(v_1,v_2) \;\;= \;\;
\begin{array}{c}
\alpha(v_1,v_2)+\alpha(v_2,v_1)
\end{array}
\end{equation}
and $h_{(2)} = 2\cdot 1$. Hence $\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
\end{array}}{6pt}}V^*$ is the space of covariant symmetric 2-tensors and the Young projector $\rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
\end{array}}{6pt}}$ is the usual projector on symmetric 2-tensors.
\item
We have \begin{equation}\label{eq:Young_(2,1)}
\rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
1 & \multicolumn{1}{c}{} \\
\cline{1-1}
\end{array}}{12pt}}\beta(v_1,v_2,v_3)\;\;= \;\; \left \lbrace
\begin{array}{c}
\;\ \beta(v_1,v_2,v_3) - \beta(v_2,v_1,v_3)\\
+ \beta(v_1,v_3,v_2) - \beta(v_3,v_1,v_2)
\end{array}\right \rbrace.
\end{equation}
Further, $h_{(2,1)} = 3\cdot 1\cdot 1$. Hence the following is equivalent:
\begin{itemize}
\item
$
\beta \in \bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
1 & \multicolumn{1}{c}{} \\
\cline{1-1}
\end{array}}{12pt}} V^*$,
\item
$\rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
1 & \multicolumn{1}{c}{} \\
\cline{1-1}
\end{array}}{12pt}}\beta(v_1,v_2,v_3) \;\;= \;\; 3\, \beta(v_{1},v_{2},v_{3})$,
\item
$\beta\in V^* \otimes \Sym^2 V^*$ and
such that the completely symmetric part $\beta(v,v,v)$ vanishes.
\end{itemize}
Such tensors are the algebraic prototype of the
covariant derivative $\nabla_{x_1}\kappa(x_2,x_3)$ of a symmetric Killing 2-tensor at a given point of some Riemannian manifold.
\item We have $ \rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\gamma(v_1,v_2,v_3,v_4) =$ \begin{equation}\label{eq:Young_(2,2)}
\left \lbrace \begin{array}{c}\;\ \gamma(v_1,v_2,v_3,v_4) - \gamma(v_3,v_2,v_1,v_4) - \gamma(v_1,v_4,v_3,v_2) + \gamma(v_3,v_4,v_1,v_2)\\
+ \gamma(v_2,v_1,v_3,v_4) - \gamma(v_3,v_1,v_2,v_4) - \gamma(v_2,v_4,v_3,v_1) + \gamma(v_3,v_4,v_2,v_1)\\
+ \gamma(v_1,v_2,v_4,v_3) - \gamma(v_4,v_2,v_1,v_3) - \gamma(v_1,v_3,v_4,v_2) + \gamma(v_4,v_3,v_1,v_2)\\
+ \gamma(v_2,v_1,v_4,v_3) - \gamma(v_4,v_1,v_2,v_3) - \gamma(v_2,v_3,v_4,v_1) + \gamma(v_4,v_3,v_2,v_1)\\
\end{array}
\right \rbrace.
\end{equation}
Further, $h_{(2,2)} = 3\cdot 2\cdot 2\cdot 1$. Therefore the following is equivalent:
\begin{itemize}
\item $\gamma \in \bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}V^*$,
\item
$
S_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\gamma(v_1,v_2,v_3,v_4) \;\;= \;\; 12\, \gamma(v_1,v_2,v_3,v_4)
$
\item $\gamma\in \Sym^2(V^*)\otimes\Sym^2(V^*)$ and $\gamma(u,u,u,v) = 0$ for all
$u,v\in V$.
\end{itemize}
Then $\gamma$ is called a symmetrized algebraic curvature tensor (see~\cite[Example~4.10]{Sch}). Via the isomorphism $\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}V^* \cong \bbS^\star _{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 3 \\
\cline{1-2}
2 & 4 \\
\cline{1-2}
\end{array}}{12pt}}V^*$ this is the algebraic prototype of the symmetrization
\begin{equation}\label{eq:symmetrized_curvature_tensor}
\forall x_1,x_2,x_3,x_4\in T_p\tilde M:\; \gamma(x_1,x_2,x_3,x_4) := \cyclic_{12} \tilde R(x_1,x_3,x_2,x_4)
\end{equation} of the curvature tensor $\tilde R$ of some Riemannian manifold.
\end{enumerate}
\end{example}
\subsection{A generalisation of the wedge product}
\label{se:product}
Let $U$, $V$ and $W$ be irreducible representations of the general linear
group $\GL(n)$. If $W$ occurs with multiplicity one in the decomposition of $U\otimes V$, then there exists an invariant bilinear map
$\owedge \colon U\times V \to W$ which is unique up to a factor by Schurs Lemma. In the cases relevant for our article,
the underlying Young frame of all three involved partitions has at most two
rows. Let $\lambda = (\lambda_1,\lambda_2)$, $\mu=(\mu_1,\mu_2)$ and $\nu =
(\nu_1,\nu_2)$ be partitions of length at most two. Further, suppose that
$\lambda_1 \leq \nu_1\leq \lambda_1 + \mu_1$, $\lambda_2\leq \nu_2$ and that the length of these partitions satisfies $|\nu|= |\lambda| + |\mu|$.
Let $T_1$, $T_2$ and $T_3$ be tableaus of shape $\lambda$, $\mu$ and $\nu$, respectively.
Then $\bbS_{T_3}V^*$ occurs with multiplicity one in the decomposition of the tensor product
$\bbS_{T_1}\,V^* \otimes \bbS_{T_2}\,V^*$ according to the
Littlewood-Richardson rules. Hence, because of Schurs lemma there is up to
a factor a unique invariant bilinear map
\begin{equation}
\owedge \colon \bbS_{T_1}\,V^* \times \bbS_{T_2}\,V^*\to \bbS_{T_3}V^*.
\end{equation}
In order to reduce the occurence of annoying factors in
our formulas, we make the following specific choice of $\owedge$: suppose for simplicity that all three
tableaus are normal.
Then $\lambda \owedge \mu(v_1,\cdots, v_{\nu_1 + \nu_2})$ is obtained by applying the Young symmetrizer
$\rmS_{T_3}$ to the product $\lambda(v_1,\cdots,v_{\lambda_1},v_{\nu_1+1},\cdots,v_{\nu_1 + \lambda_2})
\mu(v_{\lambda_1+1},\cdots, v_{\nu_1},v_{\nu_1 + \lambda_2 + 1},\cdots,v_{\nu_1 + \nu_2})$
and dividing by the product $h_\lambda h_\mu$ of all hook numbers of the two
frames of shape $\lambda$ and $\mu$, this way generalising the standard
definition of the wedge product $\omega\wedge \eta := \frac{n!}{p!q!} \mathrm{Alt}(\omega\otimes
\eta)$ of $p$- and $q$-forms $\omega$ and $\eta$ from
differential geometry. Since our plain notation $\owedge$ is still ambiguous, we describe explicitly the products used in the sequel:
\bigskip
\begin{definition}\label{de:product}
\begin{enumerate}
\item We keep to the standard notation
\begin{equation}
\lambda_1\bullet \lambda_2(v_1,v_2) \;\;:=\;\; \lambda_1(v_1)\lambda_2(v_2) + \lambda_1(v_2)\lambda_2(v_1)
\end{equation}
for the symmetrized tensor product of 1-forms $\lambda_1$ and $\lambda_2$ (cf.~\cite{HMS}).
\item Given a 1-form $\lambda$ and a symmetric 2-tensor $\alpha$, we set
\begin{equation}\label{eq:product_of_a_one_form_and_a_symmetric_2-tensor}
\lambda\owedge \alpha(v_1,v_2,v_3) \;\;:=\;\; -\alpha\owedge \lambda (v_1,v_2,v_3)\;\;:=\;\;\frac{1}{2}\, \rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
1 & \multicolumn{1}{c}{} \\
\cline{1-1}
\end{array}}{12pt}}\, \lambda(v_2)\alpha(v_1,v_3).
\end{equation}
\item We define the Cartan product
\begin{equation}\label{eq:def_curvature_tensor_associated_with_two_2-forms}
\omega_1\owedge \omega_2(v_1,v_2,v_3,v_4)\;\;:=\;\; \frac{1}{4}\, \rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4\\
\cline{1-2}
\end{array}}{12pt}}\, \omega_1(v_1,v_3)\omega_2(v_2,v_4)
\end{equation}
of alternating 2-forms $\omega_1$ and $\omega_2$.
\item Given a 1-form $\lambda$ and $\beta\in \bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
1 & \multicolumn{1}{c}{} \\
\cline{1-1}
\end{array}}{12pt}}V^*
$, we define
\begin{equation}\label{eq:noch_ein_produkt}
\lambda \owedge \beta(v_1,v_2,v_3,v_4)\;\;:=\;\;- \beta\owedge\lambda(v_1,v_2,v_3,v_4)\;\;:=\;\;-\frac{1}{3}\,\rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4\\
\cline{1-2}
\end{array}}{12pt}}\, \lambda(v_1)\beta(v_2,v_3,v_4).
\end{equation}
\item
We define a variant of the Kulkarni-Nomizu product
\begin{equation}\label{eq:def_Kulkarni_Nomizu_product}
\alpha_1\owedge \alpha_2(v_1,v_2,v_3,v_4)\;\;:=\;\;\frac{1}{4}\,\rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4\\
\cline{1-2}
\end{array}}{12pt}}\, \alpha_1(v_1,v_2)\alpha_2(v_3,v_4)
\end{equation}
for symmetric 2-tensors $\alpha_1$ and $\alpha_2$.
\end{enumerate}
\end{definition}
\section{The main results}
Although we are mainly interested in the Riemannian case, we consider more
generally a pseudo Riemannian manifold $(M,g)$ of dimension $n$ with tangent
bundle $TM$, Levi-Civita connection $\nabla$ and curvature tensor
$R$. Further, we will use the standard notation $\langle x,y\rangle := g(x,y)$ for $x,y\in T_pM$.
The first and second (standard) variable of the prolongation of the Killing equation are defined by
\begin{align}\label{eq:kappa_prolong_1}
\kappa^1(x_1,x_2,x_3) \;:=\; & \rmP_{\scaleto{
\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
1 & \multicolumn{1}{c}{\;\;\;}\\
\cline{1-1}
\end{array}}{12pt}} \nabla_{x_1}\kappa(x_2,x_3),\\
\label{eq:kappa_prolong_2}
\kappa^2(x_1,x_2,x_3,x_4) \;:=\; & \rmP_{\scaleto{\begin{array}{|c|c|}\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4\\
\cline{1-2}
\end{array}}{12pt}}\nabla^2_{x_1,x_2}\kappa(x_3,x_4).
\end{align}
In Section~\ref{se:prolongation} we will show that the triple $(\kappa,\kappa^1,\kappa^2)$ is in fact closed under componentwise covariant derivative,
which yields the prolongation of the Killing equation for symmetric 2-tensors.
\bigskip
\begin{definition}
The (Riemannian) cone $\hat M$ is the trivial fibre bundle $\tau \colon M\times \R_+ \to M$ whose total space is equipped with the metric tensor
\begin{equation}\label{eq.cone}
\hat g := r^2\, g + \d r^2.
\end{equation}
\end{definition}
Then we see $M$ as a (pseudo) Riemannian submanifold of $\hat M$ via $\iota \colon M\to
\hat M,p\mapsto (p,1)$. Every symmetrized algebraic curvature tensor $S$ on $\hat M$
defines a symmetric 2-tensor $\kappa := \frac{1}{2} \pd_r\lrcorner\pd_r \lrcorner S$ on $M$, i.e
\begin{equation}\label{eq:kanonische_konstruktion_von_killingtensoren}
\kappa(x,y) := \frac{1}{2} S(\pd_r|_p,\pd_r|_p,\d_p \iota\, x,\d_p \iota\, y)
\end{equation}
for all $p\in M$ and $x,y\in T_pM$. For a construction in the inverse
direction, let a symmetric 2-tensor $\kappa$ on $M$ be given:
\bigskip
\begin{definition}
The symmetrized algebraic curvature tensors defined by
\begin{align}
\label{eq:def_C_kappa}
C^\kappa &\;\;:=\;\; \kappa^2 + \kappa \owedge g,\\
\label{eq:def_S_kappa}
S^\kappa&\;\;:=\;\; r^2\, \kappa\owedge \d r\bullet \d r + r^3\,
\kappa^1\owedge \dr + r^4\, C^\kappa
\end{align}
will be called the associated symmetrized algebraic curvature tensors on $M$ and $\hat M$, respectively.
\end{definition}
We remark without proof that $S^\kappa$ has the alternative description $\rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4\\
\cline{1-2}
\end{array}}{12pt}}\hat \nabla^2_{v_1,v_2}r^4\tau^*\kappa(v_3,v_4)$
.
\bigskip
\begin{theorem}\label{th:main_1}
Let $M$ be a pseudo Riemannian manifold of dimension at least two with Levi-Civita
connection $\nabla$ and curvature tensor $R$. The definitions given in~\eqref{eq:kanonische_konstruktion_von_killingtensoren},~\eqref{eq:def_C_kappa} and~\eqref{eq:def_S_kappa} yield a 1-1 correspondence between
\begin{enumerate}
\item symmetric Killing 2-tensors $\kappa$ on $M$ which satisfy the following nullity conditions
\begin{align}\label{eq:curv_cond_1}
R_{x,y}\kappa\;\;=\;\;& - x\wedge y \cdot \kappa\;,\\
\label{eq:curv_cond_2}
R_{x,y} \nabla \kappa\;\;=\;\;& - x\wedge y\cdot \nabla \kappa
\end{align}
for all $p\in M$ and $x,y\in T_pM$;
\item pairs $(\kappa,C)$ of symmetric 2-tensors $\kappa$ and symmetrized algebraic curvature tensors $C$ on $M$ satisfying
\begin{align}\label{eq:alg_curv_tensor_1}
C(x,\,\cdot\,,\,\cdot\,,\,\cdot\,)\;\;=\;\;& \nabla_x\nabla\kappa + 2\,\kappa\owedge x^\sharp,\\
\label{eq:alg_curv_tensor_2}
\nabla_x C \;\;=\;\;& - \nabla\kappa\owedge x^\sharp
\end{align}
for all $p\in M$ and $x \in T_pM$;
\item parallel symmetrized algebraic curvature tensors on the Riemannian cone $\hat M$.
\end{enumerate}
\end{theorem}
We would like to remark that there are always trivial solutions to~(a),~(b) and~(c) of the previous theorem coming from the metric tensor.
For example a constant multiple $\kappa := c\, g$ of the metric tensor is in accordance with~\eqref{eq:curv_cond_1},~\eqref{eq:curv_cond_2}. Further, it is easy to see that the requirement that the dimension of $M$
has to be strictly larger than one is necessary, see Remark~\ref{re:main_3}.
\bigskip
\begin{example}\label{ex:constant_curvature}
If $M$ is simply connected and of constant (non-vanishing) sectional curvature, then we can realise $M$
as a standard model in the pseudo Euclidean space $(V^{n+1},\langle\ ,\ \rangle)$. Further,
possibly after reversing the sign of the metric of $M$ and scaling by a positive
constant, we can assume that $M$ is the generalised unit sphere $\{v\in V|\langle v,v\rangle = 1\}$. Then the cone
over $M$ is $V\setminus\{0\}$ and the nullity
conditions~\eqref{eq:curv_cond_1} and~\eqref{eq:curv_cond_2} become
tautological. Thus algebraic curvature tensors on $V$ correspond with elements of $\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}V^*$ via~\eqref{eq:symmetrized_curvature_tensor}. We obtain
from $(a)\Leftrightarrow (c)$ that symmetric Killing 2-tensors on $M$ are in
1-1 correspondence with algebraic curvature tensors on $V$ (the flat case is handeled similarly).
Thus we have recovered the main result of~\cite{McMS} in the
special case of 2-tensors.
\end{example}
\bigskip
\begin{remark}\label{re:main_2}
If a pair $(\kappa,C)$ satisfies Equations~\eqref{eq:alg_curv_tensor_1}-\eqref{eq:alg_curv_tensor_2},
then $C = C^\kappa$ and, vice versa, in dimension larger than one also $\kappa$ is determined by $C$, see~\eqref{eq:def_kappaC}-\eqref{eq:def_modified_Ricci_trace_und_tilde_scalC}.
\end{remark}
Further, every Riemannian manifold $(M,g)$ can also be seen as a pseudo Riemannian
manifold with negative definite metric tensor and vice versa
by considering $(M,-g)$. Hence the following remark addresses in fact compact
Riemannian manifolds:
\bigskip
\begin{remark}\label{re:main_1}
On a compact manifold with negative definite metric tensor there are
no solutions to the previous theorem besides the trivial ones.
\end{remark}
Here the compactness is essential, as becomes clear from considering the
hyperbolic space.
The proofs of Theorem~\ref{th:main_1} and the following two remarks are implicitly contained in
Propositions~\ref{p:prolong},~\ref{p:main} and~\ref{p:get_rid} below.
\subsection{Characterization of Sasakian manifolds and spheres}
Next, we would like to discuss the relevance of Theorem~\ref{th:main_1} in the Riemannian case. When does a Riemannian
manifold admit tensors fitting (a),~(b) or~(c) other than the trivial solutions?
Recall that there are several ways to define a Sasakian manifold
$M$. One possibility is via the existence of a Kähler
structure $I$ on $\hat M$ (cf.~\cite[Proposition~1.1.2]{BG}). The
corresponding parallel 2-from $\omega(u,v) := \langle I\, u,v\rangle$ is
called the Kähler form. Then the 1-form $\eta := \iota^*(\pd_r\lrcorner
\omega)$ on $M$ (defined by $\eta(x) := \omega(\pd_r|_p,\d_p\iota x)$ for all $p\in M$ and $x\in T_pM$) is called the characteristic form of the Sasakian manifold $M$.
More specifically, one considers 3-Sasakian manifolds. Here the cone carries a Hyperkähler structure $\{I_1,I_2,I_3\}$.
The corresponding Kähler forms $\{\omega_1,\omega_2,\omega_3\}$ induce 1-forms
$\eta_i := \iota^*(\pd_r\lrcorner \omega_i)$ on $M$ as before called the characteristic forms
of the 3-Sasakian manifold $M$ (cf.~\cite[Proposition~1.2.2]{BG}).
Examples of Sasakian and 3-Sasakian manifolds are round spheres in odd dimension and dimension 3 {\rm mod} 4, respectively.
In Section~\ref{se:sasaki} we will use S.~Gallots before mentioned theorem in
order to classify complete Riemannian manifolds whose
cone carries a parallel algebraic curvature tensor by means of the holonomy principle. We obtain our second main result:
\bigskip
\begin{theorem}\label{th:main_2}
The classification of complete and simply connected Riemannian manifolds $(M,g)$ with respect to Theorem~\ref{th:main_1} is as follows.
\begin{itemize}
\item On a round sphere the space of symmetric Killing
2-tensors is isomorphic to the linear space of algebraic curvature tensors
in dimension $n+1$, see Example~\ref{ex:constant_curvature}.
\item Suppose that $(M,g)$ is a 3-Sasakian manifold which is not of constant sectional curvature.
Let $\{\eta_1,\eta_2,\eta_3\}$ denote its characteristic forms related to Kähler forms $\{\omega_1,\omega_2,\omega_3\}$
on the cone. Here the symmetrized tensor products $\eta_i\bullet\eta_j$ are symmetric Killing 2-tensors which correspond to
the parallel symmetrized algebraic curvature tensors $\omega_i\owedge\omega_j$ on the cone.
The pair $(\kappa,C^\kappa)$ with $\kappa := \eta_i\bullet\eta_j$ and $C^\kappa := \frac{1}{4} \d\eta_i\owedge\d\eta_j$ is a solution to~\eqref{eq:alg_curv_tensor_1}-\eqref{eq:alg_curv_tensor_2}. Together with the metric tensor we obtain a 7-dimensional space of symmetric 2-tensors matching for example~\eqref{eq:curv_cond_1},\eqref{eq:curv_cond_2} of Theorem~\ref{th:main_1}~a).
\item Suppose that $(M,g)$ is a Sasakian manifold which is neither 3-Sasakian
nor of constant sectional curvature one. Let $\eta$ denote its
characteristic 1-form. Similar as in the 3-Sasakian case, the set of Killing
tensors fitting into Theorem~\ref{th:main_1} is the 2-dimensional space spanned by the square $\eta \otimes \eta $ of the characteristic form and the metric tensor.
\item Otherwise, there are only the obvious solutions to (a),~(b) or~(c)
coming from the metric tensor.
\end{itemize}
\end{theorem}
We observe that all Killing tensors mentioned in the above theorem are linear combinations of the metric tensor and symmetric products of Killing vector fields.
Further, we recall that spheres and Sasakian manifolds are irreducible.
\bigskip
\begin{corollary}\label{co:main_1}
An even (odd) dimensional simply connected complete Riemannian manifold $M$ of
dimension at least two admits a non-trivial Killing tensor matching~\eqref{eq:curv_cond_1},\eqref{eq:curv_cond_2} of Theorem~\ref{th:main_1}~a) if and only if $M$ is the unit sphere (a Sasakian manifold).
\end{corollary}
Due to the fact that a Sasakian vector field has constant length,
the square of the characteristic form of a Sasakian manifold has
constant trace. The same is true for the symmetric products of the characteristic forms of a
3-Sasakian manifold. Conversely, the Killing tensor $\kappa$ associated with a
parallel curvature tensor $S$ of $\R^{n+1}$ on an $n$-dimensional round sphere has $\d\, \trace
\kappa \neq 0$ unless the algebraic Ricci-tensor of $S$ is a multiple of
the Euclidean scalar product (i.e. unless $S$ has the algebraic
properties of the curvature tensor of an Einstein manifold). Therefore:
\bigskip
\begin{corollary}\label{co:main_2}
A simply connected complete Riemannian manifold $M$ carries a symmetric 2-tensor $\kappa$ matching~\eqref{eq:curv_cond_1},\eqref{eq:curv_cond_2} of Theorem~\ref{th:main_1}~a)
and such that $\d\, \trace \, \kappa \neq 0$ if and only if $M$ is a
round sphere.
\end{corollary}
Recall that one of the original goals of the paper~\cite{Ga} was to investigate the following differential
equation $(E_2)$ of order three on functions on Riemannian manifolds,
\begin{equation}\label{eq:Gallots_Gleichung}
\left . \begin{array}{c} \nabla^3_{y_1,y_2,y_3}f +
2\,\d f(y_1) \langle y_2,y_3\rangle +
\cyclic_{23}\langle y_1,y_2 \rangle\,
\d f(y_3) \end{array} \right .\;\;=\;\;0.
\end{equation}
Let $\kappa$ be the symmetric 2-tensor
\begin{equation}\label{eq:Gallots_kappa}
\kappa^f(x,y) := f\,\langle x ,y \rangle + \frac{1}{4} \nabla^2_{x,y} f.
\end{equation}
We will show in Section~\ref{se:Gallot} that the pair $(\kappa,C^\kappa)$ satisfies the the conditions of Theorem~\ref{th:main_1}~(b). Hence we can see the following result in the
context of symmetric Killing 2-tensors:
\bigskip
\begin{corollary}\label{co:Gallot}{\cite[Corollary~3.3]{Ga}}
Suppose that $M$ is a complete Riemannian manifold.
If there exists a non-constant solution to~\eqref{eq:Gallots_Gleichung},
then the universal covering of $M$ is the Euclidean sphere of radius one.
\end{corollary}
We conjecture that the differential equation $(E_p)$ considered in Chapter~4 of the same paper~\cite{Ga} is related to symmetric Killing tensors of valence $p$ for all $p\geq 1$ in a similar way.
\section{The associated symmetrized algebraic curvature tensor on the cone}
\label{se:prolongation_on_the_cone}
Let $M$ be a (pseudo) Riemannian manifold and $\hat M$ its cone. The canonical projection $\tau:\hat M\to M$
is obtained by forgetting the second component of a point $(p,t)\in \hat M$.
Further, the natural inclusion
\[
\iota \colon M \to \hat M,p\mapsto (p,1)
\]
exhibits $M$ as a Riemannian submanifold of $\hat M$.
The tangent bundle of $\hat M$ splits as $\tau^*T \hat M = T M \oplus \R$.
Vectors tangent to $M$ are called horizontal vectors.
In the following $x,y,\ldots$ denote horizontal vectors and $u,v,\ldots$
arbitrary tangent vectors of $\hat M$. The Levi-Civita connection is given by (cf.~\cite[Ch.~1]{Ga})
\begin{align}\label{eq:Levi-Civita_connection_of_the_cone_1}
& \hat\nabla_{\partial_r} \partial_r\;\;=\;\;0\;,\\
\label{eq:Levi-Civita_connection_of_the_cone_3}
& \hat\nabla_x y\;\;=\;\;\nabla_x y - r\, \langle x,y\rangle \partial_r,\\
\label{eq:Levi-Civita_connection_of_the_cone_2}
&\hat\nabla_{\partial_r} x\;\;=\;\;\hat\nabla_x {\partial_r}\;\;=\;\;\frac{1}{r} x.
\end{align}
The last two equations are precisely the equations of Gauß and Weingarten
describing the Levi-Civita connection on the euclidean sphere of radius $r$ where $\pd_r$ becomes the outward pointing normal vector field.
Equation~\eqref{eq:Levi-Civita_connection_of_the_cone_1} implies that the vector field $\frac{x}{r}$ is parallel along the curve $r\mapsto (p,r)$ for each $x\in T_pM$ , i.e.
\begin{equation}\label{eq:parallele_vektorfelder_laengs_radialer_kurven}
\hat\nabla_{\pd_r}\frac{x}{r} = 0.
\end{equation}
Further, it follows that
\begin{align}\label{eq:hat_nabla_diff_r}
& \hat\nabla_{\partial_r}\d r\;\;=\;\;0\;,\\
\label{eq:hat_nabla_x_dr}
& \hat\nabla_x\d r\;\;=\;\;r\, x^\sharp
\end{align}
(where $x^\sharp$ denotes the dual 1-form $\langle x,\,\cdot\,\rangle $\,. Let $\gamma$ be an arbitrary covariant $k$-tensor of $M$.
\begin{itemize}
\item Given $x\in T_pM$ we define a covariant tensor $x\lrcorner \gamma$ of valence $k-1$ by setting
\begin{equation}\label{eq:x_einsetzen}
x\lrcorner \gamma (y_1,\cdots,y_{k-1}) := \gamma (x,y_1,\cdots,y_{k-1}).
\end{equation}
\item Given some endomorphism $A$ of $T_pM$ we define $A\cdot \gamma$ by
\begin{equation}\label{eq:derivation}
A\cdot \gamma (x_1,\cdots,x_k) := - \sum_{i=1}^k \gamma(x_1,\cdots,A x_i,\cdots, x_k).
\end{equation}
\end{itemize}
If $\gamma$ is a covariant $p$-tensor on $M$, the pullback $\tau^*\gamma$ will
also be denoted by $\gamma$. As a consequence of the previous we have:
\bigskip
\begin{lemma}
Let $\gamma$ and $\kappa$ be an arbitrary and a symmetric covariant $k$-tensor, respectively, on $M$.
\begin{align}\label{eq:hat_nabla_kappa_r}
& \hat\nabla_{\partial_r}\gamma\;\;=\;\;- \frac{k}{r} \gamma\;,\\
\label{eq:hat_nabla_gamma_x}
& \hat\nabla_{x}\gamma\;\;=\;\;\nabla_x \gamma - \frac{1}{r} \d r\otimes x \cdot \gamma,\\
\label{eq:hat_nabla_kappa_x}
& \hat\nabla_{x}\kappa\;\;=\;\;\nabla_x \kappa - \frac{1}{r}\d r \bullet x \lrcorner \kappa.
\end{align}
\end{lemma}
\bigskip
\begin{remark}\label{re:reduction}
Every basis of $T_pM$ can be extended to a basis of $T_{(p,r)}\hat M$ using the radial vector $\pd_r|_{(p,r)}$.
Thus, the principal $\GL_{n+1}(\R)$ fiber bundle of frames of the vector bundle $T\hat M$
admits a natural reduction to the principal $\GL_n(\R)$ fiber bundle of frames of the pullback vector bundle $\tau^*TM$. In particular, $\nabla$ induces a connection on
every tensor bundle of $T\hat M$. Therefore the latter decomposes into a direct sum of $\nabla$-parallel subbundles
according to the branching rules for the Lie algebra pair $\gl_n(\R)\subset \gl_{n+1}(\R)$.
\end{remark}
For the vector bundle $\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\, T^*\hat M$ of symmetrized algebraic curvature tensors of the cone these
branching rules are given as follows:
\bigskip
\begin{lemma}\label{le:branching_rules}
The vector bundle $\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\, T^*\hat M$ admits a decomposition into three direct summands
\begin{equation}\label{eq:glndec}
\tau^*\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
\end{array}}{6pt}}\,T^*M\oplus \tau^*\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
1 & \multicolumn{1}{c}{\;\;\;} \\
\cline{1-1}
\end{array}}{12pt}}\,T^*M \oplus \tau^*\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\, T^*M
\end{equation}
which is parallel with respect to the Levi-Civita connection of $M$. Suitable projection maps are given as follows:
\begin{eqnarray}
\label{eq:projection_1}
\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\, T^*\hat M\to \tau^*\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
\end{array}}{6pt}}\,T^*M, & & S \mapsto \alpha^S \;\;:= \;\; \frac{1}{2}\pd_r\lrcorner \pd_r \lrcorner S\;,\\
\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\, T^*\hat M\to \tau^*\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
1 & \multicolumn{1}{c}{\;\;\;}\\
\cline{1-1}
\end{array}}{12pt}}T^*M \;,& & S \mapsto \beta^S \;\;:= \;\; \iota^* \pd_r \lrcorner S\;,\\
\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\, T^*\hat M \to \tau^*\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4\\
\cline{1-2}
\end{array}}{12pt}}T^*M \;,& & S\mapsto \gamma^S \;\;:= \;\; \iota^*S.
\end{eqnarray}
Then every triple $(\alpha,\beta,\gamma)$ defines the symmetrized algebraic curvature tensor
\begin{equation}\label{eq:S_explizit}
S := \alpha \owedge\d r \bullet \d r + \beta \owedge\d r + \gamma.
\end{equation}
\end{lemma}
\begin{proof}
It is straightforward to verify~\eqref{eq:S_explizit}. Hence the map which projects from $\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\, T^*\hat M $ to $\tau^*\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
\end{array}}{6pt}}\,T^*M \oplus\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
1 & \multicolumn{1}{c}{\;\;\;} \\
\cline{1-1}
\end{array}}{12pt}}\,T^*M \oplus \bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\,T^ *M $ along $\tau$ is surjective. In order to demonstrate its injectivity, suppose $\alpha^S = 0$, $\beta^S = 0$ and $\gamma^S = 0$.
Then $S(u_1,u_2,u_3,u_4)$ vanishes whenever at least two of the vectors $u_i$ are horizontal by means of the symmetries of a curvature tensor.
On the other hand, if at most one vector is horizontal, then at least three
are proportional to $\partial_r$ and thus $S(u_1,u_2,u_3,u_4) = 0$, since $S$
satisfies the corresponding exchange rule of order one between rows. This proves the decomposition~\eqref{eq:glndec}.
\end{proof}
Therefore symmetrized algebraic curvature tensors of $T_{(p,t)}\hat M$ should be seen as triples
\[
(\alpha,\beta,\gamma) \in \bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
\end{array}}{6pt}}\,T_p^*M \oplus \bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
1 & \multicolumn{1}{c}{\;\;\;} \\
\cline{1-1}
\end{array}}{12pt}}\, T_p^*M \oplus \bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\, T_p^*M.
\]
In particular, the triple
\begin{equation}\label{eq:zerlegung_von_kappa_dach}
\left ( \alpha,\beta,\gamma \right )
\;\;=\;\; \left (r^2\, \kappa, r^3\,\kappa^1, r^4\,C^\kappa\right )
\end{equation}
corresponds with the associated algebraic curvature tensor on the cone $S^\kappa$ defined in~\eqref{eq:def_S_kappa}.
\subsection{Symmetric Killing 2-tensors whose associated symmetrized algebraic curvature tensor is parallel}
It is natural to ask under which conditions the symmetrized algebraic curvature
tensor $S^\kappa$ on the cone $\hat M$ associated with a symmetric Killing 2-tensor
$\kappa$ on $M$ is parallel with respect to the Levi-Civita connection of
$\hat M$. More general, we wish to understand the relation between parallel
sections of $\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\, T^*\hat M$ and symmetric Killing 2-tensors on $M$.
Let $S$ be a symmetrized algebraic curvature tensor on the cone
seen as a triple $(\alpha,\beta,\gamma)$ according to~\eqref{eq:glndec}
and $S|_M$ denote the restriction of $S$ to $M$ via the natural inclusion $\iota \colon M \to \hat M$.
Hence $S|_M$ is the triple $(\alpha|_M,\beta|_M,\gamma|_M)$ where $\alpha|_M$, $\beta|_M$ and $\gamma|_M$ are sections of
$\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
\end{array}}{6pt}}\,T^*M $, $\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
1 & \multicolumn{1}{c}{\;\;\;} \\
\cline{1-1}
\end{array}}{12pt}}\, T^*M$ and $\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\, T^*M$, respectively.
The following lemma shows that parallelity of $S$ along the radial direction does
not have anything to do with the Killing equation:
\bigskip
\begin{lemma}\label{le:parallel_in_the_radial_direction}
A section $S = (\alpha,\beta,\gamma)$ of $\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\, T^*\hat M$
is parallel along the radial direction $\hat\nabla_{\pd_r} S = 0$ if and only if $S = (r^2\,\alpha|_M,r^3\, \beta|_M,r^4\,\gamma|_M)$.
\end{lemma}
\begin{proof}
Recall that the radial vector field $\pd_r$ is covariantly constant in the radial direction according to~\eqref{eq:Levi-Civita_connection_of_the_cone_1} and
that $\frac{x}{r}$ is a parallel vector field along the radial curve $r\mapsto (p,r)$ for each $x\in T_pM$, see~\eqref{eq:parallele_vektorfelder_laengs_radialer_kurven}.
Comparing this with~\eqref{eq:S_explizit}, we see that a triple $(\alpha,\beta,\gamma)$ is covariantly constant in the radial direction $\hat\nabla_{\pd_r} S = 0$ if and only if $S = (r^2\, \alpha|_M,r^3\, \beta|_M,r^4\, \gamma|_M)$.
\end{proof}
Therefore it remains to understand the condition $\hat\nabla_x S = 0$ for horizontal vectors $x$.
For this we recall that the vector bundle $T\hat M = \tau^*TM \oplus \R$ carries two relevant
connections: there is the Levi-Civita connection $\hat \nabla$ of the metric
cone and also the Levi Civita connection of $M$ according to Remark~\ref{re:reduction}. Because
of~\eqref{eq:Levi-Civita_connection_of_the_cone_3},~\eqref{eq:Levi-Civita_connection_of_the_cone_2}
the difference tensor $\Delta := \hat \nabla - \nabla $ satisfies
\begin{equation}\label{eq:Delta}
\begin{array}{l} \Delta(x,y) = - r\langle x,y \rangle,\\
\Delta(x,\pd_r|_{(p,r)}) = \frac{1}{r} x
\end{array}
\end{equation}
for all $p\in M$, $r>0$ and $x\in T_pM$.
\bigskip
\begin{lemma}\label{le:parallel_in_the_horizontal_direction}
Let $S$ be a symmetrized algebraic curvature tensor of $T_{(p,r)}\hat M$ described as a triple $(\alpha,\beta,\gamma)$ according to~\eqref{eq:glndec}.
For each horizontal vector $x\in T_pM$ the difference $\hat \nabla_xS - \nabla_xS$
is given by the triple
\begin{equation}\label{eq:second_fundamental_form_of_bbP}
\left ( \begin{array}{lll}
- \frac{1}{r}\, x\lrcorner \beta & & \\
- \frac{1}{r}\, x\lrcorner \gamma & \; + 2\, r\, \alpha \owedge x^\sharp &\\
&\mbox{\ } + r\, \beta\owedge x^\sharp &
\end{array} \right ).
\end{equation}
\end{lemma}
\begin{proof}
We have $\hat \nabla_xS - \nabla_xS = x\lrcorner \Delta\cdot S$, where $\Delta$ is defined by~\eqref{eq:Delta} and $\cdot$ is the usual action of endomorphisms on tensors by algebraic derivation, see~\eqref{eq:derivation}. Hence,
\begin{align*}
x\lrcorner \Delta S(\partial_r,\partial_r,y_1,y_2) &\;\;=\;\; \frac{- 2}{r}\underbrace{\langle \partial_r|_p,\partial_r|_p \rangle}_{=1} S( \partial_r,x,y_1,y_2).
\end{align*}
This yields the first component in~\eqref{eq:second_fundamental_form_of_bbP}. For the second, we use that the cyclic sum of $y\lrcorner S$ vanishes for fixed $y$.
Hence $ 2\, S(x,z,y,z) = 2\, S(y,z,x,z) = - S(y,x,z,z) = - S(z,z,x,y)$. We get
\begin{align*}
& x\lrcorner \Delta S(\partial_r,y_1,y_2,y_3) &\;\;=\;\;& \left \lbrace\begin{array}{cc}- \frac{1}{r} S(x,y_1,y_2,y_3) & + r\, \langle x,y_1 \rangle S(\partial_r|_p,\partial_r|_p,y_2,y_3)\\
+ r\, \langle x,y_2 \rangle S(\partial_r|_p,y_1,\partial_r|_p,y_3) & + r\, \langle x,y_3\rangle S(\partial_r|_p,y_1,y_2,\partial_r|_p) \end{array} \right \rbrace\\
& &\;\;=\;\;& \left \lbrace\begin{array}{cc}-\frac{1}{r} S(x,y_1,y_2,y_3) & + r\, \langle x,y_1 \rangle S(\partial_r|_p,\partial_r|_p,y_2,y_3)\\
- \frac{1}{2} r\, \langle x,y_2 \rangle S(\partial_r|_p,\partial_r|_p,y_1,y_3) & - \frac{1}{2}r\, \langle x,y_3 \rangle S(\partial_r|_p,\partial_r|_p,y_1,y_2) \end{array} \right \rbrace\\
& &\;\;=\;\;& - \frac{1}{r} S(x,y_1,y_2,y_3) + r\, \rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
1 \\
\cline{1-1}
\end{array}}{12pt}}\, \langle x, y_1\rangle \alpha(y_2,y_3)\\
& & \;\;\stackrel{\eqref{eq:product_of_a_one_form_and_a_symmetric_2-tensor}}{=}\;\; & - \frac{1}{r} \gamma(x,y_1,y_2,y_3) + 2\,r\, \alpha\owedge x^\sharp (y_1,y_2,y_3)
\end{align*}
for all $x,y_1,\ldots,y_3\in T_pM$. For the last component: on the one side, we have
\begin{align*}
x\lrcorner \Delta S(y_1,y_2,y_3,y_4)\;\;=\;\;& \left \lbrace \begin{array}{c}\ \;r\, \langle x,y_1\rangle S(\partial_r|_p,y_2,y_3,y_4)
+ r\, \langle x,y_2\rangle S(y_1,\partial_r|_p,y_3,y_4)\\
+ r\, \langle x,y_3\rangle S(y_1,y_2,\partial_r|_p,y_4)
+ r\, \langle x,y_4\rangle S(y_1,y_2,y_3,\partial_r|_p)\end{array}\right \rbrace\\
\end{align*}
for all $x,y_1,\ldots,y_4\in T_pM$. On the other side,
\begin{align*}
3\, \beta \owedge x^\sharp (x,y_1,y_2,y_3,y_4)\;\;\stackrel{\eqref{eq:noch_ein_produkt}}{=}\;\; &
\rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\, \langle x, y_1 \rangle \beta (y_2,y_3,y_4) \\
\;\;=\;\; & \cyclic_{12}\cyclic_{34}\left ( \begin{array}{c}\langle x,y_1\rangle S(\partial_r|_p,y_2,y_3,y_4) \\
- \langle x,y_3\rangle S(\partial_r|_p,y_2,y_1,y_4)\\
- \langle x,y_1\rangle S(\partial_r|_p,y_4,y_3,y_2)\\
+ \langle x,y_3\rangle S(\partial_r|_p,y_4,y_1,y_2)\end{array}\right ).
\end{align*}
The first and the last line of the above term yield
\begin{equation*}
\cyclic_{12}\cyclic_{34}\left ( \begin{array}{c}\langle x,y_1\rangle S(\partial_r|_p,y_2,y_3,y_4) \\
+ \langle x,y_3\rangle S(\partial_r|_p,y_4,y_1,y_2)\end{array}\right ) \;\;=\;\;
\cyclic_{12}\cyclic_{34}\left ( \begin{array}{c}\langle x,y_1\rangle S(\partial_r|_p,y_2,y_3,y_4) \\
+ \langle x,y_3\rangle S(y_1,y_2,\partial_r|_p,y_4)\end{array}\right )\\
\end{equation*}
which gives $\frac{2}{r} x\lrcorner \Delta S(y_1,y_2,y_3,y_4)$. The two lines in the middle are by the Bianchi identity
\begin{align*}
- \cyclic_{12}\cyclic_{34} \left ( \begin{array}{c}
\langle x,y_3\rangle S(\partial_r|_p,y_2,y_1,y_4)\\
+ \langle x,y_1\rangle S(\partial_r|_p,y_4,y_3,y_2)
\end{array}\right ) &\;\;=\;\;& \cyclic_{12}\cyclic_{34} \left ( \begin{array}{c}
\langle x,y_3 \rangle\big ( S(y_1,y_2,\partial_r|_p,y_4) + S(\partial_r|_p,y_1,y_2,y_4)\big )\\
+ \langle x,y_1\rangle \big (S(y_3,y_4,\partial_r|_p,y_2) + S(\partial_r|_p,y_3,y_4,y_2)\big )
\end{array}\right ).
\end{align*}
Using the Bianchi identity in the form mentioned further above, this yields
\begin{align*}
\cyclic_{12}\cyclic_{34} \left ( \begin{array}{c}
\langle x,y_3 \rangle\big ( S(y_2,y_1,\partial_r|_p,y_4) - \frac{1}{2}S(y_1,y_2,\partial_r|_p,y_4)\big )\\
\langle x,y_1\rangle \big (S(y_4,y_3,\partial_r|_p,y_2) - \frac{1}{2} S(y_3,y_4,\partial_r|_p,y_2)\big )
\end{array}\right ),
\end{align*}
which adds $\frac{1}{r}x\lrcorner \Delta S(y_1,y_2,y_3,y_4)$ to the Young symmetrizer. We conclude that
\begin{align*}
\rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\,\langle x, y_1 \rangle \partial_r\lrcorner S(y_2,y_3,y_4)\;\;=\;\;\frac{3}{r}x\lrcorner \Delta S(y_1,y_2,y_3,y_4).
\end{align*}
This yields the last component of~\eqref{eq:second_fundamental_form_of_bbP}.
\end{proof}
The following proposition characterizes Killing tensors whose associated
algebraic curvature tensor is parallel on the cone. This can be seen as
the analogue of~\cite[Lemma~3.2.1]{Se} for symmetric Killing 2-tensors.
\bigskip
\begin{proposition}\label{p:prolong}
Let $S$ be a symmetrized algebraic curvature tensor on the cone seen as a triple $(\alpha,\beta,\gamma)$
according to~\eqref{eq:glndec}. The following assertions (a) - (d) are equivalent:
\begin{enumerate}
\item $\hat\nabla S = 0$.
\item The pair $(\kappa,C)\;:=\;(\alpha|_M,\gamma|_M)$ satisfies~\eqref{eq:alg_curv_tensor_1}-\eqref{eq:alg_curv_tensor_2}, $\nabla\kappa = \beta|_M$ and $S =S^\kappa$.
\item There exists a symmetric Killing 2-tensor $\kappa$ such that $S$ is the associated
symmetrized algebraic curvature tensor $S^\kappa$ and the following equations hold:
\begin{align}
\label{eq:prolong_1_speziell}
\nabla_x \kappa\;\, &\;\;=\;\;x\lrcorner \kappa^1,\\
\label{eq:prolong_2_speziell}
\nabla_x\kappa^1 &\;\;=\;\;x\lrcorner \kappa^2 + x\lrcorner (g\owedge \kappa) - 2\, \kappa\owedge x^\sharp ,\\
\label{eq:prolong_3_speziell}
\nabla_x \kappa^2 &\;\;=\;\;- g\owedge x\lrcorner \kappa^1 - \kappa^1\owedge x^\sharp .
\end{align}
\item There exists a symmetric Killing 2-tensor $\kappa$ such that $S = S^\kappa$ and $S^\kappa|_M$ is $\hat\nabla$-parallel.
\end{enumerate}
\end{proposition}
\begin{proof}
For $(a)\Rightarrow (b)$: if $\hat\nabla_x S|_{(p,1)} = 0$, then~\eqref{eq:second_fundamental_form_of_bbP} implies
\begin{equation*}
\left ( \begin{array}{l}
\nabla_x \alpha \\
\nabla_x\beta\\
\nabla_x\gamma
\end{array} \right )\;\;=\;\;\left ( \begin{array}{lll}
x\lrcorner \beta & & \\
x\lrcorner \gamma & - 2\, \alpha\owedge x^\sharp &\\
&\quad - \beta \owedge x^\sharp &
\end{array} \right )
\end{equation*}
for all $p\in M$ and $x\in T_pM$. From this the first two assertions of~(b)
follow, i.e. $S = S^\kappa$ along $r = 1$. Further, $\hat\nabla S = 0$ yields
that $S$ has the same radial dependence as $S^\kappa$ according to Lemma~\ref{le:parallel_in_the_radial_direction}.
Therefore we conclude that $S = S^\kappa$.
For $(b)\Rightarrow (c)$: The first equation~\eqref{eq:prolong_1_speziell} of~(c) is a rephrasing of $\nabla\kappa = \beta|_M$.
Clearly, this implies that $\kappa$ is Killing. Further,
\[
C(x_1,x_2,x_3,x_4) \;\stackrel{\text{Example}~\ref{ex:Young_symmetrizer}~\text{(c)}}{=}\; \rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4\\
\cline{1-2}
\end{array}}{12pt}} C(x_1,x_2,x_3,x_4).
\]
Moreover, a straightforward consideration shows that
\begin{align*}
\rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4\\
\cline{1-2}
\end{array}}{12pt}} \rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
3 & 4 \\
\cline{1-2}
2 & \multicolumn{1}{c}{\;\;\;} \\
\cline{1-1}
\end{array}}{12pt}} \langle x_1, x_2\rangle \kappa(x_3,x_4)\;\;=\;\;\rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4\\
\cline{1-2}
\end{array}}{12pt}} \langle x_1, x_2\rangle \kappa(x_3,x_4).
\end{align*}
Therefore applying the Young projector $\rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4\\
\cline{1-2}
\end{array}}{12pt}}$ to~\eqref{eq:alg_curv_tensor_1} and recalling that the
numbers $h_{(2,1)}$ and $h_{(2,2)}$ are 3 and 12, respectively, the
definitions given in~\eqref{eq:product_of_a_one_form_and_a_symmetric_2-tensor},~\eqref{eq:def_Kulkarni_Nomizu_product}
and~\eqref{eq:kappa_prolong_2} imply that $C = C^\kappa$. Substituting~\eqref{eq:prolong_1_speziell} this
gives~\eqref{eq:prolong_2_speziell}. In the same way~\eqref{eq:alg_curv_tensor_2}
implies \eqref{eq:prolong_3_speziell}.
For $(c)\Rightarrow (d)$: Applying~\eqref{eq:second_fundamental_form_of_bbP}
to $S := S^\kappa$ we obtain that
\begin{equation}\label{eq:second_fundamental_form_of_bbP_im_spezialfall}
\hat\nabla_xS^\kappa|_{(p,1)}\;\;=\;\;\left ( \begin{array}{llll}
\nabla_x\kappa & - x\lrcorner \kappa^1& &\\
\nabla_x \kappa^1 & - x\lrcorner \kappa^2 & - x\lrcorner (g\owedge \kappa) & + 2\, \kappa\owedge x^\sharp\\
\nabla_{x}\kappa^2 & & + \ \ \ g\owedge \nabla_x\kappa & +\ \ \kappa^1\owedge x^\sharp
\end{array} \right ).
\end{equation}
Hence~\eqref{eq:prolong_1_speziell}-\eqref{eq:prolong_3_speziell} imply that $\hat\nabla_xS^\kappa|_{(p,1)} = 0$ for all $p\in M$ and $x\in T_pM$.
For $(d)\Rightarrow (a)$: we have to show that $\hat\nabla S^\kappa =0 $. Covariant constancy of $S^\kappa$
in the radial direction follows automatically from the structure of~\eqref{eq:zerlegung_von_kappa_dach} according to Lemma~\ref{le:parallel_in_the_radial_direction}.
Further, we have by assumption $\hat\nabla_x S^\kappa|_{(p,1)} = 0$ for all $p\in M$ and $x\in T_pM$. It remains to show that $\hat\nabla_x S^\kappa|_{(p,r)} = 0$ for all $r>0$.
According to Lemma~\ref{le:parallel_in_the_horizontal_direction}, the tensorial difference $\Delta := \hat\nabla - \nabla$ of the two connections $\hat\nabla$ and $\nabla$ satisfies
\begin{equation*}
x\lrcorner \Delta S^\kappa|_{(p,r)} \;\;=\;\;\left ( \begin{array}{lll}
- r^2\, x\lrcorner \nabla\kappa& & \\
- r^3\, x\lrcorner C^\kappa & + \ 2\, r^3\, \kappa \owedge x^\sharp&\\
&\quad \ \ r^4\, \kappa^1\owedge x^\sharp &
\end{array} \right )
\end{equation*}
Therefore, the scaling of the components of $x\lrcorner \Delta S^\kappa$ in the radial parameter $r$ is the same as for $S^\kappa$. Hence
$\nabla_xS^\kappa|_{(p,r)} = - x\lrcorner \Delta S^\kappa|_{(p,r)}$ for all $p\in M$ and $r>0$ as soon as this holds for $r = 1$. Therefore $\hat\nabla_x S^\kappa|_{(p,r)} = 0$ for all $r>0$ as soon as this is true for $r=1$.
\end{proof}
\section{Prolongation of the Killing Equation}\label{se:prolongation}
We will show through direct calculations that the triple $(\kappa,\kappa^1,\kappa^2)$ is closed
under componentwise covariant derivative for a Killing tensor $\kappa$. More precisely, we consider the following curvature expressions
\begin{align}\label{eq:def_F21}
F^1(\kappa;x_1,x_2,x_3,x_4) &\;\;:=\;\; \begin{array}{cc} \frac{1}{2}\; R_{x_1,x_2}\kappa(x_3,x_4)
- \frac{1}{4}\; \cyclic_{34}\cyclic_{12} R_{x_3,x_1}\kappa(x_2,x_4),
\end{array}\\
\label{eq:def_F22}
F^2(\kappa;x_1,\cdots,x_5) &\;\;:=\;\;\scaleto{\rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}}}{24pt} \left (
\begin{array}{c}
\nabla_{x_4}\big (R_{x_1,x_5}\cdot \kappa(x_2,x_3) + 2\, R_{x_2,x_5} \cdot \kappa(x_1,x_3)\big )\\
+ R_{x_1,x_5}\cdot \nabla_{x_4}\kappa(x_2,x_3) + R_{x_2,x_5} \cdot \nabla_{x_4}\kappa(x_1,x_3)
\end{array}\right )
\end{align}
(where $\cyclic$ denote the cyclic sum). Obviously these expressions depend tensorial on $\kappa$ and $(\kappa,\nabla \kappa)$,
respectively. Then we will see that the equations of the prolongation of the
Killing equation
\begin{align}\label{eq:prolong_1}
\nabla_{x_1}\kappa (x_2,x_3)& \;\;=\;\; \kappa^1(x_1,x_2,x_3),\\
\label{eq:prolong_2}
\nabla_{x_1}\kappa^1(x_2,x_3,x_4) & \;\;=\;\; \kappa^2(x_1,x_2,x_3,x_4) + F^1(\kappa;x_1,x_2,x_3,x_4),\\
\label{eq:prolong_3}
\nabla_{x_1}\kappa^2(x_2,x_3,x_4,x_5) & \;\;=\;\; F^2(\kappa;x_1,x_2,x_3,x_4,x_5)
\end{align}
hold for every Killing tensor $\kappa$. In particular,
the triple $(\kappa,\kappa^1,\kappa^2)$ is parallel with respect to a linear
connection which is immediately clear
from~\eqref{eq:prolong_1}-\eqref{eq:prolong_3}, the Killing connection.
We already know from Lemma~\ref{le:branching_rules} that this triple can be seen as an algebraic curvature tensor
on the direct sum $TM \oplus \R$. Because $\frac{1}{3}\binom{n+2}{2}\binom{n+1}{2}$ is the dimension of the linear space of algebraic curvature tensors in dimension $n+1$, we recover the well known result that this number
is an upper bound for the dimension of the space of symmetric Killing 2-tensors on any Riemannian manifold (cf.~\cite[Theorem~4.3]{Th}).
The prolongation of the Killing equation for abitrary symmetric tensors
using the adjoint Young symmetrizers and projectors is given
in~\cite{HTY}. However in comparison with~\eqref{eq:prolong_1}-\eqref{eq:prolong_3} that approach has the disadvantage of the appearance of an additional Young
symmetrizer (see~\cite[(7)-(9)]{HTY}) complicating the formulas unnecessarily.
The first equation of the prolongation~\eqref{eq:prolong_1} is a consequence
of Example~\ref{ex:Young_symmetrizer}~(b). The other two are derived as follows.
\subsection{The second equation of the prolongation}
We consider the curvature expression $F^{1}(\kappa)$ defined in~\eqref{eq:def_F21} and show that the second equation of the prolongation~\eqref{eq:prolong_2} of the Killing equation holds.
First note that $x_1\mapsto F^1(\kappa;x_1)$ defines a 1-form on $M$ with values in the vector bundle $\bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
3 & 4 \\
\cline{1-2}
2 & \multicolumn{1}{c}{} \\
\cline{1-1}
\end{array}}{12pt}} T^*M$
and that $F^1(\kappa)$ is a tensor in $\kappa$. Further, recall that the curvature endomorphism $R_{x,y}$ acts on $\kappa$ via algebraic
derivation $R_{x,y}\cdot \kappa$. Therefore, we could allow $R$
to be an arbitrary algebraic curvature tensor to obtain a tensorial bilinear expression $F^1(\kappa,R)$.
Then we have $F^1(\kappa,R) = \tilde F^1(R \cdot \kappa)$ for some tensorial expression in the curvature tensor applied to $\kappa$.
In oder to establish~\eqref{eq:def_F21}, we need the following intermediate result:
\bigskip
\begin{lemma}
Let $\kappa$ be Killing. Then
\begin{eqnarray}\label{eq:Killing_alternativ_*}\cyclic_{12}\nabla^2_{x_1,x_2}\kappa(x_3,x_4) &=& \cyclic_{34} \nabla^2_{x_3,x_4}\kappa(x_1,x_2)
+ 2\, \cyclic_{34}\cyclic_{12} R_{x_3,x_1}\kappa(x_2,x_4)
\end{eqnarray}.
\end{lemma}
\begin{proof} On the one hand,
\begin{eqnarray*}
\nabla^2_{x_1,x_2}\kappa(x_3,x_4) &=& - \nabla^2_{x_1,x_3}\kappa(x_2,x_4) - \nabla^2_{x_1,x_4}\kappa(x_2,x_3)\\
&=& \left \lbrace\begin{array}{c} - \big (\nabla^2_{x_3,x_1}\kappa(x_2,x_4) + \nabla^2_{x_4,x_1}\kappa(x_2,x_3) \big )\\
+ R_{x_3,x_1}\kappa(x_2,x_4) + R_{x_4,x_1}\kappa(x_2,x_3) \end{array} \right \rbrace\\
&=& \left \lbrace\begin{array}{cccc}
\nabla^2_{x_3,x_4}\kappa(x_1,x_2) &+ \nabla^2_{x_3,x_2}\kappa(x_1,x_4) &+ \nabla^2_{x_4,x_3}\kappa(x_1,x_2) &+ \nabla^2_{x_4,x_2}\kappa(x_1,x_3)\\
+ R_{x_3,x_1}\kappa(x_2,x_4) & &+ R_{x_4,x_1}\kappa(x_2,x_3) & \end{array} \right \rbrace.
\end{eqnarray*}
On the other hand,
\begin{eqnarray*}
\nabla^2_{x_1,x_2}\kappa(x_3,x_4) - R_{x_1,x_2}\kappa(x_3,x_4) &=& \nabla^2_{x_2,x_1}\kappa(x_3,x_4)\\
&\;\;=\;\;& \left .\begin{array}{c} - \nabla^2_{x_2,x_3}\kappa(x_1,x_4) - \nabla^2_{x_2,x_4}\kappa(x_1,x_3)
\end{array} \right .\\
&=& \left \lbrace \begin{array}{c} - \nabla^2_{x_3,x_2}\kappa(x_1,x_4) - \nabla^2_{x_4,x_2}\kappa(x_1,x_3)\\
+ R_{x_3,x_2}\kappa(x_1,x_4) + R_{x_4,x_2}\kappa(x_1,x_3)\end{array} \right \rbrace
\end{eqnarray*}
The result follows by adding the two expressions.
\end{proof}
\bigskip
\begin{proposition} \label{p:prolong_4}
If $\kappa$ is Killing, then~\eqref{eq:prolong_2} holds.
\end{proposition}
\begin{proof}
\begin{eqnarray*}
\rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\nabla^2_{x_1,x_2}\kappa(x_3,x_4) &\;\;=\;\;& \cyclic_{12}\; \rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
4 & 3 \\
\cline{1-2}
2 & \multicolumn{1}{c}{\;\;\;} \\
\cline{1-1}
\end{array}}{12pt}} \nabla^2_{x_1,x_2}\kappa(x_3,x_4) - \nabla^2_{x_3,x_2}\kappa(x_1,x_4)\\
& \stackrel{\eqref{eq:def_Killing}}{=} & \cyclic_{12} \rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
4 & 3 \\
\cline{1-2}
2 & \multicolumn{1}{c}{\;\;\;} \\
\cline{1-1}
\end{array}}{12pt}}\big (\nabla^2_{x_1,x_2}\kappa(x_3,x_4) + \nabla^2_{x_3,x_4}\kappa(x_1,x_2) + \nabla^2_{x_3,x_1}\kappa(x_2,x_4)\big )\\
& \stackrel{Example~\ref{ex:Young_symmetrizer}~(b)}{=} &\cyclic_{12}\big ( 3\, \nabla^2_{x_1,x_2}\kappa(x_3,x_4) + \rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
4 & 3 \\
\cline{1-2}
2 & \multicolumn{1}{c}{\;\;\;} \\
\cline{1-1}
\end{array}}{12pt}}\nabla^2_{x_3,x_4}\kappa(x_1,x_2)\big ).
\end{eqnarray*}
Further, \begin{eqnarray*}
\cyclic_{12}\; \rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
4 & 3 \\
\cline{1-2}
2 & \multicolumn{1}{c}{\;\;\;} \\
\cline{1-1}
\end{array}}{12pt}} \nabla^2_{x_3,x_4}\kappa(x_1,x_2)
&\;\;=\;\;& \cyclic_{12}\cyclic_{34} \big (\nabla^2_{x_3,x_4}\kappa(x_1,x_2) - \nabla^2_{x_3,x_2}\kappa(x_1,x_4) \big )\\
&\stackrel{\eqref{eq:def_Killing}}{=} & 3\; \cyclic_{34} \nabla^2_{x_3,x_4}\kappa(x_1,x_2)\\
& \stackrel{\eqref{eq:Killing_alternativ_*}}{=} & 3\; \cyclic_{12} \; \nabla^2_{x_1,x_2}\kappa(x_3,x_4) - 3\big (\cyclic_{12}\cyclic_{34} R_{x_3,x_1}\kappa(x_2,x_4)\big ).
\end{eqnarray*}
Therefore
\begin{eqnarray*}
\rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}} \nabla^2_{x_1,x_2}\kappa (x_3,x_4) - \frac{1}{2} \cyclic_{12}\nabla^2_{x_1,x_2}\kappa(x_3,x_4) & =&
\frac{1}{4}\; \cyclic_{12}\cyclic_{34} R_{x_3,x_1}\kappa(x_2,x_4).
\end{eqnarray*}
Using the Ricci-identity,~\eqref{eq:prolong_2} follows.
\end{proof}
From the previous we obtain a Weitzenböck formula. For this recall the natural action of the curvature tensor on a covariant symmetric 2-tensor $\kappa$,
\begin{equation}\label{eq:def_of_R*R}
q(R) \kappa(x_3,x_4)\;:=\;- \sum_{i=1}^nR_{x_3,e_i}\cdot \kappa(x_4,e_i) + R_{x_4,e_i}\cdot \kappa(x_3,e_i).
\end{equation}
This is the zeroth-order term appearing in the definition of the Lichnerowicz
Laplacian on symmetric 2-tensors
\begin{equation}\label{eq:Lichnerowicz_Laplacian}
\Delta \kappa\;:=\;\nabla^*\nabla \kappa + q(R)\kappa
\end{equation}
(cf.~\cite[1.143]{Be}). Further, if $\kappa$ is Killing, then the differential
$\d\, \trace \, \kappa$ of
its trace and the divergence $\delta\, \kappa$ are related by
\begin{equation}\label{eq:divergenz_versus_differential_der_spur}
\d\, \trace \, \kappa\;\;=\;\;2\,\delta\,\kappa.
\end{equation}
\bigskip
\begin{corollary}
Let $\kappa$ be a symmetric Killing 2-tensor. Then \begin{eqnarray}\label{eq:Wb}
\nabla^*\nabla \kappa &\;\;=\;\;& q(R)\kappa - \nabla^2\trace \, \kappa .
\end{eqnarray}
\end{corollary}
\begin{proof}
On the one hand, taking the trace on~\eqref{eq:prolong_2} with respect to $x_3,\, x_4$ and $x_1,\, x_2$ gives
\begin{eqnarray*}
\trace_{3,4} \big (\rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}} \nabla^2\kappa (x_1,x_2,x_3,x_4)\big ) &=& - \nabla^*\nabla \kappa(x_1,x_2) - \frac{1}{4}\;\sum_{i=1}^n
\left \lbrace\begin{array}{c}
2\, \underbrace{R_{e_i,e_i}\kappa(x_3,x_4)}_{=0}\\
+ R_{x_1,e_i}\kappa(e_2,x_i) + R_{x_1,e_i}\kappa(e_i,x_2)\\ + R_{x_2,e_i}\kappa(e_i,x_1) + R_{x_2,e_i}\kappa(e_i,x_1) \end{array} \right \rbrace\\
&=& - \nabla^*\nabla \kappa(x_1,x_2) + \frac{1}{2} q(R)\kappa(x_1,x_2)\;,
\end{eqnarray*}
and
\begin{eqnarray*}
\trace_{1,2} \big (\rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\nabla^2\kappa (x_1,x_2,x_3,x_4)\big ) &=& \nabla^2_{x_3,x_4 }\trace \, \kappa
+ \frac{1}{4}\;\left \lbrace\begin{array}{c}
2\, \underbrace{R_{x_3,x_4}\kappa(e_i,e_i)}_{=0}\\
+ R_{x_3,e_i}\kappa(x_4,e_i) + R_{x_4,e_i}\kappa(x_3,e_i)\\ + R_{x_3,e_i}\kappa(x_4,e_i) +
R_{x_4,e_i}\kappa(x_3,e_i)\end{array} \right \rbrace\\
&=& \nabla^2_{x_3,x_4 }\trace \, \kappa - \frac{1}{2} q(R)\kappa(x_3,x_4).
\end{eqnarray*}
On the other hand, $\trace_{3,4} \big ( \rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\nabla^2\kappa \big )(u,v) = \trace_{1,2} \big ( \rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
1 & 2 \\
\cline{1-2}
3 & 4 \\
\cline{1-2}
\end{array}}{12pt}}\nabla^2\kappa \big )(u,v)$ by the symmetries of a curvature tensor.
\end{proof}
\subsection{The third equation of the prolongation}
We will now consider the term $F^{2}(\kappa)$ defined in~\eqref{eq:def_F22} and show that~\eqref{eq:prolong_3} holds.
By definition, $x_1\mapsto F^2(\kappa;x_1)$ is a 1-form on $M$ with values in
the vector bundle $\bbS_{\scaleto{\begin{array}{|c|c|c|}
\cline{1-2}
2 & 4 \\
\cline{1-2}
3 & 5 \\
\cline{1-2}
\end{array}}{12pt}}T^*M$. The expression $F^2(\kappa)$ depends
linearly on $\kappa$. If we allow $R$ to be an arbitrary algebraic curvature
tensor, then we obtain a bilinear expression $F^2(\kappa,R)$. Via this
interpretation we have $F^2(\kappa,R) = \tilde
F^2(R\cdot\kappa,R\cdot\nabla\kappa)$ for some expression $\tilde F$ which depends linearly on the pair
$(R\cdot\kappa,R\cdot\nabla\kappa)$.
It is also clear from~\eqref{eq:def_F22} that $F^2(\kappa)$ is a (linear) tensor in
$(\kappa,\nabla\kappa)$ and that $F^2(\kappa,R)$ is a bilinear tensor
in the pairs $(\kappa,\nabla\kappa)$ and $(R,\nabla R)$. More precisely:
\bigskip
\begin{lemma}\label{le:ausmultiplizieren}
Let $M$ be Riemannian manifold $M$ with Levi Civita connection $\nabla$. Let
$\kappa$ and $R$ be a symmetric Killing 2-tensor and an algebraic curvature tensor on $M$.
We have
\begin{equation}\label{eq:F22_ausmultipliziert}
F^2(\kappa,R;x_1,\ldots,x_5) \;\;=\;\; \scaleto{\rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}}}{24pt}\left (
\begin{array}{c}
\nabla_{x_4} R_{x_1,x_5}\cdot\kappa(x_2,x_3) + 2\, \nabla_{x_4}R_{x_2,x_5}\cdot \kappa(x_1,x_3)\big )\\
+ 2\, R_{x_1,x_5}\cdot\nabla_{x_4}\kappa(x_2,x_3) + 3\, R_{x_2,x_5}\cdot\nabla_{x_4}\kappa(x_1,x_3)\\
- \nabla_{R_{x_1,x_5} x_4}\cdot\kappa(x_2,x_3) - 2 \nabla_{R_{x_2,x_5}x_4}\cdot\kappa(x_1,x_3)
\end{array}\right ).
\end{equation}
\end{lemma}
\begin{proof}
The product rule for the covariant
derivative of an endomorphism-valued tensor field $A$ acting on $\alpha$ yields
\begin{eqnarray}
\nabla_x \big (A\cdot \alpha(y_2,y_3) \big ) &\;\;=\;\;& \begin{array}{c}\nabla_x A \cdot \alpha(y_2,y_3)
+ A\cdot \nabla_x \alpha(y_2,y_3) - \nabla_{A\,x} \alpha(y_2,y_3)\end{array},
\end{eqnarray}
since the endomorphism should in effect not act on the covariant derivative slot. Applying this to~\eqref{eq:def_F21} we obtain~\eqref{eq:F22_ausmultipliziert}.
\end{proof}
We will prove the third equation of the prolongation of the Killing equation:
\bigskip
\begin{proposition}\label{p:prolong_3}
If $\kappa$ is Killing, then~\eqref{eq:prolong_3} holds with $F^2(\kappa)$ defined via~\eqref{eq:def_F22}.
\end{proposition}
\begin{proof}
Using the Killing equation~\eqref{eq:def_Killing} together with the fact that
\begin{equation}\label{eq:klaro}
\rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}} \big ( \begin{array}{c}
\nabla^{3)}_{x_2,x_4,x_5}\kappa(x_1,x_3) + \nabla^{3)}_{x_4,x_2,x_5}\kappa(x_1,x_3)
\end{array}\big )\;\;=\;\;0,
\end{equation}
we conclude hat
\begin{eqnarray*}
\scaleto{\rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}}}{32pt} \left ( \begin{array}{c}
+ \frac{1}{2}R_{x_2,x_4} \nabla_{x_5}\kappa(x_1,x_3)\\ + \nabla_{x_4}\big (R_{x_2,x_5} \kappa (x_1,x_3)\big )\\
+ \frac{1}{2} \begin{array}{c} \nabla_{x_4}\big (R_{x_3,x_5} \kappa(x_1,x_2)\big ) \end{array}
\end{array}\right ) &=& \scaleto{\rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}}}{24pt} \left ( \begin{array}{l}
\big (\nabla^{3)}_{x_2,x_4,x_5} + \nabla^{3)}_{x_4,x_2,x_5} -
\nabla^{3)}_{x_4,x_5,x_2}\big )\kappa(x_1,x_3)\\
- \nabla^{3)}_{x_4,x_5,x_3}\kappa(x_1,x_2)\\
\end{array}\right )\\
&\stackrel{\eqref{eq:klaro}}{=}& \rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}}\left ( \begin{array}{l}
- \nabla^{3)}_{x_4,x_5,x_2}\kappa(x_1,x_3) - \nabla^{3)}_{x_4,x_5,x_3}\kappa(x_1,x_2) \\
\end{array}\right )\\
&\stackrel{\eqref{eq:def_Killing}}{=}& \rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}} \nabla^{3)}_{x_4,x_5,x_1}\kappa(x_2,x_3)
\end{eqnarray*}
Thus the Ricci identity
\begin{equation*}
\nabla^{3)}_{x_1,x_4,x_5}\kappa(x_2,x_3) - \nabla^{3)}_{x_4,x_5,x_1}\kappa(x_2,x_3)\;\;=\;\;R_{x_1,x_4}\nabla_{x_5}\kappa(x_2,x_3) + \nabla_{x_4} \big (R_{x_1,x_5} \kappa (x_2,x_3)\big )
\end{equation*}
shows that
\begin{eqnarray*}
\nabla_{x_1}\rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}}\nabla^2_{x_4,x_5}\kappa(x_2,x_3) &\;\;=\;\;&\scaleto{\rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}}}{32pt}\left (
\begin{array}{c}
R_{x_1,x_4}\nabla_{x_5}\kappa(x_2,x_3)
+ \nabla_{x_4} \big ( R_{x_1,x_5} \kappa(x_2,x_3)\big )\\
+ \nabla_{x_4}\big (R_{x_2,x_5} \kappa(x_1,x_3) \big)\\
+ \frac{1}{2}\left ( \begin{array}{c}\nabla_{x_4}\big (R_{x_3,x_5} \kappa(x_1,x_2)\big )
+ R_{x_2,x_4} \nabla_{x_5}\kappa(x_1,x_3)\end{array} \right )
\end{array}\right )
\end{eqnarray*}
The skew-symmetry $R_{u,v} = - R_{v,u}$ implies that
\begin{eqnarray*}
\frac{1}{2}\rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}}\nabla_{x_4} \big (R_{x_3,x_5} \kappa(x_1,x_2)\big ) &=& \rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}}\nabla_{x_4}\big (R_{x_2,x_5}\kappa(x_1,x_3)\big ),
\end{eqnarray*}
\begin{eqnarray*}
\frac{1}{2}\rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}} R_{x_2,x_4} \nabla_{x_5} \kappa(x_1,x_3) &=& \rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}} R_{x_2,x_5} \nabla_{x_4} \kappa(x_1,x_3)
\end{eqnarray*}
and the symmetry of $\kappa$ gives
\begin{eqnarray*}
\rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}} R_{x_1,x_4}\nabla_{x_5} \kappa(x_2,x_3) &=& \rmS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
4 & 5 \\
\cline{1-2}
\end{array}}{12pt}}R_{x_1,x_5}\nabla_{x_4}\kappa(x_2,x_3).
\end{eqnarray*}
This establishes that~\eqref{eq:prolong_3} holds with $F^2(\kappa)$ defined
by~\eqref{eq:def_F22}.
\end{proof}
\subsection{Prolongation of symmetric Killing 2-tensors in spaces of constant
curvature}
Let $M$ be a (pseudo) Riemannian manifold of constant sectional
curvature. Since the following arguments are purely local, we can assume that
$M$ is a (generalised) sphere as considered in
Example~\ref{ex:constant_curvature}. Then the cone is a flat (pseudo)
Euclidean space, hence Proposition~\ref{p:prolong} shows that here the
dimension of the space of Killing tensors is the maximal possible. In particular,~\eqref{eq:prolong_1_speziell}-\eqref{eq:prolong_3_speziell} describe
the prolongation of symmetric Killing 2-tensors on $M$. This enables us
to calculate $F^1(\kappa,R_1)$ and $F^2(\kappa,R_1)$ with $R_1(x,y)\;:=\;-
x\wedge y$ on every pseudo Riemannian manifold $M$ as follows.
\bigskip
\begin{lemma}\label{le:curvature_terms_for_S1}
We have
\begin{align}\label{eq:F21_speziell}
x\lrcorner F^1(R_1,\kappa)\;\;=\;\;& x\lrcorner (g\owedge \kappa) \ - 2\, \kappa\owedge x^\sharp ,\\
\label{eq:F22_speziell}
x\lrcorner F^2(R_1,\kappa)\;\;=\;\;& \ \ \ - g\owedge x\lrcorner \kappa^1 - \kappa^1 \owedge x^\sharp .
\end{align}
for every symmetric 2-tensor $\kappa$ on $M$.
\end{lemma}
\begin{proof}
We have already remarked that $F^1$ is a linear tensor in $R$ and that $F^2$ depends
tensorial linear on the 1-jet of $R$. Therefore, and because $\nabla R_1 = 0$ on every
pseudo Riemannian manifold, it suffices to show that~\eqref{eq:F21_speziell} and~\eqref{eq:F22_speziell} hold on a generalised sphere $M$. Further, comparing the equations of the prolongation of the
Killing equation~\eqref{eq:prolong_2}-\eqref{eq:prolong_3} with~\eqref{eq:prolong_1_speziell}-\eqref{eq:prolong_3_speziell},
we see that~\eqref{eq:F21_speziell},\eqref{eq:F22_speziell} hold for every Killing tensor. Moreover, since both $F^1$ and $F^2$
depend tensorial linear on the 1-jet of $\kappa$ and because moreover every 1-jet of a symmetric 2-tensor on $M$ can be extended
to a Killing tensor according to Proposition~\ref{p:prolong}, these identities
automatically hold for all symmetric 2-tensors $\kappa$.
\end{proof}
For a proof of the following in the Riemannian case see~\cite[Lemme~1.2]{Ga},
the indefinite case does not need an extra argument.
\bigskip
\begin{lemma}\label{le:krümmungstensor_des_kegels}
The curvature tensor of $\hat M$ is a horizontal tensor. When seen as a
$(1,3)$-tensor it does not depend on the radial component $r$ and is given by
\begin{equation}
\label{eq:hat_R_1}
\hat R_{x,y}z\;\;=\;\;R_{x,y}z + x\wedge y (z)
\end{equation}
for all $p\in M$ and $x,y,z\in T_pM$.
\end{lemma}
We obtain the following extension of Proposition~\ref{p:prolong}:
\bigskip
\begin{proposition}\label{p:main}
The following conditions are equivalent:
\begin{enumerate}
\item
The algebraic curvature tensor $S^\kappa$ associated with $\kappa$ is parallel on the cone $\hat M$;
\item $\kappa$ is Killing and the curvature conditions~\eqref{eq:curv_cond_1} and~\eqref{eq:curv_cond_2} hold.
\end{enumerate}
\end{proposition}
\begin{proof}
For $(a) \Rightarrow (b)$: since $\hat \nabla S^\kappa = 0$, the curvature
endomorphism $\hat R_{x,y}$ annihilates $S$ for all $p\in M$ and $x,y\in T_pM$. Because $\hat R$ is horizontal, we conclude that $\hat R$ annihilates
the components of~\eqref{eq:zerlegung_von_kappa_dach}. In particular, $\hat R_{x,y}\cdot \kappa = 0$ and $\hat R_{x,y} \cdot \nabla\kappa = 0$ for all $p\in M$ and $x,y\in T_pM$.
Using~\eqref{eq:hat_R_1}, we see that both~\eqref{eq:curv_cond_1} and~\eqref{eq:curv_cond_2} hold.
For $(b)\Rightarrow (a)$: by assumption,~\eqref{eq:curv_cond_1}
and~\eqref{eq:curv_cond_2} hold. The expression $F^1(\kappa,R)$ is explicitly described in~\eqref{eq:def_F21}. From this one sees that it depends in fact only on the action $R_{x,y}\cdot \kappa$ of
the given algebraic curvature tensor $R$ on $\kappa$. Therefore~\eqref{eq:curv_cond_1} implies that
$F^1(R,\kappa) = F^1(R_1,\kappa)$. Similarly, the explicit
description~\eqref{eq:def_F22} of $F^2(\kappa,R)$ implies that the latter expression depends only on the actions
$R_{x,y}\cdot\kappa$ and $R_{x,y}\cdot \nabla \kappa$ of the algebraic curvature tensor $R$ on $\kappa$ and $\nabla\kappa$.
We obtain from~\eqref{eq:curv_cond_1} and~\eqref{eq:curv_cond_2} that $F^2(\kappa,R) = F^2(\kappa,R_1)$. Then we conclude from Lemma~\ref{le:curvature_terms_for_S1} that $F^1(\kappa,R)$ and $F^2(\kappa,R)$ are given by~\eqref{eq:F21_speziell} and~\eqref{eq:F22_speziell}. Substituting this into~\eqref{eq:prolong_1}-\eqref{eq:prolong_3} we see that~\eqref{eq:prolong_1_speziell}-\eqref{eq:prolong_3_speziell} hold. Hence the result follows from Proposition~\ref{p:prolong}.
\end{proof}
\section{Proof of Theorem~\ref{th:main_1}}
\label{se:implications}
Because of Propositions~\ref{p:prolong} and~\ref{p:main}, it remains to prove the following proposition:
\bigskip
\begin{proposition}\label{p:get_rid}
Suppose that $\dim(M) \geq 2$.
\begin{enumerate}
\item
If the conditions~\eqref{eq:alg_curv_tensor_1}-\eqref{eq:alg_curv_tensor_2} hold for a pair $(\kappa,C)$, then $\kappa$ is
Killing. Further, $\kappa = \kappa^C$ where
\begin{equation}\label{eq:def_kappaC}
\kappa^C \;:=\; - \frac{1}{2}\widetilde \Ric^C + \frac{\widetilde \scal^C}{2(n - 1)}g
\end{equation}
and
\begin{equation}\label{eq:def_modified_Ricci_trace_und_tilde_scalC}
\widetilde \Ric^C \;\;:=\;\; \Ric^C + \frac{1}{4}\nabla^2\scal^C\ \ \ \text{and}\ \ \
\widetilde \scal^C \;\;:=\;\; \scal^C - \frac{1}{4} \nabla^*\nabla\scal^C.
\end{equation}
\item Suppose that $M$ is compact with negative definite metric tensor. If ~\eqref{eq:alg_curv_tensor_1}-\eqref{eq:alg_curv_tensor_2} hold for
a pair $(\kappa,C)$, then there exists a constant $c$ such that $\kappa = c\, g$.
\end{enumerate}
\end{proposition}
In the previous proposition and hence also in Theorem~\ref{th:main_1} the condition $\dim(M) > 1$ can not be neglected:
\bigskip
\begin{remark}\label{re:main_3}
On the real line $\R$ the pair $(\kappa,C)$ where $\kappa$ is a quadratic form with affine linear
coefficients and $C\;:=\;0$ matches~\eqref{eq:alg_curv_tensor_1}-\eqref{eq:alg_curv_tensor_2} of Theorem~\ref{th:main_1}.
But $\kappa$ is neither Killing nor $\kappa = \kappa^C$ holds.
\end{remark}
For the proof of Proposition~\ref{p:get_rid}, we make the following general assumption for the rest of this section:
\begin{center}
\flushleft
\begin{small}
\bf Let a pair $(\kappa,C)$ be given such that~\eqref{eq:alg_curv_tensor_1} and~\eqref{eq:alg_curv_tensor_2} hold for some constant $c$ different from zero.
\end{small}
\end{center}
The proof of the previous proposition requires several lemmas.
\bigskip
\begin{lemma}\label{le:Bianchis_cyclic_sum_is_parallel}
The cyclic sum $\cyclic_{123} \nabla_{y_1} \kappa (y_2,y_3)$ is parallel, i.e.
\begin{equation}\label{eq:Bianchis_cyclic_sum_is_parallel}
\cyclic_{123} \nabla^2_{x,y_1} \kappa (y_2,y_3)\;\;=\;\;0
\end{equation}
for all $x,y_1,y_2,y_3\in T_pM$ and $p\in M$.
\end{lemma}
\begin{proof}
We have
\[
\cyclic_{123} C(x,y_1,y_2,y_3)\;\;=\;\;0
\]
for every symmetrized algebraic curvature tensor. Further, $\cyclic_{123}\beta(x_1,x_2,x_3) = 0$ for every $\beta\in \bbS_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
1 \\
\cline{1-1}
\end{array}}{12pt}}T_pM^*$. Therefore,
\begin{align*}
\cyclic_{123}\, \rmP_{\scaleto{\begin{array}{|c|c|}
\cline{1-2}
2 & 3 \\
\cline{1-2}
1 \\
\cline{1-1}
\end{array}}{12pt}}\,\langle x, y_1\rangle \kappa (y_2,y_3)\;\;=\;\;0.
\end{align*} Thus~\eqref{eq:Bianchis_cyclic_sum_is_parallel} follows from~\eqref{eq:alg_curv_tensor_1}.
\end{proof}
\bigskip
\begin{lemma}
The algebraic Ricci tensor of $C$ and its scalar trace are given by
\begin{align}\label{eq:Ricci_spur_von_C_1}
\Ric^C(x,y) &\;\;=\;\;\nabla^2_{x,y}\trace \, \kappa + 2(\langle x,y\rangle \trace \, \kappa - \kappa(x,y))\\
\label{eq:Ricci_spur_von_C_2}
&\;\;=\;\;- \nabla^*\nabla \kappa(x,y) + 2(n-1) \kappa(x,y),\\
\label{eq:skalare_spur_von_C}
\scal^C &\;\;=\;\;-\nabla^*\nabla\trace \, \kappa + 2(n-1)\trace \, \kappa.
\end{align}
\end{lemma}
\begin{proof}
For~\eqref{eq:Ricci_spur_von_C_1}, take a suitable trace of~\eqref{eq:alg_curv_tensor_1}. From this~\eqref{eq:skalare_spur_von_C} follows immediately.
\end{proof}
Let $\delta\kappa$ and $\trace \, \kappa$ denote the divergence and trace of $\kappa$.
\bigskip
\begin{lemma}
We have
\begin{align}\label{eq:kovariante_Ableitung_des_Ricci_tensors_von_C}
- 3 \nabla_{x} \Ric^C(y_1,y_2) &\;\;=\;\;2 \cyclic_{12}\left \lbrace \begin{array}{c}\big (\langle x,y_1\rangle \d \, \trace \, \kappa(y_2) - \nabla_{y_1} \kappa(y_2,x)\\
+\langle x,y_1\rangle \delta\kappa(y_2) \end{array} \right \rbrace + 4 \nabla_x \kappa(y_1,y_2),\\
\label{eq:differential_der_skalaren_spur_von_C}
- 3\d \scal^C &\;\;=\;\;8\big (\d\, \trace \, \kappa + \delta\kappa \big ).
\end{align}
\end{lemma}
\begin{proof}
Recalling the
definition of the product $\owedge$ from~\eqref{eq:noch_ein_produkt}, the
negative of r.h.s. of~\eqref{eq:alg_curv_tensor_2} is explicitly given as a
Young symmetrized expression
\begin{align*}
& \langle x,x_1\rangle \nabla_{x_2} \kappa(x_3,x_4) - \langle x,x_3\rangle \nabla_{x_2} \kappa(x_1,x_4) - \langle x,x_1\rangle \nabla_{x_4} \kappa(x_3,x_2) + \langle x,x_3\rangle \nabla_{x_4} \kappa(x_1,x_2)\\
& \langle x,x_2\rangle \nabla_{x_1} \kappa(x_3,x_4) - \langle x,x_3\rangle \nabla_{x_1} \kappa(x_2,x_4) - \langle x,x_2\rangle \nabla_{x_4} \kappa(x_1,x_3) + \langle x,x_3\rangle \nabla_{x_4} \kappa(x_1,x_2)\\
& \langle x,x_1\rangle \nabla_{x_2} \kappa(x_3,x_4) - \langle x,x_4\rangle \nabla_{x_2} \kappa(x_1,x_3) - \langle x,x_1\rangle \nabla_{x_3} \kappa(x_4,x_2) + \langle x,x_4\rangle \nabla_{x_3} \kappa(x_1,x_2)\\
& \langle x,x_2\rangle \nabla_{x_1} \kappa(x_3,x_4) - \langle x,x_4\rangle \nabla_{x_1} \kappa(x_2,x_3) - \langle x,x_2\rangle \nabla_{x_3} \kappa(x_1,x_4) + \langle x,x_4\rangle \nabla_{x_3} \kappa(x_1,x_2)
\end{align*}
Taking the trace with respect to $x_3$ and $x_4$ yields
\begin{align*}
& \langle x,x_1\rangle \nabla_{x_2} \kappa(e_i,e_i) - \langle x,e_i\rangle \nabla_{x_2} \kappa(x_1,e_i) - \langle x,x_1\rangle \nabla_{e_i} \kappa(e_i,x_2) + \langle x,e_i\rangle \nabla_{e_i} \kappa(x_1,x_2)\\
& \langle x,x_2\rangle \nabla_{x_1} \kappa(e_i,e_i) - \langle x,e_i\rangle \nabla_{x_1} \kappa(x_2,e_i) - \langle x,x_2\rangle \nabla_{e_i} \kappa(x_1,e_i) + \langle x,e_i\rangle \nabla_{e_i} \kappa(x_1,x_2)\\
& \langle x,x_1\rangle \nabla_{x_2} \kappa(e_i,e_i) - \langle x,e_i\rangle \nabla_{x_2} \kappa(x_1,e_i) - \langle x,x_1\rangle \nabla_{e_i} \kappa(e_i,x_2) + \langle x,e_i\rangle \nabla_{e_i} \kappa(x_1,x_2)\\
& \langle x,x_2\rangle \nabla_{x_1} \kappa(e_i,e_i) - \langle x,e_i\rangle \nabla_{x_1} \kappa(x_2,e_i) - \langle x,x_2\rangle \nabla_{e_i} \kappa(x_1,e_i) + \langle x,e_i\rangle \nabla_{e_i} \kappa(x_1,x_2)\\
\end{align*}
Thus~\eqref{eq:kovariante_Ableitung_des_Ricci_tensors_von_C} follows from~\eqref{eq:alg_curv_tensor_2}. Taking the trace of this yields~\eqref{eq:differential_der_skalaren_spur_von_C}.
\end{proof}
\bigskip
\begin{lemma}
We have
\begin{align}\label{eq:erste_vorstufe_zur_Bianchi_identity}
& 2\,\cyclic_{123} \nabla_{x_1} \kappa (x_2,x_3)\;\;=\;\;\left \lbrace \begin{array}{c} 3\nabla^3_{x_1,x_2,x_3}\trace \, \kappa + 6\, \d \, \trace \, \kappa(x_1) \langle x_2,x_3\rangle\\
+ 2\,\cyclic_{2,3}\langle x_1,x_2\rangle \delta\kappa(x_3)\\
+ 2\,\cyclic_{2,3}\langle x_1,x_2 \rangle\, \d \, \trace \, \kappa(x_3) \end{array} \right \rbrace.
\end{align}
\end{lemma}
\begin{proof}
Solving~\eqref{eq:kovariante_Ableitung_des_Ricci_tensors_von_C} for the cyclic sum of $\nabla \kappa$ yields:
\begin{align*}
& 2\,\cyclic_{123} \nabla_{x_1} \kappa (x_2,x_3)\;\;=\;\;\left \lbrace \begin{array}{c} 3\, \nabla_{x_1} \Ric^C(x_2,x_3) + 6\, \nabla_{x_1} \kappa(x_2,x_3)\\
+ 2\,\cyclic_{23}\langle x_1,x_2\rangle \delta \kappa(x_3)\\
+ 2\,\cyclic_{23}\langle x_1,x_2 \rangle\, \d \, \trace \, \kappa(x_3) \end{array} \right \rbrace.
\end{align*}
Substituting now the formula~\eqref{eq:Ricci_spur_von_C_1} for the algebraic Ricci tensor of $C$
gives~\eqref{eq:erste_vorstufe_zur_Bianchi_identity}.
\end{proof}
\bigskip
\begin{lemma}\label{le:M_is_irreducible}
If $\kappa$ is not a constant multiple of the metric tensor, then $M$ is irreducible.
\end{lemma}
\begin{proof}
Assume by contradiction that $M$ is not irreducible. Then $M$ is the product $M_1 \times M_2$ of two Riemannian manifolds of dimensions at least one.
Let $(p,q)\in M_1 \times M_2$ and $(x,y)\in T_pM_1\oplus T_qM_2$.
Since $x$ and $y$ are tangent to different factors, the covariant derivatives commute $\nabla^2_{y,x}\kappa = \nabla^2_{x,y}\kappa$. Using~\eqref{eq:Bianchis_cyclic_sum_is_parallel}, we thus have
\begin{align*}
0 &\stackrel{\eqref{eq:Bianchis_cyclic_sum_is_parallel}}{=} \nabla^2_{y,x}\kappa(x,x)\;\;=\;\; \nabla^2_{x,y}\kappa(x,x)\stackrel{\eqref{eq:Bianchis_cyclic_sum_is_parallel}}{=} - 2\,\nabla^2_{x,x}\kappa(x,y)
\end{align*}
for all $x\in T_pM_1$ and $y\in T_qM_2$. Hence, the exchange rule $C(x,x,x,y) = 0$ implies from~\eqref{eq:alg_curv_tensor_1} that
\begin{align*}
0 \;\;=\;\; - \langle x, x\rangle \kappa (x,y) + \underbrace{\langle x, y\rangle \kappa (x,x)}_{=0}
\end{align*}
We conclude that $ \kappa (x,y) = 0$, i.e. $\kappa = \kappa_1 + \kappa_2$ where $\kappa_1$ and $\kappa_2$ are the restrictions of
$\kappa$ to $T_pM_1\times T_pM_1$ and $T_qM_2\times T_qM_2$, respectively. Further, the symmetry $C(x,y,x,y) = C(y,x,x,y)$
and the commuting of the covariant derivatives
$\nabla^2_{x,y}\kappa(x,y) = \nabla^2_{y,x}\kappa(x,y)$ yields
\begin{equation*}
\langle x, x\rangle \kappa (y,y)\;\;=\;\;\langle y, y\rangle \kappa (x,x)
\end {equation*}
Hence $\kappa(x,x) = \kappa(y,y)$ for all unit vectors $x\in T_pM_1$ and $y\in T_qM_2$.
Thus there exists a function $f \colon M_1\times M_2 \to \R_+$ such that
$\kappa_1(x,x) = f(p,q) = \kappa_2(y,y)$ for all unit vectors $x$ and $y$, i.e. $\kappa = f\langle\ ,\ \rangle$. Inserting this into~\eqref{eq:Bianchis_cyclic_sum_is_parallel} implies that $(n + 2)\d f$ is parallel, hence
$\nabla \d f = 0$. Substituting this into~\eqref{eq:erste_vorstufe_zur_Bianchi_identity} yields
\begin{align*}
& \cyclic_{123}\, \d_{x_1}f \langle x_2,x_3 \rangle\;\;=\;\;3\,n\, \langle x_2,x_3\rangle \d_{x_1}f + (n-1) \cyclic_{23}\langle x_1,x_2 \rangle\, \d_{x_3}f.
\end{align*}
Evaluating this equation with $x_1 = x_2 = x_3$ implies that $(n - 1) \d f = 0$. Therefore if $n > 1$ then $\kappa$ is a constant multiple of the metric tensor. We conclude that $M$ is irreducible.
\end{proof}
\bigskip
\begin{corollary}\label{co:contracted_Bianchi_identity}
If $\dim(M) \geq 2$, then the contraction~\eqref{eq:divergenz_versus_differential_der_spur} of the Killing equation holds.
\end{corollary}
\begin{proof}
We can assume that $\kappa$ is not a constant multiple of the metric tensor,
since otherwise~\eqref{eq:divergenz_versus_differential_der_spur} is obvious. Then, on
the one hand, $M$ is irreducible by the previous lemma and hence the holonomy
group acts fixed point free on tangent vectors, because $\dim(M)\geq 2$. On
the other hand, since the contraction
of~\eqref{eq:Bianchis_cyclic_sum_is_parallel} implies that the 1-form
$\lambda\;:=\;\d \, \trace \, \kappa - 2 \delta \kappa$ is parallel, the latter is fixed under the holonomy group at each point. We conclude that $\lambda = 0$.
\end{proof}
\bigskip
\begin{corollary}
If $\dim(M) \geq 2$, then we have
\begin{align}\label{eq:zweite_vorstufe_zur_Bianchi_identity}
2 \cyclic_{123} \nabla_{y_2} \kappa (y_1,y_3) &\;\;=\;\;\left \lbrace \begin{array}{c} 3\nabla^3_{y_1,y_2,y_3}\trace \, \kappa + 6\, \d \, \trace \, \kappa(y_1) \langle y_2,y_3\rangle + 3\,\cyclic_{23}\langle y_1,y_2 \rangle\, \d \, \trace \, \kappa(y_3) \end{array} \right \rbrace,\\
\label{eq:vergleich_der_spur_von_kappa_mit_der_skalaren_kruemmung_von_C}
\d\scal^C &\;\;=\;\;-4\, \d\, \trace \, \kappa.
\end{align}
\end{corollary}
\begin{proof} This follows by substituting~\eqref{eq:divergenz_versus_differential_der_spur} into~\eqref{eq:erste_vorstufe_zur_Bianchi_identity} and~\eqref{eq:differential_der_skalaren_spur_von_C}, respectively.
\end{proof}
Let $R_1$ denote the algebraic curvature tensor defined by
\begin{equation}
R_1(x,y,z)\;:=\;- \langle x , z \rangle y + \langle y , z \rangle x
\end{equation}
for all $x,y,z\in T_pM$. If $g$ is positive definite, then this is the curvature tensor of a round
sphere of radius one.
Recall that the $1$-nullity of a pseudo Riemannian manifold is the subspace of $T_pM$ defined by
\begin{equation}\label{eq:curvature_constancy}
\scrC_1(T_pM)\;:=\;\{x\in T_pM| R(x,y,z)= R_1(x,y,z)\}.
\end{equation}
Hence a vector $x$ belongs to $\scrC_1(T_pM)$ if and only if the sectional curvature of any two-plane in $T_pM$ containing $x$ is equal to one (see~\cite{Gray1},~\cite{MoSe1}).
For example, a Sasakian vector field $\xi$ belongs at each point to the
$1$-nullity distribution.
In general, $\scrC_1(TM)\;:=\;\bigcup_{p\in M}\scrC_1(T_pM)$ is not a subbundle of $TM$. However, for a symmetric space $\nabla R = 0$ implies that $\scrC_1(TM)$ is invariant under parallel translation and hence a parallel vector subbundle of $TM$.
\bigskip
\begin{lemma}\label{le:curvature_constancy}
If $\dim(M) \geq 2$, then the gradient of $\trace \, \kappa$ belongs to the 1-nullity $\scrC_1(TM)$ everywhere.
\end{lemma}
\begin{proof}
Let $f := \trace \, \kappa$. Since l.h.s. of~\eqref{eq:zweite_vorstufe_zur_Bianchi_identity} is completely
symmetric in $u_1,u_2,u_3$, the same is true for the right hand side of that
equation. Hence
\begin{align*}
& 3\, \big (\nabla^3_{y_1,y_2,y_3}f + 2\, \langle y_2,y_3\rangle \d f(y_1)
+ \cyclic_{23}\langle y_1,y_2\rangle \d f(y_3) \big )\\
&\;\;=\;\; \cyclic_{123} \big (\nabla^3_{y_1,y_2,y_3}f + 2\, \langle y_2,y_3\rangle \d f(y_1)
+ \cyclic_{23}\langle y_1,y_2\rangle \d f(y_3) \big )\\
&\;\;=\;\;\cyclic_{123}\big (\nabla^3_{y_1,y_2,y_3}f + 4\,\langle y_1,y_2\rangle \d f(y_3)\big )\\
&\;\;=\;\;\nabla^3_{y_1,y_2,y_3}f + \nabla^3_{y_3,y_1,y_2}f + \nabla^3_{y_2,y_3,y_1}f + 4\,\cyclic_{123}\,\langle y_1,y_2\rangle \d f(y_3).
\end{align*}
Using the symmetry of the Hessian $\nabla\d f$, we have
\begin{align*}
\nabla^3_{y_3,y_1,y_2}f &\;\;=\;\;\nabla^2_{y_1,y_3}\d f(y_2) + R_{y_3,y_1} \d f(y_2)\;\;=\;\;\nabla^3_{y_1,y_2,y_3}f + R_{y_3,y_1} \d f(y_2),\\
\nabla^3_{y_2,y_3,y_1}f &\;\;=\;\; \nabla^3_{y_2,y_1,y_3}f\;\;=\;\;\nabla^2_{y_1,y_2,y_3}f + R_{y_2,y_1} \d f(y_3).
\end{align*}
Further,
\begin{equation*}
6\, \langle y_2,y_3\rangle \d f(y_1)
+ 3\,\cyclic_{23}\langle y_1,y_2\rangle \d f(y_3) - 4\,\cyclic_{123}\,\langle y_1,y_2\rangle \d f(y_3)\;\;=\;\;2\, \langle y_2,y_3\rangle
\d f(y_1) - \cyclic_{23}\langle y_1,y_2\rangle \d f(y_3)
\end{equation*}
We conclude that
\begin{align*}
& 2\, \langle y_2,y_3\rangle
\d f(y_1) - \cyclic_{23}\langle y_1,y_2\rangle \d f(y_3)
\;\;=\;\;R_{y_3,y_1} \d f(y_2) + R_{y_2,y_1} \d f(y_3) .
\end{align*}
Setting $y_1 := x$ and $y_2 := y_3 := y$ we obtain that
\begin{equation*}
2\, \d f(R(x,y,y))\;\;=\;\;2\, \langle y,y\rangle
\d f(x) - 2\, \langle x,y\rangle \d f(y).
\end{equation*}
This shows that the gradient of $\trace \, \kappa$ belongs to $\scrC_1(TM)$.
\end{proof}
\begin{proof} [Proof of Proposition~\ref{p:get_rid}~(a)]
Suppose that~\eqref{eq:alg_curv_tensor_1}-\eqref{eq:alg_curv_tensor_2} hold for a pair $(\kappa,C)$ and that $\dim(M)\geq 2$.
We show that the cyclic sum $\cyclic_{xyz} \nabla_x\kappa(y,z)$ vanishes at
each point $p\in M$. We distinguish two cases:
If there exists an open neighbourhood around $p$ where $\trace \, \kappa$ is constant, then $\kappa$ is Killing on this neighbourhood according
to~\eqref{eq:zweite_vorstufe_zur_Bianchi_identity}.
Suppose there exists a point $p\in M$ such that $\d \, \trace \, \kappa|_p\neq 0$.
In particular, $\kappa$ is not a constant multiple of the metric tensor on any open neighbourhood of $p$.
Therefore each neighbourhood of $p$ in $M$ is irreducible according to Lemma~\ref{le:M_is_irreducible}. Hence we can assume that $M$ is simply connected and irreducible. Further, recall that the cyclic sum $\cyclic_{xyz}\nabla_x\kappa(y,z)$ is a parallel symmetric 3-tensor according to Lemma~\ref{le:Bianchis_cyclic_sum_is_parallel}. Therefore if the holonomy group of $M$ at $p$ acts transitively on the unit sphere of $T_pM$,
then the corresponding homogeneous polynomial function $f(x) := \nabla_x\kappa(x,x)$ with $x\in T_pM$ is constant on vectors of equal length.
Since the degree of $f$ is odd, we see that $f(x) = f(-x) = - f(x)$ for all $x\in T_pM$ . Therefore $\cyclic_{xyz} \nabla_x\kappa(y,z)|_p = 0$.
If, by contradiction, we assume that the holonomy group acts non-transitively, then $M$ is locally symmetric, $\nabla R = 0$,
according to Bergers classification of holonomy groups~\cite[5.21]{Ba}. Hence
the $1$-nullity~\eqref{eq:curvature_constancy} is a parallel subbundle of
$TM$. Since $\scrC_1(T_pM)\neq \{0\}$ because of
Lemma~\ref{le:curvature_constancy}, it is non-trivial and thus $\scrC_1(TM) = TM$ by the irreducibility of $M$. This implies that $M$ has constant
sectional curvature, in particular the holonomy group is $\SO(n)$ which acts
transitively on the unit sphere of $T_pM$. But this is contrary to our assumptions.
So far we have shown that $\cyclic_{xyz}\nabla_x\kappa(y,z)|_p = 0$ at
all points $p$ of $M$ where $\d \, \trace \, \kappa|_p \neq 0$ or $\trace \, \kappa$ is constant on an open neighbourhood of $p$. Obviously the set of these points is dense in $M$. Since moreover the set points where the cyclic sum $\cyclic_{xyz}\nabla_x\kappa(y,z)$ vanishes is closed, we conclude that $\kappa$ is Killing.
We finish the proof of~(a) of Proposition~\ref{p:get_rid} by showing that automatically $\kappa = \kappa^C$: we use~\eqref{eq:Ricci_spur_von_C_1} and~\eqref{eq:vergleich_der_spur_von_kappa_mit_der_skalaren_kruemmung_von_C} to obtain that
\begin{align*}
\widetilde \Ric^C(x,y) \stackrel{\eqref{eq:def_modified_Ricci_trace_und_tilde_scalC}}{=} \Ric^C(x,y) + \frac{1}{4} \nabla^2_{x,y}\scal^C
\stackrel{\eqref{eq:Ricci_spur_von_C_1},\eqref{eq:vergleich_der_spur_von_kappa_mit_der_skalaren_kruemmung_von_C}}{=} 2 \big (\langle x,y\rangle \trace \, \kappa - \kappa(x,y)\big ).
\end{align*}
Hence
\begin{equation*}
\widetilde\scal^C\;\;=\;\;2(n - 1)\trace \, \kappa.
\end{equation*}
Inserting the previous into~\eqref{eq:def_kappaC} shows that $\kappa = \kappa^C$. This finishes the proof of Part~(a) of Proposition~\ref{p:get_rid}.
\end{proof}
\begin{proof} [Proof of Proposition~\ref{p:get_rid}~(b)]
Let a pair $(\kappa,C)$ with~\eqref{eq:alg_curv_tensor_1}-\eqref{eq:alg_curv_tensor_2} on a compact manifold $M$
with negative definite metric tensor $g$ be given. We claim that $\kappa$ is a constant multiple of $g$:
If $M$ is a 1-dimensional sphere, then the space of algebraic curvature tensors is trivial.
Hence~\eqref{eq:prolong_2_speziell} and~\eqref{eq:prolong_3_speziell} together
show that the third covariant derivative of the pullback $\pi^*\kappa$ of
$\kappa$ under the canonical covering $\pi:\R\to M$ vanishes.
This implies polynomial growth of $\pi^*\kappa$. But $\pi^*\kappa$ is periodic, hence the previous forces that $\pi^*\kappa$ is constant,
i.e. $\kappa$ is trivial. Hence we can assume that $\dim(M)\geq 2$. Using that $\kappa$ is Killing, we conclude
from~\eqref{eq:zweite_vorstufe_zur_Bianchi_identity} that $f\;:=\;\trace \, \kappa$ satisfies Gallots equation~\eqref{eq:Gallots_Gleichung}. Taking in this equation the trace with respect to $y_1$ and $y_2$, we obtain for the rough Laplacian
\begin{equation}\label{eq:Laplacian_of_f}
\nabla^*\nabla \ df \;\;=\;\; (n + 3) \d f.
\end{equation}
Since on a compact manifold with negative definite metric the rough Laplacian $\nabla^*\nabla$ is a strictly negative operator, we
conclude that $\d\, \trace \, \kappa = 0$, i.e. $\trace \, \kappa$ is a constant $c$.
Using~\eqref{eq:Ricci_spur_von_C_1} and~\eqref{eq:Ricci_spur_von_C_2} we thus see that
\begin{equation*}
\nabla^*\nabla \big (n\, \kappa - c\, g\big )\;\;=\;\;2\, \, (n\, \kappa
- c\,g\big ).
\end{equation*} Using again the compactness of $M$,
we conclude that $\kappa = \frac{c}{n}\, g$.
This shows that $\kappa$ is a constant multiple of the metric tensor.
\end{proof}
\section{Proof of Theorem~\ref{th:main_2}}
\label{se:sasaki}
For the proof of Theorem~\ref{th:main_2}, it remains to analyze the space of parallel algebraic curvature tensors on the cone over a Riemannian manifold.
This is purely a question of holonomy. Gallot showed that the cone over a complete Riemannian manifold is irreducible unless the universal covering of $M$ is isometric to a round sphere. Moreover, the cone $\hat M^{n+1}$ can never be Einstein unless it is Ricci-flat, since the curvature tensor $\hat R$ is purely horizontal according to~\eqref{eq:hat_R_1}. In the following we can assume that $M$ is simply connected and complete.
The remaining possible holonomy algebras are according to Bergers classification~\cite[5.21]{Ba}: $\so(n + 1)$, $\u(m)$ or $\su(m)$
for $n + 1 = 2 m$ and $m \ge 2$, $\sp(m)$ for $n + 1 = 4 m$ and $m \ge 2$ or $\mathfrak{g}_2$ for $n + 1= 7$
or $\spin_7$ for $n + 1= 8$.
Holonomy algebras $\su(m)$, $\u(m)$ and $\sp(m)$ of the cone
correspond to (Einstein)-Sasakian and 3-Sasakian manifolds, respectively. For a Kähler
manifold with Kähler form $\omega$, the Cartan product $\omega\odot\omega$
clearly is a non-trivial parallel algebraic curvature tensor and similar every
Hyperkähler manifold with Hyperkähler structure $\{\omega_1,\omega_2,\omega_3\}$
admits six linearly independent non-trivial parallel algebraic curvature
tensors $\omega_i \odot \omega_j$.
By the holonomy principle every parallel section
of the vector bundle of algebraic curvature tensors
on $\hat M$ corresponds to an element of the vector space of algebraic
curvature tensors on some tangent space that is fixed by the holonomy
algebra. For the proof of Theorem~\ref{th:main_2}, we thus have to show that
\begin{itemize}
\item the actions of $\frakg_2$, $\spin_7$ and $\so(n+1)$ on algebraic curvature tensors in dimensions $7$, $8$ and $n+1$ each only have a one-dimensional trivial component,
\item the trivial component of $\su(m)$ and $\u(m)$ acting on $\scrC(\R^{2m})$ is 2-dimensional,
\item and the trivial component of $\sp(m)$ on $\scrC(\R^{4m})$ is seven-dimensional.
\end{itemize}
By irreducibility of $\hat M$, every symmetric 2-tensor is a multiple of the metric, i.e. the space of traceless symmetric 2-tensors $\Sym^2(V^*)_0$ remains irreducible over the holonomy algebra. The next two lemmas will show that the space of algebraic curvature tensors $\scrC(T)$ remains irreducible over $\spin_7$ or $\lieG2$.
This finishes the proof of the classification of Riemannian manifolds matching the statement of Theorem~\ref{th:main_2}.
The following result can be found in~\cite[(4.7)]{KQ}:
\bigskip
\begin{lemma} \label{lem:sasaki:g2_trivial}
Let $V$ be an irreducible $\so(7)$ representation that contains a
$\lieG2$-trivial subspace, then there is a $k \in \I{N}$ such that $V$ is
the $k$-fold Cartan product of the spin representation $\Delta$. That is $V = \Delta^{\odot k}$.
In particular, the $\so(7)$-representation space space of Weyl tensors $\scrW(V)$ yields no further trivial component when restricted to $\lieG2$.
\end{lemma}
The result for $\spin_7 \subset \so(8)$ is similar. Let $\Delta^{\pm}$ be
the spin representations of $\so(8)$. Take a non-vanishing element $s \in
\Delta^+$ and consider its isotropy algebra $\lie{h} = \set{X\suchthat X s =
0} \subset \so(8)$. Then $\lie{h} = \spin_7$ is isomorphic to the holonomy
algebra of a manifold $M$ with $\Spin_7$ holonomy \cite[Chapter IV,
§10]{LM}.
Note that changing orientation on $M$ interchanges the spin representations
$\Delta^+$ and $\Delta^-$ \cite[Theorem~3.6.1]{Jo}.
\bigskip
\begin{lemma} \label{lem:sasaki:spin7_trivial}
Let $V$ be an irreducible $\so(8)$ representation that contains a
$\spin_7$-trivial subspace, then there is a $k \in \I{N}$ such that $V$ is
the $k$-fold Cartan product of the spin representation $\Delta^+$. That is
$V = {\Delta^+}^{\odot k}$.
\end{lemma}
\begin{proof}
The fundamental representations of $\so(8)$ are the standard representation
$\hat\kappa \colon\ \so(8) \to \End(\I{R}^8)$, the adjoint representation
$\ad \colon\ \so(8) \to \End \parens{\Exterior^2 \I{R}^8}$ and the spinor
representations $\hat\kappa^{\pm} \colon \so(8) \to \End(\I{R}^8)$.
Furthermore the triality automorphism $\sigma$ of $\so(8)$ is an outer automorphism
of order three that interchanges the representations $\hat\kappa$ and
$\hat\kappa^\pm$, that is $\hat\kappa \circ \sigma = \hat\kappa^-$ and $\hat\kappa^- \circ \sigma =
\hat\kappa^+$, while $\ad \circ \sigma$ is equivalent to $\ad$~\cite[Chapter 1 §8]{LM},~\cite[§20.3]{FH}.
The image of the standard embedding of $\so(7) \subset \so(8)$ is the
isotropy algebra
\begin{equation}
\so(7)\;\;=\;\;\set{X \in \so(8) \suchthat \hat\kappa(X) e_1 = 0}
\end{equation}
while representatives of the other two conjugacy classes of $\spin_7$ are
given as
\begin{equation}
\spin^\pm_7
\;\;=\;\;\set{X \in \so(8) \suchthat \hat\kappa^{\pm}(X) e_1 = 0}.
\end{equation}
As described above, taking $\hat\kappa^+$ yields the holonomy algebra of a manifold
with holonomy $\Spin_7$.
Because $\hat\kappa = \hat\kappa^+ \circ \sigma$ it immediately follows that $\so(7) =
\sigma^{-1}(\spin^+_7)$. By that, the branching rules of an $\so(8)$-representation
$\tau \colon\ \so(8) \to \End(V)$ to the subalgebra
$\spin^+_7$ are the same as the branching rules of $\tau \circ \sigma$ to
$\so(7)$. The latter rules are stated in.
It follows from~\cite[(25.34),(25.35)]{FH} that the highest weight
of an $\so(8)$-irreducible representation $V$ containing a non-trivial
$\so(7)$-trivial subspace is necessarily of the form $(k,0,0,0)$ so $V$ is the
$k$-fold Cartan product $T^{\odot k}$ of the standard representation.
On the other hand, the automorphism $\sigma$ permutes the fundamental weights
and hence maps simple roots to simple roots. By \cite[Theorem~5.5
(a)]{Kn} and the formula given in \cite[Theorem~5.5 (d)]{Kn},
$\sigma$ maps every highest weight of a representation to a highest weight.
Hence, the automorphism commutes with the Cartan product.
This proves that every $\so(8)$-irreducible representation $V$ containing
a $\spin^+_7$-trivial subspace has to be ${\Delta^+}^{\odot k}$.
\end{proof}
It is left to calculate the multiplicity of the $H$-trivial
subrepresentation in $\scrC(\hat T) $ for the remaining holonomy
algebras and compare it to the number of parallel sections obtained by taking
Cartan products of $S_\circ$ and the Kähler forms $\omega_I$, $\omega_J$ and
$\omega_K$. This can be done by the branching rules given in \cite{Ho}.
For this we have to consider the complexification
$\scrC(\hat T) \otimes \I{C} = \scrC(\hat T\otimes \C)$ which is a $\GL(n+1, \I{C})$ representation.
If the holonomy $H$ of the cone $\hat M$ is $\U(m)$ or
$\SU(m)$ with $n+1 = 2m$, then the complexifications of $H$-representations
are $\GL(m,\I{C})$- or $\SL(m,\I{C})$-representations where
$\GL(m, \I{C}) \subset \Sp(2m, \I{C}) \subset \GL(2m, \I{C})$. If $H = \Sp(m)$
with $4m=n+1$, then complexifications of $\Sp(m)$-representations are
$\Sp(2m,\I{C})$-representations where $\Sp(2m,\I{C}) \subset \GL(2m,\I{C})
\subset \Sp(4m,\I{C}) \subset \GL(4m,\I{C})$. The Branching rules for $\GL(m, \I{C})\subset\Sp(2m,\I{C})$ and
$\Sp(2m, \I{C}) \subset \GL(2m, \I{C})$ can be found in~\cite{Ho}. From these one easily deduces the next two lemmas:
\bigskip
\begin{lemma} \label{lem:sasaki:um}
For $m \ge 2$, consider the embeddings $\GL(m,\I{C}) \subset \Sp(2m,\I{C})
\subset \GL(2m,\I{C})$. There are two copies of the trivial $\GL(m,\I{C})$-representation contained in
$\scrC(\hat T\otimes \C)$. Furthermore, if $m>2$, any other
non-trivial one-dimensional $\GL(m,\I{C})$-representation is not
contained in $\scrC(\hat T)$ so that this branching rule is
the same for $\SL(m,\I{C})$.
\end{lemma}
\bigskip
\begin{lemma} \label{lem:sasaki:spm}
For $m \ge 2$ and $\Sp(2m,\I{C}) \subset \GL(2m,\I{C}) \subset \Sp(4m,\I{C})
\subset \GL(4m,\I{C})$ there are seven copies of the trivial $\Sp(2m,\I{C})$-representation contained in $\scrC(\hat T\otimes \C)$.
\end{lemma}
Moreover, the assumption $m>2$ in the second part of Lemma~\ref{lem:sasaki:um} is not really restrictive. Namely every 3-dimensional Einstein-Sasakian manifold is 3-Sasakian in accordance with $\SU(2) \simeq \Sp(1)$. Further, every 3-dimensional 3-Sasakian manifold is of constant sectional curvature one according to~\cite[Proposition~1.1.2~(iii)]{BG} and hence locally isometric to the 3-dimensional standard sphere. This finishes the proof of Theorem~\ref{th:main_2}.
\section{Proof of Corollary~\ref{co:Gallot}}
\label{se:Gallot}
Let us first recall Gallots original proof of Corollary~\ref{co:Gallot} (cf.~\cite[Corollary~3.3]{Ga}). For every function $f$ on $M$ we
define the function $F(p,r)\;:=\;r^2 f(p)$ and the symmetric two-tensor $q :=
\frac{1}{2}\hat\nabla^2 F$ on the cone $\hat M$. Then we can recover the original function via
\begin{equation}\label{eq:kanonische_konstruktion_von_funktionen}
f(p)\;\;=\;\;q(\partial_r|_{(p,1)},\partial_r|_{(p,1)}).
\end{equation}
Further, we know that~\eqref{eq:Gallots_Gleichung} holds if and only if $\hat
\nabla q = 0$. As mentioned already before, he also showed that the cone over
$M$ is irreducible unless the universal covering of $M$ is a Euclidean sphere $\rmS^n$. If the cone is irreducible, then
its holonomy group acts irreducible on the tangent spaces of $\hat M$. Therefore, $q$ is a constant multiple of the metric tensor of $\hat M$ according to Schurs Lemma, i.e. $f$ is constant.
To put Gallots result in order with the theory developed in this article, we consider the symmetric 2-tensor $\kappa$ defined by~\eqref{eq:Gallots_kappa},
the symmetrized algebraic curvature tensor $S := q \owedge \hat g$ on $\hat M$ and its pullback to $M$
\begin{equation}
C\;\;:=\;\;f\, \langle\ ,\ \rangle \owedge \langle\ ,\ \rangle + \frac{1}{2}
\nabla^2 f \owedge \langle\ ,\ \rangle.
\end{equation}
If~\eqref{eq:Gallots_Gleichung} holds, then $q$ is $\hat\nabla$-parallel by Gallots original calculations and hence $S$ shares this property.
Further, it is straightforward that $\kappa$ defined by~\eqref{eq:Gallots_kappa} is related to $S$ via~\eqref{eq:kanonische_konstruktion_von_killingtensoren} or that
the pair $(\kappa,C)$ is a solution to~\eqref{eq:alg_curv_tensor_1}-\eqref{eq:alg_curv_tensor_2}.
Either way, it follows from Theorem~\ref{th:main_2} that $M$ is a Sasakian
manifold or a sphere. To show that $M$ is a sphere, according to Corollary~\ref{co:main_2} it suffices to show that the trace of $\kappa$ is not constant unless $f$ is: we have
\begin{align*}
&\trace \, \kappa\;\;=\;\;n\, f - \frac{1}{4} \nabla^*\nabla f,\\
&\d\, \trace \, \kappa\;\;=\;\;n\, \d f - \frac{1}{4} \d\, \nabla^*\nabla f.
\end{align*}
Evaluating~\eqref{eq:Gallots_Gleichung} we see that
\[
\d\, \trace \, \kappa\;\;=\;\;\frac{3\, n+1}{2} \d f.
\]
Hence, if $\d f\neq 0$, then we know from Corollary~\ref{co:main_2} that $M$ is a
sphere.\qed
\paragraph{}ACKNOWLEDGMENTS. We would like to thank Uwe Semmelmann for some helpful critic comments on the form of this article. We would also like to express our gratitude
to Gregor Weingart for sharing his ideas about Killing tensors.
\ \ \
\bibliographystyle{amsplain}
| 12,506
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Margaret Dean
Margaret Dean – President MD West Area Team
Margaret is the current President of the Area Team. Prior to becoming President, Margaret held positions on the Area Team and on the Cumberland, MD Aglow Community Lighthouse. Being a part of Aglow has been a life changer for Margaret especially since completing and facilitating the Game Changer/Life Changer series taught by Graham Cooke for Aglow. Understanding her identity in Christ along with grasping the truth that HE lives in her has changed everything from her marriage of 45 years to husband, Ralph, and daughter, Angie, to recognizing that everyone she touches is her “neighbor.” Jesus said, “That we are to love God with all our heart, mind, soul, and strength and our “neighbor” as our self.“ Who is our neighbor? The person who is next to Margaret at any given moment. Margaret believes that the Game/Life Changer series differently changes the way we play the game of life!
| 197,399
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Understanding the financial treatment of casual harvest workers
Chris Campbell
Tax Senior Manager
20 October 2017
If you are a farming employer, you’ve very likely experienced challenges in the financial treatment of ‘casual’ employees, an area of particular relevance at this time of the year when many short term workers are employed to assist on farms during the harvest season.
As an employer, most Farmers will be aware of the concept of Real Time Information, which requires details of all payments made to employees to be sent to HM Revenue & Customs (HMRC) on or before the date of payment to the employee. This is normally sent electronically by a Full Payment Submission on each payroll run, along with other information such as hours worked and pay frequency.
Special concessions
Farm businesses can benefit from a special concession available for casual workers who are not family members and work outdoors harvesting perishable crops, or as casual beaters for a shoot. This is a very specific concession which recognises the particular needs of the farming sector and the practicalities of declaring payments for what, in some cases, could be a significant number of short term workers. However, if the farming concession does not apply, then the bottom line is that the worker needs to go on the payroll and the Real Time Information submissions need to be made as normal.
Under the concession for farming businesses, it is unnecessary to deduct Income Tax under Pay As You Earn (PAYE) from a harvest casual or casual beater in either of two potential scenarios. One is where the worker is employed for one day or less (provided they are paid off afterwards and have no contract for further employment with the business); and the other is where the worker is employed for less than two weeks, has not worked for the business since the start of the tax year and is paid at a rate of pay below the PAYE threshold (currently £221 per week).
National Insurance need not be deducted if the worker earns below the Primary Threshold for employees and the Secondary Threshold for employers (both currently £157 per week).
It is important to note that, even if no deductions are required, the farming business must still retain a record of the payments made including the worker’s full name, address, date of birth, National Insurance number, gender and amounts paid. This will need to be held on file in case of HM Revenue & Customs enquiry.
Regardless of whether the tax concession applies to your workers, other regulatory factors such as ensuring that the worker is entitled to work in the UK, and compliance with minimum wage legislation must be adhered to. When checking if an employee has the right to work in the UK, the employer should examine the original documentation and retain a copy on file, as the penalty charged for employing workers not entitled to work in the UK can be significant, with potential adverse effects on the cashflow of your business.
The rules around the tax requirements for short term workers are complex, and it is important to seek professional advice to avoid unnecessary penalties and fines for non-compliance. Our agricultural and farming experts are here to help, and should you need further advice, please visit.
| 217,957
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Comprehensive information about Kuvempu including biographical information, facts, literary works, and more. Kuppali Venkatappa. This educational Kuvempu resource has information about the author's life, works, quotations, articles and essays, and more.
Poems are below...
Articles about Kuvempu or articles that mention Kuvempu.
Here are a few random quotes by Kuvempu.
See also:
All Kuvempu Quotes
| 266,801
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TITLE: Relation between $k[x,y]/(y^3 - x^5)$ and its integral closure.
QUESTION [1 upvotes]: I am struggling with the following problem:
Let $R = k[x,y]/(y^3 - x^5)$ and $S$ be its normalization. What is the dimension of the $k$-vector space $S/R$?
I started computing $S$, but most likely there are some dimension theory theorem that can help me to compute the dimension of $S/R$ without actually computing $S$. Thanks in advance!
REPLY [4 votes]: We can embed $R$ into $k[t]$ by means of $y\mapsto t^5$ and $x\mapsto t^3$.
We identify $R$ with $k[t^3,t^5]$ which is the span of $1$, $t^3$, $t^5$, $t^6$, $t^8$, $t^9$, $t^{10},\ldots$. The normalisation is $k[t]$, the integral closure of $R$. The missing monomials are $t$, $t^2$, $t^4$, $t^7$
so the dimension you seek is $4$.
| 162,615
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TITLE: Could you explain the meaning of the sentence
QUESTION [0 upvotes]: My native language is not English and can't realize what does the first sentence on the picture sentence mean. In particular, "each angle and the $x$ axis are components". Could you explain it in other words?
Thank you!
REPLY [1 votes]: They are talking about connected components here. The set $X$ consists of the $x$-axis, along with infinitely many "angles". An angle in this case means one horizontal ray, meeting a line segment from $(0,1)$ at an angle. Here is a hastily-drawn picture of one such "angle":
As you can see from the drawing in your book, $X$ consists of infinitely many such angles, all joined together at $(0,1)$.
If you remove $(0,1)$ from this space, then each of those angles will be connected omponents of the space that is left.
| 94,713
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One of our Facilitators cooks up the food on the public BBQ and warmly greets everyone, casually introducing people to each other.
The attendance can be anything from 4 people to 15 on a busy day.
We sometimes kick a football around or play bowls or just sit / stand in the sun or shade and chat in the open park space.
Conversation can range from our jobs, families, hobbies or interests and no-one has to say anything if they wish, just enjoy the food and sun.
The background in age is from 25 right up to 60, from all walks of life, race and confidence. Some men even bring their children or partners or dogs (I love the idea that one day I will bring my son again when he can understand why we are there and it be considered "normal").
It is not a men’s group, we do not stand or sit in a circle and it is not an AA-meeting eg. I would not get up and say “Hi I’m Terry and I have depression, anxiety and PTSD.”
If it feels natural some will share their stories and when Terry attends he often tells his in casual conversation. Generally this can lead to a hugely trusting space.
Some incredibly deep, real and powerful but relaxed discussions have come out of our Meetups and friendships formed. As well as many laughs too.
Finally anything discussed is completely confidential and non-clinical.
If you are keen to come along click here for a map of our Meetups currently running across Australia.
Hi Cathy,
Lovely to hear from you.
Check out the Facebook page for the upcoming Meetup BBQs and exact details of where we meet.
Hope your son got to come along.
Cheers,
Terry
Hi my name is Cathy and I have a son who is 36 years old who has no friends and suffers very bad with depression and anxiety he would like to come to the BBq at castle hill tomorrow were abouts do you meet up in the park please
Hey Prakesh!
Great to hear from you – we hope told more regional VIC Meetup BBQs soon. Feel free to come along to Echuca if you get the chance.
Cheers!
Just heard about Mr Perfect. I live in Maldon, VIC postcode 3463. Perhaps the closest is Echuca. There is a Men’s Shed in Maldon. The guys are quite friendly but I didn’t think they would be into a deep conversation.
Hi Ryan!
I have sent you an email with more details and yes, all donations are tax deductible (we have DGR status).
Cheers!
| 281,775
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Dow and S&P remain near all-time highs even as many stocks plunge.
The Rally Will End When Share Buybacks Slow Down
Higher interest rates could reduce share buybacks, eliminating a key source of support for current stock prices.
Volatile Day on Wall Street
Volatility has been rising and could portend a bigger selloff ahead.
| 303,325
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The Surprising Connection Between Weight Loss and Knee Pain
Being able to maintain a healthy weight brings with it a lot of benefits, such as a decreased risk of heart disease, high blood pressure, type 2 diabetes, and more. And, since knee pain is one of the most common side-effects of carrying extra pounds around, losing even a modest amount of weight can significantly decrease the amount of stress that is placed on your knees.
How is that possible?
Well, a huge portion of it has to do with the fact that your knees bear the brunt of your body weight. The amount of additional pressure that carrying around extra weight places on our knees is pretty darn significant. Those extra pounds also increase the likelihood of developing osteoarthritis – in fact, women who are overweight are four times more likely (and men, five times more likely) to develop osteoarthritis than their healthier counterparts.
However, if you’re experiencing chronic knee pain that is due to weighing more than you should, don’t despair – there are things you can do to ease the burden on your knees and reduce your risk of developing osteoarthritis. So, what are these “things”?
Well, the best thing you can do is shed some of those extra pounds because every pound of weight lost correlates to a four pound reduction of load to the knee joint. That means that by losing just 10 pounds you can reduce the pounds per step that your knees have to support by 40 pounds – over the course of a mile, that 10 pound weight loss translates into each of your knees being subjected to 48,000 fewer pounds of pressure. So, even though 10 pounds may not sound like a lot, it’s pretty clear that the benefits begin to add up quite quickly – after all, the more you decrease the wear and tear that your knees are enduring, the lower your risk of developing osteoarthritis in the future. On top of all of that, by working to lower your weight by 5% to 10%, you can also lower your risk of having a stroke or developing heart disease.
If you’re struggling with knee pain – regardless of the cause – please don’t hesitate to give us a call to schedule an appointment. Dr. Hurlbut – along with our awesome staff of physical therapists – work with our patients to develop plans of care to help address your particular areas of concern.
sources: healthline.com; mayoclinic.org
| 126,380
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\begin{document}
\title[]{Density and Fractal Property of the Class of Oriented Trees}
\author{Jan Hubi\v cka}
\address{Department of Applied Mathematics (KAM), Charles University,
Ma\-lo\-stransk\'e n\'am\v est\'i 25, Praha 1, Czech Republic}
\curraddr{}
\email{hubicka@kam.mff.cuni.cz}
\urladdr{}
\author{Jaroslav Ne\v set\v ril}
\address{Computer Science Institute of Charles University (IUUK),
Charles University, Ma\-lo\-stransk\'e n\'am\v est\'i 25, Praha 1, Czech
Republic}
\curraddr{}
\email{nesetril@iuuk.mff.cuni.cz}
\urladdr{}
\author{Pablo Oviedo}
\address{Departament de Matem\` atiques, Universitat Polit\` ecnica de Catalunya, Barcelona, Spain}
\curraddr{}
\email{pablo.oviedo@estudiant.upc.edu}
\urladdr{}
\thanks{The first author was supported by project 18-13685Y of the Czech Science Foundation (GA\v CR) and by Charles University project Progres Q48. The third author was supported by the Spanish Research Agency under project MTM2017-82166-P.}
\keywords{}
\subjclass{05C05, 05C38, 05C20, 06A06}{}
\begin{abstract}
We show a density theorem for the class of finite proper trees ordered by the homomorphism order, where a proper tree is an oriented tree which is not homomorphic to a path. We also show that every interval of proper trees, in addition to being dense, is in fact universal. We end by considering the fractal property in the class of all finite digraphs.
This complements the characterization of finite dualities of finite digraphs.
\end{abstract}
\maketitle
\section{Introduction}
\label{section1}
In this note we consider finite directed graphs (or digraphs) and countable partial orders. A homomorphism between two digraphs $f:G_1\rightarrow G_2$ is an arc preserving mapping from $V(G_1)$ to $V(G_2)$. If such homomorphism exists we write $G_1\leq G_2$. The relation $\leq$, called the homomorphism order, defines a quasiorder on the class of all digraphs which, by considering equivalence classes, becomes a partial order. A core of a digraph is its minimal homomorphic equivalent subgraph.
In the past three decades the richness of the homomorphism order of graphs and digraphs has been extensively studied \cite{llibre}. In 1982, Welzl showed that undirected graphs, with one exception, are dense \cite{origdens}. Later in 1996, Ne\v set\v ril and Zhu characterized the gaps and showed a density theorem for the class of finite oriented paths \cite{pathhomomorphism}. We contribute to this research by showing a density theorem for the class of oriented trees. We say that an oriented tree is \emph{proper} if its core is not a path.
\begin{theorem}
\label{1}
Let $T_1$ and $T_2$ be two finite oriented trees satisfying $T_1<T_2$. If $T_2$ is a proper tree, then there exists a tree $T$ such that $T_1<T<T_2$.
\end{theorem}
This result was claimed by Miroslav Treml around 2005, but never published. Our proof of Theorem \ref{1} is new and simple, and leads to further consequences. In particular, we can show the following strengthening. Let us say that a partial order is universal if it contains every countable partial order as a suborder.
\begin{theorem}
\label{2}
Let $T_1$ and $T_2$ be two finite oriented trees satisfying $T_1<T_2$. If $T_2$ is a proper tree, then the interval $[T_1,T_2]$ is universal.
\end{theorem}
Recently, it has been shown that every interval in the homomorphism order of finite undirected graphs is either universal or a gap \cite{fractal}. As consequence of Theorem \ref{2}, this property, called fractal property, seems to be present in other classes of digraphs. In fact, we have shown the following result related to the class of finite digraphs.
\begin{theorem}
\label{3}
Let $G$ and $H$ be two finite digraphs satisfying $G<H$, where the core of $H$ is connected and contains a cycle. Then the interval $[G,H]$ is universal.
\end{theorem}
The proof of Theorem \ref{3} will appear in the full version of this note.
\section{Preliminaries}
\label{section2}
We follow the notation used in Hell and Ne\v set\v ril's book \cite{llibre}.
A \emph{digraph} $G$ is an ordered pair of sets $(V,A)$ where $V=V(G)$ is a set of elements called \emph{vertices} and $A=A(G)$ is a binary irreflexive relation on $V$. The elements $(u,v)$, denoted $uv$, of $A(G)$ are called \emph{arcs}.
A \emph{path} is a digraph consisting in a sequence of different vertices $\{v_0,\dots,v_k\}$ together with a sequence of different arcs $\{e_1,\dots,e_k\}$ such that $e_i$ is an arc joining $v_{i-1}$ and $v_i$ for each $i=1,\dots,k$. A \emph{cycle} its defined analogously but with $v_0=v_k$. A \emph{tree} is a connected digraph containing no cycles. The \emph{height} of a tree is the maximum difference between forward and backward arcs of a subpath in it.
A \emph{homomorphism} from a digraph $G$ to a digraph $H$ is a mapping $f:V(G)\rightarrow V(H)$ such that $uv\in E(G)$ implies $f(u)f(v)\in E(H)$. It is denoted $f:G\rightarrow H$. If there exists a homomorphism from $G$ to $H$ we write $G\rightarrow H$, or equivalently, $G\leq H$. We shall write $G<H$ for $G\leq H$ and $H\nleq G$. The \emph{interval} $[G,H]$ consists in all digraphs $X$ such that $G\leq X\leq H$. A \emph{gap} is an interval in which there is no digraph $X$ such that $G<X<H$.
The relation $\leq$ is clearly a quasiorder which becomes a partial order by choosing a representative for each equivalence class, in our case the so called core. A \emph{core} of a digraph is its minimal homomorphic equivalent subgraph.
Given two partial orders $(\mathcal{P}_1,\leq_1)$ and $(\mathcal{P}_2,\leq_2)$, an \emph{embedding} from $(\mathcal{P}_1,\leq_1)$ to $(\mathcal{P}_2,\leq_2)$ is a mapping $\Phi:\mathcal{P}_1\rightarrow \mathcal{P}_2$ such that for every $a,b\in \mathcal{P}_1$, $a\leq b$ if and only if $\Phi(a)\leq \Phi(b)$. If such a mapping exists we say that $(\mathcal{P}_1,\leq_1)$ can be embedded into $(\mathcal{P}_2,\leq_2)$.
Finally, a partial order is \emph{universal} if every countable partial order can be embedded into it.
\section{Density Theorem}
In order to prove Theorem \ref{1} we shall construct a tree $\dt$ from a given proper tree $T_2$ which will satisfy $T_1<\dt<T_2$ for every tree $T_1<T_2$.
Given a tree $T$, a vertex $u\in V(T)$ and a set of vertices $S\subseteq V(T)$, the \emph{plank} from $u$ to $S$, denoted $P(u,S)$, is the subgraph induced by the vertices of every path which starts with $u$ and contains some vertex $v\in S$.
Let $T_2$ be the core of a proper tree. Then there exists a vertex $x\in V(T_2)$ such that $x$ is adjacent to at least three different vertices, name them $u,v,w$. Without loss of generality we shall assume that $ux$ and $wx$ are arcs. Let $X'\subseteq V(T_2)$ be the set of vertices, different from $u$ and $w$, which are adjacent to $x$. Note that $X'$ is not empty since $v\in X'$. Let $X=P(x,X')$, $U=P(x,\{u\})\backslash\{x\}$ and $W=P(x,\{w\})\backslash\{x\}$. Observe that $U\sqcup X\sqcup W\sqcup \{ux,wx\}=T_2$. See Figure~\ref{figure.T2}.
\begin{figure}
\centering
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[scale=0.5]{T2}
\caption{Tree $T_2$.}
\label{figure.T2}
\end{minipage}
\begin{minipage}{0.6\textwidth}
\centering
\includegraphics[scale=0.5]{D1.pdf}
\caption{Tree $\mathcal{D}_1(T)$.}
\label{figure.D1}
\end{minipage}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{Dn}
\caption{Tree $\dt$. Observe the enumeration of the vertices and planks of each tree $\mathcal{D}_1(T)$.}
\label{figure.Dn}
\end{figure}
Now, let $\mathcal{D}_1(T_2)$ be the tree from Figure~\ref{figure.D1}, where $W$ and $W'$ are copies of the plank $W\subset T_2$, $U$ is a copy of $U\subset T_2$, and $X$ and $X'$ are copies of $X\subset T_2$.
Finally, let $\mathcal{D}_n(T_2)$ be a tree consisting in $n$ consecutive trees $\mathcal{D}_1(T_2)$ whose planks $W'$ are identified with the planks $W$ of the following trees. See Figure~\ref{figure.Dn}. We shall refer to the vertices $w_i,a_i,u_i,x_i,b_i,x'_i\in \dt$ for $i=1,\dots,n$ as \emph{labelled vertices}.
\begin{lemma}
\label{labelled.vertices}
Let $T_1$ and $T_2$ be finite oriented trees such that $T_2$ is a proper tree and $T_2\nrightarrow T_1$. If there exists a homomorphism $f:\dt\rightarrow T_1$, then every labelled vertex of $\dt$ is mapped to a different vertex of $T_1$.
\end{lemma}
\begin{proof}
Assume that $T_2$ is a core and consider a homomorphism $f:\dt\rightarrow T_1$. Observe that two consecutive labelled vertices can not be mapped via $f$ to the same vertex since it would imply that $T_1$ contains a loop. Now, observe that if any pair of labelled vertices of distance two are mapped to the same vertex, it will induce a homomorphism $T_2\rightarrow T_1$. This follows from the construction of $\dt$. See Figure~\ref{figure.Dn}. Finally, if two labelled vertices at distance greater or equal to three are mapped to the same vertex, it would imply that $T_1$ contains a cycle, which is a contradiction since $T_1$ is a tree. We conclude that every labelled vertex has to be mapped to a different vertex of $T_1$.
\end{proof}
A digraph $G$ is \emph{rigid} if it is a core and the only automorphism $f:G\rightarrow G$ is the identity. We shall use the following fact.
\begin{fact}
\label{rigid}
The core of a tree is rigid.
\end{fact}
\begin{proof}[Proof of Theorem \ref{1} (sketch)]
Assume that $T_2$ is a core. Let $n>|V(T_1)|$ and consider the tree $\dt$. It is clear that $\dt\rightarrow T_2$. It can also be checked that $T_2\nrightarrow \dt$ (here we might use Fact \ref{rigid}). To see that $\dt\nrightarrow T_1$ observe that by Lemma \ref{labelled.vertices} every labelled vertex in $\dt$ has to be mapped to a different vertex in $T_1$, but the number of labelled vertices in $\dt$ is greater than $|V(T_1)|$. Thus, $T_1<T_1+\dt<T_2$.
We end by joining $T_1$ with $\dt$ by a proper and long enough zig-zag. The method is similar to the one used in the proof of the density theorem for paths \cite{pathhomomorphism}.
\end{proof}
\section{Fractal property for proper trees}
\label{section3}
\begin{proof}[Proof of Theorem \ref{2} (sketch)]
Let $n>|V(T_1)|+2|V(T_2)|$ and consider the tree $\dt$. We know by Theorem \ref{1} that $T_1<T_1+\dt<T_2$.
Let $T$ be the core of $\dt$. By Lemma \ref{labelled.vertices} every labelled vertex in $\dt$ has to be mapped to a different vertex in $T$. Since $n>|V(T_1)|+2|V(T_2)|$, it follows that $T$ has at least $|V(T_1)|$ labelled vertex. Let $y$ and $z$ be the initial and ending labelled vertex of $T$ respectively. Let $T'$ be the tree obtained from $T$ by adding two new vertices $y'$ and $z'$ and joining $y'$ to $y$ and $z'$ to $z$ by a proper zig-zag of length 5 or 6 so $y'$ and $z'$ have the same level, as shown in Figure~\ref{figure.gadget}. Finally let $T''$ be the tree obtained by joining $T_1$ with $T'$ by a proper and long enough zig-zag.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{gadget}
\caption{This is an example of how $T'$ might look. The vertices $y$ and $z$ might be different from the ones in the figure but they must be labelled vertices of $\dt$.}
\label{figure.gadget}
\end{figure}
Now, we shall construct an embedding $\Phi$ from the homomorphism order of the class of oriented paths, which we know is a universal partial order \cite{univepaths}, into the interval $[T_1,T_2]$.
Given an oriented path $P$, let $\Phi(P)$ be the tree obtained by replacing each arc $v_1v_2$ in $P$ by a copy of $T''$ identifying $v_1$ with $y'$ and $v_2$ with $z'$. Observe that $T_1<\Phi(P)<T_2$. It is clear that any homomorphism $f:P_1\rightarrow P_2$ induces a homomorphism $g:\Phi(P_1)\rightarrow \Phi(P_2)$ by identifying arcs with copies of $T''$. To see the opposite, observe that since $T$ is rigid by Fact \ref{rigid}, every copy of $T$ in $\Phi(P_1)$ must be map via the identity to some copy of $T$ in $\Phi(P_2)$. It follows that adjacent copies of $T''$ in $\Phi(P_1)$ must be mapped to adjacent copies of $T''$ in $\Phi(P_2)$. Hence, each homomorphism $g:\Phi(P_1)\rightarrow \Phi(P_2)$ induces a homomorphism $f:P_1\rightarrow P_2$.
\end{proof}
\section{Fractal property for finite digraphs}
\label{section4}
We say that a class of digraphs $\Vec{\mathcal{G}}$ has the \emph{fractal property} if every interval in the homomorphism order $(\Vec{\mathcal{G}},\leq)$ is either universal or a gap. The fractal property was introduced by Ne\v set\v ril \cite{nesetril} and it has been shown recently that the class of finite undirected graphs (or symmetric digraphs) has the fractal property \cite{fractal}. In this note, we have shown that the class of proper trees has also the fractal property (as consequence of Theorem \ref{2}). However, the class of finite digraphs, and even the class of oriented trees, is more complicated.
Ne\v set\v ril and Tardif characterised all gaps in the homomorphism order of finite digraphs \cite{dual}. It was shown that for every tree $T$ there exists a digraph $G$ such that $[G,T]$ is a gap, and that all gaps have this form. Theorem \ref{1} contributes to this result by implying that if $[G,T]$ is a gap and $T$ is a proper tree, then $G$ must contain a cycle.
The characterisation of universal intervals in the homomorphism order of finite digraphs seems to be complicated. Related to this issue, we have stated Theorem \ref{3}. Its proof combines some techniques already used \cite{llibre,fractal} with some extra arguments, and will appear in the full version of this note. This result together with Theorem \ref{2} imply that the class of finite digraphs whose cores are not paths has the fractal property. However, intervals of the form $[G,P]$ where the core of $P$ is a path remain to be studied and characterised.
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The Met Office is warning motorists to take care overnight and into tomorrow morning because of heavy fog.
The forecasters say mist will form on Monday evening, lasting until mid-morning in some areas - adding please be aware of the risk of disruption to transport.
The yellow alert has been issued across the entire county from 7pm tonight.
The Chief Forecaster said: ."
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\begin{document}
\title{Invariant and hyperinvariant subspaces for amenable operators}
\title{Invariant and hyperinvariant subspaces for amenable operators}
\author[L. Y. Shi]{Luo Yi Shi}
\address{Department of Mathematics\\Tianjin Polytechnic University\\Tianjin 300160\\P.R. CHINA}
\email{sluoyi@yahoo.cn}
\author[Y. J. Wu]{YU Jing Wu}
\address{Tianjin Vocational Institute \\Tianjin 300160\\P.R. CHINA}
\email{wuyujing111@yahoo.cn}
\author[Y.Q. Ji]{You Qing Ji}
\address{Department of Mathematics\\Jilin University\\Changchun 130012\\P.R. CHINA}
\email{jiyq@jlu.edu.cn}
\thanks{Supported by NCET(040296), NNSF of China(10971079) and
the Specialized Research Fund for the Doctoral Program
of Higher Education(20050183002) }
\date{7.04. 2010}
\subjclass[2000]{47C05 (46H35 47A65 47A66 47B15)}
\keywords{Amenable; Invariant subspaces; Hyperinvariant subspaces; Reduction property}
\begin{abstract}There has been a long-standing conjecture in
Banach algebra that every amenable operator is similar to a normal operator. In this paper, we study the
structure of amenable operators on Hilbert spaces. At first, we show that the conjecture is equivalent to every
non-scalar amenable operator has a non-trivial hyperinvariant subspace and equivalent to every amenable operator
is similar to a reducible operator and has a non-trivial invariant subspace; and then, we give two
decompositions for amenable operators, which supporting the conjecture.
\end{abstract}
\maketitle
\section{Introduction}
Throughout this paper, $\mathfrak{H}$ denotes a complex separable infinite-dimension Hilbert space and
$\mathfrak{B}(\mathfrak{H})$ denotes the bounded linear operators on $\mathfrak{H}$. For an algebra
$\mathfrak{A}$ in $\mathfrak{B}(\mathfrak{H})$, we write $\mathfrak{A}'$ for the commutant of $\mathfrak{A}$
(i.e., $\mathfrak{A}'=\{B\in\mathfrak{B}(\mathfrak{H}), BA=AB ~~\textup{for all} A\in \mathfrak{A}\})$ and
$\mathfrak{A}''$ for the double commutant of $\mathfrak{A}$ (i.e., $\mathfrak{A}''=(\mathfrak{A}')'$). We also
write Lat$\mathfrak{A}$ for the collection of those subspaces which are invariant for every operator in
$\mathfrak{A}$. We say $\mathfrak{A}$ is {\it completely reducible} if for every subspace $M$ in
Lat$\mathfrak{A}$ there exists $N$ in Lat$\mathfrak{A}$ such that $\mathfrak{H}=M\dot{+}N$ (i.e., $M\cap
N=\{0\}$ and $\mathfrak{H}$ is the algebraic direct sum of $M$ and $N$); $\mathfrak{A}$ is {\it reducible} if
for every subspace $M$ in Lat$\mathfrak{A}$ we have $M^\bot$ (the orthogonal complement of $M$) in
Lat$\mathfrak{A}$; $\mathfrak{A}$ is {\it transitive} if Lat$\mathfrak{A}=\{\{0\}, \mathfrak{H}\}$. If
$T\in\mathfrak{B}(\mathfrak{H})$, we say that a subspace $M$ of $\mathfrak{H}$ is a {\it hyperinvariant
subspace} for $T$ if $M$ is invariant under each operator in $\mathfrak{A}_T'$; $M$ is a {\it reducible
subspace} for $T$ if $M, M^\bot\in$ Lat$T$.
The concept of amenable Banach algebras was first introduced by B. E. Johnson in \cite{BE1972}. Suppose that
$\mathfrak{A}$ is a Banach algebra. A {\it Banach $\mathfrak{A}$-bimodule} is a Banach space $X$ that is also an
algebra $\mathfrak{A}$-bimodule for which there exists a constant $K>0$ such that $||a\cdot x||\leq K
||a||||x||$ and $||x\cdot a||\leq K ||a||||x||$ for all $a\in \mathfrak{A}$ and $x\in X$. We note that $X^*$,
the dual of $X$, is a Banach $\mathfrak{A}$-bimodule with respect to the dual actions
$$ [a \cdot f](x)=f(x \cdot a), [f \cdot a](x)=f(a \cdot x), a\in \mathfrak{A}, x\in X,
f\in X^*.$$
Such a Banach $\mathfrak{A}$-bimodule is called a {\it dual
$\mathfrak{A}$-bimodule}.
A {\it derivation} $D : \mathfrak{A}\rightarrow X$ is a continuous linear map such that $D(ab) =
a \cdot D(b) + D(a)\cdot b$, for all $a, b \in \mathfrak{A}$. Given $x\in X$, the {\it inner derivation}
$\delta_x : \mathfrak{A}\rightarrow X$, is defined by $\delta_x (a) = a\cdot x- x \cdot a$.
According to Johnson¡¯s
original definition, a Banach algebra $\mathfrak{A}$ is {\it amenable} if every
derivation from $\mathfrak{A}$ into the dual $\mathfrak{A}$-bimodule
$X^*$ is inner for all Banach $\mathfrak{A}$-bimodules $X$. If $T\in \mathfrak{B}(\mathfrak{H})$, denote the
norm-closure of span$\{T^k: k\in \{0\}\cup \mathbb{N}\}$ by $\mathfrak{A}_T$, where $\mathbb{N}$ is the set of
natural numbers, $T$ is said to be an {\it amenable operator}, if $\mathfrak{A}_T$ is an amenable Banach
algebra. Ever since its introduction, the concept of amenability has played an important role in research in
Banach algebras, operator algebras and harmonic analysis. There has been a long-standing conjecture in the
Banach algebra community, stated as follows:
\begin{conj}\label{conj 1}
A commutative Banach subalgebra of $\mathfrak{B}(\mathfrak{H})$ is amenable if and only if it is similar to a
$C^*$-algebra.
\end{conj}
One of the
first result in this direction is due to Willis \cite{W1995}. Willis showed that if $T$ is an amenable compact
operator, then $T$ is similar to a normal operator. In \cite{G2006} Gifford studied the reduction property for
operator algebras consisting of compact operators and showed that if such an algebra is amenable then it is
similar to a $C^*$-algebras. In the recent papers \cite{F2005}, \cite{F2007} Farenick, Forrest and Marcoux
showed that if $T$ is similar to a normal operator, then $\mathfrak{A}_T$ is amenable if and only if
$\mathfrak{A}_T$ is similar to a $C^*$-algebra and the spectrum of $T$ has connected complement and empty
interior; If $T$ is a triangular operator with respect to an orthonormal basis of $\mathfrak{H}$, then
$\mathfrak{A}_T$ is amenable if and only if $T$ is similar to a normal operator whose spectrum has connected
complement and empty interior. For further details see \cite{F2005} and \cite{F2007}.
In this paper, we give the characterization of the structure of amenable operators. At first, we use the
reduction theory of von Neumann to give two equivalent descriptions for Conjecture 1.1; and then, we give two
decompositions for amenable operators, which supporting the Conjecture 1.1.
\vskip1cm
\section{An equivalent formulation of the conjecture 1.1}
In this section we use the reduction theory of von Neumann to give two equivalent descriptions for Conjecture
1.1. We obtain that every amenable operator is similar to a normal operator if and only if every non-scalar
amenable operator has a non-trivial hyperinvariant subspace if and only if every amenable operator is similar to
a reducible operator and has a non-trivial invariant subspace.
In order to proof the main theorem, we need to introduce von Neumann's reduction theory \cite{S1967} and some
lemmas.
Let
$\mathfrak{H_1}\subseteq\mathfrak{H_2}\subseteq\cdots\subseteq\mathfrak{H_\infty}$
be a sequence of Hilbert spaces chosen once and for all,
$\mathfrak{H_n}$ having the dimension $n$. Let $\mu$ be a finite
positive regular measure defined on the Borel sets of a separable
metric space $\wedge$, and let $\{E_n\}_{n=1}^\infty$ be a
collection of disjoint Borel sets of $\wedge$ with union $\wedge$.
Then the symbol
$$\int_\wedge^\oplus \mathfrak{H}(\lambda)\mu(d\lambda)$$
denotes the set of all functions $f$ defined on $\wedge$ such that
(1)$f(\lambda)\in \mathfrak{H}_n\subseteq\mathfrak{H}_\infty$ if
$\lambda\in E_n$;
(2)$f(\lambda)$ is a $\mu$-measurable function with values in
$\mathfrak{H}_\infty$;
(3) $\int_\wedge^\oplus |f(\lambda)|^2\mu(d\lambda)<\infty$.
We put
(4)$(f, g)=\int_\wedge^\oplus (f(\lambda), g(\lambda))\mu(d\lambda).$ \\
The set of functions thus defined is
called the {\it direct integral Hilbert space with measure $\mu$ and dimension sets $\{E_n\}$} and denoted by
$\mathfrak{H}=\int_\wedge^\oplus \mathfrak{H}(\lambda)\mu(d\lambda)$.
An operator on $\mathfrak{H}$ is said to be {\it decomposable} if
there exists a strongly $\mu$-measurable operator-value function
$A(\cdot)$ defined on $\wedge$ such that
$A(\lambda)$ is a
bounded operator on the space $\mathfrak{H}(\lambda)=\mathfrak{H}_n$ when $\lambda\in E_n$, and for all
$f\in\mathfrak{H}$, $(Af)(\lambda)=A(\lambda)f(\lambda)$. We write $A=\int_\wedge^\oplus
A(\lambda)\mu(d\lambda)$ for the equivalence class corresponding to $A(\cdot)$. If $A(\lambda)$ is a scalar
multiple of the identity on $\mathfrak{H}(\lambda)$ for almost all $\lambda$, then $A$ is called {\it diagonal}.
The collection of all diagonal operator is called the {\it diagonal algebra} of $\wedge$. In \cite{S1967}I.3,
Schwartz showed that an operator $A$ on Hilbert space $\mathfrak{H}=\int_\wedge^\oplus
\mathfrak{H}(\lambda)\mu(d\lambda)$ is decomposable if and only if $A$ belong to the commutant of the diagonal
algebra of $\wedge$. And $||A||=\mu-ess.sup_{\lambda\in\wedge}||A(\lambda)||$.
In \cite{A 1977}, Azoff, Fong and Gilfeather used von Neumann's reduction theory to define the reduction theory
for non-selfadjoint operator algebras: Fix a partitioned measure space $\wedge$ and let $\mathfrak{D}$ be the
corresponding diagonal algebra. Given an algebra $\mathfrak{A}$ of decomposable operators. Each operator
$A\in\mathfrak{A}$ has a decomposition $A=\int_\wedge^\oplus A(\lambda)\mu(d\lambda)$. Chosse a countable
generating set $\{A_n\}$ for $\mathfrak{A}$. let $\mathfrak{A}(\lambda)$ be the strongly closed algebra
generated by the $\{A_n(\lambda)\}$. $\mathfrak{A}\sim\int_\wedge^\oplus \mathfrak{A}(\lambda)\mu(d\lambda)$ is
called the {\it decomposition of} $\mathfrak{A}$ {\it respect to} $\mathfrak{D}$. A decomposition
$\mathfrak{A}\sim\int_\wedge^\oplus \mathfrak{A}(\lambda)\mu(d\lambda)$ of an algebra is said to be {\it
maximal} if the corresponding diagonal algebra is maximal among the abelian von Neumman subalgebras of
$\mathfrak{A}'$. The following lemma is a basic result in \cite{A 1977} which will be used in this paper.
\begin{lem}\label{Fong}(\cite{A 1977}, Theorem 4.1)
Let $\mathfrak{A}\sim\int_\wedge^\oplus
\mathfrak{A}(\lambda)\mu(d\lambda)$ be a decomposition of a
reductive algebra. Then almost all of $\{\mathfrak{A}(\lambda)\}$
are reducible. In particular, if the decomposition is maximal, then
almost all of the algebras $\{\mathfrak{A}(\lambda)\}$ are
transitive.
\end{lem}
In \cite{G2006} Gifford studied the reduction property for operator
algebras and obtained the following result:
\begin{lem}\label{lem 1}(\cite{G2006} Lemma 4.4, Lemma 4.12)
If $\mathfrak{A}$ is a commutative amenable operator algebra, then
$\mathfrak{A}',\mathfrak{A}''$ are complete reducible and there
exists $M\geq 1$ so that for any idempotent $p\in\mathfrak{A}''$
$||p||\leq M$.
\end{lem}
Assume $\mathfrak{A}$ is a operator algebra, let $P(\mathfrak{A})$ denote the idempotents in $\mathfrak{A}$ and
$\mathcal{P}(\mathfrak{A})$ denote the strongly closed algebra generated by $P(\mathfrak{A})$. We get the
following lemma:
\begin{lem}\label{lem 2}
If $\mathfrak{A}$ is a commutative amenable operator algebra, then $\mathcal{P}(\mathfrak{A''})$ is similar to
an abelian von Neumann algebra.
\end{lem}
\begin{proof}
By Lemma \ref{lem 1} and \cite[ Corollary 17.3]{D1988}, it follows that there exists $X\in
\mathfrak{B}(\mathfrak{H})$ such that $XpX^{-1}$ is selfadjoint for each $p\in P(\mathfrak{A''})$. Hence
$\mathcal{P}(\mathfrak{A''})$ is similar to a abelian von Neumann algebra.
\end{proof}
\begin{lem}\label{lem 3}(\cite{F2005})
Let $\mathfrak{A}$ and $\mathfrak{B}$ be Banach algebras and suppose
that $\varphi: \mathfrak{A}\longrightarrow\mathfrak{B}$ is a
continuous homomorphism with $\varphi(\mathfrak{A})$ dense in
$\mathfrak{B}$. If $\mathfrak{A}$ is amenable, then $\mathfrak{B}$
is amenable.
\end{lem}
\begin{nota}\label{nota}
From Lemma \ref{lem 2},\ref{lem 3} we always assume that $\mathcal{P}(\mathfrak{A}_T'')$ is a abelian von
Neumann algebra, and $\mathfrak{A}_T'$ is a reducible operator algebra in this section.
\end{nota}
Now we will proof the main result of this section:
\begin{thm}\label{thm1}
The following are equivalent:
\textup{(1)} Every amenable operator is similar to a normal operator;
\textup{(2)} Every non-scalar amenable operator has a non-trivial hyperinvariant subspace;
\textup{(3)} Every amenable Banach algebra which is generated by an operator is similar to a $C^*$-algebra.
\end{thm}
\begin{proof}
$(1)\Leftrightarrow(3)$ and $(1)\Rightarrow(2)$ is clear by
\cite{F2005}. Therefore, in order to establish the theorem it
suffices to show the implications $(2)\Rightarrow(1)$.
Assume (2), by Lemma \ref{lem 2} choose a maximal decomposition for $\mathfrak{A}_T'\sim \int_\wedge^\oplus
\mathfrak{A}_T'(\lambda)\mu(d\lambda)$ respect to the diagonal algebra $\mathcal{P}(\mathfrak{A}_T'')$.
Assume $T\sim \int_\wedge^\oplus T(\lambda)\mu(d\lambda)$ is the
decomposition for $T$. Let $\{p_n\}_{n=1}^\infty$ denote the all
rational polynomials. Then $p_n(T)\sim \int_\wedge^\oplus
p_n(T)(\lambda)\mu(d\lambda)$ is decomposable for all $n$ and there
exists a measurable
$E\subseteq\wedge$ such that $\mu(\wedge-E)=0$ and for any $\lambda\in
E$ we have
$ p_n(T)(\lambda)= p_n(T(\lambda))$ and
$||p_n(T)(\lambda)||\leq ||p_n(T)||$ by \cite[Lemma I.3.1, I.3.2]{S1967}. Define a mapping $\varphi_\lambda :
\mathfrak{A}_{T}\rightarrow \mathfrak{A}_{T(\lambda)}$ by $\varphi_\lambda(p_n(T))=p_n(T(\lambda))$ for each
rational polynomial $p_n$ and $\lambda\in E$. Note that $||p_n(T(\lambda))||\leq ||p_n(T)||$ for each rational
polynomial $p_n$ and furthermore, $\{p_n(T)\}$ is dense in $\mathfrak{A}_{T}$. Hence, $\varphi_\lambda$ is
well-defined and $\varphi_\lambda$ a continuous homomorphism with $\varphi(\mathfrak{A}_T)$ dense in
$\mathfrak{A}_{T(\lambda)}$. By Lemma \ref{lem 3}, $T(\lambda)$ is amenable for almost all $\lambda$.
Now for almost all $\lambda$, $T(\lambda)$ is amenable and $\mathfrak{A}_T'(\lambda)\subseteq
\mathfrak{A}_{T(\lambda)}'$ and $\mathfrak{A}_T'(\lambda)$ is transitive by Lemma \ref{Fong}. Thus almost all of
$T(\lambda)$ are scalar operators, i.e. $T$ is a normal operator.
\end{proof}
\begin{cor}
Every amenable operator is similar to a normal operator if and only
if there exists a non-trivial idempotent in the double-commutant of
every non-scalar amenable operator.
\end{cor}
\begin{rem}\label{rem1}
In \cite{F2005}, Farenick, Forrest and Marcoux showed that if $T\in\mathfrak{B}(\mathfrak{H})$ is amenable and
similar to a
normal operator $N$, then the spectrum of $N$ has connected complement and empty
interior. According to \cite[Theorem 1.23]{R2003}, $N$ is a reducible operator. Hence, there exists an
invertible operator $X\in\mathfrak{B}(\mathfrak{H})$ such that $\mathfrak{A}_{XTX^{-1}}''$ is a reducible
algebra. The following theorem give the equivalent description for Conjecture 1.1 from the existence of
invariant subspace for amenable operators.
\end{rem}
\begin{thm}\label{thm11}
The following are equivalent:
\textup{(1)} Every amenable operator is similar to a normal operator;
\textup{(2)} For every amenable operator $T\in\mathfrak{B}(\mathfrak{H})$, there exists an invertible operator
$X\in\mathfrak{B}(\mathfrak{H})$ such that $\mathfrak{A}_{XTX^{-1}}''$ is a reducible algebra and $T$ has a
non-trivial invariant subspace.
\end{thm}
\begin{proof}
$(1)\Rightarrow(2)$ is clear by Remark \ref{rem1}.
$(2)\Rightarrow(1)$ is trivial modifications adapt the proof of theorem 2.6.
\end{proof}
\begin{rem}
According to theorem 2.6, \ref{thm11}, we obtain that the Conjecture \ref{conj 1} for operator algebra which is
generated by an operator is equivalent to the following statements:
(1) Every amenable operator $T$ has a non-trivial invariant subspace and renorm $\mathfrak{H}$ with an
equivalent Hilbert space norm so that under this norm Lat$\mathfrak{A}_T$ becomes orthogonally complemented;
(2) Every non-scalar amenable operator has a non-trivial hyperinvariant subspace.
\end{rem}
\vskip1cm
\section{Decomposition of Amenable operators}
In this section, we get two decompositions for amenable operators and prove that the two decompositions are the
same which supporting Conjecture 1.1.
At first, we summarize some of the details of multiplicity theory for abelian von Neumann algebras. For the
most part, we will follow \cite{D1996}. If $A$ is an operator on a Hilbert space $\mathfrak{K}$ and $n$ is a
cardinal number, Let $\mathfrak{K}^n$ denote the orthogonal direct sum of $n$ copies of $\mathfrak{K}$, and
$A^{(n)}$ be the operator on $\mathfrak{K}^n$ which is the direct sum of $n$ copies of $A$. Whenever
$\mathfrak{A}$ is an operator algebra on $\mathfrak{K}$, $\mathfrak{A}^{(n)}$ denotes the algebra $\{A^{(n)},
A\in\mathfrak{A}\}$. An abelian von Neumann algebra $\mathfrak{B}$ is of {\it uniform multiplicity $n$} if it is
(unitary equivalent to) $\mathfrak{A}^{(n)}$ for some maximal abelian von Neumann algebra $\mathfrak{A}$. By
\cite{D1996}, for any abelian von Neumann algebra $\mathfrak{A}$, there exists a sequens of regular Borel
measures $\{\mu_n\}$ on a sequens of separable metric space $\{X_n\}$ such that $\mathfrak{A}$ is unitary
equivalent to $\sum_{n=1}^\infty\oplus \mathfrak{B}_n\oplus\mathfrak{B}_\infty$, where $\mathfrak{B}_n$ is a von
Neumman algebra which has uniform multiplicity $n$ for all $1\leq n\leq \infty$. For further details see
\cite[II.3]{D1996}.
\begin{prop}\label{prop1}
Suppose that $T$ is amenable operator and $\mathfrak{A}_T'$ contains a subalgebra which is similar to an abelian
von Neumman algebra with no direct summand of uniform multiplicity infinite, then $T$ is similar to a normal
operator.
\end{prop}
\begin{proof}
For the sake of simplicity, we assume $\mathfrak{A}_T'$ contains a subalgebra $\mathfrak{B}$ which is an abelian
von Neumman subalgebra with no direct summand of uniform multiplicity infinite. Trivial modifications adapt the
proof to the more general case.
By \cite[II.3]{D1996}, there exists a sequens of regular Borel measures $\{\mu_n\}$ on a sequens of separable
metric space $\{X_n\}$ such that $\mathfrak{B}$ is unitarily equivalent to $\sum_{n=1}^\infty\oplus
\mathfrak{B}_n$, where $\mathfrak{B}_n$ is a von Neumman algebra which has
uniform multiplicity $n$ for all $n$. Hence, $T=\sum_{n=1}^\infty\oplus T_n$,
where $T_n\in \mathfrak{B}_n'$. It suffices to show that $T_n$ is similar to a normal operator for all $n$, then
by \cite[Corollary 26]{F1977}, it follows that $T$ is similar to a normal operator.
Since $T\in\mathfrak{B}_n'$, according to \cite[Theorem 7.20]{R2003}, for any $1\leq n<\infty$ there exists a
unitary operator $U_n\in\mathfrak{B}_n'$ such that
$$U_nT_n(U_n)^{-1}= \left[\begin{array}{ccccc}
N_{11}&N_{12}&\cdots&\cdots&N_{1n}\\
0&N_{22}&\cdots&\cdots&N_{2n}\\
0&0&\ddots&\ddots&\vdots\\
\vdots&\vdots&\ddots&\ddots&\vdots\\
0&0&\cdots&0&N_{nn}\\
\end{array}\right]
$$
where $N_{ij}$ is a normal operator for all $1\leq i,j\leq n$. By \cite[ Proposition 3.1]{Y}, it follows that $T_n$ is similar to $\oplus_{i=1}^{n}
N_{ii}$, i.e. $T_n$ is similar to a normal operator for all $n$.
\end{proof}
\begin{cor}\label{cor1}
Assume $T$ is amenable operator, then there exists hyperinvariant subspaces $M, N$ of $T$ such that $T$ has the
form $T=T_1\dot{+}T_2$ respect to the space decomposition $\mathfrak{H}=M\dot{+}N$, where $T_1, T_2$ are
amenable operators, $T_1$ is similar to a normal operator and $\mathcal{P}(\mathfrak{A}_{T_2}'')$ is similar to
an abelian von Neumman algebra with uniform multiplicity infinite.
\end{cor}
The proof of the following lemma is straightforward and we omit
it.
\begin{lem}\label{lem4}
Suppose that $\mathfrak{A}$ is a completely reductive operator algebra and $p\in P(\mathfrak{A}')$. Then
$p\mathfrak{A}$ is a completely reductive operator algebra on $\textup{Ran} p$.
\end{lem}
We are in need of the following propositions before we can address the main theorem of this section.
\begin{prop}\label{prop6}
Assume that $T$ is a amenable operator and there exists a space decomposition $\mathfrak{H}=M\dot{+}N$ such
that $T$ has the matrix form $T= \left[\begin{array}{cc}
T_{1}&\\
&T_{2}
\end{array}\right
]
\begin{matrix}
\mbox{$M$}\\
\mbox{$N$}\\
\end{matrix}
$. Then $T$ is similar to a normal operator if and only if $T_1$ and $T_2$ are similar to normal operators.
\end{prop}
\begin{proof}
Assume that $T$ has the matrix form $$T= \left[\begin{array}{cc}
T_{1}&T_{12}\\
&\widetilde{T_{2}}
\end{array}\right
]
\begin{matrix}
\mbox{$M$}\\
\mbox{$M^{\bot}$}\\
\end{matrix}$$
respect to the space decomposition $\mathfrak{H}=M\oplus M^{\bot}$. By \cite[Lemma 2.8]{Y}, there exists an
invertible operator $S=\left[\begin{array}{cc}
I&S_{12}\\
&I
\end{array}\right
]
\begin{matrix}
\mbox{$M$}\\
\mbox{$M^{\bot}$}\\
\end{matrix}$ such that $S^{-1}TS=\left[\begin{array}{cc}
T_{1}&\\
&\widetilde{T_{2}}
\end{array}\right
]
\begin{matrix}
\mbox{$M$}\\
\mbox{$M^{\bot}$}\\
\end{matrix}$. Assume that $S$ has the matrix form
$S=\begin{array}{cc}
\begin{array}{cc}M&M^{\bot}
\end{array}\\
\left[\begin{array}{cc}
I&\\
&S_1
\end{array}\right
]\end{array}
\begin{matrix}
\\
\mbox{$M$}\\
\mbox{$N$}\\
\end{matrix}
$, we obtain that $T_2=S_1\widetilde{T_{2}}S_1^{-1}$. By \cite[propsition 6.5]{H1978}, we get that $T$ is
similar to a normal operator if and only if $T_1$ and $T_2$ are similar to normal operators.
\end{proof}
\begin{prop}\label{prop5}
Suppose that $T$ is an amenable operator, $M_1\in$
Lat$\mathfrak{A}_T'$ and $M_2\in$ Lat$\mathfrak{A}_T''$. Then
$M_1+M_2$ is closed.
Moreover, if $T|_{M_1}$ and $T|_{M_2}$ are similar to normal operators, then $T|_{M_1+M_2}$ is similar to a
normal operator.
\end{prop}
\begin{proof}
Let $N_0=M_1\cap M_2$, according to Lemma \ref{lem4}, there exists $N\in$ Lat$\mathfrak{A}_{T}''$ such that
$M_2=N_0\dot{+}N$. Choose $q\in P(\mathfrak{A}_{T}')$, such that $\textup{Ran} q=N$. By the assumption,
$M_1\in$ Lat$\mathfrak{A}_T'$. Hence $qM_1\subset M_1\cap N=\{0\}$. Therefore $M_1\subset (I-q)\mathfrak{H}$. We
see that $M_1+M_2=M_1\dot{+}N$ is closed. This establishes the first statement of the proposition.
Since $T|_{M_2}$ is similar to a normal operator, by proposition \ref{prop6}, we get that $T|_N$ is similar to a
normal operator. By the assumption $T|_{M_1}$ is similar to a normal operator, using proposition \ref{prop6}
again, we obtain that $T|_{M_1+M_2}=T|_{M_1\dot{+}N}$ is similar to a normal operator.
\end{proof}
Now we will obtain the main theorem of this section.
\begin{thm}\label{thm2}
Assume $T$ is an amenable operator, then there exists hyperinvariant subspaces $M_1, M_2$ of $T$ such that $T$
has the form $T=T_1\dot{+}T_2$ respect to the space decomposition $\mathfrak{H}=M_1\dot{+}M_2$
and satisfies that:
\textup{(1)} $T_1,T_2$ are amenable operators;
\textup{(2)} If $M$ is a hyperinvariant subspace of $T$ and $T|_M$ is similar to a normal operator, then
$M\subseteq M_1$, \textup{i.e.} $M_1$ is the largest hyperinvariant subspace on which $T$ is similar to a normal
operator;
\textup{(3)} For any $q\in P(\mathfrak{A}_{T_2}'')$, $T_2|_{\textup{Ran} q}$ is not similar to a normal
operator;
\textup{(4)} $\mathcal{P}(\mathfrak{A}_{T_2}'')$ is similar to an abelian von Neumman algebra with uniform
multiplicity infinite;
\textup{(5)} $\mathfrak{A}_{T}'=\mathfrak{A}_{T_1}'\dot{+}\mathfrak{A}_{T_2}'$,
$\mathfrak{A}_{T}''=\mathfrak{A}_{T_1}''\dot{+}\mathfrak{A}_{T_2}''$;
\textup{(6)} There exists no nonzero compact operator in $\mathfrak{A}_{T_2}'$.
\end{thm}
\begin{proof}
Case1. For any $p\in P(\mathfrak{A}_{T}'')$, $T|_{\textup{Ran} p}$ is not similar to normal operator. According
to the proof of Proposition \ref{prop1}, we obtain that $\mathcal{P}(\mathfrak{A}_{T}'')$ is similar to an
abelian von Neumman algebra with uniform multiplicity infinite. Let $M_1=0$.
Case2. There exists $p\in P(\mathfrak{A}_{T}'')$ such that $T|_{\textup{Ran} p}$ is similar to normal operator.
Then, by Zorn's Lemma and the same method in the proof of \cite[Corollary 26]{F1977}, we can show that there
exists an element $p_0\in P(\mathfrak{A}_{T}'')$ which is maximal with respect to the property that
$T|_{\textup{Ran} p_0}$ is similar to a normal operator. Using Proposition \ref{prop5}, $\textup{Ran} p_0$ is
the largest hyperinvariant subspace of $T$ on which $T$ is similar to a normal operator. Hence, $T$ has the form
$T=T_1\dot{+} T_2$ with respect to the space decomposition $\mathfrak{H}=\textup{Ran} p_0\dot{+} \textup{Ker}
p_0$ where $T_1$ is similar to a normal operator, $T_1,T_2$ are amenable operators. Let $M_1=\textup{Ran} p_0,
M_2=\textup{Ker} p_0$.
Next we will prove that for any $q\in P(\mathfrak{A}_{T_2}'')$, $T_2|_{\textup{Ran} q}$ is not similar to
normal operator. Then according to Proposition \ref{prop1} $\mathcal{P}(\mathfrak{A}_{T_2}'')$ is similar to an
abelian von Neumman algebra with uniform multiplicity infinite.
Indeed, if there exists $q\in P(\mathfrak{A}_{T_2}'')$ such that $T_2|_{\textup{Ran} q}$ is similar to a normal
operator and $q$ has the form $q= \left[\begin{array}{cc}
I&0\\
0&0\\
\end{array}\right]
\begin{matrix}
\mbox{$\textup{Ran} q$}\\
\mbox{$\textup{Ker} q$}\\
\end{matrix}$. Then for any $A\in \mathfrak{A}_{T}'$, $A$ has the form
$$A= \left[\begin{array}{ccc}
A_{11}&&\\
&A_{22}&\\
&&A_{33}\\
\end{array}\right]
\begin{matrix}
\mbox{$\textup{Ran} p_0$}\\
\mbox{$\textup{Ran} q$}\\
\mbox{$\textup{Ker} q$}\\
\end{matrix}.$$
Let
$$R= \left[\begin{array}{ccc}
I&&\\
&I&\\
&&0\\
\end{array}\right]
\begin{matrix}
\mbox{$\textup{Ran} p_0$}\\
\mbox{$\textup{Ran} q$}\\
\mbox{$\textup{Ker} q$}\\
\end{matrix}.$$
Then $R\in P(\mathfrak{A}_{T}'')$. By the assumption $T|_{\textup{Ran} R}$ is similar to a normal operator which
contradicts to the maximal property of $p_0$.
At last we will prove that there exists no nonzero compact operator in $\mathfrak{A}_{T_2}'$.
Indeed, if there exists a nonzero compact operator $k_0\in\mathfrak{A}_{T_2}'$, let $L_1$ denote the subspace
spanned by the ranges of all compact operators in $\mathfrak{A}_{T_2}'$, and $L_2$ the intersection of their
kernel, by \cite[Lemma 3.1]{R1993}, both $L_1, L_2$ lie in Lat$\mathfrak{A}_{T_2}'$ and
$L_1\dot{+}L_2=\textup{\textup{Ker}} p_0$. Considering the restricting $T_2|_{L_1}$, assume $T_{21}=T_2|_{L_1}$,
then $T_{21}$ is an amenable operator and $\mathfrak{A}_{T_{21}}'$ contain a sufficient set of compact
operators. By Lemma \ref{lem 1} and \cite[Theorem 9]{R1982}, $T_{21}$ is similar to a normal operator which
contradicts to the above discussion.
\end{proof}
Trivial modifications adapt the proof of Theorem \ref{thm2}, we obtain the following theorem which decomposes
amenable operators by the invariant subspaces of them. The proof is similar to Theorem \ref{thm2} and we omit
it.
\begin{thm}\label{thm22}
Assume $T$ is an amenable operator, then there exists invariant subspaces $N_1, N_2$ of $T$ such that $T$ has
the form $T=A_1\dot{+}A_2$ respect to the space decomposition $\mathfrak{H}=N_1\dot{+}N_2$
and satisfies that:
\textup{(1)} $A_1,A_2$ are amenable operators;
\textup{(2)} If $N$ is an invariant subspace of $T$ such that $N_1\subseteq N$ and $T|_N$ is similar to a normal
operator, then $N= N_1$, \textup{i.e.} $N_1$ is the maximal invariant subspace on which $T$ is similar to a
normal operator;
\textup{(3)} For any $q\in P(\mathfrak{A}_{T_2}')$, $T_2|_{\textup{Ran} q}$ is not similar to a normal operator;
\textup{(4)} If $\mathcal{P}(\mathfrak{A}_{T_2}')$ contains a subalgebra which is similar to an abelian von
Neumman algebra then the von Neumman algebra has the uniform multiplicity infinite.
\end{thm}
\begin{rem}
If the answer to Conjecture 1.1 is positive, by Theorem 2.6, every amenable is similar to a normal operator.
Then, for the above theorem $M_1=N_1=\mathfrak{H}$. That is to say, the two decompositions of theorem
3.6 and 3.7 are the same. The remainder of this section, we will prove that
the two decompositions are the same which supporting Conjecture 1.1.
\end{rem}
\begin{lem}\cite{F1977}\label{lem5}
If $T\in \mathfrak{B}(\mathfrak{H})$ is an amenable operator and there exist a one-to-one bounded linear map
$W:\mathfrak{H}\rightarrow\mathfrak{H}_2$, a bounded linear map $V:\mathfrak{H}_1\rightarrow\mathfrak{H}$ with
dense range and operators $S_1\in \mathfrak{B}(\mathfrak{H}_1)$, $S_2\in \mathfrak{B}(\mathfrak{H}_2)$ which
are similar to normal operators such that $TV=VS_1$ and $WT=S_2W$, then $T$ is similar to a normal operator.
\end{lem}
\begin{cor}
Assume $T=B_1B_2$ is an amenable operator, where $B_1,B_2$ are positive operators, then $T$ is similar to a
normal operator.
\end{cor}
\begin{proof}
Assume $B_1, B_2$ have the forms
$$B_2=\left[\begin{array}{cc}
0&\\
&\widetilde{B_2}\\
\end{array}\right],
B_1=\left[\begin{array}{cc}
B_{11}&B_{12}\\
B_{12}^*&B_{22}\\
\end{array}\right],$$ respect to the space decomposition $\mathfrak{H}=\textup{Ker} B_2\oplus(\textup{Ker} B_2)^\bot$ where $\widetilde{B_2}$ is one-to-one and $B_{11}, B_{22}$ are
positive operators. Thus $T$
has the form $T=\left[\begin{array}{cc}
0&B_{12}\widetilde{B_2}\\
0&B_{22}\widetilde{B_2}\\
\end{array}\right]$
respect to the space decomposition. Since $T$ is an amenable operator, by \cite[Lemma 2.8]{Y}, $T$ is similar to
$\left[\begin{array}{cc}
0&0\\
0&B_{22}\widetilde{B_2}\\
\end{array}\right]$. Thus without loss of generality, we may assume that $B_2$ is one-to-one.
Assume that $B_1, B_2$ has the form
$$B_1=\left[\begin{array}{cc}
\widetilde{B_1}&\\
&0\\
\end{array}\right],
B_2=\left[\begin{array}{cc}
B_{11}&B_{12}\\
B_{12}^*&B_{22}\\
\end{array}\right],$$
respect to the space decomposition $\mathfrak{H}=(\textup{Ker} B_1)^\bot\oplus \textup{Ker} B_1$ where
$\widetilde{B_1}$ is one-to-one and has dense range and $B_{11}, B_{22}$ are positive operators. Thus $T$
has the form $T=\left[\begin{array}{cc}
\widetilde{B_1}B_{11}&\widetilde{B_1}B_{12}\\
0&0\\
\end{array}\right]$
respect to the space decomposition. Since $T$ is an amenable operator, by \cite[Lemma 2.8]{Y}, $T$ is similar to
$\left[\begin{array}{cc}
\widetilde{B_1}B_{11}&0\\
0&0\\
\end{array}\right]$ and there exists an operator $S$ such that $\widetilde{B_1}B_{12}=\widetilde{B_1}B_{11}S$.
Note that $\widetilde{B_1}, B_2$ are one-to-one, hence $B_{12}=B_{11}S$, and $B_{11}$ is one-to-one. Thus
without loss of generality, we may assume that $B_1$ has dense range and $B_2$ is one-to-one.
Note that $B_1^{\frac{1}{2}}B_2B_1^{\frac{1}{2}}, B_2^{\frac{1}{2}}B_1B_2^{\frac{1}{2}}$ are positive operators
and $TB_1^{\frac{1}{2}}=B_1^{\frac{1}{2}}B_1^{\frac{1}{2}}B_2B_1^{\frac{1}{2}}$ and
$B_2^{\frac{1}{2}}T=B_2^{\frac{1}{2}}B_1B_2^{\frac{1}{2}}B_2^{\frac{1}{2}}$, by Lemma \ref{lem5}, $T$ is similar
to a normal operator.
\end{proof}
\begin{thm}
The two decompositions for an amenable operator in Theorem \ref{thm2}, \ref{thm22} are the same.
\end{thm}
\begin{proof}
According to Theorem \ref{thm2}, \ref{thm22}, and Proposition \ref{prop5}, it is suffices to proof that
$N_1\in
Lat \mathfrak{A}_T'$.
In fact, if not. $T$ has the form $T= \left[\begin{array}{cc}
T_1&\\
&T_2\\
\end{array}\right]
\begin{matrix}
\mbox{$N_1$}\\
\mbox{$N_2$}\\
\end{matrix}$
and there exists
$S= \left[\begin{array}{cc}
0&0\\
Y&0\\
\end{array}\right]
\begin{matrix}
\mbox{$N_1$}\\
\mbox{$N_2$}\\
\end{matrix}\in \mathfrak{A}_T'$ where $Y\neq 0$.
Note that $S$ and $T$ have the form $$S= \left[\begin{array}{ccc}
0&0&0\\
\tilde{Y}&0&0\\
0&0&0\\
\end{array}\right]
\begin{matrix}
\mbox{$N_1$}\\
\mbox{$\overline{\textup{Ran} Y}$}\\
\mbox{$N_2\ominus\overline{\textup{Ran} Y}$}\\
\end{matrix},
T= \left[\begin{array}{ccc}
T_1&0&0\\
&T_{21}&T_{22}\\
&T_{23}&T_{24}\\
\end{array}\right]
\begin{matrix}
\mbox{$N_1$}\\
\mbox{$\overline{\textup{Ran} Y}$}\\
\mbox{$N_2\ominus\overline{\textup{Ran} Y}$}\\
\end{matrix},$$
where $\tilde{Y}$ has dense range. Note that $TS=ST$, we get that $T_{23}=0$. Since $T$ is amenable, by
\cite[Lemma 2.8]{Y} there exists an operator $B: N_2\ominus\overline{\textup{Ran} Y}\rightarrow
\overline{\textup{Ran} Y}$ such that
$$\left[\begin{array}{ccc}
I&0&0\\
&I&B\\
&&I\\
\end{array}\right]
\left[\begin{array}{ccc}
T_1&0&0\\
&T_{21}&T_{22}\\
&&T_{24}\\
\end{array}\right]
\left[\begin{array}{ccc}
I&0&0\\
&I&-B\\
&&I\\
\end{array}\right]
=\left[\begin{array}{ccc}
T_1&0&0\\
&T_{21}&0\\
&&T_{24}\\
\end{array}\right].$$ Moreover,
$$\left[\begin{array}{ccc}
I&0&0\\
&I&B\\
&&I\\
\end{array}\right]
\left[\begin{array}{ccc}
0&0&0\\
\tilde{Y}&0&0\\
0&0&0\\
\end{array}\right]
\left[\begin{array}{ccc}
I&0&0\\
&I&-B\\
&&I\\
\end{array}\right]=
\left[\begin{array}{ccc}
0&0&0\\
\tilde{Y}&0&0\\
0&0&0\\
\end{array}\right].$$
Hence, we can assume that $Y$ has dense range. Using $T$ is amenable again, there exists
$L=\left[\begin{array}{cc}
0&X\\
0&0\\
\end{array}\right]
\begin{matrix}
\mbox{$N_1$}\\
\mbox{$N_2$}\\
\end{matrix}\in \mathfrak{A}_T'$, where $X\neq 0$, by \cite[ lemma 4.11]{G2006}.
Similar to the decomposition to $S$ and $T$, we get that $S$, $L$ and $T$ have the form
$$S=
\left[\begin{array}{ccc}
0&0&0\\
\tilde{Y_1}&0&0\\
\tilde{Y_2}&0&0\\
\end{array}\right],
L= \left[\begin{array}{ccc}
0&0&\tilde{X}\\
0&0&0\\
0&0&0\\
\end{array}\right],
T= \left[\begin{array}{ccc}
T_1&0&0\\
&T_{31}&T_{32}\\
&T_{33}&T_{34}\\
\end{array}\right],$$
respect to the space decomposition $\mathfrak{H}=N_1\oplus \textup{Ker} X\oplus(N_2\ominus \textup{Ker} X)$,
where $\tilde{X}$ is one-to-one, and $\tilde{Y_1},\tilde{Y_2}$ has dense range. Note that $LT=TL$, we get that
$T_{33}=0$.
Using $T$ is amenable again, there exists an operator
$C: N_2\ominus \textup{Ker} X\rightarrow \textup{Ker} X$ such that
$$\left[\begin{array}{ccc}
I&0&0\\
&I&C\\
&&I\\
\end{array}\right]
\left[\begin{array}{ccc}
T_1&0&0\\
&T_{31}&T_{32}\\
&&T_{34}\\
\end{array}\right]
\left[\begin{array}{ccc}
I&0&0\\
&I&-C\\
&&I\\
\end{array}\right]
=\left[\begin{array}{ccc}
T_1&0&0\\
&T_{31}&0\\
&&T_{34}\\
\end{array}\right]$$
$$\left[\begin{array}{ccc}
I&0&0\\
&I&C\\
&&I\\
\end{array}\right]
\left[\begin{array}{ccc}
0&0&\tilde{X}\\
0&0&0\\
0&0&0\\
\end{array}\right]
\left[\begin{array}{ccc}
I&0&0\\
&I&-C\\
&&I\\
\end{array}\right]
=\left[\begin{array}{ccc}
0&0&\tilde{X}\\
0&0&0\\
0&0&0\\
\end{array}\right]$$
and
$$\left[\begin{array}{ccc}
I&0&0\\
&I&C\\
&&I\\
\end{array}\right]
\left[\begin{array}{ccc}
0&0&0\\
\tilde{Y_1}&0&0\\
\tilde{Y_2}&0&0\\
\end{array}\right]
\left[\begin{array}{ccc}
I&0&0\\
&I&-C\\
&&I\\
\end{array}\right]
=\left[\begin{array}{ccc}
0&0&0\\
\tilde{Y_1}+C\tilde{Y_2}&0&0\\
\tilde{Y_2}&0&0\\
\end{array}\right].$$
Moreover, $\tilde{Y_2}T_1=T_{34}\tilde{Y_2}, T_1\tilde{X}=\tilde{X}T_{34}$, and $T_1$ is similar to a normal
operator, by Lemma \ref{lem5}, $T_{34}$ is similar to a normal operator, which contracts to Theorem \ref{thm22}.
\end{proof}
\begin{cor}
Assume $T$ is an amenable operator, then $M$ is a maximal invariant subspace such that $T|_M$ is similar to a
normal operator if and only if $M$ is the largest invariant subspace such that $T|_M$ is similar to a normal
operator.
\end{cor}
\begin{cor}
Assume $T$ is an amenable operator and which is quasisimilar to a compact operator, then $T$ is similar to a
normal operator.
\end{cor}
\begin{proof}
Suppose, $TV=VK, WT=KW$ with $V,W$ injective operators having dense ranges and $K$ is a compact operator. Then
$TVKW=VKWT$. Let $C=VKW$, $C\in \mathfrak{A}_{T}'$, and $C$ is a compact operator. According to Theorem
\ref{thm2}, $C$ has the form $\left[\begin{array}{cc}
C_1&\\
&0\\
\end{array}\right]$ respect to the space decomposition in the Theorem. If $Cx=0$, $VWTx=Cx=0$, thus $Tx=0$. It
follows that there is no part of $T_2$, i.e. $T$ is similar to a normal operator.
\end{proof}
\vskip1cm
\section{(Essential) operator valued roots of abelian analytic functions}
In this section, we will study the structure of an operator which is an (essential) operator valued roots of
abelian analytic functions and then
we get that if such an operator is also amenable, then it is similar to a normal operator.
In \cite{G1974} Gilfeather introduce the concept of operator valued roots of abelian analytic functions as
follows: Let $\mathfrak{A}$ is an abelian von Neumann algebra and $\psi(z)$, an $\mathfrak{A}$ valued analytic
function on a domain $\mathcal{D}$ in the complex plane. We may decompose $\mathfrak{A}$ into a direct integral
of factors such that for $A\in\mathfrak{A}$, there exists a unique $g\in L_{\infty}(\wedge, \mu)$ such that
$\mathfrak{A}=\int_\wedge^\oplus g(\lambda)I(\lambda)\mu(d\lambda)$. If $T\in\mathfrak{A}'$ and
$\sigma(T)\subseteq \mathcal{D}$, let
$$\psi(T)=(2\pi i)^{-1}\int_\wedge(T-zI)^{-1}\psi(z)dz.$$
An operator $T$ is called a (essential)roots of the abelian analytic function $\psi$, if $\psi(T)=0$(compact,
respectively). The structure of roots of a locally nonzero abelian analytic function has been given in
\cite{G1974}, in this section we main study the structure of essential roots of a locally nonzero abelian
analytic function.
\begin{lem}\label{lem6}
Assume $T\in\mathfrak{B}(\mathfrak{H})$, $f$ is a locally nonzero analytic function on the neighborhood of
$\sigma(T)$ and assume $f(T)$ is a compact operator, then $T$ is a polynomial compact operator.
\end{lem}
\begin{proof}
Let $\widehat{T}$ denote the image of $T$ in the Calkin algebra, then $\widehat{f(T)}=0$. Since $f$ is a locally
nonzero analytic function on $\sigma(T)$, there exists a polynomial $p$ such that $\widehat{p(T)}=0$. i.e. $T$
is a polynomial compact operator.
\end{proof}
\begin{thm}
Let $\psi$ be a locally nonzero abelian analytic function on $\mathcal{D}$ taking values in the von Neumann
algebra $\mathfrak{A}$. If $T$ is an essential roots of $\psi$ and is amenable, then $T$ is similar to a normal
operator.
\end{thm}
\begin{proof}
Since $\mathfrak{A}$ is an abelian von Neumann algebra, $\mathfrak{A}$ is unitarily equivalent to
$\sum_{n=1}^\infty\oplus \mathfrak{B}_n\oplus\mathfrak{B}_{\infty} $, where $\mathfrak{B}_n$ is a von Neumman
algebra which has
uniform multiplicity $n$ for all $1\leq n\leq\infty$. Note $T\in \mathfrak{A}'$ is an amenable operator, thus $T=T_1\oplus T_2$,
where $T_1$ is similar to a normal operator, and $T_1\in(\sum_{n=1}^\infty\oplus \mathfrak{B}_n)',
T_2\in\mathfrak{B}_{\infty}'$. Let $\sigma_1(\sigma_2)$ denote the continuous (atom, respectively) parts of the spectrum
of $\mathfrak{B}_{\infty}$, then $\mathfrak{B}_{\infty}=\mathfrak{C}_{\infty}\oplus\mathfrak{D}_{\infty}$,
where $\mathfrak{C}_{\infty}$ and $\mathfrak{D}_{\infty}$ are uniform multiplicity $\infty$ von Neumman
algebra and $\sigma(\mathfrak{C}_{\infty})=\sigma_1,\sigma(\mathfrak{D}_{\infty})=\sigma_2$ and $T_2=T_3\oplus
T_4$, where $T_3\in\sigma(\mathfrak{C}_{\infty})', T_4\in\sigma(\mathfrak{D}_{\infty})'$.
Assume $\psi$ is a locally nonzero abelian analytic function on $\mathcal{D}$ and
$\sigma(T)\subseteq\mathcal{D}$, then $\psi(T)=\psi(T_1)\oplus\psi(T_3)\oplus\psi(T_4)$, note that $\psi(T_3)$
is a compact operator and $\sigma(\mathfrak{C}_{\infty})=\sigma_1$, so $\psi(T_3)=0$. Since
$\mathfrak{D}_{\infty}$ are uniform multiplicity $\infty$ and $\sigma(\mathfrak{D}_{\infty})=\sigma_2$, by Lemma
\ref{lem6}, it follows that $T_4$ is direct sum of polynomial compact operators. According to \cite[Theorem
2.1]{G1974}, there exists a sequence of mutually orthogonal projections $\{P_n, Q_m\}$ in $\mathfrak{A}$ with
$I=\sum P_n+\sum Q_m$ so that $T|_{P_n}$ is finite type spectral operator and $T|_{Q_m}$ is polynomial compact
operator. By \cite[Theorem 3.5, 4.5]{Y}, we get that $T$ is similar to a normal operator.
\end{proof}
\nocite{liyk2,liyk/kua1}
| 138,225
|
NEW YORK Ice Cream specialty chain Cold Stone Creamery of Scottsdale, Ariz., is using at its Times Square location here Panasonic Information Systems' Stingray workstation, a self-contained point-of-sale terminal that functions without a back-office server.
"Stingray's all-in-one design reduces the complexity of uploading sales or downloading programming changes," said Tom Streveler, director of in-store technology for Cold Stone-parent Kahala Corp. "The 15-inch touchscreen reduces keying errors, and the processing power makes the system very responsive. In addition, by eliminating the back-office PC, we expect our franchisees to experience fewer support issues and interruptions to their store operations."
Cold Stone Creamery uses Panasonic registers, SMP software and back-office management workstations in all of its approximately 1,400 locations worldwide, with Streveler citing the technology's durability and ease of use. He said that the Stingray workstation runs SMP software but "removes the communication step between the registers and a PC, while still offering the same array of reporting options to the franchisee and our corporate office."
Elgin, Ill.-based Panasonic Information Systems is a division of the Panasonic Systems Solutions Co. unit of Panasonic Corp. of North America.
| 103,228
|
Money Management Strategies for Teens
Posted by Jill Suskind
Let’s take the “haphazard” out of our teens’ money management habits.
Many of us learned our money habits piecemeal, rather than as a whole system with all parts working together. As a teenager, when the occasion arose to contribute to charity, I sometimes donated some of my own money. When I wanted to save for something expensive, I would put all my money toward that thing until I bought it. When I needed to buy something and I had enough money, I bought it. Money management for me was completely based on my emotions.
Does that sound familiar?
The problem is, with this system, we don’t ever learn to make all the parts work at the same time.
This is where parents fail to prepare their teens for adulthood with excellent money management habits. I know I used the “Hack it together” system of money management for most of my adult life. I mean, I pretty much just hacked together a system that worked, sort of, and I added parts to it as I grew and when I had time to think about it.
When we think about it, why would we ever want our teen to suffer the consequences of poor money management habits when they are adults? It puts them at a terrible disadvantage and risk for major problems in their adulthood. It’s so unnecessary and so preventable. The internet is an excellent source of educational materials for teens in the area of money. There are summer camps where they can go learn about money and have a blast! There are books and articles that teach about compound interest and why it makes so much sense to start to save and invest at a young age. There are board games -- I highly recommend Robert Kiyasaki’s Cash Flow 101 that teaches how to think about money as a wealth builder.
In the WealthQuest for Teens Online Video, teens learn the Silo System. They learn to design a whole money management strategy that is organized around their needs, wants, dreams, and commitments. This strategy makes money management matter to them on a personal level.
| 385,155
|
Wild Wings
Window into Winter Framed Art Print Wall Art
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Embellish your walls with this Window into Winter Framed Art Print Wall Art by Marjolein Bastin and make an impressive statement in your home, cabin, lodge, ranch, or cottage. Framed, open edition art print made from the original watercolor painting depicting a charming wintery bird feeder surrounded by a border of paperwhites, holly and berries. Image size, 10.5" x 15.75", Framed size, 19" x 23.5" with a 2" rustic pine-finished moulding with an oatmeal-colored, faux burlap mat. Give your walls a comforting feeling and make your walls come to life when you hang this framed art print. This stunning wall decor will make a bold statement and suit the theme of your decor.View AllClose
- Size:
- Image Size: 10.5" x 15.75"
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| 409,576
|
TITLE: Random graphs with a hamiltonian path
QUESTION [6 upvotes]: Suppose we randomly choose a graph with $n$ vertices in the following manner: each edge is included with probability $\frac12$. Thus, each graph has the same probability of being chosen, and there are $2^{\frac{n(n-1)}{2}}$ possible graphs. Denote $H_n$ the event that such a graph contains a Hamiltonian path. Prove that
$$\lim_{n \to \infty} P(H_n) = 1,$$
that is, as we consider larger and larger graphs, the proportion of graphs that contain a Hamiltonian path approaches $1$.
As a bonus, find an efficient algorithm for finding a Hamiltonian path in a graph. The algorithm must also be precise, that is, it must find a Hamiltonian path most of the time. More concretely, we are looking for an algorithm with the following two properties:
Its average time complexity is polynomial with respect to the number of vertices. (But you should aim for as good as possible.)
Denote $S_n$ the event that the algorithm succeeds if its input is a random graph with $n$ vertices. Then,
$$\lim_{n \to \infty} P(S_n) = 1.$$
So, what have I tried so far? There is a relatively easy way to obtain the lower bound of $0.25$ that holds for all $n$. Denote $X_n[a]$ the event that there exists a Hamiltonian path starting at $a$, in a random graph with $n$ vertices. Clearly, the probability of such an event occuring is the same for all vertices, and we will denote this probability $x$. We can bound $x = P(X_n[a])$ from below as follows:
If there is at least one edge to some vertex $b$, we can consider Hamiltonian paths that start with vertices $a, b$. The probability that such a Hamiltonian path exists is $P(X_{n-1}[b]) = x_{n-1}$.
If there is no edge from $a$, there is no Hamiltonian path starting at $a$. The probability of this occuring is $\frac{1}{2^{n-1}}$.
Thus we obtain the following inequality:
$$x_n \geq (1 - \frac {1}{2^{n-1}}) \cdot x_{n-1}.$$
From this it is easy to prove that $x_n \geq \frac14$ for all $n$. To obtain a better lower bound, we can consider additional cases based on how many edges there are from $a$:
If there are $0$ adjacent vertices, there is no hope.
If there is one adjacent vertex, we can reduce to $x_{n-1}$.
If there are at least $2$ adjacent vertices, denote them $b$ and $c$. The path can go through either $b$ or $c$. Thus, by the inclusion-exclusion principle, the probability that at least one of the paths exists is
$$P(X_{n-1}[b]) + P(X_{n-1}[c]) - P(X_{n-1}[b] \land X_{n-1}[c]).$$
Thus, if we were to find a good enough upper bound for $P(X_{n-1}[b] \land X_{n-1}[c])$, the problem would be solved. However, such an upper bound eludes me, all I can come up with are lower bounds (which are not helpful).
Alternative forms of the above expression that may help:
$$P(X_{n-1}[b])\ +\ P(X_{n-1}[c] \mid \neg X_{n-1}[b])$$
In this form, we are trying to find a lower bound for the second term, which corresponds to the probability that there exists a Hamiltonian path starting at $c$, given that we know that there is no Hamiltonian path from $b$.
Also, it may be better to consider the case with more vertices adjacent to $a$, such as three. Though with more vertices comes bigger trouble when applying the inclusion-exclusion principle.
REPLY [4 votes]: For context: a result of Pósa says that a random graph with $C n \log n$ edges already contains a Hamiltonian path with probability tending to $1$, and here we have a uniformly random graph (with $\frac{n^2}{4}$ edges on average). So we can be fairly wasteful.
As far as an algorithm goes, we can take the algorithm used to prove Dirac's theorem. The hypotheses of that theorem don't hold here, but the algorithm will still work with high probability. The strategy is this:
Pick a path greedily: start at a vertex $v_1$, pick a vertex $v_2$ it is adjacent to, then pick a vertex $v_3$ adjacent to $v_2$, and so on. Repeat until you get stuck.
Turn the path into a cycle: if it has endpoints $v_1$ and $v_\ell$, find adjacent vertices $v_i$ and $v_{i+1}$ on the path such that $v_{i+1}$ is adjacent to $v_1$ and $v_i$ is adjacent to $v_\ell$. Then the cycle is $v_1, \dots, v_i, v_\ell, \dots, v_{i+1}, v_1$.
Turn the cycle into a longer path: if $v_{\ell+1}$ is a vertex not on the cycle, find a vertex $v$ on the cycle adjacent to $v_{\ell+1}$, and take the path starting next to $v$, going around the cycle, and then going to $v_{\ell+1}$.
Repeat steps 2 and 3 until the path contains all the vertices.
All of these steps are quite likely to work individually. The problem is that to analyze how likely they are to work as a whole, we'd have to check edges of the graph multiple times, which doesn't preserve independence.
So instead, we will write our uniformly random graph $G$ as the union of graphs $G_0, G_1, \dots, G_{10 \log n}$, where $G_0$ contains each edge independently with probability $\frac14$, and $G_1, \dots, G_{10 \log n}$ contain each edge independently with probability $p$, chosen so that $\frac34(1-p)^{10 \log n} = \frac12$. Then the union will contain each edge with probability $\frac12$, so it will be the uniformly random graph. If we solve for $p$, we get $p = O(\frac{1}{\log n})$, but all we'll really need is for $p$ to be asymptotically bigger than $\frac{1}{\sqrt n}$.
Then I claim that:
A greedy path chosen in $G_0$ will reach length $n - 5\log n$ with very high probability.
Doing step 2 of the algorithm in any of the $G_1, \dots, G_{10 \log n}$ will work with very high probability.
Doing step 3 of the algorithm in any of the $G_1, \dots, G_{10 \log n}$ will work with very high probability.
Then we can just look at graphs $G_0, G_1, \dots$ sequentially as we go through the algorithm. This preserves independence, and if the edges we need will be in the graph $G_i$ we're looking at, they will be in the union $G$.
For the first claim: the probability that a greedy path will get stuck while there's still $5 \log n$ vertices to pick from is $(\frac34)^{5\log n} = n^{-5\log \frac43}$, so the probability is at most $n^{1 - 5\log \frac43} \approx n^{-0.43}$ that it ever gets stuck.
For the second claim: we have at least $\frac n2$ options for vertices $v_i$ and $v_{i+1}$, and each one works with probability $p^2$. The probability that none of them work is $(1-p^2)^{n/2} \le e^{-p^2n/2}$, which approaches $0$ as $n\to \infty$. (This is where we want $p \gg \frac{1}{\sqrt n}$, so that the exponent goes to $-\infty$ as $n\to\infty$. We actually want a bit better than that, because we want all $O(\log n)$ of these steps to work, so the probability should go to $0$ faster than $\frac{1}{\log n}$. But our $p$ is actually quite a bit larger than $\frac{1}{\sqrt n}$, so we have that wiggle room.)
For the third claim: we have at least $\frac n2$ options for the vertex $v$, and each works with probability $p$. The probability that none of them work is $(1-p)^{n/2} \le e^{-pn/2}$, which approaches $0$ as $n\to \infty$.
| 155,512
|
June 26, 2013
Since last Wednesday I have gotten in a couple rides on Poe, and they were all an improvement on the day before. That pony is unflappable, and picks up on things quickly. He has a great mind set for games.
I was originally giving him a crash course on mounted games, introducing him to the equipment, and having a rider leaning and moving around on his back. I was planning to attend the Summer Sizzler on Sunday, and take him in the Green Pony division, to see what he thought of games in a real format, and try him on some hand offs. Then I was going to take Simon in intermediate and get in some good practice for nationals that is just a few quick weeks away. But the opportunity presented itself for me to go pick up my new (used) trailer that day instead. And since it was going to be a 10 hour round trip drive, minimum, I decided to get that done instead. So I stopped the crash course, and just continued the basic games introduction, and spent more time getting acquainted with Poe.
One day I decided to throw balloon race at him. Nothing else had phased him, and balloon tends to be one of those races that pushes the limits on some new ponies. So I wanted to see what he thought. I led Poe down to the riding field and, I went straight over to my old balloon board and started blowing up and attaching balloons while Poe was standing with me. He nibbled on my pockets, and laid his head against my arm while I was blowing up balloons. I even popped one smack in my face. He did not seem to notice. Once all the balloons were attached, I laid the board on the ground and walked around it to go set up some other stuff. I was leading Poe, who took a short cut and stepped over the balloons instead of skirting them. Yep, he sure seemed terrified of them.
After a brief warm up I dove right into balloon, and decided just to give it a trot past and pop one. Poe was unfazed, and after cantering back and forth until I had all the balloons popped, I don’t think he even flicked an ear. This guy was born to be a games pony.
We worked on some Litter, and again, cool as a cucumber. Flag, nothing to it. Bottle, we got that. The only races that have shown any need for adjustment at this point would be mug shuffle and ball and cone. And When I say, they show need for adjustment, that is really a stretch. In mug shuffle, he is still a tiny bit confused if he is supposed to go straight or if he should bend the poles. And in ball and cone he is a little unsure if he should step under me when I lean over. Yeah, so not really issues at all.
Neck reining is a must in games. Riders spend a lot of time with something in one of their hands, so being able to navigate with one hand is important. Last night I spent some time working on his neck reining. I started a drill I have used with all my past ponies that needed to learn this skill. I start off at a walk, weaving through the line of poles, around the end pole, and back through, and around the end and back. So on, just back and forth, turning the ends. I direct rein, neck rein and employ my legs. I eventually use less and less direct rein and continue with the neck rein and legs only.
After a few times through, Poe started to respond to only neck reining through the weave. The end turn still required leg and direct, but not as much. The reason I do this drill, is that the pony quickly picks up the pattern. Then they start to key in on the rein on the neck, signaling the weave. Eventually they pick it up on the end turn too. I intentionally keep the end turn tight. So that they learn to make tight turns, and also so that turn is harder to do with just a neck rein. More of a challenge.
Of course I am also always neck reining when I ride him, in conjunction with direct reining. He is a smart pony, and I think this will come quickly.
I am eager to push him up in speed through the games, but I am also a firm believer in taking my games training seriously and in steps. I feel that a pony needs to learn the races first and foremost. They need to learn the patterns, up-and-back, and they need to learn to stop and to go, at once, and not through gradual transitions. I feel like getting this down at a trot and easy canter is important, and then allow the pony to start picking his pace to a degree. At this point, the pony should have no idea that he is actually racing. Once the skills and patterns are pretty solid, adding some speed, and asking for a little more, degree by degree, is a good next step. Particularly with fossil pony training, where excellent manners, and the skill of stopping and standing is even more important, this slow progress is even more beneficial. Basically the pony learns his job first, and then to add in speed later. It seems to help keep their head on straight.
His mind set continues to amaze me. And I think a sport like mounted games will help keep his mind engaged and occupied. He is also a really friendly good natured pony. He is kind to both Simon and Linus, even though he is top dog. Linus, who is a very timid pony, happily lets Poe help him finish his feed, and Poe allows Linus to stay to lick the pan with him, vs running him off and keeping it all for himself. Simon seems almost enamored with Poe, and is constantly touching him. I am also happy that Poe loves attention and comes in with Simon and Linus when he sees me. He walks right up to me and lets me put on his halter and seems eager for one on one attention. He loves to put his head against me and enjoys being brushed.
I really like this pony. He is such a pleasure to be around, and I am really enjoying riding him. And training him for games is really fun!
| 37,268
|
\begin{document}
\title{Weak tangents and level sets of Takagi functions}
\author{Han Yu}
\address{Han Yu\\
School of Mathematics \& Statistics\\University of St Andrews\\ St Andrews\\ KY16 9SS\\ UK \\ }
\curraddr{}
\email{hy25@st-andrews.ac.uk}
\thanks{}
\subjclass[2010]{Primary: 28A80, 37C45 Secondary: 26A27}
\keywords{Littlewood polynomial, Takagi functions, level sets, Assouad dimension}
\date{}
\dedicatory{}
\begin{abstract}
In this paper we study some properties of Takagi functions and their level sets. We show that for Takagi functions $T_{a,b}$ with parameters $a,b$ such that $ab$ is a root of a Littlewood polynomial, there exist large level sets. As a consequence we show that for some parameters $a,b$, the Assouad dimension of graphs of $T_{a,b}$ is strictly larger than their upper box dimension. In particular we can find weak tangents of those graphs with large Hausdorff dimension, larger than the upper box dimension of the graphs.
\end{abstract}
\maketitle
\allowdisplaybreaks
\section{Introduction}
In this paper we study graphs of the following functions,
\[
T_{a,b}(x)=\sum_{n=0}^{\infty} a^nT(b^n x),
\]
where $a,b$ are real parameters $a,b$ such that $a<1,b>1, ab\geq 1$ and $T:\mathbb{R}\to\mathbb{R}$ is the tent map which has period $1$ and defined on the unit interval as follows,
\[
T(x)=
\begin{cases}
x & x\in [0,\frac{1}{2}] \\
1-x & x\in [\frac{1}{2},1].
\end{cases}
\]
Such functions $T_{a,b}$ are called the Takagi functions. Originally the Takagi function was referred to $T_{1/2,2}$ but no confusion should appear if we also call $T_{a,b}$ the Takagi functions. There has been a lot of interest in the Hausdorff and box counting dimensions of the graphs of such functions. For the box dimensions we know from \cite[Section 2]{KPY} and \cite[Theorem 2.4]{Ba} that the upper box dimension of graphs of these functions $T_{a,b}$ can be computed by the following formula
\[
B=2+\frac{\ln a}{\ln b}=1+\frac{\ln ab}{\ln b}.
\]
The Hausdorff dimensions of graphs of these functions are harder to obtain, see \cite{WXS},\cite{BBR} and the references therein for more recent results on related questions.
One of the results of this paper is about the Assouad dimension of some Takagi functions. In what follows, for a function $f:\mathbb{R}\to\mathbb{R}$, we denote the following set
\[
\Gamma_f=\{(x,y)\in\mathbb{R}^2:x\in [0,1], y=f(x)\},
\]
to be the graph of $f$ over the interval $[0,1]$.
\begin{thm}[Assouad dimension]\label{MAIN}
Let the product $ab>1$ be a root of a Littlewood polynomial of degree $k-1$, namely
\[
\sum_{n=0}^{k-1}\epsilon_n(ab)^n=0,
\]
for a sequence $\{\epsilon_n\}_{n\in \{0,\dots,k-1\}}$ over $\{0,1\}$.
Furthermore, if $b$ is an integer greater than $2$, then we have the following result,
\[
\dim_{\mathrm{A}} \Gamma_{T_{a,b}}\geq 1+\frac{1}{k}.
\]
\end{thm}
Notice that by keeping the product $ab$ unchanged and making $b$ larger, this lower bound can be larger than the upper box dimension $\frac{\ln ab}{\ln b}$ for large $b$. For example, when we choose parameters such that $ab=\frac{\sqrt{5}+1}{2}, b=8$, then $\ubox\Gamma_{T_{a,b}}\approx 1.23$ and $\Assouad\Gamma_{T_{a,b}}\geq 4/3.$
One consequence of Theorem \ref{MAIN} is that there exist large weak tangents of the graphs of Takagi functions. See Section \ref{Pre} for more details about the notions of dimensions, definition of weak tangent and some basic properties.
\begin{lma}[Weak tangent]\label{Weaktangent}
Let $a,b$ be as in the statement of Theorem \ref{MAIN}, then there exists weak tangent $E$ of $\Gamma_{T_{a,b}}$ such that
\[
\Haus E=\Assouad \Gamma_{T_{a,b}}\geq\ubox\Gamma_{T_{a,b}}.
\]
The last inequality can be strict.
\end{lma}
Theorem \ref{MAIN} follows from the existence of large level set of graphs $\Gamma_{T_{a,b}}$ and we think this result is interesting on its own.
\begin{thm}\label{LEVEL}
Let $T_{a,b}$ be as in the statement of Theorem \ref{MAIN} but we also allow $ab=1$. For each $y\in\mathbb{R}$ we define the following level set
\[
L(y)=\{x\in [0,1]: T_{a,b}(x)=y\}\times \{y\}.
\]
There exists $y\in\mathbb{R}$ such that
\[
\Haus L(y)\geq \frac{1}{k}.
\]
\end{thm}
The restriction of the product $ab$ as a certain algebraic integer seems to be really strong, however, with some effort we can show that those algebraic integers are dense in $[\frac{1}{2},2]$.
\begin{thm*}
Let $L$ be the set of algebraic integers which are roots of Littlewood polynomials namely, $x\in\mathbb{C}$ and there exist a finite sequence $\epsilon_n\in\{\pm 1\}$ such that
\[
\sum_{n=0}^{k-1}\epsilon_nx^n=0.
\]
Then $L\cap [\frac{1}{2},2]$ is dense in $[\frac{1}{2},2]$.
\end{thm*}
Proofs of the above result and its generalizations can be found in \cite{OP}, \cite{Bandt} and \cite{BY}.
\section{Discussions and future work}
In this section we give some backgrounds of Theorem \ref{MAIN} and \ref{LEVEL}. We also pose some questions which are related with the results in this paper.
\subsection{Assouad dimensions of graphs of functions}
Theorem \ref{MAIN} deals with the Assouad dimension of some Takagi functions. It is natural to think about the Assouad dimension of other nowhere differentiable functions, for example Weierstrass functions and graphs of the Wiener process. For the latter, we have the following result (\cite[Theorem 2.2]{HY}).
\begin{thm}[HY17]
The graph of the Wiener process $W(.)$ over the unit interval has the Assouad dimension equal to $2$ almost surely.
\end{thm}
We have not completely determined the Assouad dimension of any Takagi function yet. We only showed that the Assouad dimension can be strictly larger than the upper box dimension for graphs of Takagi functions. Based on the above theorem we think that the Assouad dimension of all Takagi functions should be $2$.
\begin{conj}
For $a,b\in\mathbb{R}^+$ and $ab>1$, we have that for all Takagi functions $T_{a,b}$
\[
\Assouad \Gamma_{T_{a,b}}=2.
\]
\end{conj}
\subsection{Level sets of Takagi functions}
For more details about the level sets of Takagi functions, see \cite{JL} and \cite{AK}. Notice that if we set $a=0.5, b=2$ then we can find level set of $T_{a,b}$ with Hausdorff dimension at least $0.5$. This is sharp, see \cite{AMO}. For other values of $a,b$ for example $a=(\sqrt{5}+1)/16, b=8$ we see that we can find a level set with Hausdorff dimension at least $1/3$ and we do not know whether this is sharp.
\begin{ques}
What is the largest level set of the Takagi function
$
T_{\frac{\sqrt{5}+1}{16},8}
$
in terms of the Hausdorff dimension?
\end{ques}
\section{Notation}\label{Notation}
\begin{itemize}
\item[1.] For a real number $x\in\mathbb{R}$ we use the symbol $x^+$ to denote a number $x+\epsilon$ where $\epsilon>0$ is some fixed positive number whose value can be chosen freely and we will point out the specific value of $\epsilon$ when necessary. Similarly, we use $x^-$ for a number smaller but close to $x$.
\item[2.]For a function $f:\mathbb{R}\to\mathbb{R}$, the following set
\[
\Gamma_f=\{(x,y)\in\mathbb{R}^2:x\in [0,1], y=f(x)\},
\]
is called the graph of $f$ over the interval $[0,1]$.
\item[3.] For a real number $x$, we use $\lfloor x\rfloor$ to denote the greatest integer that is not strictly larger than $x$.
\end{itemize}
\section{Preliminaries}\label{Pre}
We will now introduce some notions of dimensions which will be used in this paper. We use $N_r(F)$ for the minimal covering number of a bounded set $F$ in $\mathbb{R}^n$ with balls of side length $r>0$.
\subsection{Hausdorff dimension}
For any $s>0$ and $\delta>0$ define the following quantity:
\[
\mathcal{H}^s_\delta(F)=\inf\left\{\sum_{i=1}^{\infty}(\mathrm{diam} (U_i))^s: \bigcup_i U_i\supset F, \mathrm{diam}(U_i)<\delta\right\}.
\]
The $s$-Hausdorff measure of $F$ is
\[
\mathcal{H}^s(F)=\lim_{\delta\to 0} \mathcal{H}^s_{\delta}(F),
\]
and Hausdorff dimension of $F$ is
\[
\Haus F=\inf\{s\geq 0:\mathcal{H}^s(F)=0\}=\sup\{s\geq 0: \mathcal{H}^s(F)=\infty \}.
\]
\subsection{upper box dimension}
The upper box dimension of $F$ is
\[
\overline{\nbox}=\limsup_{r\to 0}\left(-\frac{\log N_r(F)}{\log r}\right).
\]
\subsection{Assouad dimension and weak tangents}
The \textit{Assouad dimension} of $F$ is
\begin{align*}
\Assouad F = \inf \Bigg\{ s \ge 0 \, \, \colon \, (\exists \, C >0)\, (\forall & R>0)\, (\forall r \in (0,R))\, (\forall x \in F) \\
&N_r(B(x,R) \cap F) \le C \left( \frac{R}{r}\right)^s \Bigg\}
\end{align*}
where $B(x,R)$ denotes the closed ball of centre $x$ and radius $R$.
An important tool for studying the Assouad dimension is \emph{weak tangents} introduced in \cite{MT} and \emph{microsets} in \cite{Fu}. The next definition appeared in \cite[Definition 1.1]{Fr}.
\begin{defn}
Let $X\in\mathcal{K}(\mathbb{R}^n)$ be a fixed reference set (usually the closed unit ball or cube) and let $E,F\subset\mathbb{R}^n$ be compact sets. Suppose there exists a sequence of similarity maps $T_k:\mathbb{R}^n\to\mathbb{R}^n$ such that $d_\mathcal{H}(E,T_k(F)\cap X)\to 0$ as $k\to 0$. Then $E$ is called a \emph{weak tangent} of $F$.
\end{defn}
Here $(\mathcal{K}(\mathbb{R}^n),d_{\mathcal{H}})$ is a complete metric space with the Hausdorff metric, namely, for two compact subsets $A,B\subset\mathbb{R}^n$ is defined by
\[
d_{\mathcal{H}}(A,B)=\inf\{\delta>0: A\subset B_{\delta}, B\subset A_{\delta}\},
\]
where for any compact set $C\subset\mathbb{R}^n$
\[
C_{\delta}=\{x\in\mathbb{R}^n: |x-y|<\delta \text{ for some } y\in C\}.
\]
For Lemma \ref{Weaktangent} we need the following result \cite[Proposition 5.7]{KOR}
\begin{thm}[KOR]\label{ThKOR}
Let $F$ be a compact set with $\Assouad F=s$. Then there exist a weak tangent $E$ of $F$ such that
\[
\Haus E=s.
\]
In other words, we have
\[
\Assouad F=\max\{\Haus E: E \text{ is a weak tangent of } F\}.
\]
\end{thm}
\subsection{Assouad spectrum}
\begin{defn}[Fraser and Yu \cite{JMF3}]\label{ASP}
\begin{eqnarray*}
\Assouad^{\theta} F &=& \inf \bigg\{ \alpha \ : \ (\exists C>0) \, (\exists \rho>0) \, (\forall 0<R\leq \rho) \, (\forall x \in F) \\ \\
&\,& \qquad \qquad \qquad \qquad N_{R^{1/\theta}} \big( B(x,R) \cap F \big) \ \leq \ C \left(\frac{R}{R^{1/\theta}}\right)^\alpha \bigg\}.
\end{eqnarray*}
\end{defn}
As $\theta$ ranges in $(0,1)$ the function $\Assouad^{\theta} F$ with respect to $\theta$ is called the \emph{Assouad spectrum} of $F$. For more background introduction of Assouad dimension/spectrum and how they are related to homogeneity of fractal sets see \cite{JMF}, \cite{JMF2} as well as \cite{JMF3}. It is known (\cite[Theorem 1.3]{JMF3}) that for any $\theta\in (0,1)$ we have
\[
\ubox F\leq\Assouad^{\theta} F\leq\Assouad F.\label{Di}\tag{1}
\]
\subsection{Covering by disjoint cubes}
For convenience, in this paper we will count covering number with disjoint squares rather than balls. We denote $S(a,R)$ for $a\in\mathbb{R}^2, R>0$ as the square centred at $a$ with side length $2R$ whose sides are parallel to the coordinate axis. Since we are dealing with graph of functions, the choice of axis is natural. We denote the following covering number,
\begin{eqnarray*}
& &N(F\cap S(a,R),r)=\bigg|\bigg\{(i,j)\in\mathbb{Z}^2\cap [0,\lfloor R/r \rfloor+1]^2 :\\
& &S((a-R/2+r/2+ir,a-R/2+r/2+jr),r) \cap F\neq\emptyset\bigg\}\bigg|.
\end{eqnarray*}
This is equivalent to $N_r(F\cap B(a,R))$ in the sense that there exists a constant $C>0$ such that for all $a\in F, 0<r<R<1$ we have the following inequality,
\[
C^{-1}N_r(F\cap B(a,R)) \leq N(F\cap S(a,R),r)\leq C N_r(F\cap B(a,R)).
\]
\subsection{Some properties of Takagi functions}
In this paper we will use the following result whose proof can be found in \cite{HL} and we use the version presented in \cite[Theorem 2.4]{Ba}.
\begin{lma}\label{LL2}
Let $T:\mathbb{R}\to\mathbb{R}$ be a continuous piecewise $C^1$ and periodic function. Then the following function
\[
T_{a,b}(x)=\sum_{n=0}^{\infty} a^nT(b^n x)
\]
must satisfy one of the two properties,
\item{1}:
$T_{a,b}$ is piecewise $C^1$.
\item{2}:
For a positive constant $C>0$ and any interval $J\subset\mathbb{R}$ we have the following inequality,
\[
\sup_{x,y\in J} |T_{a,b}(x)-T_{a,b}(y)|\geq C|J|^{-\frac{\ln a}{\ln b}}.
\]
Notice that if $a<1,ab>1$ then $-\frac{\ln a}{\ln b}\in (0,1)$ and we see that if $|J|<1$ then
\[
\sup_{x,y\in J} |T_{a,b}(x)-T_{a,b}(y)|\geq C|J|.
\]
\end{lma}
\begin{rem}
When $T$ is the tent map defined in the beginning of the first section, it is known that when $a<1,ab\geq 1$, the function $T_{a,b}$ is nowhere differentiable therefore only the second property of lemma \ref{LL2} can be true.
\end{rem}
\section{Large level sets, proof of Theorem \ref{LEVEL}}
We show that there exist large level sets for function $T_{a,b}$ with certain parameters $a,b$. Since $ab$ is a root of a Littlewood polynomial we see that
\[
\sum_{i=0}^k {\epsilon_i} (ab)^i=0
\]
for an integer $k\geq 1$ and some choice of $\epsilon_i\in \{\pm 1\}$.
Next we consider the first $k$ terms partial sum
\[
F_1(x)=\sum_{n=0}^{k-1}a^nT(b^n x).
\]
The derivative of the above function is not continuous at $x=m b^{-k}$ for integers $m$. Let us now assume that $x$ is an irrational number then the derivative is
\[
F'_1(x)=\sum_{i=0}^{k-1} \epsilon_i(x) (ab)^i,
\]
where $\epsilon_i(x)\in \{\pm 1\}$ depends on the $b$-nary expansion of $x$. In particular, if
\[
x=0.b_1b_2\dots
\]
then
\[
\epsilon_i(x)=
\begin{cases}
1 & b_i\in [0,b/2] \\
-1 & b_i\in (b/2,1]
\end{cases}
\]
Therefore we can find at least $2$ disjoint intervals of length $\frac{1}{2b^{k-1}}$ where $F_1'(x)=0$ and $F_1(x)=a_1$ is for a constant $a_1\geq 0$ on those two intervals. Indeed, when
\[
\sum_{i=0}^{k-1} {\epsilon_i} (ab)^i=0,
\]
we also have
\[
-\sum_{i=0}^{k-1} {\epsilon_i} (ab)^i=0.
\]
So there are at least two intervals we can find and the union is symmetric with respect to the line $\{x=0.5\}$. Then because $F_1$ is also symmetric with respect to the line $\{x=0.5\}$ we see that $F_1(.)$ takes the same value on those two intervals, say, $I_1$ and $I_2$.
We consider the next $k$ terms sum
\[
F_2(x)=\sum_{n=k}^{2k-1}a^nT(b^n x).
\]
Then we can find $b$ many intervals of length $1/2b^{2k-1}$ in $I_1,I_2$ such that the above sum stays constant $a_2\geq 0$ on those intervals. To see this, consider $I_1$, which is an interval of length $1/2b^k$. Now observe the following
\[
F_2(x)=\sum_{n=k}^{2k-1}a^nT(b^n x)=\sum_{n=0}^{k}a^{n+k}T(b^{k}b^n x)=a^{k} F_1(b^k x).
\]
Therefore the graph of $F_2$ is an affine copy, or intuitively speaking, a narrowed version of the graph of $F_1$. Then we see that there are exactly $b$ many intervals in $I_1$ of length $1/2b^{2k-1}$ such that $F_2$ equals to $a_2$ on all those intervals. Indeed, over any interval the form $[l/b^{k-1},(l+1)/b^{k-1}]$ the graph of $F_2$ has $2b$ many platforms of the same level. That is to say, we can find $2b$ many $1/2b^{2k-1}$ length intervals on which $F_2(x)=a_2$. Since $I_1$ is only a half $1/b^{k-1}$ length interval, therefore we can find $b$ many platforms over $I_1$. Here we used the mirror symmetry of $F_2$.
We can apply the above argument to $j$-th $k$-terms partial sums for each $j\geq 2$ and as a result we can find a Cantor set $C$ such that $T_{a,b}(C)=\{c\}$ for a constant $c$. This Cantor set is a self-similar set satisfying open set condition and its Hausdorff dimension is $1/k$ (the contraction ratio is $1/b^k$ and branching number is $b$, see for example \cite[Theorem 9.3]{Fa}). Thus we have proved theorem \ref{LEVEL}.
\section{Squashing and counting, proof of Theorem \ref{MAIN}}\label{SAC}
In order to deal with the Assouad dimension of graphs of Takagi functions we need to handle the following quantity
\[
N(S(x,R/2)\cap\Gamma_{T_{a,b}},r).
\]
The situation is not too bad when we want to deal with the above quantity for $\Gamma_{f+g}$ with one of the functions, say $f$, is Lipschitz continuous. The following result is a localized and quantitative version of \cite[Lemma 2.1,2.2]{FF}.
\begin{lma}\label{ADD}
If $f:[0,1]\to\mathbb{R}$ is Lipschtz continuous with Lipschitz constant $M>0$ and $g:[0,1]\to\mathbb{R}$ is continuous then we have the following inequality for $0<r<R<1$ whenever $\frac{R}{r}$ is an integer,
\begin{eqnarray*}
& &\sup_{a\in [0,1]\times\mathbb{R}}N(S(a,R/2)\cap\Gamma_{f+g},r)\geq \\ & &\frac{1}{M+2}\sup_{a\in [0,1]\times\mathbb{R}}N(S(a,R/2)\cap\Gamma_{g},r)-\frac{M+2}{\lfloor M\rfloor+2}\frac{R}{r}.
\end{eqnarray*}
\end{lma}
\begin{rem}\label{ADDr}
For the case when $R/r$ is not an integer, we can replace $r$ with a larger value
\[
r'=R \frac{1}{\lfloor R/r \rfloor}.
\]
As we will eventually choose $R/r$ to be arbitrarily large, $r'$ and $r$ are essentially the same. For any $\delta>0$, if $R/r$ is large enough the following relation holds
\[
(1-\delta)r<r'<(1+\delta)r.
\]
Then the inequality of this theorem holds with $1/(M+2)$ being replaced by some other constant which depends only on $M$.
\end{rem}
\begin{proof}[Proof of lemma \ref{ADD}]
Let $a\in\mathbb{R}^2$ and consider the square $S(a,R/2)$. For any $r<R$ we consider the following rectangles for $i=0,1,\dots,\frac{R}{r}-1$
\[
S_i=[a-R/2+ir,a-R/2+(i+1)r]\times [a-R/2,a+R/2]\subset S(a,R/2).
\]
Each rectangle $S_i$ contains the following squares for $j=0,1,\dots,\frac{R}{r}-1$
\[
S_{ij}=[a-R/2+ir,a-R/2+(i+1)r]\times [a-R/2+jr,a-R/2+(j+1)r].
\]
Now if $S_{ij}\cap\Gamma_{g}\neq\emptyset$ we colour it black, otherwise we colour it white. Let $n_i\geq 0$ denote the number of black squares among $S_{ij},j=0,1,2,\dots,\frac{R}{r}-1$. By continuity of $g$, the fact that we have $n_i$ black squares implies the following inequality
\[
\sup_{x,y\in [a-R/2+ir,a-R/2+(i+1)r]} |g(x)-g(y)|\geq (n_i-2)r.
\]
By Lipschitz property of function $f$ we see that
\[
|f(x)-f(y)|\leq M|x-y|,
\]
this implies that
\[
\sup_{x,y\in [a-R/2+ir,a-R/2+(i+1)r]} |f(x)-f(y)|\leq Mr.
\]
Then we see that
\[
\sup_{x,y\in [a-R/2+ir,a-R/2+(i+1)r]} |f(x)+g(x)-f(y)-g(y)|\geq (n_i-2-M)r.
\]
So we see that to cover the set
\[
\{(x,y+f(x))\in\mathbb{R}^2 : (x,y)\in S_i\cap\Gamma_{g}\}
\]
we need at least
\[
n_i-2-M
\]
many squares of side length $r$. Summing over all $i$ we see that to cover the set
\[
\{(x,y+f(x))\in\mathbb{R}^2 : (x,y)\in S(a,R/2)\cap\Gamma_{g}\}
\]
we need at least
\[
\sum_{i} n_i-(M+2)\frac{R}{r}
\]
many squares with side length $r$.
The next fact to notice is that the following set
\[
\{(x,y+f(x))\in\mathbb{R}^2 : (x,y)\in S(a,R/2)\cap\Gamma_g\}
\]
is contained in a $R\times (M+1)R$ rectangle. This rectangle can be covered by $\lfloor M\rfloor+2$ squares with side length $R$, so for at least one of the $\lfloor M\rfloor+2$ squares need at least
\[
\frac{\sum_{i} n_i-(M+2)\frac{R}{r}}{\lfloor M\rfloor+2}
\]
many squares with side length $r$ to cover.
It is then easy to see that
\[
\{(x,y+f(x))\in\mathbb{R}^2 : (x,y)\in S(a,R/2)\cap\Gamma_{g}\}\subset \Gamma_{f+g}.
\]
This implies that
\[
\sup_{a'\in [0,1]\times\mathbb{R}}N(S(a',R/2)\cap\Gamma_{f+g},r)\geq
\frac{\sum_{i} n_i-(M+2)\frac{R}{r}}{\lfloor M\rfloor+2},
\]
and since $\frac{R}{r}$ is integer we see that
\[
\sum_{i} n_i=N(S(a,R/2)\cap\Gamma_{g},r).
\]
It follows that
\[
\sup_{a'\in [0,1]\times\mathbb{R}}N(S(a',R/2)\cap\Gamma_{f+g},r)\geq \frac{1}{M+2}N(S(a,R/2)\cap\Gamma_{g},r)-\frac{M+2}{\lfloor M\rfloor+2}\frac{R}{r},
\]
We can take the supreme of $a$ on the right hand side of the above inequality and the lemma concludes.
\end{proof}
Now we can move on dealing with the Assouad dimension of Takagi functions. We shall need the following lemma.
\begin{lma}\label{POWERBOUND}
For any number $1<\beta<2$ and all integer $k$, there exists a sequence $\epsilon_n$ of $\pm 1$ such that $|\sum_{n=0}^{k}\epsilon_n \beta^n| \leq \frac{1}{\beta-1}$.
\end{lma}
\begin{proof}
Let $k>0$ be an integer, consider the following two functions forming an IFS (known as iterated function system, see for example \cite[chapter 9]{Fa}),
\[
f_{-1}(x)=\beta x-1,
f_{1}(x)=\beta x+1.
\]
Then for any sequence $\epsilon_n\in\{\pm 1\}$ with $n=0,1,2,\dots,k-1$ we can define an iteration by
\[
f_{\epsilon_{0}}\circ\dots\circ f_{\epsilon_{k-2}}\circ f_{\epsilon_{k-1}},
\]
and notice that
\[
f_{\epsilon_{0}}\circ\dots\circ f_{\epsilon_{k-2}}\circ f_{\epsilon_{k-1}}(0)=\sum_{n=0}^{k-1}\epsilon_n \beta^n.
\]
So as long as we can find an iteration of this IFS such that the trajectory of $0$ stays bounded by $\frac{1}{\beta-1}$ the existence of a sequence $\epsilon_n$ will follow. Now since $f_1(0)=1<\frac{1}{\beta-1}$ which is the intersection of the line $y=f_{1}(x)$ and $y=x$, we can apply function $f_{1}$ before the value exceeds $\frac{1}{\beta-1}$ and apply $f_{-1}$ before the value drops below $-\frac{1}{\beta-1}$. More precisely, we put $x_0=0$ and if we find $x_i\in (-1/(\beta-1),1/(\beta-1))$ then if $f_1(x_i)<1/(\beta-1)$ we set $x_{i+1}=f_1(x_i)$ otherwise we set $x_{i+1}=f_{-1}(x_i)$. We need to check $x_{i+1}\in (-1/(\beta-1),1/(\beta-1))$ as well. In fact, if $x_{i+1}>1/(\beta-1)$ then $f_{-1}(x_i)>1/(\beta-1)$ and this implies that $x_i>1/(\beta-1)$. If $x_i<-1/(\beta-1)$ then either $f_1(x_i)<-1/(\beta-1)$ or $f_{-1}(x_i)<-1/(\beta-1)$ in the first case we have $x_i<-1/(\beta-1)$. The later case implies that $x_i<(2-\beta)/\beta(\beta-1)$ but then $f_1(x_i)<1/(\beta-1)$. So in any case $x_{i+1}\in (-1/(\beta-1),1/(\beta-1))$ as required. This procedure gives us an sequence $\epsilon_n$, with $n=0,1,2,\dots,k-1$ for any integer fixed $k-1$ such that
\[
\left|\sum_{n=0}^{k-1}\epsilon_n \beta^n\right| \leq \frac{1}{\beta-1}.
\]
\end{proof}
Notice that when $\beta$ is a root of a Littlewood polynomial it is necessary that $0.5\leq |\beta|\leq 2$. Therefore if parameters of $T_{a,b}$ are as stated in Theorem \ref{MAIN} then $1\leq ab\leq 2$ and therefore the result of Lemma \ref{POWERBOUND} holds for $ab$. Now we have all the ingredients needed to prove Theorem \ref{MAIN} however we find it convenient to introduce the following general result.
\begin{lma}\label{SC}[Squash and count]
Suppose $T_{a,b}(x):\mathbb{R}\to\mathbb{R}$ is a function of the following form
\[
T_{a,b}(x)=\sum_{n=0}^{\infty}a^nT(b^n x),
\]
where $T(x):\mathbb{R}\to\mathbb{R}$ is a piecewise $C^1$ continuous function with period $1$ and $a>0,b>0,ab>1$. Suppose the following two conditions holds:
\item[1, \emph{(interval with slow changing)}]: There exists a positive constant $C_1>0$ such that for any integer $M>0$, there is a integer $k$ such that $J_k=(\frac{k}{b^{M+1}},\frac{k+1}{b^{M+1}})$ and for all $x_1,x_2\in J_k$ the following condition holds,
\[
\bigg|\sum_{n=0}^{M}a^nT(b^n x_1))-\sum_{n=0}^{M}a^nT(b^n x_2))\bigg|<C_1|x_1-x_2|.
\]
\item[2, \emph{(large level set)}]:
There exists a level set $L\subset [0,1]$ with lower box dimension at least $D$, namely,
\[
\exists y\in\mathbb{R}, \lbox L(y)=\lbox\{x\in [0,1]: T_{a,b}(x)=y\}\geq D.
\]
Then we have the following result,
\[
\Assouad \Gamma_{T_{a,b}}\geq D+1.
\]
\end{lma}
\begin{proof}
For any positive integer $M>0$, we can find an integer $k$ and $x_0=\frac{k}{b^{M+1}}$ such that $(x_0,x_0+\frac{1}{b^{M+1}})\subset [0,1]$ and on this subset we have the following condition for the oscillation of the following $M$-th partial sum
\[
\bigg|\sum_{n=0}^{M}a^nT(b^n x_1))-\sum_{n=0}^{M}a^nT(b^n x_2))\bigg|<C_1|x_1-x_2|,
\]
where $x_1,x_2\in (x_0,x_0+\frac{1}{b^{M+1}})$. Then we can write that
\begin{eqnarray*}
T_{a,b}(x)=\sum_{n=0}^{\infty}a^nT(b^n x)&=&\sum_{n=0}^{M}a^nT(b^n x)+\sum_{n=M+1}^{\infty}a^nT(b^n x)\\
&=& F_M(x)+G_M(x),
\end{eqnarray*}
where the functions $F_M,G_M$ are the first sum and second sum in the third expression. Then we see that the graph $\Gamma_{G_M}$ is actually a 'squashed' version of $\Gamma_{T_{a,b}}$, namely we have the following relation
\[
X_M(\Gamma_{T_{a,b}})=\Gamma_{G_M},
\]
where the linear transformation $X_M:\mathbb{R}^2\to \mathbb{R}^2$ is defined to be as follows
\[
X_M(x,y)=\left(\frac{x}{b^{M+1}},a^{M+1}y,\right).
\]
We see that since $ab>1$ this linear transformation squashes a square to a very thin rectangle for large enough $M$. Now we concentrate on the strip
\[
S=\left(x_0,x_0+\frac{1}{b^{M+1}}\right)\times\mathbb{R}.
\]
The graph $\Gamma_{G_M}$ over this strip is the squashed version of the graph $\Gamma_{T_{a,b}}$ over $[0,1]$. We want to find a $\frac{1}{b^{M+1}}$-square contained in this strip such that we need a reasonably large amount of $r$-squares to cover $\Gamma_{G_M}$, where $r>0$ is number that will be specified later. Now because of the bijective linear map $X_M$, covering a $\frac{1}{b^{M+1}}$-square with $r$-square in $\Gamma_{G_M}$ is the same thing as covering the original graph $\Gamma_{T_{a,b}}$ over $[0,1]$ inside a $1\times\frac{1}{(ab)^{M+1}}$ -rectangle with $rb^{M+1}\times\frac{r}{a^{m+1}}$-rectangles.
Now consider the level set $L$ with lower box dimension $D$ mentioned in the second condition. For some real number $y'$ we have that
\[
L=L(y')=\{x\in [0,1]: T_{a,b}(x)=y'\}\times\{y'\},
\]
then we do box counting in the $1\times\frac{1}{(ab)^{M+1}}$-rectangle containing $L$. For any $(x,y')\in L$ the graph $\Gamma_{T_{a,b}}$ intersects the middle axis of the rectangle at $(x,y')$, so by lemma \ref{LL2} we see that there exists a positive constant $C_3>0$ such that the projection of graph $\Gamma_{T_{a,b}}$ inside any $\frac{1}{(ab)^{M+1}}$-square centred in $L$ to the vertical axis has length at least
\[
\min\bigg(C_3\frac{1}{(ab)^{M+1}},\frac{1}{(ab)^{M+1}}\bigg).
\]
If $M$ is large enough we need at least $\left(\frac{1}{ab}\right)^{-D^-(M+1)}$ many $\frac{1}{(ab)^{M+1}}$-square to cover this $1\times\frac{1}{(ab)^{M+1}}$-rectangle because for covering the level set $L$ we already need that many squares. Since each square is sufficiently occupied by $\Gamma_{T_{a,b}}$ in the sense that the curve occupies at least $\min(C_3,1)$ portion of the vertical length. This means that for each such square we need some constant times
\[
\frac{1/(ab)^{M+1}}{r/a^{M+1}}=\frac{1}{rb^{M+1}}
\]
many $rb^{M+1}\times\frac{r}{a^{m+1}}$-rectangle to cover. Now we choose the following value for $r$
\[
r=\frac{1}{(ab^2)^{M+1}}=\left(\frac{1}{(ab)^{M+1}}\right)^{\frac{\ln ab^2}{\ln ab}}=\left(\frac{1}{(ab)^{M+1}}\right)^{\frac{B}{B-1}}=\left(\frac{1}{b^{M+1}}\right)^B,
\]
where $B=2+\frac{\ln a}{\ln b}$ is the upper box dimension of $\Gamma_{T_{a,b}}$. We denote $R=\frac{1}{b^{M+1}}$ and note that $r=R^{\frac{1}{\theta}}$ with $\theta=\frac{1}{B}$ and we get the following relation
\[
\sup_{a\in\mathbb{R}^2}N(S(a,R/2)\cap\Gamma_{G_M},R^{\frac{1}{\theta}})\geq \left(\frac{1}{ab}\right)^{-D^-(M+1)}\frac{1}{rb^{M+1}}=\left(\frac{R}{r}\right)^{1+D^-}.
\]
The above inequality holds for arbitrarily large $M$ and therefore it holds also for arbitrarily small $R,R^{\frac{1}{\theta}}$. This is a covering property for $\Gamma_{G_M}$ and we can translate it to a covering property for $\Gamma_{T_{a,b}}$. By using Lemma \ref{ADD} and Remark \ref{ADDr} together with condition $(1)$ we see that there exist a constant $C>0$ such that
\begin{eqnarray*}
& &\sup_{a\in [x_0,x_0+\frac{1}{b^{M+1}}]\times\mathbb{R}}N(S(a,R/2)\cap\Gamma_{T_{a,b}},r)\\
&\geq& C\sup_{a\in [x_0,x_0+\frac{1}{b^{M+1}}]\times\mathbb{R}}N(S(a,R/2)\cap\Gamma_{G_M})-C\frac{R}{r}\\
&\geq& C\left(\frac{R}{r}\right)^{1+D^-}-C\frac{R}{r}.
\end{eqnarray*}
Then by definition of the Assouad spectrum and the inequality (\ref{Di}) in the first section we see that
\[
\Assouad \Gamma_{T_{a,b}}\geq \Assouad^\frac{1}{B} \Gamma_{T_{a,b}}\geq 1+D.
\]
This concludes the proof.
\end{proof}
We can now finish the proof of Theorem \ref{MAIN}. By Lemma \ref{SC} and Theorem \ref{LEVEL} we see that it is enough to show that the Takagi functions satisfy condition $(1)$ in the statement of Lemma \ref{SC}. In fact, condition $(1)$ is satisfied by $T_{a,b}$ whenever $a<1, b\in\{2\}\cup [2,\infty], ab\in (1,2)$. We put the last step of proving Theorem \ref{MAIN} in the following lemma.
\begin{lma}\label{NONINTEGRAL}
If the parameters $a,b$ satisfy the following conditions
\[
a<1,b\in\{2\}\cup[3,\infty],2>ab>1,
\]
then the Takagi functions
\[
T_{a,b}(x)=\sum_{n=0}^{\infty}a^nT(b^n x)
\]
satisfy the condition 1 in the statement of Lemma \ref{SC}.
\end{lma}
\begin{proof}
For each integer $M>0$ we shall consider the $M$-level intervals
\[
I_M(j)=\left[\frac{j}{2b^M},\frac{j+1}{2b^M}\right],j\in\mathbb{Z}.
\]
If $b\geq 3$, we see that
\[
\frac{1}{2b^M}\geq\frac{3}{2b^{M+1}},
\]
this implies that any interval $I_M(j)$ contains at least two $(M+1)$-level intervals say
\[
I_{M+1}(l), I_{M+1}(l+1).
\]
$l, l+1$ is a pair of integers, one of them is odd and the other is even. It is easy to see that this is also true for $b=2$. Then we see that given any sequance $\omega_i\in\{\pm 1\},i=0,1,2\dots$ it is possible to choose a sequence of integers $j_i,i=0,1,\dots$ such that
\[
I_i(j_i)\subset I_{i-1}(j_{i-1})\subset[0,1],
\]
and that $j_i$ is even if and only if $\omega_i=1$.
We see that
\[
T'(b^M x)=
\begin{cases}
1 & x\in I_M(j), j \text{ is even},\\
-1 & x\in I_M(j), j \text{ is odd}.
\end{cases}
\]
So for any integer $M>0$ we can find intervals $I_M(j)\subset [0,1]$ such that
\[
D_M(x)=\sum_{n=0}^{M}a^nT(b^n x))'=\sum_{n=0}^{M}a^nb^nT'(b^n x)
\]
is constant on $I_M(j)$ and the value can take all numbers in the following set
\[
U=
\left\{t\in\mathbb{R}:\exists \epsilon_n\in\{\pm1\},n\in\{0,1,2\dots,M\},t=\sum_{n=0}^{M}\epsilon_n(ab)^n\right\}.
\]
Then by lemma \ref{POWERBOUND} we see that the result follows.
\end{proof}
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| 143,214
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\begin{document}
\title{Central limit theorems for multilevel Monte Carlo methods}
\author{H{\aa}kon Hoel}
\address[H.~Hoel]{Department of Mathematical Sciences, Chalmers
University of Technology and University of Gothenburg, SE-412 96
Gothenburg, Sweden}
\email[Corresponding author]{hhakon@chalmers.se, haakonah1@gmail.com}
\author{Sebastian Krumscheid}
\address[S.~Krumscheid]{Calcul Scientifique et Quantification de l'Incertitude
(CSQI), Institute of Mathematics, {\'E}cole Polytechnique
F{\'e}d{\'e}rale de Lausanne, CH-1015 Lausanne, Switzerland}
\email{sebastian.krumscheid@epfl.ch}
\date{\today}
\maketitle
\noindent {\bf Keywords.} Multilevel Monte Carlo, Central Limit Theorem
\begin{abstract}
In this work, we show that uniform integrability is not a necessary
condition for central limit theorems (CLT) to hold for normalized
multilevel Monte Carlo estimators, and we provide near optimal
weaker conditions under which the CLT is achieved. In particular, if
the variance decay rate dominates the computational cost rate (i.e.,
$\beta> \gamma$), we prove that the CLT always holds.
\end{abstract}
\section{Introduction}
The multilevel Monte Carlo (MLMC) method is a hierarchical sampling
method which in many settings improves the computational efficiency of
weak approximations by orders of magnitude. The method was
independently introduced in the papers~\cite{MR1629093, MR2436856} for
the purpose of parametric integration and for approximations of
observables of stochastic differential equations, respectively. MLMC
methods have since been applied with considerable success in a vast
range of stochastic problems, a collection of which can be found in
the overview~\cite{MR3349310}. In this work we present near optimal
conditions under which the normalized MLMC estimator converges in
distribution to a standard normal distribution. Our result has
applications in settings where the MLMC approximation error is
measured in terms of probability of failure rather than the classical
mean square error.
\subsection{Main result}\label{sec:mainResult}
We consider the probability space $(\Omega, \cF, \bP)$ and let
$X \in L^2(\Omega)$ be a scalar random variable for which we seek the
expectation $\E{X}$, and let
$\{X_\ell\}_{\ell=-1}^{\infty} \subset L^2(\Omega)$ be a sequence of
random variables satisfying the following:
\begin{assumption}\label{ass:mlmcRates}
There exist positive rate constants $\alpha, \beta, \gamma$ with
$\min(\beta, \gamma) \leq 2 \alpha$ and a positive constant
$c_\alpha > 0$ such that for all
$\ell \in \bN_0 \defeq \bN \cup \{0\}$
\begin{itemize}
\item[(i)] $\abs{ \E{X - X_\ell} } \leq c_\alpha e^{-\alpha \ell}$,
\item[(ii)] $V_\ell \defeq \text{Var}(\DlX) = \cO_\ell(e^{-\beta \ell})$,
\item[(iii)] $C_\ell \defeq \text{Cost}(\DlX) = \Theta_\ell(e^{\gamma \ell})$,
\end{itemize}
where $\DlX \defeq X_\ell-X_{\ell-1}$ with $X_{-1} \defeq0$. Here,
$f(x_\ell) = \cO_\ell(y_\ell)$ means that there exists a constant
$C >0$ such that $|f(x_\ell)| < C |y_\ell|$ for all $\ell \in \bN_0$,
while $f(x_\ell) = \Theta_\ell(y_\ell)$ means that there exist
constants $C>c>0$, such that $c|y_\ell| < |f(x_\ell)| < C |y_\ell|$
for all $\ell \in \bN_0$.
\end{assumption}
\begin{definition}[MLMC estimator~\cite{MR3349310,MR1629093}]\label{def:mlmc}
The MLMC estimator $\cA_{ML}\colon (0,\infty) \to L^2(\cF, \bP)$
applied to estimate the expectation of $X \in L^2(\Omega)$ based on
the collection of random variables (r.v.)
$\{X_\ell\} \subset L^2(\Omega)$ satisfying
Assumption~\ref{ass:mlmcRates} is defined by
\[
\cA_{ML}(\epsilon) = \sum_{\ell=0}^{L(\epsilon)} \sum_{i=1}^{M_\ell(\epsilon)} \frac{\DlX^i}{M_\ell(\epsilon)}\;.
\]
Here
\[
L^2(\Omega)\ni \Delta_\ell X^i = X_{\ell}^i-X_{\ell-1}^i, \quad \ell \in \bN_0, \quad i \in \bN
\]
denotes a sequence of independent r.v.~and every subsequence
$\{\Delta_\ell X^i\}_i$ consist of independent and identically
distributed (i.i.d.) r.v., the number of levels is
\[
L(\epsilon) \defeq \max\left(\ceil{\frac{\log(c_\alpha \epsilon^{-1}) }{\alpha}}, 1\right), \quad \epsilon >0,
\]
and the number of samples per level $\ell = 0,1, \ldots$ is
\begin{equation}\label{eq:Ml}
M_\ell(\epsilon) \defeq \max\left( \ceil{\epsilon^{-2} \sqrt{\frac{V_\ell}{C_\ell}} \sum_{\ell=0}^{L(\epsilon)} \sqrt{C_\ell V_\ell}}, 1 \right), \quad \epsilon>0.
\end{equation}
We will refer to
\[
\frac{\cA_{ML}(\epsilon) - \E{X_{L(\epsilon)}}}{\sqrt{\Var{\cA_{ML}(\epsilon)}}}
\]
as the normalized MLMC estimator.
\end{definition}
\subsubsection*{Notation and conventions}
When confusion is not possible, we will use the following shorthands,
\[
\cA_{ML} \defeq \cA_{ML}(\epsilon), \quad M_\ell \defeq M_\ell(\epsilon),
\quad L \defeq L(\epsilon).
\]
The following conventions will be employed throughout
\[
0\cdot (\pm \infty) = 0 \quad \text{and} \quad 0/0 = 0,
\]
and we define the monotonically increasing sequence
\begin{equation}\label{eq:SLDef}
S_k \defeq \sum_{\ell =0}^k \sqrt{V_\ell C_\ell}, \quad k \in \bN_0.
\end{equation}
Then the main result of this work can be stated as follows.
\begin{theorem}[Main result]\label{thm:mainResult}
Let $\cA_{ML}$ denote the MLMC estimator applied estimate the
expectation of $X \in L^2(\Omega)$ based on the collection of r.v.\
$\{X_\ell\} \subset L^2(\Omega)$ satisfying
Assumption~\ref{ass:mlmcRates}. Suppose that $V_0>0$ and further
that
\begin{enumerate}
\item[(i)] if $\beta = \gamma$, then $\lim_{k\to \infty} S_k = \infty$
and
\begin{equation}\label{eq:limCond}
\lim_{\ell \to \infty} \1{V_\ell>0}\E{ \frac{ \abs{\DlX - \E{\DlX}}^2}{ V_\ell} \1{\frac{ \abs{\DlX -
\E{\DlX}}^2}{V_\ell} > \nu S_{\ell}^2 \exp\pr{(2\alpha - \gamma)\ell} }} = 0 \quad \forall \nu>0,
\end{equation}
\item[(ii)] if $\gamma>\beta$, then assume that $\beta< 2\alpha$,
equality~\eqref{eq:limCond} holds and that there exists an
$\upsilon \in [\beta,2\alpha)$ such that
$\lim_{k \to \infty} S_k e^{(\upsilon - \gamma)k/2} >1$.
\end{enumerate}
Then the normalized MLMC estimator satisfies the central limit
theorem (CLT), in the sense that
\begin{equation}\label{eq:cltMain}
\frac{\cA_{ML} - \E{X_{L}}}{\sqrt{\Var{\cA_{ML}}}} \toDist \cN(0,1) \quad \text{as} \quad \epsilon \downarrow 0.
\end{equation}
\end{theorem}
The main result follows from Theorems~\ref{thm:betaLargest}
and~\ref{thm:gammaLargest} below. We note that
Theorem~\ref{thm:mainResult} implies that in settings with
$\beta>\gamma$ the (CLT) always holds for the normalized MLMC
estimator.
\begin{remark}
For the setting $\gamma>\beta$ and $\beta= 2\alpha$, which we have
not included in Theorem~\ref{thm:mainResult}, one cannot impose
reasonable assumptions to exclude $M_L = \Theta_\epsilon(1)$ and
$V_L/\Var{\cA_{ML}} = \Theta_\epsilon(1)$;
cf.~Example~\ref{ex:fail1}. This implies there are no reasonable
ways to exclude cases for which a non-negligible contribution to the
variance of the resulting MLMC estimator derives from a finite
number of samples. Therefore, the central limit theorem is not
relevant for this setting.
\end{remark}
In literature, the CLT has been proved for the MLMC method through
assuming (or verifying for the particular sequence of r.v.~considered)
either a Lyapunov condition~\cite{hoel2014implementation}, or uniform
integrability~\cite{MR3297771,MR3449315,Giorgi}, or a weaker higher
moment decay rate~\cite{MR3348197} for the sequence
$\{\1{V_\ell>0}|\DlX-\E{\DlX}|^2/V_\ell\}_{\ell\in \bN_0}$. To show
that this work extends the existing literature, we now provide an
explicit example where Theorem~\ref{thm:mainResult} is valid although
uniform integrability does not hold.
\begin{example}
Let $\{X_\ell\}_{\ell=-1}^\infty$ denote the sequence of r.v.~defined by
\[
X_\ell = \begin{cases} 0 & \ell = -1\;,\\
\sum_{k=0}^\ell e^{k/4} \1{\omega \in \Omega_k} & \ell \in \bN_0\;,
\end{cases}
\]
where $\Omega_k \in \cF$ and $\bP(\Omega_k) = (1-e^{-1})e^{-k}$ for $k\in \bN_0$
and $\Omega_j \cap \Omega_k = \emptyset$ for all $j\neq k$.
Let further
\[
X \defeq \sum_{k=0}^\infty e^{k/4} \1{\omega \in \Omega_k}.
\]
Then
\[
|\E{X- X_\ell}| = \Theta_\ell(e^{-3\ell/4})
\]
and
\[
V_\ell = \E{ \pr{e^{\ell/4} \1{\Omega_\ell} - e^{-3\ell/4}(1-e^{-1})}^2}= \Theta_\ell(e^{-\ell/2}),
\]
yielding the respective decay rates $\alpha = 3/4$ and
$\beta = 1/2$, cf.~Assumption \ref{ass:mlmcRates}. Moreover, for
any $x>1$
\[
\lim_{\ell \to \infty} \E{\frac{\abs{\DlX - \E{\DlX}}^2}{V_\ell} \1{\frac{\abs{\DlX - \E{\DlX}}^2 }{V_\ell} > x}} =1,
\]
which implies that the sequence
\[
\frac{ \abs{\DlX - \E{\DlX}}^2}{ V_\ell}, \quad \ell =0,1,\ldots
\]
is not uniformly integrable. As $V_0 > e^{-1}(1-e^{-1})>0$,
$\beta<2\alpha$, and
\[
S_k =
\begin{cases}
\Theta_k(1) & \text{if} \quad \beta>\gamma\\
\Theta_k(k) & \text{if} \quad \beta = \gamma\\
\Theta_k(e^{(\gamma-\beta)k/2}) & \text{if} \quad \beta< \gamma,
\end{cases}
\]
the CLT~\eqref{eq:cltMain} of Theorem~\ref{thm:mainResult}
holds for all settings with $\gamma \le 1/2 =\beta$.
\end{example}
\section{Theory}\label{sec:cltTheory}
In this section we derive weak assumptions under which the normalized
MLMC estimator $(\cA_{ML} -\E{X_L})/\sqrt{\Var{\cA_{ML}}}$ converges
in distribution to a standard normal as $\epsilon \to 0$. The main
tool used for verifying the CLT will be the Lindeberg condition, which
in its classical formulation is an integrability condition for
triangular arrays of independent random variables (r.v.) $Y_{nm}$,
with $n \in \bN$ and $1\le m \le k_n$; cf.~\cite{MR1609153}. However,
in the multilevel setting it is more convenient to work with
generalized triangular arrays of independent r.v.\ of the form
$Y_{\epsilon m}$, which for a fixed $\epsilon>0$ take possible
non-zero elements within the set of indices $ 1\le m \le n(\epsilon)$,
where $n\colon (0,\infty) \to \bN$ is a strictly decreasing function
of $\epsilon>0$ with $\lim_{\epsilon \downarrow 0} n = \infty$.
The following theorem is a trivial extension of~\cite{klenke} from
triangular arrays to generalized triangular arrays.
\begin{theorem}[Lindeberg-Feller Theorem]\label{thm:LindebergFeller}
For every $\epsilon >0$, let $\{Y_{\epsilon m}\}$,
$ 1\leq m \leq n(\epsilon)$ with $n\colon (0,\infty) \to \bN$ and
$\lim_{\epsilon \downarrow 0} n = \infty$ be a generalized
triangular array of independent random variables that are centered
and normalized, so that
\begin{equation}\label{eq:normCenter}
E[Y_{\epsilon m}] = 0\quad\text{and}\quad \sum_{m=1}^{n(\epsilon)} \E{Y_{\epsilon m}^2} = 1\;,
\end{equation}
respectively. Then, the Lindeberg condition:
\begin{equation}\label{eq:LFCondition1}
\lim_{\epsilon \downarrow 0}
\sum_{m=1}^{n(\epsilon)} \E{Y_{\epsilon m}^2 \1{|Y_{\epsilon m}|>\nu }} = 0\quad\forall\,\nu>0\;,
\end{equation}
holds, if and only if
\begin{equation}\label{eq:extendedClt}
\sum_{m=1}^{n(\epsilon)}
Y_{\epsilon m} \toDist \cN(0,1) \text{ as } \epsilon \downarrow 0 \quad \text{and} \quad
\lim_{\epsilon \downarrow 0} \max_{m \in \{1,2,\ldots,n(\epsilon)\}}
\E{Y_{\epsilon m}^2} = 0\;.
\end{equation}
\end{theorem}
We will refer to~\eqref{eq:extendedClt} as the \emph{extended CLT
condition}. By defining
\begin{equation}\label{eq:nDef}
n(\epsilon) \defeq \sum_{\ell=0}^L M_\ell,
\end{equation}
and
\begin{equation}
Y_{\epsilon m} \defeq \begin{cases}
\frac{ \Delta_0 X^m - \E{\Delta_0X}}{ \sqrt{\Var{\cA_{ML}}} M_0} & m \le M_0\\
\frac{\Delta_1 X^m - \E{\Delta_1 X}}{\sqrt{\Var{\cA_{ML}} } M_1} & M_0 < m \le M_0 + M_{1}\\
\qquad\vdots\\
\frac{\Delta_L X^m - \E{\Delta_L X}}{\sqrt{\Var{\cA_{ML}} } M_L} &
n(\epsilon)-M_L < m \le n(\epsilon),
\end{cases}
\end{equation}
the normalized MLMC estimator can be represented by generalized
triangular-arrays as follows:
\begin{equation}\label{eq:scaledML}
\frac{\cA_{ML} - \E{X_L}}{\sqrt{\Var{\cA_{ML}}}} = \sum_{m=1}^{n(\epsilon)} Y_{\epsilon m}\;.
\end{equation}
\begin{corollary}\label{cor:lindebergGeneral}
Let $\cA_{ML}$ denote the MLMC estimator applied to estimate the
expectation of $X \in L^2(\Omega)$ based on the collection of r.v.\
$\{X_\ell\} \subset L^2(\Omega)$ satisfying
Assumption~\ref{ass:mlmcRates}. Suppose that $\Var{\cA_{ML}}>0$ for
any $\epsilon >0$. Then the normalized MLMC
estimator~\eqref{eq:scaledML} satisfies the extended CLT
condition~\eqref{eq:extendedClt}, if and only if for any $\nu>0$,
\begin{equation}\label{eq:lindebergGeneral}
\lim_{\epsilon \downarrow 0} \sum_{\ell=0}^L
\frac{ V_\ell}{ \Var{\cA_{ML}} M_\ell } \E{ \frac{ \abs{\DlX - \E{\DlX}}^2}{V_\ell} \1{\frac{ \abs{\DlX -
\E{\DlX}}^2}{V_\ell} > \frac{\Var{\cA_{ML}} M_\ell^2}{V_\ell} \nu}} =0.
\end{equation}
\end{corollary}
\begin{proof}
For all $\epsilon>0$, the triangular array
representation~\eqref{eq:scaledML} of the MLMC estimator obviously
satisfies the centering and normalization
conditions~\eqref{eq:normCenter}, and its elements are centered and
mutually independent. By Theorem~\ref{thm:LindebergFeller}, the
extended CLT condition thus holds, if and only if Lindeberg's
condition~\eqref{eq:LFCondition1} holds. For any $\nu >0$, here
Lindeberg's condition takes the form:
\[
\begin{split}
&\lim_{\epsilon \to 0} \sum_{m=1}^{n(\epsilon)} \E{ Y_{\epsilon m}^2
\1{Y_{\epsilon m} > \nu}} \\
&= \lim_{\epsilon \to 0} \sum_{\ell=0}^L \sum_{i=1}^{M_\ell}
\Var{\cA_{ML}} \E{ \frac{ \abs{\DlX^i -
\E{\DlX}}^2}{M_\ell^2 \Var{\cA_{ML}}} \1{\frac{ \abs{\DlX^i -
\E{\DlX}}^2}{\Var{\cA_{ML}} M_\ell^2} > \nu}}\\
& = \lim_{\epsilon \downarrow 0} \sum_{\ell=0}^L \frac{V_\ell}{M_\ell\Var{\cA_{ML}}} \E{ \frac{ \abs{\DlX -
\E{\DlX}}^2}{V_\ell} \1{\frac{ \abs{\DlX -
\E{\DlX}}^2}{V_\ell} > \frac{\Var{\cA_{ML}} M_\ell^2}{V_\ell}
\nu}}.
\end{split}
\]
\end{proof}
Assumption~\ref{ass:mlmcRates} does not provide any lower bound on the
decay rate of the variance sequence $\{V_\ell\}$, and, therefore, it
alone is not sufficiently strong to ensure that Lindeberg's
condition~\eqref{eq:lindebergGeneral} holds in general. The problem is
that without any lower bound on $V_\ell$, there are asymptotic
settings where a non-negligible contribution to the variance of the
MLMC estimator derives from a finite number of samples.
\begin{example}\label{ex:fail1}
Consider the setting where $\beta< \min(\gamma, 2\alpha)$ and
positive constants $c_1,c_2$ such that
\[
c_1 e^{-\alpha \ell}\le V_\ell \le c_2 e^{-\beta\ell}
\]
Let there be an infinite subsequence $\{k_i\} \subset \bN_0$ for which
\[
V_{k_i} = \Theta_{k_i}(e^{-2\alpha k_i}) \quad \text{and} \quad S_{k_i} = \Theta_{k_i}(e^{(\gamma-2\alpha)k_i}).
\]
Then equation~\eqref{eq:Ml} implies there exists $c,C,\tilde{c},\hat{c} \in \bR_+$ such that
for all $y \in \{\epsilon>0 \mid L(\epsilon) \in \{k_i\} \}$,
\[
M_{L(y)} < C,
\]
and
\[
\hat{c} \le \max\pr{\frac{V_{L(y)}}{M_{L(y)} \Var{\cA_{ML}(y)}}, \;
\frac{M_{L(y)}^2 \Var{\cA_{ML}(y)}}{V_L(y)}} \le \tilde{c}.
\]
Hence, for any $\nu<(2\tilde{c})^{-1}$,
\[
\begin{split}
& \limsup_{\epsilon \downarrow 0} \sum_{\ell=0}^L
\frac{V_\ell}{ M_\ell \Var{\cA_{ML}} } \E{ \frac{ \abs{\DlX -
\E{\DlX}}^2}{V_\ell} \1{\frac{ \abs{\DlX -
\E{\DlX}}^2}{V_\ell} > \frac{\Var{\cA_{ML}} M_\ell^2}{V_\ell}
\nu}} \\
& \ge
\limsup_{\epsilon \downarrow 0}
\frac{V_L}{ M_L \Var{\cA_{ML}} } \E{ \frac{ \abs{\DlX -
\E{\DlX}}^2}{V_L} \1{\frac{ \abs{\DlX -
\E{\DlX}}^2}{V_L} > \frac{\Var{\cA_{ML}} M_L^2}{V_L}
\nu}}\\
& \ge
\lim_{i \to \infty}
\hat{c} \, \E{ \frac{ \abs{\Delta_{k_i} X -
\E{\Delta_{k_i} X}}^2}{V_{k_i}} \1{\frac{ \abs{\Delta_{k_i} X -
\E{\Delta_{k_i} X}}^2}{V_{k_i}} > \frac{1}{2}}} > 0.
\end{split}
\]
\end{example}
Example~\ref{ex:fail1} illustrates that Assumption~\ref{ass:mlmcRates}
is not sufficiently strong to ensure condition~\eqref{eq:lindebergGeneral}.
We therefore impose the
following additional variance decay assumptions, which can be viewed
as implicit weak lower bounds on the sequence $\{V_\ell\}$.
\begin{assumption}\label{ass:mlmcRates2}
For the rate triplet introduced in Assumption~\ref{ass:mlmcRates},
assume that $V_0 >0$ and
\begin{itemize}
\item[(i)] if $\beta = \gamma$, then $\lim_{k \to \infty} S_k = \infty$,
\item[(ii)] if $\gamma > \beta$, then we assume that $\beta <2\alpha$ and
that there exists a $\upsilon \in [\beta, 2\alpha)$ such that
\[
\liminf_{k \to \infty} S_k e^{(\upsilon -\gamma)k/2} >1.
\]
\end{itemize}
\end{assumption}
\begin{lemma}\label{lem:varEpsRelation}
Let $\cA_{ML}$ denote the MLMC estimator applied to estimate the
expectation of $X \in L^2(\Omega)$ based on the collection of r.v.\
$\{X_\ell\} \subset L^2(\Omega)$ satisfying
Assumption~\ref{ass:mlmcRates}. If also
Assumption~\ref{ass:mlmcRates2} holds, then
\begin{equation}\label{eq:condition1}
\lim_{\epsilon \downarrow 0} \frac{\Var{\cA_{ML}}}{\epsilon^2} =1\;.
\end{equation}
\end{lemma}
\begin{proof}
For any $\epsilon >0$, it follows from equation~\eqref{eq:Ml} that
\[
\begin{split}
\frac{\Var{\cA_{ML}}}{\epsilon^2} =
\sum_{\ell=0}^L \frac{V_\ell }{\epsilon^2 M_\ell}
\leq \sum_{\ell=0}^L \frac{\sqrt{V_\ell C_\ell} }{S_L}= 1\;,
\end{split}
\]
and by the mean value theorem there exists a
constant $C>0$ such that
\begin{equation}\label{eq:condition1LowerBound}
\begin{split}
\sum_{\ell=0}^L \frac{V_\ell }{\epsilon^2 M_\ell} &\ge
\sum_{\ell=0}^L \1{V_\ell >0} \frac{V_\ell }{ \sqrt{\frac{V_\ell}{C_\ell}}S_L + \epsilon^2}\\
&\ge 1- \sum_{\ell=0}^L \1{V_\ell >0}\frac{V_\ell \epsilon^2}{\frac{V_\ell}{C_\ell} S_L^2 }\\
& \ge 1 - \epsilon^2 \frac{\sum_{\ell=0}^L C_\ell}{S_L^2 }\\
&\ge 1 - C \epsilon^2\frac{ e^{\gamma L}}{S_L^2}\;.
\end{split}
\end{equation}
To complete the proof, we thus have to verify that
\begin{equation}\label{eq:condition1Closure}
\lim_{\epsilon \downarrow 0} \frac{\epsilon^2 e^{\gamma L}}{s_L^2} =0\;.
\end{equation}
If $\beta < \gamma$, then Assumption~\ref{ass:mlmcRates2}(ii) implies that
\[
\frac{\epsilon^2 e^{\gamma L}}{S_L^2} = \cO(\epsilon^{2-\upsilon/\alpha})\;,
\]
and since $\upsilon < 2 \alpha$, the claim follows in this case.
Similarly, if $\beta = \gamma$, then
$\epsilon^2 e^{\gamma L} = \cO(1)$, and the claim follows from
Assumption~\ref{ass:mlmcRates2}(i). Finally, if $\beta >\gamma$, then
the assumption $\min(\beta,\gamma) \leq 2 \alpha$
(cf.~Assumption~\ref{ass:mlmcRates}) implies $\gamma \le 2\alpha$, and
we have to consider two cases: $(I)$ $\gamma < 2 \alpha$ and $(II)$
$\gamma = 2\alpha$. For case $(I)$, equation~\eqref{eq:condition1Closure}
follows from
$ \lim_{\epsilon \downarrow 0} \epsilon^2 e^{\gamma L} = 0$ and
$S_L \ge S_0 >0$ for all $L\ge 0$.
For case $(II)$, we introduce
\[
\widehat L \defeq \max\pr{\ceil{\frac{4 \log(c_\alpha \epsilon^{-1})}{\gamma + \beta}}, \, 1 }\;.
\]
As $2\alpha = \gamma < \beta$, $\widehat L \le L$,
\[
\begin{split}
\frac{\Var{\cA_{ML}}}{\epsilon^2} &\ge
\sum_{\ell=0}^L \1{V_\ell >0} \frac{V_\ell }{
\sqrt{\frac{V_\ell}{C_\ell}} S_L + \epsilon^2} \ge \sum_{\ell=0}^{\widehat{L}} \1{V_\ell >0} \frac{V_\ell }{
\sqrt{\frac{V_\ell}{C_\ell}} S_L + \epsilon^2}
\ge 1 - C \epsilon^2\frac{ e^{\gamma \widehat{L}}}{S_L^2}\;,
\end{split}
\]
and the result follows by $\lim_{\epsilon \downarrow 0} \epsilon^2 e^{\gamma \widehat{L}} = 0$.
\end{proof}
Lemma~\ref{lem:varEpsRelation} implies that we can reformulate
Lindeberg's condition for the MLMC estimator as follows:
\begin{corollary}
Let $\cA_{ML}$ denote the MLMC estimator applied estimate the
expectation of $X \in L^2(\Omega)$ based on the collection of r.v.\
$\{X_\ell\} \subset L^2(\Omega)$ satisfying
Assumption~\ref{ass:mlmcRates}. If Assumptions~\ref{ass:mlmcRates}
and~\ref{ass:mlmcRates2} hold, then the normalized MLMC estimator
satisfies the extended CLT condition~\eqref{eq:extendedClt}, if and
only if for any $\nu >0$,
\begin{equation}\label{eq:lindebergCond2}
\lim_{\epsilon \downarrow 0} \sum_{\ell=0}^L \frac{\sqrt{V_\ell C_\ell}}{S_L} \1{V_\ell >0} \E{\frac{\abs{\DlX - \E{\DlX}}^2}{V_\ell}
\1{\frac{\abs{\DlX - \E{\DlX}}^2 }{V_\ell} > \frac{\epsilon^2 M_\ell^2}{V_\ell} \nu}} =0\;.
\end{equation}
\end{corollary}
\begin{proof}
From the proof of Lemma~\ref{lem:varEpsRelation} it follows that
there exists an $\bar \epsilon>0$ such that
\[
\frac{1}{2} \le \frac {\Var{\cA_{ML}}}{\epsilon^2} \le 1\;, \quad
\forall \epsilon \in (0, \bar \epsilon)\;.
\]
Consequently, for any $\epsilon \in(0, \bar \epsilon)$ and any $\nu>0$
we have that
\[
\begin{split}
& \sum_{\ell=0}^L \E{ \frac{ \abs{\DlX -
\E{\DlX}}^2}{\Var{\cA_{ML}} M_\ell} \1{\frac{ \abs{\DlX -
\E{\DlX}}^2}{V_\ell} > \frac{\Var{\cA_{ML}} M_\ell^2}{V_\ell} \nu}}\\
& \ge \sum_{\ell=0}^L \E{ \frac{ \abs{\DlX -\E{\DlX}}^2}{\epsilon^2 M_\ell} \1{\frac{ \abs{\DlX -
\E{\DlX}}^2}{V_\ell} > \frac{\epsilon^2 M_\ell^2}{V_\ell} \nu}}\;,
\end{split}
\]
as well as
\[
\begin{split}
& \sum_{\ell=0}^L \frac{1}{\Var{\cA_{ML}}} \E{ \frac{ \abs{\DlX -
\E{\DlX}}^2}{M_\ell} \1{\frac{ \abs{\DlX -
\E{\DlX}}^2}{V_\ell} > \frac{\Var{\cA_{ML}} M_\ell^2}{V_\ell} \nu}}\\
& \le 2\sum_{\ell=0}^L \E{ \frac{ \abs{\DlX -
\E{\DlX}}^2}{\epsilon^2 M_\ell} \1{\frac{ \abs{\DlX -
\E{\DlX}}^2}{V_\ell} > \frac{\epsilon^2 M_\ell^2}{2V_\ell} \nu}}\;.
\end{split}
\]
These upper and lower bounds imply that that Lindeberg's
condition~\eqref{eq:lindebergGeneral} is equivalent to the following
condition: for any $\nu>0$ it holds that
\[
\lim_{\epsilon \downarrow 0} \sum_{\ell=0}^L \E{ \frac{ \abs{\DlX -
\E{\DlX}}^2}{\epsilon^2 M_\ell} \1{\frac{ \abs{\DlX -
\E{\DlX}}^2}{V_\ell} > \frac{\epsilon^2 M_\ell^2}{V_\ell} \nu}}
= 0\;.
\]
Following similar steps as those leading to
inequality~\eqref{eq:condition1LowerBound}, we further note that for
sufficiently small $\epsilon>0$,
\begin{equation}\label{eq:condition2_1}
\begin{split}
&\sum_{\ell=0}^L \frac{1}{\epsilon^2 M_\ell}\E{\abs{\DlX - \E{\DlX}}^2 \1{\frac{\abs{\DlX - \E{\DlX}}^2 }{\epsilon^2 M_\ell^2} > \nu}}\\
& = \sum_{\ell=0}^L \left\{\frac{\sqrt{V_\ell C_\ell}}{S_L} \E{\frac{\abs{\DlX - \E{\DlX}}^2}{V_\ell}
\1{\frac{\abs{\DlX - \E{\DlX}}^2 }{V_\ell} > \frac{\epsilon^2
M_\ell^2}{V_\ell} \nu}} \right\} + \rho(\epsilon)\;,
\end{split}
\end{equation}
where the mapping $\rho\colon\bR_+ \to \bR$, satisfying
$\lim_{\epsilon\downarrow 0} \rho(\epsilon) =0$, can be derived as in
the proof of Lemma~\ref{lem:varEpsRelation}.
\end{proof}
In settings with $\beta>\gamma$, the geometric decay of the sequence
${\{\sqrt{C_\ell V_\ell}\}}_{\ell\ge 0}$ turns out to be sufficient to
prove that the extended CLT condition holds.
\begin{theorem}\label{thm:betaLargest}
Let $\cA_{ML}$ denote the MLMC estimator applied to estimate the
expectation of $X \in L^2(\Omega)$ based on the collection of r.v.\
$\{X_\ell\} \subset L^2(\Omega)$ satisfying
Assumption~\ref{ass:mlmcRates}. If $\beta > \gamma$ and if
Assumption~\ref{ass:mlmcRates2} holds (the only relevant condition
being $V_0>0$), then the extended CLT
condition~\eqref{eq:extendedClt} is satisfied for the normalized
MLMC estimator.
\end{theorem}
\begin{proof}
We prove this result by verifying that
condition~\eqref{eq:lindebergCond2} holds. It follows from
Assumption~\ref{ass:mlmcRates2} that
\[
S \defeq \lim_{k \to \infty} S_k = \lim_{k \to \infty}
\sum_{\ell=0}^k \sqrt{V_\ell C_\ell} \leq c \lim_{k \to \infty}
\sum_{\ell =0}^k e^{(\gamma-\beta)\ell/2} < \infty\;.
\]
Furthermore, as the sequence ${\{S_L\}}_{L\ge 0}$ is monotonically
increasing, it is contained in the bounded set $[S_0, S]$ with
$S_0 >0$. Consequently, Lindeberg's
condition~\eqref{eq:lindebergCond2} is equivalent to:
\[
\lim_{\epsilon \downarrow 0} \sum_{\ell=0}^L \sqrt{V_\ell C_\ell} \1{V_\ell>0} \E{\frac{\abs{\DlX - \E{\DlX}}^2}{V_\ell}
\1{\frac{\abs{\DlX - \E{\DlX}}^2 }{V_\ell} > \frac{\epsilon^2 M_\ell^2}{V_\ell} \nu}} =0\;,\quad\forall\,\nu>0\;.
\]
For a fixed $\nu>0$, introduce the sequence of functions
$\{f_\epsilon\}_{\epsilon>0}$, where $f_\epsilon\colon [0,S] \to [0,1]$ is
defined by
\[
f_\epsilon(x) = \begin{cases}
\E{\frac{\abs{\Delta_0X - \E{\Delta_0 X}}^2}{V_0}
\1{\frac{\abs{\Delta_0 X - \E{\Delta_0 X}}^2 }{V_0} > \frac{\epsilon^2
M_0^2}{V_0} \nu}} & \text{if } 0 \le x <S_0\;, \\
\E{\frac{\abs{\Delta_1X - \E{\Delta_1 X}}^2}{V_1}
\1{\frac{\abs{\Delta_1 X - \E{\Delta_1 X}}^2 }{V_1} > \frac{\epsilon^2
M_1^2}{V_1} \nu}} & \text{if } S_0 \le x <S_1\;,\\
\qquad \qquad \vdots \\
\E{\frac{\abs{\Delta_LX - \E{\Delta_L X}}^2}{V_L}
\1{\frac{\abs{\Delta_L X - \E{\Delta_L X}}^2 }{V_L} > \frac{\epsilon^2
M_L^2}{V_L} \nu}} & \text{if } S_{L-1} \le x < S_L\;, \\
0 & \text{if } S_L \le x \le S\;.
\end{cases}
\]
For any $\epsilon>0$, one thus has that
\[
\sum_{\ell=0}^L \sqrt{V_\ell C_\ell} \1{V_\ell>0} \E{\frac{\abs{\DlX - \E{\DlX}}^2}{V_\ell}
\1{\frac{\abs{\DlX - \E{\DlX}}^2 }{V_\ell} > \frac{\epsilon^2
M_\ell^2}{V_\ell} \nu}}
= \int_{0}^S f_\epsilon(x) dx\;.
\]
Since $|f_\epsilon| \le \1{[0,S]}$ for all $\epsilon >0$, the
dominated convergence theorem implies that
\[
\lim_{\epsilon \downarrow 0} \sum_{\ell=0}^L \sqrt{V_\ell C_\ell} \1{V_\ell>0}\E{\frac{\abs{\DlX - \E{\DlX}}^2}{V_\ell}
\1{\frac{\abs{\DlX - \E{\DlX}}^2 }{V_\ell} > \frac{\epsilon^2
M_\ell^2}{V_\ell} \nu}}
= \int_{0}^S \lim_{\epsilon \downarrow 0} f_\epsilon(x) dx\;.
\]
It remains to verify that
\begin{equation}\label{eq:fIsZero}
\int_{0}^S \lim_{\epsilon \downarrow 0} f_\epsilon(x) dx =0.
\end{equation}
Consider two cases: (I) $\gamma < 2\alpha$ and (II) $\gamma =2\alpha$.
For case $(I)$, note that
\[
\begin{split}
\lim_{\epsilon \downarrow 0} \min_{\ell \in \{0,1,\ldots,L\}} \frac{\epsilon^2 M_\ell^2}{V_\ell} \ge
\lim_{\epsilon \downarrow 0} \min_{\ell \in \{0,1,\ldots,L\}} \epsilon^{-2} \frac{S_L^2}{C_\ell}
\ge \lim_{\epsilon \downarrow 0} \epsilon^{-2} \frac{S_L^2}{C_L}
= \infty\;,
\end{split}
\]
since $C_L = \Theta_\epsilon(\epsilon^{-\gamma/\alpha})$. For any
$x \in [0,S)$, say $x \in [S_{\ell-1}, S_\ell )$ for some
$ 0 \le \ell \le L$ for which $V_\ell>0$, the dominated convergence
theorem then implies that
\begin{equation}\label{eq:domConv}
\lim_{\epsilon \downarrow 0} f_\epsilon(x) = \E{ \lim_{\epsilon
\downarrow 0} \frac{\abs{\DlX - \E{\DlX}}^2}{V_\ell}
\1{\frac{\abs{\DlX - \E{\DlX}}^2 }{V_\ell} > \frac{\epsilon^2
M_\ell^2}{V_\ell} \nu} } = 0.
\end{equation}
Consequently,
\[
\lim_{\epsilon \downarrow 0} \sum_{\ell=0}^L \sqrt{V_\ell C_\ell} \1{V_\ell>0}\E{\frac{\abs{\DlX - \E{\DlX}}^2}{V_\ell}
\1{\frac{\abs{\DlX - \E{\DlX}}^2 }{V_\ell} > \frac{\epsilon^2
M_\ell^2}{V_\ell} \nu}} = 0\;,
\]
and~\eqref{eq:fIsZero} follows. For case (II), we introduce
\[
\widehat{L} = \max\pr{ \ceil{\frac{4 \log(c_\alpha \epsilon^{-1})}{\gamma + \beta}}, \, 1 }\;.
\]
Since $\widehat{L} \le L$ and
\[
\int_{S_{\widehat{L}}}^{S} \lim_{\epsilon \downarrow 0} f_\epsilon dx \le
\lim_{\epsilon \downarrow 0} \int_{S_{\widehat{L}}}^{S_L} dx =
\lim_{\epsilon \downarrow 0} (S_L - S_{\widehat L}) = 0,
\]
we have that
\[
\int_{0}^S \lim_{\epsilon \downarrow 0} f_\epsilon(x) dx \le
\int_{0}^{S_{\widehat L}} \lim_{\epsilon \downarrow 0} f_\epsilon(x) dx.
\]
Moreover,
\[
\lim_{\epsilon \downarrow 0} \min_{\ell \in \{0,1,\ldots, \widehat L\}}
\frac{\epsilon^2 M_\ell^2}{V_\ell} \ge \lim_{\epsilon \downarrow 0} \epsilon^{-2} \frac{S_{\widehat L}^2}{C_{\widehat L}}
= \infty,
\]
since
$C_{\widehat L} = \Theta_\epsilon(\epsilon^{-4\gamma/(\gamma +
\beta)})$ and $2\gamma/(\gamma +\beta)<1$. From~\eqref{eq:domConv}
it then follows that
$\lim_{\epsilon \downarrow 0} f_{\epsilon}(x) = 0$ for all
$x \in [0, S_{\widehat L})$, so that~\eqref{eq:fIsZero} holds.
As the above argument is valid for any fixed $\nu>0$, we have
proved that Lindeberg's condition holds.
\end{proof}
We conclude the paper by the treating the case $\gamma \ge \beta$.
\begin{theorem}\label{thm:gammaLargest}
Let $\cA_{ML}$ denote the MLMC estimator applied to estimate the
expectation of $X \in L^2(\Omega)$ based on the collection of r.v.\
$\{X_\ell\} \subset L^2(\Omega)$ satisfying
Assumption~\ref{ass:mlmcRates}. Suppose that $\beta \ge \gamma$,
Assumption~\ref{ass:mlmcRates2} holds, and
\[
\lim_{\ell \to \infty} \1{V_\ell >0} \E{ \frac{ \abs{\DlX - \E{\DlX}}^2}{ V_\ell} \1{\frac{ \abs{\DlX -
\E{\DlX}}^2}{V_\ell} > e^{(2\alpha - \gamma)\ell} S_{\ell}^2\nu}} = 0
\]
holds for any $\nu >0$. Then the extended CLT
condition~\eqref{eq:extendedClt} is satisfied for the normalized MLMC
estimator.
\end{theorem}
\begin{proof}
From~\eqref{eq:Ml} and $C_\ell = \Theta_\ell(e^{-\gamma \ell})$ it
follows that there exists a $c>0$ such that
\[
\1{V_\ell>0} \frac{\epsilon^2 M_\ell^2}{V_\ell} \ge \1{V_\ell} \frac{\epsilon^{-2}S_\ell^2}{C_\ell} > \1{V_\ell} c e^{(2\alpha - \gamma)\ell} S_{\ell}^2.
\]
Consequently,
\[
\begin{split}
&\sum_{\ell=0}^L \frac{\sqrt{V_\ell C_\ell}}{S_L} \E{\frac{\abs{\DlX - \E{\DlX}}^2}{V_\ell}
\1{\frac{\abs{\DlX - \E{\DlX}}^2 }{V_\ell} > \frac{\epsilon^2 M_\ell^2}{V_\ell} \nu}}\\
&\le \sum_{\ell=0}^L \frac{\sqrt{V_\ell C_\ell}}{S_L} \E{\frac{\abs{\DlX - \E{\DlX}}^2}{V_\ell}
\1{\frac{\abs{\DlX - \E{\DlX}}^2 }{V_\ell} > \nu c e^{(2\alpha - \gamma)\ell} S_{\ell}^2}}.
\end{split}
\]
Introduce the infinite matrix $A = (a_{k\ell})$ where
\[
a_{k\ell} \defeq \1{\ell\le k} \frac{\sqrt{V_\ell C_\ell}}{S_k}, \quad k, \ell \in \bN_0
\]
and the sequence
\[
x_\ell \defeq \1{V_\ell >0} \E{\frac{\abs{\DlX - \E{\DlX}}^2}{V_\ell}
\1{\frac{\abs{\DlX - \E{\DlX}}^2 }{V_\ell} > \nu c e^{(2\alpha - \gamma)\ell} S_{\ell}^2}}, \quad \ell \in \bN_0.
\]
Since
\[
\lim_{k \to \infty} a_{k\ell} =0 \quad \forall \ell \in \bN_0, \quad \lim_{k \to \infty } \sum_{\ell=0}^\infty a_{k\ell} =1, \quad
\text{and} \quad \sum_{\ell=0}^\infty |a_{k\ell}| \le 1 \quad \forall k \in \bN_0,
\]
the matrix $A$ defines a regular
summability method and the Silverman--Toeplitz theorem~\cite{toeplitz1911allgemeine,kreyszig1978introductory} yields that
\[
\begin{split}
&\lim_{\epsilon\downarrow 0} \sum_{\ell=0}^L \frac{\sqrt{V_\ell C_\ell}}{S_L} \E{\frac{\abs{\DlX - \E{\DlX}}^2}{V_\ell} = \1{\frac{\abs{\DlX - \E{\DlX}}^2 }{V_\ell} > \nu c e^{(2\alpha - \gamma)\ell} S_{\ell}^2}}\\
& = \lim_{k \to \infty} \sum_{\ell=0}^\infty a_{k\ell} x_\ell = \lim_{\ell \to \infty } x_\ell =0\;,
\end{split}
\]
which completes the proof.
\end{proof}
| 117,540
|
TITLE: Is it true that $\frac{d}{dt}\int_S \mathbf{B} \cdot d \mathbf{a}$ goes to zero if the amperian loop delimiting $S$ contracts indefinitely?
QUESTION [1 upvotes]: I suppose to have an ordinary magnetic field: in the answer I'm not interested to involve Dirac delta: the integral goes to zero. I want to focus on another point: an infinitesimal physical quantity can have a finite time derivative? Of course derivative of zero is zero, but this flux is never strictly zero, and this trouble me because the step
$$
\frac{d}{dt}\int_S \mathbf{B} \cdot d \mathbf{a} \to 0
$$
(when the surface connected to the amperian loop can be taken indefinitely small) is used when we exploit Maxwell equations to fix boundary conditions on the discontinuity between two media. Maybe I'm getting flustered in the slightest thing, but this confuse me and I can't get to the bottom of this problem. How could I see clearly this passage?
REPLY [0 votes]: The previous answers and comments inspired to me what follow: maybe this is the simplest (and so the best) way to see why $\frac{d}{dt} \int_S \mathbf{B} \cdot d \mathbf{a}$ vanishes when the loop contracts indefinitely.
I can simply consider a small surface that doesn't vary in time, but it is essential take into account the possibility that the magnetic field (and consequently the flux) does vary. Now, if surface $d\mathbf{a}$ is very small we can ignore the integral and simply write $\Phi = \mathbf{B} \cdot d \mathbf{a}$ (in case we are on a media discontinuity we will broke into $\mathbf{B}_1 \cdot d \mathbf{a}_1 + \mathbf{B}_2 \cdot d \mathbf{a}_2$, this has no effects on next reasoning) from wich
$\dot{\Phi} = \dot{\mathbf{B}}\cdot d \mathbf{a} + \mathbf{B} \cdot d \dot{\mathbf{a}} = \dot{\mathbf{B}}\cdot d \mathbf{a}$ (remember that the surface is constant in time). Because of the smallness of $d\mathbf{a}$ this can be read $\dot{\Phi}(t) \cong 0$. So the answer is yes: it goes to zero because previous steps show that time derivative too is pushed to zero by the infinitesimal surface.
| 168,205
|
The recent improvement in the condition of the market especially in the housing market encourages the homeowners to invest in renovation of home that thy hop would increase the value of their homes.
According to the survey conducted by he Joint Center for Housing Studies at Harvard University, for the past one and half years the trend of spending on home improvement has stated and it is predicted that it would increase almost 20% and reach nearly $151 billion by the end of the fourth quarter. The report also reveals that numerous homeowners decide to renovate their kitchen or basement from the idea that the finished and bigger one would make their home more comfortable one. On the other hand, individuals willing to sell their home, remodeling it with the hope that it would boost the price of their home.
It is worthy to mention over here that the report publishes six wonderful suggestions that one need to consider before investing in home renovation project.
To measure the effectiveness of home improvement, Richard Borger who serves as the president of the Appraisal Institute suggests that estimation should require that would provide the homeowners an idea of returnable amount. In most of the cases, he adds, the renovation offer the homeowners near about 50 percent return on their new investment. According to the survey of remodeling magazine, prices of home are rising so fast that it now becomes easier to recover the expenses spent on home renovation, regardless of the upgradation.
It is certain that installing latest features like appliances and cabinets increases the value of home. One should require to be enough conscious while selecting the appliances and finishes that is suitable to the neighborhood.
Borges also states that another popular home improvement plan that proves less effective is addition of an extra room in home that expands the size of home. He is of the opinion that projects that includes splitting down an exterior wall that requires moving windows. Doors and several other features cost much better than modifying an attic into a bedroom that he believes utilizes existing space of the room. He stated that it is difficult to recover the expenses of costly improvement.
One should consider all types of buyers when planned to modernize the home. Borger explained that in case one has decided to add a swimming pool that the person hopes would increase the value of the home and makes it easier to sell the house. But the reality is just the opposite. Installing the pool make home quite hard and tough to sell as most of the possible buyers would deny taking the extra maintenance cost.
The most important part of the home improvement is obviously consulting an expert who has profound knowledge in the field. Usually an expert charges $500 to $1000 to deliver their advices. However, one can take the assistance of real estate agent who offers their services for free with the hope that homeowners would take their help when they decided to sell it.
Changing the master bedroom into a bigger one or changing the downstairs closet into a half bath proves to be good investments. However, these may not be aesthetic enough. According to Remodeling magazine, homeowners can regain the maximum amount of money from the improvement that costs less than $5,000. The magazine informs that it was almost 85.6 percent of the cost.
| 11,386
|
: Wardrobes
Flat Pack Helping YouFlat pack furniture by its very nature is easy to store, transport and deliver through difficult entrances compared to ready built furniture and will be with us a lon
Frequently Asked Questions
The FAQ section on this site will develop and re-strucutre over time, initially there will be a simple list of questions, but as it expand
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| 45,852
|
TITLE: Reduce the difference between three numbers
QUESTION [0 upvotes]: Is there a way to know the maximum possible reduction between three numbers $(a>b>c)$, so that all three numbers are closer to each other but with the constraint that combined subtraction allowed from the three numbers is limited to a number $'M'$ and combined addition allowed from all three numbers is limited to a number $'N'$.
So for an easy example, if $a=9, b=6, c=5$ and combined addition allowed $(N)=7$ & combined subtraction allowed $(M)=0$ then the three numbers can become $a'=(9+0)=9, b'=(6+3)=9, c'=(5+4)=9$. Which is the closest they can get given the constraints $\rightarrow$ 3 was added to $'b'$ and 4 to $'c'$ making the total $(N)=7$.
Can there be a general way to do this for any $(a,b,c)$? Or some analytical formula which can give maximum bound all three numbers will remain within given $M \,\, \& \,\,N$.
REPLY [1 votes]: You can solve the problem via linear programming as follows. Let $M_a, M_b, M_c$ be the amounts added, and let $N_a, N_b, N_c$ be the amounts subtracted. Let $U$ represent $\max(a',b',c')$ and $L$ represent $\min(a',b',c')$. The problem is to minimize $U-L$ subject to:
\begin{align}
L \le a + M_a - N_a &\le U \\
L \le b + M_b - N_b &\le U \\
L \le c + M_c - N_c &\le U \\
M_a + M_b + M_c &\le M \\
N_a + N_b + N_c &\le N \\
M_a, M_b, M_c &\ge 0 \\
N_a, N_b, N_c &\ge 0 \\
\end{align}
| 210,621
|
The last time I made oatmeal cookies, I mentioned that I’m not a huge fan of raisins, so I typically put chocolate or peanut butter chips in my oatmeal cookies. I forgot about one seriously important exception — Raisinettes. I don’t hate raisins, but they really do nothing for me. Raisinettes, on the other hand? That’s a WHOLE different game. And it’s a game that goes awesome with oatmeal cookies. Typically I use old fashioned oats in my cookies, but all I had on hand was quick oats, so the recipe is a bit different this time. Equally as delicious!
You will need:
1 cup unsalted butter, softened
1 cup packed light brown sugar
1/2 cup sugar
2 eggs
2 teaspoons vanilla extract
1 1/4 cups all-purpose flour
1/2 teaspoon baking soda
1 teaspoon salt
3 cups quick cooking oats
1 1/2 cups Raisinettes
Preheat the oven to 325. In a large bowl, cream together the butter and sugars until smooth. Beat in eggs one at a time, then stir in vanilla. Combine the flour, baking soda, and salt; stir into the creamed mixture until just blended. Mix in the quick oats and combine. Fold in Raisinettes. Drop by heaping spoonfuls onto ungreased baking sheets. Bake for 12 minutes, until golden on the edges. Allow cookies to cool for a few minutes before transferring to a wire rack to cool completely.
Super sweet, chewy, and delicious– boyfriend approved and requested again, so these will certainly become a staple in my house 🙂
Love the chocolate covered raisins! It’s the best of both worlds!
| 206,158
|
The laboratory is a part of the BioPharm Lab Complex in Sofia Tech park. The primary mission of the laboratory is to support the utilization of the existing biodiversity in Bulgaria – medicinal and aromatic plants, which are of high quality and contain active ingredients, thus stimulating sustainable development and economic growth.
- Application of green extraction technologies for utilization of Medicinal Aromatic Plants (MAPs).
- Fractioning and purification of extracts, investigation of the content of active ingredients and their biological activity.
- Determination of activity profiles and development of standard procedures and documentation for phytochemical production, oriented to small and medium sized enterprises (SMEs).
- Development of standardization and control procedures for MAPs-based products, with the purpose to guarantee the consumer’s interests.
- Isolation of individual biologically active compounds, structure elucidation and synthesis of natural compounds analogues, possessing structural diversity and bioactivity (identification of promising compounds for development of drug candidates).
Current projects:
- Development of a method for quantitative determination of pyrrolizidine alkaloids in dry plant material (herbs)
- Characterization and standardization of the various types of essential oil of Melissa officinalis (Lemon Balm oil);
- Development of new approaches for verifying the authenticity of Rose and Lavender oils;
- Development of methods for monitoring of natural products by high performance thin layer chromatography (HPTLC);
- Development of a method for the determination of allergens in essential oils and cosmetic formulations.
Completed projects:
Preparation of bioactive extracts from medicinal plants for the creation of innovative products
Services
- Development of procedures for extraction of biologically active components from medicinal plants. Preparation of pilot quantities of standardized bioactive extracts accompanied by analytical and others protocols
- Development of methods for analysis of mixtures of natural compounds, components for incorporation in cosmetic products, and final pharmaceutical formulations.
- Development of methodologies for analysis of natural products by high-performance thin-layer chromatography (HPTLC).
- Development of methods and procedures for synthesis, as well as obtaining of specified quantities of organic compounds ordered by pharmaceutical companies.
- Development of methods for the control of compounds in plant extracts (eg caffeic acid, rosmarinic acid, quercetin, etc.);
- Authentication of rose oil by GC-MS analysis.
Equipment
- GC with QTOF MS / MS detector
- GC with FID and MS detectors
- Triple Quadrupole Mass Spectrometer – LCMS-8045
- Liquid Chromatograph with Diode Array Detector and Single Quadrupole Mass Spectrometer – LCMS-2020
- Liquid Chromatograph with Diode Array Detector and Fluorescence Detector – Nexera X2.
- Apparatuses for automatic application, development, functionalization and densitometry of TLC / HPTLC plates;
- Flash chromatographs;
- Multi-purpose 5 liter reactor;
- Laboratory equipment for extraction of natural products and synthesis of organic compounds (reactors, rotary vacuum evaporators, vacuum installations, etc.)
Management
Dissertation (Dr. rer. Nat.) at the Technical University of Leuna-Merseburg; 2002 “Doctor of chemical sciences” HAC; 1989-1990 Scholarship Foundation Alexander von Humboldt; 1992 and 1998-2001 researcher at the University of Zurich;
Numerous lectures at foreign universities and international conferences; Numerous articles in international journals;
Many completed contracts in the EU framework programs, international funds, etc; contracts with pharmaceutical companies. Contracts in the EU framework programs, international funds, etc .; contracts with pharmaceutical companies.
Sofia, bul. Tsarigradsko Shousse 111
Laboratory Complex, Floor 2
Phone: + 359 889 900 614
Laboratory enquiry for your project:
| 257,288
|
\begin{document}
\maketitle
\begin{abstract}
We characterize all $(n-2)$-dimensional linear subspaces of $\mathbb{P}^{n}$ such that the induced linear projection, when restricted to the rational normal curve, gives a Galois morphism. We give an explicit description of these spaces as a disjoint union of locally closed subvarieties in the Grassmannian $\mathbb{G}(n-2,n)$.
\end{abstract}
\section{Introduction}
Given an embedding $X\hookrightarrow\mathbb{P}^n$ of a $d$-dimensional smooth projective variety into projective space and a linear subspace $W\in\mathbb{G}(n-d-1,n)$, it is natural to ask about the monodromy of the linear projection $\pi:X\dashrightarrow\mathbb{P}^d$ from $X$ with center $W$, and in particular if it induces a Galois extension of function fields. The monodromy of linear projections from varieties embedded in projective space has been the focus of several articles (see for example \cite{Cuk} and \cite{Pirola}), and in the last decade the question of when a linear projection is Galois has been intensely studied under the name of \emph{Galois embeddings}.
The formal definition of Galois embedding was introduced by Yoshihara in \cite{Yoshi}. We recall that if $X$ is a smooth projective variety of dimension $d$ and $D$ is a very ample divisor on $X$ that induces an embedding $\varphi_D:X\to\mathbb{P}^n$, then the embedding is said to be \emph{Galois} if there exists a linear subspace $W\in\mathbb{G}(n-d-1,n)$ such that $W\cap\varphi_D(X)=\varnothing$ and the linear projection $\pi_W:\mathbb{P}^n\dashrightarrow\mathbb{P}^d$ defined by $W$ restricts to a morphism $\pi:X\to\mathbb{P}^d$ that induces a Galois extension of function fields $\pi^*(k(\mathbb{P}^d))\subseteq k(X)$. The calculation of Galois subspaces for different varieties has been an object of interest in the past few years, and especially in the case of curves and abelian varieties. See, for instance, \cite{Auff}, \cite{Cuk}, \cite{Pirola}, \cite{Takahashi}, \cite{Yoshi} and the references therein. The purpose of this article is to classify all linear subspaces of $\mathbb{P}^n$ that induce a Galois morphism for an embedding of $\mathbb{P}^1$ into $\mathbb{P}^n$, thereby answering a question asked by Yoshihara in his list of open problems \cite{Yoshiproblems}. This question was already addressed by Yoshihara himself for $n=3$ in \cite[Prop. 4.1]{Yoshi2}, and our article can be seen as a generalization of his results to arbitrary dimension.
In the case of $\mathbb{P}^1$, the only embedding of $\mathbb{P}^1$ into $\mathbb{P}^n$ such that the image is not contained in a hyperplane, modulo a linear change of coordinates, is the Veronese embedding $\nu_n:\mathbb{P}^1\to\mathbb{P}^n$ which is always Galois as we will see. Given a linear projection $\pi_W:\mathbb{P}^n\dashrightarrow\mathbb{P}^1$ with center $W$, one can ask about the Galois closure of the field extension induced by $\pi=\pi_W\circ \nu_n:\mathbb{P}^1\to\mathbb{P}^1$. The Galois group of this closure is the monodromy group of $\pi$, and is related to several hard and interesting problems; see for example \cite{Adrianov}, \cite{Cuk}, \cite{GS}, \cite{Konig}, \cite{Muller} and \cite{Muller2} among many others.
By \cite[Theorem 2.2]{Yoshi}, an embedding of $X$ into $\mathbb{P}^n$ is Galois if and only if there exists a finite group of automorphisms $G$ of $X$ such that $|G|=(D^n)$, there exists a $G$-invariant linear subspace $\mathcal{L}$ of $H^0(X,\mathcal{O}_X(D))$ of dimension $d+1$ such that $G$ acts on $\mathbb{P}\mathcal{L}$ as the identity, and such that the linear system $\mathcal{L}$ has no base points. In practice, given a group $G$ and a $G$-invariant divisor $D$, one must find the subspace $\mathcal{L}$; this is the approach we will take in this article.
Before stating our main theorem, we define the linear subspaces we will be working with. For $a,b,c,d\in k$ we define the homogeneous linear polynomial in the variables $x_0,\ldots,x_n$
$$L_{a,c}:=\sum_{i=0}^n\binom{n}{i}(-1)^ia^ic^{n-i}x_i.$$
If $\alpha\in k$ and $n=2m$ is even, we define the polynomials
$$A_{a,b,c,d}:=\sum_{i+j+l=m}\binom{m}{i,j,l}(-1)^i(ad+bc)^i(ab)^j(cd)^lx_{i+2l}$$
$$B_{a,b,c,d}^\alpha:=\sum_{i=0}^{n}\binom{n}{i}(-1)^i(\alpha^ma^{n-i}c^i+b^{n-i}d^i)x_i.$$
We note that the hyperplane defined by $L_{a,c}$ only depends on the class of $(a,c)$ in $\mathbb{P}^1$ when $(a,c)$ is non-zero, and we will therefore denote it by $H_{[a:c]}$. The linear subspace defined by $A_{a,b,c,d}=B_{a,b,c,d}^\alpha=0$ for $(a,b,c,d)\neq(0,0,0,0)$ only depends on the class of $(a,b,c,d)$ in $\mathbb{P}^3$ and on $\alpha$ and will be denoted by $V_{[a:b:c:d]}^\alpha$.
\begin{theorem}\label{main}
Let $k$ be a field, let $\nu_n:\mathbb{P}^1\to\mathbb{P}^n$ be the Veronese embedding of degree $n\geq3$ over $k$, and assume that the characteristic of $k$ does not divide $n$ and that $k$ contains a primitive $n$th root of unity $\zeta_n$. Then we have the following:
\begin{enumerate}
\item For each $([a:c],[b:d])\in(\mathbb{P}^1\times\mathbb{P}^1)\backslash\Delta$,
$$W_{[a:c],[b:d]}:=H_{[a:c]}\cap H_{[b:d]}$$
is a Galois subspace of $\mathbb{P}^n$ for $\nu_n$ with cyclic Galois group that is conjugate to the group generated by $[x:y]\mapsto[\zeta_nx:y]$. This gives a 2-dimensional locally closed subvariety $\mathcal{C}_n$ of $\mathbb{G}(n-2,n)$.
\item If $n=2m$ is even, in addition to the previous item, for each representative $\alpha\in k^\times/(k^\times)^2\mu_m(k)$ we have the Galois subspaces $V_{[a:b:c:d]}^\alpha$ for $[a:b:c:d]\in\mathbb{P}^3$ and $ad-bc\neq0$. Here the Galois group is the dihedral group of order $2m$ and is conjugate to the group generated by $[x:y]\mapsto[\zeta_nx:y]$ and $[x:y]\mapsto[\alpha y:x]$. Thus we obtain families $\mathcal{D}_m^\alpha$ which over $\overline{k}$ give a 3-dimensional locally closed subvariety $\mathcal{D}_m$ of $\mathbb{G}(n-2,n)$.
\end{enumerate}
The above Galois subspaces are all disjoint from $\nu_n(\mathbb{P}^1)$. Moreover, if $n\notin\{4,12,24,60\}$, these are the only Galois subspaces of $\mathbb{P}^n$ for $\nu_n$ disjoint from $\nu_n(\mathbb{P}^1)$.
\end{theorem}
The exceptional cases $n\in\{2,4,12,24,60\}$ are addressed in the following proposition:
\begin{proposition}\label{exceptional}
We have the following families of Galois subspaces disjoint from $\nu_n(\mathbb{P}^1)$:
\begin{enumerate}
\item For $n=2$, every element of $\mathbb{G}(0,2)\backslash \nu_2(\mathbb{P}^1)$ gives a Galois subspace for $\nu_2(\mathbb{P}^1)$.
\item For $n=4$, apart from $\mathcal{C}_4$, for every $\beta\in k^\times/(k^\times)^2$ there is a family $\mathcal{K}^\beta\subseteq\mathbb{G}(2,4)$ that corresponds to Galois subspaces whose Galois group is the Klein four group. Over $\overline{k}$ these give a $3$-dimensional locally closed subvariety $\mathcal{K}$ of $\mathbb{G}(2,4)$.
\end{enumerate}
If, moreover, $-1$ is the sum of two squares in $k$, then:
\begin{enumerate}
\item[(3)] For $n=12$, apart from $\mathcal{C}_{12}$ and the spaces $\mathcal{D}_{12}^\alpha$, there exists a 3-dimensional family $\mathcal{A}_4\subseteq\mathbb{G}(10,12)$ corresponding to Galois subspaces with Galois group $A_4$.
\item[(4)] For $n=24$, apart from $\mathcal{C}_{24}$ and the spaces $\mathcal{D}_{24}^\alpha$, there exists a 3-dimensional family $\mathcal{S}_4\subseteq\mathbb{G}(22,24)$ corresponding to Galois subspaces with Galois group $S_4$.
\item[(5)] For $n=60$, if $5$ is a square in $k$, then apart from $\mathcal{C}_{60}$ and the spaces $\mathcal{D}_{60}^\alpha$, there exists a 3-dimensional family $\mathcal{A}_5\subseteq\mathbb{G}(58,60)$ corresponding to Galois subspaces with Galois group $A_5$.
\end{enumerate}
\end{proposition}
The previous two results exhaust the list of possible Galois subspaces in the case they are disjoint from $\nu_n(\mathbb{P}^1)$. The case when the Galois subspace is not disjoint from $\nu_n(\mathbb{P}^1)$ will be analyzed in Section \ref{Nondisjoint} where we give a complete explicit classification of all Galois subspaces for the Veronese embedding in any degree in the case that the base field is algebraically closed. In particular, we show that for $n\geq60$ there are $n+\lfloor\frac{n}{2}\rfloor+2$ disjoint families of Galois subspaces in $\mathbb{G}(n-2,n)$, and the biggest family is of dimension $n$. See Table \ref{classification} for the full classification.
\section{Preliminaries}
In this section we recall some classical results, along with results we will need for Galois embeddings. Let $\text{End}_n(\mathbb{P}^1)$ denote the set of endomorphisms of $\mathbb{P}^1$ of degree $n$. We see that $\mbox{Aut}(\mathbb{P}^1)$ acts on $\text{End}_n(\mathbb{P}^1)$ via composition. Let
$$\nu_n:\mathbb{P}^1\to\mathbb{P}^n$$
denote the $n$th Veronese embedding.
\begin{lemma}\label{Bij}
There is a bijective correspondence between $\text{End}_n(\mathbb{P}^1)/\text{Aut}(\mathbb{P}^1)$ and all linear subspaces $W\in\mathbb{G}(n-2,n)$ such that $W\cap \nu_n(\mathbb{P}^1)=\varnothing$.
\end{lemma}
\begin{proof}
Let $x,y$ be the coordinates on $\mathbb{P}^1$. Then a degree $n$ endomorphism $f$ is given by $[p(x,y):q(x,y)]$ where $p$ and $q$ are homogeneous polynomials of degree $n$ without common factors. Write
$$p(x,y)=a_0x^n+a_1x^{n-1}y+\cdots+a_ny^n$$
$$q(x,y)=b_0x^n+b_1x^{n-1}y+\cdots+b_ny^n.$$
We then send $f$ to the linear subspace
$$W_f:=\{a_0x_0+\cdots+a_nx_n=b_0x_0+\cdots+b_nx_n=0\}\in\mathbb{G}(n-2,n).$$
If $W_f\cap \nu_n(\mathbb{P}^1)\neq\varnothing$, then there exists $[x:y]\in\mathbb{P}^1$ such that
$$a_0x^n+\cdots+a_ny^n=b_0x^n+\cdots+b_ny^n=0.$$
This, however, implies that the polynomials $p(x,y)$ and $q(x,y)$ share a common factor, and thus $f$ is not of degree $n$. Therefore $W_f\cap \nu_n(\mathbb{P}^1)=\varnothing$.
Now take $W\in\mathbb{G}(n-2,n)$ such that $W\cap \nu_n(\mathbb{P}^1)=\varnothing$, let $L_0,L_1\in k[x_0,\ldots,x_n]_1$ be two defining equations for $W$, and consider the morphism $\pi=[L_0:L_1]\circ \nu_n:\mathbb{P}^1\to\mathbb{P}^1$. We first note that the sections $\nu_n^*L_0,\nu_n^*L_1$ do not have common roots, since that would contradict the fact that $W\cap \nu_n(\mathbb{P}^1)=\varnothing$. Secondly, these forms have degree $n$ since $\nu_n(\mathbb{P}^1)$ is of degree $n$ in $\mathbb{P}^n$. We see that different defining equations for $W$ differ by an action of $\text{PGL}(2,k)$, and this translates to composing the morphism $\pi$ by an automorphism of $\mathbb{P}^1$.
\end{proof}
The following corollary is clear from the previous proof:
\begin{corollary}
The previous correspondence gives a bijection between Galois morphisms of degree $n$ from $\mathbb{P}^1$ to $\mathbb{P}^1$ modulo the action of $\text{Aut}(\mathbb{P}^1)$, and Galois subspaces of $\mathbb{P}^n$ for $\nu_n$, disjoint from $\nu_n(\mathbb{P}^1)$.
\end{corollary}
This shows that characterizing Galois subspaces of $\mathbb{P}^n$ for $\nu_n$ is essentially the same as characterizing Galois morphisms from $\mathbb{P}^1$ to itself, and therefore the same as characterizing rational subfields of $k(t)$ that give Galois extensions.
Let $G$ be a finite group acting (effectively) on $\mathbb{P}^1$ such that its order is not divisible by the characteristic of the ground field $k$. By a close study of the Riemann-Hurwitz formula and the ramification points that may appear for this action, one can conclude that $G$ is either $\mathbb{Z}/n\mathbb{Z}$ for any $n\in\mathbb{N}$, the dihedral group $D_m$ of order $2m$ for any $m\in\mathbb{N}$, $A_4$, $S_4$ or $A_5$. Over the complex numbers, these last three groups appear as the (holomorphic) automorphism groups of the Platonic solids which can be inscribed in the Riemann sphere. Not all of these groups always appear, however, for an arbitrary field. Indeed, we have the following (cf. \cite[Prop. 1.1]{Beau}) which shows the conditions that need to be met in Proposition \ref{exceptional}:
\begin{proposition}
\begin{enumerate}
\item $\text{PGL}(2,k)$ contains $\mathbb{Z}/n\mathbb{Z}$ and the dihedral group $D_n$ if and only if $k$ contains $\zeta+\zeta^{-1}$ for some primitive $n$th root of unity $\zeta$.
\item $\text{PGL}(2,k)$ contains $A_4$ and $S_4$ if and only if $-1$ is the sum of two squares in $k$.
\item $\text{PGL}(2,k)$ contains $A_5$ if and only if $-1$ is the sum of two squares and $5$ is a square in $k$.
\end{enumerate}
\end{proposition}
By \cite[Theorem 4.2]{Beau}, two isomorphic finite groups of automorphisms are conjugate by automorphisms of the projective line, except for $G\simeq\mathbb{Z}/2\mathbb{Z}$, $G$ the Klein four group, or $G$ the dihedral group. We will look at this further on.
Let $D\in\mbox{Div}(\mathbb{P}^1)$ be a divisor of positive degree (which for $\mathbb{P}^1$ is equivalent to being very ample), and let $\varphi_D:\mathbb{P}^1\to\mathbb{P}^n$ be the associated embedding, where $n=\deg(D)$. Since $D$ is linearly equivalent to $n[1:0]$, we have that $\varphi_D$ is the Veronese embedding
$$[x:y]\mapsto[x^n:x^{n-1}y:\cdots:xy^{n-1}:y^n]$$
of $\mathbb{P}^1$ in $\mathbb{P}^n$ and its image is a rational normal curve. We are interested in characterizing all Galois subspaces of this embedding, thereby answering a question raised by Yoshihara (cf. \cite[(4)(b)(ii)]{Yoshiproblems}).\\
Let $D=\sum_{i}^rm_i[p_i:q_i]$, and consider the rational map
$$f_D:=\prod_{i}^r(q_ix-p_iy)^{-m_i}.$$
A trivial calculation shows that
\begin{eqnarray}\nonumber L(D)&:=&\{h\in k(\mathbb{P}^1):\text{div}(h)+D\geq0\}\\
\nonumber&=&\langle f_Dx^n,f_Dx^{n-1}y,\ldots,f_Dxy^{n-1},f_Dy^n\rangle\\
\nonumber &=&f_DH^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(n))
\end{eqnarray}
Therefore given a finite group of automorphisms $G$ of $\mathbb{P}^1$ that preserve the divisor $D$, in order to find the Galois subspace of $\mathbb{P}^n$ associated to the group $G$ we must understand the action of $G$ on $L(D)$ by pullback. Indeed, we are looking for a 2-dimensional subspace $\mathcal{L}\subseteq L(D)$ such that $G$ acts as the identity on $\mathbb{P}\mathcal{L}$.
In what follows we will divide our analysis according to the group that is acting.
\section{Cyclic group action}
Let $n\geq3$, assume that $k$ contains a primitive $n$th root of unity $\zeta_n$, and consider the order $n$ morphism
$$C_n:[x:y]\mapsto[\zeta_nx:y].$$
By \cite[Theorem 4.2]{Beau}, we have that every cyclic subgroup of $\text{PGL}(2,k)$ is conjugate to $\langle C_n\rangle$. Take
$$M:=\left(\begin{array}{cc}a&b\\c&d\end{array}\right)\in\text{PGL}(2,k).$$
We have that
$$T_n:=MC_nM^{-1}=\left(\begin{array}{cc}ad\zeta_n-bc&ab(1-\zeta_n)\\cd(\zeta_n-1)&ad-bc\zeta_n\end{array}\right)$$
and $T_n$ leaves the divisor $D:=n[a:c]=nM([1:0])$ invariant. By the previous section, we have that
$$L(D)=\langle f_Dx^n,f_Dx^{n-1}y,\ldots,f_Dxy^{n-1},f_Dy^n\rangle$$
where
$$f_D=(cx-ay)^{-n}.$$
We see that
$$1=(cx-ay)^{n}f_D\in L(D)$$
and is trivially invariant under the action of $T_n$. Notice that the automorphism
$$R:=\left(\begin{array}{cc}d&-b\\-c&a\end{array}\right)$$
sends $[b:d]$ to $[0:1]$ and $[a:c]$ to $[1:0]$; moreover $RT_n$ does the same. This implies that $RT_n$ is a multiple of $R$, and upon further inspection it is easy to see that this multiple is an $n$th root of unity. Therefore
$$(dx-by)^{n}f_D\in L(D)$$
is also invariant under composition by $T_n$. This means that the subspace of $L(D)$ generated by
$$1=f_D\sum_{i=0}^n\binom{n}{i}(-1)^ia^ic^{n-i}x^{n-i}y^i$$
$$(dx-by)^{n}f_D=f_D\sum_{i=0}^n\binom{n}{i}(-1)^ib^id^{n-i}x^{n-i}y^i$$
is the eigenspace of $T_n$ for the eigenvalue 1. Therefore, using the notation in the introduction, we obtain the Galois subspaces
$$H_{[a:c],[b:d]}:=\{L_{a,c}=L_{b,d}=0\}.$$
These are the only Galois subspaces (disjoint from $\nu_n(\mathbb{P}^1)$) with cyclic Galois group and give a family $\mathcal{C}_n\subseteq\mathbb{G}(n-2,n)$; we will verify that the dimension of this family is 2 in Section \ref{ex}.
\section{Dihedral group action}
Let $n=2m\geq6$ be even, assume that $k$ contains a primitive $n$th root of unity $\zeta_n$, and set $\zeta_m:=\zeta_n^2$. For $\alpha\in k^\times/(k^\times)^2\mu_m(k)$ consider the morphisms
$$C_m:[x:y]\mapsto[\zeta_mx:y]$$
$$I_\alpha:[x:y]\mapsto[\alpha y:x].$$
By \cite[Theorem 4.2]{Beau}, we have that every subgroup of $\text{PGL}(2,k)$ isomorphic to the dihedral group of order $2m$ is conjugate to $\langle C_m,I_\alpha\rangle$ for some $\alpha$, and the conjugacy classes of the dihedral group in $\text{PGL}(2,k)$ are parametrized by $\alpha\in k^\times/(k^\times)\mu_m(k)$.
Taking the same matrix
$$M=\left(\begin{array}{cc}a&b\\c&d\end{array}\right)\in\text{PGL}(2,k)$$
as before, we see that
$$T_m=MC_mM^{-1}=\left(\begin{array}{cc}ad\zeta_m-bc&ab(1-\zeta_m)\\cd(\zeta_m-1)&ad-bc\zeta_m\end{array}\right)$$
$$J_\alpha:=MI_\alpha M^{-1}=\left(\begin{array}{cc}bd-\alpha ac& \alpha a^2-b^2\\d^2-\alpha c^2&\alpha ac-bd\end{array}\right).$$
Now we take the divisor $D=m[a:c]+m[b:d]$ which is invariant under these two transformations. We set
$$f_D=(cx-ay)^{-m}(dx-by)^{-m},$$
and use the basis $\{f_Dx^n,f_Dx^{n-1}y,\ldots,f_Dxy^{n-1},f_Dy^n\}$ of $L(D)$. Using the same idea as the previous section, we see that
$$1=(cx-ay)^{m}(dx-by)^{m}f_D\in L(D)$$
is fixed by the action of $T_m$ and $J_\alpha$. With a little work, it is not difficult to show that
$$\left(\alpha^m(cx-ay)^n+(dx-by)^n\right)f_D$$
is also fixed by both transformations. Therefore after expanding, we obtain that the group generated by $T_m$ and $J_\alpha$ acts with the trivial representation on the space generated by
$$f_D\sum_{i+j+l=m}\binom{m}{i,j,l}(-1)^i(ad+bc)^i(ab)^j(cd)^lx^{i+2l}y^{i+2j}$$
$$f_D\sum_{i=0}^{n}\binom{n}{i}(-1)^i(\alpha^ma^{n-i}c^i+b^{n-i}d^i)x^{i}y^{n-i}.$$
This implies that the linear space
$$V_{[a:b:c:d]}^\alpha:=\{A_{a,b,c,d}=B_{a,b,c,d}^\alpha=0\}$$
is a Galois subspace disjoint from $\nu_n(\mathbb{P}^1)$ with dihedral Galois group. This gives a family $\mathcal{D}_m^\alpha\subseteq\mathbb{G}(n-2,n)$ whose dimension we will verify to be 3 in the next section.
\section{Exceptional cases} \label{ex}
We will now look at the remaining cases (i.e. $n\in\{2,4,12,24,60\}$); assume here that $k$ is algebraically closed. For $n=2$, each linear projection composed with the Veronese embedding is either of degree 2 or 1, depending on whether the center of projection is disjoint from $\nu_2(\mathbb{P}^1)$ or not. In either case, the induced field extension is of the same degree, and is therefore Galois. Therefore every element of $\mathbb{G}(0,2)=\mathbb{P}^2$ induces a Galois morphism.
For $n\in\{4,12,24,60\}$, we invite the interested reader to explicitly calculate the Galois subspaces as done above in the cyclic and dihedral cases. Although tedious, the calculations are elementary. We will proceed to show that in each of these cases, for the Klein four group $K_4$, $A_4$, $S_4$ and $A_5$ there is a locally closed 3-dimensional family of Galois subspaces in the respective Grassmannian.
We note that $\mbox{PGL}(2,k)$ is 3-dimensional, and the conjugacy class of a given subgroup $G\leq\text{PGL}(2,k)$ is parametrized by $\mbox{PGL}(2,k)/N_G$, where $N_G$ is the normalizer of $G$. If $C_G$ is the centralizer of $G$ in $\text{PGL}(2,k)$, we have a natural homomorphism $\varphi:N_G\to\text{Aut}(G)$ whose kernel is $C_G$. In particular, if $G$ is finite, then $C_G$ is of finite index in $N_G$. By the table on page 27 of \cite{Beau}, we have that the centralizers of $K_4$, $D_n$, $A_4$, $S_4$ and $A_5$ in $\text{PGL}(2,k)$ are all finite, and therefore the normalizers are also finite. In the case of $\mathbb{Z}/n\mathbb{Z}$, the centralizer is 1-dimensional and therefore the normalizer is as well. By \cite[Theorem 4.2]{Beau}, since we are assuming $k$ is algebraically closed, there is only one conjugacy class of each of finite subgroup of $\text{PGL}(2,k)$.
\begin{proposition}
The families $\mathcal{C}_n$ are 2-dimensional and the families $\mathcal{D}_{m}$ are 3-dimensional for all $n,m\geq 3$. For $K_4$, $A_4$, $S_4$ and $A_5$ we obtain families of Galois subspaces $\mathcal{K}\subseteq\mathbb{G}(2,4)$, $\mathcal{A}_4\subseteq\mathbb{G}(10,12)$, $\mathcal{S}_4\subseteq\mathbb{G}(22,24)$ and $\mathcal{A}_5\subseteq\mathbb{G}(58,60)$, respectively, that are all 3-dimensional.
\end{proposition}
This completes the proof of Theorem \ref{main} and Proposition \ref{exceptional}.
\section{Non-disjoint Galois subspaces}\label{Nondisjoint}
For this section we will assume that $k$ is algebraically closed. Let $W\in\mathbb{G}(n-2,n)$ be a subspace that is not disjoint from $\nu_n(\mathbb{P}^1)$. Projection from $W$, restricted to $\nu_n(\mathbb{P}^1)$, can give a Galois morphism as in the case of $W=\{x_0=x_2=0\}$.
However, we see by Lemma \ref{Bij} and its proof that if $W\in\mathbb{G}(n-2,n)$ and $W\cap \nu_n(\mathbb{P}^1)\neq\varnothing$, then $\pi_W \nu_n:\mathbb{P}^1\to\mathbb{P}^1$ is of degree less than $n$, where $\pi_W:\mathbb{P}^n\dashrightarrow\mathbb{P}^1$ is the projection with center $W$. As a matter of fact, by the same lemma, there exists a unique $\widetilde{W}\in\mathbb{G}(m-2,m)$ such that $\pi_{\widetilde{W}} \nu_m=\pi_W \nu_n$ (modulo automorphisms of $\mathbb{P}^1$), where $m=\deg(\pi_W \nu_n)$. We therefore obtain a morphism
$$\Phi_{n,m}:\frak{X}_{n,m}:=\{W\in\mathbb{G}(n-2,n):\deg(\pi_W\nu_n)=m\}\to\mathbb{G}(m-2,m).$$
\begin{proposition}
The fibers of $\Phi_{n,m}$ are irreducible and of dimension $n-m$. In particular $\Phi_{n,m}$ is injective if and only if $n=m$.
\end{proposition}
\begin{proof}
Let us first calculate the dimension of the fibers. We see that the image of $\Phi_{n,m}$ consists of those subspaces that are disjoint from $\nu_m(\mathbb{P}^1)$. Indeed, we have a map $i_{m,n}:\mathbb{G}(m-2,m)\hookrightarrow\mathbb{G}(n-2,n)$ that takes a subspace defined by two equations and sends it to the subspace of dimension $n-2$ defined by the same two equations; it is easy to see that $\Phi_{n,m}i_{m,n}$ is the identity on those subspaces of $\mathbb{G}(m-2,m)$ that are disjoint from $\nu_m(\mathbb{P}^1)$.
In particular, if $F$ is a general fiber of $\Phi_{n,m}$, then
$$\dim F=\dim\frak{X}_{n,m}-2(m-1).$$
Take $W\in\frak{X}_{n,m}$ and let it be the intersection of two hyperplanes $H_1,H_2\in\mathbb{G}(n-1,n)$. Write
$$\nu_n^*H_1=p_1+\cdots+p_n$$
$$\nu_n^*H_2=q_1+\cdots+q_n$$
where the above points are not necessarily different. Then since $W\in\frak{X}_{n,m}$, we have that exactly $n-m$ of the $p_i$ (counted with multiplicity) are equal to $n-m$ of the $q_i$ (counted with multiplicity). We have a rational map
$$\text{Sym}^{n-m}(\mathbb{P}^1)\times\text{Sym}^{m}(\mathbb{P}^1)\times\text{Sym}^{m}(\mathbb{P}^1)\dashrightarrow\mathbb{G}(n-2,n)$$
where
$$(D_1,D_2,D_3)\mapsto\text{span}\left\{\nu_n(D_1+D_2)\right\}\cap\text{span}\left\{\nu_n(D_1+D_3)\right\}.$$
Let $\frak{Y}_{n,m}$ be the set of all $(D_1,D_2,D_3)\in\text{Sym}^{n-m}(\mathbb{P}^1)\times\text{Sym}^{m}(\mathbb{P}^1)\times\text{Sym}^{m}(\mathbb{P}^1)$ such that $D_1+D_2\neq D_1+D_3$ and $\text{supp}(D_2)\cap\text{supp}(D_3)=\varnothing$. Then we have a morphism
$$\theta:\frak{Y}_{n,m}\to\mathbb{G}(n-2,n)$$
whose image is $\frak{X}_{n,m}$. Since the natural intersection morphism
$$(\mathbb{G}(n-1,n)\times\mathbb{G}(n-1,n))\backslash\Delta\to\mathbb{G}(n-2,n)$$
has 2-dimensional fibers, we obtain that the fibers of the map $\theta$ are 2-dimensional. This implies that
$$\dim\frak{X}_{n,m}=n+m-2.$$
We now conclude that a general fiber of $\Phi_{n,m}$ has dimension
$$n+m-2-2(m-1)=n-m.$$
A more careful analysis shows that if $V\in\mathbb{G}(m-2,m)$ and is disjoint from $\nu_m(\mathbb{P}^1)$, and $(D_1,D_2,D_3)\in\theta^{-1}\Phi_{n,m}^{-1}(V)$ is any preimage, then
$$\Phi_{n,m}^{-1}(V)=\theta(\text{Sym}^{n-m}(\mathbb{P}^1)\times\{D_2\}\times\{D_3\})$$
and is therefore irreducible.
\end{proof}
With respect to determining when an element $W\in\mathbb{G}(n-2,n)$ that is not disjoint from $\nu_n(\mathbb{P}^1)$ gives a Galois subspace, it is clear that we have the following:
\begin{proposition}
If $W\in\mathbb{G}(n-2,n)$ is such that $\deg(\pi_W\nu_n)=m$, then $W$ is a Galois subspace for $\nu_n$ if and only if $\Phi_{n,m}(W)$ belongs to one of the families presented in Theorem \ref{main} and Proposition \ref{exceptional}.
\end{proposition}
We will define $\Phi_{n,1}(\mathcal{C}_1)$ to be the set of all $W\in\mathbb{G}(n-2,n)$ such that $\pi_W\nu_n$ is an automorphism of $\mathbb{P}^1$; by the previous analysis it is easy to see that this family is $(n-1)$-dimensional.
\begin{theorem}
Table \ref{classification} gives the list of all families of Galois subspaces for $\nu_n:\mathbb{P}^1\to\mathbb{P}^n$.
\end{theorem}
In particular, we see that the largest family is $\Phi_{n,2}^{-1}(\mathcal{C}_2)$ which is $n$-dimensional.
\begin{table}[]
\begin{tabular}{|c|c|c|c|}
\hline& Families of Galois subspaces & Dimension of families & Number of families \\\hline&&&\\
$n=2,3$&$\Phi_{n,1}^{-1}(\mathcal{C}_1)$& $n-1$&$n$\\
&$\Phi_{n,k}^{-1}(\mathcal{C}_k)$, $k=2,\ldots,n$ &$n-k+2$&\\&&&\\\hline&&&\\
$4\leq n\leq 11$&$\Phi_{n,1}^{-1}(\mathcal{C}_1)$& $n-1$&$n+\lfloor\frac{n}{2}\rfloor-1$\\
&$\Phi_{n,k}^{-1}(\mathcal{C}_k)$, $k=2,\ldots,n$& $n-k+2$&\\
&$\Phi_{n,2k}^{-1}(\mathcal{D}_k)$, $k=3,\ldots,\lfloor\frac{n}{2}\rfloor$&$n-2k+3$&\\
&$\Phi_{n,4}^{-1}(\mathcal{K})$&$n-1$&\\&&&\\\hline&&&\\
$12\leq n\leq 23$&$\Phi_{n,1}^{-1}(\mathcal{C}_1)$& $n-1$&$n+\lfloor\frac{n}{2}\rfloor$\\
&$\Phi_{n,k}^{-1}(\mathcal{C}_k)$, $k=2,\ldots,n$& $n-k+2$&\\
&$\Phi_{n,2k}^{-1}(\mathcal{D}_k)$, $k=3,\ldots,\lfloor\frac{n}{2}\rfloor$&$n-2k+3$&\\
&$\Phi_{n,4}^{-1}(\mathcal{K})$&$n-1$&\\
&$\Phi_{n,12}^{-1}(\mathcal{A}_4)$&$n-9$&\\&&&\\\hline&&&\\
$24\leq n\leq 59$&$\Phi_{n,1}^{-1}(\mathcal{C}_1)$& $n-1$&$n+\lfloor\frac{n}{2}\rfloor+1$\\
&$\Phi_{n,k}^{-1}(\mathcal{C}_k)$, $k=2,\ldots,n$& $n-k+2$&\\
&$\Phi_{n,2k}^{-1}(\mathcal{D}_k)$, $k=3,\ldots,\lfloor\frac{n}{2}\rfloor$&$n-2k+3$&\\
&$\Phi_{n,4}^{-1}(\mathcal{K})$&$n-1$&\\
&$\Phi_{n,12}^{-1}(\mathcal{A}_4)$&$n-9$&\\
&$\Phi^{-1}(\mathcal{S}_4)$&$n-21$&\\&&&\\\hline&&&\\
$n\geq60$&$\Phi_{n,1}^{-1}(\mathcal{C}_1)$& $n-1$&$n+\lfloor\frac{n}{2}\rfloor+2$\\
&$\Phi_{n,k}^{-1}(\mathcal{C}_k)$, $k=2,\ldots,n$& $n-k+2$&\\
&$\Phi_{n,2k}^{-1}(\mathcal{D}_k)$, $k=3,\ldots,\lfloor\frac{n}{2}\rfloor$&$n-2k+3$&\\
&$\Phi_{n,4}^{-1}(\mathcal{K})$&$n-1$&\\
&$\Phi_{n,12}^{-1}(\mathcal{A}_4)$&$n-9$&\\
&$\Phi_{n,24}^{-1}(\mathcal{S}_4)$&$n-21$&\\
&$\Phi_{n,60}^{-1}(\mathcal{A}_5)$&$n-57$&\\&&&\\\hline
\end{tabular}
\caption{List of all families of Galois subspaces}
\label{classification}
\end{table}
\clearpage
| 15,097
|
\begin{document}
\markboth{X. M. Aretxabaleta, M. Gonchenko, N. L. Harshman, S. G. Jackson, M. Olshanii, and G. E. Astrakharchik}{Two-ball billiard predicts digits of the number PI in non-integer numerical bases}
\title{Two-ball billiard predicts digits of the number PI \\
in non-integer numerical bases}
\author{X. M. Aretxabaleta}
\affiliation{Departament de F\'{\i}sica, Universitat Polit\`{e}cnica de Catalunya, 08034 Barcelona, Spain}
\author{Marina Gonchenko}
\affiliation{Universitat de Barcelona, Barcelona, Spain}
\author{N.L. Harshman}
\affiliation{Department of Physics, American University, 4400 Massachusetts Ave.\ NW, Washington, DC 20016, USA}
\author{Steven Glenn Jackson}
\affiliation{Department of Mathematics, University of Massachusetts Boston, Boston, MA 02125, USA}
\author{Maxim Olshanii}
\affiliation{Department of Physics, University of Massachusetts Boston, Boston, MA 02125, USA}
\author{G. E. Astrakharchik}
\affiliation{Departament de F\'{\i}sica, Universitat Polit\`{e}cnica de Catalunya, E08034 Barcelona, Spain}
\begin{abstract}
We calculate the number PI from the problem of two billiard balls colliding with a wall (Galperin billiard method).
We provide a complete explicit solution for the balls' positions and velocities as a function of the collision number and time.
The relation between collision number and time can be further simplified in the limit of large base $b$ or mantissa length $N$.
We find new invariants of motion.
Also, we recover previously known ones for which we provide a simple physical explanation in terms of the square of the angular momentum and demonstrate that they coincide with the action invariant derived within the adiabatic approximation close to the return point.
We show that for general values of the parameters the system is integrable and for some special values of the parameters it is maximally superintegrable.
The portrait of the system is close to a circle in velocity-velocity and velocity-inverse position coordinates and to a hyperbola in position-time variables.
A differential equation describing the heavy-ball trajectory close to the point of return is derived and results in a parabolic trajectory.
A generalization of the system to finite-size balls (hard rods) is provided.
We propose to treat the possible error in the last digit as a systematic error.
Examples of integer and non-integer bases are considered.
In the intriguing case of expressing $\pi$ in the base $b=\pi$, the generated expression is different from a finite number, $\pi = 1\times \pi^1$, and instead is given
by an infinite representation, $\pi = 3 + 1/\pi^2 + 1/\pi^3+\cdots$.
The difference between finite and infinite representation is similar to that of $1 = 0.999(9)$ in the decimal system.
Finally we note that the finite representation is not unique in the base of the golden number.
\end{abstract}
\date{\today}
\maketitle
\tableofcontents
\newpage
\section{Introduction}
The invention of numbers was probably one of the most influential discoveries in the history of science leading to the foundation and development of mathematics.
In many ancient cultures, the symbols for the first digits correspond to a graphical representation of counting.
In Babylonian, Roman and Japanese numerals, digit ``1'' contains one counting object, digit ``2'' two objects, digit ``3'' three objects, see Fig.~\ref{Fig:numerals}.
After counting, the next important concept is that of positioning system (position of a digit defines its value) and the number base representation.
Throughout history different bases were used, including the modern decimal system and the sexagesimal one introduced in Babylon around second millennium BC.
Its legacy can still be found in modern units of time, with 60 seconds in one minute and 60 minutes in one hour.
\begin{figure}[ht]
\includegraphics[width=0.4\columnwidth]{FigRomanJapaneaseNumbers.pdf}
\caption{}
\label{Fig:numerals}
\end{figure}
It was realized that some numbers, referred to as {\em irrational}, cannot be written as a simple ratio of integer numbers and in this sense they are the most difficult to be calculated precisely.
Probably, the most important and fascinating irrational numbers are $\sqrt{2}$, $\pi$, Euler's constant $e$ and golden ratio $\varphi$.
Already in the antiquity there was a practical interest in representing some of those numbers explicitly~\cite{ArndtBook,Goyanes2007,BerggrenBook}.
In a Babylonian clay tablet from second millennium BC, the first four digits of $\sqrt{2}$ are explicitly given in sexagesimal system as $1, 24, 51, 10$.
In decimal system the error appears in the eighth digit as can be appreciated by comparing $1 + 24/60 +51/60^2+10/60^3 = 1.4142130$ with the proper value $1.4142135\dots$.
Another irrational number, $\pi$, naturally appears when calculating the ratio between a circumference of a circle and its diameter~\cite{BeckmannBook}.
In the Old Testament [1 Kings 7:23], the ratio between a circumference of a round vessel and its diameter is said to be equal to $3$.
While in many practical situations, it is sufficient to use an approximate value, it was of fundamental interest to find a method of finding the next digits.
Some other ancient estimations come from an Egyptian papyrus which implies $\pi = 256/81 = 3.160\dots$ and a Babylonian clay tablet leading to the value $25/8 = 3.125\dots$ .
Archimedes calculated the upper bound as $22/7 = 3.1428\dots$.
The fascination with the number $\pi$ makes scientists compete for the largest number of digits calculated.
Simon Newcomb (1835-1909) is quoted for having said ``{\it Ten decimal places of $\pi$ are sufficient to give the circumference of the earth to a fraction of an inch, and thirty decimal places would give the circumference of the visible universe to a quantity imperceptible to the most powerful microscope}''.
The current world record~\cite{Trueb2016} consists in calculating first 22,459,157,718,361 ($\pi^e$ trillion) digits.
Such a task manifestly goes beyond any practical purpose but can be justified by the great attractiveness of the number $\pi$ itself.
Apparently the distribution of digits is flat in different bases~\cite{Trueb2016} and it was tested that the sequence of $\pi$ digits makes a good random number generator which can be used for practical scientific and engineering computations~\cite{SHU-JU2005}.
An alternative popular idea is that, in contrast, special information might be coded in digits of $\pi$~\cite{ShumikhinBook}, or even God's name as in the plot of ``Pi'' film from 1990.
Recently, analogies between anomalies in the cosmic microwave background and patterns in the digits of $\pi$ were pointed out in ``Pi in the Sky'' article~\cite{Frolop2016}, which appeared on the 1st of April.
While the number $\pi$ elegantly arises in a large variety of trigonometric relations, integrals, series, products, continued fractions, far fewer experimental methods are known of how to obtain its digits by performing measurements according to some procedure.
A stochastic method, stemming from Comte de Buffon dates back to the eighteenth century and consists in dropping $N$ needles of length $l$ on parallel lines separated by length $L$ and experimentally determining the number of times $N_{cross}$ that those needles were crossing the lines.
The number $\pi$ can be then approximated by $\pi \approx 2l\cdot N/(L N_{cross})$ with the error in its estimation proportional to $1/\sqrt{N}$.
It means that in order to get the first $D$ digits right, one has to perform more than $100^{D}$ trials.
This makes it extremely difficult to obtain the precise value in a real-world experiment although an equivalent computer experiment can be easily performed with modern computational power.
Still, a simple mechanical system which would provide explicitly the digits of number $\pi$ was lacking for a long time.
A completely new perspective has emerged when G. A. Galperin formulated a deterministic method based on a two-ball billiard~\cite{Galperin2001}.
The scheme of the method is summarized in Fig.~\ref{Fig:balls}.
Two balls, heavy and light, move along a groove which ends with a wall.
The heavy ball collides with the stationary light ball and the number of collisions $\Pi$ is counted for different mass ratio of the heavy to light ball.
It was shown by Galperin that the number of collisions is intimately related to the number $\pi$, providing the first digits of the irrational number.
Thus, for equal masses, $M = m$, the number of collisions is $\Pi = 3$, which corresponds to the first digit of number $\pi$. For masses $M = 100 m$ the number of collisions is $\Pi = 31$, giving the first two digits.
The case of $M = 10 000 m$ results in $\Pi = 314$ thus providing three digits and so on.
To a certain extent, finding digits of number $\pi$ became conceptually as simple and elegant as enumerating the counting objects like Roman or Japanese digits shown in Fig.~\ref{Fig:numerals}.
\begin{figure}[ht]
\includegraphics[width=0.5\columnwidth]{FigBalls.pdf}
\caption{Schematic picture of a billiard system, consisting of a heavy ball $M$, light ball $m$ and a wall}
\label{Fig:balls}
\end{figure}
The problem of the ergodic motion of two balls within two walls was posed in the book by Sinai\cite{SinaiBook}.
He showed that the configuration space of the system is limited to a triangle and, thus, the problem is equivalent to a billiard with the same opening angle.
Also Sinai used ``billiard variables'' such that the absolute value of the rescaled velocity is conserved and the product of the vector of the rescaled velocities with the vector $(\sqrt{M},\sqrt{m})$ is constant.
The number of collisions in a ``gas of two molecules'' was given in the book by Galperin and Zemplyakov~\cite{GalperinBook} in 1990, although no relation to the digits of $\pi$ was given at that moment.
Similarly, Tabachnikov in his 1995 book\cite{TabachnikovBook} argued that the number of collisions is always finite and provided the same expression for it.
In the 90s in Galperin's seminars the way to extract digits of $\pi$ from the billiard was explained.
In year 2001 Galperin published a short article on that procedure in Russian\cite{Galperin2001} and in year 2003 in English~\cite{Galperin2003}.
This fascinating problem was given as a motivating example in the introduction of
another book by Tabachnikov, Ref.~\cite{TabachnikovBook2005}, illustrating trajectory unfolding.
Gorelyshev in Ref.~\cite{Gorelyshev2006} applied adiabatic approximation to the two-ball problem and found a conserved quantity, namely the action, close to the point of return.
Weidman~\cite{Weidman2013} found two invariants of motion, corresponding to ball-ball and ball-wall collisions.
He noted that the terminal collision distinguishes between even and odd digits.
Davis in Ref.~\cite{Davis2015} solved the equations of motion as a system of two linear equations for ball-ball and ball-wall collisions, finding the rotation angle from determinantal equation.
In addition to the energetic circle~\cite{Galperin2001}, defining the velocities, he expressed the balls positions as a function of the number of collisions.
A number of related systems was recently studied, including two balls in one dimension with gravity~\cite{Whelan1990}, dynamics of polygonal billiards~\cite{Gutkin1996} and a ping-pong ball between two cannonballs~\cite{Redner2004}.
In the present work we describe how Galperin's billiard method for mass ratio $M / m = b^{2N}$ generates first $N$ digits of the fractional part (i.e. digits beyond the radix point) of number $\pi$ in base-$b$ numeral system and address the underlying assumptions.
We consider the cases of integer and non-integer base systems, including a compelling case of representing number $\pi$ in a system base of $\pi$.
The article is organized as follows.
In Sec.~\ref{Sec:Galperin billiard method} we review the Galperin billiard method.
The basic model is introduced in Sec~\ref{Sec:model}.
The relation between the number of collisions and the number $\pi$ in a system with base $b$ is explained in Sec.~\ref{Sec:Number of collisions}.
Different properties of the trajectory of balls are derived and discussed in Sec.~\ref{Sec:ball trajectory}.
An iterative discrete solution to the equations of motion is given in Sec.~\ref{Sec:equations of motion}.
Minimal and maximal values of the velocities are obtained in Sec.~\ref{Sec:max:V}.
A differential equation describing the motion close to the point of return is derived and solved in Sec.~\ref{Sec:point of return}.
The analytical vs non-analytical shapes of the envelope of the trajectory are discussed in Sec.~\ref{Sec:non-analytic}.
The adiabatic approximation and action-phase variables are introduced in Sec.~\ref{Sec:adiabatic approximation} in the vicinity of the point of return.
In Sec.~\ref{Sec:invariants} it is shown that the adiabatic action invariant is preserved not only close to the point of return, but for all collisions.
In Sec.~\ref{Sec:Unfolding} the unfolding of the trajectory is explained. The integrability of the model is addressed in Sec.~\ref{Sec:integrability}, and superintegrable mass ratios are identified.
Minimal and maximal distances from the wall are obtained in Sec.~\ref{Sec:max:X} and the hyperbolic shape for the position vs.\ time for the large ball in the large mass ratio limit is derived in Sec.~\ref{Sec:xvs.t}. In Sec.~\ref{Sec:(V,1/x)circle} we show that the phase portrait of the heavy particle in velocity - inverse position coordinates takes a circular form.
In Sec.~\ref{Sec:exact solution} we provide an explicit solution for the positions and velocities as a function of the number of collision and introduce the exact relation between the moment of time $t$ and the collision number $n$.
In Sec.~\ref{Sec:approximate solution} we derive an approximate relation between $t$ and $n$ which is expressed by a simple formula.
The relation between the terminal collision and the parity of the digits is discussed in Sec.~\ref{Sec:terminal collision}.
In Sec.~\ref{Sec:physical realizations} we discuss different systems to which the Galperin method can be applied.
We generalize the point-size billiard to finite-size spheres (``hard rods'') in \ref{Sec:hard rods}.
Other equivalent systems include a single-ball billiard in a wedge (Sec.~\ref{Sec:billiard}) and a four-ball chain (\ref{Sec:four ball chain}) with appropriate masses and initial conditions.
In Sec.~\ref{Sec:error} we introduce the concept of systematic error and analyze the possible differences between digits generated by the Galperin billiard method and the usual methods of expressing the number $\pi$ in some base.
In Sec.~\ref{Sec:integer bases} we address the case of integer bases.
In Sec.~\ref{Sec:integer representation} we start by reviewing usual methods which might be employed for representing a number in an integer base.
In Sec.~\ref{Sec:degeneracy} we consider the case of identical balls which should be treated separately from the other cases due to the presence of a degeneracy.
As relevant examples of integer bases we consider the cases of decimal (Sec.~\ref{Sec:Decimal base}), binary and ternary (Sec.~\ref{Sec:binary base}) bases.
We conclude this section by discussing in Sec.~\ref{Sec:experiment} which bases are the most appropriate to carrying out a real experiment.
Section~\ref{Sec:Non-integer bases} is dedicated to non-integer base systems.
In Sec.~\ref{Sec:representation} we briefly overview how a number can be represented in such number systems.
Examples of non-integer base representation are reported in Sec.~\ref{Sec:irrational base} and include a compelling case of representing the number $\pi$ in a system base of $\pi$.
Finally, we draw the conclusions in Sec.~\ref{Sec:conclusions}.
\section{Galperin billiard method\label{Sec:Galperin billiard method}}
In this Section we introduce the Galperin billiard model and review how it can be used to calculate the digits of $\pi$.
As a general rule, we assign capital letters to the variables of the heavy ball and lower-case letters to those of the light ball.
\subsection{Model \label{Sec:model}}
The system consists of two balls of the same size but of a different mass running along a straight line and a hard wall, located at the origin.
We assume that the positions of the heavy and the light balls satisfy $X<x<0$ at any moment of time.
The system is unrestricted on one side (semifinite billard).
The heavy ball with mass $M$ is thrown against the stationary light ball of mass $m<M$ with some initial velocity $V_0>0$, its precise value being irrelevant for the total number of collisions.
The initial position of the light ball $x_0$ defines the length scale for the rest of the processes.
The mass of the heavy ball is chosen as
\begin{eqnarray}
M = b^{2N}m\;,
\end{eqnarray}
with parameter $b$ (integer or not) referred to as the {\em base} and an integer parameter $N$ which we call {\em mantissa} which eventually will define the number of obtained digits.
Once the collision sequence is started, one counts the total number of collisions denoted as $\Pi(b,N)$.
All collisions are considered to be elastic so that the total kinetic energy is conserved.
\subsection{Number of collisions\label{Sec:Number of collisions}}
It will be demonstrated in this Section that the velocities of the balls after a certain number of collisions and the total number of collisions can be found explicitly by using conservation laws.
Instead, the ball positions cannot be expressed by a simple expression although they can be obtained by integrating the equations of motion.
The kinetic energy $T$ is conserved during elastic ball-ball and ball-wall collisions,
\begin{eqnarray}
\frac{1}{2}M V^2 + \frac{1}{2}m v^2 = T \;.
\label{Eq:concervation law:Ekin}
\end{eqnarray}
For two balls, energy conservation law~(\ref{Eq:concervation law:Ekin}) can be given a geometrical interpretation as a mathematical equation defining the shape of an ellipse.
Instead of the particle coordinates $X$ and $x$ it is convenient to introduce {\em billiard variables}\cite{SinaiBook} defined as
\begin{equation}
\begin{split}
Y &= \sqrt{M} X,\\
y &= \sqrt{m} x
\end{split}
\label{Eq:billiard variables}
\end{equation}
with the scope of reducing the ellipse to a circle.
The billiard velocities (or configuration speed in Ref.~\cite{GalperinBook}) are defined as time derivative of the position~(\ref{Eq:billiard variables}) and are also scaled with the square root of balls' masses
\begin{eqnarray}
\nonumber
W&=&\frac{dY}{dt}=\sqrt{M}\frac{dX}{dt}=\sqrt{M}\cdot V \\
w&=&\frac{dy}{dt}=\sqrt{m}\frac{dY}{dt}=\sqrt{m}\cdot v
\label{Eq:billiard velocities}
\end{eqnarray}
Energy conservation law~(\ref{Eq:concervation law:Ekin}) expressed in billiard velocities~(\ref{Eq:billiard velocities}) reads as
\begin{equation}
W^2(t) + w^2(t) =
2T \; ,
\label{Eq:concervation law:circle}
\end{equation}
with the geometric position of allowed values of $W(t)$ and $w(t)$ forming a circle and the square of its radius defined by twice the kinetic energy of the system.
A vector of velocities, defined as $\mathbf{w}=(W,w)$, rotates on the circle forming some angle $\phi$ with the horizontal axis.
The goal is to find the values of the angle after the first collision, $\phi_1$, second collision, $\phi_2$, and so on which will completely define the velocities.
In order to find out how angle $\phi$ changes during collisions we will analyze how the momentum is changed.
There are two types of the collisions: (i) ball---ball (ii) ball---wall.
In the first type of collision the total momentum of the system is conserved
\begin{equation}
MV + mv = const\;,
\label{Eq:concervation law:momentum}
\end{equation}
while momentum of each individual ball is changed.
It is convenient to recast the constraint~(\ref{Eq:concervation law:momentum}) in a vector form as
\begin{equation}
MV + mv = \sqrt{M}\cdot W+\sqrt{m}\cdot w= \\
=\begin{pmatrix}
\sqrt{M}\\
\sqrt{m}
\end{pmatrix}\cdot
\begin{pmatrix}
W\\
w
\end{pmatrix} = |\mathbf{m}| \cdot |\mathbf{w}|\cos\varphi = const
\label{Eq:rotation}
\end{equation}
where we have introduced so far an unknown angle of the rotation $\varphi$ and a vector $\mathbf{m} =(\sqrt{M},\sqrt{m})$ with its elements corresponding to the square root of each mass.
Since the mass vector $\mathbf{m}$ does not change and according to Eq.~(\ref{Eq:concervation law:circle}) the absolute value of the velocity vector $\mathbf{w}$ remains invariant, one immediately realizes that $\cos\varphi$ has to remain constant.
Therefore, during a collision the angle between the two vectors changes from $\varphi$ to $-\varphi$, keeping the value of $\cos\varphi$ constant.
The angle $\phi$ between the vector $\mathbf{w}$ and the horizontal axis is incremented by $2\varphi$ during each ball-ball collision.
The specific value of $\varphi$ depends on the mass ratio, as will be discussed later.
In the second type of collisions, when the light ball hits the wall, its velocity gets inverted $w \to -w$ and the momentum of the system changes.
The vector of velocities gets flipped vertically, $\left(W,w\right)\to \left(W,-w\right)$ and the angle between the vector $\mathbf{w}$ and the horizontal axis is reflected, $\phi \to -\phi$.
Using two rules, describing ball-ball and ball-wall collisions, it is straightforward to obtain the angle $\phi_{n}$ describing the velocities after $n$ collisions.
Typical changes of the vector $\mathbf{w}$ are depicted in Fig.~\ref{Fig:energetic circle} and can be summarized as follows:
\begin{itemize}
\item[$n=0$,] before any collision has happened, the light particle is at rest, $v = w = 0$, as shown with the horizontal vector with $\phi_0 = 0$
\item[$n=1$,] after the fist ball-ball collision, vector $\mathbf{w}$ is rotated by $2\varphi$, resulting in $\phi_1 = 2\varphi$
\item[$n=2$,] after the fist ball-wall collision, vector $\mathbf{w}$ is flipped vertically, resulting in $\phi_2 = -2\varphi$
\item[$n=3$,] after the second ball-ball collision, vector $\mathbf{w}$ is rotated by $2\varphi$, resulting in $\phi_3 = 4\varphi$
\item[$n=4$,] after the second ball-wall collision, vector $\mathbf{w}$ is flipped vertically, resulting in $\phi_2 = -4\varphi$
\item[odd $n$,] after $(n+1)/2$ ball-ball collisions $\phi_n = (n+1)\varphi$
\item[even $n$,] after $n/2$ ball-wall collisions $\phi_n = -n \varphi$.
\end{itemize}
During each ball-ball collision, the velocity $V$ of the heavy ball is changed by a negative amount, eventually stopping and reversing the heavy ball.
After the angle has crossed $\pi/2$ position, the velocity of the heavy mass becomes negative (the ball is moving away from the wall) and its absolute value is increased with each consecutive collision.
Collisions continue until the closest angle to $\pi$ position is reached.
After that the iterations end, as continuing further would result in a decrease of the velocity of the heavy mass, which is physically impossible.
The total number of collisions, labeled by $\Pi$, is then given by
\begin{equation}
\Pi = \integer\left[\frac{2\pi}{\phi}\right]\;,
\label{Eq:Pi(phi)}
\end{equation}
where function $\integer[x]$ denotes the integer part of $x$, the greatest integer less than $x$ or equal to it.
\begin{figure}[]
\hspace*{\fill}
\begin{subfigure}{0.45\columnwidth}
\includegraphics[width=\linewidth]{Fig3V.pdf}
\caption{~}
\label{Fig:energetic circle:a}
\end{subfigure}
\hspace*{\fill}
\begin{subfigure}{0.45\columnwidth}
\includegraphics[width=\linewidth]{Fig4W.pdf}
\caption{~}
\label{Fig:energetic circle:b}
\end{subfigure}
\hspace*{\fill}
\caption{A characteristic example of the dependence of the vector of velocities on collision number $n$.
Left panel, vector of velocities $\mathbf{v} = (V, v)$ forming an ellipse;
right panel, vector of rescaled velocities $\mathbf{w} = (V, v)$ forming a circle.
The shown data is obtained for $b=3$ and $N=1$.
}
\label{Fig:energetic circle}
\end{figure}
Let us find out how the pivot angle $\varphi$ is related to the mass ratio.
Suppose there is a ball-ball collision which corresponds to a rotation of $\mathbf{v}$ vector from $\alpha - \varphi$ to $\alpha+\varphi$.
The change in the momentum of the heavy mass is
$-\sqrt{M}(\cos(\alpha+\varphi)+\cos(\alpha-\varphi)) = -2\sqrt{M}\cos\alpha\cos\varphi$,
while the change in the momentum of the light mass is
$\sqrt{m}(\sin(\alpha+\varphi)-\sin(\alpha-\varphi))=2\sqrt{m}\sin\varphi\cos\alpha$.
The sum of the two should be equal to zero in the ball-ball collision, resulting in a condition which is independent of the actual value of $\alpha$,
\begin{eqnarray}
\tan \varphi = \sqrt{\frac{m}{M}}\;.
\end{eqnarray}
That is the pivot angle is fully determined by the square root of the mass ratio.
Hence, the angle between the velocity vectors during a ball-ball collision will be:
\begin{equation}
\phi=2\varphi=2\cdot \arctan\left(\frac{\sqrt{m}}{\sqrt{M}}\right)=2\cdot \arctan(b^{-N})
\label{Eq:phi}
\end{equation}
Consequently, the number of collisions~(\ref{Eq:Pi(phi)}) can be explicitly evaluated as a function of parameters $b$ and $N$ as
\begin{equation}
\Pi=\integer\left[\frac{2\pi}{2\arctan(b^{-N})}\right]=\integer\left[\frac{\pi}{\arctan(b^{-N})}\right] \;.
\label{Eq:pi:acrctg}
\end{equation}
Moreover, for large base $b$ and large mantissa $N$ the argument of the inverse tangent function is small, $x = b^{-N}\ll 1$, and the inverse tangent function can expanded as $\arctan (x) \approx x$, resulting in an elegant expression
\begin{equation}
\Pi(b,N)\approx \integer\left[\frac{\pi}{b^{-N}}\right] \;.
\label{Eq:hurbilketa}
\end{equation}
This equation provides the basis for expressing the number $\pi$ in systems with integer and non-integer bases.
In Ref.~\cite{Davis2015} the new velocities after a one round of ball-ball and ball-wall collisions are found in terms of an eigenproblem of a system of two linear equations.
The eigenvalues $e^{i2\varphi}$ correspond to a rotation by angle $2\varphi$ as can be seen in Fig.~\ref{Fig:energetic circle}.
\section{Ball trajectory\label{Sec:ball trajectory}}
\subsection{Solving the equations of motion\label{Sec:equations of motion}}
In Sec.~\ref{Sec:Number of collisions}, it was shown that the total number of collisions can be explicitly obtained from the conservation laws resulting in Eq.~(\ref{Eq:hurbilketa}), and it does not depend on the exact positions of the balls and the initial velocity of the incident ball. Here we outline how the trajectory of the balls can be obtained.
First, one has to define the initial conditions (positions and velocities) and then integrate the equations of motion. Let $X_0$ and $x_0$ ($X_0<x_0<0$) be the initial coordinates of the heavy and light balls, and $V_0>0$ and $v_0=0$ be their initial velocities.
As the velocities change only at contact, the balls move with constant velocities between the collisions.
The velocities change according to the rules (the energy and momentum conservation laws) provided in Sec.~\ref{Sec:Number of collisions}.
All odd collisions, $n=2k+1$, correspond to balls hitting each other, while even collisions, $n=2k$, to the light ball hitting the wall.
Let $X_n$, $x_n$, $V_n$, $v_n$ be the coordinates and velocities of the heavy and light balls at the time $t_n$ of the $n$-th collision, respectively.
Then the time between the consecutive collisions is
\begin{equation}
\begin{split}
\tau_{2k} &= \frac{x_{2k}-X_{2k}}{V_{2k}-v_{2k}} \\
\tau_{2k-1} &= - \frac{x_{2k-1}}{v_{2k-1}},
\end{split}
\label{Eq:tau(n):exact}
\end{equation}
where $\tau_{2k} = t_{2k+1}- t_{2k}$ is the time interval passed between ball-ball $2k$ and the subsequent ball-wall $2k+1$ collision,
and $\tau_{2k-1} = t_{2k} - t_{2k-1}$ is the time interval between ball-wall $2k-1$ and ball-ball $2k$ collisions.
The time moment of the $n$-th collision can be calculated as the sum of the preceding time intervals as
\begin{equation}
t_n = \sum\limits_{l=1}^{n-1} \tau_l.
\label{Eq:t(n):exact}
\end{equation}
The solution to the equations of motion can be expressed as the following iterative formulas for a ball-ball collision ($n=2k+1, k=0,1,\dots$)
\begin{equation}
\begin{split}
X_{2k+1} &= X_{2k} + (X_{2k}-x_{2k}) \frac{V_{2k}}{v_{2k}-V_{2k}} \\
x_{2k+1} &= x_{2k} + (X_{2k}-x_{2k}) \frac{v_{2k}}{v_{2k}-V_{2k}} \\
V_{2k+1} &= \frac{M-m}{m+M} V_{2k} + \frac{2m}{m+M} v_{2k} \\
v_{2k+1} &= \frac{2M}{m+M} V_{2k}+ \frac{m-M}{m+M} v_{2k},
\end{split}
\label{Eq:equation of motion:BB}
\end{equation}
and for a ball-wall collision ($n=2k, k=1,2,\dots$)
\begin{equation}
\begin{split}
X_{2k} &=X_{2k-1}-\frac{V_{2k-1}}{v_{2k-1}} x_{2k-1} \\
x_{2k} &=0 \\
V_{2k} &=V_{2k-1} \\
v_{2k} &= -v_{2k-1}. \\
\end{split}
\label{Eq:equation of motion:BW}
\end{equation}
The iterative process stops when one of the following equivalent conditions holds: (i) $X_n>0$, (ii) $V_n$ is not monotone and starts decreasing after the point of return, see Sec.~\ref{Sec:point of return} for the discussion on the point of return, (iii) after a ball-ball collision $v_n<0$, which physically means that the light ball goes to $-\infty$.
In Fig.~\ref{Fig:positions} we display a typical example of heavy and light ball trajectories, $(t_n, X_n)$ and $(t_n, x_n)$, calculated by iteratively solving Eqs.~(\ref{Eq:equation of motion:BB}-\ref{Eq:equation of motion:BW}) for $b=2$ and different $N$.
\begin{figure}[ht]
\includegraphics[width=0.6\columnwidth]{FigTrajectory.pdf}
\caption{Distance of the heavy and light balls from the wall as a function of time for base $b=2$ and different values of $N$ (in arbitrary units).
Solid lines and solid symbols, heavy ball $X$;
dashed lines and open symbols, light ball $x$.
}
\label{Fig:positions}
\end{figure}
\subsection{Minimal and maximal velocities\label{Sec:max:V}}
In this Section we discuss which smallest or largest value of velocity and momentum the balls might have.
The characteristic value of velocity is set by the initial velocity $V_0$ of the heavy ball.
The energy conservation law~(\ref{Eq:concervation law:Ekin}) relates the velocities of the heavy and light balls,
the maximal velocity of the light ball, $v_{max}$, which is reached close to the point of return, where the heavy ball stops, $V=0$.
The maximal possible velocity of the light ball is
\begin{equation}
v_{max} \approx \sqrt{\frac{M}{m}}V_0 = b^N V_0\;.
\label{Eq:v:max}
\end{equation}
The light ball might move much faster than the heavy ball.
The larger is the mass ratio, the more pronounced is the effect.
Accordingly, the maximal momentum of the heavy $P_{max}$ and light $p_{max}$ balls are
\begin{eqnarray}
\label{Eq:P:max}
P_{max} &=& M V_0 \\
p_{max} &\approx& m v_{max} = b^{-N} P_{max}.
\label{Eq:p:max}
\end{eqnarray}
It should be noted that while $P_{max}$ is always reached as it corresponds to the initial momentum before the first collision happens, $p_{max}$ is obtained by assuming that the heavy ball completely stops.
Due to finite discretization of the angle $\varphi$, the heavy does not necessarily comes to a complete stop, although the larger the mass ratio $M/m$, the better Eqs.~(\ref{Eq:v:max},\ref{Eq:p:max}) are satisfied.
\subsection{Differential equation for motion close to the point of return\label{Sec:point of return}}
In this section we derive a differential equation describing the motion of the slow (heavy) particle close to the point of return and find its solution.
The direction of the light ball is inverted at each collision, refer to Fig.~\ref{Fig:positions}.
It is positive after the first or any odd collision, $n=2k+1$, and is negative after an even number of collisions, $n=2k$,
\begin{eqnarray}
w^{(2k)}&=& -w_{max} \sin(k\phi)\\
w^{(2k+1)}&=&w_{max} \sin(k\phi),
\label{Eq:w(n)}
\end{eqnarray}
where $w_{max}$ is the maximal value of the scaled velocity that the light ball might have.
Instead, the velocity of the heavy ball is gradually reduced and is inverted at a certain moment. Also, the velocity of the heavy ball is not changed when the light ball hits the wall
\begin{eqnarray}
W^{(2k)}&=& W_{max} \cos(k\phi)\\
W^{(2k+1)}&=&W_{max} \cos(k\phi)
\label{Eq:W(n)}
\end{eqnarray}
with $W_{max} = w_{max}$.
The point of return is signaled by the inversion of the velocity of the heavy ball and is closely defined by the condition that the cosine function in Eq.~(\ref{Eq:W(n)}) has a node, $W=0$,
\begin{equation}
k_{inversion}\phi\approx\frac{\pi}{2}\;.
\label{Eq:kinversion}
\end{equation}
We assume that the ``discretization'' $\varphi$ is rather small and the light ball makes a large number of collisions.
In this situation, Eq.~(\ref{Eq:kinversion}) predicts well the collision index $k_{inversion}$ corresponding to the point of return.
At that moment, the heavy ball stops moving and reaches the smallest distance $X_{min}$ from the wall.
Counting the number of collisions $k'$ from the return point (primed variables are counted from the point of return), $k' = k - k_{inversion}$, the trigonometric functions in Eqs.~(\ref{Eq:w(n)}-\ref{Eq:W(n)}) can be expanded leading to
\begin{eqnarray}
w^{(2k')}
&= -w_{max} \sin(\frac{\pi}{2}+k'\phi)
&= -w_{max} \cos(k'\phi)
\approx w_{max}\\
W^{(2k')}
&= W_{max} \cos(\frac{\pi}{2}+k'\varphi)
&= -W_{max} \sin(k' \varphi)
\approx -W_{max}k'\varphi
\label{Eq:w(n')}
\end{eqnarray}
In terms of the velocities we obtain
\begin{eqnarray}
v^{(2k')} &\approx& - v_{max}\\
V^{(2k')} &\approx& - V_{max}k'\varphi
\label{Eq:v(n')}
\end{eqnarray}
As at this point the absolute value of the velocity of the light ball practically does not change and the heavy ball is almost not moving, the time
\begin{eqnarray}
\tau_{inversion} = \frac{|X_{min}|}{v_{max}}
\label{Eq:tau:inversion}
\end{eqnarray}
between consecutive collisions is constant.
Velocity of the heavy ball is changed only at ball-ball collisions with time $\Delta t'$ between consecutive BB collisions equal to
\begin{eqnarray}
\Delta t' = 2\tau_{inversion} = \frac{2|X_{min}|}{v_{max}} = const\;.
\label{Eq:Delta t}
\end{eqnarray}
The time $t'$ passed after the return point is $t'=k'\Delta t'$.
The position of the heavy ball changes as
\begin{eqnarray}
\frac{\Delta X}{\Delta t'}
= -k' V_{max}\varphi
\approx
- \frac{V_{max} v_{max} \varphi}{|X_{min}|} t'
\;,
\label{Eq:return:discrete}
\end{eqnarray}
where the negative sign shows that the heavy ball goes to the left direction, away from the wall.
The heavier is the ball, the smaller is its minimal distance from the wall, $X_{min}$, and the larger is the maximal velocity of the light ball, $v_{max}$.
According to Eq.~(\ref{Eq:Delta t}) both tendencies make time $\Delta t'$ between collisions smaller.
By treating it as an infinitesimal increment $dt'$ in Eq.~(\ref{Eq:return:discrete}), we find the differential equation describing the trajectory of the heavy ball
\begin{eqnarray}
\frac{dX}{dt'}
= - \frac{V_{max} v_{max} \varphi}{|X_{min}|} t'
\;.
\label{Eq:return:differential}
\end{eqnarray}
Equation~(\ref{Eq:return:differential}) can be explicitly integrated, giving the trajectory of the heavy particle close to the return point in the continuous approximation,
\begin{eqnarray}
X(t')
= X_{min} - \frac{V_{max} v_{max} \varphi}{2X_{min}} (t')^2
\approx
X_{min} - \frac{V_{max}^2}{2|X_{min}|}(t')^2
\;.
\end{eqnarray}
where we have used Eqs.~(\ref{Eq:phi},\ref{Eq:v:max}) and made an expansion of $\phi$ assuming $M\gg m$.
The relative displacement of the heavy ball is expressed as
\begin{eqnarray}
\frac{X(t')}{|X_{min}|}
= 1 + \frac{1}{2} \left(\frac{V_{max} t'}{X_{min}}\right)^2\;.
\label{Eq:return:parabola}
\end{eqnarray}
\begin{figure}[ht]
\includegraphics[width=0.6\columnwidth]{FigParabola.pdf}
\caption{Distance of the heavy ball close to the return point, for $b=2$ and $N=3$.
Solid symbols, heavy ball $X$;
dashed line, limit of an infinitely massive ball, Eq.~(\ref{Eq:return:heavyball});
solid line, parabola defined by Eq.~(\ref{Eq:return:parabola}).
}
\label{Fig:parabola}
\end{figure}
Figure~\ref{Fig:positions} shows a characteristic example of the time evolution of the position of the heavy ball for $b=2$ and different values of $N$.
For $N=0$ there are only three collisions, the first ball-ball collision stops the first ball;
ball-wall collision inverts the velocity of the second ball, while the heavy ball remains immobile;
the last ball-ball collision sends the first ball back.
For $N=1$ no clear regular structure is visible.
For $N=2$ a parabola, predicted by Eq.~(\ref{Eq:return:parabola}), starts being formed.
For $N=3$, shown in Fig.~\ref{Fig:parabola} the parabolic dependence becomes clearly visible.
\subsection{Analytic vs non-analytic trajectory envelope\label{Sec:non-analytic}}
The trajectory is a piece-wise function, as the velocities are constant between the collisions.
The resulting ``edges'' in the trajectory are clearly visible for the small number of collisions, similarly to $N=0$ case, schematically shown in Fig.~\ref{Fig:non-analytic trajectory}a.
It is interesting to note that such trajectory is described by a {\em non-analytic} function, as it experiences kinks and the first derivative is undefined at any of the collision points.
However, when the number of collisions is large, the envelope of the trajectory becomes smooth and is described by an {\em analytic} function, which close to the point of return has a parabolic shape~(\ref{Eq:return:parabola}).
Such a parabolic dependence is schematically shown in Fig.~\ref{Fig:non-analytic trajectory}b.
In the limit of an infinitely heavy ball ($N\to\infty$ and $M\to\infty$), the trajectory again becomes non-analytic with a kink in the point where the heavy ball hits the wall,
\begin{eqnarray}
X(t') = V_{max}|t'|
\label{Eq:return:heavyball}
\end{eqnarray}
shown in Fig.~\ref{Fig:non-analytic trajectory}c with two straight lines.
\begin{figure}[htbp]
\hspace*{\fill}
\begin{subfigure}{0.2\columnwidth}
\includegraphics[width=\linewidth]{FigNonAnalytic1.pdf}
\caption{~}
\label{Fig:non-analytic trajectory:a}
\end{subfigure}
\hspace*{\fill}
\begin{subfigure}{0.2\columnwidth}
\includegraphics[width=\linewidth]{FigNonAnalytic2.pdf}
\caption{~}
\label{Fig:non-analytic trajectory:b}
\end{subfigure}
\hspace*{\fill}
\begin{subfigure}{0.2\columnwidth}
\includegraphics[width=\linewidth]{FigNonAnalytic3.pdf}
\caption{~}
\label{Fig:non-analytic trajectory:c}
\end{subfigure}
\hspace*{\fill}
\caption{
Schematic shape of the trajectory of the heavy particle close to the point of return, given by
(a) a non-analytic function for $N=0$;
(b) analytic function~(\ref{Eq:return:parabola});
(c) non-analytic function~(\ref{Eq:return:heavyball}) in $N\to\infty$ limit.
}
\label{Fig:non-analytic trajectory}
\end{figure}
\subsection{Adiabatic approximation\label{Sec:adiabatic approximation}}
From the point of view of Hamiltonian systems, the problem of two balls has two degrees of freedom, namely two positions $X$, $x$ while momenta $P = MV$ and $p=mv$ are conjugate variables.
As it was discussed in Sec.~\ref{Sec:point of return},
when the heavy ball approaches the point of return, it slows down while the light ball wildly oscillates between the heavy ball and the wall.
This separates the scales into {\em fast} and {\em slow} variables so that during a single oscillation of a light ball, the position of the heavy ball is only slightly changed.
It was argued by Kapitza~\cite{Kapitza51UFN,Kapitza51JETP} in his work on driven pendulum (Kapitza pendulum) that by averaging over the fast variables it might be possible to simplify the problem and provide a solution if the separation of scales is large enough.
In our case the parameter which defines the separation of scales is the mass ratio $M/m = b^{2N}$, so for any base $b$ by increasing $N$ the needed condition $M/m \gg 1$ is well satisfied.
The systems with different scales can be studied in the theory of adiabatic invariants\cite{ArnoldKozlovNeishtadtBook}.
It is useful to analyze the $(p,x)$ portrait of the system, corresponding to the fast variables.
A typical example is shown in Fig.~\ref{Fig:xp}.
After the ball-ball collision, (for example, $n=1$), the light ball moves with a constant momentum $p$ until it hits the wall.
This results in a horizontal line with some momentum $p$ and $0<x/x_0 < X_1/x_0$.
During the ball-wall ($n=2$) collision the momentum of the light mass is inverted, resulting in a vertical line $x=0$, $p\to -p$.
After that the light ball travels with constant momentum until it hits the heavy ball ($n=3$), corresponding to a horizontal line at $-p$ from $0<x/x_0 < X_3/x_0$.
At the next ball-ball collision, the velocity of the light particle inverts the sign but its absolute value is slightly changed due to a small but finite momentum transfer from the heavy ball.
During a single cycle (or ``period'') consisting of four collisions the light ball draws an almost closed rectangle.
The larger is the mass of the heavy ball and the smaller is its velocity, the more similar is the trajectory during a cycle to a closed rectangle.
\begin{figure}[ht]
\includegraphics[width=0.6\columnwidth]{Figxp.pdf}
\caption{(x,p) portrait for $b=10$ and $N=1$;
Red symbols, light particle during ball-wall ($x=0$) and ball-ball ($x\ne 0$) collisions.
Green thick lines, constant action curve defined by Eq.~(\ref{Eq:action}).
Blue thin lines, trajectory.
Index $n=1,2,3,\cdots$ denotes the state after $n$ collisions while primed index $n'$ correspond to an intermediate state in which the velocity of the light ball is not yet reflected.
Area, covered by the trajectory between two consecutive collusion of the same type (BB or BW) define the action~(\ref{Eq:action}).
}
\label{Fig:xp}
\end{figure}
The area covered by the light particle during a cycle in $(p,x)$ space has units of [energy $\times$ sec] and is called {\em action} $I$, defined as
\begin{eqnarray}
I = \frac{X p}{2\pi} \,,
\label{Eq:action}
\end{eqnarray}
where $p$ is the maximal momentum of the light particle and $X$ is its maximal distance from the wall (defined by the position of the heavy particle) during a single cycle.
Within the adiabatic approximation the action is conserved, implying the following relation between the momentum of the light particle $p$ and the position of the heavy particle $X$
\begin{eqnarray}
p = \frac{2\pi I}{X}.
\label{Eq:adiabatic:p(X)}
\end{eqnarray}
It is shown in Ref.~\cite{Gorelyshev2006} that for times of the order of $\varepsilon^2$, action~(\ref{Eq:action}) is conserved with accuracy $\varepsilon$ where
$\varepsilon = \sqrt{m/M}$
is treated as a small parameter.
In the same limit the Hamiltonian can be written as
\begin{eqnarray}
H = \frac{P^2}{2M} + \frac{\pi^2 I^2}{2mX^2}\;.
\label{Eq:adiabatic:H}
\end{eqnarray}
At the point of return the heavy particle has zero momentum and the energy can be expressed in terms of the minimal distance $X_{min}$ between the heavy particle and the wall,
\begin{eqnarray}
H = \frac{\pi^2 I^2}{2mX_{min}^2}\;.
\label{Eq:adiabatic:Xmin}
\end{eqnarray}
From Eqs.~(\ref{Eq:adiabatic:H}-\ref{Eq:adiabatic:Xmin}) it follows that the dependence of the momentum $P$ of the heavy particle on its inverse coordinate $1/X$ has the semicircular form~\cite{Gorelyshev2006},
\begin{eqnarray}
P = \frac{\pi I}{X_{min}} \sqrt{1 - \frac{X_{min}^2}{X^2}}\;,
\label{Eq:adiabatic:P}
\end{eqnarray}
which we will address in more detail in Sec.~\ref{Sec:(V,1/x)circle}.
Variables $I$ and $\phi_{\rm phase}$ are conjugate, and time derivative of the phase can be obtained from Hamiltonian~(\ref{Eq:adiabatic:H}) as $\dot\phi_{\rm phase} = dH / dI$.
The integration over the time gives the final phase after all collisions have happened as
$\phi_{\rm phase}^{\rm final} = \pi^2 \sqrt{M}/\sqrt{m}$
~\cite{Gorelyshev2006}.
During each cycle there are two collisions (BB and BW) and the phase changes by $2\pi$, so the total number of collisions $\Pi$ can be infer as $\phi_{\rm phase}^{\rm final} = \Pi \pi$, resulting in $\Pi = \pi / \varphi + O(\varphi)$ where $\varphi \approx \sqrt{m} /\sqrt{M}$.
This formula should be compared with Eq.~(\ref{Eq:Pi(phi)}) and, indeed, correctly relates total number of collisions $\Pi$ with $\pi$.
At the same time it is not a priori obvious that the adiabatic approximation should be precise far from the return point, that is for times $t\gg \varepsilon^2$, especially at the time of the final collision.
Indeed, it might be observed in Fig.~\ref{Fig:xp} that while action~(\ref{Eq:action}) is a good adiabatic invariant close to the return point (shown with thick green line), the first few collisions ($n=1; 3; \cdots$) are quite off.
In Sec.~\ref{Sec:invariants} it will be shown that in the present problem it is possible to find two invariants (for BB and BW collisions), which coincide close to the point of return with adiabatic invariant (action) given by Eq.~(\ref{Eq:action}), and, in particular, this clarifies why the adiabatic approximation leads to the correct number of collisions even if the region of applicability of the approximation is violated.
Finally we note that the time dependence of the phase $\phi_{\rm phase}(t)$ is related to the time dependence of the collision number $n(t)$ according to $\phi_{\rm phase}(t_n) = \pi n(t_n)$.
In the continuous limit of many collisions, the phase increases as an inverse tangent function, as shown in Fig.~\ref{Fig:n} of Sec.~\ref{Sec:approximate solution} below.
\subsection{Action invariants\label{Sec:invariants}}
We have seen in Sec.~\ref{Sec:adiabatic approximation}, that action~(\ref{Eq:action}) is an adiabatic invariant and is not changed in the vicinity of the return point.
Consequently, the iterative solution~(\ref{Eq:equation of motion:BB}-\ref{Eq:equation of motion:BW}) to the equations of motion should also preserve the action $I$.
The change of coordinates within one cycle of collisions is obtained by applying consequently ball-ball~(\ref{Eq:equation of motion:BB}) and ball-wall~(\ref{Eq:equation of motion:BW}) movements, and it can be straightforwardly verified that the action remains constant for a cycle which starts and ends with a ball-wall collision, $n=2k$,
\begin{equation}
X_{2k} v_{2k} = \frac{\pi I}{m} = const\;.
\label{Eq:invariant:BW}
\end{equation}
Importantly, action~(\ref{Eq:invariant:BW}) is {\em always} conserved on a ball-wall cycle and not only close to the point of return.
The absolute value of the velocity $v_{2k}$ of the light particle is constant on the paths BB-BW and BW-BB between consecutive collisions, while the edge position of the light particle is displaced from $x_{2k-1}$ to $x_{2k+1}$, see Fig.~\ref{Fig:xp}.
As a result, the trajectory during a single BB-BW-BB loop is not closed, and it is rather natural that the exact invariant~(\ref{Eq:invariant:BW}) should be satisfied for some value of $x$, lying between $x_{2k-1}$ and $x_{2k+1}$.
The position of the heavy particle $X$ coincides with $X_{2k-1} = x_{2k-1}$ and $X_{2k+1}=x_{2k+1}$ and $x = X_{2k}$, indeed, lies between $x_{2k-1}$ and $x_{2k+1}$.
One also might note that a simple average position $x = (x_{2k-1}+x_{2k+1})/2$ does not lead to an invariant.
It was discovered by Weidman~\cite{Weidman2013} that there also exists a second invariant which remains unchanged during a ball-ball collision, $n=2k+1$, given by
\begin{equation}
X_{2k+1} (V_{2k+1}-v_{2k+1}) = \frac{\pi I}{m} = const\;.
\label{Eq:invariant:BB}
\end{equation}
Furthermore, it can be shown\cite{Weidman2013} from the equations of motion~(\ref{Eq:equation of motion:BB}-\ref{Eq:equation of motion:BW}) that the action $I$, entering into Eqs.~(\ref{Eq:invariant:BW}-\ref{Eq:invariant:BB}), can be expressed in terms of the initial conditions as
\begin{equation}
I = \frac{|x_0| V_0 m}{\pi}
\label{Eq:action invariant}
\end{equation}
Figure~\ref{Fig:Xv} shows the $(X,v)$ portrait of the system.
By using Eq.~(\ref{Eq:action invariant}), the ball-wall invariant~(\ref{Eq:invariant:BW}) reduces to an elegant expression, $X_{2k} v_{2k} = - x_0 V_0$ shown with a solid line.
It can be appreciated that the ball-wall invariant is, indeed, conserved for any BW collision.
Instead, the mirrored line, $X_{2k} v_{2k} = + x_0 V_0$, describes correctly the ball-ball process only close to the point of return.
Indeed, there it was derived within the adiabatic approximation, as given by Eq.~(\ref{Eq:adiabatic:p(X)}).
Formally, the error arising within the adiabatic approach can be explicitly seen from the BB invariant~(\ref{Eq:invariant:BB}), which reduces to $X_{2k+1}v_{2k+1} = const$ only when $V_{2k+1} = 0$, that is exactly at the point of return.
\begin{figure}[ht]
\includegraphics[width=0.6\columnwidth]{FigInvariant1.pdf}
\caption{$(X,v)$ portrait for $b=10$ and $N=1$;
Red symbols, heavy particle during ball-ball ($v>0$) and ball-ball ($v < 0$) collisions.
Green thick lines, ball-wall invariant~(\ref{Eq:invariant:BW}), corresponding to action~(\ref{Eq:action invariant}).
}
\label{Fig:Xv}
\end{figure}
Finally, it is worth recalling that during ball-ball collision, $x$ and $X$ coordinates obviously coincide and BB invariant~(\ref{Eq:invariant:BB}) is applicable not only for the coordinate of the heavy particle, but also to the light one,
\begin{equation}
x_{2k+1} (V_{2k+1}-v_{2k+1}) = \frac{2\pi I}{m} = const\;.
\label{Eq:invariant:BB:x}
\end{equation}
\subsection{Unfolding the trajectory\label{Sec:Unfolding}}
The trajectory of the balls in the phase space can be given a simple geometrical interpretation which also clarifies one more time how the number of collisions is related to the opening angle\cite{Galperin2003,TabachnikovBook2005}.
The original particle coordinates are restricted to the region $0\leq |x| \leq |X|$, where boundary $x=0$ corresponds to the light ball hitting the wall and $x=X$ the ball-ball collision, see Fig.~\ref{Fig:unfolding}a.
The opening angle is $45\degree$ but the reflections do not obey the laws of optics, as the incident angle differs from the angle of reflection and neither is the velocity $V$ conserved.
Instead, the opening angle in billiard coordinates $Y$ and $y$ is equal to $\varphi$, see Fig.~\ref{Fig:unfolding}b.
Now the absolute value of the scaled velocity is conserved and reflections obey the optical laws.
In this way the original two-body problem is mapped to a problem of a single ball moving in a wedge with opening angle $\varphi$ with specular reflections from the mirrors.
It was demonstrated in Sec.~\ref{Sec:Number of collisions} that the vector $\mathbf{m} =(\sqrt{M},\sqrt{m})$ is preserved during any ball-ball collision and ball-wall collisions do not affect the trajectory of the heavy ball.
The boundary line of the configuration space corresponds to the collision condition $X=x$, written in billiard coordinates as $Y / \sqrt{M} - y / \sqrt{m} = 0$.
The scalar product of its normal vector $(1/\sqrt{M}, -1/\sqrt{m})$ and the vector $\mathbf{m}$ is equal to zero.
It means that in the billiard coordinates, the vector $\mathbf{m}$ is tangent to the boundary line and if the the ricocheting trajectory in the wedge is unfolded, it results in a straight line.
In Fig.~\ref{Fig:unfolding} we show a typical example.
It provides a simple geometrical interpretation for the number of collisions as the number of times the opening angle can fit into the maximal angle of $180$ degrees or $\pi$ radian.
\begin{figure}[]
\includegraphics[width=0.4\columnwidth]{FigXx.pdf}
\includegraphics[width=0.4\columnwidth]{FigYy.pdf}
\includegraphics[width=0.8\columnwidth]{FigUnfolded.pdf}
\caption{
The trajectory in different phase spaces,
(a) original coordinates $0\leq |x| \leq |X|$,
(b) variables of billiard in a wedge, $0\leq |y| \leq |Y|\tan\varphi$,
(c) unfolded trajectory.
The parameters are $b=2$ and $N=1$ and correspond to time-dependent data shown in Fig.~\ref{Fig:positions}.
}
\label{Fig:unfolding}
\end{figure}
\subsection{Liouville integrability, superintegrability, and maximal superintegrability\label{Sec:integrability}}
Our system can be shown to be \emph{Liouville integrable}, i.e.\ it possesses as many exact constants (first integrals) of motion in involution as it has degrees of freedom.
For our two-degree-of-freedom system, this implies an existence of only one constant of motion in addition to the total energy \footnote{Generally, a notion of \emph{involution} between two observables---vanishing of the Poisson bracket between them---requires a Hamiltonian reformulation of the laws of dynamics of the system. However, for the two-dimensional systems, the only zero Poisson bracket required is the one between the additional conserved quantity and the Hamiltonian; the latter is simply automatic.}.
To identify this conserved quantity, let us consider the system in the billiard coordinates~(\ref{Eq:billiard variables}). It is represented by a two-dimensional particle of a unit mass moving a wedge of an opening $\varphi = \arctan(\sqrt{m/M})$ as shown in Fig.~\ref{Fig:unfolding}.
In between the collisions, the angular momentum, $L=(Y w - y W)$, is conserved.
Upon a ball-wall or a ball-ball collision, the angular momentum changes sign.
However, its square,
\begin{equation}
L^2 = (Y w - y W)^2 = m M (X v - x V)^2
\label{LSq}
\end{equation}
remains invariant throughout the evolution.
At the instances of a ball-wall collision, where $x=0$, the invariant~(\ref{LSq}) is proportional to the square of the adiabatic invariant (\ref{Eq:invariant:BW}):
\begin{align*}
L^2 \Big|_{2k} = m M (X_{2k} v_{2k})^2 = \pi^2 b^{2N} I^2\,\,.
\end{align*}
Likewise, on a ball-ball collision, the angular momentum square assumes a value proportional to the corresponding adiabatic invariant~(\ref{Eq:invariant:BB}):
\begin{align*}
L^2 \Big|_{2k+1} = m M \left(X_{2k+1} (v_{2k+1}-V_{2k+1})\right)^2 = \pi^2 b^{2N} I^2\,\,,
\end{align*}
with the same coefficient of proportionality.
For a discrete set of mass ratios with a commensurate opening angle ($\varphi = \pi/q$),
\begin{align}
&
\frac{m}{M} = \tan^2(\pi/q)
\label{kaleidoscopes}
\\
&
n=3,\,4,\,5,\,\ldots
\nonumber
\,\,,
\end{align}
a third constant of motion appears, promoting our system to \emph{superintegrable}; i.e. it will have more functionally independent constant of motion than degrees of freedom. In fact, two-dimensional superintegrable systems are also always \emph{maximally superintegrable}: they have a maximally allowed number of functionally independent conserved quantities that is one less the dimensionality of the phase space, coordinates and velocities combined.
For bounded orbits, the superintegrability manifests itself as a reduction of the dimensionality of the phase space available from a given initial condition. Maximal superintegrability results in closed one-dimensional orbits. For unbounded orbits, the manifestation of the maximal superintegrability is more subtle, but still---as we will see below---tangible.
For the mass ratios~(\ref{kaleidoscopes}), the wedge in the billiard coordinates $(y,Y)$ depicted in Fig.~\ref{Fig:unfolding}b acquires an opening of $\pi/q$, with $q \ge 3$ being an integer. In this case, sequences of reflections about the cavity walls form
a finite group with order $2q$ known as the reflection group $I_{2}(q)$. As it has been shown in Ref.~\cite{olshanii2015_105005},
in such a situation a new constant of motion can be constructed: it is represented by the first nontrivial invariant (or Chevalley) polynomial of the group \cite{chevalley1955_778,mehta1988_1083}, evaluated on the momentum vector. The constant of
motion $J$ produced by this construction in our case is as follows:
\begin{align}
\begin{split}
J
&
= \frac{1}{2 M^{q/2}}\left((W+iw)^{q}+(W-iw)^{q}\right)
\\
&
= \frac{1}{2}
\left(
(V + i \tan(\pi/q) v)^{q} + (V - i \tan(\pi/q) v)^{q}
\right)
\end{split}
\label{J}
\,\,.
\end{align}
Some notable examples include
\begin{align*}
&
q=3
&
\frac{m}{M} = 3
&&
J = V^3-9 V v^2
&
\\
&
q=4
&
\frac{m}{M} = 1
&&
J = V^4 - 6 V^2 v^2 +v^4
&
\\
&
q=5
&
\frac{m}{M} = 5 - 2 \sqrt{5}
&&
J = V^5 - 10(5-2\sqrt{5}) V^3 v^2 + 25(9-4\sqrt{5}) V v^4
&
\\
&
q=6
&
\frac{m}{M} = \frac{1}{3}
&&
J = V^6 - 5 V^4 v^2 + \frac{5}{3} V^2 v^4 - \frac{1}{27} v^6
&
\,\,.
\end{align*}
Observe that in these examples and in general, even(odd) $q$, produces an even(odd) constant of motion $J$, with respect to
the $V\to -V,\,v\to -v$ inversion. This difference between the even and odd cases, will lead to a difference between the maximal superintegrability manifestations between these two cases.
To discuss the consequences of the maximal superintegrability, we will enlarge, temporarily, the set of the initial conditions considered, allowing for a nonzero initial velocity of the light particle.
Generally, the allowed sets of incident (in) velocities, i.e.\ the states where no collisions occurred in the past, would require positive initial velocities ordered according to
\begin{equation}
V_{\text{in}} > v_{\text{in}} > 0 \;.
\label{incident_state}
\end{equation}
Likewise, an outgoing state (out), i.e.\ a state that does not lead to any collisions in the future, is characterized by negative final velocities ordered according to
\begin{equation}
V_{\text{out}} < v_{\text{out}} < 0\;.
\label{outgoing_state}
\end{equation}
It can be shown that the conservation of energy and the observable $J$,
\begin{eqnarray*}
T_{\text{out}}& =& T_{\text{in}}\\
J_{\text{out}}& =& J_{\text{in}}\;\,
\end{eqnarray*}
both being a function of the velocities \emph{only}, restricts the set of the allowed outgoing velocity pairs produced by the given incident pair, to one value only \footnote{This can be shown, in particular, by observing that (a) the outgoing pair $(W_{\text{out}},\,w_{\text{out}})$ is an image of the incident pair, $(W_{\text{out}},\,w_{\text{out}})$, upon application of one of the elements of the group, and that (b) the condition (\ref{outgoing_state}) defines a particular \emph{chamber} of this group. However, by construction, there is only one point of an orbit of a group per chamber.}.
Notice that in this case, the outgoing velocities do not depend on the order of collisions: depending on the initial coordinates $X_0<x_0<0$, the first collision in the chain can be represented by either a ball-wall or a ball-ball collision. This independence can be regarded as a classical (as opposed to quantum) manifestation of the so-called Yang-Baxter property \cite{gaudin1983_book_english,sutherland2004_book} for the three-body system where the wall is considered a third, infinitely massive body.
In contrast to the superintegrable mass ratios, a generic mass ratio produces two different outcomes, depending on the order of collisions. Notice that two qualitatively different trajectories may even originate from two infinitely close initial conditions. In the maximally superintegrable case of integer $q$, these two trajectories collapse to a single one-dimensional line. This phenomenon can be regarded as an unbounded orbit analogue of the closing the orbits in the bounded case.
The actual sets of the outgoing velocities are very different in the even and in the odd cases. In the even case, the initial velocities are simply inverted:
\begin{equation}
q=\text{even} \to
\begin{array}{l}
\quad V_{\text{out}} = - V_{\text{in}}\\
\quad v_{\text{out}} = - v_{\text{in}}
\end{array}
\;.
\end{equation}
Indeed, since the energy and, in this case, the observable $J$ are even functions of the velocities,
the above connection protects the conservation laws. The odd case is much more involved. One can show that
\begin{equation}
q=\text{odd} \to
\begin{array}{l}
\quad V_{\text{out}} = - \cos(\pi/q) V_{\text{in}} - \tan(\pi/q) \sin(\pi/q) v_{\text{in}}\\
\quad v_{\text{out}} = - \cos(\pi/q) ( V_{\text{in}} - v_{\text{in}})
\end{array}
\;.
\end{equation}
Remark that the case $v_{\text{in}} = V_{\text{in}}$, where $v_{\text{out}}$ vanishes, may be regarded as a generalization of a notion of a \emph{Galilean Cannon} \cite{olshanii2016_161001060}: a system of balls that arrives at the wall with the same speed and transfers all the energy to the far-most one in the end.
\subsection{Minimal and maximal distances\label{Sec:max:X}}
In this section we discuss what the extreme positions of the balls are.
The characteristic unit of length is set by the initial position of the light ball, $x_0$.
The largest distance to the wall corresponds to the asymptotically large separations $|X|\to\infty$ and $|x|\to\infty$ which are asymptotically approached after the terminal collision, except for the very special situation when the final velocity of the light ball is equal to zero (this situation happens for $N=0$).
The minimal distance of the light ball is $x=0$ reached at any ball-wall collision.
The closest position $X_{min}$ of the heavy ball to the wall is reached at the collision in which the heavy ball inverts its velocity, while the light ball achieves its maximal velocity approximated by Eq.~(\ref{Eq:v:max}).
As can be seen from Eq.~(\ref{Eq:adiabatic:Xmin}) derived within the adiabatic approximation, $X_{min}$ is inversely proportional to the action $I$.
According to the discussion in Sec.~\ref{Sec:invariants}, the action is conserved during the whole length of the processes and its value is given by Eqs.~(\ref{Eq:v:max}), (\ref{Eq:invariant:BW}) and~(\ref{Eq:action invariant}).
As a result, the minimal distance of the heavy ball can be expressed as
\begin{equation}
X_{min}
= \frac{V_0}{v_{max}}x_0
= \sqrt{\frac{m}{M}}x_0
= \frac{x_0}{b^N}\;.
\label{Eq:X:min}
\end{equation}
Expression~(\ref{Eq:X:min}) is approximate
and it can be made exact by using the ball-wall invariant~(\ref{Eq:invariant:BB}).
From a practical point of view, even in its simple form it works rather precisely.
For example for $b=10$ one finds $X_{min} = 0.0998$ instead of $1/10$ already for $N=1$ and the accuracy is further improved as $N$ is increased.
The curvature of parabola~(\ref{Eq:return:parabola}) close to the point of return can be expressed using Eq.~(\ref{Eq:X:min}) in a simple form
\begin{eqnarray}
\frac{X(t')}{|X_{min}|}
= 1 + \frac{b^{2N}}{2}\left(\frac{t'}{t_0}\right)^2
= 1 + \frac{1}{2}\frac{M}{m}\left(\frac{t'}{t_0}\right)^2
\;,
\end{eqnarray}
where $t_0 = |x_0| / V_0$ is the characteristic timescale for the period between the first collision and the reflection.
The larger is the mass ratio, the ``sharper'' is the trajectory close to the point of return.
\subsection{Position as a function of time: hyperbolic shape\label{Sec:xvs.t}}
Here we will demonstrate that a hyperbolic curve describes the positions of the light ball at BB collisions and of the heavy ball both at BB and BW collisions.
In the description of a billiard in a wedge, the trajectory is bound to the phase space $0\leq |y| \leq |Y| \tan(\varphi)$, as shown in Fig.~\ref{Fig:unfolding}.
The collisions happen when either $y=0$, i.e. when the light particle hits the wall (BW collision) or when $y=Y$ and the light particle hits the heavy one (BB collision).
The unfolded trajectory is formed by reflecting the wedge, so that its angle $\varphi$ is preserved.
The collisions in the unfolded trajectory occur when the straight line intersects one of the mirrors, corresponding to an angle $n\varphi$ with $n$ the number of the collision.
For any intersection its distance from the origin is the same in unfolded picture and that of the billiard in a wedge.
In particular, for a ball-ball collision, this distance is equal to $\sqrt{Y^2(t)+y^2(t)}$.
Instead, in the moment of a ball-wall collision, the light ball has coordinate $y(t)=0$ and this distance is equal to the position of the heavy ball $Y(t)$.
The minimal possible distance $Y_{min}$ of the heavy ball from the wall corresponds to the point of return, which is located on the vertical line directly above the origin.
The projection to the horizontal axis is given by $Wt'$, where $t'$ is the time counted from the point of return and $W$ is the constant velocity, equal to the initial velocity of the heavy ball $W = W_{max}$.
Catheti $Y_{min}$, $W_{max}t'$ and hypotenuse $Y(t)$ forming a right-angled triangle are related as
$Y^2(t) = Y^2_{min} + (W_{max}t')^2$.
The same expression written in terms of the original coordinate $X(t')$ and velocity $V$ leads to the hyperbolic relation
\begin{eqnarray}
\left(\frac{X(t')}{X_{min}}\right)^2 - \left(\frac{t'}{X_{min}/V_{max}}\right)^2 = 1\;,
\label{Eq:hyperbola:BW}
\end{eqnarray}
exactly satisfied for any ball-wall collision.
Here $X_{min}$ is given by Eq.~(\ref{Eq:X:min}).
Instead, for a ball-ball collision both coordinates of the heavy and light particles are equal, $X=x$, and lie on a hyperbola of a slightly smaller semi-axis
\begin{eqnarray}
\left(\frac{X(t')}{\sqrt{\frac{M}{M+m}}X_{min}}\right)^2 - \left(\frac{t'}{X_{min}/V_{max}}\right)^2 = 1\;,
\label{Eq:hyperbola:BB}
\end{eqnarray}
In the limit of large mass, Eqs.~(\ref{Eq:hyperbola:BW}-\ref{Eq:hyperbola:BB}) coincide.
\begin{figure}[ht]
\includegraphics[width=0.6\columnwidth]{FigHyperbola.pdf}
\caption{Distance of the heavy ball close to the return point, for $b=2$ and $N=3$.
Solid symbols, heavy ball $X$;
dashed line, limit of an infinitely massive ball, Eq.~(\ref{Eq:return:heavyball});
solid line, parabola defined by Eq.~(\ref{Eq:hyperbola:BB}).
}
\label{Fig:hyperbola}
\end{figure}
We compare predictions of Eq.~(\ref{Eq:hyperbola:BB}) with the exact results in Fig.~\ref{Fig:hyperbola}.
Close to the return point, parabolic dependence~(\ref{Eq:return:parabola}) shown in Fig.~\ref{Fig:parabola} is recovered.
Instead, far from the return point, the limit of an infinitely massive ball, Eq.~(\ref{Eq:return:heavyball}), is satisfied.
The minimal possible distance $X_{min}$ is actually reached only if there is a crossing of the unfolded trajectory at the vertical line above the origin (see Fig.~\ref{Fig:unfolding}), otherwise the actual minimal distance is larger.
\subsection{Circle in ($V$,$1/X$) variables \label{Sec:(V,1/x)circle}}
Within the adiabatic approximation, introduced in Sec.~\ref{Sec:adiabatic approximation}, the $(P, 1/X)$ portrait has a semicircular shape given by Eq.~(\ref{Eq:adiabatic:P}) with the coefficient of proportionality linear in the action $I$.
In Sec.~\ref{Sec:invariants} it was verified that some of the predictions of the adiabatic theory actually remain exact even far from the point of return, effectively expanding the limits of its applicability.
In particular, the action $I$ is conserved for any ball-wall collision throughout the whole process, and its value can be expressed in terms of the initial position of the light ball $x_0$ and the initial velocity of the heavy ball $V_0$ according to Eq.~(\ref{Eq:action invariant}).
This suggests that the portraits in ($P$, $1/X$) and ($V$, $1/X$) coordinates are close to ellipses.
A straightforward way to see it is to use the Hamiltonian~(\ref{Eq:adiabatic:H}) obtained within the adiabatic approximation,
\begin{eqnarray}
\frac{MV^2}{2} + \frac{\pi^2 I^2}{2mX^2}
= \frac{MV_0^2}{2}
\label{Eq:H:(V,X)}
\end{eqnarray}
where we extended its validity to any BW collision, in particular to collisions happening far from the point of return, $X\to\infty$.
Equation~(\ref{Eq:H:(V,X)}) can be recast in the form of an ellipse for $(V,1/X)$ coordinates as
\begin{eqnarray}
\frac{1}{b^{2N}}
\left(\frac{x_0}{X}\right)^2
+
\left(\frac{V}{V_0}\right)^2
=
1\;.
\label{Eq:ellipse:(V,X)}
\end{eqnarray}
Figure~\ref{Fig:ellipse} shows an example of the trajectory in $(V,1/X)$ coordinates.
The first collision happens at $V/V_0 = 1$ and $x_0/X = 1$ corresponding to the initial velocity and the initial (large) distance from the wall.
As the collisions go on, the heavy ball comes closer to the wall until it inverts its velocity at the point $V=0$, which corresponds to the point of return.
At this moment the heavy ball is located at the closest distance to the wall.
It might be appreciated that Eq.~(\ref{Eq:X:min}) describing this distance is quite precise from the practical point of view.
The case illustrated in Fig.~\ref{Fig:ellipse} corresponds to the binary base, $b=2$, and for mantissa length $N=1;2;3;4$ the heavy ball is expected to come closer to the wall by a factor of $2;4;8;16$ compared to the initial position of the light ball.
Once the point of return is passed, the heavy ball has a negative velocity which increases in absolute value up to $V/V_0 \approx -1$, while the ball moves far away from the ball $x_0/X\to 0$.
\begin{figure}[ht]
\includegraphics[width=0.5\columnwidth]{FigEllipse.pdf}
\caption{
$(V,1/X)$ portrait for $b=2$ and $N=0,1,2,3,4$ (from bottom to top).
Solid symbols, ball-wall collision;
open symbols, ball-ball collision.
Solid lines, ellipse~(\ref{Eq:ellipse:(V,X)}).
Dashed horizontal lines, inverse of the minimal distance~(\ref{Eq:X:min}).
}
\label{Fig:ellipse}
\end{figure}
Overall, the shapes obtained are quite similar to ellipses predicted by Eq.~(\ref{Eq:ellipse:(V,X)}).
The ``discretization'' becomes smaller as $N$ is increased.
The pairs with same velocity $V$ but different values of $X$ correspond to light ball-wall collisions (velocity of the heavy ball is not changed) and ball-ball collisions, shown in Fig.~\ref{Fig:ellipse} with closed and open symbols.
The points which correspond to the ball-wall collisions lie exactly on the top of the ellipse due to presence of ball-wall invariant~(\ref{Eq:invariant:BW}).
Instead, for ball-ball collision there is some shift, with a different sign for the heavy ball moving towards the wall or away from it.
A similar effect was observed in Fig.~\ref{Fig:Xv} and it originates from an additional contribution containing the velocity of the heavy ball in the ball-ball invariant~(\ref{Eq:invariant:BB}).
At the point of return this correction vanishes and the adiabatic theory becomes fully applicable.
\subsection{Exact solution\label{Sec:exact solution}}
A peculiarity of this problem is that it allows an explicit solution, so that the velocities and positions can be explicitly expressed as a function of the collision number.
According to Sinai\cite{SinaiBook}, the billiard variables~(\ref{Eq:billiard variables}-\ref{Eq:billiard velocities}) reduce the rescaled velocities to a circle, so that the velocities of the light and the heavy balls at collision $n$ is fully defined by the angle $\phi_n$.
This angle is flipped at any BW collision and flipped and increased at each BB collision, leading to a simple dependence of $\phi_n$ on $n$.
We note that action invariants relate the position of the balls with the momentum of the heavy ball, thus permitting to extract particle positions once their velocities are known.
The time $\tau_n$ passed between collision $n$ and the next one can be inferred from the balls positions and their velocities according to Eq.~(\ref{Eq:tau(n):exact}).
The exact solution for the positions, velocities and period of time between the collisions
can be explicitly written as a function of the collision number,
The explicit solution can be summarized as follows
\begin{eqnarray}
\phi_n &= &(-1)^{n+1}2\arctan\left(\frac{1}{b^N}\right) \integer\left[\frac{n+1}{2}\right], \\
\phi_{2k} &= & - 2k\arctan\left(\frac{1}{b^N}\right), \;\;\; k=0, 1, \ldots,\\
V_n &= &V_0 \cos \phi_n,\\
v_n &= &V_0b^N \sin \phi_n,\\
X_{2k} &= &-\frac{x_0}{b^N \sin \phi_{2k}}, \;\;\; k= 1, 2, \ldots,\\
X_{2k+1} &= &\frac{x_0}{b^N \sin \phi_{2k}-\cos \phi_{2k}}, \;\;\; k=0, 1, \ldots,\\
x_{2k} & = & 0, \;\;\; k= 1, 2, \ldots,\\
x_{2k+1} &= & X_{2k+1}, \;\;\; k=0, 1, \ldots,\\
\tau_{2k-1} &= & - \frac{x_0}{V_0b^N\sin\phi_{2k}(b^N\sin\phi_{2k} + \cos\phi_{2k})}, \;\;\; k= 1, 2, \ldots,\\
\tau_{2k} &=& - \frac{x_0}{V_0b^N\sin\phi_{2k}(b^N\sin\phi_{2k} - \cos\phi_{2k})}, \;\;\; k= 1, 2, \ldots.
\end{eqnarray}
Here the collision number is $n = 0, 1, \ldots$ with $n=0$ corresponding to the initial conditions.
It is also useful to express the quantities as a function of time.
The exact relation between collision number $n$ and the moment of time $t(n)$ when the collision happened can be obtained by summation of the time intervals $\tau_n$ passed between collisions, see Eq.~(\ref{Eq:t(n):exact}).
Although the result of the summation can be explicitly written in terms of $q$-digamma function, its presentation is quite cumbersome, and we prefer to keep the exact function $t(n)$ as a sum, while in the next section we provide an elegant approximate expression.
\subsection{Approximate solution\label{Sec:approximate solution}}
It is also of interest to obtain an explicit relation in the continuous limit of many collisions, $b^N\gg 1$.
In this case one can neglect cosine function in $\tau$ obtaining the following simple expression for the inverse of time $\tau_n$ between consecutive collisions,
\begin{eqnarray}
\frac{t_0}{\tau_n}
\approx b^{2N}\sin^2(n/b^N)
\approx b^{2N}\cos^2(n'/b^N)
\label{Eq:tau(n):approx}
\end{eqnarray}
where the characteristic time
\begin{eqnarray}
t_0 = \left|\frac{x_0}{V_0}\right|
\end{eqnarray}
defines the appropriate units of time, i.e.\ how long the heavy ball would take to hit the wall in the absence of the light ball.
Figure~\ref{Fig:tau} shows an example of how the inverse time $t_0 / \tau_n$ depends on collision number.
In the considered case with $b=10$ and $N=1$, the total number of collisions is $\Pi = 31$.
The first and last collisions are ``slow'', the time between them is large and the inverse time goes to zero.
Close to the return point the oscillations are fast and the inverse time reaches its maximal value, where the time between collisions is by factor of $b^{2N}$ faster than the characteristic time $t_0$ of the whole process.
The parabolic shape of the trajectory derived in Sec.~\ref{Sec:point of return} was obtained by ignoring dependence of $\tau$ on $n$, see Eq.~(\ref{Eq:tau:inversion}), and corresponds to the dashed horizontal line.
\begin{figure}[ht]
\includegraphics[width=0.5\columnwidth]{FigTau.pdf}
\caption{
Inverse of the time between adjacent collisions for $b=10$ and $N=1$.
Symbols, exact results;
solid blue line, Eq,~(\ref{Eq:tau(n):approx});
dashed green line, $b^{2N}$.
}
\label{Fig:tau}
\end{figure}
Integrating Eq.~(\ref{Eq:tau(n):approx}) close to the point of return and treating the number of collisions after the return point $n' = n - \Pi /2$ as a continuous variable we get
\begin{eqnarray}
t_{n'} /t_0
= 1 + \int\limits_0^{n'} \frac{dn}{b^{2N}\cos^2(n/b^N)}
= 1 + \frac{1}{b^N}\tan(n'/b^N).
\label{Eq:t:approx}
\end{eqnarray}
Figure~\ref{Fig:n} shows the dependence of the number of collisions $n$ on time $t$ for $b=10$ and $N=1$. The function has an inclination point at $t_0$ with the value of the point of return $n_{inversion}\approx \Pi/2$. The final collision takes place for a large $t$ and the function tends to $\Pi$ as $t\to +\infty$. This function is related to the phase $\phi_{\rm phase}$, since $\phi_{\rm phase}(t)=\pi n(t)$.
\begin{figure}[ht]
\includegraphics[width=0.5\columnwidth]{Fign.pdf}
\caption{
The collision number $n$ as a function of time $t$ for the same parameters as in Fig.~\ref{Fig:tau}.
Symbols, exact results;
solid blue line, Eq.~(\ref{Eq:t:approx});
dashed line, inversion collision approximated by $n_{inversion}\approx \Pi/2$.
}
\label{Fig:n}
\end{figure}
\subsection{Terminal collision\label{Sec:terminal collision}}
The last collision defines if the number of collisions is an odd or an even number.
Depending on its value, the corresponding digit of $\pi$ is either odd or even.
Physically, its parity depends if the last collision was ball-wall with no more ball-ball impacts or if it was a ball-ball collision.
In Ref.~\cite{Davis2015} it is shown that an even number of collisions occurs, $\Pi=2k$, when
$2k\varphi < \pi < (2k+1)\varphi$.
\section{Physical realizations\label{Sec:physical realizations}}
\subsection{Finite-size balls\label{Sec:hard rods}}
The pair $(X, x)$ of positions generates a configuration point, and the set of all configuration points form the configuration space\cite{GalperinBook}.
For point-size balls, it is bounded by the position of the wall, $|X|> 0$ and $|x|>0$, and the condition of the impenetrability of the balls, preserving their order, $0<|x|<|X|$.
More realistically, the real balls must have some finite size which we denote as $R$ and $r$ for the radii of the heavy and light balls, respectively.
Still, we argue that if all collisions are elastic, the problem can be effectively reduced to the previous one of point-size balls.
One might note that finite-size impenetrable balls have a smaller configuration space, schematically shown in Fig.~\ref{Fig:excluded volume}, which contains an {\em excluded volume}\cite{Girardeau60}.
The configuration space of finite-size balls is $|x|>r$ and $|X|>|x|+r+R$.
In other words, mapping which removes the excluded volume,
\begin{eqnarray}
x' &=& x + r\\
X' &=& X + R + r,
\label{Eq:excluded volume}
\end{eqnarray}
reduces the problem of finite-size hard spheres to the problem of point-like objects, via a simple scaling which does not affect the balls' velocities.
As sphere is a three-dimensional object, sometimes finite width one-dimensional balls are referred to as {\em hard rods}.
\begin{figure}[ht]
\includegraphics[width=0.45\columnwidth]{FigExcludedA.pdf}
\includegraphics[width=0.45\columnwidth]{FigExcludedB.pdf}
\caption{Configuration space for (a) two point balls (b) balls of size $r$ and $R$.
Mapping~(\ref{Eq:excluded volume}) translates configuration space (a) into (b).
}
\label{Fig:excluded volume}
\end{figure}
\subsection{Billiard\label{Sec:billiard}}
The restricted domain of the available phase space (half of a quadrant) together with the specular reflection laws makes the system consisting of two identical balls and a wall mappable to a problem of a {\em billiard} with opening angle of $45^\circ$.
In a billiard, the balls move in straight lines and collide with the boundaries (mirrors), where the incident and reflected angles are equal\cite{KozlovTreshchevBook}.
It might be shown\cite{Galperin2003,TabachnikovBook2005} that billiard variables~(\ref{Eq:billiard variables}) change the opening angle to $\phi$ and have a special property which is that the reflections result in a straight trajectory.
This unfolding creates a straight-line trajectory which intersects a certain number of lines, each of them rotated by the angle $\phi$.
Each intersection corresponds to a single collision and the total number of intersections defines the total number of collisions $\Pi$.
Altogether, this picture provides an intuitive visualization of the relation between $\pi$, corresponding to the angle of $180^\circ$, and the number of collisions.
\subsection{Four-ball chain\label{Sec:four ball chain}}
Another physical system which conceptually is related to the present system consisting of two balls and a wall, is a problem of four balls on a line.
The action of the wall consists in reflecting the mass $m$ ball with the same absolute value of the velocity, $v \to -v$.
The same effect can be achieved by replacing the rigid wall by another ball of mass $m$, moving with velocity $-v$.
During an elastic collision, both balls will exchange their velocities.
In order to make the system completely symmetric, one has also to add an additional heavy ball, resulting in $M -m -m - M$ chain.
The distance between $1-2$ and $3-4$ balls must be the same, while $2-3$ distance can be arbitrary chosen.
Finally, the initial velocities should be chosen such $v_2 = v_3$ and $v_1 - v_2 = v_3 - v_4$.
\section{Systematic error\label{Sec:error}}
Any real experimental procedure should contain an error analysis.
For example, the stochastic method of Buffon provides not only an approximate value of $\pi$, but also the statistical error associated with it.
After $N$ trials of dropping the needle, $\pi$ is estimated as an average value while the statistical error is
$\varepsilon_{stat} = \sigma / \sqrt{N-1}$, where $\sigma$ is the variance.
Although in each experiment the realizations are different, the statistical error can be estimated and its value can be controllably reduced by increasing the number of trials.
In the present study we do not report results of a real experiment, in which the number of collisions will be limited by friction, non-perfect elasticity of collisions, etc.
Nevertheless, the relation~(\ref{Eq:hurbilketa}) between the number of collisions and the Galperin billiard relies on the Taylor expansion of inverse tangent function in Eq.~(\ref{Eq:pi:acrctg}) and on taking its integer part, and these might induce a certain error to the final result.
Accuracy of the approximations used is reported in Fig.~\ref{Fig:error} as a function of the base $b$ and mantissa $N$.
For completeness, here we consider $N$ not limited to integer values but as a continuous variable $N\ge 0$ and the base $b>1$.
The analyzed data gives error $\varepsilon$ limited to two values $\varepsilon = 1$ (light color) and $\varepsilon = 0$ (black).
It becomes evident that for large $N$ the approximate formula always works correctly, while for small $N$ there appears a complicated structure as a function of $b$.
For large system base (for example, decimal $b=10$ and hexadecimal $b=16$ cases) formula~(\ref{Eq:hurbilketa}) works correctly for any length of mantissa apart from $N=0$ case, which in any case should be treated separately due to degeneracy as will be discussed in Sec.~\ref{Sec:degeneracy}.
\begin{figure}[]
\includegraphics[width=0.6\columnwidth]{FigAccuracy.pdf}
\caption{
Difference between the exact number of collisions~(\ref{Eq:pi:acrctg}) and the approximation~(\ref{Eq:hurbilketa}), which relates it to the digits of pi,
as a function of base $b$ and mantissa $N$.
Two possible values are denoted with the dark (0) and light (1) colors.
}
\label{Fig:error}
\end{figure}
The error $\varepsilon$ is a complicated non-analytic function of $N$ and $b$, as can be perceived from Fig.~\ref{Fig:error}.
It turns out that for some integer bases expressions~(\ref{Eq:pi:acrctg}) and~(\ref{Eq:hurbilketa}) lead to different results.
Namely, the error is $\varepsilon = 1$ for integer bases $b=6 ; 7; 14$ and $N=1$.
It means that for the mentioned combinations, Galperin billiard method does not provide the digits of $\pi$ exactly, as there is an error of $\varepsilon = 1$ in the last digit.
The cardinality of irrational numbers is greater than that of the integer numbers.
For irrational numbers it is possible to find examples where the error is different from zero for different values of $N$ and the same value of the base $b$.
Namely, $\varepsilon=1$ for $b=3.7823797$ and $N=1, 2, 3, 4$ and $6$.
In general, it is clear that the closer is the base to $b=1$ the worse is the description, and for a larger number of values of $N$ Galperin billiard gives digits different from $\pi$.
We propose to treat a possible difference between~(\ref{Eq:pi:acrctg}) and~(\ref{Eq:hurbilketa}) as a
{\em systematic error}, so that the final result of each ``measurement'' is $\varepsilon / b^N$ with $\varepsilon \le 1$.
That is, the approximation of $\pi$ in a base $b$ can be expressed from the number of collisions $\Pi(b,N)$ as
\begin{equation}
\pi_b = \frac{\Pi(b,N)}{b^N} \pm \frac{\varepsilon}{b^N}\;.
\label{Eq:error}
\end{equation}
Such a classification is closer to a spirit of a real measurement, where different effects might contribute to the error.
Another advantage of the proposed idea of introducing the concept of the systematic error, is that it solves the problem of the number of digits which are predicted correctly using Galperin billiard.
It was noted by Galperin in Ref.~\cite{Galperin2003} (see also Ref.~\cite{TabachnikovBook2005}) that if there is a string of nines, that might lead to a situation when more than one digit is different.
In a similar sense, the numbers $0.999$ and $1.000$ differ by all four digits.
If instead, one allows an error of $0.001$, both numbers become compatible.
Indeed, from a practical point of view (suppose
we calculate perimeter of a circle knowing its radius), the use of the incorrect value would lead to a relative error of 0.001 and not to completely incorrect result as all the original digits are different.
In the next sections we consider the cases of integer and non-integer bases.
\section{Integer bases\label{Sec:integer bases}}
Equation~(\ref{Eq:pi:acrctg}) has a profound mathematical meaning, as the number of collisions $\Pi(b,N)$ provides the first $N$ digits of the fractional part (i.e. digits beyond the radix point) of the number $\pi$ in base $b$.
It might be immediately realized that as the number of collisions is obviously an integer number, its integer base representation can be chosen to be finite.
In Sections~\ref{Sec:integer bases}-\ref{Sec:Non-integer bases} we use number of collisions $\Pi(b, N)$, as given by Eq.~(\ref{Eq:pi:acrctg}), to approximate the digits of $\pi$ for different integer bases $b$, then $(\Pi/b^N)_b$ yields the base-$b$ representation of $\pi$ with $N$ digit beyond the radix point.
\subsection{Representing a number in integer bases\label{Sec:integer representation}}
Let $b>1$ be an integer number.
Any positive number $x$ has the integer expansion in base $b$, i.e. can be represented in powers of $b$ as
\begin{equation}
\label{Eq:integer}
x =(a_n a_{n-1} \ldots a_0.a_{-1}a_{-2} \ldots)_b = \sum\limits_{i=-\infty}^n a_i b^i,
\end{equation}
where
$n=\integer [\log_b x]$ and $a_i=\{0,1, \ldots, b-1\}$ are the digits in the corresponding numeral system and we use form $x_b$ to denote the representation of number $x$ in base $b$.
For bases with $b>10$, the symbols $A, B, \ldots$ are commonly used to denote $10, 11, \ldots$.
In order to obtain the digits $a_i$, one can use the following iterative process: $a_i=\integer [ r_i/b^i ]$, $i\leq n$ with $r_n=x$ and $r_{j-1}=r_j-a_j\cdot b^j$, $j\leq n-1$. Multiplying the base-$b$ representation~(\ref{Eq:integer}) by $b^i$ shifts the radix point by $i$ digits. Thus, approximation~(\ref{Eq:hurbilketa}) gives the integer part and first $N$ digits of the fractional part of $\pi$ in base $b$.
The most frequently used integer systems are decimal ($b=10$) and binary $b=2$ systems.
Occasionally, also ternary $b=3$, octal ($b=8$), hexadecimal ($b=16$) and others systems are used.
Importantly, for integer bases, finite representations are unique, while infinite representations might be not unique.
For example, the finite number $1_{10}$ in the decimal base can be written as $1.000(0)_{10}=0.999(9)_{10}$.
\subsection{Degenerate case of equal masses and submultiple angles\label{Sec:degeneracy}}
Before considering in detail the representation in bases $b=10; 2; 3$ reported in Tables~\ref{table:b=10},\ref{table:b=2},\ref{table:b=3}, we note that $N=0$ case is universal as the mass ratio $M/m = b^N = 1$ does not depend on the base $b$.
In other words, the digit in front of the radix point always correspond to the same number.
The Eq.~(\ref{Eq:pi:acrctg})
formally gives 4 collisions, which is different from the physically correct number of 3 collisions.
The reason for such a difference comes from a degeneracy between the third and fourth collision.
While for $N>0$, the direction of the light ball is always inverted in the last two collisions ($\phi\to -\phi$), for $N=0$ the light ball completely stops exactly at the third collision.
In physical sense there is no difference between $v_3 = -0$ and $v_4 = +0$ velocities.
Thus, Eqs.~(\ref{Eq:pi:acrctg}-\ref{Eq:hurbilketa}) are applicable only for $N\geq1$ while $N=0$ is a special case and it should be treated separately.
The analogous result takes place in the case of the angle $\phi$ being submultiple of $\pi$, i.e. when the ratio $\pi/\phi$ is an integer number. The number of collisions is not given correctly by~(\ref{Eq:pi:acrctg}) as the last collision is degenerate as well.
\subsection{Decimal base \label{Sec:Decimal base}}
For the most natural case of the decimal base system, $b=10$, the number of collisions $\Pi(10, N)$ is given in Table~\ref{table:b=10}.
It is easy to follow, how Galperin billiard generates digit of $\pi$.
For $N=0$, Eq.~(\ref{Eq:pi:acrctg}) results in the first digit of $\pi$ approximated by 4, while due to degeneracy discussed in Sec.~\ref{Sec:degeneracy}, physically there are 3 collisions.
For $N=1$, there are $31$ collisions, resulting in expression with 1 digit after the radix point, $3.1$.
From $N=2$, the number of collisions in 314 giving the number $\pi$ with 2 digits after the radix point.
One can see that the billiard method correctly approximates the number $\pi$ as 3 plus $N$ more digits in the decimal base.
Conceptually, one might ask if there is a difference between the number of collisions Eq.~(\ref{Eq:pi:acrctg}) which depend on $\arctan(b^{-N})$ rather than $b^{-N}$, as in Eq.~(\ref{Eq:hurbilketa}).
It turns out that the base $b=10$ is large enough (see Fig.~\ref{Fig:error}) so that there is no any difference in the integer part of the expansion.
\begin{table}[ht]
\centering
\caption{Number of collisions $\Pi(10,N)$ given by Eq.~(\ref{Eq:pi:acrctg}) for the decimal base, $b=10$.
The first column reports the value of mantissa $N$.
The second column is the resulting number of collisions in the decimal base.
The third column is the number $\pi$ with $N$ digits in the fractional part in the decimal representation.
The fourth column gives the systematic error according to Eq.~(\ref{Eq:error}).
}
\label{table:b=10}
\begin{tabular}{c|l|l|l}
N & $\Pi(10,N)_{10}$ & $(\Pi(10,N)/10^N)_{10}$ & $(1/10^N)_{10}$
\\ \hline
0 & 4 & 4 & 1 \\
1 & 31 & 3.1 & 0.1\\
2 & 314 & 3.14 & 0.01\\
3 & 3141 & 3.141 & 0.001\\
4 & 31415 & 3.1415 & 0.0001\\
5 & 314159 & 3.14159 & 0.00001\\
6 & 3141592 & 3.141592 & 0.000001\\
7 & 31415926 & 3.1415926 & 0.0000001\\
8 & 314159265 & 3.14159265 & 0.00000001\\
9 & 3141592653 & 3.141592653 & 0.000000001\\
10 & 31415926535 & 3.1415926535 & 0.0000000001
\end{tabular}
\end{table}
\subsection{Binary and ternary bases \label{Sec:binary base}}
Other important examples of number systems include the binary ($b=2$) and ternary ($b=3$) base systems.
The binary system lies in the core of modern computers which operate with {\em bits} $0, 1$.
Interestingly, base-$3$ computer named Setun was built 1958 under leadership of mathematician Sergei Sobolev and operated with {\em trits}, $0,1,2$.
Table~\ref{table:b=2} reports the number of collisions $\Pi(2,N)$ obtained for $b=2$ base.
By expressing the number of collisions in binary base using zeros and ones, one obtains the representation of the number $\pi$ in binary base.
In the ternary base, the number of collisions are written using the three allowed digits, $0,1,2$, see Table~\ref{table:b=3}.
\begin{table}[t]
\centering
\caption{
Number of collisions $\Pi$ given by Eq.~(\ref{Eq:pi:acrctg}) for the binary base, $b=2$.
The first column reports the value of mantissa $N$.
The second column is the resulting number of collisions in the decimal base.
The third column is the number of collisions written in the binary representation.
The fourth column is the binary representation of the number $\pi$ with $N$ digits in the fractional part.
The fifth column gives the systematic error according to Eq.~(\ref{Eq:error}).
}
\label{table:b=2}
{
\begin{tabular}{l|l|l|l|l}
N & $\Pi(2,N)_{10}$ & $\Pi(2,N)_{2}$ & $(\Pi(2,N)/2^N)_{2}$
& $(1/2^N)_{2}$
\\ \hline
0 & 4 & 100 & 100 & 1\\
1 & 6 & 110 & 11.0 & 0.1\\
2 & 12 & 1100 & 11.00 & 0.01\\
3 & 25 & 11001 & 11.001 & 0.001\\
4 & 50 & 110010 & 11.0010 & 0.0001\\
5 & 100 & 1100100 & 11.00100 & 0.00001\\
6 & 201 & 11001001 & 11.001001 & 0.000001\\
7 & 402 & 110010010 & 11.0010010 & 0.0000001\\
8 & 804 & 1100100100 & 11.00100100 & 0.00000001\\
9 & 1608 & 11001001000& 11.001001000 & 0.000000001\\
10 & 3216 & 110010010000& 11.0010010000 & 0.0000000001
\end{tabular}
}
\end{table}
\begin{table}[t]
\centering
\caption{
Number of collisions $\Pi$ given by Eq.~(\ref{Eq:pi:acrctg}) for the ternary base, $b=3$.
The first column reports the value of mantissa $N$.
The second column is the resulting number of collisions in the decimal base.
The third column is the number of collisions written in the binary representation.
The fourth column is the ternary representation of the number $\pi$ with $N$ digits in the fractional part.
The fifth column gives the systematic error according to Eq.~(\ref{Eq:error}).
}
\label{table:b=3}{
\begin{tabular}{l|l|l|l|l}
N & $\Pi(3,N)_{10}$ & $\Pi(3,N)_{3}$ & $(\Pi(3,N)/3^N)_{3}$
& $(1/3^N)_{3}$\\ \hline
0 & 4 & 11 & 11 & 1\\
1 & 9 & 100 & 10.0 & 0.1\\
2 & 28 & 1001 & 10.01 & 0.01\\
3 & 84 & 10010 & 10.010 & 0.001\\
4 & 254 & 100102 & 10.0102 & 0.0001\\
5 & 763 & 1001021 & 10.01021 & 0.00001\\
6 & 2290 & 10010211 & 10.010211 & 0.000001\\
7 & 6870 & 100102110 & 10.0102110 & 0.0000001\\
8 & 20611 & 1001021101 & 10.01021101 & 0.00000001\\
9 & 61835 & 10010211012 & 10.010211012 & 0.000000001\\
10 & 185507 & 100102110122 & 10.0102110122 & 0.0000000001
\end{tabular}
}
\end{table}
\subsection{Best bases for a possible experiment\label{Sec:experiment}}
As concerning the effects of the friction and other sources of energy dissipation, it is easier to perform experiments for small base $b$.
While for $N=0$ (the mass ratio is $M/m = 1$ independently of $b$) there are 3 collisions which can be easily observed with identical balls, for larger $N$ the number of collisions grows exponentially fast.
The decimal system has a rather ``large'' base $b=10$ which already for $N=1$ results in 31 collisions and $N=2$ even in 314 collisions.
It might be notoriously hard to create a clean system in which such a large number of collisions can be reliably measured.
For the binary base $b=2$ and $N=1$ the number of collisions to be observed is much smaller, $3; 6; 12; 25; \ldots$, see Table~\ref{table:b=2}, making such system more suitable for an experimental observation.
\section{Non-integer bases \label{Sec:Non-integer bases}}
As anticipated above, the Galperin billiard method should provide digits of $\pi$ in an arbitrary base $b$, even for a non-integer one.
In this Section we consider a number of examples.
\subsection{Representing a number in a non-integer base\label{Sec:representation}}
For a non-integer base $b>1$, any positive number $x$ can be written in the base-$b$ representation according to
\begin{equation}
x_b=d_n...d_2d_1d_0.d_{-1}d_{-2}\ldots\;,
\label{Eq:non-integer base:x_b}
\end{equation}
where digits $d_i$ can take only non-negative integer values smaller than non-integer base, $d_i<\lceil b \rceil$ ($\lceil x \rceil$ stands for the least integer which is greater than or equal to $x$), and
\begin{equation}
x = \sum\limits_{i=-\infty}^n d_i b^i\;.
\label{Eq:non-integer base:x}
\end{equation}
Unlike the integer bases, for a non-integer base $b$, even finite fractions might have different $b$ representations.
For example, in the golden mean $\varphi\approx 1.61803$ base, due to the equality $\varphi^2=1+\varphi$, one has $100_\varphi = 11_\varphi$.
With Eqs.~(\ref{Eq:non-integer base:x_b}-\ref{Eq:non-integer base:x}) we can find at least one representation for $x$.
\subsection{Number systems with irrational bases}\label{Sec:irrational base}
Some notable examples of non-integer bases include the fundamental cases of the bases with the golden mean $b=\phi$, the natural logarithm $b=e$ and a curious situation when the number $\pi$ is used itself as a base, $b=\pi$.
The number of collisions $\Pi(b,N)$ is obviously an integer number, and it always can be written with a finite representation in any integer base system.
Contrarily, in a non-integer base $b$ it is a common situation that an integer number needs an infinite representation which corresponds to the number.
In Table~\ref{table_phi} we give two different representations for $\pi_\varphi$, since its integer part $100_\varphi=11_\varphi$.
The fourth and fifth columns in Table report the $\varphi$-representation with the integer part $100_\varphi$ and $11_\varphi$, respectively.
The allowed digits for both representations are $0$ and $1$.
Table~\ref{table_e} reports the resulting number of collisions $\Pi(e, N)$ in base $e$ with the allowed digits $0, 1$ and $2$. We get the representation $\pi=(10.1010020200\ldots)_e=e+e^{-1}+e^{-3}+2 e^{-6} + 2 e^{-8}+\ldots$. One can see the influence of the error of the computation by the Galpelin billiard which in the case of base $b=e$ is $1/e^N=0.00\ldots01_e$. Due to this error, the last digit may be incorrectly predicted by the method. Especially, when the last digit is the maximum allowed ($2$ for $b=e$, see the cases $N=4$ and $N=9$ in Table~\ref{table_e}), then the two last digit may be incorrect.
\begin{table}[t]
\centering
\caption{Number of collisions $\Pi(\varphi,N)$ given by (\ref{Eq:pi:acrctg}) for $b=\varphi$.
The first column is $N$.
The second column is $\Pi(\varphi,N)$ in the decimal base.
The third column is the integer part of $\Pi(\varphi,N)$ written in the base $\pi$.
The fourth column is the number $\pi$ with $N$ digits in the fractional part (Type I) in the base $\pi$.
The fifth column is the number $\pi$ with $N$ digits in the fractional part (Type II) in the base $\pi$.
The sixth column gives the systematic error according to Eq.~(\ref{Eq:error}).
}
\label{table_phi}{
\begin{tabular}{l|l|l|l|l|l}
N & $\Pi(\varphi, N)_{10}$ & $\Pi(\varphi, N)_{\varphi}$ & $(\Pi(\varphi, N)/\varphi^N)_{\varphi}$ (I) & $(\Pi(\varphi, N)/\varphi^N)_{\varphi}$ (II) & $(1/\varphi^N)_{\varphi}$ \\ \hline
0 & 4 & 101. & 100. & 11. & 1\\
1 & 5 & 1000. & 100.0 & 11.0 & 0.1\\
2 & 8 & 10001. & 100.01 & 11.01 & 0.01\\
3 & 13 & 100010. & 100.010 & 11.010 & 0.001\\
4 & 21 & 1000100. & 100.0100 & 11.0100 & 0.0001\\
5 & 34 & 10001000. & 100.01001 & 11.01001 & 0.00001\\
6 & 56 & 100010010. & 100.010010 & 11.010010 & 0.000001\\
7 & 91 & 1000100101. & 100.0100101 & 11.0100101 & 0.0000001\\
8 & 147 & 10001001010. & 100.01001010 & 11.01001010 & 0.00000001\\
9 & 238 & 100010010100. & 100.010010101 & 11.010010101 & 0.000000001\\
10 & 386 & 1000100101010. & 100.0100101010 & 11.0100101010 & 0.0000000001\\
\end{tabular}
}
\end{table}
\begin{table}[t]
\centering
\caption{Number of collisions $\Pi(e,N)$ given by (\ref{Eq:pi:acrctg}) and approximation of $\pi$ for $b=e$.
The first column is $N$.
The second column is $\Pi(e,N)$ in the decimal base.
The third column is the integer part of the number of collisions $\Pi(e,N)$ written in the base $e$.
The fourth column is the number $\pi$ with $N$ digits in the fractional part in the base $e$.
The fifth column gives the systematic error according to Eq.~(\ref{Eq:error}).
}
\label{table_e}{
\begin{tabular}{l|l|l|l|l}
N & $\Pi(e,N)_{10}$ & $\Pi(e,N)_{e}$ & $(\Pi(e,N)/e^N)_{e}$ & $(1/e^N)_{e}$ \\ \hline
0 & 4 & 11. & 11. & 1\\
1 & 8 & 100. & 10.0 & 0.1\\
2 & 23 & 1010. & 10.10 & 0.01\\
3 & 63 & 10101. & 10.101 & 0.001\\
4 & 171 & 101002. & 10.1002 & 0.0001\\
5 & 466 & 1010100. & 10.10100 & 0.00001\\
6 & 1267 & 10101001. & 10.101001 & 0.000001\\
7 & 3445 & 101010020. & 10.1010020 & 0.0000001\\
8 & 9364 & 1010100201. & 10.10100201 & 0.00000001\\
9 & 25456 & 10101002012. & 10.101002012 & 0.000000001\\
10 & 69198 & 101010020200. & 10.1010020200 & 0.0000000001\\
\end{tabular}
}
\end{table}
In Table~\ref{table_pi} we show approximations of $\pi$ in the base $b=\pi$
The allowed digits in this base are $0, 1, 2$ and $3$.
The Galperin billiard does not provide an integer-number representation for the number $\pi$ even in this case, as instead of the ``natural'' possibility $\pi = 10_\pi$ one obtains an infinitely long representation
$$\pi = (3.0110211100\ldots)_{\pi}=3+\pi^{-2}+\pi^{-3}+2\pi^{-5}+\pi^{-6} + \pi^{-7} + \pi^{-8} +\ldots.$$
This non-unique representation is similar to the infinite representation $0.999(9)\ldots$ of $1$ in the decimal system.
Another peculiarity of this base is that for $N=1$ (line marked with bold in Table~\ref{table_pi}) the number of collisions $\Pi(\pi, 1)$ in the Galperin billiard, given by Eq.~(\ref{Eq:pi:acrctg}), does not coincide with expression~(\ref{Eq:hurbilketa}) which is used to transcribe $N$ digits of $\pi$ in the base $b$.
The resulting difference in the last digit can be interpreted as a the systematic error $\varepsilon=1$ in the spirit of Section~\ref{Sec:error}.
\begin{table}[t]
\centering
\caption{Number of collisions $\Pi(\pi,N)$ given by (\ref{Eq:pi:acrctg}) for $b=\pi$.
The first column is $N$.
The second column is the number of collisions in the decimal base.
The third column is the integer part of the number of collisions $\Pi(\pi,N)$ written in the base $\pi$.
The fourth column is the number $\pi$ with $N$ digits in the fractional part in the base $\pi$.
The fifth column gives the systematic error according to Eq.~(\ref{Eq:error}). The case $N=1$ is emphasized since there is the difference 1 between $\Pi(\pi,1)$ by~(\ref{Eq:pi:acrctg}) and the approximation~(\ref{Eq:hurbilketa}).
}
\label{table_pi}{
\begin{tabular}{l|l|l|l|l}
N & $\Pi(\pi,N)_{10}$ & $\Pi(\pi,N)_{\pi}$ & $(\Pi(\pi, N)/\pi^N)_{\pi}$ & $(1/\pi^N)_{\pi}$ \\ \hline
0 & 4 & 10. & 10. & 1\\
\textbf{1} & \textbf{10} & \textbf{100.} & \textbf{10.0} & 0.1\\
2 & 31 & 301. & 3.01 & 0.01\\
3 & 97 & 3010. & 3.010 & 0.001\\
4 & 306 & 30110. & 3.0110 & 0.0001\\
5 & 961 & 301102. & 3.01102 & 0.00001\\
6 & 3020 & 3011021. & 3.011021 & 0.000001\\
7 & 9488 & 30110210. & 3.0110210 & 0.000001\\
8 & 29809 & 301102110. & 3.01102110 & 0.0000001\\
9 & 93648 & 3011021110. & 3.011021110 & 0.00000001\\
10 & 294204 & 30110211100. & 3.0110211100 & 0.000000001\\
\end{tabular}
}
\end{table}
The accuracy of the approximation to $\pi$ obtained by Galperin's billiard can estimated by Eq.~(\ref{Eq:error}) which we treat as systematic error of the method.
It was demonstrated by Sinai~\cite{SinaiBook}, that in presence of the second wall, the motion is not ergodic for a rational angle~(\ref{Eq:phi}), i.e. when it can be written as $\varphi = 2\pi r/s$ with $r$ and $s$ some integer numbers.
\section{Conclusions\label{Sec:conclusions}}
To summarize, we have studied how the digits of the number $\pi$ are generated in a simple mechanical system consisting of one heavy ball, one light ball and a wall (Galperin's billiard method).
The number base $b$ and mantissa length $N$ define the mass ratio according to $M/m = b^{2N}$.
We obtain for the first time, to our best knowledge, the complete explicit solution for the balls' positions and velocities as a function of the collision number and time.
Also we find that the adiabatic approximation works not only close to the return point but also for any ball-wall collision, even far from the wall.
The action $I = P X / (2\pi)$ is an adiabatic invariant and its value is preserved during ball-wall collisions.
The number $\pi$ is intimately related to the Galperin billiard and it has a number of ``round'' properties.
We find that the portraits of the system in $(P,1/X)$ and $(V,1/X)$ coordinates have a shape close to a circular one.
Another circle appears in $(V,v)$ coordinates and corresponds to the energy conservation law.
Instead a hyperbolic shape appears in the $(X,t)$ plane.
We derive a differential equation which describes the trajectory close to the point of return.
Its solution is a parabola
$X(t')/X_{min} = 1 + (V_0 t' / X_{min})^2$
with the width defined by the initial velocity $V_0$ and the minimal distance from the wall $X_{min}$ of the heavy particle.
With a good precision the minimal and maximal values of the positions and velocities are approximated by $X_{min} = b^{-2N}X_0$ and $v_{max} = b^{2N}V_0$.
We note that the behavior of the trajectories close to the return point is changed from a non analytic for few collisions to an analytic one (envelope in the shape of a parabola) for many collisions,
and again a non analytic (of type $|x|$) in the limit of a large mass ratio.
Examples of integer bases $b$, including decimal, binary and ternary, are considered.
We argue that smaller bases (for example, $b=2;3$) are the easiest to be realized in an experiment and show how the Galperin billiard can be generalized to finite-size balls (hard rods).
We show that the dependence of a possible error in the last digit as a function of $b$ and $N$ has a complicated form, with the error disappearing in the limit of $b^N\to\infty$.
We propose to treat the possible error in the last digit as a systematic error.
In particular this resolves the problem of the correct number of obtained digits.
We consider non-integer bases including an intriguing case of expressing $\pi$ in the base $\pi$.
The generated expression is different from a finite number, $\pi = 1\times \pi^1$, and instead is given by an infinite representation, $\pi = 3 + 1/\pi^2 + 1/\pi^3+\cdots$.
The difference between finite and infinite representation is similar to that of $1 = 0.999(9)$ in the decimal system.
Finally we note that finite representation is not unique in the base of the golden number $\phi$.
\section*{Acknowledgment}
The research leading to these results received funding from the MICINN (Spain) Grant No. FIS2014-56257-C2-1-P.
MG has been partially supported by Juan de la Cierva-Formaci\'on Fellowship FJCI-2014-21229, the Russian Scientific Foundation Grant~14-41-00044 and the MICIIN/FEDER grants MTM2015-65715-P and MTM2016-80117-P (MINECO/FEDER, UE).
MO acknowledges financial support from the National Science Foundation grant PHY-1607221 and the US-Israel Binational Science Foundation grant 2015616.
| 95,466
|
TITLE: Why is decomposing rational functions by assigning selected numerical values to x mathematically consistent?
QUESTION [4 upvotes]: Possible Duplicate:
How does partial fraction decomposition avoid division by zero?
Say you have the rational function:
$\frac{x^2 + 1}{(x-1)(x-2)(x-3)}$
This means that the function is undefined when x is equal to 1, 2, or 3.
Then to decompose it, you can equate that function to:
$\frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x-3}$
If you clear the fraction then you will get:
x^2 + 1 = A(x-2)(x-3) + B(x-1)(x-3) + C(x-1)(x-2)
Then if you let x = 1,2,3 you can find A,B, and C but why are you even allowed to do that? From the beginning don't we define the domain of the function to be all real numbers besides 1,2, and 3? So why can we can go against the domain of the function to solve for the coefficients ?
REPLY [2 votes]: Once we are looking at the equation $x^2 + 1 = A(x-2)(x-3) + B(x-1)(x-3) + C(x-1)(x-2)$ , we are no longer solving the initial problem -- rather, we are ascertaining that the polynomials on the left hand and right hand sides are equal for all values of $x$, and as such we can figure out what the values of $A$, $B$, and $C$ are.
Traditionally, to the best of my knowledge, this is done by simply expanding out the polynomial on the right hand side, then getting three linear equations in three unknowns between the coefficients of $x^2$, $x$, and the constants $(x^0)$. However, because polynomials are perfectly nice and continuous, etc., and because a degree $n$ polynomial can always be precisely specified by $n+1$ points with different $x$ values, then the values you find by constraining the above quadratic with 3 points will be the same as if you used the 'traditional' method, in every case. Is this sufficient?
REPLY [2 votes]: When you clear the denominator, you are left with an equality of polynomial functions, rather than simply rational functions. Even though the domain of the original rational function excludes $1,2,3,$ the polynomials are equal iff they are equal as functions on the entire domain.
| 103,077
|
My great grand uncle, Hassan Al-Karmi, passed away last night at the age of 101. He was a great man that I am very proud to be related to. He was an incredible author, thinker, poet, speaker, educator, and father to us all.
:
- [The International Writers Magazine] Remembering Hassan Al Karmi by Marwan Asmar
- [Petra News Agency] The passing of a Palestinian Scolar Hassan Al-Karmi (Arabic)
- [The Guardian] Hasan Karmi: Palestinian intellectual and broadcaster passionate about the suffering of his people
- [Jordan Times] Hasan Said Karmi — ‘a soul burning for Palestine’
- Wikipedia Entry
- [The Independent] Hasan Karmi by Donald Macintyre
- [The Guardian] Letter by Sir James Craig
| 306,297
|
Why Michy Batshuayi is the internet's new favourite footballer
Michy Batshuayi has made an immediate impact since arriving at Chelsea. The 23-year-old is slowly building a following of new fans, regardless of their footballing allegiances, and it's all down to his performances online.
The Belgian became Antonio Conte's first summer signing when he completed his £33m move from Marseille in the wake of Belgium's Euro 2016 exit at the hands of Wales.
With Diego Costa in such fine form for the club, Batshuayi's first team appearances have been limited.
While Batshuayi has yet to break through into the Chelsea first team, he is already making quite a name for himself on social media.
We're all too used to the banal and predictable posts from professional footballers on Twitter.
As Troy Deeney recently observed: "If they win it's 'Great team performance, the fans were great,' If they lose it's 'Not quite at it today, we'll get it right next time, the fans were great.' I just sit there and laugh."
It's become common place for footballers to be mocked on social media with a familiar variety of memes and GIFs, but it's quite rare for a footballer to reply to the mockery.
But Batshuayi is a refreshing break from the norm.
Batshuayi started in Chelsea's 4-2 victory over Leicester City in the EFL Cup on Tuesday night, spurning a number of opportunities to get his name on the scoresheet.
When Batshuayi had a close range shot saved by goalkeeper, Ron-Robert Zieler, one internet wag posted, in reference to Batshuayi's 'Batsman' nickname, a meme of Batman with his head in his hands along with the caption "Batshuayi after that miss".
Clearly not one to be upstaged by an online troll, Batshuayi response with this brilliant tweet.
The 22-year-old's cheeky retort was shared over 2,000 times as fans celebrated Batshuayi's 'unreal tweeting.'
Referring to himself as 'Batsman', the former Marseille striker regularly replies to fans on Twitter - even when the tweets are bizarre and even lewd.
The £33m striker has made four appearances for the club since completing his move in June, all of which have come from the substitutes bench.
Batshuayi scored his first Chelsea goal in the club's 2-1 victory over Watford and the 22-year-old tweeted the culprit again to enquire whether he reacted in more suitable fashion this time around.
Having struck up a relationship with Nathaniel Chalobah, Batshuayio congratulated his new friend on breaking an unusual record on Tuesday night.
The England under-21 international made his Chelsea 2,190 days after he first appeared on the Chelsea bench back in 2010.
The previous record wait for a Chelsea debut was Nick Colgan who had to wait five years to make his first appearance for the Blues.
Earlier this month, as gamers across the world eagerly awaited the release of the new demo for Fifa 17, the young striker took the game's makers to task over his underwhelming rating.
EA Sports' Twitter account responded with a cheeky dig at the Belgian suggesting he needs to use the game's practice mode to improve his rating.
Batshuayi responded with the reply: '*downloading PES*', referring to Fifa 17's video game rivals, Pro Evolution Soccer.
The post was retweeted over 30,000 times prompting Pro Evolution Soccer's official Twitter account to weigh into the exchange.
Ahead of Chelsea's visit to Arsenal on Saturday evening, there have been suggestions that Conte could employ both Batshuayi alongside Diego Costa up front.
And the pair already a good working relationship if the Belgian's Twitter page is anything to go by.
While Batshuayi is yet to start a Premier League under Antonio Conte, the Italian has professed to have been impressed with the Belgian since he arrived at Chelsea.
"Batshuayi is a young player but he has great potential for me: technique, great talent, two feet and also he is very fast," said Conte.
"He can improve and become a strong forward. He is very young with great talent and I am happy for him and the other players."
| 170,249
|
>>.
Still no DC offering… At this point, maybe they’ll wait for the Supreme Court before lighting up additional regions?
Good news of course, but what about iPhone support? I’m aware that AirPlay to an Apple TV is possible, but even though I have an Apple TV and an iPhone I’d honestly rather use the ChromeCast, largely due to the HDMI-CEC auto-input switching… and the fact that I can do whatever I want with my phone once I hand the service off…
They stopped dead with lighting up new areas several months ago. I’m in one of the cities that was supposed to get service in the fall. I think all we do now is wait for that Supreme Court ruling.
| 68,547
|
TITLE: tr(ab) = tr(ba)?
QUESTION [34 upvotes]: It is well known that given two Hilbert-Schmidt operators $a$ and $b$ on a Hilbert space $H$, their product is trace class and $tr(ab)=tr(ba)$. A similar result holds for $a$ bounded and $b$ trace class.
The following attractive statement, however, is false:
Non-theorem:
Let $a$ and $b$ be bounded operators on $H$. If $ab$ is trace class , then $ba$ is trace class and $tr(ab)=tr(ba)$.
The counterexample is $a=\pmatrix{0&0&0\\0&0&1\\0&0&0}\otimes 1_{\ell^2(\mathbb N)}$, $b=\pmatrix{0&1&0\\0&0&0\\0&0&0}\otimes 1_{\ell^2(\mathbb N)}$.
I'm guessing that the following is also false, but I can't find a counterexample:
Non-theorem?:
Let $a$ and $b$ be two bounded operators on $H$. If $ab$ and $ba$ are trace class, then $tr(ab)=tr(ba)$.
REPLY [20 votes]: This follows from proposition 7.3 in "Trace-class operators and commutators" by N.J. Kalton. The theorem actually proves something stronger, that for $AB-BA$, if you arrange the eigenvalues with algebraic multiplicity so that $|\lambda_n|\geq |\lambda_{n+1}|$, then $$\sum_{n=1}^{\infty}\frac{\lambda_1+\cdots+\lambda_{n}}{n}<\infty$$
which implies trace zero, but the converse is false.
| 200,472
|
\begin{document}
\setcounter{page}{1}
\begin{center}
{\Large\bf A Study of @-numbers.}
\vspace{3mm}
{ \bf Abiodun E. Adeyemi}
Department of Mathematics, University of Ibadan, \\
Ibadan, Oyo state, Nigeria \\
e-mail: \url{elijahjje@yahoo.com}
\vspace{2mm}
\end{center}
\vspace{0mm}
\begin{abstract}
This paper deals more generally with @-numbers defined as follows: Call `\textit{alpha number}' of order $(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2$, (denote its family by @$_{(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2; \mathcal{A}\subset \mathbb{N}}$) any $n\in\mathcal{A}\subset \mathbb{N}$ satisfying $\sigma_{\underline{\alpha}}(n) = \alpha n^{\bar{\alpha}}$ where $\sigma_{\underline{\alpha}}(n)$ is sum of divisors function and $\alpha\in\mathbb{H}$, the set of \textit{quaternions}. Specifically, if integer $n$ is such that $\alpha=\alpha_1/\alpha_2,\ \alpha_1,\alpha_2\in\mathbb{Z}^+$ with $1\leq\max(\alpha_1, \alpha_2) \le \omega(n),$ $\ \le \tau (n), \ < n$ (where $\omega(n)$ is the number of distinct prime factors of $n$, $\tau (n)$ is the number of factors of $n$), then $n$ is respectively called strong, weak or very weak alpha number. We give some examples and conjecture that there is no odd strong alpha number of order $(1,1)$. The truthfulness of this assertion implies that there is no odd perfect and certain odd multi-perfect numbers. We give all the strong even alpha numbers of order $(1,1)$ below $10^5$ and then show that there is no odd strong alpha number of order $(1,1)$ below $10^5$, using some of our results motivated by some results of Ore and Garcia. With computer search this bound can easily be surpassed. In this paper, using Rossen, Schonfield and Sandor's inequalities, in addition to the aforementioned definition, we also bound the quotient $\alpha_1/\alpha_2 =\alpha$ of order $(1,1)$, though a very weak bound. Some areas for future research are also pointed out as recommendations.
\end{abstract}
{\bf Keywords:} Perfect numbers, multi-perfect numbers, harmonic divisors number.\\
{\bf 2010 Mathematics Subject Classification:} 11N25, 11Y70.
\section{Introduction}
\qquad Throughout, let $\sigma_x (n)$, $\omega(n)$ and $\tau
(n)$ represent the sum of divisors function of $n$, the number of its distinct prime divisors, and the number of its distinct positive divisors, respectively. We define, for every positive integer $n$ and quaternion $x$, $$\sigma_x (n):=\sum_{d\in\mathbb{N}, d\mid n}d^x$$ and recall the definitions $\omega (n):=\sum_{ p \ prime, \\\ p\mid n}1 \ \ \ and \ \ \tau(n):=\sum_{d\in\mathbb{N}, d\mid n} 1.$
Note that traditionally, when $x=1$, we drop the subscript and simply write $\sigma (n)$ (called the sigma function) which now represents the sum of the factors of $n$ including $n$ itself. For example, the sum of positive divisors of $n=p^{\alpha}$ is $\sigma(p^\alpha)=1+p+...+p^\alpha$ while its $\omega(n)=1$, and $\tau(n)=\alpha +1$. Also, recall that any multiply perfect number $n$ satisfies $\sigma(n)=kn$ (where case $k=2$ specifically defines the perfect numbers such as 6 and 28, detailed in \cite{san1}, \cite{san2}) while any Ore's harmonic number \cite{{ore1}} satisfies $$\sigma(n)=(n\tau(n))/H(n),$$ where $H(n)$ is the harmonic mean of integer $n$. Now, we define the following numbers which extend both the multiply perfect numbers and Ore's harmonic numbers: Let $n\in\mathcal{A}\subset\mathbb{N}$ satisfy \begin{equation}\label{eqn1}
\sigma_{\underline{\alpha}}(n) = \frac{\alpha_1}{\alpha_2}n^{\bar{\alpha}}
\end{equation}
where $(\alpha_1 ,\alpha_2 )$ is a pair of arbitrary but co-prime integers. Then, we call $n$:
\begin{itemize}
\item[(i)]{a strong alpha number} of order $(\underline{\alpha},\bar{\alpha})$” and denote its family by $@_{(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2;\mathcal{A}\subset\mathbb{N}}$, if $n$ further satisfies $2\leq\max ( \alpha_1 ,\alpha_2 )\leq\omega(n)$.
\item[(ii)]{a weak alpha number} of order $(\underline{\alpha},\bar{\alpha})$” and denote its family by $\bar{@}_{(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2;\mathcal{A}\subset\mathbb{N}}$, if $n$ further satisfies $2\leq\omega(n) < \max ( \alpha_1 ,\alpha_2 )\leq\tau(n)$.
\item[(iii)]{a very weak alpha number} of order $(\underline{\alpha},\bar{\alpha})$” and denote its family by $\bar{\bar{@}}_{(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2;\mathcal{A}\subset\mathbb{N}}$, if $n$ further satisfies $2\leq\tau(n) < \max (\alpha_1 ,\alpha_2) < n$.
\end{itemize}
In continuation, suppose $n\in\mathcal{A}\subset\mathbb{N}$ satisfies, instead of (1),
\begin{equation}\label{eqn1}
\lfloor|\sigma_{\underline{\alpha}}(n)|\rfloor = \frac{\alpha_1}{\alpha_2}\lfloor |n^{\bar{\alpha}}|\rfloor
\end{equation}
where $\gcd(\alpha_1 ,\alpha_2)=1$, $\lfloor . \rfloor$ is the usual floor function and $| . |$ is the modulus function. Then, we call $n$:
\begin{itemize}
\item[(i)]{a strongly floored alpha number} of order $(\underline{\alpha},\bar{\alpha})$” and denote its family by $@^{\lfloor \rfloor}_{(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2;\mathcal{A}\subset\mathbb{N}}$, if $n$ further satisfies $2\leq\max(\alpha_1, \alpha_2)\leq\omega(n)$.
\item[(ii)]{a weakly floored alpha number} of order $(\underline{\alpha},\bar{\alpha})$” and denote its family by $\bar{@}^{\lfloor \rfloor}_{(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2;\mathcal{A}\subset\mathbb{N}}$, if $n$ further satisfies $2\leq\omega(n) < \max(\alpha_1, \alpha_2)\leq\tau(n)$.
\item[(iii)]{a very weakly floored alpha number} of order $(\underline{\alpha},\bar{\alpha})$” and denote its family by $\bar{\bar{@}}^{\lfloor \rfloor}_{(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2;\mathcal{A}\subset\mathbb{N}}$, if $n$ further satisfies $2\leq\tau(n) < \max (\alpha_1 ,\alpha_2) < n$.
\end{itemize}
And if $n\in\mathcal{A}\subset\mathbb{N}$ satisfies, instead of (1),
\begin{equation}\label{eqn1}
\lceil|\sigma_{\underline{\alpha}}(n)|\rceil = \frac{\alpha_1}{\alpha_2}\lceil |n^{\bar{\alpha}}|\rceil
\end{equation}
where $(\alpha_1 ,\alpha_2 )=1$, $\lceil . \rceil$ is the usual ceiling function and $| . |$ is the modulus function. Then, we call $n$:
\begin{itemize}
\item[(i)]{a strongly ceiled alpha number} of order $(\underline{\alpha},\bar{\alpha})$” and denote its family by $@^{\lceil \rceil}_{(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2;\mathcal{A}\subset\mathbb{N}}$, if $n$ further satisfies $2\leq\max(\alpha_1, \alpha_2)\leq\omega(n)$.
\item[(ii)]{a weakly ceiled alpha number} of order $(\underline{\alpha},\bar{\alpha})$” and denote its family by $\bar{@}^{\lceil \rceil}_{(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2;\mathcal{A}\subset\mathbb{N}}$, if $n$ further satisfies $2\leq\omega(n) < \max(\alpha_1, \alpha_2)\leq\tau(n)$.
\item[(iii)]{a very weakly ceiled alpha number} of order $(\underline{\alpha},\bar{\alpha})$” and denote its family by $\bar{\bar{@}}^{\lceil \rceil}_{(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2;\mathcal{A}\subset\mathbb{N}}$ if $n$ further satisfies $2\leq\tau(n) < \max(\alpha_1, \alpha_2) < n$.
\end{itemize}
However, if $n\in\mathcal{A}\subset\mathbb{N}$ satisfies $\sigma_{\underline{\alpha}}(n) = \alpha n^{\bar{\alpha}}$ instead of (1), (2) and (3), we call $n$ \textit{a partial alpha number} of order $(\underline{\alpha},\bar{\alpha})$” and denote its family by $@^\star_{(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2;\mathcal{A}\subset\mathbb{N}}$.\\
Note that the major interest yet in number theory is the existence of even and odd of such numbers, as the application of special numbers is still a puzzle, though a golden challenge in Mathematics (see \cite{rao},\cite{san2}, \cite{tat}). Truly, even and odd alpha numbers exist, as we shall soon see various examples in this paper; but no odd strong alpha number, in particular, of order $(1,1)$ is known. So a main conjecture in this paper states that there is no such odd number. The truthfulness of this conjecture implies the odd perfect number conjecture and also implies that there is no certain odd multiply perfect numbers and Ore's harmonic number.
The following are formerly the conjectures which generalize the odd perfect number conjecture and other related conjectures concerning multiply perfect numbers:
\begin{con}
$@_{(1,1); 2\mathbb{N}+1}=\emptyset.$
\end{con}
\begin{con}
Let $N_e(x,n)$, $\bar{N}_e(x,n)$ and $\bar{\bar{N}}_e(x,n)$ respectively count the number of strong, weak and very weak even alpha numbers $n$ that do not exceed real $x$. Then $N_e(x,n)\rightarrow\infty$, $\bar{N}_e(x,n)\rightarrow\infty$, and $\bar{\bar{N}}_e(x,n)\rightarrow\infty$ as $x\rightarrow\infty$.
\end{con}
\begin{con}
Let $\bar{N}_o(x,n)$ and $\bar{\bar{N}}_o(x,n)$ respectively count the number of weak and very weak odd alpha numbers $n$ that do not exceed real $x$. Then $\bar{N}_o(x,n)\rightarrow\infty$ and $\bar{\bar{N}}_o(x,n)\rightarrow\infty$ as $x\rightarrow\infty$.
\end{con}
In what follows, we begin with some examples of @-numbers.
\section{ Examples of @-numbers}
Here, we extract some @-numbers from the table of the sum of positive divisors function given below:
\\
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$n$ & $\sigma_0(n)=\tau(n)$ & $\sigma_1(n)$ & $\sigma_2(n)$ & $\sigma_{0.5}(n)$ & $\sigma_{\sqrt{-1}}(n)$ & $\lfloor |\sigma_{0.5}(n) |\rfloor$ & $\lfloor |\sigma_{\sqrt{-1}}(n) |\rfloor$ \\
\hline
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
2 & 2 & 3 & 5 & 2.4142 & 1.7692+ 0.6390i & 2 & 1 \\
\hline
3 & 2 & 4 & 10 & 2.7321 & 1.4548+ 0.8906i & 2 & 1 \\
\hline
4 & 3 & 7 & 21 & 4.4142 & 1.9527+ 1.6220i & 4 & 2 \\
\hline
5 & 2 & 6 & 26 & 3.2361 & 0.9614+ 0.9993i & 3 & 1 \\
\hline
6 & 4 & 12 & 50 & 6.5959 & 2.0049+ 2.5052i & 6 & 3 \\
\hline
7 & 2 & 8 & 50 & 3.6458 & 0.6336+ 0.9305i & 3 & 1 \\
\hline
8 & 4 & 15 & 85 & 7.2426 & 1.466+ 2.4954i & 7 & 2 \\
\hline
9 & 3 & 13 & 91 & 5.7321 & 0.8686+ 1.7007i & 5 & 1 \\
\hline
10 & 4 & 18 & 130 & 7.8126 & 1.0624+ 2.3822i & 7 & 2 \\
\hline
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\
\hline
24 & 8& 60 & 850 & 19.787 & -0.0899+ 4.936i & 19 & 4 \\
\hline
25 & 3 & 31 & 651 & 8.236 & -0.0356+ 0.922i & 8 & 0 \\
\hline
26 & 4 & 42 & 850 & 11.118 & -0.0623+1.068i & 11 & 1 \\
\hline
27 & 4 & 40 & 820 & 10.928 & -0.1200+ 1.547i & 10 & 1 \\
\hline
28 & 6 & 56 & 1050 & 16.03 & -0.2719+ 2.845i & 16 & 2 \\
\hline
29 & 2 & 30 & 842 & 6.385 & 0.0025-0.224i & 6 & 0 \\
\hline
30 & 8 & 72 & 1300 & 21.344 & -0.5759+ 4.412i & 21 & 4 \\
\hline
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\
\hline
\end{tabular}
\begin{center}
Table 1.
\end{center}
\textbf{Examples of strong alpha numbers of order }$(1,1)$:\\
\\
\begin{tabular}{|c|c|c|c|c|}
\hline
$@_{(1,1)}$ & $\sigma_1(n)$ & $\alpha_1$ & $\alpha_2$ & $\omega(n)$ \\
\hline
$n = 6=2\cdot 3$& 12 & 2 & 1 & 2 \\
\hline
$n = 28=2^2\cdot 7$& 56 & 2 & 1 & 2 \\
\hline
$n = 523776 =2^9\cdot 3\cdot 11\cdot 31$ & 1571328 & 3 & 1 & 4 \\
\hline
$n = 707840=2^8\cdot 5\cdot 7\cdot 79$ & 1962240 & 3 & 1 & 4 \\
\hline
\end{tabular}
\begin{flushleft}
Table 2.
\end{flushleft}
\textbf{Examples of weak alpha numbers of order }$(1,1)$:\\
\\
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$\bar{@}_{(1,1)}$ & $\sigma_1(n)$ & $\alpha_1$ & $\alpha_2$ & $\omega(n)$ & $\tau(n)$ \\
\hline
$n = 24=2^3\cdot 3$ & 60 & 5 & 2 & 2 & 8 \\
\hline
$n = 11172 = 2^2\cdot 3\cdot 7^2\cdot 19$ & 31920 & 20 & 7 & 4 & 36 \\
\hline
$n = 544635^\ast = 3^2\cdot 5\cdot 7^2\cdot 13\cdot 19$ & 1244860 & 16 & 7 & 5 & 72 \\
\hline
$n = 931095^\ast = 3^4\cdot 5\cdot 11^2\cdot 19$ & 1931160 & 56 & 27 & 4 & 60 \\
\hline
$n = 6517665 = 3^4\cdot 5\cdot 7\cdot 11^2\cdot 19$ & 15449280 & 64 & 27 & 5 & 120 \\
\hline
\end{tabular}
\begin{flushleft}
Table 3.
\end{flushleft}
\begin{tiny}
Thanks to Professor J. Shallit for pointing our attention (in a private communication) to the examples of odd alpha numbers in asterisk on Table 3 which enabled us to refine the conjecture of this paper concerning the existence of odd alpha numbers.
\end{tiny}
\section{Results on @-numbers:}
Then, we proceed to offer the following results which were inspired by the work of Garcia \cite{Ga} and Ore \cite{ore2}, and which shall find application later in searching for alpha numbers.
\begin{theorem}\label{t1} All perfect, multiply perfect and Ore's harmonic number are alpha numbers of order $(1,1)$.
\end{theorem}
\begin{proof} Let a positive integer $n$ be $k$-fold multi perfect if $\sigma(n)=kn$. Also observe that $\sigma(n)< nn$ for every integer $n>1$, since $\tau(n)<n$ and each $d_i\mid n$ is such that $1\le d_i \le n$ and $n>\min (d_1, ... d_z)$. Thus there is no $n$-fold multi perfect number. So,
Theorem 3.1 is an implication of the definitions of multiply perfect numbers, Ore's harmonic numbers and alpha numbers since every multiply perfect number and Ore's harmonic number satisfy (1).
\end{proof}
\begin{theorem}\label{t2} A power of prime can neither be strong, weak, nor very weak alpha number of order $(1,1)$.
\end{theorem}
\begin{proof}It suffices to show that there is no $n=p^\alpha$ satisfying $\dfrac{\sigma(n)}{n}=\dfrac{\alpha_1}{\alpha_2}$ where arbitrary integers $\alpha_1$ and $\alpha_2$ are co-prime, and $\alpha_1< n$. Note that, $\sigma(p^\alpha)=1+p+...+p^\alpha$ is co-prime to $p$, so to $p^\alpha$, therefore the fraction $\sigma(n)/n$ is reduced. Should this equal $\alpha_1/\alpha_2$, then $\sigma(n)$ divides $\alpha_1$. In turn, this leads to $\sigma(p^\alpha)=1+p+...+p^\alpha\le p^\alpha$, a contradiction.
\end{proof}
\begin{theorem}\label{t3} No square-free odd integer is a strong or weak alpha number of order $(1,1)$.
\end{theorem}
\begin{proof} We show that $\max(\alpha_1, \alpha_2) \not<\tau(n)$ for any square-free odd integer $n$ satisfying (1).
On the contrary, suppose a positive odd integer $n$ of the canonical prime factorisation form $n=\prod_{i=1}^{\omega(n)} p_i$ satisfies (1), then $$\alpha_2=\frac{\alpha_1\prod_{i=1}^{\omega(n)} p_i}{\sigma(\prod_{i=1}^{\omega(n)} p_i)}=\frac{\alpha_1\prod_{i=1}^{\omega(n)} p_i}{\prod_{i=1}^{\omega(n)} \sigma(p_i)}=\frac{\alpha_1\prod_{i=1}^{\omega(n)} p_i}{2^{\omega(n)}\prod_{i=1}^{\omega(n)} q_i}$$
where each $q_i$ is an integer greater than $1$, since $\sigma$ is multiplicative. Now that $\alpha_1$ and $\alpha_2$ are integers and each $p_i$ is an odd prime, it is necessary that $2^{\omega(n)} \mid \alpha_1$. Moreover, the $\min\lbrace q_1, q_2,...q_{\omega(n)}\rbrace$ must divide $\alpha_1$ since $1<\min\lbrace q_1, q_2,...q_{\omega(n)}\rbrace<\min\lbrace p_1, p_2,...p_{\omega(n)}\rbrace$. Thus, $\alpha_1> \tau(n)=2^{\omega(n)}$, a contradiction. Hence, Theorem 3.3.
\\
\end{proof}
An alternative proof of Theorem 3.3 can again proceed using the multiplicative property of sigma-function from $$\alpha_1=\frac{\alpha_2\sigma(\prod_{i=1}^{\omega(n)} p_i) }{\prod_{i=1}^{\omega(n)} p_i}=\alpha_2\cdot 2^{\omega(n)}\frac{\prod_{i=1}^{\omega(n)}\frac{\sigma(p_i)}{2}}{\prod_{i=1}^{\omega(n)} p_i}$$ where one could infer that each $p_i\mid \alpha_2$ or $p_i\mid {\prod_{i=1}^{\omega(n)}{\sigma(p_i)}/{2}}\in\mathbb{N}$, by Euclid's lemma, and specifically $$\max \lbrace p_1, p_2,...p_{\omega(n)}\rbrace\nmid \prod_{i=1}^{\omega(n)}{\sigma(p_i)}/{2}\in\mathbb{N}.$$ Hence $\max \lbrace p_1, p_2,...p_{\omega(n)}\rbrace$ must divide $\alpha_2$ since $\alpha_1$ is an integer. Thus implies that $\alpha_1 > 2^{\omega(n)}=\tau(n)$, also a contradiction.
\begin{theorem}\label{t4} There is no strong odd alpha number $n$ of order $(1,1)$ in the canonical form $\prod_{i=1}^{\omega(n)}p_i^{n_i}$, with each $n_i\ge 2$ and each $\sigma(p_i^{n_i})$ prime.
\end{theorem}
\begin{proof}
It is sufficient to show that $\alpha_2 >\omega(n)$ for such $n$. Now, suppose contrary that such odd $n$ exists with its attached condition $n_i\ge 2$ and $\sigma(p_i^{n_i})$ is prime for each $i$. Thus
\begin{equation}
|\lbrace p_1, p_2, ..., p_{\omega(n)} \rbrace|= | \lbrace \sigma(p_1^{n_1}), \sigma(p_2^{n_2}), ..., \sigma(p_{\omega(n)}^{n_{\omega(n)}}) \rbrace|,
\end{equation} where $|X|$ represent the cardinality of set $X$. Now by condition (4) and the fact that $\omega(\sigma(p_i^{n_i}))=1$ implies $\omega(n)= \omega(\sigma(n))$, $n$ through (1), satisfies
\begin{equation}
\alpha_2=\alpha_1\frac{\prod_{i=1}^{\omega(n)}p_i^{n_i}}{\prod_{i=1}^{\omega(n)}\sigma(p_i^{n_i})}\ge \min \lbrace p_1, ..., p_{\omega(n)}\rbrace^{\Omega(n) -\omega(\sigma(n))} \ge 3^{\Omega(n) -\omega(\sigma(n))},
\end{equation}
where $\Omega(n):=\sum_{ p \ prime, \\\ p^v\mid n}1$.\\
Thus (5) implies $$\alpha_2 \ge 3^{\Omega(n) -\omega(n)} \ge 3^{\omega(n)} >\omega(n),$$
since $n_i\ge 2 \ \forall \ i$. Thus yield the required contradiction.\\
\end{proof}
An alternative approach to establishing Theorem 3.4 is to consider $n_i\ge 2 \ \forall \ i$ which implies $\Omega(n)-\omega(n)\ge \frac{\Omega(n)}{2}$. This further implies $$\log_3\prod_{i=1}^{\omega(n)}3^{n_i -1}\ge \log_3\prod_{i=1}^{\omega(n)}3^{\frac{n_i}{2}}\ge \log_3\prod_{i=1}^{\omega(n)}n_i +1.$$
One can then conclude through
$$\alpha_2=\alpha_1\frac{\prod_{i=1}^{\omega(n)}p_i^{n_i}}{\prod_{i=1}^{\omega(n)}\sigma(p_i^{n_i})}\ge 3^{\Omega(n) -\omega(n)} \ge \prod_{i=1}^{\omega(n)}3^{n_i -1}\ge\prod_{i=1}^{\omega(n)}3^{\frac{n_i}{2}}\ge \prod_{i=1}^{\omega(n)}n_i +1 =\tau(n)$$ which yields a contradiction, since necessarily $\max(\alpha_1, \alpha_2) < \omega(n)$.\\
\textbf{Remark}: An improvement of Theorem 3.4 is to exclude the condition $\sigma(p_i^{n_i})$ prime for every $i$ in the the statement of the theorem.
\begin{theorem}\label{t5} Except for perfect numbers, no integer $n$ with prime factor decomposition $p_1^\alpha p_2$ is a strong alpha number of order $(1,1)$.
\end{theorem}
\begin{proof} Let $n=p_1^\alpha p_2$ with distinct primes $p_1$ and $p_2$ be a strong alpha number of order $(1,1)$. Then there exist $\alpha_1, \alpha_2\in\mathbb{Z}^+$ with $(\alpha_1, \alpha_2)=1$ and $2\le \max(\alpha_1, \alpha_2)\le \omega(n)=2$ such that
\begin{equation}
\alpha_2=\frac{\alpha_1 \cdot p_1^\alpha\cdot p_2}{(p_1^\alpha +...+p_1 +1)(p_2+1)}.
\end{equation}
Then we consider the possibilities $p_2=2$ and $p_2\not=2$:\\
For case $p_2=2$, it is necessary that $$\alpha_2=\frac{2\alpha_1\cdot p_1^\alpha}{3(p_1^\alpha +...+p_1 +1)}$$ and $p_1^\alpha +...+p_1 +1\mid 2\alpha_1$ since $(p_1^\alpha +...+p_1 +1, p_1^\alpha)=1.$ Thus follows that $p_1^\alpha +...+p_1 +1\le 2\alpha_1\le 4$ since $\alpha_1$ is $2$ at maximum. But $p_1^\alpha +...+p_1 +1\geq 3^\alpha +...+3 +1\not<4$. Thus necessarily $\alpha=1$, $p_1=3$, and consequently $n=6$, a perfect number.\\
For case $p_2\not=2$, from (6), we write
\begin{equation}
\alpha_2=\frac{p_1^\alpha\cdot p_2}{(p_1^\alpha +...+p_1 +1)((p_2+1)/\alpha_1)}
\end{equation} and notice that the fact that $\sigma(p_1^\alpha p_2) > p_1^\alpha p_2$ implies, through (6), that $1\le \alpha_2< \alpha_1\le 2$, hence $\alpha_1=2$ and $\alpha_2=1$.\\
Equation (7) thus implies
\begin{equation}
p_2= p_1^\alpha +...+p_1 +1,
\end{equation}
since $p_2$ should necessarily factor $p_1^\alpha +...+p_1 +1$ while $p_1^\alpha +...+p_1 +1$ also divides $p_2$, a consequence of the fact that $(p_1^\alpha +...+p_1 +1, p_1^\alpha)=1$ and $(({p_2+1})/{2}, p_2)=1$.
This further implies
$$ p_1^\alpha =\frac{p_2+1}{2} =\frac{p_1^\alpha +...+p_1 +2}{2}\ge \frac{3^\alpha +...+ 3 +2}{2},$$ since $\alpha_2=1$ for (7). Thus $p_1\mid 2$, since necessarily $p_1\mid ({p_1^\alpha +...+p_1 +2})/{2}$. Thus implies $p_1=2$. Recalling Euclid's theorem which states that every number of the form $2^k\sigma(2^k)$ is perfect if $\sigma(2^k)$ is prime, it therefore follows that $n=p_1^\alpha p_2$ is a perfect number. Therefore, Theorem 3.4 holds.
\\
\end{proof}
The next result under this section, although only provides a weak bound on the quotient $\alpha_1/\alpha_2$ of any alpha number $n$, is a direct consequence of the following important Lemmas connecting arithmetic functions such $\phi(n)$ and $\sigma(n)$:
\begin{lemma} \emph{(J.B. Rosser and L. Schenfield \cite{ros})}\\
If $n\geq 3$, then $\frac{n}{\phi(n)}<e^\gamma\log\log n +\frac{0.6483}{\log\log n}$ where $\gamma$ is the \emph{Euler constant}.
\end{lemma}
\begin{lemma} \emph{(J. s\'{a}ndor \cite{san})}\\
There is a constant $C>0$ such that $\frac{n}{\phi(n)}<C\cdot\log\log\phi(n) \ \forall \ n>3.$
\end{lemma}
\begin{theorem}\label{t6} For any alpha number $n\ge 3$ of order $(1,1)$, $${\alpha_1}/{\alpha_2} < e^\gamma\log\log n +\frac{0.6483}{\log\log n},$$
where $\gamma$ is the \emph{Euler constant}, and moreover, there exists a constant $C>0$ such that $${\alpha_1}/{\alpha_2}< C\log\log \phi(n)$$.
\end{theorem}
\begin{proof}
It is sufficient to show that $\frac{\sigma(n)}{n}<\frac{n}{\phi(n)}$ using Euler product formula $\phi(n)=n\prod_{p\mid n}(1-\frac{1}{p})$ and $\sigma(n=\prod_{i=1}^{\omega(n)}p_i^{n_i})=\prod_{i=1}^{\omega(n)}\frac{p^{n_i+1}-1}{p-1}$, and then applying Lemma 3.6 and Lemma 3.7 into the definition of alpha numbers.
\\
\end{proof}
\par
\noindent
\textbf{Remark :} We remark that the above bound is very weak for strong alpha number, since quotient ${\alpha_1}/{\alpha_2}\le \omega(n)$ necessarily, so it is presented for further study.
\begin{theorem}\label{t7} Any strong alpha number $n$ of order $(\underline{\alpha}\in\mathbb{N},\bar{\alpha}\in\mathbb{N})$ with $\omega(n)\le 2$ cannot be a perfect square.
\end{theorem}
\begin{proof}
Case $\omega(n)=1$ is a direct implication of Theorem 3.2, so we only need to establish that for $\omega(n)=2$, $n=p_1^{2x}p_2^{2y}$ with distinct primes $p_1$ and $p_2$ does not satisfy
\begin{equation}
\alpha_2=\frac{\alpha_1 \cdot p_1^{2\bar{\alpha}x}\cdot p_2^{2\bar{\alpha}y}}{\sigma_{\underline{\alpha}}(p_1^{2x})\sigma_{\underline{\alpha}}(p_2^{2y})},
\end{equation}
where $\alpha_1=2$ and $\alpha_2=1$ (that is, $n$ contradicts the definition of strong alpha numbers). If on the contrary, equation (9) holds, then $n$ must satisfy $2p_1^{2\bar{\alpha}x}=\sigma{\underline{\alpha}}(p_2^{2y})$ and $p_2^{2\bar{\alpha}y}=\sigma{\underline{\alpha}}(p_1^{2x})$, or $2p_2^{2\bar{\alpha}y}=\sigma{\underline{\alpha}}(p_1^{2 x})$ and $p_1^{2\bar{\alpha}x}=\sigma{\underline{\alpha}}(p_2^{2y})$, since $(p_1^{2\bar{\alpha}x}, \sigma{\underline{\alpha}}(p_1^{2x}))=1$ and $(p_2^{2\bar{\alpha}y}, \sigma{\underline{\alpha}}(p_2^{2y}))=1$. But this contradicts the fact that both $\sigma{\underline{\alpha}}(p_1^{2x})$ and $\sigma{\underline{\alpha}}(p_2^{2y})$ are odd. Hence Theorem 3.9.
\\
\end{proof}
\vspace{0.5cm}
\begin{theorem}\label{t8} $@_{(\underline{\alpha}\in\mathbb{R}_{>0},\bar{\alpha}\in\mathbb{R}_{>\underline{\alpha}+2});\mathbb{N}}=\emptyset$.
\end{theorem}
\begin{proof} On the contrary, let $n\in@_{(\underline{\alpha}\in\mathbb{R}_{>0},\bar{\alpha}\in\mathbb{R}_{>2 +\underline{\alpha}});\mathbb{N}}\not=\emptyset$ such that $n$ with unique prime decomposition $\prod_{i=1}^{\omega(n)}p_i^{n_i}$ satisfies (1) with $2\le\max (\alpha_1,\alpha_2)\le \omega(n)$. Thus, in order to establish Theorem 3.10, it is sufficient to show that
\begin{equation}
\prod_{i=1}^{\omega(n)}\sigma_{\underline{\alpha}}(p_i^{n_i})\le\prod_{i=1}^{\omega(n)}p_i^{(n_i\bar{\alpha}-1)} \ \forall \ \ i,
\end{equation}
if $\underline{\alpha}\in\mathbb{R}_{>0}$ and $\bar{\alpha}\in\mathbb{R}_{>\underline{\alpha}+2}$, since $\sigma_x(n)$ is a multiplicative function and $n$ necessarily satisfies $\prod_{i=1}^{\omega(n)}p_i >\omega(n)$ and
\begin{equation}
\alpha_2=\alpha_1\frac{\prod_{i=1}^{\omega(n)}p_i^{(n_i\bar{\alpha}-1)} \prod_{i=1}^{\omega(n)}p_i}{\sigma_{\underline{\alpha}}(\prod_{i=1}^{\omega(n)}p_i^{n_i})}.
\end{equation}
Then, we have to verify the following assertion which implies (10):
\begin{equation}
\sigma_{\underline{\alpha}}(p_i^{n_i})\le p_i^{(n_i\bar{\alpha}-1)} \ \forall \ i
\end{equation}
where $\underline{\alpha}\in\mathbb{R}_{>0}$ and $\bar{\alpha}\in\mathbb{R}_{>\underline{\alpha}+2}$.
Clearly, by the definition of $\tau(.)$ and $\sigma_x(.)$, $$\sigma_{\underline{\alpha}}(p_i^{n_i})<p_i^{n_i\underline{\alpha}}\cdot \tau(p_i^{n_i}),$$ and obviously, $$p_i^{n_i\underline{\alpha}}\cdot \tau(p_i^{n_i})<p_i^{n_i\underline{\alpha}}\cdot 3^{n_i}\le p_i^{n_i\underline{\alpha}}\cdot p_i^{2n_i-1}.$$ Thus, setting $\bar{\alpha}=\underline{\alpha}+ 2$ in (12) implies (10). Consequently $\alpha_2 > \omega(n)$, a contradiction. Therefore Theorem 3.10 follows.\\
\end{proof}
In what follows, we formally include the following propositions without proof:\\
\begin{proposition}
Let $N(x)$ and $N^\star(x)$ respectively count the number of strong and partial alpha numbers that do not exceed real $x$. Then $N^\star(x) \gg N(x)$ for a sufficiently large $x$.
\end{proposition}
\begin{proposition}
Let $N^{\lfloor \rfloor}(x)$, $N^{\lceil \rceil}(x)$ and $N^\star(x)$ respectively count the number of strongly floored, strongly ceiled and partial alpha numbers that do not exceed real $x$. Then $N^\star(x) \gg N^{\lfloor \rfloor}(x)$ and $N^\star(x)\gg N^{\lceil \rceil}(x)$ for a sufficiently large $x$.
\end{proposition}
\begin{proposition}
Let F-alpha number, be any alpha number (strong, weak or very weak) such that the ordered pair $(\alpha_1, \alpha_2)$ in the frame $\alpha_2[\sigma_{\underline{\alpha}}(n)] = {\alpha_1}[n^{\bar{\alpha}}]$ is a pair of invertible functions $(\mathfrak{f}_{\underline{\alpha}}(n), (\mathfrak{f}_{\bar{\alpha}}(n))$, then with $\mathfrak{f}_{\underline{\alpha}}(n)$ and $\mathfrak{f}_{\bar{\alpha}}(n)$ as secret keys and $n$ itself a product of public primes, a more secured encryption and decryption can be made feasible.
\end{proposition}
\section{ \textbf{@- numbers below $10^5$}}
Here, we first obtain the even strong alpha numbers of order $(1,1)$ below $10^5$. In order to achieve this goal, we shall make use of an approach similar in nature to that of Ore \cite{ore2} and Garcia \cite{Ga}. By the fundamental theorem of arithmetic and definition of alpha numbers, it is necessary to consider $$n=2^\lambda x <10^5 : x = p_2^{n_2}p_3^{n_3}\cdots p_{\omega(n)}^{n_{\omega(n)}}$$ in the frame \begin{equation}
\alpha_2=\alpha_1\cdot \frac{n}{\sigma(n)}, \ \ 2 \le \max (\alpha_2, \alpha_1 )\le\omega(n),
\end{equation} where $p_2, p_3, ..., p_{\omega(n)}$ are distinct odd primes. By virtue of Lemmas 2, 3, 4, and 5, $\lambda \leq 13$ with $\omega(2^{13}x <10^5)< 4$. So we consider the possibility $2^{13}\mid n$ in (2): $$\alpha_2=\alpha_1\cdot \frac{2^{13}x}{3\cdot 43\cdot 127\sigma(x)}.$$
But again, the fact that $\alpha_1\leq \omega(2^{13}x <10^5)<127$ strictly implies that $127$ must divide $n$, and further implies that $n\geq 2^{13}\cdot 127>100000$, a contradiction. Thus, there is no even strong alpha number of order $1$ divisible by $2^{13}$. Continuing this way, the first $\lambda$ such that $2^\lambda\mid n < 100000$ is $6$, this gives us $$\alpha_2 =\alpha_1\cdot \frac{2^{6}x}{127\sigma(x)}.$$ Then, since $\alpha_1\leq \omega(2^{6}x<10^5)<127$ and $2^6\cdot 127^2 > 100000$, it is only possible that $127\mid n$ such that $$\alpha_2 =\alpha_1\cdot \frac{2^{6}\cdot 127y}{127\cdot 2^7\sigma(y)}=\alpha_1\cdot\frac{y}{2\sigma(y)}.$$
Now we are left with two possibilities: either integer $y=1$ such that strictly $n=2^{6}\cdot 127,$ $\alpha_1=2$ and $\alpha_2=1$, or odd $y\ge 3$, $n>2^{6}\cdot 127,$ and $2\mid \alpha_1$. The later case implies that $\alpha_1$ satisfies the following:
\begin{align*}\label{E:1}
\alpha_1
&=2\alpha_2\cdot \frac{\sigma(y)}{y} : \alpha_2\in\mathbb{N},\\
\alpha_1
& =2t , \ t=1, 2, 3, ...,\\
\\
\alpha_1
& <\omega(n=2^6 \cdot 127y<10^5)< 4, \ \textit{since} \ 2^6\cdot 127\cdot 3\cdot 5 >10^5.\\
\end{align*}
Thus implies that $\alpha_1\not>2$. Hence $y=1$, and so $n=2^{6}\cdot 127$ is an alpha number with $\alpha_1=2$ and $\alpha_2=1$. Repeating the above procedure for $\lambda = 5, 4, 3, 2$ and $\lambda=1$ yields the following even strong alpha numbers of order $(1,1)$.\\
\\
\begin{tabular}{|c|c|c|c|c|}
\hline
$@_{(1,1)}$ & $\sigma(n)$ & $\alpha_1$ & $\alpha_2$ & $\omega(n)$ \\
\hline
$n = 6=2\cdot 3$& 12 & 2 & 1 & 2 \\
\hline
$n = 28=2^2\cdot 7$& 56 & 2 & 1 & 2 \\
\hline
$n = 120 =2^3\cdot 3\cdot 5$ & 360 & 3 & 1 & 3 \\
\hline
$n = 496=2^4\cdot 31$ & 992 & 2 & 1 & 2 \\
\hline
$n = 672=2^5\cdot 3\cdot 7$ & 2016 & 3 & 1 & 3 \\
\hline
$n = 1090=2^3\cdot 3\cdot 5\cdot 7\cdot 13$ & 40320 & 4 & 1 & 5 \\
\hline
$n = 8128=2^6\cdot 127$ & 16256 & 2 & 1 & 2 \\
\hline
$n = 30240=2^5\cdot 3^3\cdot 5\cdot 7$ & 120960 & 4 & 1 & 4 \\
\hline
$n = 32760 =2^3\cdot 3^2\cdot 13\cdot 7\cdot 5$ & 131040 & 4 & 1 & 5 \\
\hline
\end{tabular}
\begin{flushleft}
Table 3.
\end{flushleft}
\textbf{Remark}: We observed that all the strong even alpha numbers between $1$ and $100000$ as recorded in the above table are purely multiply perfect numbers, so the first non-multiply perfect strong even alpha number will be $\ge 10^5$, and will be an interesting example of alpha numbers.
\\
\textbf{Odd alpha numbers of order $(1,1)$ below $10^5$.}
\begin{theorem}
There is no strong odd alpha number of order $(1,1)$ below $10^5$.
\end{theorem}
Our strategy here is to make the above method for obtaining even strong alpha numbers rigorous, and to achieve this goal, we first define $\omega_B(x):= \omega(x <B)$, and then continue by introducing the following function to conveniently extract the prime-power divisors of any positive integer, say $y$, bounded by $B>0$:\\
\begin{align*}
{\mathit{f}}_i(y)|_B
&:=\left\{
\begin{array}{ll}
y, \ \emph{ if } \ y \ \emph{is} \ \emph{prime} \ \emph{ and } \ y > {\omega_B(yx)}.\\
\\
p_j^{n_j}: j= (i-1) + k, \ p_k = \min (p_1,p_2,...,p_{\omega(y)}) > {\omega_B(p_i^{n_i}x)} \ \emph{where} \ y=\prod_{i=1}^{\omega(x)} p_i^{n_i} \\ \emph{with} \ p_i < p_{i+1} \ \forall \ i<\omega(x).\\
\\
1, otherwise.\\
\end{array}
\right.\\
\end{align*}
Then, we define the special integers that are potential alpha numbers not greater than bound $B>0$, in that they satisfy certain conditions of alpha numbers and can further be tested. These set of integers shall henceforth be regarded as `\textit{virtual alpha number}'.\\
\\
\textbf{Definition.} Let `$\mid\mid $' be such that $p^y\mid\mid x\Rightarrow p^y\mid x$ but $p^{y+1}\nmid x$. Further let $p_{\star}$ be prime such that $p_{\star}^{\lambda_\star}\mid \mid u\in\mathbb{N}\Rightarrow u\mid \bar{n}\in\mathbb{N}$, and if another prime $p_i\mid \bar{n}$, prime $p_\star<p_i$, then, we say that $$u=p_{\star}^{\lambda_\star}\cdot \prod_ {i\in I=\lbrace 1, ... , k\rbrace}({\mathit{f}}_i(\sigma(p_{\star}^{\lambda_\star})))^{\alpha_i}\le B$$
is the alpha seed of length $k$ of \textit{virtual alpha number} $\bar{n}= u\prod_{({f}_j(\sigma(u)),u)=1} {\mathit{f}}_j(\sigma(u))$ bounded by $B$ where $p_{\star}^{\lambda_\star}$ is the \textit{generator} of $u$.\\
\\
Thus alpha number $n$ relates directly with virtual alpha number $\bar{n}$, in one and only one way, as follows: $n\ge\bar{n}$. Hence, every $n<\bar{n}$ is clearly not an alpha number.
Thus (regarding every virtual alpha number $\bar{n}$ that make alpha number $n$ $:n=\bar{n}\prod p$ as being transformed), the following characteristic function is necessary and sufficient to determine the virtual alpha numbers $\bar{n}$ bounded by real $B>0$ that transformed to alpha number $n$.
\begin{align*}
\chi_\alpha(\bar{n})
&:=\left\{
\begin{array}{ll}
0; \ \emph{if it is necessary that} \ p_i^k\mid n \ \emph{where} \ p_i<p_\star \ \emph{and} \ k\ge 1, \ \emph{or if} \ \bar{n}>B.\\
\\
1; \ \emph{if} \ \emph{otherwise}.\\
\end{array}
\right.
\end{align*}
\\
\textbf{Proof of Theorem 4.1}
\begin{proof} On the contrary, suppose such odd alpha number $n<10^5$ exists. Then, by virtue of Theorem 3.2 to Lemma 3.5, there exists alpha seeds of generator $p_\star^{\lambda_\star}$ with $p_\star\in\lbrace 3, 5, 7, 11, 13\rbrace$, since $10^5<17^2\cdot 19\cdot 23< 17^2\cdot 19^3 <17\cdot 19^2\cdot 23 < ...$ Thus, we have the following table:
\\
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$p_\star$ & $3$ & $5$&$7$ &$11$ &$13$ \\
\hline
$\lambda_\star$ & $7$ & $4$ & $3$ &$2$ &$2$\\
\hline
$\omega_{10^5}(p_\star^{\lambda_\star}x)$ & $3$ & $3$ & $3$ &$3$ &$3$ \\
\hline
\end{tabular}
\\
\begin{flushleft}
Table 4.
\end{flushleft}
\end{center}
To make our work easier, we first obtain the sum of divisors of some higher powers of primes that are involved in our investigation:
\begin{center}
\begin{tabular}{|c|c|}
\hline
$x$ & $\sigma(x)$ \\
\hline
$3^7$ & $2^4\cdot 541$ \\
\hline
$3^6$ & $1093$ \\
\hline
$3^5$ & $2^2\cdot 7\cdot 13$ \\
\hline
$3^4$ & $11^2$ \\
\hline
$3^3$ & $2^3\cdot 5$ \\
\hline
$3^2$ & $13$ \\
\hline
$5^5$ & $3^3\cdot 7\cdot 31$ \\
\hline
$5^4$ & $11\cdot 71$ \\
\hline
$5^3$ & $2^2\cdot 3\cdot 13$ \\
\hline
$5^2$ & $ 31$ \\
\hline
$7^4$ & $2801$ \\
\hline
$7^3$ & $2^4\cdot 5^2$ \\
\hline
$7^2$ & $3\cdot 19$ \\
\hline
$11^2$ & $7\cdot 19$ \\
\hline
$13^3$ & $2\cdot 5\cdot 7\cdot 17$ \\
\hline
$13^2$ & $3\cdot 61$ \\
\hline
$19^2$ & $3\cdot 127$\\
\hline
$31^2$ & $3\cdot 331$\\
\hline
\end{tabular}
\begin{flushleft}
Table 5.
\end{flushleft}
\end{center}
Then we construct, using Table 5 and equation (3), a consequence of (1),the strong alpha seeds of every odd integer $<10^5$.\\ In the table, we denote the set of alpha seeds of length $2$ and bound $B>0$, by \textit{$\mathit{U}(p_{\star}^{\lambda_\star}, B, 2):=\lbrace u: u<B \ \emph{is an alpha seed}\rbrace.$ Note also that the letters $n$ and $nn$ within the table mean `none' and `not necessary' respectively}\\
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$p_{\star}^{\lambda_\star}$ & ${f}_1(\sigma(p_{\star}^{\lambda_\star}))=s_1$& ${f}_2(\sigma(p_{\star}^{\lambda_\star}))=s_2$ &${f}_1(\sigma(s_1))$ &${f}_2(\sigma(s_1))$& ${f}_1(\sigma(s_2))$ & $\mathit{U}(p_{\star}^{\lambda_\star}, 10^5)$ \\
\hline
$3^7$ & $541$& $nn$ &$nn$ &$nn$ &$nn$ & $\lbrace \rbrace$ \\
\hline
$3^6$& $1093$ & $nn$ &$nn$ &$nn$ &$nn$ &$\lbrace \rbrace$ \\
\hline
$3^5$& $7$ & $13$ & $n$ &$n$ &$7$ & $\lbrace 3^5\cdot 7\cdot 13\rbrace$ \\
\hline
$3^4$& $11^2$ & $n$ & $7$ & $19$ & $nn$ & $\lbrace \rbrace$ \\
\hline
$3^3$& $5$ & $n$ &$n$ &$n$ &$n$ & $\lbrace 3^3\cdot 5, 3^3\cdot 5^2, $ \\
& & & & & & $ 3^3\cdot 5^3, 3^3\cdot 5^4, $ \\
& & & & & & $ 3^3\cdot 5^5 \rbrace$ \\
\hline
$3^2$& $13$ & $n$ & $7$ &$n$ &$n$ &$\lbrace 3^2\cdot 13\cdot 7, $ \\
& & & & & & $ 3^2\cdot 13\cdot 7^2, $ \\
& & & & & & $ 3^2\cdot 13\cdot 7^3, $ \\
& & & & & & $ 3^2\cdot 13^2\cdot 7, $ \\
& & & & & & $ 3^2\cdot 13^2\cdot 7^2 \rbrace$ \\
\hline
$^\star 3$ & $n$ &$n$ &$n$ &$n$ & $n$& \\
\hline
$5^4$& $11$ & $71$ & $n$& $n$& $n$& $\lbrace \rbrace$ \\
\hline
$5^3$& $13$ & $n$ & $7$ &$n$ & $n$& $\lbrace5^3\cdot 13\cdot 7, $\\
& & & & & & $ 5^3\cdot 13\cdot 7^2\rbrace$ \\
\hline
$5^2$& $31$ & $n$ & $n$&$n$ &$n$ & $\lbrace5^2\cdot 31, 5^2\cdot 31^2 \rbrace$ \\
\hline
$^\star 5$& $n$ & $n$ &$n$ &$n$ &$n$ & \\
\hline
$7^3$& $5^2$ & $n$ & $11$ & $n$ &$nn$ & $\lbrace \rbrace$ \\
\hline
$7^2$& $19$ & $n$ & $5$ & $n$ & $n$& $\lbrace \rbrace$ \\
\hline
$^\star 7$ & $n$& $n$ &$n$ &$n$ &$n$ & \\
\hline
$11^2$& $7$ & $19$ &$n$ & $n$& $5$& $\lbrace \rbrace$ \\
\hline
$^\star 11$ & $n$& $n$ &$n$ &$n$ &$n$ & \\
\hline
$13^2$& $61$ & $n$ & $31$ &$n$ &$nn$ & $\lbrace \rbrace$ \\
\hline
$^\star 13$ & $7$ & $n$ & $n$&$n$ &$n$ & \\
\hline
\end{tabular}
\\
\begin{flushleft}
Table 6.
\end{flushleft}
\vspace{1cm}
For the sake of small available space, we omitted column ${f}_1(\sigma(s_2))$ (which, of course, should be filled with 'nn' through). We also considered $p_\star^{\lambda_\star}$ in asterisk at the end of Table 5.
The next table is actually meant to test the virtual alpha numbers that transformed to alpha number:\\
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
${u}$ & $\prod_{({f}_j(\sigma(u)),u)=1} {f}_j(\sigma(u))$ & $\bar{n}=u\prod {f}_j(\sigma(u))$ & $2^m\mid n$ or $3^w\mid n$ & $\chi_\alpha(\bar{n})$ \\
\hline
$3^5\cdot 7\cdot 13$ & $$ & $$ & $m>0$ & $0$ \\
\hline
$3^3\cdot 5$ & $$ & $$ & $m>0$ & $0$ \\
\hline
$3^3\cdot 5^2$ & & $$ & $m>0$ & $0$ \\
\hline
$3^3\cdot 5^3$ & & $$ & $m>0$ & $0$ \\
\hline
$3^3\cdot 5^4$ & $11\cdot 71$ & $>10^5$ & $m>0$ & $0$ \\
\hline
$3^3\cdot 5^5$ & $7\cdot 31$ & $>10^5$ & $m>0$ & $0$ \\
\hline
$3^2\cdot 13 \cdot 7$ & $$ & $$ & $m>0$ & $0$ \\
\hline
$3^2\cdot 13 \cdot 7^2$ & $$ & $$ & $m>0$ & $0$ \\
\hline
$3^2\cdot 13 \cdot 7^3$ & $5^2\cdot31$ & $>10^5$ & $m>0$ & $0$ \\
\hline
$3^2\cdot 13^2\cdot 7$ & $61\cdot 31$ & $>10^5$ & $m>0$ & $0$ \\
\hline
$3^2\cdot 13^2\cdot 7^2$ & $19\cdot 61\cdot 5$ & $>10^5$ & $m>0$ & $0$ \\
\hline
$5^3\cdot 13\cdot 7$ & $$ & & $m>0$ & $0$ \\
\hline
$5^3\cdot 13\cdot 7^2$ & $19$ & $>10^5$ & $m>0$ & $0$ \\
\hline
$5^2\cdot 31$ & $$ & $$ & $m>0$ & $0$ \\
\hline
$5^2\cdot 31^2$ & $331\cdot83$ & $>10^5$ & $m\ge0$ & $0$ \\
\hline
\end{tabular}
\\
\end{center}
\end{center}
\begin{flushleft}
Table 7.
\end{flushleft}
\vspace{1cm}
Finally for $\lambda_\star=1$, $p_\star^{\lambda_\star}\in\lbrace 3,5,7,11,13\rbrace$. So there exist $u\in\mathit{U}(10^5, p_\star^{\lambda_\star})$ in Table 7 so that the new seed $u_\star=p_\star^{\lambda_\star}u$ with $(p_\star^{\lambda_\star},u)=1$. By our initial supposition that alpha number $n$ exists, $n=p_\star^{\lambda_\star}uz$, thus implies that $$\alpha_2=\alpha_1\frac{n}{\sigma(p_\star^{\lambda_\star})\sigma(u)\sigma(z)}.$$ But $u\prod {f}_j(\sigma(u))>10^5$ or $2\mid {\mathit{f}}(\sigma(u))$ and ${\mathit{f}}(\sigma(u))\mid n$ or $p_i\mid n$ with $p_i<p_\star$, a contradiction.
\\
\end{proof}
\section{{Conclusion and Recommendation:}}
The results of this paper, particularly, Theorems 3.2 , 3.3, 3.5 and 3.7 give a rough picture of the general form for odd @-numbers of order (1,1), if at all exist. So, in order to fully establish Conjecture 1 of this paper, it suffices to establish the case of non-square-free, non-prime-power odd $n$. This is recommended for further study (see \cite{{san1}} $\&$ \cite{{san2}} for motivation). Also note that
an in-depth study of alpha numbers can be pursued further as follows:
\begin{itemize}
\item[(2)] Is there a general form for even alpha numbers analogous to Euclid-Euler form for even perfect numbers?
\item[(3)] What is the congruent form (properties) of odd alpha numbers analogous to Euler form for odd perfect numbers?
\item[(4)] Is every alpha number a practical number?
\item[(5)] What are the properties of odd alpha number $n$ (especially of those alpha numbers with non-rational complex order) in terms of size, the bounds (lower and upper), abundancy and etc?
\item[(6)] Is there any applicable relationship between function $\sigma_{\underline{\alpha}}(.)/(.)^{\bar{\alpha}}$ and the Riemann zeta function $\zeta(.)$?
\item[(7)] Is there any efficient and effective algorithm to generate alpha numbers?
\item[(8)] Can there be a counting function, say $C(x)$ generating the number of alpha number up to a
desired bound $x$ such that the number of alpha numbers in regular intervals, say $10^{0}-10^{3}$, $10^{3}-2\cdot 10^{3}$, $2\cdot 10^{3}-3\cdot 10^{3}$ and etc can be determined?
\item[(9)] Are the zeros of $\lfloor|\sigma_{\underline{\alpha}}(n)|\rfloor$ of complex order $\alpha$ in table 1 significant in any way?
\item[(10)] What other hidden identities of $\sigma_\alpha(.)$ function can be derived in order to solve
the problem of existence of alpha numbers?
\item[(11)] Let alpha number $n$ of order $(\underline{\alpha}, \bar{\alpha})$ with pair $(\alpha_1, \alpha_2)$ be denoted by $(\underline{\alpha}, \bar{\alpha}; \alpha_1, \alpha_2)- n$, and also let $\mathrm{T}_k^\alpha$ be a transformation on alpha numbers such that $$\mathrm{T}_k^\alpha : (\underline{\alpha}, \bar{\alpha}; \alpha_1, \alpha_2)- n \rightarrow(\underline{\alpha}, \bar{\alpha}; \alpha_1, \alpha_2)- m=n p_1\cdot p_2\cdots p_k.$$ For example, $$(1,1; 13,4)-2^9\cdot 3^2\cdot 31\cdot 11\ _{\overrightarrow{\mathrm{T}_1^\alpha}} \ (1,1; 7,2)-2^9\cdot 3^2\cdot 31\cdot 11\cdot 13 $$ $$ (1,1; 13,4)-2^9\cdot 3^2\cdot 31\cdot 11\ _{\overrightarrow{\mathrm{T}_2^\alpha}} \ (1,1; 4,1)-2^9\cdot 3^2\cdot 31\cdot 11\cdot 13\cdot 7$$ which transformed a weak alpha number $ 2^9\cdot 3^2\cdot 31\cdot 11 $ to strong alpha number $2^9\cdot 3^2\cdot 31\cdot 11\cdot 13\cdot 7.$ Then one might wish to determine how prevalent do such transformation occur and terminate.
\item[(12)] What formidable results can come forth from the following certain generalizations of alpha numbers (see \cite{{laa}}\cite{{san1}} \cite{san2} $\&$ \cite{{wei}} for definitions, notations and motivation)?
\end{itemize}
\begin{itemize}
\item[(I)] Let any positive integer $n$ satisfying $\sigma^\alpha_{\underline{\alpha}}(n)=(\alpha_1/\alpha_2)n^{\bar{\alpha}}$ be called $\alpha$-super @-number of order $(\underline{\alpha},\bar{\alpha})$, where $\sigma^\alpha_{\underline{\alpha}}(n)=\sigma^{\alpha-1}_{\underline{\alpha}}\sigma_{\underline{\alpha}}(n)$ is the divisor function such that $\sigma^\alpha$ is the $\alpha$th iterate of $\sigma$-function and co-prime pair integral $1 \leq \max(\alpha_1, \alpha_2)\leq \omega(n)$, $\omega(n)< \max(\alpha_1, \alpha_2)\leq \tau(n)$ and $\tau(n)< \max(\alpha_1, \alpha_2) < n$ respectively for strong, weak, very weak of such number.
\item[(II)] Let any positive integer $n$ satisfying $\sigma^\star_{\underline{\alpha}}(n) =(\alpha_1/\alpha_2)n^{\bar{\alpha}}$ be called $m$-unitary @-number, where $\sigma^\star_\alpha(n)$ is the unitary integral divisor function such that unitary divisors of $n$ are used instead of positive divisors of $n$ in the computation of $\sigma_k(n)$.
\item[(III)]Let any positive integer $n$ satisfying $\sigma_{\alpha,\infty}(n)=(\alpha_1/\alpha_2)n^{\bar{\alpha}}$ be called $m$-infinitary @-number, where $\sigma_{\alpha,\infty}(n)$ is infinitary integral divisor function such that infinitary divisors of $n$ are used instead of positive divisors of $n$ in the computation of $\sigma_{\underline{\alpha}}(n)$.
\item[(IV)] Let any positive integer $n$ satisfying $\sigma_{\alpha
,e}(n))=(\alpha_1/\alpha_2)n^{\bar{\alpha}}$ be called m-exponential @-number, where $\sigma_{\alpha,e}(n)$ is the integral exponential divisor function such that exponential divisors of $n$ are used instead of positive divisors of $n$ in the computation of $\sigma_{\underline{\alpha}}(n)$.
\item[(V)] Let any positive integer $n$ satisfying $\sigma_{\underline{\alpha}}(n))=\frac{\alpha_1}{\alpha_2}(n+k)^{\bar{\alpha}}:k<\omega(n)$ be called near @-number, where $\sigma_{\alpha}(n)$ is the sum of divisor function.
\end{itemize}
\textbf{Acknowledgement:}
Thanks to Professor V. A. Babalola, Professor Terence Tao, and Professor J. S$\acute{a}$ndor for their timely support concerning arXiv documentation of this paper.
| 18,032
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This year I subbed at 9 different schools, for about 30 classes, and got a bunch of hugs, some respect, and the finger. Here are some random thoughts about my year as a substitute teacher...
- You can learn a lot about fashion while subbing.
- I so wanted to LOVE teaching middle school, but most middle school kids LOVE having a sub so they can especially goof off. Consequently, I fell back in love with teaching 2nd-4th grades. In those classrooms, I get hugs, notes, free art, and respect.
- Lucked out one day while subbing at West Glades Middle. There was a special anti-bullying assembly starring Brooks Gibbs and he drove home the "sticks and stones" message really well. When he said, "Words are just words," Jane the writer's ears pricked up in a "Oh, really????" kind of way as I thought about all of those times I carefully selected just the right word. But I get what he meant in the context of his presentation.
- I was totally unaware that "I don't feel like doing this" was an option. One day while subbing for an 8th grade science teacher, I handed out worksheets about skin cancer and most 8th graders handed them back incomplete before the end of class because they simply weren't in the mood.
- Yo, teachers, this can be a lifesaver: When a unisex name and an androgynous appearance have you confused as to the gender of a student, refrain from using pronouns and simple call that student by their proper name. I once used "he" when referring to a girl...and trust me, you do not want to suffer through making that error.
- They have Interior Design 101!!
- Some science classrooms have a backdoor which opens to a common faculty area where there is a near-private clean bathroom. If you have ever subbed at West Glades Middle, you know what I am talking about. This year I had a renewed enthusiasm for subbing science.
- While subbing for 9th grade English, I discovered that the answers to worksheets are readily available on the internet and that if not watched closely, students will access the answers and cheat.
- Keep your iPhone in your pocket or out of site.
- In middle and high schools, cell phones are a clear obstruction to learning even though kids will tell you music helps them study and that their teacher says it's ok.
- Kids always try to guess my age. Most guess younger, much younger, and I am flattered. Middle schoolers have referred to me as "the cool surfer sub" and a third grader has drawn me as a sprightly young thing.
- There are silver linings even to the most challenging substitute teaching gigs. While the morning that an 8th grade student at Forest Glen Middle School told me to basically "F--- off" was a tearful disaster, later that day, one of my students played the violin for me. That was also emotional, but I did not bawl. So, clearly, the day ended on a high note!
- I have come to realize that I still suck at deciphering traditional poetry.
- High school love is measured in cupcakes, teddy bears, and balloons.
- The student body can be a wonderful source for socio-cultural enlightenment. Not only am I there to teach, but I come to learn! Hence..."The Vocabulary Lesson".
- If you teach middle school, you will undoubtedly meet Mr. Dick Page, and just know that during class, when a student discovers a drawing of a penis in a text book, there will be 5-10 minutes of unruly behavior that follows.
happy summer y'all.
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We’re an HBS student organization dedicated to fostering an educational, responsible and inclusive environment in which students can learn about the cannabis industry and contribute to building it in an equitable way.
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Don´t hesitate to get in touch if you want to be involved in any way!
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TITLE: Prove there are infinitely many *primitive* solutions to $x^2 + y^2 = z^4$
QUESTION [0 upvotes]: For x, y, z $\in \mathbb N $ where $\gcd(x,y) = 1$
These solutions must also be primitive.
If we let $ w = z^2 $ so that $ x^2 + y^2 = w^2$
I have that for r, s $\in \mathbb N$ where $\gcd(r, s) = 1$ (coprime),
with opposite parity:
Let $ x = r^2 - s^2\\
y = 2rs \\
w = r^2 + s^2 $
Now for $t, u \in \mathbb N$ where $\gcd(t, u) = 1$ (coprime),
with opposite parity:
let $ r = t^2 - u^2\\
s = 2tu\\
z = t^2 + u^2 $
Where $r,s,z$ are coprimes with $r$ odd and $s$ even (by definition).
So we have:
$ x = ( t^2 - u^2)^2 -(2tu)^2\\
y = 2( t^2 - u^2)(2tu)\\
w = z^2 = ( t^2 - u^2)^2 + (2tu)^2 $
Hence $x, y$ and $z^2 $ is a primitive solution.
Any mistakes here?
REPLY [1 votes]: Notice
$$\begin{cases}
X(t) = t^4 - 6t^2 + 1\\
Y(t) = 4t(t^2-1)\\
Z(t) = t^2+1
\end{cases}
\quad\text{ satisfy }\quad X(t)^2 + Y(t)^2 = Z(t)^4
$$
and the algebraic identity
$$(1-320t^2)X(16t) + (1280t^2 - 29)t Y(16t) = 1$$
implies $X(16t)$ and $Y(16t)$ are relative prime for any integer $t$
as long as neither $X(16t)$ nor $Y(16t)$ vanish.
Since $X(16t)$ and $Y(16t)$ have only finitely many roots, we have infinite many primitive solutions of the original equation in the form $(x,y,z) = (X(16t),Y(16t),Z(16t)$.
For example,
$$\begin{array}{rcrcl}
t = 1 &\quad\longrightarrow\quad& 64001^2 + 16320^2 &=& 257^4\\
t = 2 &\quad\longrightarrow\quad& 1042433^2 + 130944^2 &=& 1025^2\\
t = 3 &\quad\longrightarrow\quad& 5294593^2 + 442176^2 &=& 2305^2
\end{array}$$
Note
The $X(t), Y(t)$ and $Z(t)$ here is really a special case of the parametrization
given in the question. We have fixed the parameter $u$ there to be $1$.
It is easy to verify $$\gcd(X(t),Y(t)) = 1\quad\text{ as a polynomial over }\mathbb{Q}$$
Equivalently, this means there are $m(t), n(t) \in \mathbb{Q}[t]$ such that
$$m(t) X(t) + n(t) Y(t) = 1$$
What I'm doing is filling the last gap in the question. I choose $t$ in such a way to force $m(t)$ and $n(t)$ to be integers. This turn the solution given in question to a primitive one.
| 165,378
|
How often do we question why an athlete performs the same pre-competition routine, or carries out the same number of reps during training? These may simply be harmless superstitions and routines, however, they do potentially have the ability to manifest into something detrimental to performance.
The question remains as to whether it is the ritualistic and repetitive behaviours that athletes are regularly exposed to that potentially cause the onset of this debilitating condition.
What is OCD?
OCD is considered to be an extremely crippling and debilitating disorder and causes extreme psychological distress for the sufferer. It has been a topic of extensive research in psychology with Antony et al. (1998) stating that 80% of people in the general population will experience obsessions and compulsions from time to time. Obsessions are unwanted thoughts, images or urges that repeatedly occur despite efforts in trying to resist them. These thoughts may include recurrent doubts about whether actions are being performed correctly, this often leads to an impairment in functioning as a result and can cause significant anxiety and distress for the individual.
Are Athletes at Risk?
However, Obsessive and compulsive behaviours in athletic populations is a largely under researched area in the literature. It is not uncommon for us to witness top athletes adhering to rituals and superstitious behaviours which they may be using in a bid to control their anxiety. However, examining the type of mental states that it takes for a sprinter to burst out of the blocks at just the right moment or to dive perfectly in unison with another person has proven difficult.
With such precision required, are athletes more prone to developing OCD than the general population? Dr. Judy L. Van Raalte (a Springfield University sports psychologist who works with college and professional athletes) believes that athletes are not quite prone to the development of the crippling disorder, but the perfectionism and obsessiveness that tends to make up their personalities does leave them vulnerable to the onset of these behaviours. “That obsessive compulsive person is awesome up to a point. You’re willing to train, to commit, and it feels comfortable to you, it doesn’t tire you out” stated Dr. Raalte. However, just like anyone else, athletes do have weaknesses that can be triggered by stress, a theory known as the diathesis-stress model of mental illness. Elite athletes confront distinct, sport-specific challenges, which can wreak mental havoc when presented in moments of extreme pressure.
Research that has been conducted in this area has regarded over-exercising and obsessive compulsive tendencies:
– Kagan (1987) compared chronic joggers on the compulsiveness inventory scale with the results indicating that the frequency of jogging was positively associated with a compulsive profile.
– Davis (1997) highlighted that high personal standards in perfectionism pose a risk for compulsive exercise.
Although it remains to be seen how this relationship transfers to athletes, as for exercisers, control may be a key factor. However, the role of anxiety may be more highly associated with that of the athlete as they are often under greater pressure to perform.
High-Profile Examples
In the past, there has been talk of several cases of top-tier athletes having to withdraw from Olympic competitions due to debilitating OCD symptoms (Aldhous, 2009). For example, the high profile case of Canadian diver Kelly MacDonald is an example of how these routine obsessions and compulsions can inevitably wreak havoc with an athlete’s career. It was originally reported that MacDonald had been “side-lined with injuries” when she appeared absent from the 2008 Beijing Olympic Games. However, in 2009, it was revealed that obsessive-compulsive disorder had prevented her from competing and that her pre-dive routine of “clearing her throat, tapping her leg and blinking at certain steps on her approach had eventually left her stuck on the diving board, unable to calm down.” Ironically, MacDonald’s actions were likely to have been beneficial at some point, hence making it difficult for her coaches to spot the problem.
There is a lot about a sport that is out of the athlete’s control, this can be influenced by a variety of unpredictable factors, from how well they slept the night before, to the order in which they compete. In order to compensate, some athlete’s may develop strict routines or adopt superstitions, which can provide some familiarity and comfort and help them to control their pre-competitive nerves. A famous example is that of grand slam tennis queen Serena Williams, who believes that much of her winning ways are the result of closely followed routines. These quirks include tying her shoelaces a specific way, along with bouncing the ball five times before her first serve and twice before her second. However, it is only when an athlete’s rituals begin to interfere with their daily life and training that obsession has drifted into dangerous territory.
Conclusions
An obsessive nature may keep an athlete training after others call it a day, yet it is these traits that may also be beneficial in helping to keep anxiety under control during extreme pressure situations. However, a vital question that needs to be addressed is when is it that these traits veer into illness and become of a destructive nature to the athlete as opposed to being advantageous.
It is vital for coaches and sports psychologists working closely with athletes to keep an eye out for any obsessive or compulsive behaviours that may be getting a little out of hand in order for intervention to take place before they begin to have an adverse effect on performance or perhaps in the daily living of the athlete.
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\begin{document}
\title{Extremal quantum protocols}
\author{Giacomo Mauro D'Ariano}
\affiliation{QUIT group, Dipartimento di Fisica ``A. Volta'', and INFN Sezione di Pavia, via Bassi
6, 27100 Pavia, Italy.}
\author{Paolo Perinotti}
\affiliation{QUIT group, Dipartimento di Fisica ``A. Volta'', via Bassi
6, 27100 Pavia, Italy.}
\author{Michal Sedl\'ak}
\affiliation{QUIT group, Dipartimento di Fisica ``A. Volta'', via Bassi
6, 27100 Pavia, Italy.}
\affiliation{Institute of Physics, Slovak Academy of Sciences, D\'ubravsk\'a cesta 9, 845 11 Bratislava, Slovakia}
\date{ \today}
\begin{abstract}
Generalized quantum instruments correspond to measurements where the
input and output are either states or more generally quantum
circuits. These measurements describe any quantum protocol including
games, communications, and algorithms. The set of generalized
quantum instruments with a given input and output structure is a
convex set. Here we investigate the extremal points of this set for
the case of finite dimensional quantum systems and generalized
instruments with finitely many outcomes. We derive algebraic
necessary and sufficient conditions for extremality.
\end{abstract}
\maketitle
\section{Introduction}
Experiments in quantum theory can be modeled through quantum networks
that provide the natural description of an arbitrary quantum
procedure, corresponding to a causal sequence of steps. The most basic
building blocks of quantum networks are state preparations, state
transformations (channels and state reductions) and measurements.
Provided we have a quantum network, we can isolate open sub-circuits,
whose connections constitute the whole network. Any optimization
problem in quantum theory can be seen as the search for the most
suitable sub-circuit for a specified purpose. For example, for
discrimination of states we need to optimize a measurement, or for
discrimination of channels we need to optimize the network into which
the channel is inserted. Open sub-circuits provide a representation
for the most general quantum protocol, where the gates represent the
sequence of operations performed by the agent that is
communicating, computing or applying a strategy for a quantum game.
From a more abstract point of view any sub-circuit represents the most
general input-output map that can be achieved via a quantum circuit,
that is
called generalized quantum instrument (GQI)\cite{comblong}. GQIs then
provide the mathematical description for any quantum protocol
including games, communications, and algorithms. It is possible to
uniquely associate \cite{architecture} a positive operator to any
deterministic GQI---corresponding to a sub-circuit that does not
provide outcomes---in the same way as a positive operator is
associated to any channel through the Choi-Jamio\l kowski
correspondence. More generally, it is possible to associate a set of
positive operators to any GQI \cite{memoryeff} in such a way that each
operator corresponds to a possible measurement outcome and summarizes
the probabilistic input-output behavior of the GQI as a sub-circuit,
conditionally on the outcome. The advantage of this description comes
from neglecting the implementation details that are irrelevant for the
input-output behavior of the GQI within a quantum network, like
arbitrary transformations on ancillary systems, etc. The set of GQIs
with the same input and output types is convex, since a random choice
of two different GQIs provides a convex combination of the
corresponding two input-output maps.
It is thus clear that the description of quantum maps through GQIs
\cite{comblong}
in optimization problems is convenient for two reasons. The first one
is that this approach gets rid of many irrelevant parameters, and the
second one is that the optimization problems are reduced to convex
optimization on suitably defined convex sets. Applications of GQIs in
optimization problems can be found in
\cite{cloning,covnet,tomo,infodist,learning,memory,ucomp,zimandiscr}.
The theory of GQIs was alternatively introduced\cite{comblong} as a
theory of higher order quantum functions, spawning interest in the
investigation of more computational consequences of the properties of
GQIs\cite{watrous,gutoski,costa1}. A similar approach to general
affine functions on convex subsets of state spaces was recently
published\cite{jenco}, explicitly inspired to the concept of GQIs and
quantum combs (namely singleton GQIs).
As a special case of GQIs, we have the elementary examples of states,
channels, and POVMs. The analysis of the extremality conditions for
states is trivial, and can be found in any textbook of quantum theory.
Algebraic extremality conditions for channels were provided in Ref.
\cite{choi}, while the conditions for POVMs
were derived later
in Refs. \cite{storm,partha,dlp,pellonpaa,chriribella1}. Other special cases of GQIs
are quantum combs \cite{architecture}, corresponding to deterministic GQIs, or quantum testers \cite{memoryeff,comblong}, which are GQIs with outputs
that are probability distributions.
While all GQIs could be decomposed into states, channels, and measurements,
it is much more practical to consider the corresponding networks as a whole.
Optimization tasks in quantum
information processing can be rephrased in terms of optimization of a
certain \GQI with respect to some particular figure of merit, which
is often a convex function on the set of GQIs and the maximum is achieved on an extremal
point of this set. Moreover, also for those problems that resort to convex
optimization or minimax problems, numerical optimization is enhanced
by the possibility of generating arbitrary extremal elements. For this
purpose, having an algebraic characterization is a crucial step.
In the present paper we consider the convex sets of GQIs, and characterize their extremal points for the case of
finite dimensional quantum systems and the instruments that have finitely many outcomes.
As special cases we obtain the extremality conditions for POVMs, channels,testers or instruments.
The paper is organized as follows. In section \ref{sec:combs} we
introduce the theoretical framework we use to describe quantum networks. In section \ref{sec:extgqi} we formulate the necessary and sufficient
condition of extremality for \GQI. Sections \ref{sec:extest}, \ref{sec:excombs}, and \ref{sec:exinst} study the implications of the extremality
condition in the case of quantum testers, quantum channels, and quantum instruments, respectively. Finally, the summary of the results is placed in
section \ref{sec:conclusion}.
\section{Theory of quantum networks}
\label{sec:combs}
Let us summarize some pieces of the theoretical framework of quantum networks introduced in \cite{comblong} that we will use.
An arbitrary quantum network $\mathcal{R}$ can be formally understood as a quantum memory channel \cite{architecture}, whose inputs and outputs are
labeled by even or odd numbers from $0$ to $2N-1$, respectively. The Hilbert spaces associated with these inputs and outputs can be in general
different and we denote them by $\Hs_i$ $i=0,\ldots,2N-1$. As it was shown already in \cite{architecture} deterministic quantum network $\mathcal{R}$
is fully characterized by its Choi-Jamiolkowski operator, i.e. a deterministic quantum $N$-comb $R$.
\begin{Def}
A deterministic quantum $N$-comb on $\Hs_0,\ldots,\Hs_{2N-1}$ is a positive operator $R\equiv R^{(N)}\in
\mathcal{L}(\Hs_0\otimes\cdots\otimes\Hs_{2N-1})$, which obeys the following normalization conditions
\begin{eqnarray}
\Tr_{2n-1}R^{(n)}&=&\one_{2n-2}\otimes R^{(n-1)} \quad 0\leq n\leq N \nonumber\\
\Tr_{1}R^{(1)}&=&\one_{0},
\label{normcascade}
\end{eqnarray}
where the operators $R^{(n)}$ are defined recursively.
\end{Def}
Positive operators $T\in \mathcal{L}(\Hs_0\otimes\cdots\otimes\Hs_{2N-1})$, such that $T\leq R$ for some deterministic quantum comb $R$, are called
non-deterministic quantum $N$-combs. An arbitrary probabilistic quantum network, whose different outcomes are indexed by $i=1,\ldots,M$ is described by a
collection of non-deterministic quantum $N$-combs $\{T_i\}_{i=1}^M$ defined as follows.
\begin{Def}
Generalized quantum $N$-instrument is a collection $\{T_i\}_{i=1}^M$ of non-deterministic quantum $N$-combs
that sum up to a deterministic quantum comb
\begin{eqnarray}
\sum_{i=1}^M T_i&=&R. \label{norminst}
\end{eqnarray}
\end{Def}
A realization theorem can be proved\cite{comblong}, providing the
interpretation of GQIs as the appropriate mathematical representation
for the most general quantum network, because any GQI can be
implemented through a quantum circuit as in Fig. \ref{f:gqisnet}, and
viceversa any quantum circuit possibly involving measurements
corresponds to a GQI.
\begin{figure}
\includegraphics[width=0.7\textwidth]{fig-4.eps}
\caption{\label{f:gqisnet} The circuit implementation of a general
GQI shows that GQIs correspond to the most general quantum
network\cite{comblong}, and viceversa one can prove that any
quantum circuit possibly involving measurements corresponds to a
GQI. The transformations ${\conv V}_i$ are isometries, and $M_i$
denotes a POVM. Notice that the measurement can always be
postponed to the very last step.}
\end{figure}
For $M=1$ the corresponding network is deterministic and the set of generalized quantum instruments coincides with deterministic quantum combs. On the
other hand, if $N=1$ a generalized quantum instrument is a collection of completely positive maps forming a channel, which is usually called an
instrument.
Another special case of generalized quantum instruments is provided by quantum testers.
\begin{Def}
A quantum $N$-tester is a generalized quantum $(N+1)$-instrument on $\Hs_0,\ldots,\Hs_{2N+1}$ with one-dimensional Hilbert spaces $\Hs_0$,
$\Hs_{2N+1}$.
\end{Def}
Quantum testers are analogous to the concept of Positive Operator Valued Measures (POVM) as they allow to express probability distributions for
arbitrary tests on quantum combs.
As we will show our analysis of the extremal points of the set of generalized quantum instruments provides necessary and sufficient conditions for
extremality, and leads to specific new conditions also for all the above mentioned special cases.
\section{Extremality condition for generalized quantum instruments}
\label{sec:extgqi}
In this section we shall apply the method of perturbations to find
extremal generalized quantum instruments. The perturbation method was
also used to determine extremal channels \cite{choi} and POVMs
\cite{dlp}. However, the application of the perturbation method to
GQIs does not come as a straightforward generalization of previous
results, because the richer structure of the normalization constraints
for GQIs requires a radically different analysis.
Let us consider arbitrary generalized quantum $N$-instrument $\{T_i\}_{i=1}^M$.
We denote by $\esp{V}_i$ the support of the operator $T_i$. The support of the sum of positive operators is the span of the supports of the summed
operators. Thus, the support of the normalization $R=\sum_{i=1}^M T_i$ is $\Hs_R\equiv \operatorname{Span}\{\esp{V}_i\}_{i=1}^{M}$. A set of operators
$\{ D_i\}_{i=1}^{M}$ is called a valid perturbation of \GQI $\{ T_i\}_{i=1}^{M}$ if and only if $\{ T_i\pm D_i\}_{i=1}^{M}$ are valid GQIs. Existence
of a perturbation has two major implications. First, the positivity of $T_i\pm D_i$ requires $D_i$ to be hermitian and to have support only in
$\esp{V}_i$. This is proved by the following lemma.
\begin{lemma}
Suppose that operators $T, D$ fulfill $T\geq0$, $D^\dag=D$. If $T\geq\pm D$, then $\Supp(D)\subseteq\Supp(T)$.
\end{lemma}
\begin{Proof}
The statement of the lemma can be equivalently formulated as
$\Ker(T)\subseteq\Ker(D)$. This can be proved considering the general
decomposition of a vector $\ket{\psi}$ as $\alpha\ket{\Psi_S}+\beta\ket{\Psi_K}$ where
$\Psi_S\in\Supp(T)$ and $\Psi_K\in\Ker(T)$. Then we have
\begin{eqnarray}
|\alpha|^2\bra{\Psi_S}(T\pm D)\ket{\Psi_S} &\pm & 2\Re\alpha^{*} \beta\bra{\Psi_S}D\ket{\Psi_K}
\pm |\beta|^2\bra{\Psi_K}D\ket{\Psi_K}\geq0,
\end{eqnarray}
for all $\alpha,\beta$. Choosing $\alpha=0$ one immediately obtains
$\bra{\Psi_K}D\ket{\Psi_K}=0$, which by the polarization identity implies also $\bra{\Psi^\prime_K}D\ket{\Psi_K}=0$ for all $\Psi^\prime_K\in\Ker(T)$. The
previous inequality can thus be rewritten
as follows
\begin{equation}
|\alpha|^2\bra{\Psi_S}(T\pm D)\ket{\Psi_S}\pm2\Re\alpha^*\beta\bra{\Psi_S}D\ket{\Psi_K}\geq0,
\end{equation}
for all $\alpha,\beta$. Suitably choosing the phases of $\alpha$,
$\beta$, one has
\begin{equation}
|\alpha|^2\bra{\Psi_S}(T\pm D)\ket{\Psi_S}\mp 2|\alpha||\beta||\bra{\Psi_S}D\ket{\Psi_K}|\geq0,
\end{equation}
and for $\beta=\frac12$ and $|\alpha|>0$ we obtain
\begin{equation}
|\alpha|\bra{\Psi_S}(T\pm D)\ket{\Psi_S}\geq|\bra{\Psi_S}D\ket{\Psi_K\>}|,
\end{equation}
for all $|\alpha|$, implying that $\bra{\Psi_S}D\ket{\Psi_K}=0$ holds for all $\Psi_S$ and $\Psi_K$. This together with
$\bra{\Psi^\prime_K}D\ket{\Psi_K}=0$ $\forall \Psi^\prime_K\in\Ker(T)$ allows us to conclude that $\bra{\psi}D\ket{\Psi_K}=0$ for every $\psi\in\Hs$,
i.e. $D\ket{\Psi_K}=0$ for all $\Psi_K$. This proves that $\Ker(T)\subseteq\Ker(D)$, or,
equivalently, $\Supp(D)\subseteq\Supp(T)$. \qed
\end{Proof}
As a consequence if we write operators $T_i$ in their spectral form $T_i = \sum_k \lambda^{(i)}_k \ket{v^{i}_k}\bra{v^{i}_k}$ then arbitrary hermitian
operator $D_i$ with support in $\esp{V}_i$ can be written as
\begin{eqnarray}
D_i = \sum_{n,m} D^{(i)}_{nm} \ket{v^{i}_n}\bra{v^{i}_m},
\end{eqnarray}
where $D^{(i)}_{nm}$ is a hermitian matrix with $r^2_i\equiv(\dim{\esp{V}_i})^2$ real parameters. We form a basis
$\bas{H}_i\equiv\{Q^{(i)}_j\}_{j=1}^{r^2_i}$ of hermitian operators with support in $\esp{V}_i$ and we define $\bas{D}_M:=\bigcup_{i=1}^M\bas{H}_i$.
The second consequence of requiring valid perturbed \GQI $\{T_i\pm D_i\}_{i=1}^M$ is that, due to the normalization condition (\ref{norminst}) the
perturbed \GQI has to sum up to deterministic $N$-combs $R_{\pm}$, which can be stated as
\begin{eqnarray}
\sum_{i=1}^M D_i = \Delta ,
\label{perturbcond1}
\end{eqnarray}
where $\Delta\equiv \pm (R_{\pm}-R)$ is an operator expressible as a difference of two deterministic quantum $N$-combs.
Using the parametrization of deterministic quantum combs developed in Appendix \ref{secparamcombs} it is clear that $\Delta$ lies in
$\esop{W}_C$, the subspace of operators spanned by the basis
\begin{eqnarray}
\bas{D}_{(N)}&\equiv&\{E^{(2N-1)}_i\otimes F^{(2N-2)}_j, \one_{2N-1,2N-2}\otimes E^{(2N-3)}_i \otimes F^{(2N-4)}_j, \ldots,
\one_{2N-1,\ldots,2}\otimes E^{(1)}_i \otimes F^{(0)}_j \} \nonumber
\end{eqnarray}
where $\{E^{(k)}_i\}_{i=2}^{d_k^2}$ is a basis of traceless hermitian operators on $\Hs_k$, and $\{F^{(k)}_j\}_{j=1}^{d^2_k\ldots d^2_0}$ is basis of all
hermitian operators acting on $\Hs_k\otimes\Hs_{k-1}\otimes\cdots \otimes\Hs_0$. On the other hand, due to the positivity requirement for $R_{\pm}=R\pm\Delta$, $\Delta$ must be a hermitian operator with support in $\Hs_R$. Let us call $\esop{W}_S$ the subspace of hermitian operators with support in $\Hs_R$. Thus, the allowed perturbations of the normalization lie in the intersection $\esop{W}_I\equiv \esop{W}_S \cap \esop{W}_C$. The
relation between non-existence of a valid perturbation and the requirements on the operators $D_1,\ldots,D_M,\Delta$ is expressed by the following
theorem.
\begin{theorem}
\label{th1}
A generalized quantum $N$-instrument $\{ T_i\}_{i=1}^{M}$ acting on $\Hs_{2N-1}\otimes\cdots\otimes\Hs_0$ is extremal if and only if $\bas{D}_M \cup
\bas{D}_{(N)}$ is an linearly independent set of operators.
\end{theorem}
\begin{Proof}
We are going to prove the theorem by showing the equivalence of the negated statements, i.e. \GQI is not extremal if and only if the basis $\bas{D}_M
\cup\bas{D}_{(N)}$ is linearly dependent. It is easy to show that if a \GQI is not extremal then the basis $\bas{D}_M \cup\bas{D}_{(N)}$ is linearly
dependent. If a point of a convex set is not extremal then there exists a bidirectional perturbation to it. Hence, there exists a set of operators
$D_i$ such that $\{ T_i\pm D_i\}_{i=1}^{M}$ is a valid \GQI. In particular, due to at least one operator $D_i$ being non-zero we have Eq.
(\ref{perturbcond1}), which after expanding the LHS in $\bas{D}_M$ and RHS in $\bas{D}_{(N)}$ proves the linear dependence of basis
$\bas{D}_M\cup\bas{D}_{(N)}$.
In order to prove the converse statement we are going to show that if the basis $\bas{D}_M \cup\bas{D}_{(N)}$ is linearly dependent then there exists a
valid perturbation of the considered \GQI and hence it is not extremal. Linear dependence of $\bas{D}_M \cup\bas{D}_{(N)}$ means there exists a
non-zero vector consisting of all coefficients $D^{(i)}_{nm}, s_k$ such that
\begin{eqnarray}
\sum_{i,n,m} D^{(i)}_{nm} \ket{v^{i}_n}\bra{v^{i}_m} + \sum_k s_k G_k = 0,
\label{prooflindep}
\end{eqnarray}
where $D^{(i)}_{nm}$ are for each $i$ hermitian matrices, $\ket{v^{i}_n}$ are eigenvectors of $T_i$ and $G_k$ are basis elements of $\bas{D}_{(N)}$.
Let us recall that the basis $\bas{D}_{(N)}$ is by construction linearly independent, so all $D^{(i)}_{nm}$ can not be zero simultaneously.
We rewrite the equation (\ref{prooflindep}) as:
\begin{eqnarray}
\sum_{i,n,m} D^{(i)}_{nm} \ket{v^{i}_n}\bra{v^{i}_m}=\sum_k -s_k G_k\equiv \Delta.
\label{prooflindep1}
\end{eqnarray}
For each $i$ the operators on the LHS of (\ref{prooflindep1}) have support in the subspace $\esp{V}_i$. All subspaces $\esp{V}_i$ are included in the
support of the normalization $R$. Thus, the operator on the LHS of (\ref{prooflindep1}) belongs to an operator subspace $\esop{W}_S$. Since the RHS of
(\ref{prooflindep1}) is from subspace $\esop{W}_C$ it is clear that $\sum_k s_k G_k\in \esop{W}_S\cap \esop{W}_C=\esop{W}_I$. This implies that for
suitably small $\varepsilon$ the operator $R\pm \varepsilon \Delta$ is positive as well as all operators $T_i\pm \varepsilon D_i$. Thus, we have found
a valid perturbation of the \GQI $\{T_i\}_{i=1}^M$ showing that it is not extremal, which concludes the proof. \qed
\end{Proof}
\section{Extremality of quantum testers}
\label{sec:extest}
In this section we focus our attention to quantum testers, which can be used to solve problems like discrimination of quantum channels, or optimization
of quantum oracle calling algorithms and others, because they describe achievable probability distributions for all possible experiments with given
resources. More precisely, we consider quantum $N$-testers and we try to identify the extremal points of this set. We start by the analysis of
$1$-testers, also called Process-POVMs \cite{ziman}. A $1$-tester with $M$ outcomes is defined by positive operators $\{ T_i\}_{i=1}^{M}$ acting on
$\Hs_2\otimes\Hs_1$, which satisfy the normalization condition
\begin{eqnarray}
\sum_{i=1}^M T_i = \one_2 \otimes \rho_1,
\label{norm1tester}
\end{eqnarray}
where $\rho$ is a state on $\Hs_1$\cite{note1}. As before we denote by
$\esp{V}_i$ the supports of operators $T_i$. Let us denote the support
of $\rho$ by $\Hs_\rho$ and by $r=\dim{\Hs_\rho}$ the rank of $\rho$.
The $1$-tester $\{ T_i\}_{i=1}^{M}$ on $\Hs_2\otimes\Hs_1$ can be
considered as a valid $1$-tester on $\Hs_2\otimes\Hs_1^\prime$ for
arbitrary $\Hs_1^\prime$ that includes $\Hs_\rho$ (e.g.
$\Hs_1^\prime=\Hs_\rho$).
\subsection{Extremality condition for $1$-testers}
\label{cond1test}
In the following we express the general extremality condition from Theorem \ref{th1} for $1$-testers and we propose a slightly different extremality
condition, which is easier to check. The set $\bas{D}_{(N)}$ from Theorem \ref{th1} is in this case formed by the operators $\{\one_2\otimes
E^{(1)}_i\}_{i=2}^{d_1^2}$, where $\{E^{(1)}_i\}_{i=2}^{d_1^2}$ is a basis of trace zero hermitian operators on $\Hs_1$.
\begin{corollary}
A quantum $1$-tester $\{ T_i\}_{i=1}^{M}$ is extremal if and only if there exists only a trivial solution of an equation
$\sum_{i=1}^M \sum_{n,m} D^{(i)}_{nm} \ket{v^{i}_n}\bra{v^{i}_m}+ \sum_{j=1}^{d_1^2-1} s_j \one_2\otimes E^{(1)}_j = 0,$
where $\forall i$ $D^{(i)}_{nm}$ are hermitian matrices and $s_j$ are real numbers.
\end{corollary}
Since the normalization of the perturbed tester must be supported inside the support of the original normalization, it is natural that,
$\mathbb{D}_{I}\equiv\{\one_2\otimes\sigma_l\}_{l=1}^{r^2 -1}$, the basis of trace zero operators supported under the original normalization
$\one_2\otimes\rho_1$ can be used in the Theorem \ref{th1} instead of $\bas{D}_{(N)}$.
\begin{theorem}
\label{th2}
A quantum $1$-tester $\{ T_i\}_{i=1}^{M}$ is extremal if and only if the equation
\begin{eqnarray}
\sum_{i=1}^M \sum_{n,m} D^{(i)}_{nm} \ket{v^{i}_n}\bra{v^{i}_m}+ \sum_{l=1}^{r^2-1} s_l \one_2\otimes\sigma_l = 0,
\label{perturbcond2}
\end{eqnarray}
where $D^{(i)}_{nm}$ are for each $i$ hermitian matrices and $s_l$ are real numbers, has only a trivial solution.
\end{theorem}
Actually, the basis $\mathbb{D}_{I}$ of the subspace $\esop{W}_I$ can be always used in the Theorem \ref{th1} and the proof still holds. However, for
$N\neq 1,2$ it is often easier to specify $\bas{D}_{(N)}$ rather than $\mathbb{D}_{I}$.
As we said for $1$-testers $\mathbb{D}_{I}$ is formed by trace zero operators supported under $\rho$ tensored with unity on $\Hs_2$ and this will help
us to get more insight to $1$-testers. The extremality condition for $1$-tester $\{ T_i\}_{i=1}^{M}$ from Theorem \ref{th2} allows us to give the
following bound
\begin{eqnarray}
\sum_{i=1}^M r_i^2 + r^2-1 \leq (r d_2)^2 ,
\label{bound1}
\end{eqnarray}
on the ranks $r_i$ of the operators $T_i$. The bound is derived by counting the number of elements of $\bas{D}_M \cup \bas{D}_{I}$ and realizing that
these operators should be linearly independent hermitian operators acting only on $\Hs_2\otimes\Hs_\rho$. From the bound (\ref{bound1}) it is clear
that the extremal tester can have the highest possible number of outcomes if $r=d_1$ and the ranks $r_i$ are as close to one as possible. Assuming all
$r_i$ are rank $1$ we get the bound on the number of elements of the extremal quantum $1$-tester
\begin{eqnarray}
M\leq d_1^2(d_2^2-1)+1 ,
\label{boundnumber}
\end{eqnarray}
\subsection{Classification of extremal $1$-testers}
\label{secclassif}
Let us now answer the question, which normalizations $\one\otimes\rho$ allow existence of extremal testers. For this purpose let us define a
superoperator $\xi_{\rho,U}$ that acts on linear operators on $\Hs_2\otimes\Hs_1$ as
\begin{eqnarray}
\xi_{\rho,U}(T_i)\equiv d_1 (\one\otimes\sqrt{\rho}\; U) \;T_i \; (\one\otimes U^\dagger \sqrt{\rho}).
\label{transftesters}
\end{eqnarray}
For any state $\rho$ with full rank (i.e. $r=d_1$) and any unitary $U$ acting on $\Hs_1$, the superoperator $\xi_{\rho,U}$ is invertible and preserves
positivity of operators. Using $\xi_{\rho,U}$ we can formulate the following theorem.
\begin{theorem}
\label{th3}
Suppose we have a full rank state $\rho$, a unitary operator $U$ and a $1$-tester $\{ T_i\}_{i=1}^{M}$ on $\Hs_2\otimes\Hs_1$ with $\sum_{i=1}^M
T_i=\one\otimes \frac{1}{d_1}\one$. Then the tester $\{ T^\prime_i\equiv \xi_{\rho,U}(T_i)\}_{i=1}^{M}$ on $\Hs_2\otimes\Hs_1$ has normalization
$\sum_{i=1}^M T^\prime_i=\one\otimes\rho$ and is extremal if and only if $1$-tester $\{ T_i\}_{i=1}^{M}$ is extremal.
\end{theorem}
\begin{Proof}
First, let us note that the form of $\xi_{\rho,U}$ guarantees positivity of $T^\prime_i$ and leads to the normalization
\begin{eqnarray}
\sum_{i=1}^M T_i^\prime=\xi_{\rho,U}(\one\otimes\frac{1}{d_1}\one)=\one\otimes\rho.
\end{eqnarray}
Now we prove that the tester $\{ T^\prime_i\}_{i=1}^{M}$ is extremal if the original tester $\{ T_i\}_{i=1}^{M}$ was. Let us stress that for any
extremal $1$-tester its normalization $\one\otimes\rho$ is (up to multiplication) the only operator of the form $\one\otimes X$ that is in the span of
the operators $D_i$. This holds, because the span of the operators $D_i\in\mathcal{L}(\esp{V}_i)\subseteq\mathcal{L}(\Hs_2\otimes\Hs_\rho)$ covers $\one\otimes\rho$ and
it is independent from $r^2-1$ dimensional subspace of traceless hermitian operators of the form $\one\otimes X$ due to linear independence
(\ref{perturbcond2}). Superoperator $\xi_{\rho,U}$ is invertible so it preserves linear independence. In our case this means that the basis
$\bas{H}^\prime_i$ of hermitian operators derived from $\xi_{\rho,U}(\ket{v^{i}_n}\bra{v^{i}_m})$ is linearly independent and spans the whole space of
hermitian operators that have support in the support of $T^\prime_i$. Moreover, due to extremality of the original tester $\{ T_i\}_{i=1}^{M}$
($\sum_{i=1}^M T_i=\one\otimes\frac{1}{d_1}\one$) and the invertibility of $\xi_{\rho,U}$ we can conclude that also $\one\otimes\rho=\xi_{\rho,U}(\one\otimes\frac{1}{d_1}\one)$ is the only operator of the form $\one\otimes X$ that is in the span of $\bas{D}^\prime_M=\cup_{i=1}^M \bas{H}^\prime_i$.
Let us assume that the tester $\{ T^\prime_i\}_{i=1}^{M}$ is not extremal even though the original tester $\{ T_i\}_{i=1}^{M}$ was extremal. In other
words we assume that $\bas{D}^\prime_M$ is linearly dependent with traceless operators $\{\one_2\otimes\sigma_l\}_{l=1}^{r^2 -1}$. As a consequence
there must exist a traceless operator of the form $\one\otimes X$ in the span $D^\prime_i$. However, this is a contradiction, because the only operator
of such form is $\one\otimes\rho$ and has trace one. We conclude that the transformed tester $\{ T^\prime_i\}_{i=1}^{M}$ must be extremal.
In fact, the same argumentation can be used to prove that $\{T_i\}_{i=1}^{M}$ is extremal if $\{ T^\prime_i\}_{i=1}^{M}$ was, because $\xi_{\rho,U}$ is
invertible. Hence, for arbitrary extremal tester $\{ T^\prime_i\}_{i=1}^{M}$ with normalization $\one\otimes\rho$ using $(\xi_{\rho,U})^{-1}$ one
obtains extremal tester $\{ T_i\}_{i=1}^{M}$ with normalization $\one\otimes\frac{1}{d_1}\one$.
\qed
\end{Proof}
The theorem \ref{th3} is very useful, because to classify all extremal $1$-testers it suffices to classify extremal $1$-testers with normalization
$\one\otimes\frac{1}{d_1}\one$. More precisely, using $\xi^{-1}_{\rho,\one}$ each extremal tester is in one to one correspondence with an extremal
tester with normalization $\one\otimes\frac{1}{r}\Pi_\rho$, where $\Pi_\rho$ is a projector onto a support of $\rho$. This tester can be considered as
a tester on $\Hs_2\otimes\Hs_{\rho}$, where its normalization is of the above mentioned form $\one\otimes\frac{1}{d_1}\one$.
Thus, we can formulate the following corollary of theorem \ref{th3}.
\begin{corollary}
Extremal $1$-testers with $M$ outcomes exist either for all normalizations $\one\otimes\rho$ with given rank $r$ of $\rho$ or for none of them.
\end{corollary}
Let us now relate the set $\extest$ of extremal quantum testers with
normalization $\one\otimes\frac{1}{d_1}\one$ to the set
$\textit{P}(\Hs_2\otimes\Hs_1)$ of extremal POVMs on
$\Hs_2\otimes\Hs_1$. Namely, each extremal tester $\{ T_i\}_{i=1}^{M}$
with normalization $\one\otimes\frac{1}{d_1}\one$ defines an extremal
POVM $\{E_i=d_1 T_i\}_{i=1}^M$. This follows directly from the
extremality condition for quantum testers (\ref{perturbcond2}), which
necessarily requires the basis of hermitian operators with supports on
$\esp{V}_i$ to be linearly independent. This is exactly the necessary
and sufficient condition for the extremality of the POVM \cite{dlp}
$\{E_i\}_{i=1}^M$. Apart from the multiplicative difference in
normalization, we will prove later that extremal quantum testers with
normalization $\one\otimes\frac{1}{d_1}\one$ are a proper subset of
extremal POVMs on $\Hs_2\otimes\Hs_1$. On the other hand there are
extremal POVMs on $\Hs_2\otimes\Hs_1$, which cannot be rescaled to
form an extremal tester. One example are informationally complete
POVMs on $\Hs_2\otimes\Hs_1$ with $(d_1 d_2)^2$ outcomes. Their
existence was proved in \cite{dlp} for any dimension, but they have
too many outcomes to form an extremal $1$-tester (see Eq.
(\ref{boundnumber})).
\subsection{Extremal $1$-testers with rank one normalization}
Having a tester with rank one normalization $\rho=\ket{\phi}\bra{\phi}$ implies that all the elements of the tester have the form $T_i=E_i\otimes\rho$,
where $E_i$ is positive operator acting on $\Hs_2$. Let us note that these testers correspond to preparation of a pure state $\rho$ and performing a
POVM $\{E^T_i\}_{i=1}^M$. Since the support of $\rho$ is one-dimensional, there are no traceless operators with support in $\Hs_\rho$. Thus, the
extremality condition (\ref{perturbcond2}) is in this case equivalent to linear independence requirement
\begin{eqnarray}
0&=& \sum_{i=1}^M \sum_{n,m} D^{(i)}_{nm} \;\ket{w^{i}_n}\ket{\phi}\bra{w^{i}_m}\bra{\phi}=
\left( \sum_{i=1}^M \sum_{n,m} D^{(i)}_{nm} \ket{w^{i}_n}\bra{w^{i}_m} \right) \otimes\ket{\phi}\bra{\phi} \;
\Rightarrow D^{(i)}_{nm}=0 \quad \forall i,n,m \nonumber
\end{eqnarray}
for the basis of hermitian operators on the supports of $E_i$. This is
precisely the necessary and sufficient condition of the extremality of
the POVM \cite{dlp} with elements $E_i$. Thus, the quantum tester
$\{T_i=\ket{\phi}\bra{\phi}\otimes E_i\}_{i=1}^{M}$ is extremal if and
only if POVM $\{E_i\}_{i=1}^M$ is extremal. In particular, the number
of outcomes of the extremal quantum tester in this case cannot exceed
$d_2^2$, which is the number given by the bound (\ref{boundnumber})
and by the maximal number of elements of an extremal POVM
\cite{dlp} as well. On the other hand, a single outcome extremal
POVM $\{E_1=\one\}$ leads to an extremal $1$-tester
$\{T_1=\one\otimes\rho \}$ for arbitrary pure state normalization
$\rho$.
\begin{remark}
Actually, the only extremal single outcome $1$-testers are those with pure state normalization.
\end{remark}
\subsection{Extremal qubit $1$-testers}
For qubit tester ($d_1=d_2=2$) the rank $r$ of the normalization $\rho$ can be either one or two. If $\rho$ is a pure state ($r=1$) then the previous
section tells us that such extremal testers are in one to one correspondence with the extremal qubit POVMs, which can have at most four outcomes.
Hence, to classify all extremal qubit testers (based on section \ref{secclassif}) it remains to investigate qubit testers with normalization
$\rho=\one\otimes\frac{1}{2}\one$. We will identify extremal testers with two outcomes. Then we discuss the case $2< k \leq 13$
(see bound \ref{boundnumber}) and we propose some ways how to construct such testers
\subsubsection{Two outcome testers}
Considering the ranks $r_1$, $r_2$ of the two parts of the tester, there are only three possibilities compatible with bound (\ref{bound1}):
$i)$ $(r_1,r_2)=(1,3)$, $ii)$ $(r_1,r_2)=(2,2)$, $iii)$ $(r_1,r_2)=(2,3)$, where we assume without loss of generality that $r_1\leq r_2$. As we already
mentioned the supports of the tester operators $T_i$ necessarily have to obey conditions for extremal POVMs on $\Hs_2\otimes\Hs_1$. In particular,
operators $T_i$ cannot have intersecting supports (see corollary $3$ in ref \cite{dlp}). This rules out $(r_1,r_2)=(2,3)$ case.
Let us now consider the case $i)$ $(r_1,r_2)=(1,3)$. In this case $T_1$ necessarily equals $\frac{1}{2}$ projector onto a pure state, because otherwise
the rank of $T_2=\frac{1}{2}\one\otimes\one-T_1$ would not be three. Consequently, we can write the tester as
\begin{eqnarray}
T_1&=&\frac{1}{2}\ket{\phi_1}\bra{\phi_1}; \label{qtester13}\\
T_2&=&\frac{1}{2}(\one\otimes\one-\ket{\phi_1}\bra{\phi_1})=\frac{1}{2}\sum_{i=2}^4\ket{\phi_i}\bra{\phi_i}, \nonumber
\end{eqnarray}
where vectors $\ket{\phi_i}\; i=1,\ldots,4$ form an orthonormal basis of $\Hs_2\otimes\Hs_1$. As we show in the appendix \ref{appr13}, the only
two-outcome testers of the above form that are not extremal are those with $\ket{\phi_1}$ being a product state.
Looking on how the considered type of testers transforms under superoperator $\xi_{\rho,\one}$ from equation (\ref{transftesters}) one can easily
conclude that also for arbitrary rank two normalization $\rho$ the two outcome testers with $(r_1,r_2)=(1,3)$ are extremal if and only if
$\ket{\phi_1}$ is not a product state.
The case $ii)$ $(r_1,r_2)=(2,2)$ has some similarities to the previous one. Since $T_1, T_2$ are both rank two and their sum is $\frac12
\one\otimes\one$, then they both must be equal to $\frac12 P_i$, where $P_i$ are orthogonal projectors. Consequently, we can write the tester as
\begin{eqnarray}
T_1&=&\frac{1}{2}P_1=\frac{1}{2}(\ket{\phi_1}\bra{\phi_1}+\ket{\phi_2}\bra{\phi_2}); \nonumber\\
T_2&=&\frac{1}{2}P_2=\frac{1}{2}(\ket{\phi_3}\bra{\phi_3}+\ket{\phi_4}\bra{\phi_4}); \label{qtester22}
\end{eqnarray}
where vectors $\ket{\phi_i}\; i=1,\ldots,4$ form an orthonormal basis of $\Hs_2\otimes\Hs_1$. As we show in the appendix \ref{appr22} this type of
tester is not extremal only if $P_1=\one\otimes\ket{v}\bra{v}$ for some $\ket{v}\in\Hs_1$ or if the states $\ket{\phi_1}$, $\ket{\phi_3}$ can be chosen
as $\ket{\phi_1}=\ket{w}\otimes\ket{v}$, $\ket{\phi_3}=\ket{w^{\perp}}\otimes\ket{v}$ for some states $\ket{w}\in\Hs_2$, $\ket{v}\in\Hs_1$.
For arbitrary rank two normalization $\rho$ the conditions on extremality of this type of tester are very similar, but with $P_1,P_2$ playing the role
of projectors onto the support of $T_1, T_2$.
\subsubsection{$M$-outcome testers}
The analysis of extremal qubit testers for more than two outcomes is
very involved. For this reason, we provide only some examples how one
can construct them. Extremal qubit $1$-testers with $3$ or $4$
outcomes can be easily obtained by taking the extremal $2$-outcome
tester from Eq. (\ref{qtester22}) and splitting either one or both its
parts into rank one operators. Obviously this operation reduces the
subspace achievable by linear combination of operators with support on
$T_i$, thus the linear independence with the operators
$\sigma_i\otimes\one$ remains untouched and the tester obtained in
this way is extremal. A different approach allows us to generate
examples of extremal testers with up to $M\leq 10$ as follows. Let us
consider an extremal $2$-outcome tester from Eq. (\ref{qtester13}) and
let us split its element $T_2$ into $T^{\prime}_2,\ldots,
T^{\prime}_M$ in such a way that $\{2 T^{\prime}_i\}_{i=2}^M$ is an
extremal POVM on the support of $T_2$. By setting $T^{\prime}_1=T_1$
we obtain an extremal tester $\{T^{\prime}_i\}_{i=1}^M$, because we
are only restricting the operator span of allowed perturbations of the
elements $T_i$ and perturbations of
$T_2^{\prime},\ldots\,T_M^{\prime}$ are independent by construction.
Finally, one can use the technique of Heinossari and Pellonp\"a\"a
\cite{heinosaari1} (see Proposition 4) to construct extremal qubit
$1$-testers with $4\leq M \leq 13$ rank $1$ elements. The construction
generates $M+1$ outcome tester from the $M$ outcome tester until the
linear independence of rank $1$ elements with the operators
$\sigma_i\otimes\one$ can be kept (i.e. $M\leq 13$).
\section{Extremality of quantum channels}
\label{sec:excombs}
The aim of this section is to show how our general criterion from
Theorem \ref{th1} in the case of channels ($N=1$, $M=1$) relates to
known conditions of extremality. For channels mapping from
$\mathcal{L}(\Hs_0)$ to $\mathcal{L}(\Hs_1)$, we have
$\bas{D}_{(N)}=\{\sigma_a\otimes\one, \sigma_a\otimes\sigma_b\}$,
where $\{\sigma_a\}_{a=2}^{d_1^2}$, $\{\sigma_b\}_{b=2}^{d_0^2}$ are
basis of trace zero hermitian operators on $\Hs_1$, $\Hs_0$,
respectively. Suppose we want to test whether a channel $\mathcal E$
with Choi-Jamiolkowski operator $E$ is extremal. If we take the
spectral decomposition of $E=\sum_m |K_m\kk \bb
K_m|$ then the eigenvectors $|K_m\kk$ correspond through
isomorphism \cite{dket} $|A\kk=A\otimes\one
|I\kk$ (here $|I\kk\equiv\sum_i
\ket{i}\otimes\ket{i}\in\Hs_0^{\otimes 2}$) to Kraus operators $K_m$
of a minimal Kraus representation of channel $\mathcal E$. The
well-known Choi extremality condition \cite{choi} writes
\begin{eqnarray}
\sum_{m,n} \alpha_{mn} K_m^\dagger K_n=0 \quad\Leftrightarrow \alpha_{mn}=0\quad\forall m,n.
\label{eq:choi}
\end{eqnarray}
On the other hand, according to our Theorem \ref{th1} the condition
for extremality of channel $\mathcal E$ is that
\begin{align}
\sum_{m,n} &\alpha_{mn} |K_m\kk \bb K_n|+ \sum_a \beta_a \sigma_a\otimes\one + \sum_{a,b} \gamma_{ab} \sigma_a\otimes\sigma_b=0 ,\nonumber\\
&\Leftrightarrow \alpha_{mn}=0,\ \forall m,n,\quad \beta_a=0\ \forall a,\quad\gamma_{ab}=0\ \forall a,b.
\label{our}
\end{align}
We will now prove the following theorem
\begin{theorem}
The conditions in Eq.~\eqref{eq:choi} and Eq.~\eqref{our} are equivalent.
\end{theorem}
\begin{Proof} In order to prove that the condition in
Eq.~\eqref{eq:choi} implies the condition in Eq.~\eqref{our}, it is
sufficient to suppose that Eq.~\eqref{our} holds, and to take the
partial trace on the Hilbert space $\Hs_1$. We then get
\begin{equation}
\sum_{m,n=1}^{rank E} \alpha_{mn} K_m^T K_n^*=0\nonumber,
\end{equation}
which by condition Eq.~\eqref{eq:choi} implies $\alpha_{mn}=0$ for
all $m,n$. Finally, by linear independence of $\{\sigma_a\otimes
I,\sigma_a\otimes\sigma_b\}$, this also implies
$\beta_a=0=\gamma_{ab}$ for all $a$ and $b$. Conversely, one can
write $|K_m\kk \bb K_n|$ as
\begin{equation}
|K_m\kk \bb K_n|= \frac1{d_1}I_1 \otimes K^T_m K_n^* +\Delta_{mn},
\label{partra}
\end{equation}
where $\Tr_1[\Delta_{mn}]=0$ for all $m,n$. This implies that
$\Delta_{mn}$ belongs to the span of $\bas D_{(N)}$ for all $m,n$.
Let us suppose that Choi's condition Eq.~\eqref{eq:choi} is not
satisfied. Then there exist nontrivial coefficients $\zeta_{mn}$
such that $\sum_{m,n}\zeta_{m,n}K^T_m K^*_n=0$. If we then take
$\beta_a$, $\gamma_{ab}$ such that
\begin{equation}
\sum_{mn}\zeta_{mn}\Delta_{mn}=\sum_a\beta_a\sigma_a\otimes I+\sum_{ab}\gamma_{ab}\sigma_a\otimes \sigma_b,
\end{equation}
we have
\begin{equation}
\sum_{mn}\zeta_{mn}|K_m\kk\bb K_n|-\sum_a\beta_a\sigma_a\otimes I-\sum_{ab}\gamma_{ab}\sigma_a\otimes \sigma_b=0,
\end{equation}
in contradiction with Eq.~\eqref{our}.
\end{Proof}
\section{Extremality of quantum instruments}
\label{sec:exinst}
In contrast to a channel ($N=1$, $M=1$), which is specified by its
Choi-Jamiolkowski operator, an instrument ($N=1$, $M\geq 1$) is
characterized by a collection of Choi-Jamiolkowski operators
$\{N_i\}_{i=1}^M \subseteq \mathcal{L}(\Hs_1\otimes\Hs_0)$, which sum up to
Choi-Jamiolkowski operator of some channel $R$. The set
$\bas{D}_{(N)}=\{\sigma_a\otimes\one, \sigma_a\otimes\sigma_b\}$ from
Theorem \ref{th1} is the same as for channels, because it depends only
on $N$, the number of teeth of GQI, but not on $M$ the number of
outcomes of the instrument. We can take the spectral decompositions of
all the Choi-Jamiolkowski operators of the instrument $N_i=\sum_m
|K^{(i)}_m\kk \bb K^{(i)}_m|$ and we can write the necessary and
suffiecient condition of extremality as follows.
\begin{corollary}
Instrument $\{N_i\}_{i=1}^M \subseteq \mathcal{L}(\Hs_1\otimes\Hs_0)$ is extremal if and only if equation
\begin{eqnarray}
\sum_{i,m,n}\alpha^i_{mn} |K^{(i)}_m\kk \bb K^{(i)}_n|+ \sum_a \beta_a \sigma_a\otimes\one
+\sum_{a,b} \gamma_{ab} \sigma_a\otimes\sigma_b&=&0 \label{excondinst}
\end{eqnarray}
cannot be satisfied for non-trivial coefficients $\alpha^i_{mn},
\beta_a, \gamma_{ab}$.
\end{corollary}
Counting the terms in Eq. (\ref{excondinst}) that have to be linearly
independent elements of $\mathcal{L}(\Hs_1\otimes\Hs_0)$, we can
obtain a simple restriction on the ranks of the elements of the
extremal instrument.
\begin{corollary}
An extremal instrument $\{N_i\}_{i=1}^M \subseteq
\mathcal{L}(\Hs_1\otimes\Hs_0)$ satisfies the following inequality
\begin{eqnarray}
\sum_{i} r_i^2 \leq (d_0)^2,
\end{eqnarray}
where $r_i$ denotes the rank of $N_i$ and $d_0=\dim \Hs_0$.
\end{corollary}
We will now prove a theorem that provides an equivalent, but more
practical, extremality condition for quantum instruments.
\begin{theorem}
An instrument $\{\mathcal N_i\}_{i=1}^M$ with Choi-Jamio\l kowski
operators $\{N_i=\sum_{m}|K_m^{(i)}\kk\bb K^{(i)}_m|\}_{i=1}^M$ is
extremal if and only if the operators $\{K_m^{(i)\dagger}
K^{(i)}_n\}$ are linearly independent.\label{theoinst}
\end{theorem}
\begin{Proof}
Suppose that the operators $\{K_m^{(i)\dagger} K^{(i)}_n\}$ are
linearly independent. Then if Eq.~\eqref{excondinst} is satisfied,
also its partial trace over space $\Hs_1$ is satisfied, namely
\begin{equation}
\sum_{i,m,n}\alpha^i_{mn} {K^{(i)}_m}^T{K^{(i)}_n}^*=0,
\end{equation}
which implies $\alpha^{(i)}_{mn}=0$ for all $i,m,n$ and consequently
also $\beta_a=0$ for all $a$ and $\gamma_{ab}=0$ for all $a,b$.
Conversely, consider the extremality condition in
Eq.~\eqref{excondinst} along with the following generalization of
Eq.~\eqref{partra}
\begin{equation}
|K_m^{(i)}\kk\bb K_n^{(i)}|=\frac1{d_1}I_1\otimes {K^{(i)}_m}^T{K_n^{(i)}}^*+\Delta^{(i)}_{mn},
\end{equation}
where the operators $\Delta^{(i)}_{mn}$ belong to the span of $\bas
D_{(N)}$. If the operators $\{K_m^{(i)\dagger} K^{(i)}_n\}$ are not
linearly independent, then there are non-trivial coefficients
$\zeta^{(i)}_{mn}$ such that
$\sum_{i,m,n}\zeta^{(i)}_{mn}{K_m^{(i)}}^T{K^{(i)}_n}^*=0$. Then,
taking $\beta_a$ and $\gamma_{ab}$ such that
\begin{equation}
\sum_{i,m,n}\zeta^{(i)}_{mn}\Delta^{(i)}_{mn}=\sum_a\beta_a\sigma_a\otimes I+\sum_{ab}\gamma_{ab}\sigma_a\otimes \sigma_b,
\end{equation}
we have
\begin{equation}
\sum_{i,m,n}\zeta^{(i)}_{mn}|K^{(i)}_m\kk\bb K^{(i)}_n|-\sum_a\beta_a\sigma_a\otimes I-\sum_{ab}\gamma_{ab}\sigma_a\otimes \sigma_b=0,
\end{equation}
in contradiction with Eq.~\eqref{excondinst}.
\end{Proof}
\subsection{Extremality of Von Neuman-L\"uders instruments}
Let us now consider instruments of the following type
\begin{equation}
\mathcal N_i(\rho)=\sqrt P_i\rho\sqrt P_i,
\label{sqrtinst}
\end{equation}
where $P_i$ is a POVM. Then, by theorem \ref{theoinst}, the instrument
is extremal if and only if the POVM $\{P_i\}_{i=1}^M$ is linearly
independent. Indeed, the set $\{K^{(i)\dagger}_mK^{(i)}_n\}$ in this
case is provided precisely by $\{P_i\}_{i=1}^M$.
In particular, von Neuman-L\"uders instruments are extremal. Indeed,
every such instrument $\{ N_i\}_{i=1}^d$ is of the form of
Eq.~\eqref{sqrtinst} with $P_i=\Pi_i$ where
$\Pi_i\Pi_j=\delta_{ij}\Pi_i$. Using the last constraint it is easy to
prove that if $X=\sum_i \alpha_i \Pi_i=0$ then
$\Pi_jX=\alpha_j\Pi_j=0$ and consequently $\alpha_j=0$.
Since there exist POVMs that are not extremal, but have linearly
independent elements, one can easily construct examples of extremal
instruments, with non-extremal POVMs. For example, this is the case
with $d=2$ and $P_1=1/2|0\>\<0|$, $P_2=1/2|0\>\<0|+|1\>\<1|$.
\section{Conclusions}
\label{sec:conclusion}
The aim of this paper was to characterize the extremal points of the set of generalized quantum instruments (GQIs). Our main result is represented by
theorem \ref{th1}, which links extremality of the considered \GQI with linear independence of a set of operators. An important special case of GQIs are
Quantum testers. For quantum $1$-testers we derived necessary and sufficient criterion of extremality that differs from the application of general
theorem \ref{th1} and can be tested more easily. As a consequence of the criterion, we obtained a bound (\ref{bound1}) on the ranks of elements of the
extremal $1$-tester. We showed that the subsets of extremal $1$-testers with a fixed normalization are isomorphic if they have the same rank of the
normalization. This implies that to classify all extremal $1$-testers it suffices to study extremal $1$-testers with a completely mixed normalization
($\one_2\otimes\rho_1=\frac1{d_1}I_{21}$). We completely characterized qubit $1$-testers with $1$ and $2$ outcomes and provided techniques to construct
extremal qubit testers with up to $13$ outcomes, which is the maximal number allowed by the bound (\ref{boundnumber}).
In section \ref{sec:excombs} we apply our extremality condition from
theorem \ref{th1} to channels. The resulting condition is different
from the well known criterion of Choi \cite{choi}, even though we
prove it to be equivalent. The section \ref{sec:exinst} presents the
first characterization of the extremality of
instruments.
In particular, we show that instruments of the type defined in
Eq.~\eqref{sqrtinst} for POVMs $\{P_i\}_{i=1}^M$ with linearly independent
elements are extremal quantum instruments.
More generally, any quantum instrument determines not only a POVM,
when the quantum output is ignored, but also a quantum channel, when
the classical outcome is ignored. A natural question is then what
combinations of extremality can exist when we consider an instrument
along with the POVM and channel it defines. In appendix
\ref{app:examples} we present examples of instruments for seven out of
the eight possibilities. The question whether non-extremal instruments
exist, such that they determine extremal POVMs and extremal channels
is left as an open problem.
\section*{Acknowledgments}
We thank the anonymous referee for a question that stimulated us to write appendix \ref{app:examples}.
This work has been supported by the European Union through FP$7$ STREP project COQUIT and by the Italian Ministry of Education through grant PRIN 2008
Quantum Circuit Architecture.
\appendix
\section{Parametrization of the set of deterministic quantum combs}
\label{secparamcombs}
Suppose we want to choose such parametrization of the set of hermitian
operators in which the subset of deterministic quantum combs would
simply correspond to positive operators that have some parameters
fixed (e.g. to zero). Let us consider a quantum $N$-combs $R\in
\mathcal{L}(\Hs_{2N-1}\otimes\ldots\otimes\Hs_0)$. For each of the
Hilbert spaces $\Hs_k$ $k\in\{0,..,2N-1\}$ we choose a basis of
hermitian operators on $\Hs_k$ $\{E^{(k)}_a\}_{a=1}^{d_k^2}$ such that
$E^{(k)}_1=\one$ and all the other elements have zero trace. Taking
the tensor product of the basis elements
for all the Hilbert spaces $\Hs_k$ we obtain
a basis of hermitian operators $\{ E^{(2N-1)}_{a_{2N-1}}\otimes\ldots\otimes E^{(0)}_{a_{0}} \}$
on $\Hs_{2N-1}\otimes\ldots\otimes\Hs_{0}$.
Let us now use this basis to illustrate the normalization cascade requirements on the quantum $1$-combs i.e. Choi operators of quantum channels. In
this case a quantum channel mapping from ${\mathcal L}(\Hs_0)$ to ${\mathcal L}(\Hs_1)$ is represented via Choi-Jamiolkowski isomorphism by a positive
operator $R\in \mathcal{L}(\Hs_1\otimes\Hs_0)$, which has to fulfil equation $\Tr_1{R}=\one_0$. Using our basis arbitrary $R$ can be written as
\begin{eqnarray}
R&=&\sum_{a_1=1}^{d^2_1}\sum_{a_0=1}^{d^2_0} c_{a_1 a_0}\, E^{(1)}_{a_1}\otimes E^{(0)}_{a_0} \nonumber\\
&=&c_{11}\one_1\otimes\one_0 +\one_1\otimes\sum_{a_0=2}^{d^2_1} c_{1 a_0} E^{(0)}_{a_0}+ \label{chanpar1}\\
& &+\sum_{a_1=2}^{d^2_1}E^{(1)}_{a_1}\otimes\sum_{a_0=1}^{d^2_0} c_{a_1 a_0} E^{(0)}_{a_0} \nonumber
\end{eqnarray}
Let us now look how the three terms of the RHS of Eq. (\ref{chanpar1}) contribute to $\Tr_1(R)$. The first two terms do contribute, whereas the
remaining one does not. The requirement of $\Tr_1(R)=\one_0$ translates into the following equations $c_{11}=\frac{1}{d_1}, \; c_{1i}=0 \;\forall
i=2,\ldots,d_0$ for parameters $c_{a_1 a_0}$. Thus, each quantum $1$-comb (Choi operator of a channel) can be written as
\begin{eqnarray}
R=\frac{1}{d_1} \one_{10}+\sum_i E^{(1)}_i\otimes A_i,
\label{param1}
\end{eqnarray}
where
$A_i$ are arbitrary hermitian operators on $\Hs_0$.
Previous statements can be easily generalized to the case of general quantum combs. We shall first illustrate the relation of expansions for $R^{(n)}$
and $R^{(n-1)}$ and then write the expansion of general quantum $N$-comb. In our basis $R^{(n)}$ can be written as:
\begin{eqnarray}
R^{(n)}&=&\sum_{a_{2n-1},a_{2n-2}} E^{(2n-1)}_{a_{2n-1}}\otimes E^{(2n-2)}_{a_{2n-2}}\otimes L^{a_{2n-1},a_{2n-2}}, \nonumber \\
&=&\one_{2n-1,2n-2}\otimes L^{1,1}+\one_{2n-1}\otimes\sum_{j=2}^{d^2_{2n-2}} E^{(2n-2)}_{j}\otimes L^{1,j}+ \nonumber\\
& &+\sum_{i=2}^{d^2_{2n-1}}E^{(2n-1)}_{i}\otimes\sum_{j=1}^{d^2_{2n-2}} E^{(2n-2)}_{j}\otimes L^{i,j} \label{relrntorn1}
\end{eqnarray}
where
\begin{eqnarray}
L^{a_{2n-1},a_{2n-2}}=\sum_{a_{2n-3} \cdots a_0} c_{a_{2n-1}\cdots a_0}\,E^{(2n-3)}_{a_{2n-3}}\otimes\cdots\otimes E^{(0)}_{a_0} \nonumber
\end{eqnarray}
and we expanded the two sums in the same way as in (\ref{chanpar1}).
The normalization cascade (\ref{normcascade}) requires\cite{note2}
that
\begin{eqnarray}
L^{1,1}&=&\frac{1}{d_{2n-1}}R_{(n-1)}\;,\quad L^{1,j}=0 \; \forall j
\end{eqnarray}
and the operator $\sum_{j=1}^{d^2_{2n-2}} E^{(2n-2)}_{j}\otimes L^{i,j}$ can be an arbitrary operator on $\Hs_{2n-2}\otimes\cdots\otimes\Hs_0$.
As a result
\begin{eqnarray}
R_{(n)}&=&\frac{1}{d_{2n-1}}\one_{2n-1,2n-2}\otimes R_{(n-1)}+\sum_{i=2}^{d^2_{2n-1}} E^{(2n-1)}_i\otimes B_i, \nonumber\\
\label{recursiveform}
\end{eqnarray}
where
$B_i\in {\mathcal L}(\Hs_{2n-2}\otimes\cdots\otimes\Hs_0)$.
Using the above relation recursively we can write the parametrization of the general deterministic $N$-comb as
\begin{eqnarray}
R_{(N)}&=&\frac{1}{d_{2N-1}d_{2N-3}\ldots d_1}\one_{2N-1,\ldots,0}+ \nonumber \\
& &+\sum_{i=2} E^{(2N-1)}_i\otimes B^{(2N-2)}_i+ \nonumber \\
& &+\one_{2N-1,2N-2}\otimes \sum_{i=2} E^{(2N-3)}_i \otimes B^{(2N-4)}_i+ \ldots+ \nonumber \\
& &+ \one_{2N-1,\ldots,2}\otimes \sum_{i=2} E^{(1)}_i \otimes B^{(0)}_i, \label{param2}
\end{eqnarray}
where $B^{(k)}_i$ are arbitrary hermitian operators acting on
$\Hs_k\otimes\Hs_{k-1}\otimes\cdots \otimes\Hs_0$. Let us denote the
basis for hermitian operators $B_i^{(k)}$ as
$\{F^{(k)}_j\}_{j=1}^{d^2_k\ldots d^2_0}$. Consequently, the basis
used for the variable part (i.e. all terms in (\ref{param2}) except
the first) of the quantum comb is $\{E^{(2N-1)}_i\otimes F^{(2N-2)}_j,
\one_{2N-1,2N-2}\otimes E^{(2N-3)}_i \otimes F^{(2N-4)}_j, \ldots,
\one_{2N-1,\ldots,2}\otimes E^{(1)}_i \otimes F^{(0)}_j \}$ and we
denote it as $\bas{D}_{(N)}$. The operator basis $\mathbb{R}_{(N)}$
sufficient to expand arbitrary deterministic comb is then formed by
$\{\one_{2N-1,\ldots,0}\}\cup \bas{D}_{(N)}$.
\section {Two outcome qubit $1$-testers}
Suppose we have a two-outcome qubit $1$-tester $\{T_i=\frac{1}{2}P_i\}_{i=1}^2$ with normalization $\one_2\otimes \frac12 \one_1$ and $P_i$ being
orthogonal projectors. Equivalently to Theorem \ref{th2} we can say that the two outcome $1$-tester is extremal if and only if $\esp{V}_\sigma \cap
\esp{V}_T = 0$, where
$\esp{V}_\sigma=\operatorname{Span} \{\one\otimes\sigma_k\}_{k=x,y,z}$ and
$\esp{V}_T$
is the direct sum of the two subspaces of hermitian operators with support in $P_1$ and in $P_2$, respectively.
The non existence of the intersection of $\esp{V}_\sigma$ and $\esp{V}_T$ can be stated also as the impossibility to fulfill the following equation
\begin{eqnarray}
\sum_{i=1}^4 \lambda_i \ket{\phi_i}\bra{\phi_i}=\one\otimes(n_x\sigma_x + n_y\sigma_y + n_z\sigma_z),
\label{perturbc22}
\end{eqnarray}
where the left hand side of Eq. (B1) represents a generic element in $\esp{V}_T$ and the right hand side a generic element of $\esp{V}_\sigma$.
The set of $\{|\phi_i\>\}_{i=1}^4$ forms an orthonormal basis of vectors
belonging to $\Supp(P_1)\cup\Supp(P_2)$, and without loss of generality we can take $n^2_x + n^2_y + n^2_z = 1$. This guarantees that the RHS has the spectral decomposition of the following form
\begin{eqnarray}
\one\otimes\ket{v}\bra{v}-\one\otimes\ket{v^{\perp}}\bra{v^{\perp}}\nonumber
\end{eqnarray}
with two $+1$ eigenvalues and two $-1$ eigenvalues and vector $\ket{v}$ that can be arbitrary thanks to freedom in $n_x,n_y,n_z$.
Moreover projectors $P_i$ can be written as $P_1=\sum_{i=1}^{r_1}\ket{\phi_i}\bra{\phi_i}$ and $P_2=\sum_{i=r_1+1}^4\ket{\phi_i}\bra{\phi_i}$.
Let us now investigate the circumstances under which the equation can be fullfilled i.e. the tester is not extremal.
\subsection{Case $(r_1,r_2)=(1,3)$}
\label{appr13}
This type of tester must have the form
$\{T_1=\frac{1}{2}\ket{\phi_1}\bra{\phi_1},T_2=\frac{1}{2}\sum_{i=2}^4\ket{\phi_i}\bra{\phi_i}\}$.
The LHS of Eq. (\ref{perturbc22}) must have the same eigenvalues as
the RHS. Without loss of generality we can assume
$\lambda_1=\lambda_2=-\lambda_3=-\lambda_4$, because we can suitably
relabel $\ket{\phi_2},\ket{\phi_3},\ket{\phi_4}$. Hence, we have
\begin{eqnarray}
\ket{\phi_1}\bra{\phi_1}+ \ket{\phi_2}\bra{\phi_2}&=&\one\otimes \ket{e}\bra{e} \nonumber\\
\ket{\phi_3}\bra{\phi_3}+ \ket{\phi_4}\bra{\phi_4}&=&\one\otimes \ket{e^{\perp}}\bra{e^{\perp}},
\label{perturb13}
\end{eqnarray}
where $e=v$ or $e=v^{\perp}$ depending on $\lambda_1=+1$ or $\lambda_1=-1$, respectively. In both cases Eq. (\ref{perturb13}) implies that the qubit
$1$-tester of the form $T_1=\frac{1}{2}\ket{\phi_1}\bra{\phi_1}$ $ T_2=\frac{1}{2}(\one\otimes\one-\ket{\phi_1}\bra{\phi_1})$ is not extremal if and
only if $\ket{\phi_1}=\ket{f}\otimes\ket{e}$ is a product vector.
\subsection{Case $(r_1,r_2)=(2,2)$}
\label{appr22}
In this case the tester has the form $\{T_1=\frac{1}{2}P_1=\frac{1}{2}(\ket{\phi_1}\bra{\phi_1}+\ket{\phi_2}\bra{\phi_2})$,
$T_2=\frac{1}{2}P_2=\frac{1}{2}(\ket{\phi_3}\bra{\phi_3}+\ket{\phi_4}\bra{\phi_4})$. In order to fulfill the equation (\ref{perturbc22}) two
$\lambda_i$'s must be equal to $+1$ and two to $-1$. Thus, $\lambda_1$, $\lambda_2$ have either same signs or different signs. If
$\lambda_1=\lambda_2=\pm 1$ then the equation (\ref{perturbc22}) can be fulfilled if and only if $P_1=\one\otimes\ket{e}\bra{e}$, where $e=v$ for
$\lambda_{1,2}=1$ or $e=v^{\perp}$ for $\lambda_{1,2}=-1$. If $\lambda_1=-\lambda_2$ then we can assume without loss of generality that $\ket{\phi_3}$,
$\ket{\phi_4}$ are labeled so that $\lambda_1=\lambda_3$. We have
\begin{eqnarray}
\ket{\phi_1}\bra{\phi_1}+\ket{\phi_3}\bra{\phi_3}=\one\otimes\ket{e}\bra{e},
\label{pomeq1}
\end{eqnarray}
where $e=v$ or $e=v^\perp$ depending on $\lambda_1=1$ or $\lambda_1=-1$, respectively. This may hold only if $\ket{\phi_1}=\ket{f}\otimes\ket{e}$ and
$\ket{\phi_3}=\ket{f^\perp}\otimes\ket{e}$ for some vector $\ket{f}\in\Hs_2$. Due to equation (\ref{pomeq1})
$\ket{\phi_2}\bra{\phi_2}+\ket{\phi_4}\bra{\phi_4}=\one\otimes\ket{e^\perp}\bra{e^\perp}$ and $\ket{\phi_2}=\ket{h}\otimes\ket{e^\perp}$,
$\ket{\phi_4}=\ket{h^\perp}\otimes\ket{e^\perp}$ for some $\ket{h}\in\Hs_2$.
Thus, if $\lambda_1=-\lambda_2$ the tester is not extremal if and only if
\begin{eqnarray}
P_1&=&\ket{f}\bra{f}\otimes\ket{e}\bra{e}+\ket{h}\bra{h}\otimes\ket{e^\perp}\bra{e^\perp} \nonumber \\
P_2&=&\ket{f^\perp}\bra{f^\perp}\otimes\ket{e}\bra{e}+\ket{h^\perp}\bra{h^\perp}\otimes\ket{e^\perp}\bra{e^\perp}. \nonumber
\label{pomeq2}
\end{eqnarray}
for some $\ket{e}\in\Hs_1$, $\ket{f},\ket{h}\in\Hs_2$\cite{note3}. The
form of projectors $P_1$, $P_2$ can be equivalently stated as the
existence of a product vector $\ket{f}\otimes\ket{e}$ in the support
of $P_1$ such that $\ket{f^\perp}\otimes\ket{e}$ belongs to the
support of $P_2$. From our derivation it should be clear that if
$P_1\neq \one\otimes\ket{e}\bra{e}$ for any $\ket{e}\in\Hs_1$ and
$P_1$ does not have the above mentioned form then $\{T_1,T_2\}$ is an
extremal qubit two-outcome tester.
\section{Extremality of an instrument and the POVM and the channel
derived from it}
\label{app:examples}
The present Appendix addresses the question about possible
combinations of extremality of an instrument and the POVM and channel
derived from it. We show feasibility of seven out of the eight
possible combinations, by providing an example for each of them. In
the following table, we enumearte the possible combinations, and we
define them by writing $+$ if the object in the corresponding column
(channel, POVM, instrument) is extremal, and $-$ otherwise.
\bigskip
\begin{tabular}{|c|c|c|c|}
\hline
Combination & Instrument & Channel & POVM \\
\hline
1 & + & + & + \\
2 & + & + & - \\
3 & + & - & + \\
4 & + & - & - \\
5 & - & + & + \\
6 & - & + & - \\
7 & - & - & + \\
8 & - & - & - \\
\hline
\end{tabular}
\bigskip
The existence of an instrument corresponding to combination number 5 is left as an
open problem. Here is a list of examples for each of the remaining
combinations.
\noindent {\bf Combination 1:}
The identical transformation is the most simple example of this kind. Also
constant mapping to a fixed pure state has the desired properties of
extremality.
\noindent {\bf Combination 2:}
Consider an instrument with two outcomes mapping a single qubit into
two-qubits.
First, we define
\begin{eqnarray}
P_0&:=&\frac13 \ket{0}\bra{0}+\frac23 \ket{1}\bra{1} \nonumber \\
P_1&:=&\frac23 \ket{0}\bra{0}+\frac13 \ket{1}\bra{1} \nonumber \\
W&:=&\ket{0}\bra{1}-\ket{1}\bra{0} \nonumber
\end{eqnarray}
and we define the Kraus operators of the instrument as follows
\begin{eqnarray}
M_0&:=&\sqrt{P_0} \otimes \frac{1}{\sqrt{2}} (\ket{0} + \ket{1}) \nonumber \\
M_1&:=&\frac{1}{\sqrt{2}} \sqrt{P_1} \otimes \ket{0} + \frac{1}{\sqrt{2}} W\sqrt{P_1} \otimes \ket{1}, \nonumber
\end{eqnarray}
where for each outcome $i=0,1$ we have only a single Kraus operator.
One can easily verify, that the induced POVM $\{P_0,P_1\}$ is not
extremal, but linear independence of its elements guarantees
extremality of the instrument. In order to check extremality of the
induced channel one needs to take the minimal Kraus representation and
check Choi's linear independence condition.
\noindent {\bf Combination 3:} The L\"uders instrument of a Von
Neumann measurement is an extremal instrument, which induces extremal
POVM and a non extremal channel.
\noindent {\bf Combination 4:} This desired type of instrument can be
constructed as in Eq.~\eqref{sqrtinst} with a POVM $\{P_i\}_{i=1}^M$, whose
elements are linearly independent and commute. As a simple example one
can take the qubit POVM $\{P_0, P_1\}$ defined in Combination 2.
\noindent {\bf Combination 6:} This type of instrument can be
constructed as follows. One takes an extremal channel, whose minimal
dilation has $N$ (more than one) Kraus operators $K_i$. Using these
operators we define two instruments with $N$ outcomes differing only
in the choice of Kraus operators that correspond to each outcome (e.g.
$M^{(1)}_i=K_i, M^{(2)}_i=K_{\sigma(i)}$, where $\sigma$ is a
permutation). Taking a convex combination of the two instruments
provides the desired example.
\noindent {\bf Combination 7:} This type of instrument can be
constructed as a convex combination of two instruments, which induce
the same POVM, but different channels. One takes for example the
instrument $\{\mathcal N_i \}_{i=1}^M$ as in Eq.~\eqref{sqrtinst} for an
extremal POVM $\{P_i\}_{i=1}^M$, and mixes it with the same instrument, which
in addition applies an unitary channel $U$ on the quantum output.
Obviously, the induced channel differs, while the induced POVM remains
the same.
\noindent {\bf Combination 8:} For the construction of this example it
is sufficient to take a convex combination of two instruments, which
induce different POVMs, and different channels.
| 60,621
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TITLE: If $a$ and $b$ have order $n$ and $m,$ and say $a^p=b^q$ for some integers $p$ and $q$. How many elements are there of the form $a^sb^t$?
QUESTION [2 upvotes]: If $a$ and $b$ have order $n$ and $m$, and say $a^p=b^q$ for some integers $p, q$ with $0<p\leq n$ and $0<q\leq m$. How many elements are there of the form $a^sb^t$?
I tried solving a few examples. I took $n=4$ and $m=5$, and assumed $a^2=b^4$. Now, all elements of the form $a^2b^s$ reduce to some power of $b$. Also, all elements of the form $a^pb^s$ reduce to some product of a power of $a$ and power of $b$, where the power of $a$ is computed under arithmetic modulo $2$.
So I think there will be $pq$ distinct elements of the form $a^sb^t$.
Can we get a proof of this result?
REPLY [1 votes]: Your result does not hold. I will give you a hint to help you find the actual result yourself:
If $a$ and $b$ have order $n$ and $m$, and say $a^p=b^q$ for some integers $0<p\leq n$ and $0<q\leq m$, then every element can be written uniquely in the form $a^sb^t$ where $0\leq s<n$ and $0\leq t<q$.
(Note the "$0\leq t<q$" and not "$0\leq t<m$".)
| 200,444
|
TITLE: How can extra (non-curled up) dimensions be hidden from us?
QUESTION [1 upvotes]: Wikipedia says:
If extra dimensions exist, they must be hidden from us by some
physical mechanism. One well-studied possibility is that the extra
dimensions may be "curled up" at such tiny scales as to be effectively
invisible to current experiments.
I'm just curious, could there be any plausible "hiding" mechanism except the above?
Is there any theory dealing with such other possibilities?
REPLY [3 votes]: There are models where the extra dimensions don't need to be curled up.
The main issue with extra dimensions is, 'why don't the particles/fields we interact with travel in those directions?' We have extremely good limits on standard model particles (electrons, photons) travelling in extra dimensions.
However, it is possible to imagine a string inspired scenario where standard model fields are confined to a 'brane'--that is, some 3 dimensional spatial surface living in a higher dimensional space. Then the particles we interact with simply are not free to travel in the extra dimension.
Then the issue becomes confining gravity to the brane. Already the experimental constraints on gravity are much weaker than the standard model fields--this is the essence behind the proposal of Arkhani-Hamed, Dimopolous, and Dvali (ADD) (http://arxiv.org/abs/hep-ph/9803315). In that proposal the extra dimensions are still curled, but are allowed to be much larger than the limits coming from observations of standard model particles.
There are other models though where gravity looks four dimensional to us but the extra dimensions are not infinite. Probably the most famous example is the Randall Sundrum model (http://arxiv.org/abs/hep-ph/9905221). There, the extra dimensions are 'warped,' and the warping has the net effect of having gravity be localized near our brane. They are not compact, however. Another example is the model of Dvali, Gabadadze, and Porrati (DGP), with a brane living in an infinite dimensional flat space-time (http://arxiv.org/abs/hep-th/0005016). The way gravity is confined in that case is that there are two gravitational constants, a "4d" one living on the brane and a "5d" one living in the bulk, and the relative sizes of the constants is chosen so that on short distances on the brane the gravitational force is dominated by the 4d gravitational constant.
None of these models have any observational evidence in their favor. In the case of the DGP model there are also various theoretical problems--there is superluminal propagation around certain backgrounds, and there is a "ghost instability" around the background you would have wanted to use for cosmology.
| 177,474
|
TITLE: Why is every conjugacy class of an abelian group composed of a single element?
QUESTION [0 upvotes]: I know that if a group is generated by a single element then the group is abelian but does this mean that if a group is abelian then its conjugacy class is composed of a single element?
REPLY [2 votes]: For any group $G$, the conjugacy class $[g]$ of any $g \in G$ is defined to be the set of all elements of $G$ which are conjugate to $g$, that is
$[g] = \{ xgx^{-1} \mid x \in G \}; \tag 1$
if $G$ is abelian, then
$xgx^{-1} = xx^{-1}g = e g = g \tag 2$
where $e \in G$ is the identity element; thus, by virtue of (2) we easily see that
$[g] = \{g\}, \tag 3$
the singleton set whose only element is $g$.
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Eddie Jones backs young England squad to 'be great' and beat Wales after World Cup pain
TWICKENHAM is set for a "humungous" likely Six Nations decider today when England and Wales square up again for the sequel to their dramatic World Cup clash.
Eddie Jones is potentially one win away from claiming the Six nations title with England
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The championship's two remaining unbeaten teams collide with England knowing they will take the trophy for the first time in five seasons if they take revenge on Wales and France lose to Scotland on Sunday. Coach Eddie Jones is urging his team to seize the moment.
"We have got an average age of 24, most of the bench wouldn't know rugby before the 90s, but it is a fantastic and exciting opportunity for these players to be great now," said Jones.
"We've got the ability, we've got the talent, we've got the desire. For our players the time is now."
But title rivals Wales arrive with no fear having turned England over in their own backyard in September.
"We're treating this as a cup final. Whoever wins will go a long way to winning the Six Nations title," said Wales kicking coach Neil Jenkins.
"It's do or die for us and if we lose we are out. Then England have a Triple Crown and will go to Paris fancying their chances of a Grand Slam. But if we win, we face Italy for the title and we would always back ourselves against any side in Cardiff.
"Wales-England would be massive even if we had both lost all our games. But with so much at stake, it's humungous."
Wales have the big-match pedigree and are the more experienced unit but England possess the player of the championship so far in Billy Vunipola, and the 20st No8 is primed for another storming performance today.
"By nature and by size, Billy can create havoc due to his explosiveness with ball in hand and any coach would be foolhardy not to utilise that to its best potential," said England defence coach Paul Gustard.
"Billy is massively important to a team because of what he can bring and the momentum he can create with his ball-carrying. We want that to continue at the weekend but we might use him a bit differently. We'll see."
England centre Henry Slade makes his comeback three months after breaking his leg for Exeter against Newcastle today.
The Premiership will break new ground later with London Irish taking on leaders Saracens in New York.
| 221,043
|
\begin{document}
\begin{abstract}
We show that smooth well formed weighted complete intersections have finite
automorphism groups, with several obvious exceptions.
\end{abstract}
\maketitle
\section{Introduction}
\label{section:intro}
Studying algebraic varieties, it is important to understand their automorphism groups.
In some particular cases these groups have especially nice structure. For instance,
recall the following classical result due to H.\,Matsumura and P.\,Monsky (cf.~\cite[Lemma 14.2]{KS58}).
\begin{theorem}[{see~\cite[Theorems~1 and~2]{MM63}}]
\label{theorem:MM}
Let $X\subset\P^N$, $N\ge 3$, be a smooth hypersurface of degree $d\ge 3$.
Suppose that $(N,d)\neq (3,4)$.
Then the group~\mbox{$\Aut(X)$} is finite.
\end{theorem}
The following beautiful generalization of Theorem~\ref{theorem:MM} was proved
by O.\,Benoist.
\begin{theorem}[{\cite[Theorem~3.1]{Be13}}]
\label{theorem:Benoist}
Let $X$ be a smooth
complete intersection of dimension at least $2$ in
$\P^N$ that is not contained in a hyperlplane.
Suppose that $X$ does not coincide with~$\P^N$, is not a
quadric hypersurface in $\P^N$, and is not a $K3$ surface.
Then the group~\mbox{$\Aut(X)$} is finite.
\end{theorem}
The goal of this paper is to generalize Theorems~\ref{theorem:MM}
and~\ref{theorem:Benoist} to the case of smooth weighted complete intersections.
We refer the reader to~\cite{Do82} and~\cite{IF00} (or to~\S\ref{section:preliminaries} below) for
definitions and basic properties of weighted projective spaces and complete intersections therein.
Our main result is as follows.
\begin{theorem}
\label{theorem:automorphisms}
Let $X$ be a smooth well formed
weighted complete intersection of dimension~$n$.
Suppose that either $n\ge 3$, or $K_X\neq 0$.
Then the group~\mbox{$\Aut(X)$} is finite unless
$X$ is isomorphic either to $\P^n$ or to a quadric hypersurface in~$\P^{n+1}$.
\end{theorem}
Under a minor additional assumption (cf. Definition~\ref{definition:cone}
below) one can make the assertion of
Theorem~\ref{theorem:automorphisms} more precise.
\begin{corollary}
\label{corollary:automorphisms}
Let $X\subset\P$ be a smooth well formed
weighted complete intersection of dimension $n$ that
is not an intersection with a linear cone.
Suppose that either $n\ge 3$, or~\mbox{$K_X\neq 0$}.
Then the group~\mbox{$\Aut(X)$} is finite unless
$X=\P\cong\P^n$ or $X$ is a quadric hypersurface in~\mbox{$\P\cong\P^{n+1}$}.
\end{corollary}
Note that if $X$ is not an intersection with a linear cone,
then the assumption of Theorem~\ref{theorem:automorphisms}
is equivalent to the requirement
that $X$ is not one of the weighted complete intersections
listed in Table~\ref{table:K3} below.
We refer the reader to \cite{HMX}, \cite[Theorem~1.1.2]{KPS18}, and \cite[Theorem~1.2]{CPS19} for other results concerning finiteness
of automorphism groups.
Theorem~\ref{theorem:automorphisms} is mostly implied by the results of
\cite{Fle81} (see Theorem~\ref{theorem:Flenner} below). However, some cases are not
covered by \cite{Fle81} and have to be classified and treated separately
(see Proposition~\ref{proposition:low-coindex}(iv)
and Lemma~\ref{lemma:low-coindex}).
To deduce Corollary~\ref{corollary:automorphisms}
from Theorem~\ref{theorem:automorphisms}, we
need the the following assertion that is well known to
experts but for which we did not manage to find a proper reference (and which we find interesting on its own).
We will say that a weighted complete intersection~\mbox{$X\subset\P=\P(a_0,\ldots,a_N)$}
of multidegree~\mbox{$(d_1,\ldots,d_k)$} is \emph{normalized} if the inequalities~\mbox{$a_0\le\ldots\le a_N$} and $d_1\le\ldots\le d_k$ hold.
\begin{proposition}[{cf.~\cite[Lemma~18.3]{IF00}}]
\label{proposition:unique-embedding}
Let $X\subset\P(a_0,\ldots,a_N)$
and~\mbox{$X'\subset\P(a'_0,\ldots,a'_{N'})$}
be normalized quasi-smooth well formed weighted complete intersections
of multidegrees $(d_1,\ldots,d_k)$ and $(d'_1,\ldots,d'_{k'})$, respectively, such that $X$ and~$X'$ are
not intersections with linear cones. Suppose that $X\cong X'$ and $\dim X\ge 3$.
Then~\mbox{$N=N'$}, $k=k'$, $a_i=a'_i$ for every $0\le i\le N$, and $d_j=d'_j$ for every $1\le j\le k$.
\end{proposition}
When the first draft of this paper was completed,
A.\,Massarenti informed us that a result essentially similar
to Theorem~\ref{theorem:automorphisms} was proved earlier
in \cite[Proposition~5.7]{ACM}.
Note however that in \cite[\S5]{ACM} the authors work
with smooth weighted complete intersections subject to certain
strong additional assumptions (see \cite[Assumptions~5.2]{ACM} for details).
Throughout the paper we work over an algebraically closed field $\CC$ of characteristic zero.
\smallskip
The plan of our paper is as follows.
In~\S\ref{section:preliminaries} we recall some preliminary facts on weighted complete intersections.
In~\S\ref{section:linear-normality} we show uniqueness of presentation of a variety as a weighted complete intersection,
that is, we prove Proposition~\ref{proposition:unique-embedding}. Finally, in~\S\ref{section:automorphisms} we prove Theorem~\ref{theorem:automorphisms}
and Corollary~\ref{corollary:automorphisms}.
\smallskip
We are grateful to
B.\,Fu, B. van~Geemin, A.\,Kuznetsov, A.\,Massarenti,
and T.\,Sano for useful discussions.
We also thank the referee for his suggestions regarding the first draft of the
paper.
\section{Preliminaries}
\label{section:preliminaries}
In this section we recall the basic properties of weighted complete intersections. We refer the reader
to~\cite{Do82} and~\cite{IF00} for more details. Some properties of smooth
weighted complete intersections can be also found in the earlier paper~\cite{Mo75}.
Let $a_0,\ldots,a_N$ be positive integers. Consider the graded algebra~\mbox{$\CC[x_0,\ldots,x_N]$},
where the grading is defined by assigning the weights $a_i$ to the variables~$x_i$.
Put
$$
\P=\P(a_0,\ldots,a_N)=\mathrm{Proj}\,\CC[x_0,\ldots,x_N].
$$
The weighted projective space $\P$ is said to be \emph{well formed} if the greatest common divisor of any $N$ of the weights~$a_i$ is~$1$. Every weighted projective space is isomorphic to a well formed one, see~\cite[1.3.1]{Do82}.
A subvariety $X\subset \P$ is said to be \emph{well formed}
if~$\P$ is well formed and
$$
\mathrm{codim}_X \left( X\cap\mathrm{Sing}\,\P \right)\ge 2,
$$
where the dimension of the empty set is defined to be $-1$.
We say that a subvariety $X\subset\P$ of codimension $k$ is a \emph{weighted complete
intersection of multidegree $(d_1,\ldots,d_k)$} if its weighted homogeneous ideal in $\CC[x_0,\ldots,x_N]$
is generated by a regular sequence of $k$ homogeneous elements of degrees $d_1,\ldots,d_k$.
This is equivalent to
the requirement that the codimension of (every irreducible component of)
the variety~$X$
equals the (minimal)
number of generators of the weighted homogeneous ideal of~$X$,
cf.~\cite[Theorem~II.8.21A(c)]{Ha77}.
Note that $\P$ can be thought of as a weighted complete
intersection of codimension $0$ in itself; this gives us a smooth Fano variety if and only if~\mbox{$\P\cong\P^N$}.
\begin{definition}[{see~\cite[Definition 6.3]{IF00}}]
\label{definition: quasi-smoothness}
Let $p\colon \mathbb A^{N+1}\setminus \{0\}\to \P$ be the natural projection to the weighted projective space. A subvariety $X\subset \P$
is called \emph{quasi-smooth} if $p^{-1}(X)$ is smooth.
\end{definition}
Note that a smooth well formed weighted complete intersection is always quasi-smooth,
see~\cite[Corollary~2.14]{PrzyalkowskiShramov-Weighted}.
The following definition describes weighted complete intersections
that are to a certain extent analogous to complete intersections
in a usual projective space that are contained in a hyperplane.
\begin{definition}[{cf. \cite[Definition~6.5]{IF00}}]
\label{definition:cone}
A weighted complete intersection~$X\subset\P$
is said to be \emph{an intersection
with a linear cone} if one has $d_j=a_i$ for some~$i$ and~$j$.
\end{definition}
\begin{remark}
\label{remark:cone}
A general quasi-smooth well formed weighted complete intersection
is isomorphic to a quasi-smooth well formed weighted complete intersection
that is not an intersection with a linear cone,
cf. \cite[Remark~5.2]{PrzyalkowskiShramov-Weighted}.
Note however that this does not hold without the
generality assumption.
For instance, a general weighted complete intersection
of bidegree~\mbox{$(2,4)$} in $\P(1^{n+2},2)$ is isomorphic to
a quartic hypersurface in $\P^{n+1}$, while certain
weighted complete intersections of this type are isomorphic to
double covers of an $n$-dimensional quadric branched over
an intersection with a quartic.
\end{remark}
Given a subvariety $X\subset\P$,
we denote by~$\O_X(1)$ the restriction of the
sheaf~\mbox{$\O_{\P}(1)$} to~$X$, see~\mbox{\cite[1.4.1]{Do82}}.
Note that the sheaf~\mbox{$\O_{\P}(1)$} may be not invertible.
However,
if $X$ is well formed, then $\O_X(1)$ is a well-defined divisorial sheaf on $X$.
Furthermore, if $X$ is well formed and smooth,
then $\O_X(1)$ is a line bundle on~$X$.
\begin{lemma}[{\cite[Remark~4.2]{Okada2}, \cite[Proposition~2.3]{PST}, cf. \cite[Theorem 3.7]{Mo75}}]
\label{lemma:Okada}
Let~$X$ be a quasi-smooth
well formed weighted complete intersection of dimension at least~$3$.
Then the class of the divisorial sheaf $\mathcal{O}_{X}(1)$ generates
the group $\Cl(X)$ of classes of Weil divisors on~$X$. In particular, under the additional assumption that $X$ is smooth,
the class of the line bundle $\mathcal{O}_{X}(1)$ generates
the group~$\Pic(X)$.
\end{lemma}
One can describe the canonical class of a weighted complete intersection.
For a weighted complete intersection $X$ of multidegree $(d_1,\ldots,d_k)$
in~$\P$, define
\begin{equation*}\label{eq:i-x}
i_X=\sum a_j-\sum d_i.
\end{equation*}
Let $\omega_X$ be the dualizing sheaf on $X$.
\begin{theorem}[{see~\cite[Theorem 3.3.4]{Do82}, \cite[6.14]{IF00}}]
\label{theorem:adjunction}
Let $X$ be a quasi-smooth
well formed weighted complete intersection.
Then
$$
\omega_X=\O_X\left(-i_X\right).
$$
\end{theorem}
Using the bounds on numerical invariants of smooth weighted complete intersections
found in \cite[Theorem~1.3]{ChenChenChen}, \cite[Theorem~1.1]{PrzyalkowskiShramov-Weighted}, and~\cite[Corollary~5.3(i)]{PST},
one can easily obtain the classically
known lists of all smooth Fano weighted complete intersections of small dimensions. Namely, we
have the following.
\begin{lemma}
\label{lemma:small-dimension}
Let $X$ be a smooth well formed Fano weighted complete intersection of dimension at most $2$
in $\P$ that is not an intersection with a linear cone.
Then $X$ is one of the varieties listed
in Table~\ref{table:dim 12}.
\end{lemma}
\begin{center}
\begin{longtable}{||c|c|c|}
\hline
No. & $\P$ & Degrees \\
\hline
\hline
\multicolumn{3}{|c|}{dimension $1$}\\
\hline
\hline
1.1 & $\P^2$ & $2$ \\
\hline
1.2 & $\P^1$ & $\varnothing$ \\
\hline
\hline
\multicolumn{3}{|c|}{dimension $2$}\\
\hline
\hline
2.1 & $\P(1^2,2,3)$ & $6$ \\
\hline
2.2 & $\P(1^3,2)$ & $4$ \\
\hline
2.3 & $\P^3$ & $3$ \\
\hline
2.4 & $\P^4$ & $2,2$ \\
\hline
2.5 & $\P^3$ & $2$ \\
\hline
2.6 & $\P^2$ & $\varnothing$ \\
\hline
\caption[]{Fano weighted complete intersections in dimensions
$1$ and $2$}\label{table:dim 12}
\end{longtable}
\end{center}
\begin{remark}\label{remark:dP}
Let $X$ be a smooth well formed Fano weighted complete intersection of dimension $2$.
If we do not assume that $X$ is not an intersection with a linear cone, we cannot
use the classification provided by Lemma~\ref{lemma:small-dimension}.
However, Lemma~\ref{lemma:small-dimension} applied together with Remark~\ref{remark:cone}
shows that if $i_X=1$, then $X$ is a del Pezzo surface of (anticanonical) degree at
most $4$.
\end{remark}
Recall that the \emph{Fano index} of a Fano variety $X$ is defined as the maximal integer $m$ such that the canonical
class $K_X$ is divisible by $m$ in the Picard group of $X$.
Theorem~\ref{theorem:adjunction}
and Lemmas~\ref{lemma:Okada} and~\ref{lemma:small-dimension}
imply the following.
\begin{corollary}\label{corollary:index}
Let $X$ be a smooth Fano
well formed weighted complete intersection of dimension at least $2$.
Then the Fano index of~$X$ equals~$i_X$.
\end{corollary}
\begin{proof}
Suppose that $\dim X=2$.
Note that the Fano index is constant in the family of smooth weighted complete intersections
of a given multidegree in a given weighted projective space.
Similarly, $i_X$ is constant in such a family.
Thus, by Remark~\ref{remark:cone}
we may assume that~$X$ is not an intersection with a linear cone.
Now the assertion follows from the classification
provided in Lemma~\ref{lemma:small-dimension}.
If $\dim X\ge 3$, then we apply
Theorem~\ref{theorem:adjunction} together with Lemma~\ref{lemma:Okada}.
\end{proof}
Note that the assertion of Corollary~\ref{corollary:index} fails in dimension $1$:
if $X$ is a conic in $\P^2$, then~\mbox{$i_X=1$}, while the Fano index of $X$ equals~$2$.
\begin{lemma}
\label{lemma:extension}
Let $X\subset\P(a_0,\ldots,a_N)$ be a smooth well formed weighted complete
intersection of multidegree $(d_1,\ldots,d_k)$. Then
a general weighted complete intersection $X'$
of multidegree $(d_1,\ldots,d_k)$ in
$\P(1,a_0,\ldots,a_N)$ is smooth and well formed,
and $i_{X'}=i_X+1$.
\end{lemma}
\begin{proof}
Straightforward.
\end{proof}
For the converse of Lemma~\ref{lemma:extension}, see \cite[Theorem~1.2]{PST}.
Similarly to Lemma~\ref{lemma:small-dimension}, we can classify
three-dimensional smooth well formed Fano weighted complete intersections that
are not intersections with a linear cone
(see~\cite[Table 2]{PSh18}). This together with
Lemma~\ref{lemma:extension} allows us to
classify smooth well formed weighted complete intersections of
dimension up to $2$ with trivial canonical class.
\begin{lemma}
\label{lemma:K3}
Let $X$ be a smooth well formed
weighted complete intersection of dimension at most $2$
in $\P$ that is not an intersection with a linear cone. Suppose that $K_X=0$.
Then~$X$ is one of the varieties listed
in Table~\ref{table:K3}.
\end{lemma}
\begin{center}
\begin{longtable}{||c|c|c|}
\hline
No. & $\P$ & Degrees \\
\hline
\hline
\multicolumn{3}{|c|}{dimension $1$}\\
\hline
\hline
1.1 & $\P(1,2,3)$ & $6$ \\
\hline
1.2 & $\P(1,1,2)$ & $4$ \\
\hline
1.3 & $\P^2$ & $3$ \\
\hline
1.4 & $\P^3$ & $2,2$ \\
\hline
\hline
\multicolumn{3}{|c|}{dimension $2$}\\
\hline
\hline
2.1 & $\P(1^3,3)$ & $6$ \\
\hline
2.2 & $\P^3$ & $4$ \\
\hline
2.3 & $\P^4$ & $2,3$ \\
\hline
2.4 & $\P^5$ & $2,2,2$ \\
\hline
\caption[]{Calabi--Yau weighted complete intersections in dimensions $1$ and $2$}\label{table:K3}
\end{longtable}
\end{center}
\begin{remark}
\label{remark:families}
Note that each of the four families of elliptic curves listed in Table~\ref{table:K3} in fact contains all elliptic curves up to isomorphism.
This is not the case for $K3$ surfaces. For instance, a general member of the family~2.2 in Table~\ref{table:K3} does not appear in the family~2.1 due to degree reasons.
\end{remark}
\section{Uniqueness of embeddings}
\label{section:linear-normality}
In this section we prove Proposition~\ref{proposition:unique-embedding}.
Let us start with a couple of facts that are well known
and can be proved similarly to their
analogs for complete intersections in the usual projective space. However,
we provide their proofs for the reader's convenience.
The proof of the following was suggested to us by A.\,Kuznetsov.
\begin{lemma}
\label{fact: splitting}
Let $X\subset\P=\P(a_0,\ldots,a_N)$ be a subvariety.
Let $C_X^*$ be a complement of the affine cone over $X$ to its vertex, and let~\mbox{$p\colon C_X^*\to X$} be the projection.
Then one has
$$
p_*(\O_{C^*_X})=\bigoplus\limits_{m\in\ZZ} \O_X(m).
$$
\end{lemma}
\begin{proof}
Denote the polynomial ring $\CC[x_0,\ldots, x_N]$ by $R$ and let
$$
U=\big(\Spec\, R\big)\setminus \{0\}\cong\A^{N+1}\setminus\{0\}.
$$
Let
$$
Y=\Spec_\PP\left(\bigoplus\limits_{m\ge 0} \O_{\PP}(m)\right)
$$
be the relative spectrum.
Since $R\cong\Gamma\left(\bigoplus_{m\ge 0} \O_\PP(m)\right)$,
one obtains the map
$$
Y\to \Spec\, R\cong\A^{N+1}
$$
which is a weighted blow up of the origin
(with weights $a_0,\ldots,a_N$). Cutting out the origin one gets the isomorphism
$$
U\cong \Spec_\PP \left(\bigoplus_{m\in\ZZ} \O_{\PP}(m)\right).
$$
Consider the fibered product
$$
C_X^*\cong X\times_{\PP} U.
$$
Taking into account that~\mbox{$\O_\PP(m)|_X=\O_X(m)$}
by definition, we
obtain an isomorphism
$$
C_X^*\cong \Spec_X \left(\bigoplus_{m\in\ZZ} \O_{X}(m)\right),
$$
and the assertion of the lemma
follows.
\end{proof}
For a subvariety $X\subset\P(a_0,\ldots,a_N)$, we define
the Poincar\'e series
$$
P_X(t)=\sum\limits_{m\ge 0} h^0(X,\O_X(m)) t^m.
$$
The proof of the following fact was kindly shared with us by T.\,Sano.
\begin{proposition}[{see \cite[Theorem~3.4.4]{Do82}, \cite[Lemma 2.4]{PST}, \cite[Theorem~3.2.4(iii)]{Do82}}]
\label{proposition:Poincare}
Let $X\subset\P(a_0,\ldots,a_N)$ be a
weighted complete intersection
of mul\-ti\-deg\-ree~\mbox{$(d_1,\ldots,d_k)$}.
The following assertions hold.
\begin{itemize}
\item[(i)]
One has
$$
P_X(t)=\frac{\prod_{j=1}^k (1-t^{d_j})}{\prod_{i=0}^N (1-t^{a_i})}.
$$
\item[(ii)]
One has $H^i(X, \O_X(m))=0$ for all $m\in\ZZ$ and all~\mbox{$0<i<\dim X$}.
\end{itemize}
\end{proposition}
\begin{proof}
Denote
the graded polynomial ring $\CC[x_0,\ldots, x_N]$ by $R$,
denote the weighted homogeneous ideal~\mbox{$(f_1,\ldots,f_k)$} that defines $X$ by $I$, and put $S=R/I$,
so that~\mbox{$X\cong\Proj S$}.
From regularity of the sequence $f_1,\ldots,f_k$ it easily follows that
\begin{equation}\label{eq:S-Poincare}
\sum_{m\ge 0} \dim(S_m)t^m=\frac{\prod_{j=1}^k (1-t^{d_j})}{\prod_{i=0}^N (1-t^{a_i})},
\end{equation}
where $S_m$ is the $m$-th graded component of $S$.
Denote the affine cone
$$
\Spec\, S\subset \Spec\, R\cong \Aff^{N+1}
$$
over $X$ by $C_X$. By construction, one has an isomorphism
of graded algebras
\begin{equation}\label{eq:S-vs-CX}
S\cong H^0\left(C_X,\O_{C_X}\right).
\end{equation}
Following Lemma~\ref{fact: splitting} denote the complement of $C_X$ to its vertex $P$ by $C^*_X$.
Consider the local cohomology groups
$H^\bullet_P(C_X,\O_{C_X})$, see, for instance,~\cite[p.~2, Definition]{Ha67}.
We have the exact sequence
$$
\ldots\to H^i_P\left(C_X,\O_{C_X}\right)\to H^i\left(C_X,\O_{C_X}\right)\to H^i\left(C^*_X,\O_{C^*_X}\right)\to H^{i+1}_P\left(C_X,\O_{C_X}\right)\to \ldots,
$$
see~\cite[Corollary 1.9]{Ha67}.
Since $C_X$ is a complete intersection in the affine space, it is Cohen--Macaulay.
Therefore, by \cite[Proposition~3.7]{Ha67} and \cite[Theorem 3.8]{Ha67} one has
$$
H^i_P\left(C_X,\O_{C_X}\right)=0
$$
for all $i<\dim C_X=\dim X+1$.
Hence, we obtain
an isomorphism
\begin{equation}\label{eq:CX-vs-CXstar}
H^i\left(C_X,\O_{C_X}\right)\cong H^i\left(C^*_X,\O_{C^*_X}\right)
\end{equation}
for all $i<\dim X$.
Finally, denote the natural projection
$C^*_X\to X$ by $p$. Then
\begin{equation}
\label{eq:CXstar-vs-pCXstar}
H^i(C^*_X,\O_{C^*_X})\cong H^i(X, p_*\O_{C^*_X})
\end{equation}
for all $i$.
On the other hand,
by Lemma~\ref{fact: splitting} we have
\begin{equation}
\label{eq:pCXstar}
p_*(\O_{C^*_X})=\bigoplus\limits_{m\in\ZZ} \O_X(m).
\end{equation}
For $i=0$, we combine the isomorphisms~\eqref{eq:S-vs-CX}, \eqref{eq:CX-vs-CXstar},
\eqref{eq:CXstar-vs-pCXstar}, and~\eqref{eq:pCXstar} to obtain an
isomorphism
of graded algebras
\begin{equation*}
S\cong \bigoplus\limits_{m\ge 0} H^0\left(X,\O_X(m)\right).
\end{equation*}
This together with~\eqref{eq:S-Poincare} gives assertion~(i).
For $0<i<\dim X$, we combine the isomorphisms~\eqref{eq:CX-vs-CXstar},
\eqref{eq:CXstar-vs-pCXstar}, and~\eqref{eq:pCXstar} to obtain an
isomorphism
\begin{equation*}
H^i\left(C_X,\O_{C_X}\right)\cong \bigoplus\limits_{m\in\ZZ} H^i\left(X,\O_X(m)\right).
\end{equation*}
Since $C_X$ is an affine variety, we have $H^i(C_X,\O_{C_X})=0$ for all $i>0$,
see for instance~\mbox{\cite[Theorem~III.3.5]{Ha77}}.
This proves assertion~(ii).
\end{proof}
Proposition~\ref{proposition:Poincare}(i) implies
the following property
that can be considered as an analog of linear normality for usual complete intersections.
\begin{corollary}
\label{corolary:linear-normality}
Let $X\subset\P$ be a quasi-smooth well formed weighted complete intersection.
Then the restriction map
$$
H^0\big(\P,\O_{\P}(m)\big)\to H^0\big(X, \O_X(m)\big)
$$
is surjective for every $m\in\ZZ$.
\end{corollary}
\begin{proof}
The dimension of the image of the restriction map is computed by the coefficient
in the Poincar\'e series~\eqref{eq:S-Poincare}.
On the other hand, by
Proposition~\ref{proposition:Poincare}(i)
this coefficient also equals the dimension of~\mbox{$ H^0(X, \O_X(m))$}.
\end{proof}
To proceed, we will need an elementary observation.
\begin{lemma}[{see~\cite[Lemma~18.3]{IF00}}]\label{lemma:ratio}
Let $N$ and $N'$ be positive integers, and $k$ and~$k'$ be non-negative integers.
Let $a_0\le\ldots\le a_N$, $a_0'\le\ldots\le a_{N'}'$,
$d_1\le\ldots\le d_k$, and~\mbox{$d_1'\le\ldots\le d_{k'}'$} be positive integers.
Suppose that
\begin{equation}\label{eq:ratio}
\frac{\prod_{j=1}^k (1-t^{d_j})}{\prod_{i=0}^N (1-t^{a_i})}=
\frac{\prod_{j'=1}^{k'} (1-t^{d'_{j'}})}{\prod_{i'=0}^{N'} (1-t^{a'_{i'}})}
\end{equation}
as rational functions in the variable $t$.
Suppose that $a_i\neq d_j$ for all $i$ and $j$, and
$a_{i'}'\neq d_{j'}'$ for all $i'$ and $j'$.
Then $N=N'$, $k=k'$, $a_i=a'_i$ for every $0\le i\le N$, and $d_j=d'_j$ for every~\mbox{$1\le j\le k$}.
\end{lemma}
\begin{proof}
Note that numerators and denominators of the rational functions in the left and the right hand sides of~\eqref{eq:ratio}
may have common divisors (for instance, if some $d_j$ is divisible by some $a_i$, or another way around). To prove the
assertion, we will keep track of the numbers that are roots of either the numerator or the denominator, but not both of them.
Observe that equality~\eqref{eq:ratio} is equivalent to the equality obtained
from~\eqref{eq:ratio} by interchanging the collections $\{a_i\}$, $\{a_j'\}$ with $\{d_s\}$, $\{d_r'\}$, respectively.
Thus we may assume that~$d_k$ is the maximal number among $a_N$, $a_{N'}'$, $d_k$, and $d_{k'}'$.
By assumption we know that~\mbox{$a_N<d_k$}. Let $\zeta$ be a primitive $d_k$-th root of unity.
Then $\zeta$ is a root of the numerator of the left hand side of~\eqref{eq:ratio}
but not the root of its denominator. Hence $\zeta$ is a root of the numerator~$\nu(t)$ of
the right hand side of~\eqref{eq:ratio} as well.
Since~\mbox{$d_{j'}'\le d_k$} for all $j'$, we see that~$\nu(t)$
is divisible by $1-t^{d_k}$. Cancelling the factor~\mbox{$1-t^{d_k}$} from~\eqref{eq:ratio}, we complete the proof
of the lemma by induction.
\end{proof}
Now we prove the main result of this section.
\begin{proof}[Proof of Proposition~\ref{proposition:unique-embedding}]
By Lemma~\ref{lemma:Okada},
the group $\Cl(X)$ is generated by the class of the line bundle
$\O_X(1)$, while the group $\Cl(X')$ is generated by the class of the line
bundle~$\O_{X'}(1)$.
Therefore, we see from Proposition~\ref{proposition:Poincare}(i) that
$$
\frac{\prod_{j=1}^k (1-t^{d_j})}{\prod_{i=0}^N (1-t^{a_i})}=P_X(t)=P_{X'}(t)=
\frac{\prod_{j'=1}^{k'} (1-t^{d'_{j'}})}{\prod_{i'=0}^{N'} (1-t^{a'_{i'}})}.
$$
Since neither $X$ nor $X'$ is an intersection with a linear cone,
the required assertion follows from Lemma~\ref{lemma:ratio}.
\end{proof}
\begin{remark}
The assertion of Proposition~\ref{proposition:unique-embedding} also
holds for smooth Fano weighted complete intersections of dimension~$2$.
This follows from their explicit classification, see Lemma~\ref{lemma:small-dimension}.
Note however that the assertion fails in dimension $1$. Indeed,
a conic in $\P^2$ is isomorphic to~$\P^1$ (which can be considered
as a complete intersection of codimension~$0$ in itself).
\end{remark}
\begin{remark}
We point out that the
assumption that $X$ is Fano is essential for the validity of
Proposition~\ref{proposition:unique-embedding} in dimension $2$. For instance,
there exist smooth quartics in~$\P^3$ that also have a structure of a double cover
of $\P^2$ branched in a sextic curve, see e.g.~\mbox{\cite[Proof of Theorem~4]{MM63}}.
Similarly, by Remark~\ref{remark:families} the assertion of Proposition~\ref{proposition:unique-embedding}
fails for elliptic curves.
\end{remark}
We do not know if the assertion of Proposition~\ref{proposition:unique-embedding} holds
in dimension $2$ in the case when~$X$ and $X'$ are quasi-smooth del Pezzo surfaces.
We point out that the
assumption that~$X$ alone is quasi-smooth is not enough for this.
Indeed, the weighted projective plane~\mbox{$\P(1,1,2)$}
(which can be considered as a quasi-smooth well formed weighted complete
intersection of codimension~$0$ in itself)
can be embedded as a quadratic cone into~$\P^3$ (which is not quasi-smooth).
In the proof of Theorem~\ref{theorem:automorphisms}, we will need a classification of
weighted complete intersections of large Fano index.
\begin{proposition}[{cf.~\cite[Theorem~2.7]{PSh18}}]
\label{proposition:low-coindex}
Let $X\subset \PP$ be a smooth well formed Fano weighted complete intersection of dimension $n\ge 2$. Then
\begin{itemize}
\item[(i)] one has $i_X\le n+1$;
\item[(ii)] if $i_X=n+1$, then $X\cong\P^n$;
\item[(iii)] if $i_X=n$, then $X$ is isomorphic to a quadric in~$\P^{n+1}$;
\item[(iv)] if $i_X=n-1$ and $n\ge 3$, then $X$ is isomorphic to a hypersurface
of degree $6$ in $\P=\P(1^n, 2,3)$, or to
a hypersurface of degree $4$ in $\P=\P(1^{n+1}, 2)$, or to
a cubic hypersurface in $\P=\P^{n+1}$, or to an intersection
of two quadrics in $\P=\P^{n+2}$.
\end{itemize}
\end{proposition}
\begin{proof}
Recall that $i_X$ equals the Fano
index of $X$ by Corollary~\ref{corollary:index}.
By~\mbox{\cite[Corollary 3.1.15]{IP99}}, we know that $i_X\le n+1$; if $i_X=n+1$,
then $X$ is isomorphic to
$\P^n$; and if $i_X=n$, then $X$ is isomorphic to a quadric in $\P^{n+1}$. This proves
assertions~(i), (ii), and~(iii).
Now suppose that $i_X=n-1$ and $n\ge 3$.
Note that $\Pic(X)\cong\mathbb{Z}$ by Lemma~\ref{lemma:Okada}.
Thus it follows from the classification of smooth Fano varieties of Fano index $n-1$
(see~\cite{Fu80-84} or~\mbox{\cite[Theorem~3.2.5]{IP99}})
that $X$ is isomorphic either to one of the weighted complete intersections
listed in assertion~(iv), or to a linear section
of the Grassmannian~\mbox{$\mathrm{Gr}(2,5)$} in its Pl\"ucker embedding.
It remains to show that a weighted complete intersection $X$ cannot be isomorphic to
a linear section of the Grassmannian $\mathrm{Gr}(2,5)$. Suppose that $X$ is isomorphic
to such a variety. Then $n\le 6$.
If~\mbox{$n=3$}, then the Fano index of $X$ equals $2$ and its anticanonical
degree is equal to~$40$.
If~\mbox{$n=4$}, then the Fano index of $X$ equals $3$ and its anticanonical
degree is equal to~$405$.
In both cases we see from Remark~\ref{remark:cone} that there exists a
smooth weighted complete intersection $X$ with the same $n$ and $i_X$ that
is not an intersection with a linear cone.
This is impossible by a classification of
smooth Fano weighted complete intersections of dimensions~$3$ and~$4$,
see~\cite[Table 2]{PSh18} (cf.~\cite[\S12]{IP99})
and~\cite[Table~1]{PrzyalkowskiShramov-Weighted}, respectively.
Therefore, we see that $5\le n\le 6$. Recall that
$$
H^4\big(\mathrm{Gr}(2,5),\mathbb{Z}\big)\cong\mathbb{Z}^2.
$$
Thus, Lefschetz hyperplane section theorem implies that
$$
H^4(X,\mathbb{Z})\cong\mathbb{Z}^2.
$$
On the other hand, since $X$ is a
weighted complete intersection of dimension greater than~$4$,
one has $H^4(X,\mathbb{Z})\cong\mathbb{Z}$ by the Lefschetz-type theorem for complete intersections
in toric varieties, see~\cite[Proposition~1.4]{Ma99}.
The obtained contradiction completes the proof of assertion~(iv).
\end{proof}
Proposition~\ref{proposition:unique-embedding} allows us to prove more precise
classification results
concerning Fano weighted complete intersections (which we will not use directly
in our further proofs).
\begin{corollary}
\label{corollary:low-coindex}
Let $X\subset \PP$ be a smooth well formed Fano weighted complete intersection of dimension $n\ge 2$
that is not an intersection with a linear cone. Then
\begin{itemize}
\item[(i)] if $i_X=n+1$, then $X=\P=\P^n$;
\item[(ii)] if $i_X=n$, then $X$ is a quadric in $\P=\P^{n+1}$;
\item[(iii)] if $i_X=n-1$, then $X$ is either a hypersurface
of degree $6$ in $\P=\P(1^n, 2,3)$, or
a hypersurface of degree $4$ in $\P=\P(1^{n+1}, 2)$, or
a cubic hypersurface in $\P=\P^{n+1}$, or an intersection
of two quadrics in $\P=\P^{n+2}$.
\end{itemize}
\end{corollary}
\begin{proof}
Assertions (i) and (ii) follow from
assertions (ii) and (iii) of Proposition~\ref{proposition:low-coindex}, respectively,
applied together with Proposition~\ref{proposition:unique-embedding}.
If $n=2$, assertion (iii) follows from Lemma~\ref{lemma:small-dimension}.
If $n\ge 3$, assertion (iii) follows from
Proposition~\ref{proposition:low-coindex}(iv) and
Proposition~\ref{proposition:unique-embedding}.
\end{proof}
\begin{remark}
An alternative way to prove
Corollary~\ref{corollary:low-coindex} (which in turn can be used
to deduce Proposition~\ref{proposition:low-coindex})
is by induction on dimension using the classification of smooth well formed Fano weighted
complete intersections of low dimension
(say, one provided by Lemma~\ref{lemma:small-dimension}) together
with~\mbox{\cite[Theorem~1.2]{PST}}.
\end{remark}
\section{Automorphisms}
\label{section:automorphisms}
In this section we prove Theorem~\ref{theorem:automorphisms}.
Let $\P=\P(a_0,\ldots,a_N)$ be a weighted projective space.
For any subvariety $X\subset\P$, we denote by
$\Aut(\P;X)$ the stabilizer of $X$ in $\Aut(\P)$. We denote by
$\Aut_{\P}(X)$
the image of~\mbox{$\Aut(\P;X)$} under the restriction map
to $\Aut(X)$. In other words, the group $\Aut_{\P}(X)$
consists of automorphisms of $X$ induced by automorphisms of $\P$.
We start with a general result that is well known to experts
(see for instance~\mbox{\cite[Lemma~3.1.2]{KPS18}})
and that was pointed out to us by A.\,Massarenti.
\begin{lemma}\label{lemma:LAG}
Let $X$ be a normal variety, let $A$ be a very ample Weil divisor
on $X$, and let $[A]$ be the class of $A$ in $\Cl(X)$. Denote by
$\Aut(X; [A])$ the stabilizer of $[A]$ in $\Aut(X)$.
Then $\Aut(X; [A])$ is a linear algebraic group.
\end{lemma}
\begin{corollary}\label{corollary:LAG}
Let $X$ be a quasi-smooth well formed weighted complete intersection
of dimension $n$.
Suppose that either $n\ge 3$, or $K_X\neq 0$.
Then $\Aut(X)$ is a linear algebraic group.
\end{corollary}
\begin{proof}
Note that the divisor class $K_X$ is $\Aut(X)$-invariant.
Moreover, if $K_X\neq 0$, then either~$K_X$ or $-K_X$ is ample
by Theorem~\ref{theorem:adjunction}.
On the other hand, if $n\ge 3$, then $\Cl(X)\cong\mathbb{Z}$
by Lemma~\ref{lemma:Okada}, so that an ample generator of $\Cl(X)$
is $\Aut(X)$-invariant. In both cases we see that $\Aut(X)$ preserves
some ample (and thus also some very ample) divisor class on $X$.
Hence $\Aut(X)$ is a linear algebraic group by Lemma~\ref{lemma:LAG}.
\end{proof}
The following lemma will not be used in the proof of
Theorem~\ref{theorem:automorphisms}, but will allow us to prove its (weaker) analog that applies to a slightly wider class of smooth weighted complete
intersections, see Corollary~\ref{corollary:Flenner}(ii) below.
\begin{lemma}\label{lemma:non-ruled}
Let $X$ be a subvariety of $\P$. Then $\Aut_{\P}(X)$ is a linear algebraic group.
\end{lemma}
\begin{proof}
The group $\Aut(\P)$ is obviously a linear algebraic group.
The stabilizer $\Aut(\P;X)$ of $X$ in $\Aut(\P)$
and the kernel $\Aut(\P;X)_{\mathrm{id}}$
of its action on $X$ are cut out in $\Aut(\P)$ by
algebraic equations, so that they are linear algebraic groups.
The group~\mbox{$\Aut(\P;X)_{\mathrm{id}}$} is a normal subgroup of~\mbox{$\Aut(\P;X)$}.
Therefore, the
group
$$
\Aut_{\P}(X)\cong\Aut(\P;X)/\Aut(\P;X)_{\mathrm{id}}
$$
is a linear algebraic group as well, see \cite[Theorem~6.8]{Borel}.
\end{proof}
\begin{corollary}\label{corollary:non-ruled}
Let $X$ be a smooth irreducible subvariety of $\P$. Suppose that $K_X$ is numerically effective.
Then the group~\mbox{$\Aut_{\P}(X)$} is finite.
\end{corollary}
\begin{proof}
By Lemma~\ref{lemma:non-ruled}, the group $\Aut_{\P}(X)$ is a linear algebraic group.
Therefore, if~\mbox{$\Aut_{\P}(X)$} is infinite, then it contains a subgroup
isomorphic either to $\CC^{\times}$ or to $\CC^+$,
which implies that~$X$ is covered by rational curves.
On the other hand, since $K_X$ is numerically effective,
$X$ can't be covered by rational
curves, see~\cite[Theorem 1]{MM86}.
\end{proof}
The main tool we use in the proof of Theorem~\ref{theorem:automorphisms}
is the following result from~\cite{Fle81}.
\begin{theorem}[{see \cite[Satz 8.11(c)]{Fle81}}]
\label{theorem:Flenner}
Let $X$ be a smooth weighted complete intersection of dimension $n\ge 2$.
Then
$$
H^n\big(X, \Omega^1_X\otimes\mathcal{O}_X(-i)\big)=0
$$
for every integer $i\le n-2$.
\end{theorem}
\begin{remark}
Actually, the assertion of \cite[Satz 8.11(c)]{Fle81} gives more vanishing results
and holds under the weaker assumption that $X$ is quasi-smooth. However, we do not want to go into
details with the definition of the sheaf $\Omega_X^1$ here, and in any case we will need smoothness of
$X$ on the next step.
\end{remark}
Theorem~\ref{theorem:Flenner} allows us to prove finiteness of various automorphism groups.
\begin{corollary}
\label{corollary:Flenner}
Let $X$ be a smooth well formed weighted complete intersection
of dimension~\mbox{$n\ge 2$}.
Suppose that $i_X\le n-2$.
The following assertions hold.
\begin{itemize}
\item[(i)] One has $H^0(X, T_X)=0$.
\item[(ii)] The group $\Aut_{\P}(X)$ is finite.
\item[(iii)] If either $\dim X\ge 3$ or $K_X\neq 0$,
then the group $\Aut(X)$ is finite.
\end{itemize}
\end{corollary}
\begin{proof}
By Theorem~\ref{theorem:Flenner}
we have
$$
H^n\big(X, \Omega^1_X\otimes\mathcal{O}_X(-i_X)\big)=0.
$$
Recall that $\omega_X=\mathcal{O}_X(-i_X)$ by Theorem~\ref{theorem:adjunction}.
Thus assertion~(i) follows from Serre duality.
Assertion~(ii) follows from assertion~(i), because
$\Aut_{\P}(X)$ is a linear algebraic group
by Lemma~\ref{lemma:non-ruled}.
Similarly, assertion~(iii) follows from assertion~(i), because
the automorphism group of any variety
subject to the above assumptions
is a linear algebraic group by Corollary~\ref{corollary:LAG}.
\end{proof}
Recall that the smooth Fano threefold $V_5$ of Fano index $\dim V_5-1=2$
that is defined as an intersection of the Grassmannian
$\mathrm{Gr}(2,5)\subset\P^9$
in its Pl\"ucker embedding with a linear section of codimension
$3$ has infinite automorphism group $\Aut(V_5)\cong\mathrm{PGL}_2(\CC)$,
see~\mbox{\cite[Proposition~4.4]{Mukai-CurvesK3Fano}}
or \cite[Proposition~7.1.10]{CheltsovShramov}.
The next lemma shows that such a situation is impossible
for smooth weighted complete intersections.
\begin{lemma}
\label{lemma:low-coindex}
Let $X\subset \PP$ be a smooth well formed weighted complete intersection of
dimension~\mbox{$n\ge 2$}. Suppose that $i_X=n-1$. Then
the group $\Aut(X)$ is finite.
\end{lemma}
\begin{proof}
If $n=2$, the assertion follows from Remark~\ref{remark:dP} and
the properties of automorphism groups of smooth del Pezzo
surfaces, see for instance \cite[Corollary~8.2.40]{Dolgachev-CAG}.
Thus, we assume that $n\ge 3$ and
use the classification provided by Proposition~\ref{proposition:low-coindex}(iv).
If $X$ is isomorphic to an intersection
of two quadrics in~\mbox{$\P=\P^{n+2}$} or to a cubic hypersurface in~\mbox{$\P=\P^{n+1}$},
then the assertion follows from
Theorem~\ref{theorem:Benoist} (in the latter case one can also
use Theorem~\ref{theorem:MM}).
Now suppose that $X$ is isomorphic either to a
hypersurface of degree $4$ in~\mbox{$\P=\P(1^{n+1}, 2)$}, or
to a hypersurface
of degree $6$ in $\P=\P(1^n, 2,3)$.
The argument in these cases is similar to that in the proof of \cite[Lemma~4.4.1]{KPS18}.
Denote by~$H$ the ample divisor such that~\mbox{$-K_X\sim (n-1)H$}.
Then there exists
an $\Aut(X)$-equivariant double cover~\mbox{$\phi\colon X\to Y$},
where in the former case $Y\cong\P^n$ and $\phi$ is given by the linear system
$|H|$, while in the latter case~\mbox{$Y\cong\P(1^{n},2)$} and $\phi$ is given by the linear system
$|2H|$. Let $H'$ be the ample Weil divisor generating the group
$\Cl(Y)\cong\ZZ$, and let
$B\subset Y$ be the branch divisor of $\phi$. In the former
case one has $B\sim 4H'$, and in the latter case one has $B\sim 6H'$.
Note that in the latter case~$H'$ is not Cartier, but $2H'$ is;
note also that in this case $\phi$ is branched over the singular point
of $Y$ as well.
In both cases $B$ is smooth. Furthermore, it follows from adjunction
formula that either $K_B$ is ample, or $K_B\sim 0$, or $B$ is a
(smooth well formed) Fano weighted hypersurface of dimension $n-1\ge 3$ and
Fano index $i_B\le n-3$.
Since the double cover $\phi$ is $\Aut(X)$-equivariant,
we see that the quotient of the group~\mbox{$\Aut(X)$} by its normal subgroup
of order $2$ generated by the Galois involution of~$\phi$
is isomorphic to a subgroup of the stabilizer $\Aut(Y;B)$ of
$B$ in $\Aut(Y)$.
Since $B$ is not contained in any divisor linearly equivalent to
the very ample divisor $2H'$, we conclude that~\mbox{$\Aut(Y;B)$}
acts faithfully on $B$, see for
instance~\cite[Lemma~2.1]{CPS19}.
Hence
$$
\Aut(Y;B)\cong\Aut_Y(B).
$$
On the other hand, the group $\Aut_Y(B)$ is finite by
Corollary~\ref{corollary:Flenner}(ii); alternatively, one can apply
Corollaries~\ref{corollary:non-ruled} and~\ref{corollary:Flenner}(iii).
This means that the group $\Aut(X)$ is finite as well.
\end{proof}
Now we prove our main results.
\begin{proof}[Proof of Theorem~\ref{theorem:automorphisms}]
First suppose that $n=1$. We may assume
that $K_X$ is ample. In this case the finiteness
of $\Aut(X)$ is well-known, see for
instance~\mbox{\cite[Exercise~IV.5.2]{Ha77}}.
Now suppose that $n\ge 2$.
If $i_X\le n-2$, then the group $\Aut(X)$ is finite by
Corollary~\ref{corollary:Flenner}(iii).
If~\mbox{$i_X=n-1$}, then
the group $\Aut(X)$ is finite by Lemma~\ref{lemma:low-coindex}.
Finally, if~\mbox{$i_X\ge n$}, then we know from
Proposition~\ref{proposition:low-coindex}
that $X$ is isomorphic either to~$\P^n$ or to
a quadric hypersurface in~$\P^{n+1}$.
\end{proof}
Corollary~\ref{corollary:automorphisms} immediately follows
from Theorem~\ref{theorem:automorphisms} and
Proposition~\ref{proposition:unique-embedding}.
| 39,640
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\begin{document}
\maketitle
\begin{abstract}
In this paper, we analyze the $s$-dependence of the solution $u_s$ to the fractional Poisson equation $(-\Delta)^s_{\Omega}u_s=f$ in an open bounded set $\Omega\subset\R^N$. Precisely, we show that the solution map $(0,1)\rightarrow L^2(\Omega)$,~ $s\mapsto u_s$ is continuously differentiable. Moreover, when $f=\lambda_su_s$, we also analyze the one-sided differentiability of the first nontrivial eigenvalue of $(-\Delta)^s_{\Omega}$ regarded as a function of $s\in(0,1)$.
\end{abstract}
{\footnotesize
\begin{center}
\textit{Keywords.} Fractional Poisson equation, Continuously differentiable, eigenvalue.
\end{center}
}
\section{Introduction}\label{introduction}
In a bounded domain $\Omega\subset\R^N~(N\geq2)$ with $C^{1,1}$ boundary, we consider the following nonlocal Poisson problem
\begin{equation}\label{poisson-problem-for-regional-fractional-laplacian}
(-\Delta)^s_{\Omega}u_s=f\quad\text{in}\quad\Omega,
\end{equation}
where, $s\in(0,1)$ and $f\in L^{\infty}(\Omega)$ with $\int_{\Omega}f\ dx=0$. Here, $(-\Delta)^s_{\Omega}$ stands for the regional fractional Laplacian define for every function $u\in L^{\infty}(\Omega)$ as the following singular integral
\begin{equation}
(-\Delta)^s_{\Omega}u(x)=C_{N,s}P.V.\int_{\Omega}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\ dy,\qquad x\in\Omega,
\end{equation}
where $P.V.$ is a commonly used abbreviation for ''in the principal value sense'' and the normalized constant $C_{N,s}$ is defined by
\begin{equation}\label{bound-of-constant-c-n-s}
C_{N,s}=s(1-s)\pi^{-\frac{N}{2}}2^{2s}\frac{\Gamma(\frac{N+2s}{2})}{\Gamma(2-s)}\in\Big(0,4\Gamma\Big(\frac{N}{2}+1\Big)\Big],
\end{equation}
$\Gamma$ being the usual Gamma function. The bounds in \eqref{bound-of-constant-c-n-s} can be found in \cite[page $8$]{terracini2018s}. As known, $(-\Delta)^s_{\Omega}$ represents the infinitesimal generator of the so-called \textit{censored stable L\'{e}vy process}, that is, a stable process in which the jumps between $\Omega$ and its complement are forbidden (see e.g. \cite{andreu2010nonlocal,bogdan2003censored,chen2018dirichlet,guan2006integration,guan2006reflected,warma2015fractional} and the references therein).
The study of the $s$-dependence of the solution of the Poisson problem involving nonlocal operators has recently received quite some interest. This kind of study allows a well understanding of the asymptotic behavior of the solution at $s\in(0,1)$. For instance, in \cite{biccari2018poisson}, the authors analyzed the limit behavior as $s\rightarrow1^-$ of the solution to the fractional Poisson equation $(-\Delta)^su_s=f_s$, $x\in\Omega$ with homogeneous Dirichlet boundary conditions $u_s\equiv0$, $x\in\R^N\setminus\Omega$ and provided continuity in a weak setting. We also refer to \cite{deblassie2005alpha} where the equicontinuity concerning eigenfunctions and eigenvalues of $(-\Delta)^s$ in $\Omega$ was studied for $s$ belonging in a compact subset of $(0,1)$. A small order asymptotics of eigenvalues of the operators $(-\Delta)^s$ and $(-\Delta)^s_{\Omega}$ has been studied recently in \cite{chen2019dirichlet,feulefack2020small,temgoua2021eigenvalue}.
Very recently, Jarohs, Salda$\tilde{\text{n}}$a and Weth \cite{jarohs2020new} established the $C^1$-regularity of the map $(0,1)\rightarrow L^{\infty}(\Omega)$, $s\mapsto u_s$, where $u_s$ is given as the unique weak solution to the fractional Poisson problem $(-\Delta)^su_s=f$, $x\in\Omega$ with the homogeneous Dirichlet boundary data $u_s\equiv0$, $x\in\R^N\setminus\Omega$, where $\Omega$ is a bounded domain with $C^2$ boundary and $f\in C^{\alpha}(\ov\Omega)$ for some $\alpha>0$. The main advantage in their analysis relies on the representation formula of the solution $u_s$ by mean of Green function which allows obtaining several important and powerful estimates.
The purpose of the present paper is to analyze the $C^1$-regularity of the map $(0,1)\rightarrow L^2(\Omega)$, $s\mapsto u_s$, where $u_s$ is the unique solution of the fractional Poisson problem \eqref{poisson-problem-for-regional-fractional-laplacian}. The major difficulty in our development stems from the fact that, contrary to \cite{jarohs2020new}, we do not have an explicit representation of the solution $u_s$ in terms of Green function for every $s\in(0,1)$. This is due to the fact that for being $s\in(0,\frac{1}{2}]$, the fractional Poisson problem $(-\Delta)^s_{\Omega}u_s=f$, $x\in\Omega$ with Dirichlet boundary conditions $u_s\equiv g$, $x\in\partial\Omega$ remains ill-posed since there is no boundary term for $(-\Delta)^s_{\Omega}$. A series of fruitful results have been recently developed to explain this fact. We refer to \cite{bogdan2003censored} and the references therein for the probabilistic approach. A similar result has been recently established by Chen and Wei \cite{chen2020non-existence} from a purely analytic point of view. \\
The paper is organized as follows. In Section \ref{preliminary} we present some preliminaries that will be useful throughout this article. In Section \ref{differentiability-for-general-value-s}, we discuss the differentiability of the map $s\mapsto u_s$ in $(0,1)$. Finally, in Section \ref{eigenvalue-problem-case}, we derive a one-sided differentiability of the first nontrivial eigenvalue in $(0,1)$. \\
\textbf{Acknowledgements:} This work is supported by DAAD and BMBF (Germany) within project 57385104. The author is grateful to Tobias Weth, Mouhamed Moustapha Fall, and Sven Jarohs for helpful discussions.
\section{Preliminary and functional setting}\label{preliminary}
In this section, we introduce some preliminary properties that will be useful in this work. First of all, throughout the end of the paper, $d_{A}:=\sup\{|x-y|:x,y\in A\}$ is the diameter of $A\subset\R^N$ and $B_r(x)$ denotes the open ball centered at $x$ with radius $r$. We also denote by $|A|$ the $N$-dimensional Lebesgue measure of every set $A\subset\R^N$. \\
Now, for all $s\in(0,1)$ the usual fractional Sobolev space $H^s(\Omega)$ is defined by
\begin{align*}
H^s(\Omega)=\{u\in L^2(\Omega):|u|^2_{H^s(\Omega)}<\infty\},
\end{align*}
where
\begin{align*}
|u|_{H^s(\Omega)}:=\Big(\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\ dxdy\Big)^{1/2},
\end{align*}
is the so-called Gagliardo seminorm of $u$. Moreover, $H^s(\Omega)$ is a Hilbert space endowed with the norm
\begin{align*}
\|u\|_{H^s(\Omega)}:=(\|u\|^2_{L^2(\Omega)}+|u|^2_{H^s(\Omega)})^{1/2}.
\end{align*}
Also, we notice the following continous embedding $H^t(\Omega)\hookrightarrow H^s(\Omega)$ for $t>s$, see e.g. \cite[Proposition 2.1]{di2012hitchhiker's}.
It is also useful to recall that the space $C^{\infty}(\overline{\Omega})$ is dense in $H^s(\Omega)$ (see \cite[Corollary 2.71]{demengel2012functional}). We recall that $C^{\infty}(\ov\Omega)$ denotes the restriction of all $C^{\infty}(\R^N)$ functions on $\ov\Omega$. Moreover, we define the space $\mathbb{X}^s(\Omega)$ consists of functions in $H^s(\Omega)$ orthogonal to constants i.e.,
\begin{align*}
\mathbb{X}^s(\Omega):=\Big\{u\in H^s(\Omega):\int_{\Omega}u\ dx=0\Big\}.
\end{align*}
Clearly, $\mathbb{X}^s(\Omega)$ is a Hilbert space (with the norm $\|\cdot\|_{\mathbb{X}^s(\Omega)}:=|\cdot|_{H^s(\Omega)}$ equivalent to the usual one in $H^s(\Omega)$) for which every function $u\in\mathbb{X}^s(\Omega)$ satisfies the following fractional Poincar\'{e} inequality
\begin{equation}\label{Poincare-inequality}
\|u\|^2_{L^2(\Omega)}\leq\gamma_{N,s,\Omega}|u|^2_{H^s(\Omega)}\quad\text{with}~~\gamma_{N,s,\Omega}=|\Omega|^{-1}d^{N+2s}_{\Omega}.
\end{equation}
We notice also that the space of functions $\phi\in C^{\infty}(\overline{\Omega})$ with $\int_{\Omega}\phi\ dx=0$ is dense in $\mathbb{X}^s(\Omega)$. For simplicity, we set $C^{\infty}_0(\overline{\Omega}):=\{u\in C^{\infty}(\overline{\Omega}):\int_{\Omega}\phi\ dx=0\}$.
The inner product and the norm in $L^2(\Omega)$ will be denoted by $\langle\cdot,\cdot\rangle_{L^2(\Omega)}$ and $\|\cdot\|_{L^2(\Omega)}$ respectively.
Now, let $\cE_s$ be the quadratic form define on $H^s(\Omega)$ by
\begin{equation*}
(u,v)\mapsto\cE_s(u,v)=\frac{C_{N,s}}{2}\int_{\Omega}\int_{\Omega}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2s}}\ dxdy.
\end{equation*}
We have the following.
\begin{defi}
Let $f\in L^{\infty}(\Omega)$ with $\int_{\Omega}f\ dx=0$. We say that $u_s\in\mathbb{X}^s(\Omega)$ is a weak solution of \eqref{poisson-problem-for-regional-fractional-laplacian} if
\begin{equation}\label{weak-formulation-direct-method}
\cE_s(u_s,\phi)=\int_{\Omega}f\phi\ dx,\quad\forall\phi\in\mathbb{X}^s(\Omega).
\end{equation}
\end{defi}
The existence and uniqueness of weak solution of the Poisson problem \eqref{poisson-problem-for-regional-fractional-laplacian} in $\mathbb{X}^s(\Omega)$ is guaranteed by Riesz representation theorem.
Let $u_s\in\mathbb{X}^s(\Omega)$ be the unique weak solution of \eqref{poisson-problem-for-regional-fractional-laplacian}. Then, thanks to \cite{temgoua2021eigenvalue}, there exists a constant $c_1>0$ independent of $s$ such that
\begin{equation}\label{uniformly-boundedness-of-weak-solution}
\|u_s\|_{L^{\infty}(\Omega)}\leq c_1.
\end{equation}
Very recently, Fall \cite{fall2020regional} established boundary regularity for any weak solution to \eqref{poisson-problem-for-regional-fractional-laplacian}. Presicely, among other results, he proved that $u_s\in C^{\beta}(\overline{\Omega})$ and
\begin{equation}\label{boundary-regularity-for-regional}
\|u_s\|_{C^{\beta}(\overline{\Omega})}\leq C(\|u_s\|_{L^2(\Omega)}+\|f\|_{L^p(\Omega)})
\end{equation}
with
\begin{align*}
\beta:=2s-\frac{N}{p}
\end{align*}
for every $s\in(0,1)$ and $p>\frac{N}{2s}$. Moreover if $s\in(\frac{1}{2},1)$ so that $\beta=2s-\frac{N}{p}>1$, he also obtained boundary H\"{o}lder regularity for the gradient of the form $\nabla u_s\in C^{\beta-1}(\overline{\Omega})$ with
\begin{align}\label{boundary-regularity-for-the-gradient}
\|\nabla u_s\|_{C^{\beta-1}(\overline{\Omega})}\leq C(\|u_s\|_{L^2(\Omega)}+\|f\|_{L^p(\Omega)}).
\end{align}
From now on and without loss of generality, we fix $p$ such that $p>\frac{N}{s}$. Moreover, the constant $C$ appearing in \eqref{boundary-regularity-for-regional} is continuous at $s\in[s_0,1)$ for some $s_0\in(0,1)$ see \cite[Theorem 1.1]{fall2020regional}. The same conclusion holds for \eqref{boundary-regularity-for-the-gradient} provided that $s\in[s_0,1)$ for some $s_0\in (\frac{1}{2},1)$ see \cite[Remark 1.4]{fall2020regional}. Hence, by taking into account \eqref{uniformly-boundedness-of-weak-solution} we obtain from \eqref{boundary-regularity-for-regional} and \eqref{boundary-regularity-for-the-gradient} uniform bound with respect to $s$ on $C^{\beta}$ and $C^{\beta-1}$ norm of $u_s$ and $\nabla u_s$ respectively as follows
\begin{equation}\label{boundary-uniform-bound}
\|u_s\|_{C^{\beta}(\overline{\Omega})}\leq c_2\quad\quad\text{and}\quad\quad\|\nabla u_s\|_{C^{\beta-1}(\overline{\Omega})}\leq c_3.
\end{equation}
As a direct advantage of the above boundary regularity, we derive in the next proposition, higher Sobolev regularity for solution of \eqref{poisson-problem-for-regional-fractional-laplacian}.
\begin{prop}\label{higher-sobolev-regularity}
Let $f\in L^{\infty}(\Omega)$ with $\int_{\Omega}f\ dx=0$ and let $u_{s+\sigma}\in H^{s+\sigma}(\Omega)\cap L^{\infty}(\Omega)$ be the unique weak solution of problem \eqref{poisson-problem-for-regional-fractional-laplacian} with $s$ replaced by $s+\sigma$. Then $u_{s+\sigma}\in H^{s+\epsilon}(\Omega)$ for some $\epsilon>0$ and
\begin{equation}\label{b1}
\|u_{s+\sigma}\|_{H^{s+\epsilon}(\Omega)}\leq K\quad\quad\text{for all}~\sigma\in(-s_0,s_0)
\end{equation}
for some $s_0>0$.
\end{prop}
\begin{proof}
Let $u_{s+\sigma}\in H^{s+\sigma}(\Omega)\cap L^{\infty}(\Omega)$ be the unique weak solution of \eqref{poisson-problem-for-regional-fractional-laplacian} with $s$ replaced by $s+\sigma$ for all $\sigma\in(-s_0,s_0)$ for some $s_0>0$. Then, \\
$(i)$ If $2s\leq1$ then from \eqref{boundary-regularity-for-regional} we have that
\begin{align*}
\int_{\Omega}\int_{\Omega}\frac{|u_{s+\sigma}(x)-u_{s+\sigma}(y)|^2}{|x-y|^{N+2(s+\epsilon)}}\ dxdy
&\leq\|u_s\|^2_{C^{2(s+\sigma)-\frac{N}{p}}(\overline{\Omega})}\int_{\Omega}\int_{\Omega}|x-y|^{4(s+\sigma)-\frac{2N}{p}-N-2(s+\epsilon)}\ dxdy\\
&\leq d^{4\sigma}_{\Omega}\|u_s\|^2_{C^{2s-\frac{N}{p}}(\overline{\Omega})}\int_{\Omega}\int_{B_{d_{\Omega}}(x)}|x-y|^{2s-\frac{2N}{p}-N-2\epsilon}\ dydx\\
&\leq\frac{\max\{d^{-4s_0}_{\Omega},d^{4s_0}_{\Omega}\}|S^{N-1}||\Omega|\|u_s\|^2_{C^{2s-\frac{N}{p}}(\overline{\Omega})}}{2(s-\epsilon-\frac{N}{p})}d^{2(s-\epsilon-\frac{N}{p})}_{\Omega}<\infty
\end{align*}
provided that $0<\epsilon<s-\frac{N}{p}$. This shows that $u_{s+\sigma}\in H^{s+\epsilon}(\Omega)$ with uniform bound in $\sigma\in (-s_0,s_0)$ provided that $0<\epsilon<s-\frac{N}{p}$. \\
$(ii)$ If $2s>1$ then from \eqref{boundary-regularity-for-the-gradient} we have that $\nabla u_{s+\sigma}\in L^{\infty}(\overline{\Omega})\subset L^{\infty}(\Omega)$ and therefore $\nabla u_{s+\sigma}\in L^2(\Omega)$. Since also $u_{s+\sigma}\in L^2(\Omega)$, we deduce that $u_{s+\sigma}\in H^1(\Omega)$. Hence,
\begin{align}
&u_{s+\sigma}\in H^s(\Omega)\cap H^1(\Omega)\quad\quad\quad\quad\text{for all}~~\sigma\in[0,s_0)\label{1}\\
&u_{s+\sigma}\in H^{\alpha(s)}(\Omega)\cap H^1(\Omega)\quad\quad\quad\text{for all}~~\sigma\in(-s_0,0]\label{2}
\end{align}
for some $\alpha(s)<<s$ depending only on $s$. Applying the well-known Gagliardo-Nirenberg interpolation inequality (see e.g. \cite[Theorem 1]{brezis2018gagliardo}), we find that $u_{s+\sigma}\in H^r(\Omega)$ with
\begin{itemize}
\item [$(a)$] $r=\theta s+(1-\theta)\cdot1~~\text{for all}~\theta\in (0,1)$ in the case \eqref{1}, and
\begin{align}\label{interpolation-inequality}
\|u_{s+\sigma}\|_{H^r(\Omega)}\leq C(\theta,s,\Omega)\|u_{s+\sigma}\|^{\theta}_{H^s(\Omega)}\|u_{s+\sigma}\|^{1-\theta}_{H^1(\Omega)}.
\end{align}
\item[$(b)$] $r=\theta \alpha(s)+(1-\theta)\cdot1~~\text{for all}~\theta\in (0,1)$ in the case \eqref{2}, and
\begin{align}\label{interpolation-inequality'}
\|u_{s+\sigma}\|_{H^r(\Omega)}\leq C(\theta,s,\Omega)\|u_{s+\sigma}\|^{\theta}_{H^{\alpha(s)}(\Omega)}\|u_{s+\sigma}\|^{1-\theta}_{H^1(\Omega)}.
\end{align}
\end{itemize}
Let us focus on the situation $(a)$. By choosing in particular $\theta=\frac{1}{2}$, then $r=\frac{s}{2}+\frac{1}{2}=s+\frac{1-s}{2}$ and we have that $u_{s+\sigma}\in H^{s+\frac{1-s}{2}}(\Omega)$. From this, we conclude that $u_{s+\sigma}\in H^{s+\epsilon}(\Omega)$ for every $0<\epsilon<\frac{1-s}{2}$. To complete the proof, it remains to show that the RHS of \eqref{interpolation-inequality} is uniform for $\sigma$ sufficiently small.
From \eqref{uniformly-boundedness-of-weak-solution} and \eqref{boundary-uniform-bound} we have that
\begin{equation}\label{b2}
\|u_{s+\sigma}\|_{H^1(\Omega)}\leq C_1\quad\quad\text{for all}~~\sigma~~\text{sufficiently small}.
\end{equation}
On the other hand, since $s<s+\sigma$, then from \cite[Proposition 2.1]{di2012hitchhiker's} there exists $c>0$ depending only on $s$ and $N$ such that
\begin{equation}\label{b3}
|u_{s+\sigma}|_{H^{s}(\Omega)}\leq c|u_{s+\sigma}|_{H^{s+\sigma}(\Omega)}.
\end{equation}
Using now $u_{s+\sigma}$ as a test function in \eqref{poisson-problem-for-regional-fractional-laplacian} with $s$ replace by $s+\sigma$ and integrating over $\Omega$, one has
\begin{align}\label{b4}
|u_{s+\sigma}|^2_{H^{s+\sigma}(\Omega)}=\frac{2}{C_{N,s+\sigma}}\int_{\Omega}fu_{s+\sigma}\ dx\leq\frac{2}{C_{N,s+\sigma}}\|f\|_{L^{\infty}(\Omega)}\|u_{s+\sigma}\|_{L^{\infty}(\Omega)}\leq c\quad\text{as}~~\sigma\to0^+.
\end{align}
thanks to \eqref{uniformly-boundedness-of-weak-solution} and the continuity of the map $s\mapsto C_{N,s}$. This, together with \eqref{b3} yield
\begin{equation}
\|u_{s+\sigma}\|_{H^s(\Omega)}\leq C_2\quad\quad\text{for}~~\sigma~~\text{sufficiently small}.
\end{equation}
From this, one gets $u_{s+\sigma}\in H^{s+\epsilon}(\Omega)$ with uniform bound in $\sigma\in[0,s_0)$.
In situation $(b)$, a similar argument as above yields $u_{s+\sigma}\in H^{s+\tilde{\epsilon}}(\Omega)$ for some $\tilde{\epsilon}>0$ depending on $s$.
Now, by combining $(i)$ and $(ii)$, we conclude the proof.
\end{proof}
This higher Sobolev regularity will be of a capital interest in the rest of the paper.\\
Next, we recall the following decay estimate regarding the logarithmic function. For all $r,\epsilon_0>0$, there holds that
\begin{equation}\label{logarithmic-decays}
|\log|z||\leq\frac{1}{e\epsilon_0}|z|^{-\epsilon_0}~~\text{if}~~ |z|\leq r~~~~\text{and}~~~~|\log|z||\leq\frac{1}{e\epsilon_0}|z|^{\epsilon_0}~~ \text{if}~~|z|\geq r.
\end{equation}
We end this section by recalling the following.
\begin{prop}(\cite[Lemma 6.6]{jarohs2020new})\label{diff-of-the-curve}
Let $I\subset\R$ be an open interval, $E$ be a Banach space and $\gamma:I\to E$ be a curve with the following properties
\begin{itemize}
\item [$(i)$] $\gamma$ is continuous.
\item [$(ii)$] $\partial_s^+\gamma(s):=\lim\limits_{\sigma\to0^+}\frac{\gamma(s+\sigma)-\gamma(s)}{\sigma}$ exists in $E$ for all $s\in I$.
\item [$(iii)$] The map $I\to E,~s\mapsto\partial_s^+\gamma(s)$ is continuous.
\end{itemize}
Then $\gamma$ is continuously differentiable with $\partial_s\gamma=\partial_s^+\gamma$.
\end{prop}
\section{Differentiability of the solution map in $(0,1)$}\label{differentiability-for-general-value-s}
In this section, we are concerned with the regularity of the map $(0,1)\rightarrow L^2(\Omega), s\mapsto u_s$, with being $u_s$ the unique weak solution of \eqref{poisson-problem-for-regional-fractional-laplacian}. In order to obtain the regularity of the solution $u_s$, regarded as function of $s$, our strategy consist to bound uniformly the difference quotient $\frac{u_{s+\sigma}-u_s}{\sigma}$ in the Hilbert space $H^s(\Omega)$ with respect to $\sigma$, after what, due to compactness, we therefore reach our goal.
The main result of this section is the following.
\begin{thm}
Let $f\in L^{\infty}(\Omega)$ with $\int_{\Omega}f\ dx=0$ and let $u_s\in\mathbb{X}^s(\Omega)$ be the unique weak solution of \eqref{poisson-problem-for-regional-fractional-laplacian}. Then the map
\begin{equation*}
(0,1)\rightarrow L^2(\Omega),\quad s\mapsto u_s
\end{equation*}
is of class $C^1$ and $w_s:=\partial_su_s$ uniquely solves in the weak sense the equation
\begin{equation}\label{s-derivative-equation}
(-\Delta)^s_{\Omega}w_s=M^s_{\Omega}u_s\quad\text{in}\quad\Omega,
\end{equation}
where for every $x\in\Omega$,
\begin{equation*}
M^s_{\Omega}u(x)=-\frac{\partial_sC_{N,s}}{C_{N,s}}f(x)+2C_{N,s}P.V.\int_{\Omega}\frac{(u(x)-u(y))}{|x-y|^{N+2s}}\log|x-y|\ dy.
\end{equation*}
\end{thm}
\begin{proof}
The proof is devided into three steps.\\
\textbf{Step 1.} We prove that the solution map $(0,1)\to L^2(\Omega), s\mapsto u_s$ is continuous.
Fix $s_0\in(0,1)$. Let $\delta\in(0,s_0)$ and $(s_n)_n\subset(s_0-\delta,1)$ be a sequence such that $s_n\to s_0$. We want to show that
\begin{equation}\label{z1}
u_{s_n}\to u_{s_0}\quad\text{in}~~L^2(\Omega)\quad\text{as}\quad\quad n\to\infty.
\end{equation}
Put $s':=\inf_{n\in\N}s_n>s_0-\delta$. Then,
\begin{align}\label{z2}
\nonumber|u_{s_n}|^2_{H^{s'}(\Omega)}&=\int_{\Omega}\int_{\Omega}\frac{(u_{s_n}(x)-u_{s_n}(y))^2}{|x-y|^{N+2s'}}\ dxdy\\
\nonumber&=\int_{\Omega}\int_{\Omega}\frac{(u_{s_n}(x)-u_{s_n}(y))^2}{|x-y|^{N+2s_n}}|x-y|^{2(s_n-s')}\ dxdy\\
&\leq d^{2(s_n-s')}_{\Omega}|u_{s_n}|^2_{H^{s_n}(\Omega)}\leq c|u_{s_n}|^2_{H^{s_n}(\Omega)}.
\end{align}
Now, since $u_{s_n}$ is the unique weak solution to $(-\Delta)^{s_n}_{\Omega}u_{s_n}=f$ in $\Omega$, then it follows that (we use $u_{s_n}$ as a test function)
\begin{align}\label{z3}
|u_{s_n}|^2_{H^{s_n}(\Omega)}=\frac{2}{C_{N,s_n}}\int_{\Omega}fu_{s_n}\ dx\leq\frac{2}{C_{N,s_n}}\|f\|_{L^2(\Omega)}\|u_{s_n}\|_{L^2(\Omega)}\leq \frac{2c_{s'}}{C_{N,s_n}}\|f\|_{L^2(\Omega)}|u_{s_n}|_{H^{s'}(\Omega)}
\end{align}
thanks to fractional Poincar\'{e} inequality \eqref{Poincare-inequality}. Using that $s\mapsto C_{N,s}$ is continuous, then it follows from \eqref{z3} and \eqref{z2} that
\begin{equation}\label{z4}
|u_{s_n}|_{H^{s'}(\Omega)}\leq c\quad\quad\text{as}\quad n\to\infty.
\end{equation}
This means that $(u_{s_n})_n$ is uniformly bounded in $H^{s'}(\Omega)$. Then there is $u_{*}\in H^{s'}(\Omega)$ such that after passing to a subsequence,
\begin{equation}\label{z5}
\begin{aligned}
&u_{s_n}\rightharpoonup u_{*}\quad\text{weakly in}~~H^{s'}(\Omega),\\
&u_{s_n}\rightarrow u_{*}\quad\text{strongly in}~~L^2(\Omega),\\
&u_{s_n}\rightarrow u_{*}\quad\text{a.e. in}~~\Omega.
\end{aligned}
\end{equation}
In particular, $\int_{\Omega}u_{*}\ dx=0$. We wish now to show that $u_{s_0}\equiv u_{*}$. To this end, we first prove that $u_{*}\in H^{s_0}(\Omega)$.
By Fatou's Lemma, we have
\begin{align}\label{z6}
\nonumber|u_{*}|^2_{H^{s_0}(\Omega)}&=\int_{\Omega}\int_{\Omega}\frac{(u_{*}(x)-u_{*}(y))^2}{|x-y|^{N+2s}}\ dxdy\leq\liminf_{n\to\infty}\int_{\Omega}\int_{\Omega}\frac{(u_{s_n}(x)-u_{s_n}(y))^2}{|x-y|^{N+2s_n}}\ dxdy\\
&=\liminf_{n\to\infty}|u_{s_n}|^2_{H^{s_n}(\Omega)}=\frac{2}{C_{N,s_0}}\|u_{*}\|_{L^2(\Omega)}\|f\|_{L^2(\Omega)}<\infty.
\end{align}
This implies that $u_{*}\in H^{s_0}(\Omega)$. We recall that in \eqref{z6}, we have used \eqref{z3} and \eqref{z5}.
On the other hand, for all $\phi\in C^{\infty}_0(\ov\Omega)$, we have
\begin{align*}
\langle f,\phi\rangle_{L^2(\Omega)}=\int_{\Omega}f\phi\ dx&=\lim\limits_{n\to\infty}\cE_{s_n}(u_{s_n},\phi)=\lim\limits_{n\to\infty}\int_{\Omega}u_{s_n}(-\Delta)^{s_n}_{\Omega}\phi\ dx\\
&=\int_{\Omega}u_{*}(-\Delta)^{s_0}_{\Omega}\phi\ dx=\cE_{s_0}(u_{*},\phi).
\end{align*}
This shows that $u_{*}\in H^{s_0}(\Omega)$ with $\int_{\Omega}u_{*}\ dx=0$ distributionaly solves the Poisson problem $(-\Delta)^{s_0}_{\Omega}u_{*}=f$ in $\Omega$. Recalling that $u_{s_0}\in H^{s_0}(\Omega)$ with $\int_{\Omega}u_{s_0}\ dx=0$ is the unique weak (distributional) solution to the Poisson problem $(-\Delta)^{s_0}_{\Omega}u_{s_0}=f$ in $\Omega$, we find that $u_{*}\equiv u_{s_0}$, as wanted.\\
\textbf{Step 2.} We show that the solution map $(0,1)\to L^2(\Omega),~s\mapsto u_s$ is right differentiable.\\
Fix $s\in (0,1)$ and define
\begin{equation}\label{v-sigma}
v_{\sigma}=\frac{u_{s+\sigma}-u_s}{\sigma}.
\end{equation}
Here, $u_{s+\sigma}$ is the unique weak solution of \eqref{poisson-problem-for-regional-fractional-laplacian} with $s$ replaced by $s+\sigma$. We wish first to study the asymptotic behavior of $v_{\sigma}$ as $\sigma\rightarrow0^+$.
For all $\phi\in C^{\infty}_0(\overline{\Omega})$,
\begin{align*}
\cE_s(u_s,\phi)=\int_{\Omega}f\phi\ dx=\cE_{s+\sigma}(u_{s+\sigma},\phi)=\cE_s(u_s-u_{s+\sigma},\phi)+\cE_s(u_{s+\sigma},\phi)
\end{align*}
that is
\begin{align}\label{n0}
\cE_s(u_s-u_{s+\sigma},\phi)=\cE_{s+\sigma}(u_{s+\sigma},\phi)-\cE_s(u_{s+\sigma},\phi).
\end{align}
Now,
\begin{align}\label{n1}
\nonumber&\cE_{s+\sigma}(u_{s+\sigma},\phi)-\cE_s(u_{s+\sigma},\phi)\\
\nonumber&=\frac{C_{N,s+\sigma}}{2}\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s+2\sigma}}\ dxdy\\
\nonumber&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\frac{C_{N,s}}{2}\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}\ dxdy\\
\nonumber&=\frac{C_{N,s+\sigma}-C_{N,s}}{2}\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s+2\sigma}}\ dxdy\\
\nonumber&+\frac{C_{N,s}}{2}\int_{\Omega}\int_{\Omega}\Big(\frac{1}{|x-y|^{N+2s+2\sigma}}-\frac{1}{|x-y|^{N+2s}}\Big)(u_{s+\sigma}(x)-u_{s+\sigma}(y))(\phi(x)-\phi(y))\ dxdy\\
\nonumber&=\frac{1}{C_{N,s+\sigma}}\times(C_{N,s+\sigma}-C_{N,s}) \cE_{s+\sigma}(u_{s+\sigma},\phi)\\
\nonumber&\ \ \ \ \ \ \ +\frac{C_{N,s}}{2}\int_{\Omega}\int_{\Omega}\Big(|x-y|^{-2\sigma}-1\Big)\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}\ dxdy\\
\nonumber&=\frac{1}{C_{N,s+\sigma}}\times(C_{N,s+\sigma}-C_{N,s})\int_{\Omega}f\phi\ dx\\
&\ \ \ \ \ \ \ +\frac{C_{N,s}}{2}\int_{\Omega}\int_{\Omega}\Big(|x-y|^{-2\sigma}-1\Big)\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}\ dxdy.
\end{align}
Next, we write
\begin{align}\label{n2}
\nonumber|x-y^{-2\sigma}-1&=\exp(-2\sigma\log|x-y|)-1=-2\sigma\log|x-y|\int_{0}^{1}\exp(-2t\sigma\log|x-y|)\ dt\\
&=-2\sigma\psi_{\sigma}(x,y)\log|x-y|\quad\text{with}~~\psi_{\sigma}(x,y)=\int_{0}^{1}\exp(-2t\sigma\log|x-y|)\ dt.
\end{align}
Plugging \eqref{n2} into \eqref{n1}, we get
\begin{align}\label{n3}
\nonumber&\cE_{s+\sigma}(u_{s+\sigma},\phi)-\cE_s(u_{s+\sigma},\phi)\\
\nonumber&=\frac{1}{C_{N,s+\sigma}}\times(C_{N,s+\sigma}-C_{N,s})\int_{\Omega}f\phi\ dx\\
&-\sigma C_{N,s}\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}\psi_{\sigma}(x,y)\log|x-y|\ dxdy.
\end{align}
Equations \eqref{n0} and \eqref{n3} yield
\begin{align}\label{n4}
\nonumber\cE_s(v_{\sigma},\phi)&=-\frac{1}{C_{N,s+\sigma}}\times\frac{C_{N,s+\sigma}-C_{N,s}}{\sigma}\int_{\Omega}f\phi\ dx\\
&+ C_{N,s}\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}\psi_{\sigma}(x,y)\log|x-y|\ dxdy
\end{align}
for all $\phi\in C^{\infty}_0(\ov\Omega)$. Now, by density, there is $\phi_n\in C^{\infty}_0(\ov\Omega)$ such that $\phi_n\to v_{\sigma}$ in $H^{s+\epsilon}(\Omega)$ for $\epsilon>0$. Moreover, from \eqref{n3},
\begin{align}\label{n5}
\nonumber\cE_s(v_{\sigma},\phi_n)&=-\frac{1}{C_{N,s+\sigma}}\times\frac{C_{N,s+\sigma}-C_{N,s}}{\sigma}\int_{\Omega}f\phi_n\ dx\\
&+ C_{N,s}\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))(\phi_n(x)-\phi_n(y))}{|x-y|^{N+2s}}\psi_{\sigma}(x,y)\log|x-y|\ dxdy.
\end{align}
Using Cauchy-Schwarz inequality,
\begin{align}\label{n6}
\nonumber&|\cE_s(v_{\sigma},\phi_n)-\cE_s(v_{\sigma},v_{\sigma})|\\
\nonumber&\leq\frac{C_{N,s}}{2}\int_{\Omega}\int_{\Omega}\frac{|v_{\sigma}(x)-v_{\sigma}(y)||(\phi_n(x)-\phi_n(y))-(v_{\sigma}(x)-v_{\sigma}(y))|}{|x-y|^{N+2s}}\ dxdy\\
&\leq\frac{C_{N,s}}{2}|v_{\sigma}|_{H^s(\Omega)}|\phi_n-v_{\sigma}|_{H^s(\Omega)}\to0\quad\quad\text{as}~~n\to\infty.
\end{align}
This implies that
\begin{equation}\label{n7}
\cE_s(v_{\sigma},\phi_n)\to\cE_s(v_{\sigma},v_{\sigma})\quad\quad\text{as}~~n\to\infty.
\end{equation}
Since also $\phi_n\to v_{\sigma}$ in $L^{2}(\Omega)$, thanks to Poincar\'{e} ineqality, then
\begin{equation}\label{n8}
\int_{\Omega}f\phi_n\ dx\to\int_{\Omega}fv_{\sigma}\ dx.
\end{equation}
On the other hand, using that
\begin{equation}\label{n9}
|\psi_{\sigma}(x,y)|\leq\max\{1,\exp(-2\sigma\log|x-y|)\}
\end{equation}
then applying again Cauchy-Schwarz inequality, one can also show that
\begin{align}\label{n10}
\nonumber&\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))(\phi_n(x)-\phi_n(y))}{|x-y|^{N+2s}}\psi_{\sigma}(x,y)\log|x-y|\ dxdy\\
&\to\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))(v_{\sigma}(x)-v_{\sigma}(y))}{|x-y|^{N+2s}}\psi_{\sigma}(x,y)\log|x-y|\ dxdy~~~~\text{as}~~n\to\infty.
\end{align}
Combining \eqref{n7}, \eqref{n8} and \eqref{n10}, then from \eqref{n5} it follows that
\begin{align}\label{n11}
\nonumber\cE_s(v_{\sigma},v_{\sigma})&=-\frac{1}{C_{N,s+\sigma}}\times\frac{C_{N,s+\sigma}-C_{N,s}}{\sigma}\int_{\Omega}fv_{\sigma}\ dx\\
&+ C_{N,s}\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))(v_{\sigma}(x)-v_{\sigma}(y))}{|x-y|^{N+2s}}\psi_{\sigma}(x,y)\log|x-y|\ dxdy.
\end{align}
From \eqref{n11}, we write
\begin{align}\label{n12}
\nonumber\frac{C_{N,s}}{2}|v_{\sigma}|^2_{H^s(\Omega)}&=\cE_s(v_{\sigma},v_{\sigma})\leq\frac{1}{C_{N,s+\sigma}}\Big|\frac{C_{N,s+\sigma}-C_{N,s}}{\sigma}\Big|\int_{\Omega}|f||v_{\sigma}|\ dx\\
&+ C_{N,s}\int_{\Omega}\int_{\Omega}\frac{|u_{s+\sigma}(x)-u_{s+\sigma}(y)||v_{\sigma}(x)-v_{\sigma}(y)|}{|x-y|^{N+2s}}|\psi_{\sigma}(x,y)||\log|x-y||\ dxdy.
\end{align}
Since the map $(0,1)\ni s\mapsto C_{N,s}$ is of class $C^1$, then
\begin{align}\label{n13}
\frac{1}{C_{N,s+\sigma}}\Big|\frac{C_{N,s+\sigma}-C_{N,s}}{\sigma}\Big|\leq\frac{|\partial_sC_{N,s}|}{C_{N,s}}+o(1)\quad\quad\text{as}~~\sigma\to0^+.
\end{align}
Now, H\"{o}lder inequality and Poincar\'{e} inequality (see \eqref{Poincare-inequality}) yield
\begin{align}\label{n14}
\int_{\Omega}|f||v_{\sigma}|\ dx\leq\|f\|_{L^2(\Omega)}\|v_{\sigma}\|_{L^2(\Omega)}\leq C(N,s,\Omega,\|f\|_{L^2(\Omega)})|v_{\sigma}|_{H^s(\Omega)}.
\end{align}
On the other hand, from \eqref{n9}, we have
\begin{align*}
&\int_{\Omega}\int_{\Omega}\frac{|u_{s+\sigma}(x)-u_{s+\sigma}(y)||v_{\sigma}(x)-v_{\sigma}(y)|}{|x-y|^{N+2s}}|\psi_{\sigma}(x,y)||\log|x-y||\ dxdy\\
&\leq\int_{\Omega}\int_{\Omega}\frac{|u_{s+\sigma}(x)-u_{s+\sigma}(y)||v_{\sigma}(x)-v_{\sigma}(y)|}{|x-y|^{N+2s}}|\log|x-y||\ dxdy\\
&\ \ \ \ +\int_{\Omega}\int_{\Omega}\frac{|u_{s+\sigma}(x)-u_{s+\sigma}(y)||v_{\sigma}(x)-v_{\sigma}(y)|}{|x-y|^{N+2s+2\sigma}}|\log|x-y||\ dxdy.
\end{align*}
To estimate the first term on the RHS of the above inequality, we use Cauchy-Schwarz inequality together with the Logarithmic decay \eqref{logarithmic-decays}:
\begin{align}\label{n15}
\nonumber&\int_{\Omega}\int_{\Omega}\frac{|u_{s+\sigma}(x)-u_{s+\sigma}(y)||v_{\sigma}(x)-v_{\sigma}(y)|}{|x-y|^{N+2s}}|\log|x-y||\ dxdy\\
\nonumber&\leq\Big(\int_{\Omega}\int_{\Omega}\frac{|u_{s+\sigma}(x)-u_{s+\sigma}(y)|^2}{|x-y|^{N+2s}}|\log|x-y||^2\ dxdy\Big)^{1/2}|v_{\sigma}|_{H^s(\Omega)}\\
\nonumber&\leq\Bigg(\frac{1}{(e\epsilon_0)^2}\int_{\Omega}\int_{\Omega}\frac{|u_{s+\sigma}(x)-u_{s+\sigma}(y)|^2}{|x-y|^{N+2s}}(|x-y|^{-2\epsilon_0}+|x-y|^{2\epsilon_0})\ dxdy\Bigg)^{1/2}|v_{\sigma}|_{H^s(\Omega)}\\
\nonumber&\leq c(\epsilon_0)\Bigg(\int_{\Omega}\int_{\Omega}\frac{|u_{s+\sigma}(x)-u_{s+\sigma}(y)|^2}{|x-y|^{N+2s+2\epsilon_0}}\ dxdy+d^{2\epsilon_0}_{\Omega}|u_{s+\sigma}|^2_{H^s(\Omega)}\Bigg)^{1/2}|v_{\sigma}|_{H^s(\Omega)}\\
&=c(\epsilon_0)\Big(|u_{s+\sigma}|^2_{H^{s+\epsilon_0}(\Omega)}+d^{2\epsilon_0}_{\Omega}|u_{s+\sigma}|^2_{H^s(\Omega)}\Big)^{1/2}|v_{\sigma}|_{H^s(\Omega)}.
\end{align}
By Proposition \ref{higher-sobolev-regularity} there exist $K_1, K_2>0$ independent on $\sigma$ such that
\begin{equation}\label{n16}
|u_{s+\sigma}|_{H^{s+\epsilon_0}(\Omega)}\leq K_1\quad\quad\text{and}\quad\quad |u_{s+\sigma}|_{H^s(\Omega)}\leq K_2\quad\text{for}~~\sigma~~\text{sufficiently small}.
\end{equation}
Combining this with \eqref{n15}, we obtain that
\begin{equation}\label{n20}
\int_{\Omega}\int_{\Omega}\frac{|u_{s+\sigma}(x)-u_{s+\sigma}(y)||v_{\sigma}(x)-v_{\sigma}(y)|}{|x-y|^{N+2s}}|\log|x-y||\ dxdy\leq c|v_{\sigma}|_{H^s(\Omega)}.
\end{equation}
By a similar argument as above, we also obtain the following bound
\begin{equation}\label{n21}
\int_{\Omega}\int_{\Omega}\frac{|u_{s+\sigma}(x)-u_{s+\sigma}(y)||v_{\sigma}(x)-v_{\sigma}(y)|}{|x-y|^{N+2s+2\sigma}}|\log|x-y||\ dxdy\leq c|v_{\sigma}|_{H^s(\Omega)}.
\end{equation}
Combining \eqref{n13}, \eqref{n14}, \eqref{n20}, and \eqref{n21}, we find from \eqref{n12} that
\begin{equation}\label{n22}
|v_{\sigma}|_{H^s(\Omega)}\leq c.
\end{equation}
In other words, $v_{\sigma}$ is uniformly bounded in $H^s(\Omega)$ with respect to $\sigma$. Therefore, after passing to a subsequence, there is $w_s\in H^s(\Omega)$ such that
\begin{equation}\label{n23}
\begin{aligned}
&v_{\sigma}\rightharpoonup w_s\quad\text{weakly in}~~H^s(\Omega),\\
&v_{\sigma}\rightarrow w_s\quad\text{strongly in}~~L^2(\Omega),\\
&v_{\sigma}\rightarrow w_s\quad\text{a.e. in}~~\Omega.
\end{aligned}
\end{equation}
In particular, $\int_{\Omega}w_s\ dx=0$ since does $v_{\sigma}$. \\
To obtain the right-differentiability of the solution map $s\mapsto u_s$, it suffices to show that $w_s$ is unique as a limit of the whole sequence $v_{\sigma}$.
First of all, from \eqref{n4}, thanks to Proposition \ref{higher-sobolev-regularity} and Dominated Convergence Theorem, we deduce that $w_s$ solves
\begin{align}\label{n29}
\nonumber\cE_s(w_s,\phi)&=-\frac{\partial_sC_{N,s}}{C_{N,s}}\int_{\Omega}f\phi\ dx\\
&+ C_{N,s}\int_{\Omega}\int_{\Omega}\frac{(u_s(x)-u_s(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}\log|x-y|\ dxdy
\end{align}
for all $\phi\in C^{\infty}_0(\ov\Omega)$.
Let denote by $v_{\sigma_k}$ the corresponding subsequence of $v_{\sigma}$ for which \eqref{n23} holds. Consider now another subsequence $v_{\sigma_i}$ with $v_{\sigma_i}\to \overline{w}_s$ for some $\overline{w}_s\in H^s(\Omega)$ with $\int_{\Omega}\overline{w}_s\ dx=0$. We wish to prove that prove that $w_s=\overline{w}_s$. Let set $W_s=w_s-\overline{w}_s$. Then, in particular, $\int_{\Omega}W_s\ dx=0$. Moreover, for all $\phi\in C^{\infty}_0(\ov\Omega)$,
\begin{align}\label{n30}
\nonumber\cE_s(W_s,\phi)&=\cE_s(w_s-\overline{w}_s,\phi)=\cE_s(w_s,\phi)-\cE_s(\overline{w}_s,\phi)=\cE_s(w_s,\phi)-\lim\limits_{i\to\infty}\cE_s(v_{\sigma_i},\phi)\\
\nonumber&=\cE_s(w_s,\phi)-\lim\limits_{i\to\infty}\cE_s(v_{\sigma_i}-w_s+w_s,\phi)=-\lim\limits_{i\to\infty}\cE_s(v_{\sigma_i}-w_s,\phi)\\
&=-\lim\limits_{i\to\infty}\lim\limits_{k\to\infty}\cE_s(v_{\sigma_i}-v_{\sigma_k},\phi).
\end{align}
From \eqref{n4}, one gets
\begin{align*}
\cE_s(v_{\sigma_i}-v_{\sigma_k},\phi)&=\Bigg(-\frac{1}{C_{N,s+\sigma_i}}\times\frac{C_{N,s+\sigma_i}-C_{N,s}}{\sigma_i}+\frac{1}{C_{N,s+\sigma_k}}\times\frac{C_{N,s+\sigma_k}-C_{N,s}}{\sigma_k}\Bigg)\int_{\Omega}f\phi\ dx\\
&+ C_{N,s}\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma_i}(x)-u_{s+\sigma_i}(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}\psi_{\sigma_i}(x,y)\log|x-y|\ dxdy\\
&-C_{N,s}\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma_k}(x)-u_{s+\sigma_k}(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}\psi_{\sigma_k}(x,y)\log|x-y|\ dxdy.
\end{align*}
Using that $s\mapsto C_{N,s}$ is of class $C^1$, the fact that $s\mapsto u_s$ is continuous and Proposition \ref{higher-sobolev-regularity}, the Dominated Convergence Theorem yields
\begin{equation}
\lim\limits_{i\to\infty}\lim\limits_{k\to\infty}\cE_s(v_{\sigma_i}-v_{\sigma_k},\phi)=0.
\end{equation}
Consequently, one gets from \eqref{n30} that
\begin{equation}\label{n31}
\cE_s(W_s,\phi)=0.
\end{equation}
By density, \eqref{n31} also holds with $\phi$ replaced by $W_s$, that is,
\begin{equation}\label{n32}
\cE_s(W_s,W_s)=0.
\end{equation}
This implies that $W_s=const$. Morever, since also $\int_{\Omega}W_s\ dx=0$, then we get that $W_s=0$, that is, $w_s=\overline{w}_s$ as wanted. In conclusion, the solution map $s\mapsto u_s$ is right-differentiable with $\partial_s^+u_s=w_s$, solving uniquely \eqref{n29}.\\
\textbf{Step 3.} We establishe the continuity of the map $s\mapsto\partial_s^+u_s=w_s$. The proof of this is similar to that in \textbf{Step 1} and we include it for the sake of completeness.
Fix again $s_0\in(0,1)$. Let $\delta\in(0,s_0)$ and $(s_n)_n\subset(s_0-\delta,1)$ be a sequence such that $s_n\to s_0$. By putting also $s':=\inf_{n\in\N}s_n>s_0-\delta$, then as in \eqref{z2} one get that
\begin{equation}\label{w1}
|w_{s_n}|^2_{H^{s'}(\Omega)}\leq c|w_{s_n}|^2_{H^{s_n}(\Omega)}.
\end{equation}
Now, from \eqref{n29}, it follows that
\begin{align}\label{n35}
\nonumber\cE_{s_n}(w_{s_n},w_{s_n})&=-\frac{\partial_{s_n}C_{N,s_n}}{C_{N,s_n}}\int_{\Omega}fw_{s_n}\ dx\\
&+ C_{N,s_n}\int_{\Omega}\int_{\Omega}\frac{(u_{s_n}(x)-u_{s_n}(y))(w_{s_n}(x)-w_{s_n}(y))}{|x-y|^{N+2s_n}}\log|x-y|\ dxdy,
\end{align}
that is,
\begin{align}\label{n36}
\nonumber|w_{s_n}|^2_{H^{s_n}(\Omega)}&\leq\Big|\frac{2\partial_{s_n}C_{N,s_n}}{C^2_{N,s_n}}\Big|\int_{\Omega}|f||w_{s_n}|\ dx\\
&+ 2\int_{\Omega}\int_{\Omega}\frac{|u_{s_n}(x)-u_{s_n}(y)||w_{s_n}(x)-w_{s_n}(y)|}{|x-y|^{N+2s_n}}|\log|x-y||\ dxdy.
\end{align}
Now, using that $s\mapsto C_{N,s}$ is of class $C^1$, then
\begin{align}\label{n37}
\Big|\frac{2\partial_{s_n}C_{N,s_n}}{C^2_{N,s_n}}\Big|\leq\Big|\frac{2\partial_{s_0}C_{N,s_0}}{C^2_{N,s_0}}\Big|+o(1)\quad\text{as}~~n\to\infty.
\end{align}
By H\"{o}lder inequality and Poincar\'{e} inequality \eqref{Poincare-inequality}, we get
\begin{align}\label{n38}
\int_{\Omega}|f||w_{s_n}|\ dx\leq c|w_{s_n}|_{H^{s_n}(\Omega)}
\end{align}
On the other hand, Cauchy-Schwarz inequality together with \eqref{logarithmic-decays} yield
\begin{align}\label{n39}
\nonumber&\int_{\Omega}\int_{\Omega}\frac{|u_{s_n}(x)-u_{s_n}(y)||w_{s_n}(x)-w_{s_n}(y)|}{|x-y|^{N+2s_n}}|\log|x-y||\ dxdy\\
&\leq\Big(c_{\epsilon_0}|u_{s_n}|^2_{H^{s_n+\epsilon_0}(\Omega)}+\tilde{c}_{\epsilon_0}|u_{s_n}|^2_{H^{s_n}(\Omega)}\Big)|w_{s_n}|_{H^{s_n}(\Omega)}.
\end{align}
By Proposition \ref{higher-sobolev-regularity}, we find that
\begin{equation}
|u_{s_n}|_{H^{s_n+\epsilon_0}(\Omega)}\leq C_1\quad\quad\text{and}\quad\quad|u_{s_n}|_{H^{s_n}(\Omega)}\leq C_2\quad\text{as}~~n\to\infty.
\end{equation}
Taking this into account, we get from \eqref{n39} that
\begin{equation}\label{w2}
\int_{\Omega}\int_{\Omega}\frac{|u_{s_n}(x)-u_{s_n}(y)||w_{s_n}(x)-w_{s_n}(y)|}{|x-y|^{N+2s_n}}|\log|x-y||\ dxdy\leq C|w_{s_n}|_{H^{s_n}(\Omega)}
\end{equation}
for $n$ sufficiently large.
It follows from \eqref{n37}, \eqref{n38}, \eqref{w2} and \eqref{w1} that
\begin{align}\label{n40}
|w_{s_n}|_{H^{s'}(\Omega)}\leq|w_{s_n}|_{H^{s_n}(\Omega)}\leq c\quad\text{for}~~n~~\text{sufficiently large}.
\end{align}
This means that $w_{s_n}$ is uniformly bounded in $H^{s'}(\Omega)$ with respect to $n$. Therefore, up to a subsequence, there is $w_{*}\in H^{s'}(\Omega)$ such that
\begin{equation}\label{n41}
\begin{aligned}
&w_{s_n}\rightharpoonup w_{*}\quad\text{weakly in}~~H^{s'}(\Omega),\\
&w_{s_n}\rightarrow w_{*}\quad\text{strongly in}~~L^2(\Omega),\\
&w_{s_n}\rightarrow w_{*}\quad\text{a.e. in}~~\Omega.
\end{aligned}
\end{equation}
In particular, we have $\int_{\Omega}w_{*}\ dx=0$. Next, we show that $w_{*}\equiv w_{s_0}$.
By Fatou's Lemma,
\begin{align*}
|w_{*}|_{H^{s_0}(\Omega)}&=\Big(\int_{\Omega}\int_{\Omega}\frac{(w_{*}(x)-w_{*}(y))^2}{|x-y|^{N+2s_0}}\ dxdy\Big)^{1/2}\\
&\leq\liminf_{n\to\infty}\Big(\int_{\Omega}\int_{\Omega}\frac{(w_{s_n}(x)-w_{s_n}(y))^2}{|x-y|^{N+2s_n}}\ dxdy\Big)^{1/2}\\
&=\liminf_{n\to\infty}|w_{s_n}|_{H^{s_n}(\Omega)}\leq C<\infty.
\end{align*}
This implies that $w_{*}\in H^{s_0}(\Omega)$. Notice that we have used \eqref{n36}, \eqref{n37}, \eqref{n38}, \eqref{n39} and Proposition \ref{higher-sobolev-regularity}.
On the other hand, for all $\phi\in C^{\infty}_0(\ov\Omega)$, we have
\begin{align}\label{w3}
\nonumber&\cE_{s_0}(w_{*},\phi)=\int_{\Omega}w_{*}(-\Delta)^{s_0}_{\Omega}\phi\ dx=\lim\limits_{n\to\infty}\int_{\Omega}w_{s_n}(-\Delta)^{s_n}_{\Omega}\phi\ dx=\lim\limits_{n\to\infty}\cE_{s_n}(w_{s_n},\phi)\\
&=-\lim\limits_{n\to\infty}\frac{\partial_{s_n}C_{N,s_n}}{C_{N,s_n}}\int_{\Omega}f\phi\ dx+\lim\limits_{n\to\infty} C_{N,s_n}\int_{\Omega}\int_{\Omega}\frac{(u_{s_n}(x)-u_{s_n}(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s_n}}\log|x-y|\ dxdy.
\end{align}
In \eqref{w3}, we have used \eqref{n29}. Now, by \textbf{Step 1} and Proposition \ref{higher-sobolev-regularity}, one obtains from \eqref{w3} that
\begin{align}\label{n42}
\cE_{s_0}(w_{*},\phi)=-\frac{\partial_{s_0}C_{N,s_0}}{C_{N,s_0}}\int_{\Omega}f\phi\ dx
+ C_{N,s_0}\int_{\Omega}\int_{\Omega}\frac{(u_{s_0}(x)-u_{s_0}(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s_0}}\log|x-y|\ dxdy
\end{align}
for all $\phi\in C^{\infty}_0(\ov\Omega)$. \\
Since by \textbf{Step 2} $w_{s_0}\in H^{s_0}(\Omega)$ with $\int_{\Omega}w_{s_0}\ dx=0$ is the unique solution to \eqref{n42}, then one finds that $w_{*}\equiv w_{s_0}$. This yields the continuity of the map $s\mapsto w_s$ and this concudes the proof of \textbf{Step 3}.\\
In summary, from \textbf{Steps 1, 2} and \textbf{3}, we have shown that
\begin{itemize}
\item [$(i)$] $s\mapsto u_s$ is continuous;
\item [$(ii)$] $\partial_s^+u_s$ exists in $L^2(\Omega)$ for all $s\in (0,1)$;
\item [$(iii)$] The map $(0,1)\to L^2(\Omega),~s\mapsto\partial_s^+u_s$ is continuous.
\end{itemize}
Therefore, by Proposition \ref{diff-of-the-curve}, we conclude that the solution map $(0,1)\to L^2(\Omega),~s\mapsto\partial_s^+u_s$ is continuously differentiable with $\partial_su_s=\partial_s^+u_s$. Moreover, from \eqref{n29}, we have that $w_s=\partial_su_s$ solves in weak sense the equation
\begin{equation}
(-\Delta)^s_{\Omega}w_s=M^s_{\Omega}u_s\quad\quad\text{in}~~\Omega
\end{equation}
with
\begin{equation*}
M^s_{\Omega}u(x)=-\frac{\partial_sC_{N,s}}{C_{N,s}}f(x)+2C_{N,s}P.V.\int_{\Omega}\frac{(u(x)-u(y))}{|x-y|^{N+2s}}\log|x-y|\ dy,~~x\in\Omega.
\end{equation*}
\end{proof}
\section{Eigenvalues problem case}\label{eigenvalue-problem-case}
The aim of this section is to study \eqref{poisson-problem-for-regional-fractional-laplacian} when $f=\lambda_su_s$ i.e., the eigenvalues problem
\begin{equation}\label{eigenvalues-problem}
(-\Delta)^s_{\Omega}u_s=\lambda_su_s\quad\text{in}\quad\Omega.
\end{equation}
More precisely, we discuss the $s$-dependence of the map $s\mapsto \lambda_{1,s}$ where $\lambda_{1,s}$ is the first nontrivial eigenvalue of $(-\Delta)^s_{\Omega}$. We notice that equation \eqref{eigenvalues-problem} is understood in weak sense. Here and throughout the end of this section, we fix $\Omega$ as a bounded domain with $C^{1,1}$ boundary.
Let
\begin{equation}
0<\lambda_{1,s}\leq\lambda_{2,s}\leq\cdots\leq\lambda_{k,s}\leq\cdots,
\end{equation}
be the sequence of eigenvalues (counted with multiplicity) of $(-\Delta)^s_{\Omega}$ in $\Omega$ with corresponding eigenfunctions
$$\phi_{1,s}, \phi_{2,s},\dots, \phi_{k,s},\dots.$$
It is known that the system $\{\phi_{i,s}\}_i$ form an $L^2$-orthonormal basis. Variationnaly, we have \begin{equation}\label{k-th-eigenvalue-of-regional-fractional-laplacian}
\lambda_{k,s}:=\inf_{V\in V^s_k}\sup_{\phi\in S_{V}}\cE_s(\phi,\phi),
\end{equation}
where $V^s_k:=\{V\subset\mathbb{X}^s(\Omega):\dim V=k\}$ and $S_{V}:=\{\phi\in V:\|\phi\|_{L^2(\Omega)}=1\}$ for all $V\in V^s_k$. However, when $k=1$ then $\lambda_{1,s}$ is simply characterized by (see e.g., \cite[Theorem 3.1]{del2015first})
\begin{equation}\label{first-eigenvalue-of-regional-fractional-laplacian}
\lambda_{1,s}:=\inf\{\cE_s(\phi,\phi):\phi\in\mathbb{X}^s(\Omega),\|\phi\|_{L^2(\Omega)}=1\}.
\end{equation}
In this section, we wish to study the differentiability of the map $(0,1)\ni s\mapsto \lambda_{1,s}$. As first remark, we know that the first nontrivial eigenvalue of $(-\Delta)^s_{\Omega}$ is not in general \textit{simple}. Therefore, the main focus here is one-sided differentiability.\\
In what follows, we discuss the right differentiability of the map $s\mapsto\lambda_{1,s}$. Here and throughout the end of this Section, we use respectively $\lambda_s$ and $u_s$ for $\lambda_{1,s}$ and $u_{1,s}$ to alleviate the notation.
\begin{thm}\label{differentiability-of-eigenvalues}
Regarded as function of $s$, $\lambda_s$ is right differentiable on $(0,1)$ and
\begin{equation}\label{C-1-regularity-of-eigenvalues}
\partial^+_s\lambda_s:=\lim\limits_{\sigma\rightarrow0^+}\frac{\lambda_{s+\sigma}-\lambda_s}{\sigma}=\inf\{J_s(u):u\in \M_s\}
\end{equation}
where
\begin{equation}
J_s(u)=\frac{\partial_sC_{N,s}}{C_{N,s}}\lambda_s-C_{N,s}\int_{\Omega}\int_{\Omega}\frac{(u(x)-u(y))^2}{|x-y|^{N+2s}}\log|x-y|\ dxdy
\end{equation}
and $\M_s$ the set of $L^2$-normalized eigenfunctions of $(-\Delta)^s_{\Omega}$ corresponding to $\lambda_s$.
Moreover, the infimum in \eqref{C-1-regularity-of-eigenvalues} is attained.
\end{thm}
We now collect some partial results needed for the proof of Theorem \ref{differentiability-of-eigenvalues}. In the sequel, we prove the following two lemmas in the same spirit as Theorem 1.3 and Lemma 2.1 in \cite{deblassie2005alpha}.
\begin{lemma}\label{continuity-of-quadratic-form-with-respect-to-s}
Let $\phi,\psi\in C^{\infty}(\overline{\Omega})$. Then, regarded as function of $s$,
\begin{equation*}
\cE_s(\phi,\psi):(0,1)\rightarrow\R
\end{equation*}
is continuous on $(0,1)$.
\end{lemma}
\begin{proof}
It suffices to show that
\begin{equation*}
\lim\limits_{\alpha\rightarrow s}\cE_{\alpha}(\phi,\psi)=\cE_s(\phi,\psi).
\end{equation*}
Let $\alpha\in(s-\delta,s+\delta)$ where $\delta=\frac{1}{4}\min\{1-s,s\}$. Then,
\begin{align*}
\frac{(\phi(x)-\phi(y))(\psi(x)-\psi(y))}{|x-y|^{N+2\alpha}}&\leq\frac{|\phi(x)-\phi(y)||\psi(x)-\psi(y)|}{|x-y|^{N+2\alpha}}\\&\leq\|\nabla\phi\|_{L^{\infty}(\Omega)}\|\nabla\psi\|_{L^{\infty}(\Omega)}\frac{1}{|x-y|^{N+2\alpha-2}}\\&\leq C(\phi,\psi)\max\Big\{\frac{1}{|x-y|^{N+2(s-\delta)-2}},\frac{1}{|x-y|^{N+2(s+\delta)-2}}\Big\}\\&=:g_{s,\delta}(x,y).
\end{align*}
Using polar coordinates, it is not difficult to see that $g_{s,\delta}$ is integrable on $\Omega\times\Omega$ with
\begin{equation*}
\int_{\Omega}\int_{\Omega}|g_{s,\delta}(x,y)|\ dxdy\leq C(\phi,\psi)\Big(\frac{|\Omega||S^{N-1}|}{2(1-s+\delta)}d^{2(1-s+\delta)}_{\Omega}+\frac{|\Omega||S^{N-1}|}{2(1-s-\delta)}d^{2(1-s-\delta)}_{\Omega}\Big),
\end{equation*}
and therefore, applying Lebesgue's Dominated Convergence Theorem, we get that
\begin{equation*}
\lim\limits_{\alpha\rightarrow s}\cE_{\alpha}(\phi,\psi)=\cE_s(\phi,\psi),
\end{equation*}
as needed.
\end{proof}
\begin{lemma}\label{right-continuity-of-eigenvalues}
Let $k\geq1$ and $\lambda_{k,s}$ the $k$-th eigenvalue of $(-\Delta)^s_{\Omega}$ in $\Omega$. Then, regarded as function of $s$, $\lambda_{k,s}$ is continuous on $(0,1)$ for all $k\in\N$.
\end{lemma}
\begin{proof}
The proof is divided in two steps. First one shows the limsup inequality. The second step is to obtain the reverse inequality i.e., the liminf inequality.\\
\textbf{\textit{Step 1}.} We show that
\begin{equation}\label{limit-sup-for-continuity}
\limsup_{\alpha\rightarrow s}\lambda_{k,\alpha}\leq\lambda_{k,s}.
\end{equation}
Let $\epsilon>0$ and $k\geq1$. Using that $C^{\infty}_0(\overline{\Omega})$ is dense in $\mathbb{X}^s(\Omega)$, there exist $\phi_1,\dots,\phi_k\in C^{\infty}_0(\overline{\Omega})$ such that
\begin{equation}\label{density-consequence}
|\langle\phi_{i,s},\phi_{j,s}\rangle_{L^2(\Omega)}-\langle\phi_{i},\phi_{j}\rangle_{L^2(\Omega)}|\leq\frac{\epsilon}{8k^2}\quad\text{and}\quad|\cE_s(\phi_{i,s},\phi_{j,s})-\cE_s(\phi_{i},\phi_{j})|\leq\frac{\epsilon}{8k^2},
\end{equation}
for all $1\leq i,j\leq k$. Now, from Lemma \ref{continuity-of-quadratic-form-with-respect-to-s}, there is $\beta_0>0$ such that for all $\alpha\in (s-\beta_0,s+\beta_0)$,
\begin{equation}\label{c}
|\cE_{\alpha}(\phi_i,\phi_j)-\cE_s(\phi_i,\phi_j)|\leq\frac{\epsilon}{8k^2}.
\end{equation}
According to \eqref{density-consequence}, we also have
\begin{equation*}
|\langle\phi_{i},\phi_j\rangle_{L^2(\Omega)}|<\frac{\epsilon}{8k^2}~(i\neq j)\quad\text{and}\quad 1-\frac{\epsilon}{8k^2}<\|\phi_i\|^2_{L^2(\Omega)}<1+\frac{\epsilon}{8k^2},
\end{equation*}
and therefore as in \cite[Section $2$]{deblassie2005alpha}, the familly $\{\phi_i\}_{i=1,\dots,k}$ is linearily independent.
As a consequence, we have by setting in particular $V=\text{span}\{\phi_1,\dots,\phi_k\}$ that
\begin{equation}\label{d}
\lambda_{k,\alpha}\leq\sup_{\phi\in S_V}\cE_{\alpha}(\phi,\phi)\leq\cE_{\alpha}(\phi,\phi)+\frac{\epsilon}{4}.
\end{equation}
Now, for $\phi\in S_V$, there is a sequence of real numbers $\{a_i\}_{i=1,\dots,k}\subset\R$ satisfying $\sum_{i=1}^{k}a^2_i=1$ such that $\phi=\sum_{i=1}^{k}a_i\phi_i$. Using this and \eqref{c}, we get
\begin{align*}
|\cE_{\alpha}(\phi,\phi)-\cE_s(\phi,\phi)|\leq\sum_{i=1}^{k}\sum_{j=1}^{k}|a_i||a_j||\cE_{\alpha}(\phi_i,\phi_j)-\cE_s(\phi_i,\phi_j)|\leq\frac{\epsilon}{4},
\end{align*}
i.e.,
\begin{equation*}
\cE_{\alpha}(\phi,\phi)\leq\cE_s(\phi,\phi)+\frac{\epsilon}{4}.
\end{equation*}
Consequently, we have with \eqref{d} that
\begin{equation}\label{e}
\lambda_{k,\alpha}\leq\cE_s(\phi,\phi)+\frac{\epsilon}{2}.
\end{equation}
Now, by letting $\psi=\sum_{i=1}^{k}a_i\phi_{i,s}$ and by using \eqref{density-consequence}, we can follows the argument above to show that
\begin{equation}
|\cE_s(\psi,\psi)-\cE_s(\phi,\phi)|<\frac{\epsilon}{4}
\end{equation}
i.e.,
\begin{equation}
\cE_s(\phi,\phi)<\cE_s(\psi,\psi)+\frac{\epsilon}{4}.
\end{equation}
Combining this with \eqref{e} and by using also the monotonicity of $\{\lambda_{i,s}\}_i$, we see that
\begin{align*}
\lambda_{k,\alpha}&\leq\cE_{\alpha}(\phi,\phi)+\frac{\epsilon}{2}\leq\cE_s(\psi,\psi)+\frac{3\epsilon}{4}\\&\leq\sum_{i=1}^{k}a^2_i\lambda_{i,s}+\frac{3\epsilon}{4}\leq\lambda_{k,s}\sum_{i=1}^{k}a^2_i+\frac{3\epsilon}{4}=\lambda_{k,s}+\frac{3\epsilon}{4}.
\end{align*}
Since $\epsilon$ was chosen arbitrarily, we therefore have
\begin{equation*}
\limsup_{\alpha\rightarrow s}\lambda_{k,\alpha}\leq\lambda_{k,s},
\end{equation*}
as claimed.\\
\textbf{\textit{Step 2}.} We show that
\begin{equation}\label{limit-inf-for-continuity}
\liminf_{\alpha\rightarrow s}\lambda_{k,\alpha}\geq\lambda_{k,s}.
\end{equation}
To this end, we set $\lambda^{*}_{k,s}:=\liminf_{\alpha\rightarrow s}\lambda_{k,\alpha}$ and let $\alpha_n\in(0,1)$ be a sequence such that $\alpha_n\to s$ and $\lambda_{k,\alpha_n}\rightarrow\lambda^{*}_{k,s}$ as $n\rightarrow\infty$. We now choose a system of $L^2$-orthonormal eigenfunctions $\phi_{1,\alpha_n}, \dots, \phi_{k,\alpha_n}$ associated to $\lambda_{1,\alpha_n}, \dots, \lambda_{k,\alpha_n}$.
By Proposition \ref{higher-sobolev-regularity}, we have that for $n$ sufficiently large,
\begin{align}\label{f}
\phi_{j,\alpha_n}~~\text{is uniformly bounded in}~~H^s(\Omega)~~\text{for}~~j=1,\dots,k.
\end{align}
Therefore, after passing to a subsequence, there exists $e_{j,s}\in H^s(\Omega)$ such that
\begin{equation*}
\begin{aligned}
&\phi_{j,\alpha_n}\rightharpoonup e_{j,s}\quad\text{weakly in}~~H^s(\Omega),\\
&\phi_{j,\alpha_n}\rightarrow e_{j,s}\quad\text{strongly in}~~L^2(\Omega),~~\text{for}~~j=1, \dots, k,\\
&\phi_{j,\alpha_n}\rightarrow e_{j,s}\quad\text{a.e. in}~~\Omega,
\end{aligned}
\end{equation*}
which therefore imply that $\int_{\Omega}e_{j,s}\ dx=0.$ Thus, $e_{j,s}\in\mathbb{X}^s(\Omega).$~ Furthermore, by strong convergence in $L^2(\Omega)$, it follows also that $e_{1,s},\dots, e_{k,s}$ form an $L^2$-orthonormal system.
Moreover, for every $j=1,\dots,k,$ we have
\begin{align*}
\lambda^{*}_{j,s}\langle e_{j,s},\phi\rangle_{L^2(\Omega)}&=\lim\limits_{n\rightarrow\infty}\lambda_{j,\alpha_n}\langle\phi_{j,\alpha_n},\phi\rangle_{L^2(\Omega)}=\lim\limits_{n\to\infty}\langle \phi_{j,\alpha_n},(-\Delta)^{\alpha_n}_{\Omega}\phi\rangle_{L^2(\Omega)}\\
&=\langle e_{j,s},(-\Delta)^{s}_{\Omega}\phi\rangle_{L^2(\Omega)}=\cE_s(e_{j,s},\phi)
\end{align*}
i.e.,
$$ \cE_s(e_{j,s},\phi)=\lambda^{*}_{j,s}\langle e_j,\phi\rangle_{L^2(\Omega)}~~\text{for all}~~\phi\in C^{\infty}_0(\overline{\Omega}),$$
and by density,
$$ \cE_s(e_{j,s},\phi)=\lambda^{*}_{j,s}\langle e_{j,s},\phi\rangle_{L^2(\Omega)}~~\text{for all}~~\phi\in\mathbb{X}^s(\Omega).$$
Therefore, $(\lambda^{*}_{j,s})_{j\in\{1,\dots,k\}}$ is an increasing sequence of eigenvalues of $(-\Delta)^s_{\Omega}$ with corresponding eigenfunctions $(e_{j,s})_{j\in\{1,\dots,k\}}.$ Now, by choosing in particular $V=\text{span}\{e_{1,s},e_{2,s},\dots,e_{k,s}\}$, we have from \eqref{k-th-eigenvalue-of-regional-fractional-laplacian} that
\begin{equation}\label{uper-bound-of-lambda-k}
\lambda_{k,s}\leq\sup_{\phi\in S_V}\cE_s(\phi,\phi).
\end{equation}
Moreover, for all $\phi\in S_V,$ there exists a family of numbers $(c_j)_{j\in\{1,\cdots,k\}}\subset\R$ satisfying $\sum_{j=1}^{k}c^2_j=1$ such that $\phi=\sum_{j=1}^{k}c_je_{j,s}$. From this, we get that
\begin{align*}
\cE_s(\phi,\phi)&=\cE_s\Big(\sum_{j=1}^{k}c_je_{j,s},\sum_{j=1}^{k}c_je_{j,s}\Big)=\sum_{i,j=1}^{k}c_ic_j\lambda^{*}_{j,s}\langle e_{i,s},e_{j,s}\rangle_{L^2(\Omega)}\\
&=\sum_{j=1}^{k}c^2_j\lambda^{*}_{j,s}\leq\max_{j\in\{1,\dots,k\}}\lambda^{*}_{j,s}\sum_{j=1}^{k}c^2_j=\max_{j\in\{1,\dots,k\}}\lambda^{*}_{j,s}.
\end{align*}
Hence, from \eqref{uper-bound-of-lambda-k}, we have that
\begin{equation*}
\lambda_{k,s}\leq\max_{j\in\{1,\dots,k\}}\lambda^{*}_{j,s}\leq\lambda^{*}_{k,s},
\end{equation*}
which therefore implies that
\begin{equation*}
\liminf_{\alpha\rightarrow s}\lambda_{k,\alpha}=\lambda^*_{k,s}\geq\lambda_{k,s}.
\end{equation*}
Combining both \textbf{\textit{Steps 1}} and \textbf{\textit{2}} we conclude that
\begin{equation}
\lim\limits_{\alpha\rightarrow s}\lambda_{k,\alpha}=\lambda_{k,s}
\end{equation}
as wanted.
\end{proof}
Below, we now give the proof of Theorem \ref{differentiability-of-eigenvalues}.
\begin{proof}[Proof of Theorem \ref{differentiability-of-eigenvalues}]
By Lemma \ref{right-continuity-of-eigenvalues} and Proposition \ref{higher-sobolev-regularity}, we deduce that the function $u_{s+\sigma}$ is uniformly bounded in $H^s(\Omega)$ with respect to $\sigma$. Therefore after passing to a subsequence, there is $w_s\in H^s(\Omega)$ such that
\begin{equation}\label{m1}
\begin{aligned}
&u_{s+\sigma}\rightharpoonup w_s\quad\text{weakly in}~~H^s(\Omega),\\
&u_{s+\sigma}\rightarrow w_s\quad\text{strongly in}~~L^2(\Omega),\\
&u_{s+\sigma}\rightarrow w_s\quad\text{a.e. in}~~\Omega.
\end{aligned}
\end{equation}
We wish now to show that $w_s$ is also an eigenfunction corresponding to $\lambda_s$. First of all, from \eqref{m1}, we have in particular that $\|w_s\|_{L^2(\Omega)}=1$ and $\int_{\Omega}w_s\ dx=0$.
Next, we claim that
\begin{enumerate}
\item [$(a)$] $\cE_{s+\sigma}(u_{s+\sigma},\phi)\to \cE_s(w_s,\phi)$
\item[$(b)$] $\lambda_{s+\sigma}\int_{\Omega}u_{s+\sigma}\phi\ dx\to \lambda_s\int_{\Omega}w_s\phi\ dx$
\end{enumerate}
as $\sigma\to0^+$ for all $\phi\in C^{\infty}_0(\ov\Omega)$.
We start by proving $(b)$. We write
\begin{align*}
\int_{\Omega}(\lambda_{s+\sigma}u_{s+\sigma}-\lambda_sw_s)\phi\ dx=\lambda_s\int_{\Omega}(u_{s+\sigma}-w_s)\phi \ dx+(\lambda_{s+\sigma}-\lambda_s)\int_{\Omega}u_{s+\sigma}\phi\ dx.
\end{align*}
From the above decomposition and thanks to \eqref{m1} and Lemma \ref{right-continuity-of-eigenvalues}, we deduce claim $(b)$.
Regarding $(a)$, we have
\begin{align}\label{m2}
\nonumber&|\cE_{s+\sigma}(u_{s+\sigma},\phi)-\cE_s(w_s,\phi)|\\
\nonumber&\leq\Big|\frac{C_{N,s+\sigma}-C_{N,s}}{2}\Big|\Big|\int_{\Omega}\int_{\Omega}\frac{(w_s(x)-w_s(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}\ dxdy\Big|\\
\nonumber&+\frac{C_{N,s+\sigma}}{2}\Big|\int_{\Omega}\int_{\Omega}\frac{((u_{s+\sigma}(x)-u_{s+\sigma}(y))-(w_s(x)-w_s(y)))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}\ dxdy\Big|\\
\nonumber&+\frac{C_{N,s+\sigma}}{2}\Big|\int_{\Omega}\int_{\Omega}\Big(|x-y|^{-2\sigma}-1\Big)\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}\ dxdy\Big|\\
&=:I_{\sigma}+II_{\sigma}+III_{\sigma}.
\end{align}
Since $w_s, \phi\in H^s(\Omega)$ and $s\mapsto C_{N,s}$ is of class $C^1$, then from Cauchy-Schwarz inequality, we get that
\begin{equation}\label{m3}
I_{\sigma}\leq c|C_{N,s+\sigma}-C_{N,s}|\to0\quad\quad\text{as}~~\sigma\to 0^+.
\end{equation}
Now, using \eqref{bound-of-constant-c-n-s} and the fact that $u_{s+\sigma}\rightharpoonup w_s$ weakly in $H^s(\Omega)$, one gets
\begin{equation}\label{m4}
II_{\sigma}\to0\quad\quad\text{as}~~\sigma\to0^+.
\end{equation}
On the other hand, recalling \eqref{bound-of-constant-c-n-s}, \eqref{n2} and \eqref{n9}, we have
\begin{align*}
III_{\sigma}&\leq \sigma c\int_{\Omega}\int_{\Omega}\frac{|u_{s+\sigma}(x)-u_{s+\sigma}(y)||\phi(x)-\phi(y)|}{|x-y|^{N+2s}}|\log|x-y||\ dxdy\\
&+\sigma c\int_{\Omega}\int_{\Omega}\frac{|u_{s+\sigma}(x)-u_{s+\sigma}(y)||\phi(x)-\phi(y)|}{|x-y|^{N+2s+2\sigma}}|\log|x-y||\ dxdy.
\end{align*}
Arguing as in Section \ref{differentiability-for-general-value-s}, one obtains
\begin{equation}\label{m5}
III_{\sigma}\leq \sigma c\to0\quad\quad\text{as}~~\sigma\to0^+.
\end{equation}
From \eqref{m3}, \eqref{m4} and \eqref{m5}, it follows from \eqref{m2} that
\begin{equation}
\cE_{s+\sigma}(u_{s+\sigma},\phi)\to\cE_s(w_s,\phi)\quad\quad\text{as}~~\sigma\to 0^+,
\end{equation}
yielding claim $(a)$.
Finally, using that $u_{s+\sigma}$ is solution to
\begin{equation}
\cE_{s+\sigma}(u_{s+\sigma},\phi)=\lambda_{s+\sigma}\int_{\Omega}u_{s+\sigma}\phi\ dx
\end{equation}
for all $\phi\in C^{\infty}_0(\ov\Omega)$, one deduces from claims $(a)$ and $(b)$ that $w_s$ is solution to
\begin{equation}\label{m5'}
\cE_s(w_s,\phi)=\lambda_s\int_{\Omega}w_s\phi\ dx
\end{equation}
and from this, one concludes that $w_s$ with $\|w_s\|_{L^2(\Omega)}=1, \int_{\Omega}w_s\ dx=0$ is an eigenfunction corresponding to $\lambda_s$.
Coming back to the proof of \eqref{C-1-regularity-of-eigenvalues}, since $u_{s+\sigma}\in H^{s+\sigma}(\Omega)\subset H^s(\Omega)$, one can use it as an admissible function in the definition of $\lambda_s$ to get
\begin{equation}\label{m6}
\lambda_s\leq\cE_s(u_{s+\sigma},u_{s+\sigma})=\frac{C_{N,s}}{2}\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))^2}{|x-y|^{N+2s}}\ dxdy.
\end{equation}
Now, from \eqref{m6}, we have
\begin{align*}
&\lambda_{s+\sigma}-\lambda_s=\cE_{s+\sigma}(u_{s+\sigma},u_{s+\sigma})-\lambda_s\\
&\geq
\frac{C_{N,s+\sigma}}{2}\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))^2}{|x-y|^{N+2(s+\sigma)}}\ dxdy-\frac{C_{N,s}}{2}\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))^2}{|x-y|^{N+2s}}\ dxdy\\
&=\frac{C_{N,s+\sigma}-C_{N,s}}{2}\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))^2}{|x-y|^{N+2s+2\sigma}}\ dxdy\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\frac{C_{N,s}}{2}\int_{\Omega}\int_{\Omega}(|x-y|^{-2\sigma}-1)\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))^2}{|x-y|^{N+2s}}\ dxdy\\
&=\frac{C_{N,s+\sigma}-C_{N,s}}{C_{N,s+\sigma}}\lambda_{s+\sigma} +\frac{C_{N,s}}{2}\int_{\Omega}\int_{\Omega}(|x-y|^{-2\sigma}-1)\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))^2}{|x-y|^{N+2s}}\ dxdy
\end{align*}
Hence,
\begin{align}\label{q}
\nonumber\liminf_{\sigma\rightarrow0^+}\frac{\lambda_{s+\sigma}-\lambda_s}{\sigma}&\geq\lim\limits_{\sigma\rightarrow0^+}\frac{C_{N,s+\sigma}-C_{N,s}}{\sigma}\frac{\lambda_{s+\sigma}}{C_{N,s+\sigma}}\\
&+\lim\limits_{\sigma\rightarrow0^+}\frac{C_{N,s+\sigma}}{2}\int_{\Omega}\int_{\Omega}\frac{(u_{s+\sigma}(x)-u_{s+\sigma}(y))^2}{|x-y|^{N+2s}}\frac{|x-y|^{-2\sigma}-1}{\sigma}\ dxdy.
\end{align}
Next, from \eqref{n2}
\begin{align*}
\frac{|x-y|^{-2\sigma}-1}{\sigma}=\frac{\exp(-2\sigma\log|x-y|)-1}{\sigma}=-2\log|x-y|\int_{0}^{1}\exp(-2\sigma t\log|x-y|)\ dt.
\end{align*}
Therefore,
\begin{equation}\label{mean-value-theorem}
\frac{|x-y|^{-2\sigma}-1}{\sigma}\rightarrow-2\log|x-y|\quad\text{as}\quad\sigma\rightarrow0^+.
\end{equation}
Using this and recalling \eqref{logarithmic-decays}, we apply Lebesgue's Dominated Convergence Theorem in \eqref{q}, thanks to Proposition \ref{higher-sobolev-regularity}, to get that
\begin{align}\label{m7}
\liminf_{\sigma\rightarrow0^+}\frac{\lambda_{s+\sigma}-\lambda_s}{\sigma}\geq\frac{\partial_sC_{N,s}}{C_{N,s}}\lambda_s
-C_{N,s}\int_{\Omega}\int_{\Omega}\frac{(w_{s}(x)-w_{s}(y))^2}{|x-y|^{N+2s}}\log|x-y|\ dxdy.
\end{align}
that is
\begin{equation}\label{lim-inf-gamma-sigma}
\liminf_{\sigma\rightarrow0^+}\frac{\lambda_{s+\sigma}-\lambda_s}{\sigma}\geq J_s(w_s)\geq\inf\{J_s(u):u\in \M_s\}.
\end{equation}
We now show the reverse inequality i.e.,
\begin{equation}\label{lim-sup-gamma-sigma-}
\limsup_{\sigma\rightarrow0^+}\frac{\lambda_{s+\sigma}-\lambda_s}{\sigma}\leq\inf\{J_s(u):u\in \M_s\}.
\end{equation}
Thanks to Proposition \ref{higher-sobolev-regularity}, we have that $u_s\in H^{s+\sigma}(\Omega)$ for $\sigma$ sufficiently small. Combining this with $\int_{\Omega}u_s\ dx=0$ and $\|u_s\|_{L^2(\Omega)}=1$, we can use $u_s$ as an admissible function for $\lambda_{s+\sigma}$ to get
\begin{align*}
\frac{\lambda_{s+\sigma}-\lambda_s}{\sigma}&\leq\frac{\cE_{s+\sigma}(u_s,u_s)-\cE_s(u_s,u_s)}{\sigma}\\&=\frac{C_{N,s+\sigma}}{2\sigma}\int_{\Omega}\int_{\Omega}\frac{(u_s(x)-u_s(y))^2}{|x-y|^{N+2(s+\sigma)}}\ dxdy-\frac{C_{N,s}}{2\sigma}\int_{\Omega}\int_{\Omega}\frac{(u_s(x)-u_s(y))^2}{|x-y|^{N+2s}}\ dxdy\\&=\frac{C_{N,s+\sigma}-C_{N,s}}{2\sigma}\int_{\Omega}\int_{\Omega}\frac{(u_s(x)-u_s(y))^2}{|x-y|^{N+2s}}\ dxdy\\&\quad\quad+\frac{C_{N,s+\sigma}}{2}\int_{\Omega}\int_{\Omega}\frac{(u_s(x)-u_s(y))^2}{|x-y|^{N+2s}}\frac{|x-y|^{-2\sigma}-1}{\sigma}\ dxdy.
\end{align*}
By letting $\sigma\rightarrow0^+$ and
applying Lebesgue's Dominated Convergence Theorem and considering once again \eqref{mean-value-theorem} and Proposition \ref{higher-sobolev-regularity}, we have that
\begin{align*}
\limsup_{\sigma\rightarrow0^+}\frac{\lambda_{s+\sigma}-\lambda_s}{\sigma}\leq J_s(u_s).
\end{align*}
Since the above inequality does not depends on the choice of $u_s\in \M_s$, we have that
\begin{align}\label{lim-sup-gamma-sigma}
\limsup_{\sigma\rightarrow0^+}\frac{\lambda_{s+\sigma}-\lambda_s}{\sigma}\leq\inf\{J_s(u):u\in \M_s\}.
\end{align}
Putting together \eqref{lim-inf-gamma-sigma} and \eqref{lim-sup-gamma-sigma} we infer that
\begin{equation}\label{differentiability-for-s-neq-1/2}
\lim\limits_{\sigma\rightarrow0^+}\frac{\lambda_{s+\sigma}-\lambda_s}{\sigma}=\inf\{J_s(u):u\in \M_s\}\equiv\partial^+_s\lambda_s.
\end{equation}
Finally, from Proposition \ref{higher-sobolev-regularity} we easily conclude that the infimum in \eqref{C-1-regularity-of-eigenvalues} is achieved.
\end{proof}
\begin{remark}
By the same argument as above, one can establish the left differentiability of the map $(0,1)\ni s\mapsto \lambda_{1,s}$. Due to the non-simplicity of $\lambda_{1,s}$, the right and left derivative $\partial^+_s\lambda_{1,s}$ and $\partial^-_s\lambda_{1,s}$ may not coincide.
\end{remark}
\bibliographystyle{ieeetr}
| 176,496
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\begin{document}
\title{Automorphism groups of pseudoreal Riemann surfaces}
\author{Michela Artebani}
\address{
Departamento de Matem\'atica, \newline
Universidad de Concepci\'on, \newline
Casilla 160-C,
Concepci\'on, Chile}
\email{martebani@udec.cl}
\author{Sa\'ul Quispe}
\address{
Departamento de Matem\'atica y Estad\'istica, \newline
Universidad de La Frontera, \newline
Casilla 54-D,
Temuco, Chile}
\email{saul.quispe@ufrontera.cl}
\author{Cristian Reyes}
\address{
Departamento de Matem\'atica, \newline
Universidad de Concepci\'on, \newline
Casilla 160-C,
Concepci\'on, Chile}
\email{creyesm@udec.cl}
\subjclass[2000]{14H45, 14H37, 30F10, 14H10}
\keywords{Pseudoreal Riemann surface, field of moduli, NEC group}
\thanks{The authors have been supported
by Proyectos FONDECYT Regular N. 1130572, 1160897, 3140050 and
Proyecto Anillo CONICYT PIA ACT1415.
The third author has been supported by CONICYT PCHA/Mag\'isterNacional/2014/22140855.}
\begin{abstract}
A smooth complex projective curve is called pseudoreal if it is isomorphic to its conjugate but is not definable over the reals.
Such curves, together with real Riemann surfaces, form the real locus of the moduli space $\mathcal M_g$.
This paper deals with the classification of pseudoreal curves according to the structure of their automorphism group.
We follow two different approaches existing in the literature: one coming from number theory,
dealing more generally with fields of moduli of projective curves, and the other from complex geometry,
through the theory of NEC groups.
Using the first approach, we prove that the automorphism group of a pseudoreal Riemann surface $X$ is abelian if
$X/Z({\rm Aut}(X))$ has genus zero, where $Z({\rm Aut}(X))$ is the center of ${\rm Aut}(X)$.
This includes the case of $p$-gonal Riemann surfaces, already known by results of Huggins
and Kontogeorgis.
By means of the second approach and of elementary properties of group extensions,
we show that $X$ is not pseudoreal if the center of $G={\rm Aut}(X)$ is trivial and either ${\rm Out}(G)$ contains no involutions
or ${\rm Inn}(G)$ has a group complement in ${\rm Aut}(G)$. This extends and gives an elementary proof (over $\cc)$
of a result by D\`ebes and Emsalem.
Finally, we provide an algorithm, implemented in MAGMA, which classifies the automorphism groups
of pseudoreal Riemann surfaces of genus $g\geq 2$, once a list of all groups acting for such genus,
with their signature and generating vectors, are given.
This program, together with the database provided by J. Paulhus in \cite{Pau15},
allowed us to classifiy pseudoreal Riemann surfaces up to genus $10$, extending previous results
by Bujalance, Conder and Costa.
\end{abstract}
\maketitle
\tableofcontents
\section*{Introduction}
Let $X\subset \mathbb{P}_{\mathbb{C}}^n$ be a smooth complex projective curve
defined as the zero locus of homogeneous polynomials $p_1,\dots,p_r\in \mathbb{C}[x_0,\dots,x_n]$
and let $\overline{X} $ be its conjugate, i.e.
the zero locus of the polynomials obtained conjugating the coefficients of the polynomials $p_i$.
The curve $X$ is called {\em pseudoreal} if it is isomorphic to $\overline{X}$ but is not isomorphic to a curve defined
by polynomials with real coefficients.
Since any compact Riemann surface can be embedded in projective space as a smooth complex curve, and
the definition only depends on the isomorphism class of the curve, this allows to define
the concept of pseudoreal compact Riemann surface.
Equivalently, pseudoreal Riemann surfaces can be defined as Riemann surfaces
carrying anticonformal automorphisms but no anticonformal involutions.
The disjoint union of the locus of pseudoreal Riemann surfaces with the locus of real Riemann surfaces
is the fixed locus of the natural involution $X\to \overline X$ on the moduli space of curves $\mathcal M_g$.
In literature, one can find two main approaches to the study of pseudoreal Riemann surfaces:
a number-theoretical approach and an approach through NEC groups.
The first approach deals, more generally, with the problem of deciding whether the field of moduli of a curve is a field of definition. In this setting, pseudoreal curves are complex curves having $\mathbb{R}$ as field of moduli, but not as a field of definition.
A fundamental tool in this approach is a classical theorem by A. Weil (Theorem \ref{T1}), which provides a necessary and sufficient condition for a projective variety defined over a field $L$, to be definable over a subfield $K\subseteq L$ when the extension is Galois. More recently, P. D\`ebes and M. Emsalem proved that $X/{\rm Aut}(X)$ can be always defined over the field of moduli of $X$ and that $X$ has the same property when a suitable model $B$ of $X/{\rm Aut}(X)$ over the subfield $K\subseteq L$ has a $K$-rational point (see \cite[Corollary 4.3 (c)]{DebEms99}).
Such result has been applied by B. Huggins to complete the classification of pseudoreal hyperelliptic curves \cite[Proposition 5.0.5]{Hugg05} and later by A. Kontogeorgis in \cite{Kon09} in the case of $p$-gonal curves. Unfortunately, the result of D\`ebes-Emsalem is not easy to apply as soon as the genus of $X\slash{\rm Aut}(X)$ is bigger than zero.
A second approach, specific of compact Riemann surfaces, is through the theory of Fuchsian groups, and more generally of {\em non-euclidean crystallographic (NEC) groups}, which are discrete subgroups $\Delta$ of the full automorphism group of the hyperbolic plane $\mathbb{H}$ such that $\mathbb{H}/\Delta$ is a compact Klein surface.
It follows from the uniformization theorem that giving a compact Riemann surface $X$ of genus $g\geq 2$ containing a group isomorphic to $G$ in its full automorphism group is equivalent to give an epimorphism $\varphi:\Gamma\to G$, where $\Gamma$ is a NEC group, such that $\ker(\varphi)$ is a torsion free Fuchsian group.
The automorphism group ${\rm Aut}(X)$ of $X=\mathbb{H}/\ker(\varphi)$ will then contain $\varphi(\Gamma^+)=G^+$,
where $\Gamma^+$ is the canonical Fuchsian subgroup of $\Gamma$.
The Riemann surface $X$ is pseudoreal if and only if ${\rm Aut}(X)$ is an index two subgroup of its full automorphism group ${\rm Aut}(X)^{\pm}$ such that ${\rm Aut}(X)^{\pm}\backslash {\rm Aut}(X)$ contains no involutions.
This idea allowed D. Singerman to prove the existence of pseudoreal Riemann surfaces of any genus $g>1$ (Theorem \ref{existencia}). Moreover, it has been used by E. Bujalance, M. Conder and A.F. Costa in \cite{BuConCo10} and \cite{BuCo14} to classify the full automorphism groups of pseudoreal Riemann surfaces up to genus $4$.
The aim of this paper is to provide an introduction to both approaches and to show some new results on the classification of automorphism groups of pseudoreal Riemann surfaces. The paper is organized as follows.
In section $1$ we deal with the number theoretical approach. We first provide the background material,
defining the concepts of field of moduli and of pseudoreal curve.
Moreover, we introduce the theorem by D\`ebes-Emsalem and some consequences of it
when $X\slash{\rm Aut}(X)$ has genus $0$. In particular we prove the following result.
\begin{introthm}[Theorem \ref{cen}] Let $K$ be an infinite perfect field of characteristic $p\neq 2$ and let $F$ be an algebraic closure of $K$. Let $X$ be a smooth projective algebraic curve of genus $g\ge 2$ defined over $F$, $H$ the center of ${\rm Aut}(X)$ and assume that $X\slash H$ has genus $0$.
If ${\rm Aut}(X)/H$ is neither trivial nor cyclic (if $p=0$) or cyclic of order relatively prime to $p$ (if $p> 0$)
then $X$ can be defined over its field of moduli relative to the extension $F\slash K$.
\end{introthm}
This immediately implies the following.
\begin{introcor}[Corollary \ref{cencor}]
Let $K$ be a field of characteristic $0$ and let $F$ be an algebraic closure of $K$.
Let $X$ be a smooth projective algebraic curve of genus $g\ge 2$ defined over $F$ such that $X\slash Z({\rm Aut}(X))$ has genus $0$
(where $Z({\rm Aut}(X))$ is the center of ${\rm Aut}(X)$).
If $X$ can not be defined over its field of moduli $M_{F/K}(X)$, then ${\rm Aut}(X)$ is abelian.
\end{introcor}
The second section considers the NEC group approach. We recall the proof of the existence of pseudoreal Riemann surfaces in every genus
and a recent result by C.~Baginski and G.~Gromadzki \cite{BaGo10} which characterizes groups appearing as full automorphisms groups of pseudoreal Riemann surfaces.
In this setting, studying group extensions of degree two, we provide the following result, which gives an alternative
proof of \cite[Corollary 4.3 (b)]{DebEms99}
for the extension $\cc/\rr$ and new conditions on the automorphism group of a pseudoreal Riemann surface.
\begin{introthm}[Corollary \ref{comp}] Let $G$ be the conformal automorphism group of a Riemann surface $X$.
Suppose that $Z(G)=\{1\}$ and that either ${\rm Out}(G)$ has no involutions or ${\rm Inn}(G)$ has group complement in ${\rm Aut}(G)$.
Then $X$ is not pseudoreal.
\end{introthm}
Finally, we consider the maximal full automorphism groups of pseudoreal Riemann surfaces (see Theorem \ref{maximalll})
and we prove the following result.
\begin{introthm}[Theorem \ref{abeliann}] If a pseudoreal Riemann surface $X$ has maximal full automorphism group,
then its conformal automorphism group is not abelian and $X/Z({\rm Aut}(X))$ has positive genus. \end{introthm}
Section $3$ is about the classification of pseudoreal Riemann surfaces of low genus.
We develop an algorithm which allows to classify pseudoreal Riemann surfaces of a given genus $g\geq 2$
starting from the list of all groups acting on Riemann surfaces of that genus.
The algorithm has been implemented in a program written for Magma \cite{Magma} based on a program by J. Paulhus \cite{Pau15}.
By means of this program, we are able to provide the classification of pseudoreal Riemann surfaces up to genus $10$, extending previous results
in \cite{BuConCo10, BuCo14}.
\begin{introthm}[Theorem \ref{class}]
Two finite groups $G$ and $\bar G$ are the conformal and full automorphism groups of a pseudoreal Riemann surface $X$ of genus
$5 \leq g\leq 10$ if and only if $G = {\rm Aut}(X)$ and $\overline G = {\rm Aut}^{\pm}(X)$ appear in the corresponding table by genus
among Tables \ref{table5}, \ref{table6}, \ref{table7}, \ref{table8}, \ref{table9} and \ref{table10}.
\end{introthm}
\noindent {\em Aknowledgments.}
This paper grew out of the Master Thesis of Cristian Reyes Monsalve,
defended in October 2016 at Universidad de Concepci\'on (Chile).
We thank Andrea Tironi for several useful discussions and careful reading,
Antonio Laface for inspiring conversations and for his help in the design of the Magma programs,
Rub\'en Hidalgo, Jeroen Sijsling and Xavier Vidaux for key remarks which helped to improve the final version.
We especially thank Jennifer Paulhus for her kind assistance in the use of her program.
\section{The number theoretical approach}
\subsection{Fields of moduli of projective curves}
\noindent Everytime we say \emph{curve} we mean a smooth projective algebraic curve.
\begin{definition} \label{fielddefinition} Let $F$ be a field and let $X\subseteq\mathbb{P}^n_{F}$
be a curve defined as the zero locus of homogeneous polynomials with coefficients in $F$.
If $K\subset F$ is a subfield, we say that $X$ \emph{can be defined over} $K$,
or that $K$ is a \emph{field of definition} of $X$,
if there exists a curve $Y\subseteq\mathbb{P}^m_{F}$ defined by polynomials with coefficients in $K$ and isomorphic to $X$ over $F$.
If $f:X\longrightarrow Y$ is a morphism between two curves $X$ and $Y$ defined over a field $F$,
we say that \emph{$f$ is defined over $F$} if the polynomials defining $f$ have all their coefficients in $F$.
\end{definition}
If $F\slash K$ is a field extension, the group of automorphisms
\[
{\rm Aut}(F\slash K):=\{\sigma\in {\rm Aut}(F):\ \sigma|_{K}={\rm id}_K\}
\]
naturally acts on both curves and morphisms defined over $F$ in the following way.
\begin{definition} Let $F\slash K$ be a field extension and $\sigma\in {\rm Aut}(F\slash K)$.
\begin{enumerate}
\item Given a curve $X=Z(p_1,\dots,p_r)\subseteq\mathbb{P}^n_F$, where $p_1,\dots,p_r\in F[x_0,\dots,x_n]$
are homogeneous, let $X^{\sigma}:=Z(p_1^{\sigma},\dots, p_r^{\sigma})$, where
$p_i^{\sigma}$ is obtained applying $\sigma$ to the coefficients of $p_i$.
\item Given a morphism $f:X\rightarrow Y$ between curves defined over $F$, let
\[
f^{\sigma}:=\sigma\circ f\circ \sigma^{-1}:X^{\sigma}\rightarrow Y^{\sigma},
\]
where $\sigma:Z\to Z^{\sigma}$ sends $[z_0:\ldots :z_n]$ to
$[\sigma(z_0):\ldots :\sigma(z_n)]$.
\end{enumerate}
\end{definition}
\noindent A fundamental result in this area is the following
\begin{theorem}\label{T1} \textbf{(Weil's Theorem)} \cite[Theorem 1]{Weil56} Let $X$ be a curve defined over $F$ and let $F\slash K$ be a Galois extension. If for every $\rho\in {\rm Aut}(F\slash K)$ there exists an isomorphism $f_{\rho}:X\longrightarrow X^{\rho}$ defined over $F$ such that $$f_{\sigma\tau}=f_{\tau}^{\sigma}\circ f_{\sigma},\ \ \ \ \forall \sigma,\tau\in {\rm Aut}(F\slash K),$$ then there exist a curve $Y$ defined over $K$ and an isomorphism $g:X\rightarrow Y$ defined over $F$ such that $g^{\rho}\circ f_{\rho}=g, \forall \rho\in {\rm Aut}(F\slash K)$.
\end{theorem}
It is natural to ask for the smallest field of definition of a curve, this leads to the concept of field of moduli.
\begin{definition}\label{fieldmoduli}
The \emph{field of moduli} of a curve $X$ defined over $F$ {\em relative to a Galois extension} $F\slash K$ is
$$M_{F\slash K}(X):={\rm Fix}(F_K(X))\ \text{ where }\ F_K(X)=\{\sigma\in {\rm Aut}(F\slash K) : X\cong_{\overline{F}} X^{\sigma}\}.$$
\end{definition}
If $F\slash K$ is a Galois extension, then it can be easily proved that $M_{F\slash K}(X)\subseteq F'$ for any
field of definition $F'$ of $X$ such that $K\subseteq F'\subseteq F$.
Moreover, it is clear that the relative fields of moduli of two isomorphic curves over $\overline F$ are isomorphic.
The following is an easy consequence of Weil's theorem together with the fact that $M_{F\slash R}(X)=R$, where $R=M_{F\slash K}(X)$
\cite[Proposition 2.1]{DebEms99}.
\begin{proposition}\label{trivialll} If $X$ is a curve defined over a field $F$, $F\slash K$ is a Galois extension and ${\rm Aut}(X)$ is trivial, then $X$ can be defined over its field of moduli $M_{F\slash K}(X)$.
\end{proposition}
Since the generic curve of genus $g>2$ has trivial automorphism group (see \cite[Theorem 2]{Gre63}), by Proposition \ref{trivialll} we deduce that $X$ is always defined over its field of moduli relative to a Galois extension.
Moreover, for curves of genus $0$ and $1$, it is known that the field of moduli is a field of definition (see Example \ref{g1}).
By a result of H. Hammer and F. Herrlich any curve can be defined over a finite extension of its field of moduli \cite{HamHerr03}.
However there are examples of curves of any genus $g\geq 2$ with non-trivial automorphism groups whose field of moduli (relative to a certain field extension) is not a field of definition.
As Weil's theorem suggests, the structure of the automorphism group plays a key role in the definability problem over
the field of moduli.
\begin{example} \label{g1}
Let $X$ be a curve of genus $1$ defined over an algebraically closed field $F$ of characteristic $p\not=2$ and
let $F/K$ be a Galois extension.
It is well known that $X\cong_{F} C_{\lambda}$, where $C_{\lambda}\subseteq \mathbb{P}_{F}^2$ is the plane cubic defined by
$$x_1^2x_2-x_0(x_0-x_2)(x_0-\lambda x_2)=0,$$
and $\lambda\in F-\{0,1\}$. Moreover $C_{\lambda}$ and $C_{\mu}$ are isomorphic if and only if $j(\lambda)=j(\mu)$, where $j$ is the $j$-invariant \cite[Proposition 1.7, Ch. III]{Silv}. Given $\sigma\in {\rm Aut}(F/K)$ we have that $C_{\lambda}^{\sigma}=C_{\sigma(\lambda)}$.
Thus
$$M_{F\slash K}(X):={\rm Fix}(\{\sigma\in{\rm Aut}(F\slash K): j(\lambda)=j(\sigma(\lambda))=\sigma(j(\lambda))\})=K(j(\lambda)).$$
For every $\lambda\in F$, one can find a smooth plane model for $C_{\lambda}$ which is defined over $K(j(\lambda))$ \cite[Proposition 1.4]{Silv},
so $M_{F\slash K}(X)$ is also a field of definition of $X$.
\end{example}
In this paper we will focus on the case of the field extension $\cc/\rr$. More precisely, we are interested in the following concept.
\begin{definition} \label{pseudocurve} A \emph{pseudoreal curve} is a complex curve $X$ such that $M_{\mathbb{C}\slash\mathbb{R}}(X)=\mathbb{R}$
but $\mathbb{R}$ is not a field of definition for $X$.
\end{definition}
\subsection{D\`ebes-Emsalem theorem}
We recall a very useful theorem, based on Weil's theorem, that gives sufficient conditions for the definability.
Let $F\slash K$ be a Galois extension, $X$ be a smooth projective curve of genus $g\ge 2$ defined over $F$
and assume that $K$ is the field of moduli $M_{F\slash K}(X)$.
Then for every $\sigma\in {\rm Aut}(F\slash K)$ there is an isomorphism $f_{\sigma}:X\longrightarrow X^{\sigma}$.
This induces an isomorphism
\[
\varphi_{\sigma}:X\slash {\rm Aut}(X)\longrightarrow X^{\sigma}\slash {\rm Aut}(X^{\sigma})
\]
that makes the following diagram commute
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
X & X^{\sigma} \\
X\slash {\rm Aut}(X) & X^{\sigma}\slash {\rm Aut}(X^{\sigma}). \\};
\path[-stealth]
(m-1-1) edge node [left] {$\pi$} (m-2-1)
edge node [above] {$f_{\sigma}$} (m-1-2)
(m-2-1.east|-m-2-2) edge node [below] {$\varphi_{\sigma}$}
node [above] {} (m-2-2)
(m-1-2) edge node [right] {$\pi^{\sigma}$} (m-2-2);
\end{tikzpicture}
\end{center}
Composing $\varphi_{\sigma}$ with the canonical isomorphism $X^{\sigma}\slash {\rm Aut}(X^{\sigma})\longrightarrow (X\slash {\rm Aut}(X))^{\sigma}$ this gives a family of isomorphisms $$\{\overline{\varphi}_{\sigma}:X\slash {\rm Aut}(X)\longrightarrow (X\slash {\rm Aut}(X))^{\sigma}\}_{\sigma\in {\rm Aut}(F\slash K)}$$ which satisfies the condition in Weil's Theorem.
Thus there exists a curve $B$ isomorphic to $X\slash {\rm Aut}(X)$ over $F$ which is defined over $K$.
This curve $B$ is the so called \emph{canonical K-model} for $X\slash {\rm Aut}(X)$.
\begin{theorem}\label{DE} \cite[Corollary 4.3, (c)]{DebEms99} Let $F\slash K$ be a Galois extension and
$X$ be a smooth projective curve of genus $g\ge 2$ defined over $F$ such that
the order of ${\rm Aut}(X)$ is not divisible by the characteristic of $K$.
Assume that $K$ is the field of moduli of $X$ and that $F$ is the separable closure of $K$.
If the canonical $K$-model $B$ of $X\slash {\rm Aut}(X)$ has at least one $K$-rational point off the branch point set of the natural quotient $\pi:X\to B$,
then $K$ is a field of definition of $X$.
\end{theorem}
\subsection{The case when $X/{\rm Aut}(X)$ has genus zero}
In case the field has characteristic zero and the $K$-canonical model $B$ of $X/{\rm Aut}(X)$ has genus zero,
the existence of one $K$-rational point in $B$ implies that $B\cong_K \pp^1_K$, so there are infinitely many of them.
Moreover, the existence of a $K$-rational point is implied by the existence of a divisor of odd degree defined
over $K$ \cite[Lemma 5.1]{Hugg04}. These ideas led to the following beautiful result by B. Huggins.
\begin{theorem} \cite{Hugg04} \label{hug} Let $K$ be a perfect field of characteristic $p\neq 2$ and let $F$ be an algebraic closure of $K$. Let $X$ be a hyperelliptic curve defined over $F$ and let $G={\rm Aut}(X)\slash\langle i\rangle$, where $i$ is the hyperelliptic involution of $X$. If $G$ is not cyclic, or it is cyclic of order not divisible by $p$, then $X$ can be defined over its field of moduli relative to the extension $F\slash K$.
\end{theorem}
The proof relies on the fact that the quotient group ${\rm Aut}(X)\slash\langle i\rangle$ acts on $\pp_F^1$, and thus
it is isomorphic to a finite subgroup of ${\rm PGL}(2,F)$ (when $p=0$ this is isomorphic to either $C_n, D_n, A_4,A_5$ or $S_4$ \cite{MaVa80}).
The author showed that in all cases, except in the cyclic case, one can find a $K$-rational point in the canonical model $B$
of $X\slash {\rm Aut}(X)$ by studying explicitely the action of ${\rm Aut}(F/K)$ on the function field of $B$.
The case of hyperelliptic curves such that ${\rm Aut}(X)\slash\langle i\rangle$ is cyclic of order coprime to the characteristic of the
ground field has been completely described in \cite{LRS13}.
In \cite{Kon09} Kontogeorgis generalized Theorem \ref{hug} considering certain $p$-cyclic covers of the projective line,
where $p$ is prime, in any characteristic. In particular he proved the analogous of Huggins' theorem in case the subgroup generated by
the cyclic automorphism of order $p$ is normal in ${\rm Aut}(X)$.
Recently, this has been further generalized by R.~Hidalgo and S.~Quispe \cite{HidQui15} by considering
some particular subgroups of the automorphisms group of the curve, defined as follows.
\begin{definition} Let $X$ be a curve. A subgroup $H\subseteq {\rm Aut}(X)$ is \emph{unique up to conjugation}
if any subgroup $K\subseteq {\rm Aut}(X)$ isomorphic to $H$ such that the signatures of the covers
$\pi_H:X\longrightarrow X\slash H$ and $\pi_K:X\longrightarrow X\slash K$ are the same is conjugated to $H$.
\end{definition}
Observe that the subgroup $H$ generated by the hyperelliptic involution or by a cyclic automorphism of prime order $p$
with quotient $X/H$ of genus zero are unique up to conjugation. In the following, we denote by $N_G(H)$ the normalizer of
a subgroup $H$ of $G$.
\begin{theorem} \cite[Theorem 1.2]{HidQui15} Let $K$ be an infinite perfect field of characteristic $p\neq 2$ and let $F$ be an algebraic closure of $K$. Let $X$ be a smooth projective algebraic curve of genus $g\ge 2$ defined over $F$ and let $H$ be a subgroup of ${\rm Aut}(X)$ which is unique under conjugation, so that $X\slash H$ has genus $0$. If $N_{{\rm Aut}(X)}(H)\slash H$ is neither trivial nor cyclic (if $p=0$) or cyclic of order relatively prime to $p$ (if $p> 0$) then $X$ can be defined over its field of moduli relative to the extension $F\slash K$.
\end{theorem}
A similar proof led us to prove the following result, where $H$ is replaced by the center of the automorphism group.
Observe that the center is not unique up to conjugation in general (see Example \ref{cu}).
\begin{theorem}\label{cen} Let $K$ be an infinite perfect field of characteristic $p\neq 2$ and let $F$ be an algebraic closure of $K$. Let $X$ be a smooth projective algebraic curve of genus $g\ge 2$ defined over $F$, $H$ the center of ${\rm Aut}(X)$ and assume that $X\slash H$ has genus $0$.
If ${\rm Aut}(X)/H$ is neither trivial nor cyclic (if $p=0$) or cyclic of order relatively prime to $p$ (if $p> 0$)
then $X$ can be defined over its field of moduli relative to the extension $F\slash K$.
\end{theorem}
\begin{proof}
We will prove the theorem for the case $p=0$, since the proof for the case $p\neq 0$ is the same as in \cite[Theorem 1.2]{HidQui15}.
Without loss of generality we can assume $K=M_{F\slash K}(X)$ (see \cite[Proposition 2.1]{DebEms99}).
Given $\sigma\in {\rm Aut}(F\slash K)$, there exists an isomorphism $f_{\sigma}:X\longrightarrow X^{\sigma}$.
We start observing the following, where $G={\rm Aut}(X)$: \\
\noindent \emph{Claim 1.} $f_{\sigma}Z(G)f_{\sigma}^{-1}=Z(G)^{\sigma}.$
For every $a\in Z(G)$ and $b\in G^{\sigma}$ we have $$(f_{\sigma}af_{\sigma}^{-1})b=f_{\sigma}a(\underbrace{f_{\sigma}^{-1}bf_{\sigma}}_{b'\in G})f_{\sigma}^{-1}=f_{\sigma}ab'f_{\sigma}^{-1}\underbrace{=}_{a\in Z(G)}f_{\sigma}b'af_{\sigma}^{-1}=b(f_{\sigma}af_{\sigma}^{-1}).$$
This says that $f_{\sigma}Z(G)f_{\sigma}^{-1}\subseteq Z(G^{\sigma})$, which is equal to $Z(G)^{\sigma}$. The other inclusion is obtained analogously.
By Claim 1 and the fact that $f_{\sigma}Gf_{\sigma}^{-1}=G^{\sigma}$, there exist two isomorphisms $g_{\sigma}$ and $h_{\sigma}$ such that the following diagram commutes:
\[
\xymatrix{
X\ar[r]^{f_{\sigma}}\ar[d]_{\pi_1} & X^{\sigma}\ar[d]^{\pi_1^{\sigma}}\\
X/Z(G)\ar[r]^{g_{\sigma}}\ar[d]_{\pi_2} & (X/Z(G))^{\sigma}\ar[d]^{\pi_2^{\sigma}} \\
X/G\ar[r]^{h_{\sigma}} & (X/G)^{\sigma}
}
\]
\noindent \emph{Claim 2.} Without loss of generality, we can assume that the branch locus of $\pi_2$ in $X/G\cong \pp^1_F$ is $\mathcal{B}=\{0,1,\infty\}$.
\medskip
\noindent
The covering group of $\pi_2$ is isomorphic to $ G\slash Z(G)$ and it is a finite group acting on the projective line. By our hypothesis, $G\slash Z(G)$ is not a cyclic group, so it must be isomorphic to either $D_{n}, A_4, A_5$ or $S_4$ \cite{MaVa80}. In any of these four cases the branch locus of $\pi_2$ consists of $3$ points, which can be taken to be $0,1,\infty$ up to a projectivity. \\
\noindent \emph{Claim 3.} There exists an isomorphism $R:X\slash G\longrightarrow B$ to the canonical $K$-model $B$ of $X\slash G$ such that $R=R^{\sigma}\circ h_{\sigma}$.
\medskip
We will prove that given $\sigma\in {\rm Aut}(F/K)$ there exists a unique isomorphism $h_{\sigma}$ as in the diagram.
This implies that $\{h_{\sigma}\}_{\sigma\in{\rm Aut}(F\slash K)}$ satisfies Weil's Theorem \ref{T1} and gives the statement.
Let $S:=\pi_2\circ \pi_1$. Assume that there exist other isomorphisms $f_{\sigma}'$ and $h_{\sigma}'$ such that $h_{\sigma}'\circ S=S^{\sigma}\circ f_{\sigma}'$. Then $f_{\sigma}^{-1}\circ f_{\sigma}'=g\in G$ so we have
$$h_{\sigma}'\circ S=S^{\sigma}\circ f_{\sigma}'=S^{\sigma}\circ f_{\sigma}\circ g=S^{\sigma}\circ g'\circ f_{\sigma}=S^{\sigma}\circ f_{\sigma}=h_{\sigma}\circ S,$$
where in the third equality $g':=f_{\sigma}\circ g\circ f_{\sigma}^{-1}\in G^{\sigma}$ and the fourth equality follows from the fact that $S^{\sigma}$ is the quotient by $G^{\sigma}$. Thus $h_{\sigma}'=h_{\sigma}$, proving that $h_{\sigma}$ is uniquely determined by $\sigma$.
By Weil's Theorem \ref{T1} we obtain that there exists an isomorphism $R:X\slash G\longrightarrow B$ such that $B$ is a curve of genus $0$ defined over $K$ and we have the following commutative diagram $(*)$:
\[
\xymatrix{
X\ar[rr]^{f_{\sigma}}\ar[d]_S & & X^{\sigma}\ar[d]^{S^{\sigma}}\\
X/G\ar[rr]^{h_{\sigma}} \ar[rd]_R& & (X/G)^{\sigma}\ar[ld]^{R^{\sigma}} \\
& B &
}
\]
Since $g_{\sigma}$ and $h_{\sigma}$ are isomorphisms, we have that $h_{\sigma}(\mathcal{B})=\sigma(\mathcal{B})=\mathcal B\ (**)$. \\
\noindent \emph{Claim 4.} $B$ has a $K$-rational point $r$ outside the branch locus of $R\circ S$.\\
\noindent The branch divisor $D=R(0)+R(1)+R(\infty)$ satisfies
$$D^{\sigma}=\sigma(R(0))+\sigma(R(1))+\sigma(R(\infty))=R^{\sigma}(\sigma(0))+R^{\sigma}(\sigma(1))+R^{\sigma}(\sigma(\infty))$$
$$=R^{\sigma}(0)+R^{\sigma}(1)+R^{\sigma}(\infty)\overset{(*)}{=}R\circ h_{\sigma}^{-1}(0)+R\circ h_{\sigma}^{-1}(1)+R\circ h_{\sigma}^{-1}(\infty)\overset{(**)}{=}D,$$ so $B$ has a $K$-rational divisor of degree $3$. By \cite[Lemma 5.1]{Hugg04} this implies that $B$ has a $K$-rational point, and thus $B$ is isomorphic to $\pp^1_K$ over $K$. In particular, $B$ has a $K$-rational point $r$ outside the branch locus of $R\circ S$.
\medskip
By D\`ebes-Emsalem theorem (Theorem \ref{DE}\,(b)) and \emph{Claim 4} we conclude that $K$ is a field of definition of $X$.
\end{proof}
A group over its center is cyclic if and only if the group is abelian, thus we obtain the following.
\begin{corollary}\label{cencor}
Let $K$ be a field of characteristic $0$ and let $F$ be an algebraic closure of $K$.
Let $X$ be a smooth projective algebraic curve of genus $g\ge 2$ defined over $F$ such that $X\slash Z({\rm Aut}(X))$ has genus $0$.
If $X$ can not be defined over its field of moduli $M_{F/K}(X)$, then ${\rm Aut}(X)$ is abelian.
\end{corollary}
\begin{example}\label{cu}
Consider the plane quartic $X$ defined by $x^4+y^4+z^4+ax^2y^2+bxyz^2=0$, where
$ab\not=0$. By \cite[Theorem 29]{Bars05} $X$ has automorphism group isomorphic to $D_4$ and generated by
$$f: [x:y:z]\mapsto [y:x:z],\ g:[x:y:z]\mapsto [ix:-iy:z].$$
The map $g^2$ has order $2$ and generates the center of ${\rm Aut}(X)$.
The group $\langle f\rangle$ also has order $2$, and the quotients
$X\slash\langle f\rangle$ and $X\slash\langle g^2\rangle$ both
have signature $(1;[2,2,2,2])$.
Thus $Z({\rm Aut}(X))$ is not unique up to conjugation in this case.
\end{example}
\subsection{Odd signature curves}
We present here another easy criterion for the definability of a curve $X$ over its field of moduli in terms of
the signature of the covering $\pi:X\longrightarrow X\slash {\rm Aut}(X)$ provided that $X\slash {\rm Aut}(X)$ has genus $0$.
\begin{definition}
A smooth projective curve $X$ of genus $g\geq 2$ has {\em odd signature} if the signature of the covering $\pi: X \to X/{\rm Aut}(X)$
is of the form $(0; c_1,\dots,c_r)$ where some $c_i$ appears exactly an odd number of times.
\end{definition}
The following result can be proved by means of D\`ebes-Emsalem theorem and applying \cite[Lemma 5.1]{Hugg04} to the branch locus of the projection $\pi:X\longrightarrow X\slash {\rm Aut}(X)$. In case the field is finite, the result follows from \cite[Corollary 2.11]{Hugg04}.
\begin{theorem} \cite[Theorem 2.5]{ArtQui12} \label{AQ} Let $X$ be a smooth projective curve of genus $g\ge 2$
defined over an algebraically closed field $F$, and let $F\slash K$ be a Galois extension. If $H$ is a subgroup of ${\rm Aut}(X)$ unique up to conjugation and $\pi_N:X\longrightarrow X\slash N_{{\rm Aut}(X)}(H)$ is an odd signature cover, then $M_{F\slash K}(X)$ is a field of definition for $X$.
\end{theorem}
The previous theorem immediately implies that $X$ can be defined over its field of moduli anytime
the branch locus of $\pi:X\to X/{\rm Aut}(X)$ has odd cardinality. In \cite[Corollary 3.5]{ArtQui12} it has been applied
to show that non-normal $p$-gonal curves in characteristic zero, where $p$ is prime, can always be defined over their field of moduli.
\section{Pseudoreal Riemann surfaces and NEC groups}
In this section we will concentrate on complex curves, or equivalently (embedded) compact Riemann surfaces.
\subsection{Pseudoreal Riemann surfaces} Since the concepts of field of moduli and field of definition of a curve only depend on its
isomorphism class, then we can extend the definition to compact Riemann surfaces after choosing any embedding of them
in a projective space. We recall that an {\em antiholomorphic (or anticonformal)} morphism between two Riemann
surfaces $X,Y$ is a continuous map $X\to Y$ whose transition maps composed with the complex conjugation $J$ are holomorphic.
\begin{proposition} A compact Riemann surface $X$ has field of moduli
$M_{\mathbb{C}\slash\mathbb{R}}(X)=\mathbb{R}$ if and only if it has an anticonformal automorphism,
and it has field of definition $\mathbb{R}$ if and only if it has an anticonformal involution.
\end{proposition}
\begin{proof} We will identify $X$ with a smooth complex projective curve, after choosing an embedding of it in a projective space $\pp^n_\cc$.
Note that $M_{\mathbb{C}\slash\mathbb{R}}(X)=\mathbb{R}$ if and only if $X\cong_{\mathbb{C}}\overline{X}$, where we denote by
$\overline{X}$ the curve $X^{c}$ obtained applying the complex conjugation to the coefficients of the defining polynomials of $X$.
If $f:\overline{X}\longrightarrow X$ is an isomorphism and $J([x_0:\dots:x_n])=[\bar x_0:\dots:\bar x_n]$, then $f\circ J|_X:X\longrightarrow X$
is an anticonformal automorphism. Conversely, if $g$ is
an anticonformal automorphism of $X$, then $J\circ g$ is an isomorphism between $X$ and $\overline{X}$. This proves the first statement.
If $X$ has field of definition $\mathbb{R}$, then we can assume without loss of generality that $X=\overline{X}$, so that
the map $J|_X$ is an anticonformal involution of $X$.
Conversely assume that $X$ has an anticonformal involution $\tau$.
If ${\rm Aut}(\mathbb{C}\slash\mathbb{R})=\{e,c\}$, let $f_e:X\longrightarrow X^{e}=X$ be the identity and $f_{c}:=J|_X\circ \tau:X\to \overline X=X^c$.
We have $$f_{c}=(f_{c})^{e}\circ f_e,\ \ \ \ \ \ f_{c}=(f_e)^{c}\circ f_{c},\ \ \ \ \ \ f_e=(f_e)^{e}\circ f_e,$$ and we see that $$(f_{c})^{c}\circ f_{c}=(J|_X\circ \tau)^{c}\circ (J|_X\circ \tau)=((J|_X)^{c}\circ \tau^{c}\circ J|_X)\circ \tau=\tau\circ \tau={\rm Id}_X=f_e,$$
so the collection $\{f_e, f_{c}\}$ satisfies the condition of Weil's theorem. Hence $X$ can be defined over $\mathbb{R}$. \end{proof}
\begin{corollary} A Riemann surface is pseudoreal if and only if it has antiholomorphic automorphisms but no antiholomorphic involution.
\end{corollary}
Observe that with this characterization it is clear that any pseudoreal Riemann surface has non-trivial
automorphism group. In what follows we will denote by ${\rm Aut}^{\pm}(X)$ the {\em full automorphism group}
of a Riemann surface, i.e. the group containing both automorphisms and anti-holomorphic automorphisms of $X$.
To study such group we recall the language of Fuchsian and NEC groups.
\subsection{NEC groups}
\begin{definition} A \emph{NEC group} is a discrete subgroup $\Gamma$ of ${\rm PGL}(2,\mathbb{R})$, the full automorphism group of $\mathbb{H}=\{z\in\mathbb{C}:\ {\rm Im}(z)>0\}$, such that $\mathbb{H}\slash\Gamma$ is a compact surface. A \emph{Fuchsian group} is a NEC group contained in ${\rm PSL}(2,\mathbb{R})$, the group of conformal automorphisms of $\mathbb{H}$. If $\Gamma$ is a NEC group which is not a Fuchsian group, it is called a \emph{proper NEC group} and the group $\Gamma^+:=\Gamma\cap {\rm PSL}(2,\mathbb{R})$ is called the \emph{canonical Fuchsian subgroup of $\Gamma$}.
\end{definition}
Every NEC group $\Gamma$ admits a vector, called the \emph{signature of $\Gamma$}, given by
\begin{equation}\label{ssg} s(\Gamma)=(g;\pm;[m_1,\ldots, m_r],\{C_1,\ldots, C_k\}),\end{equation} where $g$ is the genus of the quotient space $\mathbb{H}\slash\Gamma$; the sign $+$ or $-$ depends on the orientability or non-orientability of the quotient space, respectively; the integers $m_i\ge 2$ are the ramification indices of the $r$ branch points of the quotient $\pi:\mathbb{H}\longrightarrow\mathbb{H}\slash\Gamma$; and the $C_i$ are $s_i$-uples of integers $$C_i=(n_{i1},\ldots, n_{is_i}),$$ such that $n_{ij}\ge 2$, where that values represents the ramification index of the quotient $\mathbb{H}\slash\Gamma$ in its $i$-th boundary component (if it has no border, we consider no $C_i$ in the signature).
Every NEC group with signature (\ref{ssg}) (for $+$ sign) has a presentation as a group given by the generators $$x_1,\ldots, x_r\ \ \ \ a_1,b_1,\ldots, a_g,b_g\ \ \ \ e_1,\ldots, e_k\ \ \ \ c_{10},\ldots, c_{1s_1},\ldots, c_{k0},\ldots, c_{ks_k}$$ which are called \emph{elliptic, hyperbolic, boundary and reflection elements}, respectively, that satisfy the relations $$x_i^{m_i}=1\ \ \ \forall i\in\{1,\ldots ,r\}$$
$$c_{ij-1}^2=c_{ij}^2=(c_{ij-1}c_{ij})^{n_{ij}}=1,\ \ c_{is_i}=e_i^{-1}c_{i0}e_i\ \ \ \forall i\in\{1,\ldots, k\}\ ,\ \ \forall j\in\{0,\ldots, s_i\}$$
$$x_1\ldots x_re_1\ldots e_k[a_1,b_1]\ldots [a_g,b_g]=1,$$
where $[a_i,b_i]=a_ib_ia_i^{-1}b_i^{-1}$. In case the signature has a $-$ sign, the relations are the same except the last one which is replaced by
$$x_1\ldots x_re_1\ldots e_kd_1^2\ldots d_g^2=1,$$
where the $d_i$ are $g$ \emph{glide reflections}, which are antiholomorphic
elements of infinite order of the NEC group (for details see \cite[Ch. 0]{BK}).
By the Riemann surfaces uniformization theorem we know that every compact Riemann surface of genus greater than $1$ is biholomorphic to $\mathbb{H}\slash\Delta$, where $\Delta$ is a Fuchsian group without torsion.
It follows with this presentation that
$${\rm Aut}(X)\cong N_{{\rm PSL}(2,\mathbb{R})}(\Delta)\slash\Delta,\quad {\rm Aut}^{\pm}(X)\cong N_{{\rm PGL}(2,\mathbb{R})}(\Delta)\slash\Delta,$$
where $N_{-}(\Delta)$ denotes the normalizer of $\Delta$ in both cases. In fact, any group $H\le {\rm Aut}(X)$ is isomorphic to some quotient group $\Gamma\slash\Delta$, where $\Gamma$ is a Fuchsian group that contains $\Delta$ as a normal subgroup of finite index.
In a similar way, a subgroup $H\le {\rm Aut}^{\pm}(X)$ is isomorphic to a quotient $\Gamma\slash\Delta$, where $\Gamma$ is a NEC group (see \cite{BK}).
With the previous notation, studying Riemann surfaces where an abstract group $H$ acts as full automorphism group with a certain signature is equivalent to
study the possible epimorphisms $\theta:\Gamma\rightarrow H$ such that $\ker(\theta)$ is a torsion free group contained in $\Gamma^+$.
Observe that $\ker(\theta)$ is a torsion free Fuchsian group if and only if $\theta$ preserves
the orders of the finite order elements of $\Gamma$ (see \cite[p.~12]{BS}).
The images by $\theta$ of the generators of $\Gamma$ form a vector with entries in $H$
which is called {\em generating vector} of $\theta$.
The following theorem relates the signature of a NEC group with the signature of its canonical Fuchsian subgroup \cite[Theorem 2]{Sin74}.
\begin{theorem} \label{5.1} Let $X\slash \Delta$ be a Riemann surface (considered as a Klein surface) and denote by $\Gamma$ the NEC group $N_{{\rm PGL}(2,\mathbb{R})}(\Delta)$ which corresponds to its full automorphism group. If $\Gamma$ is a proper NEC group which has no reflections, then its signature has the form $$(g;-;[m_1,\ldots ,m_r]),$$ and the signature of his canonical Fuchsian subgroup $\Gamma^{+}$ will be of the form
$$(g-1;+;[m_1,m_1,\ldots ,m_r,m_r]),$$
where every $m_i$ appears two times.
\end{theorem}
\subsection{Finite-extendability of Fuchsian and NEC signatures}
We now recall the concept of finitely maximal Fuchsian signature, since it will be important to decide whether a given group is the complete automorphism
group of a Riemann surface or not. In what follows we will denote by $s(G)$ the signature of a NEC group $G$ and by $d(s(G))=6g-6+2r$ the real dimension of the Teichm\"uller space of the signature $s(G)=(g;[m_1,\ldots ,m_r])$ (see \cite[p.~19]{Sin74b}).
\begin{definition} A Fuchsian group $\Delta$ is \emph{finitely maximal} if it is not contained properly in another Fuchsian group with finite index.
The signature $(g;[m_1,\ldots ,m_r])$ of a Fuchsian group $\Delta$ (a Fuchsian signature) is \emph{finitely maximal} if $d(s(\Gamma))\neq d(s(\Delta))$ for every Fuchsian group $\Gamma$ containing $\Delta$ as a proper subgroup.
\end{definition}
Almost all Fuchsian signatures are finitely maximal, and those which are not finitely maximal were identified
by L. Greenberg in \cite{Gre63} and D. Singerman in \cite{Sin72}.
They determined there the so called \emph{Singerman list} (see \cite[p.~19]{BK}), which contains the only $19$ non finitely maximal Fuchsian signatures.
Considering Theorem \ref{5.1} we are interested in some particular signatures from Singerman list.
\begin{table}[!h]
\centering\renewcommand{\arraystretch}{1.1}\setlength{\tabcolsep}{10pt}
\caption{Even signatures from Singerman list}
\label{Sing}
\begin{tabular}{| c | c | c |}
\hline
$\sigma_1$ & $\sigma_2$ & $[\sigma_2:\sigma_1]$ \\ \hline
$(2;[-])$ & $(0;[2,2,2,2,2,2])$ & $2$ \\ \hline
$(1;[t,t])$ & $(0;[2,2,2,2,t])$ & $2$ \\ \hline
$(0;[t,t,t,t])\ t\ge 3$ & $(0;[2,2,2,t])$ & $4$ \\ \hline
$(0;[t_1,t_1,t_2,t_2])\ t_1+t_2\ge 5$ & $(0;[2,2,t_1,t_2])$ & $2$ \\ \hline
\end{tabular}
\end{table}
The analogous of Singerman list for NEC groups was developed and completed in \cite{Buj82} and \cite{EstIzq06}.
We will use these lists to find NEC signatures which correspond to full automorphism groups of pseudoreal Riemann surfaces.
By \cite[Remark 1.4.7]{BS}, the signature $s(\Gamma)$ of a proper NEC group is finitely maximal if the same holds for the signature
$s(\Gamma^+)$ of its canonical Fuchsian subgroup.
In \cite[Section 4]{BaGo10}, the authors studied under which conditions a finite group $G$ with a given non finitely maximal NEC signature can act as the full automorphism group of a pseudoreal Riemann surface. The three NEC signatures they studied were $$(1;-;[k,l];\{-\}),\ \ \ \ \ \ (1;-;[k,k];\{-\}),\ \ \ \ \ \ (2;-;[k];\{-\}),$$ which are associated to the non finitely maximal Fuchsian \emph{even} signatures $$(0;[k,k,l,l]),\ \ \ (0;[k,k,k,k]),\ \ \ (1;[k,k]),$$
of the Singerman list (see Table \ref{Sing} and Theorem \ref{5.1}).
We study the missing case of the non finitely maximal NEC signature $(3;-;[-];\{-\})$, which will be needed to complete the classification
of possible automorphism groups for pseudoreal Riemann surfaces. In what follows we say that an automorphism group has an {\em essential} action
on a Riemann surface if it contains anticonformal elements.
\begin{lemma} \label{BG4} Let $\Delta$ be a NEC group with signature $(3;-;[-];\{-\})$.
There exists an epimorphism $\theta:\Delta\longrightarrow G$ onto a finite group $G$ defining an essential action on a pseudoreal Riemann surface if and only if $G$ is a non-split extension of some of its subgroups $H$ of index $2$, $G$ is generated by three elements $d',d'',d'''$ such that $d',d'',d'''\not\in H$, $(d')^2(d'')^2(d''')^2=1$ and the map $$d'\mapsto (d')^{-1},\ \ \ d''\mapsto (d')^2(d'')^{-1}(d')^{-2},\ \ \ d'''\mapsto (d''')^{-1}$$ does not induce an automorphism of $G$. Furthermore, such a group $G$ is necessarily the full automorphism group of a pseudoreal Riemann surface on which it acts.
\end{lemma}
\begin{proof}
Suppose we have an epimorphism $\theta:\Delta\longrightarrow G$
onto a finite group $G$ defining an essential action on the pseudoreal Riemann surface
$X=\mathbb{H}\slash \ker(\theta)$.
Then $H:=\theta(\Delta^{+})$ is an index $2$ subgroup of $G$,
because $G$ has anticonformal elements.
The extension $H\le G$ is non-split since otherwise $G\backslash H$ would contain anticonformal involutions,
contradicting the fact that $X$ is pseudoreal.
We have $\Delta=\langle d_1,d_2,d_3:\ d_1^2d_2^2d_3^2=1\rangle$
where the $d_i$'s are glide reflections,
so the anticonformal elements $d':=\theta(d_1), d'':=\theta(d_2)$ and $d''':=\theta(d_3)$
can not belong to $H$.
To prove the statement we need to show that the map
$$d'\mapsto (d')^{-1},\ \ \ d''\mapsto (d')^2(d'')^{-1}(d')^{-2},\ \ \ d'''\mapsto (d''')^{-1}$$
does not induce an automorphism of $G$.
To see this, observe that by \cite[p.~529-530]{Buj82} there is a NEC group $\Delta'$
with the unique signature $(0;+;[2,2,2],\{(-)\})$ containing $\Delta$ as a subgroup of index $2$.
By \cite[Proposition 4.8]{Buj82} we know that if
$$\Delta'=\langle x_1,x_2,x_3,c_1,e_1: x_1x_2x_3e_1=1,\ e_1^{-1}c_1e_1c_1=1,\ x_1^2=x_2^2=x_3^2=1,\ c_1^2=1\rangle$$
then $\Delta$ can be written as
$$\Delta=\langle d_1:=c_1x_1,\ d_2:=x_1c_1x_1x_2,\ d_3:=x_2x_1c_1x_1x_2x_3\rangle\le \Delta'.$$
If we conjugate every generator of $\Delta$ by $c_1$ we get
$$c_1^{-1}d_1c_1=d_1^{-1},\ \ \ c_1^{-1}d_2c_1=d_1^2d_2^{-1}d_1^{-2},\ \ \ c_1^{-1}d_3c_1=d_3^{-1},$$
so $\ker(\theta)$ is a normal subgroup of $\Delta'$ if and only if the images of $d_1,d_2$ and $d_3$
through $\theta$ satisfy that the map
$$d'\mapsto (d')^{-1},\ \ \ d''\mapsto (d')^2(d'')^{-1}(d')^{-2},\ \ \ d'''\mapsto (d''')^{-1}$$
induces an automorphism of $\Delta\slash\ker(\theta)=G$.
So the assertion follows, since if $\ker(\theta)$ were a normal subgroup of $\Delta'$, then $\Delta'\slash\ker(\theta)\cong {\rm Aut}^{\pm}(X)$ and it
would contain $c_1\ker(\theta)$, which is an anticonformal involution, contradicting the hypothesis that $X$ is pseudoreal.
Conversely, for a NEC group $\Delta$ with signature $(3;-;[-],\{-\})$ and a non-split extension $H\le G$ of degree $2$, we can consider the map $\theta(d_1)=d',\ \theta(d_2)=d'',\ \theta(d_3)=d'''$ which induces an epimorphism $\theta:\Delta\longrightarrow G$, defining an essential action on $X:=\mathbb{H}\slash \ker(\theta)$. We must have that $G$ is the full automorphism group of $X$, because otherwise $\ker(\theta)$ would be a normal subgroup of a NEC group $\Delta'$ with signature $(0;+;[2,2,2],\{(-)\})$, and so by the previous part of the proof, the mapping $$d'\mapsto (d')^{-1},\ \ \ d''\mapsto (d')^2(d'')^{-1}(d')^{-2},\ \ \ d'''\mapsto (d''')^{-1}$$ would define an automorphism of $G$, contradicting our assumptions. Finally, since $G$ is a non-split extension of $H$, then $G\backslash H$ contains no involutions, thus $X$ is a pseudoreal Riemann surface.
\end{proof}
\subsection{Existence for any genus}
In \cite[Theorem 1 and p.~48]{Sin80} the author proved the following result. We provide the proof since it is an interesting application
of the approach through NEC groups.
\begin{theorem}\label{existencia} There exist pseudoreal Riemann surfaces for every genus $g\ge 2$.
\end{theorem}
\proof Consider a NEC group $\Delta$ with signature $(1;-;[2^{g+1}];\{-\})$, where $g$ is even. If $x_1,\dots, x_{g+1}$ is the set of elliptic generators and $d_1$ is the glide reflection, which together generate $\Delta$, we can define an epimorphism
$\theta:\Delta\longrightarrow C_4=\langle a:a^4=1\rangle$ given by
$$\theta(x_i)=a^2,\ \ \ \forall i\in\{1,\ldots ,g+1\},\ \ \ \ \ \ \theta(d_1)=a.$$
Since $\theta$ preserves the orders of the elliptic generators then $\ker(\theta)$ is torsion free (see \cite[p.~12]{BS}),
so the quotient $X=\mathbb{H}\slash \ker(\theta)$ is a Riemann surface such that ${\rm Aut}^{\pm}(X)$ contains the group $\Delta\slash \ker(\theta)\cong C_4$. Since $\Delta$ has finitely maximal signature \cite{Buj82}, we can conclude that ${\rm Aut}^{\pm}(X)\cong C_4$. This Riemann surface $X$ has genus $g$ and has anticonformal automorphisms, but no anticonformal involutions, because $a^2$, the only element of order $2$ in $C_4$, is in the conformal part ${\rm Aut}^{+}(X)\cong \{1,a^2\}$. Thus $X$ is pseudoreal.
The proof for odd $g$ is similar, taking the NEC signature $(2;-;[2^{g-1}];\{-\})$ and considering the epimorphism $\theta:\Delta\longrightarrow C_4=\langle a:a^4=1\rangle$ given by $$\theta(x_i)=a^2,\ \ \ \forall i\in\{1,\ldots ,g-1\},\ \ \ \ \ \ \theta(d_1)=\theta(d_2)=a. \eqno\qed$$
\subsection{Full groups of pseudoreal Riemann surfaces and group extensions}
Let $\overline G$ be a group acting on a Riemann surface $X$ and containing antiholomorphic automorphisms
(i.e. acting as an essential group). Then there is an exact sequence of groups
\begin{equation}\label{ext}
1\longrightarrow G\longrightarrow \overline{G}\longrightarrow C_2\longrightarrow 0,
\end{equation}
where $G$ denotes the subgroup of holomorphic automorphisms in $\bar G$.
If $X$ is pseudoreal, then $\overline G\setminus G$ contains no order two elements, or equivalently the sequence is non-split.
On the other hand, the group $G$ contains an element of order two by Cauchy's theorem applied to $\overline G$, thus
$G$ has even order.
In \cite{BaGo10} the authors proved that these conditions are sufficient for a group $\overline G$ to act
on a pseudoreal Riemann surface.
\begin{theorem} \cite[Theorem 3.3]{BaGo10}\label{bagofull} A finite group $\overline{G}$ acts as an essential group on a
pseudoreal Riemann surface $X$ if and only if it is a non-split extension of a group of even order by the cyclic group of order $2$. For any such group,
there exists a Riemann surface $X$ having $\overline{G}$ as its full automorphism group.
\end{theorem}
As a consequence of the previous theorem, it can be easily seen that no symmetric or dihedral group
can be the full automorphism group of a pseudoreal Riemann surface, and that any cyclic group of order $4n$, $n\geq 1$,
is the full automorphism group of some pseudoreal Riemann surface.
Motivated by the previous theorem, we study the possible extensions of a group $G$ by the cyclic group $C_2$.
The most general approach to this problem is through cohomology of finite groups (see \cite[Ch.~1]{Coh04}), but we will use easier tools.
Given an extension as in (\ref{ext}), consider an element $x\in\overline{G}\backslash G$.
Since $G$ is normal in $\overline{G}$, this induces an automorphism $\phi_x$ of $G$ defined by conjugation by $x$
(from now on, we will denote by $\phi_p$ the conjugation by the element $p$).
Moreover $g=x^2\in G$, $\phi_x^2=\phi_g$ and $g$ is fixed by $\phi_x$. Let $P(G)$ be the subset of ${\rm Aut}(G)\times G$ defined by $$P(G):=\{(\phi,g)\in {\rm Aut}(G)\times G:\ \phi^2=\phi_g,\ \ \phi(g)=g\}.$$
We can define an equivalence relation on $P(G)$ by $$(\phi,g)\sim (\phi\circ \phi_h, \phi(h)gh),\ \ \ \ \forall h\in G.$$
Let $E(G)$ be the quotient set $P(G)\slash\sim$.
\begin{lemma} \label{lemma3.3.1} Given a group $G$, there exists a well defined function from the set of group extensions
$$1\longrightarrow G\longrightarrow \overline{G}\longrightarrow C_2\longrightarrow 0,$$ to $E(G)$.
\end{lemma}
\proof For any such extension we can take an element $x\in \overline{G}\backslash G$
and construct the pair $(\phi_x,x^2)$. It is an easy exercise to check that different choices of $x$ lead to equivalent pairs. \qed
\begin{lemma} \label{3.3.2} Given an element $(\phi,g)\in P(G)$, there exist an extension $$1\longrightarrow G\longrightarrow \overline{G}\longrightarrow C_2\longrightarrow 0,$$ and an element $x\in \overline{G}\backslash G$ such that $\phi=\phi_x$ and $x^2=g$.
\end{lemma}
\begin{proof} Consider the group $\overline{G}$ defined by $$\overline{G}:=(G\rtimes_F \mathbb{Z})/\langle (g^{-1},z^2)\rangle,$$ where $\mathbb{Z}$ is the cyclic group generated by $z$ and $F$ is the homomorphism induced by $$F:\mathbb{Z}\longrightarrow {\rm Aut}(G),\ \ \ \ z\mapsto \phi.$$ The subgroup $\langle (g^{-1},z^2)\rangle$ is normal in $G\rtimes_F \mathbb{Z}$, so $\overline{G}$ is a group. Clearly $G$ injects into $\overline{G}$ through $a\mapsto (a,1)$, and we have that $$\overline{G}=\{(g,1)\ ,\ g\in G\}\cup \{(g,z)\ ,\ g\in G\},$$ because for $(p,z^m)\in G\rtimes_{F}\mathbb{Z}$ we have two cases $$[(p,z^{m})]=[(pg^{\frac{m}{2}},1)]\ \ \ \ \ \ {\rm for\ even}\ m,$$ $$[(p,z^{m})]=[(pg^{\frac{m-1}{2}},z)]\ \ \ \ \ {\rm for\ odd}\ m,$$
\noindent so $|\overline{G}|=2|G|$ and we have $\overline{G}\slash G\cong C_2$. Moreover $\phi_{(1,z)}(h,1)=(\phi(h),1),$ and $(1,z)^2=(g,1),$ so we can choose $x$ as $(1,z)$.
\end{proof}
\begin{theorem}\label{theorem} There is a bijection between the set of isomorphism classes of extensions of $G$ by $C_2$, and $E(G)$.
Moreover, one such extension is split if and only if it corresponds to a pair $[(\phi,g)]$ where $g=e$,
and it is a direct product if and only if one can choose $\phi={\rm id}_G, g=e$.
\end{theorem}
\begin{proof} Given an extension as in (\ref{ext})
we can associate to it the class $[(\phi_x,x^2)]$ by Lemma \ref{lemma3.3.1}.
It is easy to see that an isomorphic extension leads to the same pair.
Conversely, by Lemma \ref{3.3.2}, we can associate to every pair $(\phi,g)\in P(G)$ an extension of $G$ defined by $$A=\left(G\rtimes_F \mathbb{Z}\right)\slash \langle (g^{-1},x^2)\rangle$$ as we did above. Every pair $(\phi\circ \phi_h,\phi(h)gh)$ equivalent to $(\phi,g)$ will give us another group $$B=(G\rtimes_{F'}\mathbb{Z})\slash \langle((\phi(h)gh)^{-1},y^2)\rangle,$$ where $\mathbb{Z}=\langle y\rangle$, $h\in G$ and $F':\mathbb{Z}\longrightarrow {\rm Aut}(G)$ is induced by $y\mapsto \phi\circ \phi_h$. An isomorphism $\alpha:A\longrightarrow B$ is induced by $\alpha(g,1)=(g,1)$, $\alpha(1,x)=(\phi(h)^{-1},y)$. It is well defined because $$\alpha(g^{-1},x^2)=(g^{-1},1)(\phi(h)^{-1},y)(\phi(h)^{-1},y)=((\phi(h)gh)^{-1},y^2)$$ and clearly $\alpha|_G={\rm Id}_G$.
The exact sequence (\ref{ext}) is split if and only if $\overline{G}\backslash G$ has an order $2$ element $p$, which gives us the desired pair $(\phi_p,e)$.
Moreover $\overline{G}\cong G\times C_2$ if and only if one can choose $\phi_p={\rm Id}_G$.
\end{proof}
\begin{corollary} Let $G$ be a finite group. Any extension of $G$ by $C_2$ is split if $Z(G)$ is trivial and one of the following holds
\begin{enumerate}
\item ${\rm Out}(G):={\rm Aut}(G)\slash {\rm Inn}(G)$ has no involutions;
\item ${\rm Inn}(G)$ has a group complement in ${\rm Aut}(G)$.
\end{enumerate}
\end{corollary}
\begin{proof} Given a group extension as in (\ref{ext}),
we can associate to it a pair $(\phi,g)$, such that $\phi^2=\phi_g$.
The class $[\phi]$ in ${\rm Out}(G)$ satisfies $[\phi]^2=[1]$, but ${\rm Out}(G)$ has no order $2$ elements, so $\phi\in {\rm Inn}(G)$.
In that case $(\phi,g)\sim (\phi\circ \phi^{-1},g')=({\rm Id}_G,g')$ for some $g'\in G$, where ${\rm Id}_G^2=\phi_{g'}$. Since $Z(G)=\{1\}$, then $g'=e$.
Thus $(\phi,g)\sim ({\rm Id}_G,e)$ so by Theorem \ref{theorem} every extension of $G$ by $C_2$ will be a direct product $G\times C_2$. This proves (i).
Let $H$ be the group complement of ${\rm Inn}(G)$ in ${\rm Aut}(G)$, that is $${\rm Aut}(G)= H\cdot {\rm Inn}(G),\ \ \ \ H\cap {\rm Inn}(G)=\{1\}.$$
If $(\phi,g)\in P(G)$, then $\phi^2=\phi_g$ and $\phi(g)=g$. We have that ${\rm Aut}(G)=H\cdot {\rm Inn}(G)$, so $\phi\in {\rm Aut}(G)$ can be written as $\phi=\varphi\circ \phi_h$ with $\varphi\in H$ and $\phi_h\in {\rm Inn}(G)$, so $(\phi,g)\sim (\varphi\circ \phi_h,g)\sim (\varphi,g')$ for some $g'\in G$. We also have $\varphi^2\in H\cap {\rm Inn}(G)=\{1\}$ so $\varphi^2=1$, but $\varphi^2=\phi_{g'}$ so $\phi_{g'}=1$, which is equivalent to $g'=e$ because $Z(G)=\{1\}$. In that case $(\varphi,g')=(\varphi,e)$ so we get the desired equality $[(\phi,g)]=[(\varphi,e)]$ and we obtain (ii) by Theorem \ref{theorem}.
\end{proof}
If we translate the previous result for pseudoreal Riemann surfaces, we get the following. The
second condition had already been proved in \cite[Corollary 4.3]{DebEms99} with different techniques.
\begin{corollary}\label{comp}
Let $G$ be the conformal automorphism group of a Riemann surface $X$.
Suppose that $Z(G)=\{1\}$ and that either ${\rm Out}(G)$ has no involutions or ${\rm Inn}(G)$ has group complement
in ${\rm Aut}(G)$. Then $X$ is not pseudoreal.
\end{corollary}
\subsection{Maximal groups}
A. Hurwitz proved that the order of the conformal automorphism group of a Riemann surface of genus $g\ge 2$ is bounded above by $84(g-1)$ (see \cite[p.~424]{Hur93}), and there are infinitely many Riemann surfaces whose conformal automorphism group attains that bound.
The first example of such groups was the order $168$ group ${\rm PSL}(2,7)$, which is the conformal automorphism group of the Klein's quartic.
In the case of pseudoreal Riemann surfaces, the Hurwitz bound is never attained because all such surfaces have conformal automorphism groups of signature $(0;[2,3,7])$, which is an odd signature.
This also follows from \cite[Theorem 5.4]{Wolfart} since such Riemann surfaces are quasiplatonic,
i.e. the genus of $X/{\rm Aut}(X)$ is zero and the quotient
$X\to X/{\rm Aut}(X)$ has at most three branch points.
For pseudoreal Riemann surfaces there is a better upper bound, as we see in the following
\begin{theorem}\label{maximalll} \cite[Theorem 5.1]{BuConCo10} If $X=\mathbb{H}\slash\Gamma$ is a pseudoreal Riemann surface of genus $g$ with full automorphism group $G$, then $|G|\le 12(g-1)$. Moreover, if $|G|=12(g-1)$ and $G=\Delta\slash\Gamma$ then the signature of $\Delta$ is $(1;-;[2,3])$.
\end{theorem}
If a pseudoreal Riemann surface $X$ has genus $g$ and full automorphism group of order $12(g-1)$,
we will say that $X$ has \emph{maximal full group}.
Using the programs in Section \ref{prog}
we found that the minimum genus for which there exists a pseudoreal Riemann surface with maximal full group is $g=14$, with conformal automorphism group ${\rm ID}(78,1)$ and full automorphism group ${\rm ID}(156,7)$.
In \cite[Theorem 5.5]{BuConCo10} the authors proved that there exists pseudoreal Riemann surfaces with maximal
full group for infinitely many genera. The groups that they got are non abelian, this inspired us to prove the next result.
\begin{theorem} \label{abeliann} If a pseudoreal Riemann surface $X$ has maximal full group,
then its conformal automorphism group is not abelian and $X/Z({\rm Aut}(X))$ has positive genus.
\end{theorem}
\begin{proof} Suppose that the conformal automorphism group $G$ of $X$ is abelian.
First observe that $G$ has order $6(g-1)$ and the Fuchsian signature associated to $G$ is $(0;[2,2,3,3])$ (Theorem \ref{maximalll} and Theorem \ref{5.1}).
By \cite[Theorem 7.1]{BuCiCo02} $G$ is cyclic since otherwise it would be a quotient of $C_2\oplus C_3\cong C_6$ (and thus cyclic).
By Table \ref{GEN2} we know that there is no conformal automorphism group of a pseudoreal Riemann surface of order $6$ in genus $2$, so we can assume $g>2$. However, in this case $6(g-1)>2g+2$, and a generator of $G$ will be an element of order $>2g+2$. By \cite[Corollary 1]{Sin74b} $X$ is not pseudoreal, contradicting the hypothesis. This proves the first half of the statement.
The second statement follows from the first one and Corollary \ref{cencor}.
\end{proof}
\section{Classification}\label{abc}
In this section we will recall what are conformal and full automorphism groups of pseudoreal
Riemann surfaces up to genus $4$ and we will extend such classification up to genus $10$. \\
\noindent \textbf{Genus 2.} By \cite[Theorem 2]{CarQuer02} we know that if $X$ is a curve of genus $2$
defined over a field of characteristic not equal to $2$, and ${\rm Aut}(X)\not\cong C_2$,
then $X$ can be defined over its field of moduli.
In particular, if $X$ is a pseudoreal Riemann surface of genus $2$, then ${\rm Aut}(X)\cong C_2$.
The latter result was also obtained in \cite[Theorem 4.1]{BuConCo10} via NEC groups and epimorphisms,
obtaining $C_4$ as the only possible full automorphism group in genus $2$ (see Table \ref{GEN2}).
An algebraic model for a pseudoreal curve of genus $2$ is Earle's example \cite{E71}
$$X\ :\ y^2=x(x^2-a^2)(x^2+ta^2x-a),$$ where $a=e^{\frac{2\pi i}{3}}$ and $t\in\mathbb{R}^{+}-\{1\}$. \\
\noindent \textbf{Genus 3.}
A hyperelliptic curve of genus three can be defined over its field of moduli unless its automorphism group is isomorphic
to $C_2\times C_2$ (see \cite[\S 4.5]{LR11}).
In the non-hyperelliptic case, in \cite[Theorem 0.2]{ArtQui12} the authors proved that if $X$
is a smooth plane quartic such that ${\rm |Aut}(X)|>4$, then $X$ can be defined over its field of moduli,
since all the other possible automorphism groups have odd signature.
Moreover, the authors proved that a pseudoreal plane quartic has automorphism group $C_2$.
The same result was obtained in \cite[Proposition 3.5]{BuCo14} via NEC groups and epimorphisms, obtaining $C_4$ and $C_4\times C_2$ as the only possible full automorphism groups in genus $3$ (see Table \ref{GEN3}).
The equation of a pseudoreal hyperelliptic curve of genus $3$ is given in \cite[p.~82]{Hugg05}
$$X\ :\ y^2=(x^2-a_1)\left(x^2+\frac{1}{\overline{a_1}}\right)(x^2-a_2)\left(x^2+\frac{1}{\overline{a_2}}\right),$$
where the coefficients must satisfy certain special conditions.
An explicit pseudoreal plane quartic is given by
$$X\ :\ y^4+y^2(x-a_1z)\left(x+\frac{1}{a_1}z\right)+(x-a_2z)\left(x+\frac{1}{\overline{a_2}}z\right)(x-a_3z)\left(x+\frac{1}{\overline{a_3}}z\right)=0,$$ where $a_1=1, a_2=1-i, a_3=2(i-1)$ \cite[Proposition 4.3]{ArtQui12}. \\
\item \textbf{Genus 4.} In \cite[Theorem 4.3]{BuCo14} the authors find that the only possible full
automorphism groups for pseudoreal Riemann surfaces are $C_4,C_8$ and the Frobenius group $F20$ (see Table \ref{GEN4}).
An algebraic model for a pseudoreal curve of genus $4$ when ${\rm Aut}^{+}(X)$ is $C_2$ with Fuchsian signature $[0;2^{10}]$
has been given by Shimura \cite{Shi72}
$$y^2=x^5+(a_1x^6-\overline{a_1}x^4)+(a_2x^7+\overline{a_2}x^3)+(a_3x^8-\overline{a_3}x^2)+(a_4x^9+\overline{a_4}x)+(x^{10}-1),$$
and has full group $C_4$, where the coefficients $a_i$ and $\overline{a_j}$ are algebraically independent over $\mathbb{Q}$. When ${\rm Aut}^{+}(X)$ is $C_4$, we have a hyperelliptic example in \cite[p.~82]{Hugg05} given by $$y^2=x(x^4-b_i)\left(x^4+\frac{1}{\overline{b_i}}\right),$$ which has full group $C_8$.
\begin{example}
We now provide the equations of a non-hyperelliptic pseudoreal curve of genus $4$
with automorphism group isomorphic to $D_5$. This example was kindly communicated to us by Jeroen Sijsling.
Let $X$ be the complete intersection of the quadric and the cubic in $\pp^3$ defined as the zero sets of the following polynomials:
\[
F_2 := (-3i + 2)(x_1^2+x_2^2+x_3^2+x_4^2) + (9i - 2)(x_1x_2+x_2x_3+x_3x_4) - 6i(x_1x_3 +x_1x_4 + x_2x_4),
\]
\[
F_3 := (-2i + 1)(x_1^2x_2 +x_2^2x_3+x_3^2x_4-x_1x_2^2-x_2x_3^2-x_3x_4^2)+ \]
\[+ 4i(x_1^2x_3 - x_1^2x_4 - x_1x_3^2 + x_1x_4^2 + x_2^2x_4 - x_2x_4^2).
\]
Observe that, since $X$ is canonically embedded, all elements of ${\rm Aut}(X)$ are induced by
projectivities in ${\rm PGL}(4,\cc)$.
The curve $X$ contains the following elements in its automorphism group
\[
T_1=\left(
\begin{array}{cccc}
-1 & 1& 0& 0\\
-1 & 0 & 1& 0\\
-1 & 0 & 0 & 1\\
-1 & 0 & 0 & 0
\end{array}
\right),\quad
T_2 = \left(
\begin{array}{cccc}
-1 & 0 & 0 & 0\\
-1 & 0 & 0 & 1\\
-1 & 0 & 1 & 0\\
-1 & 1 & 0 & 0
\end{array}
\right),
\]
which generate a subgroup $G$ of ${\rm Aut}(X)$ isomorphic to the dihedral group $D_5$:
\[
G\to D_5=\langle a,b|a^2,b^5,aba^{-1}b\rangle,\quad T_1\mapsto b,\ T_2\mapsto a.
\]
Actually, $G={\rm Aut}(X)$ and can be proved as follows.
By \cite[Proposition 2.5, II, 1]{AKuIKu90} and \cite[Table 4]{MSSV}, if $G$ were properly contained in ${\rm Aut}(X)$, then ${\rm Aut}(X)$ would be isomorphic
to the symmetric group $S_5$ and $G$ to the unique subgroup of $S_5$ isomorphic to $D_5$
(which can be generated by $(13)(45)$ and $(12345)$).
Moreover, the normalizer of $G$ in ${\rm Aut}(X)$ would be a group of
order $20$ containing an element $S$ (corresponding to $(1534)$) which acts by conjugacy on $G$ as $a\mapsto a,\ b\mapsto b^2$.
Observe that the following matrix acts by conjugacy on $T_1,T_2$ in the same way as $S$:
\[
M = \left(
\begin{array}{cccc}
1 & 0 & 0& 0\\
1 & 0 & -1 & 1\\
0 & 1 & -1 & 1\\
0 & 1 & -1 & 0.
\end{array}
\right).
\]
This implies that $A=M^{-1}S$ belongs to the centralizer of $G$ in ${\rm PGL}(4,\cc)$.
Since the trace of $T_1$ is not zero and since $T_2$ has order two, the $AT_1=T_1A$ and $AT_2=\pm T_2A$ in ${\rm GL}(4,\cc)$.
An explicit computation with the help of Magma \cite{Magma} allows to identify all such matrices $A$ (which turn out to depend on one parameter)
and to check that $MA$ never gives an automorphism of $X$.
Observe that the matrix $M$ gives an isomorphism between $X$ and its conjugate $\bar X$.
Finally, one can verify that the product of $M$ with any element of $G$ is not an involution,
showing that $X$ has no anticonformal involutions.
\end{example}
As far as the authors know, there is no explicit example of a pseudoreal Riemann surface of genus $4$
with conformal automorphism group $C_2$ with signature $(2;[2,2])$.
\begin{theorem} \label{class} Two finite groups $G$ and $\overline{G}$ are the conformal and full automorphism groups of a pseudoreal Riemann surface $X$ of genus $5\le g\le 10$ if and only if $G={\rm Aut}^{+}(X)$ and $\overline{G}={\rm Aut}^{\pm}(X)$ appear in the corresponding table by genus among Tables \ref{table5}, \ref{table6}, \ref{table7}, \ref{table8}, \ref{table9} and \ref{table10}.
\end{theorem}
\proof
The classification is obtained through a case by case analysis, carried out by means
of the Magma program which is described in the next section.
The main steps of the algorithm are the following.
\begin{enumerate}
\item Fixed a genus $5\le g\le 10$ we consider the complete list $L$ of triples $(G,s,v)$, where $G$ is a group acting on
Riemann surfaces $X$ of genus $g$, $s$ its signature and $v$ the generating vector of the action (see \S 2.2).
These data are given us by the Magma program of J. Paulhus (see \cite{Pau15}).
\item From the list $L$ we select only the triples where $G$ has even order and $s$ is an even signature (see Theorem \ref{5.1}).
\item We separate the finitely maximal and the non finitely maximal signatures.
\item In case the signature is finitely maximal,
we first consider all the non-split extensions $\overline{G}$ of $G$ by $C_2$ (Theorem \ref{bagofull}).
For any such extension we check the existence of an epimorphism $\theta:\Delta\to\overline G$, where $\Delta$ is a NEC group
whose signature is obtained from $s$ by Theorem \ref{5.1}, and $\ker(\theta)$ is a torsion free Fuchsian group.
If there is one such epimorphism, $G$ and $\bar G$ are the conformal and full automorphism groups respectively of a pseudoreal Riemann surface.
\item In case the signature is non finitely maximal, we apply Lemmas \cite[Lemma 4.1, Lemma 4.2, Lemma 4.3]{BaGo10} and Lemma \ref{BG4} to identify pseudoreal automorphism groups. \qed
\end{enumerate}
Observe that for any case in the tables of section \ref{app} there exists a pseudoreal curve with those properties.
However, finding an explicit algebraic model for such curves is a difficult problem.
In \cite{Shi72} G. Shimura provided hyperelliptic examples for any even genus $g\geq 2$ with automorphism group of order two.
Later the hyperelliptic case has been completely characterized in the following theorem by B. Huggins, which
completes previous work in \cite{BuTur02}.
\begin{theorem} \cite[Theorem 5.0.5]{Hugg05} Let $X$ be a hyperelliptic curve defined over $\mathbb{C}$ such that $M_{\mathbb{C}\slash\mathbb{R}}(X)=\mathbb{R}$. Then $X$ is pseudoreal if and only if it is isomorphic to either $y^2=f(x)$, or $y^2=g(x)$, where
$$f(x)=\prod\limits_{i=1}^r(x^n-a_i)\left(x^n+\frac{1}{\overline{a_i}}\right),\quad g(x)=x\prod\limits_{i=1}^s(x^m-b_i)\left(x^m+\frac{1}{\overline{b_i}}\right),$$
$m,n,r,s$ are non negative integers such that $2nr>5$, $sm$ is even, $r$ is odd if $n$ is odd, and $a_i, b_i$ satisfy additional conditions that can be found in \cite[Pag.~82]{Hugg05}. Moreover, these curves have automorphism groups isomorphic to $C_2\times C_n$ and $C_{2n}$, respectively.
\end{theorem}
In \cite[Theorem 2.]{E71} the author gives an example of a pseudoreal Riemann surface $X$ of genus $5$
with an order $4$ anticonformal element called $f$, which generates ${\rm Aut}^{\pm}(X)\cong C_4$
(see Figure \ref{fig:awesome_image}). There are exactly $2$ possible conformal actions
of $C_2$ on pseudoreal Riemann surfaces of genus $5$, having signatures $(3;[-])$ and $(1;[2^8])$.
Since $f^2$ has no fixed points, the conformal
action is $C_2$ with signature $(3;[-])$ (see Table \ref{table5}).
\begin{figure}[h!]
\centering
\includegraphics[width=0.3\textwidth]{earle-g5.png}
\caption{Earle's picture of his genus $5$ example} \label{fig:awesome_image}
\end{figure}
More non-hyperelliptic examples of pseudoreal curves have been introduced by B. Huggins in \cite{Hugg05}
and later by A. Kontogeorgis for $p$-gonal curves \cite{Kon09}.
In \cite{Hid09} R. Hidalgo found a nice example of a non-hyperelliptic pseudoreal curve of genus $17$ with automorphism group
$C_2^5$ which is a covering of the genus two example given by Earle.
Recently in \cite{ACHQ16} the authors constructed a tower of Riemann surfaces, going from Earle's example in genus two to
Hidalgo's example in genus $17$, which contains non-hyperelliptic examples of genus $5$ and $9$.
Finally, we apply Theorem \ref{class} to the classification of pseudoreal plane quintics.
\begin{corollary} \label{quinticaaa} Two finite groups $G$ and $\overline{G}$ are
the conformal and full automorphism groups of a pseudoreal plane quintic $X$ if and only
if $G={\rm Aut}^{+}(X)$ and $\overline{G}={\rm Aut}^{\pm}(X)$ are in a row of Table \ref{pseudoquintic}.
\begin{table}[!h]
\centering\renewcommand{\arraystretch}{1.3}\setlength{\tabcolsep}{2pt}
\medskip
\begin{tabular}{| c | c | c | c | p{3.5cm} |}
\hline
$\rm Aut^{+}(X)$ & Fuchsian signature & $\rm Aut^{\pm}(X)$ & NEC signature & Generating Vector \\ \hline
$C_4$ & $(0;4^6)$ & $C_8$ & $(1;-;[4^3])$ & $(a;a^2,a^2,a^2)$ \\ \hline
$C_2$ & $(2;[2^6])$ & $C_4$ & $(3;-;[2^3])$ & $(a,a,a;a^2,a^2,a^2)$ \\ \hline
\end{tabular}
\vspace{0.5cm}
\caption{Possible automorphism groups for pseudoreal plane quintics}
\label{pseudoquintic}
\end{table}
\end{corollary}
\vspace{-0.3cm}
\begin{proof}
In \cite{BaBa16} the authors classified the automorphism groups of plane quintics defined
over an algebraically closed field of characteristic zero, giving a smooth plane model
for every group.
This classification, together with Table \ref{table6}, gives a list of the possible conformal and
full groups of pseudoreal plane quintics. The possible conformal automorphisms groups
are $D_5, C_2, C_4$:
\begin{enumerate}
\item ${\rm Aut}(X)\cong D_{5}$, with group generators $\sigma_1([x:y:z])= [x:\epsilon_5y:\epsilon_5^2z]$ and $\sigma_2([x:y:z])= [z:y:x]$,
with a smooth plane model $$x^5+y^5+z^5+ax^2yz^2+bxy^3z=0,$$
where $a,b\in\cc$ with $(a,b)\neq (0,0)$. In this case the covering $X\longrightarrow X\slash {\rm Aut}(X)$ has signature $(0;[2^6])$.
\item ${\rm Aut}(X)\cong C_4$, with generator $\sigma([x:y:z])=[x:y:\epsilon_4z]$, with smooth plane model
$$z^4L_{1,z}+L_{5,z}=0,$$
where $L_{i,z}\in \cc[x,y]$ are homogeneous polynomials with $\deg(L_{i,z})=i$.
In this case the covering $X\longrightarrow X\slash {\rm Aut}(X)$ has signature $(0;[4^6])$.
\item ${\rm Aut}(X)\cong C_2$, with generator $\sigma([x:y:z])=[x:y:-z]$, with smooth plane model
$$z^4L_{1,z}+z^2L_{3,z}+L_{5,z}=0,$$
where $L_{i,z}\in \cc[x,y]$ are homogeneous polynomials with $\deg(L_{i,z})=i$.
In this case the covering $X\longrightarrow X\slash {\rm Aut}(X)$ has signature $(2;[2^6])$.
\end{enumerate}
We now prove that case (i) cannot occur.
Suppose we have an isomorphism $f$ between $X$ and $\bar X$.
Then $f$ must have a representation as a $3\times 3$ matrix (see \cite{BaBa16})
and it must send the fixed points of the cyclic subgroup of order $5$ of ${\rm Aut}(X)$ to the same points
for $\bar X$. Such cyclic subgroup is generated by $\sigma_1$ for both curves and its fixed points are the fundamental points,
so the matrix representing $f$ must be the composition of a permutation matrix with a diagonal matrix.
Since $[x:y:z]\mapsto [z:y:x]$ is the only non-trivial permutation allowed, then a matrix representation
for $f$, up to composing with an automorphism of $X$, is the following
$$\begin{bmatrix}
1 & 0 & 0 \\[0.3em]
0 & \epsilon_5^m & 0 \\[0.3em]
0 & 0 & \epsilon_5^n
\end{bmatrix},
$$
where $\epsilon_5$ is a primitive fifth root of unity and $m,n$ are integers.
Since $(J\circ f)^2={\rm id}$, the curve $X$ admits an anticonformal involution, thus it is not pseudoreal.
\end{proof}
\section{A Magma program} \label{prog}
In this section we will present the Magma \cite{Magma} program that we used to carry out the classification of full
automorphism groups of pseudoreal Riemann surfaces done in the previous section.
Our program relies on Jennifer Paulhus' program {\bf GenVectMagmaToGenus20}, which is available at
\begin{center}
\url{http://www.math.grinnell.edu/~paulhusj/monodromy.html}
\end{center}
and is based on the paper \cite{Pau15}.
To run our program one first needs to download the packages
{\bf genvectors.m}, {\bf searchroutines.m}, {\bf GenVectMagmaToGenus20}
and save all of them in the same folder.
To access the data in Paulhus' program, for example for genus $4$, one has to write in Magma
\begin{verbatim}
load "genvectors.m";
load "searchroutines.m";
L:=ReadData("Fullg20/grpmono04", test);
\end{verbatim}
where ``test'' is a function taking as input a permutation group,
a signature (as a vector) and a generating vector (as a vector whose entries are permutations).
For example when using the following function, the program gives the list of all triples $(G,s,v)$, where $G$ is a group of order bigger than $7$ acting
on a Riemann surface of genus $4$ with signature $s$ and generating vector $v$.
\begin{verbatim}
test:=function(G,s,Lmonod)
return Order(G) gt 7;
end function;
\end{verbatim}
Thus this program allows to analyse the automorphisms groups of all Riemann surfaces up to genus $20$,
looking for certain properties specified by the function ``test''.\footnote{The data was computed using Magma V2.19-9. In newer versions of Magma an error may be returned for some genera (at least $5$ and $9$).
See the warning in J. Paulhus' webpage.}
Observe that $G$ is not necessarily the complete automorphism group of some Riemann surface of the chosen genus (this will be one of the main issues in our program).
Given a genus $2\leq g\leq 20$ our program describes the automorphism group of all
the pseudoreal Riemann surfaces of genus $g$. More precisely it gives the full automorphism group,
the conformal automorphism group and its Fuchsian signature.
For each entry of the output there exists a pseudoreal Riemann surface of genus $g$
with such properties.
To run the program one needs to download the file \textbf{pseudoreal.m}, which is available here
\begin{center}
\url{https://www.dropbox.com/s/k786b7a2vrmt22i/pseudoreal.m?dl=0}
\end{center}
and save it in the same folder as Paulhus' programs.
The program (again in the case of genus $4$) consists of the following lines
\begin{verbatim}
load "genvectors.m";
load "searchroutines.m";
load "pseudoreal.m";
L:=ReadData("Fullg20/grpmono04", testpr);
PR(L);
\end{verbatim}
The output is a list whose entries are of the form $\langle \langle \,,\,\rangle, \langle \,,\,\rangle, [\dots]\rangle$,
where the first bracket contains the ID number of the full automorphism group, the second bracket contains the ID number
of the conformal automorphism group and the final sequence is the corresponding Fuchsian signature (the first entry is the genus of the quotient
by the conformal automorphism group).
A description of each of the functions contained in pseudoreal.m can be found in \cite{Rey16}.
\end{document}
| 49,096
|
Senior Moves KC, studied Success in Senior Real Estate at the Senior Real Estate Institute receiving our Certified Senior Housing Professional designation.
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Recent Comments
| 369,365
|
TITLE: Proving $\int_{I} f \geq 0$ for an integrable function that is positive at rationals
QUESTION [1 upvotes]: Let $I$ be a generalized rectangle in $\mathbb{R}^{n}$ and suppose $f :I \rightarrow \mathbb{R}$ is integrable. Suppose $f(x) \geq 0$ if $x$ is a point in $I$ with a rational component. Prove $\int_{\mathbf{I}} f \geq 0$ holds.
My try:
By the integrability of $f$,
$$\begin{align*} \int_{\mathbf{I}} f &= \text{sup}\{L(f, \mathbf{P}) \mid \mathbf{P} \text{ is a partition of the generalized rectangle } \mathbf{I}\} \\
&= \text{inf}\{U(f, \mathbf{P}) \mid \mathbf{P} \text{ is a partition of the generalized rectangle } \mathbf{I}\} \end{align*}.$$
Since $f(x) \geq 0$ if a component of $x$ is in $\mathbb{Q}$, for any partition $\mathbf{P} = (P_1, P_2 \ldots, P_k)$ and generalized rectangle $J$, we get
$$U(f, P_{k}) = \text{sup}\{f(x) \mid x \in J\} \cdot \text{vol } \mathbf{J} \geq 0.$$
So $\text{inf}\{U((f, P)\}, \text{sup}\{L((f, P)\} \geq 0$
and result follows.
Is my proof orrect?
REPLY [2 votes]: Note that as you have rightly pointed out, for any partition, $P$, the upper sum and lower sums are equal and upper sum is non-negative. Thus,
$$\int_I f = \lim_{\|P\|\to 0} \mathcal{U}(P,f).$$
Since each term in the limit is non-negative, the limit is non-negative. Hence $$\int_I f \geq 0.$$
| 52,703
|
\begin{document}
\newtheorem{oberklasse}{OberKlasse}
\newtheorem{definition}[oberklasse]{Definition}
\newtheorem{lemma}[oberklasse]{Lemma}
\newtheorem{proposition}[oberklasse]{Proposition}
\newtheorem{theorem}[oberklasse]{Theorem}
\newtheorem{corollary}[oberklasse]{Corollary}
\newtheorem{remark}[oberklasse]{Remark}
\newtheorem{example}[oberklasse]{Example}
\newcommand{\R}{\mathbbm{R}}
\newcommand{\N}{\mathbbm{N}}
\newcommand{\Z}{\mathbbm{Z}}
\newcommand{\C}{\mathbbm{C}}
\newcommand{\mc}{\mathcal}
\newcommand{\eps}{\varepsilon}
\renewcommand{\phi}{\varphi}
\newcommand{\cA}{{\mathcal A}}
\allowdisplaybreaks[1]
\title{Generalized Gearhart-Koshy acceleration for the Kaczmarz method}
\author{J.\ Rieger}
\date{\today}
\maketitle
\begin{abstract}
The Kaczmarz method is an iterative numerical method for solving
large and sparse rectangular systems of linear equations.
Gearhart, Koshy and Tam have developed an acceleration technique
for the Kaczmarz method that minimizes the distance to the desired solution
in the direction of a full Kaczmarz step.
The present paper generalizes this technique to an acceleration scheme that
minimizes the Euclidean norm error over an affine subspace spanned by a number
of previous iterates and one additional cycle of the Kaczmarz method.
The key challenge is to find a formulation in which all parameters of the
least-squares problem defining the unique minimizer are known,
and to solve this problem efficiently.
A numerical experiment demonstrates that the proposed affine search has the
potential to clearly outperform the Kaczmarz and the randomized Kaczmarz
methods with and without the Gearhart-Koshy/Tam line-search.
\end{abstract}
\noindent\textbf{MSC Codes:} 65F10, 65F20, 68W20\\
\noindent\textbf{Keywords:} Kaczmarz method, randomized Kaczmarz method,
acceleration, least-squares problem, computerized tomography
\section{Introduction}
The Kaczmarz method for solving systems of linear equations was initially described
and analyzed in \cite{Kaczmarz}.
It was later rediscovered in the paper \cite{Gordon:70} in the context of computerized
tomography problems, where it was used with great success.
Being a row-action method, it tends to behave well when applied to large and sparse
rectangular linear systems, see \cite{Censor}.
The performance of the Kaczmarz method depends on the fixed order in which the
method cycles through the rows of the linear system.
The randomized Kaczmarz method avoids the selection of a disadvantageous
order by selecting the rows at random.
It was proved in \cite{Strohmer} that this approach yields linear convergence
in expectation with a rate corresponding to the geometry of the problem.
\medskip
Recently, there has been a strong emphasis on the development of acceleration schemes
for the randomized Kaczmarz method.
Some are based on splittings of the set of rows into a priori known well-conditioned
blocks, see \cite{Necoara} and \cite{Needell}, while others are based on Motzkin
acceleration see \cite{Motzkin}.
The latter method selects the next row of the linear system corresponding
to the largest component of the current residual instead of cycling through the rows
in a given order.
Novel probabilistic variants of this approach select the next row with
a probability distribution proportional to or otherwise derived from the current
residual, see \cite{Bai}, \cite{Haddock}, \cite{Haddock:Needell}, \cite{Steinerberger}
and the references therein.
In a sense, these methods are greedy algorithms that aim at decreasing the
residual as fast as possible.
\medskip
The line-search proposed in \cite{Gearhart} by Gearhart and Koshy for homogeneous
and recently in \cite{Tam} by Tam for inhomogeneous linear systems in the
context of the deterministic Kaczmarz method is pursuing a greedy
strategy that is diametrically opposed to Motzkin acceleration:
It uses one full cycle of the Kaczmarz method as a search direction and minimizes
the Euclidean norm error (instead of the residual) over the corresponding line.
This is achieved using only information that is explicitly known at runtime, which
means that this acceleration is computationally inexpensive.
Example 3.24 in \cite{Bauschke} shows that in pathological situations, the Kaczmarz
method with this line-search can be slower than the plain Kaczmarz method, while a
straight-forward modification of the convergence proof in \cite{Bregman} reveals that
it is necessarily convergent.
\medskip
The aim of this paper is to generalize the Gearhart-Koshy line-search
to an acceleration scheme that minimizes the Euclidean norm error over an
affine subspace spanned by a number of previous iterates and one additional
cycle of the Kaczmarz method.
This acceleration strategy is not limited to the deterministic Kaczmarz method,
but can be applied to the randomized Kaczmarz method as well.
The key challenge is to find a formulation in which all parameters of the
least-squares problem defining the unique minimizer are known,
and to solve this problem efficiently.
It turns out that this is possible in linear time because of the particular
structure of the problem.
A numerical experiment provided in the final section of the paper demonstrates
that the proposed affine search has the potential to clearly outperform
the Kaczmarz and the randomized Kaczmarz methods with and without the
Gearhart-Koshy line-search.
\medskip
Finally, we would like to point out that the proposed method does not compete with the
above-mentioned accelerations based on splittings and Motzkin acceleration,
but it can in principle be applied to further enhance these and other methods
based on successive projections.
\section{Preliminaries}
Throughout this paper, we consider a matrix
\[A=(a_1,\ldots,a_m)^T\in\R^{m\times n}\]
with rows $a_j\in\R^n\setminus\{0\}$ and a vector $b\in\mc{R}(A)$ in the range of $A$,
and we consider the projectors
\[P_j:\R^n\to\R^n,\quad
P_j(x):=(I-\frac{a_ja_j^T}{\|a_j\|^2})x+\frac{b_j}{\|a_j\|^2}a_j,\quad j=1,\ldots,m,\]
which project any point $x\in\R^n$ to the affine subspaces
\[H_j:=\{z\in\R^n:a_j^Tz=b_j\},\quad j=1,\ldots,m.\]
Their compositions
\[P(x):=(P_m\circ\ldots\circ P_1)(x)\]
constitute a full cycle of the Kaczmarz method.
It is well-known that for any $x_0\in\R^n$, we have
$\lim_{k\to\infty}P^k(x_0)\in A^{-1}b$,
see e.g.\ \cite{Tanabe}.
\medskip
When determining the computational complexity of the Kacz\-marz method and the
accelerated variants discussed in this paper, we will denote the number of nonzero
elements of the matrix $A$ by $\nnz(A)>0$.
As the matrix $A$ is large in typical applications, we will assume that scalar
quantities such as the norms $\|a_j\|^2$ can be stored, but not the normalized rows
$a_j/\|a_j\|$.
In this situation, we can carry out one Kaczmarz cycle with $4\nnz(A)+m$ flops.
\medskip
\begin{algorithm}\label{plain:Kaczmarz}
\caption{Kaczmarz method,
originally proposed in \cite{Kaczmarz}\\
(complexity: $4\nnz(A)+m$ flops per cycle)}
\KwIn{
$A\in\R^{m\times n}$,
$b\in\R^m$,
$x_0\in\R^n$
}
\For{$k=0$ \KwTo $\infty$}{
$x_{k+1}\gets P(x_k)$\;
}
\end{algorithm}
Inspired by the papers \cite{Gearhart} and \cite{Tam}, we wish to explore
how the residual
\begin{equation}\label{res:def}
r(x):=\begin{pmatrix}
(a_1^Tx-b_1)/\|a_1\|\\
(a_2^TP_1(x)-b_2)/\|a_2\|\\
\vdots\\
(a_m^TP_{m-1}\circ\ldots\circ P_1(x)-b_m)/\|a_m\|
\end{pmatrix}
\end{equation}
can be used to speed up the Kaczmarz iteration.
Note that the quantities required for its computation are explicitly calculated
in a cycle of the Kaczmarz method.
\medskip
We begin by exploring the properties of the residual $r$,
which measures the reduction of the square distance to any solution
of the linear system in one Kaczmarz cycle and encodes information
on the angle between the vectors $x^*-x$ and $P(x)-x$.
\begin{lemma} \label{improvement}
Let $x^*\in A^{-1}b$, and let $x\in\R^n$ be arbitrary.
Then we have
\begin{align}
&\|r(x)\|^2+\|P(x)-x^*\|^2=\|x-x^*\|^2,\label{imp:1}\\
&\|r(x)\|^2+\|P(x)-x\|^2=2(x-x^*)^T(x-P(x)).\label{imp:2}
\end{align}
\end{lemma}
\begin{proof}
Since
\begin{align*}
\langle P_j(x)-x,P_j(x)-x^*\rangle
&=\langle\frac{b_j-a_j^Tx}{\|a_j\|^2}a_j,x+\frac{b_j-a_j^Tx}{\|a_j\|^2}a_j-x^*\rangle\\
&=\frac{b_j-a_j^Tx}{\|a_j\|^2}\Big(a_j^Tx+(b_j-a_j^Tx)-b_j\Big)=0,
\end{align*}
we may use the Pythagorean theorem to compute
\begin{align*}
\|x-x^*\|^2
=\|(x-P_j(x))+(P_j(x)-x^*)\|^2
=\|x-P_j(x)\|^2+\|P_j(x)-x^*\|^2.
\end{align*}
Statement \eqref{imp:1} follows from the above identity
successively applied to $x$, $P_1(x)$, $P_2\circ P_1(x)$ etc.\ in lieu of $x$,
and from the definitions of $P$ and $r$.
Now the polarization identity yields
\begin{align*}
\|r(x)\|^2+\|x-P(x)\|^2
&=\|x-x^*\|^2-\|P(x)-x^*\|^2+\|x-P(x)\|^2\\
&=2(x-x^*)^T(x-P(x)).
\end{align*}
\end{proof}
The mapping $r$ behaves indeed like a residual.
\begin{lemma} \label{res:equiv}
The following statements are equivalent:
\begin{itemize}
\item [a)] We have $Ax=b$.
\item [b)] We have $P(x)=x$.
\item [c)] We have $r(x)=0$.
\end{itemize}
\end{lemma}
\begin{proof}
If statement a) holds, then
\[P_j(x)=(I-\frac{a_ja_j^T}{\|a_j\|^2})x+\frac{b_j}{\|a_j\|^2}a_j=x,\quad
j=1,\ldots,m,\]
which implies $P(x)=x$.
Assume that statement b) holds, and let $x^*\in\R^n$ be any point with $Ax^*=b$.
If $r(x)\neq 0$, then statement \eqref{imp:1} gives
\[\|P(x)-x^*\|^2<\|x-x^*\|^2,\]
which contradicts $P(x)=x$.
Hence statement c) holds.
If statement c) holds, then statement a) follows by induction.
We clearly have $a_1^Tx=b_1$.
If $a_i^Tx=b_i$ holds for $i=1,\ldots,j$, then $P_i(x)=x$ holds for $i=1,\ldots,j$,
and
\[a_{j+1}^Tx-b_{j+1}=a_{j+1}^TP_j\circ\ldots\circ P_1(x)-b_{j+1}=r_{j+1}(x)=0.\]
By induction, we obtain $Ax=b$.
\end{proof}
\begin{remark}
A straight-forward modification of the convergence proof
in \cite{Bregman} reveals that any sequence $(x_k)_{k\in\N}$ satisfying
\begin{equation}\label{better:than:K}
\|x_{k+1}-x^*\|^2\le\|P(x_k)-x^*\|^2\quad\forall\,k\in\N,\ x^*\in A^{-1}b,
\end{equation}
converges to a solution $x^*\in A^{-1}b$, and it is clear that the typical
error estimates for cyclic projection-type methods as in Corollary 9.34 in
\cite{Deutsch} for the sequence $(P^k(x_0))_{k\in\N}$ also hold for the
sequence $(x_k)_{k\in\N}$.
Hence we will focus on generating a sequence with the above property
\eqref{better:than:K} by minimizing the errors $\|x_{k+1}-x^*\|^2$
in affine search spaces at a relatively small computational cost.
\end{remark}
\section{Acceleration by line-search}
In this section, we recover the step-size from Theorem 4.1 in \cite{Tam}
with a straight-forward geometric argument.
In addition, we quantify the error reduction in terms
of the difference between statements \eqref{imp:1} and
\eqref{gain:linesearch}.
All quantities involved in these formulas are known at runtime.
\begin{theorem}\label{greedy}
Let $x^*\in A^{-1}b$, and let $x\in\R^n$ with $P(x)\neq x$.
Then we have
\begin{align}
s^*:=\argmin_{s\in\R}\|(1-s)x+sP(x)-x^*\|^2
=\frac12+\frac{\|r(x)\|^2}{2\|P(x)-x\|^2},\nonumber\\
\|x-x^*\|^2-\|(1-s^*)x+s^*P(x)-x^*\|^2
=\frac{(\|r(x)\|^2+\|P(x)-x\|^2)^2}{4\|P(x)-x\|^2}.\label{gain:linesearch}
\end{align}
\end{theorem}
\begin{proof}
Using identity \eqref{imp:2}, we see that the strictly convex parabola
\begin{align*}
g(s)&=\|(1-s)x+sP(x)-x^*\|^2\\
&=\|x-x^*\|^2+2s(x-x^*)^T(P(x)-x)+s^2\|P(x)-x\|^2\\
&=\|x-x^*\|^2-s(\|r(x)\|^2+\|P(x)-x\|^2)+s^2\|P(x)-x\|^2
\end{align*}
has the unique minimum
\[s^*=\frac{\|r(x)\|^2+\|P(x)-x\|^2}{2\|P(x)-x\|^2}
=\frac12+\frac{\|r(x)\|^2}{2\|P(x)-x\|^2}.\]
The second statement follows from
\[g(s^*)=\|x-x^*\|^2-\frac{(\|r(x)\|^2+\|P(x)-x\|^2)^2}{4\|P(x)-x\|^2}.\]
\end{proof}
\begin{algorithm}\label{greedy:line:search}
\caption{Kaczmarz method with line-search,\\
originally proposed in \cite{Tam}\\
(complexity: $4\nnz(A)+3m+5n$ flops per cycle)}
\KwIn{
$A\in\R^{m\times n}$,
$b\in\R^m$,
$x_0\in\R^n$
}
\For{$k=0$ \KwTo $\infty$}{
compute $P(x_k)$ and $r(x_k)$ in Kaczmarz cycle from $x_k$\;
$d\gets P(x_k)-x_k$\;
$\delta\gets\|d\|^2$\;
\uIf{$\delta=0$}{\Return $x_k$\;}
$\rho\gets\|r(x)\|^2$\;
$s\gets\frac12+\frac{\rho}{2\delta}$\;
$x_{k+1}\gets x_k+sd$\;
}
\end{algorithm}
It is a straight-forward consequence of Theorem \ref{greedy}
that Algorithm \ref{greedy:line:search} is safe to use.
\begin{corollary}
Either Algorithm \ref{greedy:line:search} terminates in finite
time and returns an iterate $x_k\in A^{-1}b$, or it generates
a well-defined sequence $(x_k)_k$ that has the property \eqref{better:than:K}
and, for all $x^*\in A^{-1}b$, satisfies the identities
\begin{align}
&x_{k+1}=\argmin_{\xi\in\aff(x_k,P(x_k))}\|\xi-x^*\|^2,\label{ls:opt}\\
&\|x_k-x^*\|^2-\|x_{k+1}-x^*\|^2
=\frac{(\|r(x_k)\|^2+\|P(x_k)-x_k\|^2)^2}{4\|P(x_k)-x_k\|^2}.\label{concrete:decay:ls}
\end{align}
\end{corollary}
\begin{proof}
If Algorithm \ref{greedy:line:search} terminates after $k\in\N$ steps
and returns an iterate $x_k\in\R^n$, then the stopping criterion implies
$P(x_k)=x_k$, and Lemma \ref{res:equiv} yields $Ax_k=b$.
Otherwise, all expressions in Algorithm \ref{greedy:line:search}
are well-defined.
By Theorem \ref{greedy}, formulas \eqref{ls:opt} and \eqref{concrete:decay:ls} hold,
and formula \eqref{ls:opt} implies \eqref{better:than:K}.
\end{proof}
\section{Acceleration by affine search}
It is possible to extend the above line-search to a search in an affine subspace
spanned by several previous iterates and the latest Kaczmarz cycle,
which improves the local error reduction.
Again, all required quantities and the exact reduction are computable at runtime.
\medskip
We begin by proving a simple geometric observation that will give meaning
to the stopping criterion of the accelerated iteration.
\begin{lemma} \label{no:stop}
Let $x^*\in A^{-1}b$, and let
$x_1,\ldots,x_{\ell}\in\R^n$ be points such that the condition
\begin{equation}\label{last:opt}
x_\ell=\argmin_{\xi\in\aff(x_1,\ldots,x_\ell)}\|\xi-x^*\|^2
\end{equation}
holds.
If we have $P(x_\ell)\in\aff(x_1,\ldots,x_\ell)$, then we also have
$\|r(x_\ell)\|^2=0$, $P(x_\ell)=x_\ell$ and $Ax_\ell=b$.
\end{lemma}
\begin{proof}
By statement \eqref{imp:1}, we have
\[\|P(x_\ell)-x^*\|^2=\|x_\ell-x^*\|^2-\|r(x_\ell)\|^2.\]
If we have $P(x_\ell)\in\aff(x_1,\ldots,x_\ell)$, then condition \eqref{last:opt}
yields $\|r(x_\ell)\|^2=0$, and Lemma \ref{res:equiv} implies that $Ax_\ell=b$.
\end{proof}
The following result provides a characterization of the minimizer
\[\argmin_{\xi\in\aff(x_1,\ldots,x_\ell,P(x_\ell))}\|\xi-x^*\|^2\]
that does not use the unknown solution $x^*$ explicitly.
We formulate and prove this theorem for vectors indexed $x_1,\ldots,x_\ell$
to keep the notation simple, and we will use it later (see Corollary \ref{alg:2:cor})
for a varying number of vectors and varying indexation.
\begin{theorem} \label{affine:search:thm}
Let $x^*\in A^{-1}b$, let $x_1,\ldots,x_{\ell}\in\R^n$
be affinely independent points with \eqref{last:opt}
and $P(x_\ell)\notin\aff(x_1,\ldots,x_\ell)$.
Consider the matrices
\[V:=(x_1-x_\ell,\ldots,x_{\ell-1}-x_\ell)\in\R^{n\times(\ell-1)},\quad
M:=(V,P(x_\ell)-x_\ell)\in\R^{n\times\ell}\]
and define
\[\gamma:=\frac12(\|r(x_\ell)\|^2+\|P(x_\ell)-x_\ell\|^2).\]
Then the minimizer
\begin{equation} \label{min:s}
s^*:=\argmin_{s\in\R^\ell}\|x_\ell+Ms-x^*\|^2
\end{equation}
is the unique solution of the linear system
\begin{equation}\label{lgs}
M^TMs=\gamma e_\ell^\ell,
\end{equation}
where $e_\ell^\ell\in\R^\ell$ is the $\ell$-th unit vector, and we have
\begin{equation}\label{better}
\|x_\ell-x^*\|^2-\|x_\ell+Ms^*-x^*\|^2
=\gamma s_\ell^*
=\gamma^2\frac{\det(V^TV)}{\det(M^TM)}.
\end{equation}
\end{theorem}
\begin{remark}
For an interpretation of the identity \eqref{better}, it is
instructive to have a look at the case $\ell=2$.
Elementary computations show that whenever $P(x_2)\notin\aff(x_1,x_2)$,
the minimizer
\[s^*:=\argmin_{s\in\R^2}\|x_2+s_1(x_1-x_2)+s_2(P(x_2)-x_2)-x^*\|^2\]
satisfies
\begin{align*}
&\|x_2-x^*\|^2-\|x_2+s_1^*(x_1-x_2)+s_2^*(P(x_2)-x_2)-x^*\|^2\\
&=(1-\cos^2\angle(x_1-x_2,P(x_2)-x_2))^{-1}
\frac{(\|r(x_2)\|^2+\|P(x_2)-x_2\|^2)^2}{4\|P(x_2)-x_2\|^2}.
\end{align*}
Comparing with Theorem \ref{greedy}, we see that the planar search
outperforms the line-search by a factor $(1-\cos^2\angle(x_1-x_2,P(x_2)-x_2))^{-1}$.
\end{remark}
\begin{proof}
Since the vectors $x_1,\ldots,x_\ell,P(x_\ell)$ are affinely independent,
the Gram\-ian matrices $V^TV$ and $M^TM$ are positive definite.
The first derivatives of the strictly convex quadratic function
\[g(s):=\|x_\ell+Ms-x^*\|^2\]
are given by
\begin{align*}
\frac{dg}{ds_j}(s)&=2\Big(x_\ell+Ms-x^*\Big)^T(x_j-x_\ell),\quad
j=1,\ldots,\ell-1,\\
\frac{dg}{ds_\ell}(s)&=2\Big(x_\ell+Ms-x^*\Big)^T(P(x_\ell)-x_\ell).
\end{align*}
Using statements \eqref{imp:2} and \eqref{last:opt}, we see that the unique minimizer
$s^*$ of $g$ solves the linear equations
\begin{align}
&(x_j-x_\ell)^TMs=\langle x^*-x_\ell,x_j-x_\ell\rangle=0,\quad
j=1,\ldots,\ell-1,\label{ortho}\\
&(P(x_\ell)-x_\ell)^TMs=\langle x^*-x_\ell,P(x_\ell)-x_\ell\rangle=\gamma,
\label{not:ortho}
\end{align}
which are subsumed in the linear system \eqref{lgs}.
Using Cramer's rule, we can express
\[s_\ell^*
=\frac{\det\begin{psmallmatrix}
\langle x_1-x_\ell,x_1-x_\ell\rangle&\ldots&\langle x_1-x_\ell,x_{\ell-1}-x_\ell\rangle&0\\
\vdots&&\vdots&\vdots\\
\langle x_{\ell-1}-x_\ell,x_1-x_\ell\rangle&\ldots&\langle x_{\ell-1}-x_\ell,x_{\ell-1}-x_\ell\rangle&0\\
\langle P(x_\ell)-x_\ell,x_1-x_\ell\rangle&\ldots&\langle P(x_\ell)-x_\ell,x_{\ell-1}-x_\ell\rangle&\gamma
\end{psmallmatrix}}
{\det(M^TM)}
=\gamma\frac{\det(V^TV)}{\det(M^TM)}.\]
From system \eqref{lgs}, we infer
\[\|Ms^*\|^2=(s^*)^TM^TMs^*=\gamma s_\ell^*,\]
and together with statements \eqref{ortho} and \eqref{not:ortho}, we conclude that
\begin{align*}
g(s^*)&=\|x_\ell+Ms^*-x^*\|^2
=\|x_\ell-x^*\|^2+2(s^*)^TM^T(x_\ell-x^*)+\|Ms^*\|^2\\
&=\|x_\ell-x^*\|^2+2s_\ell^*(P(x_\ell)-x_\ell)^T(x_\ell-x^*)+\gamma s_\ell^*
=\|x_\ell-x^*\|^2-\gamma s_\ell^*.
\end{align*}
\end{proof}
Algorithm \ref{basic:affine:search} uses Theorem \ref{affine:search:thm}
after every step to reduce the size of the error $\|x_{k+1}-x^*\|^2$
in statement \eqref{better:than:K}.
\begin{remark}\label{rem:affine}
a) The parameter $\ell\in\N_1$ in Algorithm \ref{basic:affine:search} controls
how many of the previous iterates are used to span the affine search space.
When $\ell=1$, then Algorithm \ref{basic:affine:search} reduces to
Algorithm \ref{greedy:line:search}.
When $\ell>1$, the algorithm has a startup phase in which it grows the affine
basis of the search space, so that
\[V_0=[\ ],\quad
V_1=(x_0-x_1),\quad\ldots,\quad
V_{\ell-1}=(x_0-x_{\ell-1},\ldots,x_{\ell-2}-x_{\ell-1}).\]
After the startup phase, the algorithm keeps the latest $\ell$ iterates
and discards $x_{k-\ell}$, which gives
\[V_\ell=(x_1-x_\ell,\ldots,x_{\ell-1}-x_\ell),\quad
V_{\ell+1}=(x_2-x_{\ell+1},\ldots,x_\ell-x_{\ell+1}),\quad\ldots\]
When $\ell\ge n$, then the proof of Theorem \ref{alg:2:cor} reveals that
Algorithm \ref{basic:affine:search} terminates with an iterate $x_k\in A^{-1}b$,
$k\le n$.
\medskip
b) The computational complexity of one step of Algorithm \ref{basic:affine:search}
is composed in the following way:
\begin{enumerate}[(i)]
\item The Kaczmarz cycle requires $4\nnz(A)+m$ flops.
\item The computation of $d_k$, $\delta_k$ and $\rho_k$ requires $3n+2m$ flops.
\item Assembling $M$ requires $(k-j_k)n$ flops ($d_k$ is known).
\item Computing $M^TM$ requires $\frac12(k-j_k+1)(k-j_k+2)n$ flops.
\item Solving system \eqref{lgs} via LU factorization and forward and
backward substitution requires
$\frac23(k-j_k)^3+\frac72(k-j_k)^2+\frac56(k-j_k)$ flops.
\item Updating $x_k$ requires $2(k-j_k+1)n$ flops.
\end{enumerate}
After the startup phase, we have $k-j_k+1=\ell$, which gives a total computational complexity of roughly
\[4\nnz(A)+(3+3\ell+\frac12\ell^2)n+3m+\ell^3.\]
The acceleration comes at a considerable cost, mostly caused by the assembly
and by solving system \eqref{lgs}, which is not desirable.
\end{remark}
\begin{algorithm}\label{basic:affine:search}
\caption{Kaczmarz method with affine search\\
(for complexity see Remark \ref{rem:affine} part b)}
\KwIn{
$A\in\R^{m\times n}$,
$b\in\R^m$,
$x_0\in\R^n$,
$\ell\in\N_1$
}
\For{$k=0$ \KwTo $\infty$}{
compute $P(x_k)$ and $r(x_k)$ in Kaczmarz cycle from $x_k$\;
$d_k\gets P(x_k)-x_k$\;
$\delta_k\gets\|d_k\|^2$\;
\uIf{$\delta=0$}{\Return $x_k$\;}
$\rho_k\gets\|r(x_k)\|^2$\;
$\gamma_k\gets\frac12(\rho_k+\delta_k)$\;
$j_k\gets\max\{k-\ell+1,0\}$\;
$V_k\gets(x_{j_k}-x_k,\ldots,x_{k-1}-x_k)\in\R^{n\times(k-j_k)}$\;
assemble $M_k^TM_k$ from $M_k=(V_k,d_k)\R^{n\times(k-j_k+1)}$\;
solve $M_k^TM_ks_k=\gamma_k e_{k-j_k+1}^{k-j_k+1}$ for $s_k$\;
$x_{k+1}\gets x_k+M_ks_k$\;
}
\end{algorithm}
\begin{theorem} \label{alg:2:cor}
Let $x^*\in A^{-1}b$.
Either Algorithm \ref{basic:affine:search} terminates and returns an iterate
$x_k\in A^{-1}b$, or it generates
a well-defined sequence $(x_k)_k$ that satisfies
\begin{align}
&x_{k+1}=\argmin_{\xi\in\aff(x_{j_k},\ldots,x_k,P(x_k))}\|\xi-x^*\|^2,\label{cor:a}\\
&\|x_k-x^*\|^2-\|x_{k+1}-x^*\|^2=\gamma_k^2\frac{\det(V_k^TV_k)}{\det(M_k^TM_k)}
\label{cor:b}
\end{align}
for all $k\in\N$.
In particular, it has the property \eqref{better:than:K}.
\end{theorem}
\begin{proof}
We prove by induction that Algorithm \ref{basic:affine:search} either
returns a solution in finite time or generates a sequence such that
identities \eqref{cor:a} and \eqref{cor:b} and the following statements
hold for every $k\in\N$:
\begin{itemize}
\item [a)] The vectors $x_{j_k},\ldots,x_k$ are affinely independent.
\item [b)] We have $x_k=\argmin_{\xi\in\aff(x_{j_k},\ldots,x_k)}\|\xi-x^*\|^2$.
\item [c)] We have $P(x_k)\notin\aff(x_{j_k},\ldots,x_k)$.
\end{itemize}
If $k=0$, then properties a) and b) are trivially satisfied.
\medskip
Now assume that Algorithm \ref{basic:affine:search} has generated iterates
$x_0,\ldots,x_k\in\R^n$ with properties a), b).
\medskip
If Algorithm \ref{basic:affine:search} terminates and returns $x_k$,
then the stopping criterion implies that $P(x_k)=x_k$, and
Lemma \ref{res:equiv} implies $Ax_k=b$.
\medskip
If Algorithm \ref{basic:affine:search} does not terminate, then we have
$P(x_k)\neq x_k$.
Because of statement b), Lemma \ref{no:stop} implies statement c).
Since statements a), b) and c) hold for $k$, the vectors $x_{j_k},\ldots,x_k$
satisfy all assumptions of Theorem \ref{affine:search:thm}.
The linear system \eqref{lgs} with the matrix
\[M=(x_{j_k}-x_k,\ldots,x_{k-1}-x_k,P(x_k)-x_k)\]
possesses a unique solution $s^*\in\R^{k-{j_k}+1}$, and the iterate
$x_{k+1}:=x_k+Ms^*$ is well-defined.
Because of statements a) and c), we have
\[\det(V^TV)>0\quad\text{and}\quad\det(M^TM)>0,\]
so by statement \eqref{better}, we have $s_{k-{j_k}+1}^*\neq 0$.
Combining this fact with statements a) and c) yields statement a)
with $k+1$ in lieu of $k$.
Statement \eqref{min:s} implies statement \eqref{cor:a} and, since
\[\aff(x_{j_{k+1}},\ldots,x_k,x_{k+1})\subset\aff(x_{j_k},\ldots,x_k,P(x_k)),\]
also statement b) for $k+1$ in lieu of $k$.
In addition, statement \eqref{better} implies the identity \eqref{cor:b}.
\bigskip
\end{proof}
\section{Efficient updating}
The goal of this section is to simplify the solution of the linear system \eqref{lgs},
which must be solved after every Kaczmarz cycle to determine $x_{k+1}$ from
the previous iterates and the vector $P(x_k)$.
\medskip
Our first result shows that updating the submatrix $V^TV$ of the matrix
$M^TM$ from one iteration to another is straight-forward.
\begin{lemma} \label{update}
In the situation of Theorem \ref{affine:search:thm},
and denoting $x_{\ell+1}:=x_\ell+Ms^*$, we have
\begin{align*}
&\langle x_i-x_{\ell+1},x_j-x_{\ell+1}\rangle=(V^TV)_{ij}+\gamma s_\ell^*,
&&1\le i,j\le\ell-1,\\
&\langle x_i-x_{\ell+1},x_{\ell}-x_{\ell+1}\rangle=\gamma s_\ell^*,
&&1\le i\le\ell.
\end{align*}
\end{lemma}
\begin{proof}
We can express
\[x_i-x_\ell=Me_i,\quad i=1,\ldots,\ell-1.\]
For $1\le i,j\le\ell-1$, we use the identity \eqref{lgs} to obtain
\begin{align*}
&\langle x_i-x_{\ell+1},x_j-x_{\ell+1}\rangle
=\langle x_i-x_\ell-Ms^*,x_j-x_\ell-Ms^*\rangle\\
&=\langle Me_i-Ms^*,Me_j-Ms^*\rangle\\
&=e_i^TM^TMe_j-e_i^TM^TMs^*-e_j^TM^TMs^*+(s^*)^TM^TMs^*\\
&=(M^TM)_{ij}+\gamma s_\ell^*
=(V^TV)_{ij}+\gamma s_\ell^*
\end{align*}
For $1\le i<\ell$, we compute
\begin{align*}
&\langle x_i-x_{\ell+1},x_\ell-x_{\ell+1}\rangle
=\langle x_i-x_\ell-Ms^*,-Ms^*\rangle\\
&=\langle Me_i-Ms^*,-Ms^*\rangle
=-e_i^TM^TMs^*+(s^*)^TM^TMs^*=\gamma s_\ell^*,
\end{align*}
and we also obtain
\[\langle x_\ell-x_{\ell+1},x_\ell-x_{\ell+1}\rangle
=(s^*)^TM^TMs^*
=\gamma s_\ell^*.\]
\end{proof}
We will see (in the proof of Theorem \ref{algo:3:works}) that the matrices $V^TV$
generated by Algorithms \ref{basic:affine:search} and \ref{greedy:affine:search}
have the structure of the matrix $B$ defined below with known coefficients $\alpha_j$.
\begin{lemma} \label{ringed}
Let $\alpha\in\R^n$, and let
$f_1^n,\ldots,f_n^n\in\R^n$ be given by $f_j^n=\sum_{i=1}^je_i^n$,
where $e_i^n\in\R^n$ denotes the $i$-th unit vector.
Then the matrix
\[B:=\sum_{j=1}^n\alpha_jf_j^n(f_j^n)^T\]
has the structure
\[B=\begin{pmatrix}
\sum_{i=1}^n\alpha_i&\sum_{i=2}^n\alpha_i&\cdots&\sum_{i=n}^n\alpha_i\\
\sum_{i=2}^n\alpha_i&\sum_{i=2}^n\alpha_i&&\vdots\\
\vdots&&\ddots&\vdots\\
\sum_{i=n}^n\alpha_i&\cdots&\cdots&\sum_{i=n}^n\alpha_i
\end{pmatrix}.\]
If $\alpha_j\neq 0$ for $j=1,\ldots,n$, then
\[C:=\begin{pmatrix}
\alpha_1^{-1}&-\alpha_1^{-1}\\
-\alpha_1^{-1}&\alpha_1^{-1}+\alpha_2^{-1}&-\alpha_2^{-1}\\
&-\alpha_2^{-1}&\alpha_2^{-1}+\alpha_3^{-1}&-\alpha_3^{-1}\\
&&\ddots&\ddots&\ddots\\
&&&-\alpha_{n-2}^{-1}&\alpha_{n-2}^{-1}+\alpha_{n-1}^{-1}&-\alpha_{n-1}^{-1}\\
&&&&-\alpha_{n-1}^{-1}&\alpha_{n-1}^{-1}+\alpha_{n}^{-1}
\end{pmatrix}\]
is the inverse of the matrix $B$.
\end{lemma}
\begin{proof}
When $\alpha_j\neq 0$ for $j=1,\ldots,n$, then the matrix $C$ is well-defined.
Multiplying the matrices $B$ and $C$ yields the identity.
\end{proof}
In conjunction with Lemmas \ref{update} and \ref{ringed}, the next lemma shows
that the linear system \eqref{lgs} can be solved in linear time.
\begin{lemma} \label{rank:2}
Let $B\in\R^{n\times n}$ be invertible,
let $p\in\R^n$ and let $\delta>0$ and $\gamma\in\R$.
If the matrix
\[G:=\begin{pmatrix}B&p\\p^T&\delta\end{pmatrix}\]
is invertible, then we have $p^TB^{-1}p\neq\delta$, and the solution
of the linear system
\[Gx=\gamma e^n_n\]
is given by
\[G^{-1}\gamma e^n_n
=\frac{\gamma}{p^TB^{-1}p-\delta}\begin{pmatrix}B^{-1}p\\-1\end{pmatrix}.\]
\end{lemma}
\begin{proof}
Since $G$ is nonsingular, and since
\[\begin{pmatrix}B&p\\p^T&\delta\end{pmatrix}\begin{pmatrix}
B^{-1}p\\-1\end{pmatrix}=\begin{pmatrix}0\\p^TB^{-1}p-\delta\end{pmatrix}\]
holds, we have $p^TCp\neq\delta$, and the desired result follows.
\end{proof}
Lemmas \ref{update}, \ref{ringed} and \ref{rank:2} inspire Algorithm
\ref{greedy:affine:search}.
We require $\ell\ge 2$, because for $\ell=1$, when Algorithm \ref{basic:affine:search}
reduces to Algorithm \ref{greedy:line:search}, there is no data to be updated.
By $C(\alpha)$, we denote the matrix $C$ from Lemma \ref{ringed}
given by the parameter vector $\alpha$ and its dimension.
In the initial step of Algorithm \ref{greedy:affine:search},
the matrices $V_0$ and $C_0$ as well as the vectors $p_0$ and $q_0$
are empty and have to be ignored where they occur.
We split the solution $s_k$ of the linear system
$M_k^TM_ks_k=\gamma_ke_{k-j+1}$ into the vector $\overline{s_k}\in\R^{k-j}$
of the first several components and the last component $\underline{s_k}\in\R$
to exploit the structure of system \eqref{lgs}.
\begin{algorithm}\label{greedy:affine:search}
\caption{Kaczmarz method with enhanced affine search\\
(for complexity see Remark \ref{discuss:enhanced})}
\KwIn{
$A\in\R^{m\times n}$,
$b\in\R^m$,
$x_0\in\R^n$,
$\ell\in\N_2$
}
\For{$k=0$ \KwTo $\infty$}{
compute $P(x_k)$ and $r(x_k)$ in Kaczmarz cycle from $x_k$\;
$d_k\gets P(x_k)-x_k$\;
$\delta_k\gets\|d_k\|^2$\;
\uIf{$\delta_k=0$}{\Return $x_k$\;}
$\rho_k\gets\|r(x_k)\|^2$\;
$\gamma_k\gets\frac12(\rho_k+\delta_k)$\;
$j_k\gets\max\{k-\ell+1,0\}$\;
$V_k\gets(x_{j_k}-x_k,\ldots,x_{k-1}-x_k)\in\R^{n\times(k-{j_k})}$\;
$p_k\gets V_k^Td_k\in\R^{k-{j_k}}$\;
$C_k\gets C(\gamma_{j_k}\underline{s_{j_k}},\ldots,\gamma_{k-1}\underline{s_{k-1}})
\in\R^{(k-{j_k})\times(k-{j_k})}$\;
$q_k\gets C_kp_k
\in\R^{k-{j_k}}$\;
$\underline{s_k}\gets\frac{\gamma_k}{\delta-p_k^Tq_k}\in\R$\;
$\overline{s_k}\gets-\underline{s_k}q_k\in\R^{k-{j_k}}$\;
$x_{k+1}\gets x_k+V_k\overline{s_k}+\underline{s_k}d_k$\;
}
\end{algorithm}
\begin{theorem} \label{algo:3:works}
Algorithms \ref{basic:affine:search} and \ref{greedy:affine:search} generate
identical iterations.
\end{theorem}
\begin{proof}
We prove by induction that one of the following alternatives holds:
\begin{itemize}
\item [i)] Algorithms \ref{basic:affine:search}
and \ref{greedy:affine:search} both terminate in step $k$.
\item [ii)] We have
\[V_k^TV_k=\begin{cases}\sum_{i=1}^{k-j_k}\gamma_{j_k+i-1}\underline{s_{j_k+i-1}}f_i^{k-j_k}(f_i^{k-j_k})^T,&k>0\\\mathrm{the\ empty\ matrix}\ [\ ],&k=0,\end{cases}\]
where $f_1^{k-j_k},\ldots,f_{k-j_k}^{k-j_k}\in\R^{k-j_k}$
are given by $f_i^{k-j_k}=\sum_{h=1}^ie_h^{k-j_k}$,
and both algorithms compute identical $\gamma_k$, $s_k$ and $x_{k+1}$.
\end{itemize}
When $k=0$, both algorithms terminate if and only if $d_0=P(x_0)-x_0=0$.
Otherwise, both algorithms compute identical $\delta_0$, $\rho_0$ and $\gamma_0$,
and they both have $j_0=0$ and $V_0=[\ ]$.
Algorithm \ref{basic:affine:search} solves
\begin{equation}\label{l1}
\delta_0s_0=\|d_0\|^2s_0=M_0^TM_0s_0=\gamma_0,
\end{equation}
while Algorithm \ref{greedy:affine:search} has $p_0=[\ ]$, $C_0=[\ ]$ and $q_0=[\ ]$,
computes $\underline{s_0}=\delta_0^{-1}\gamma_0$ and sets $\overline{s_0}=[\ ]$.
Hence both algorithms generate identical $s_0\in\R$ and the same next iterate
\begin{equation}\label{l2}
x_1=x_0+s_0d_0\in\R^n.
\end{equation}
\medskip
Now assume that alternative ii) holds for $0,\ldots,k$.
Then both algorithms compute identical $d_{k+1}$ and $\delta_{k+1}$, and both
terminate and return $x_{k+1}$ if and only if $\delta_{k+1}=0$.
Otherwise, they compute identical $\rho_{k+1}$ and $\gamma_{k+1}$.
For the update $V_k\mapsto V_{k+1}$, we distinguish the following cases:
\begin{itemize}
\item [a)] When $k=0$, we have $j_0=j_1=0$ and $V_1=(x_0-x_1)$,
so using statements \eqref{l1} and \eqref{l2}, we find
\[V_1^TV_1=\|x_0-x_1\|^2=s_0^2\delta_0=\gamma_0s_0.\]
\item [b)] When $k>0$ and $j_{k+1}=j_k$, then we have $j_{k+1}=j_k=0$ as well as
$V_k=(x_0-x_k,\ldots,x_{k-1}-x_k)$ and
$V_{k+1}=(x_0-x_{k+1},\ldots,x_k-x_{k+1})$.
Lemma \ref{update} tells us that
\begin{align*}
&\langle x_i-x_{k+1},x_j-x_{k+1}\rangle=(V_k^TV_k)_{ij}+\gamma_k\underline{s_k},
&&0\le i,j\le k-1,\\
&\langle x_i-x_{k+1},x_k-x_{k+1}\rangle=\gamma_k\underline{s_k},
&&0\le i\le k.
\end{align*}
The induction hypothesis implies that
\begin{align*}
V_{k+1}^TV_{k+1}
&=\begin{pmatrix}V_k^TV_k&0\\0^T&0\end{pmatrix}
+\gamma_k\underline{s_k}\mathbbm{1}_{\R^{k+1}}\mathbbm{1}_{\R^{k+1}}^T\\
&=\sum_{i=1}^{k-j_k}\gamma_{j_k+i-1}\underline{s_{j_k+i-1}}
\begin{pmatrix}f_i^{k-j_k}\\0\end{pmatrix}((f_i^{k-j_k})^T,0)
+\gamma_k\underline{s_k}\mathbbm{1}_{\R^{k+1}}\mathbbm{1}_{\R^{k+1}}^T\\
&=\sum_{i=1}^{k+1-j_{k+1}}\gamma_{j_{k+1}+i-1}\underline{s_{j_k+i-1}}
f_i^{k+1-j_{k+1}}(f_i^{k+1-j_{k+1}})^T,
\end{align*}
where we have used that $\gamma_k=\gamma_{k+1}=0$ and
$\mathbbm{1}_{\R^{k+1}}=f_{k+1}^{k+1}$.
\item [c)] When $k>0$ and $j_{k+1}\neq j_k$, then we have $j_{k+1}=k-\ell+2$
and $j_k=k-\ell+1$, and we consider
$V_k=(x_{k-\ell+1}-x_k,\ldots,x_{k-1}-x_k)$ and
$V_{k+1}=(x_{k-\ell+2}-x_{k+1},\ldots,x_k-x_{k+1})$.
Again by Lemma \ref{update}, and defining $\mu\in\R$ and $v\in\R^{\ell-1}$ by
\begin{align*}
&\mu:=\langle x_{k-\ell+1}-x_{k+1},x_{k-\ell+1}-x_{k+1}\rangle\\
&v_i:=\langle x_{k-\ell+1}-x_{k+1},x_{k-\ell+i+1}-x_{k+1}\rangle,\quad
i=1,\ldots,\ell-1,
\end{align*}
we find for similar reasons that
\begin{align}
&\begin{pmatrix}\mu&v^T\\v&V_{k+1}^TV_{k+1}\end{pmatrix}
=\begin{pmatrix}V_k^TV_k&0\\0^T&0\end{pmatrix}
+\gamma_k\underline{s_k}\mathbbm{1}_{\R^{\ell}}\mathbbm{1}_{\R^{\ell}}^T\nonumber\\
&=\sum_{i=1}^{k-j_k}\gamma_{j_k+i-1}\underline{s_{j_k+i-1}}
\begin{pmatrix}f_i^{k-j_k}\\0\end{pmatrix}((f_i^{k-j_k})^T,0)
+\gamma_k\underline{s_k}\mathbbm{1}_{\R^{\ell}}\mathbbm{1}_{\R^{\ell}}^T\label{have}\\
&=\sum_{i=1}^{k+1-j_{k}}\gamma_{j_{k}+i-1}\underline{s_{j_k+i-1}}
f_i^{k+1-j_k}(f_i^{k+1-j_k})^T.\nonumber
\end{align}
The desired statement
\begin{equation}\label{want}
V_{k+1}^TV_{k+1}=\sum_{i=1}^{k+1-j_{k+1}}\gamma_{j_{k+1}+i-1}\underline{s_{j_{k+1}+i-1}}f_i^{k+1-j_{k+1}}(f_i^{k+1-j_{k+1}})^T
\end{equation}
can be verified by a component-wise comparison of equations \eqref{have} and
\eqref{want}.
\end{itemize}
Hence in all three cases, the matrix $V_{k+1}^TV_{k+1}$ has the desired representation,
and Lemma \ref{ringed} yields that
\[(V_{k+1}^TV_{k+1})^{-1}
=C(\gamma_{j_{k+1}}\underline{s_{j_{k+1}}},\ldots,\gamma_k\underline{s_k})
=C_{k+1}.\]
Since we have
\[M_{k+1}^TM_{k+1}=\begin{pmatrix}
V_{k+1}^TV_{k+1}&V_{k+1}^Td_{k+1}\\d_{k+1}V_{k+1}&\delta
\end{pmatrix},\]
Lemma \ref{rank:2} tells us that the remaining steps of Algorithm
\ref{greedy:affine:search} compute the solution
$s_k=(\overline{s_k}^T,\underline{s_k})^T$ of the linear system
\[M_{k+1}^TM_{k+1}s_{k+1}=\gamma_{k+1}e^{k+1-j_{k+1}}_{k+1-j_{k+1}}\]
and hence the same iterate $x_{k+1}$ as Algorithm \ref{basic:affine:search}.
\end{proof}
\begin{remark}\label{discuss:enhanced}
The computational complexity of one step of Algorithm \ref{greedy:affine:search}
is composed in the following way:
\begin{enumerate}[(i)]
\item The Kaczmarz cycle requires $4\nnz(A)+m$ flops.
\item The computation of $d_k$, $\delta_k$ and $\rho_k$ requires $3n+2m$ flops.
\item Assembling $V_k$ requires $(k-j_k)n$ flops.
\item Computing $p_k$ requires $2(k-j_k)n$ flops.
\item Computing $q_k$ requires $4(k-j_k)$ flops.
\item Computing $\overline{s_k}$ requires $(k-j_k)$ flops.
\item Updating $x_k$ requires $2(k-j_k+1)n$ flops.
\end{enumerate}
After the startup phase, when $k-j_k+1=\ell$, we have a total computational complexity
of roughly
\[4\nnz(A)+(3+5\ell)n+3m+5\ell.\]
Hence Algorithm \ref{basic:affine:search} arrives at the same numerical
results as Algorithm \ref{greedy:affine:search}, but it replaces the
most expensive operations (assembly of $M^TM$ at roughly $\frac12\ell^2n$ flops
and solution of system \eqref{lgs} at roughly $\ell^3$ flops) with
cheap ones (matrix-vector product with tridiagonal matrix at roughly $4\ell$ flops
and inner product at roughly $\ell$ flops).
\end{remark}
\section{Application to the random Kaczmarz method}
The random Kaczmarz method is displayed in Algorithm \ref{RK}.
We organize the iterations in epochs of $m$ projection steps.
Please refer to Algorithm \ref{RK} for the random indices used in this section.
\medskip
\begin{algorithm}\label{RK}
\caption{Random Kaczmarz method,\\
originally proposed in \cite{Strohmer}\\
(complexity: $4\nnz(A)+m$ flops per epoch with $m$ projections)}
\KwIn{
$A\in\R^{m\times n}$,
$b\in\R^m$,
$x_0\in\R^n$
}
\For{$k=0$ \KwTo $\infty$}{
$x_{k,0}\gets x_k$\;
\For{$j=0$ \KwTo $m-1$}{
draw $i_{k,j}$ uniformly at random from $\{1,\ldots.m\}$\;
$x_{k,j+1}\gets P_{i_{k,j}}(x_{k,j})$\;
}
$x_{k+1}\gets x_{x,m}$\;
}
\end{algorithm}
\begin{algorithm}\label{RK:accel}
\caption{Random Kaczmarz method with affine search\\
(for an upper bound on the complexity see Remark \ref{discuss:enhanced})}
\KwIn{
$A\in\R^{m\times n}$,
$b\in\R^m$,
$x_0\in\R^n$,
$\ell\in\N_2$
}
\For{$k=0$ \KwTo $\infty$}{
\Repeat{$x_{k,m}\neq x_k$}{
$\rho_k\gets 0$\;
$x_{k,0}\gets x_k$\;
\For{$j=0$ \KwTo $m-1$}{
draw $i_{k,j}$ uniformly at random from $\{1,\ldots.m\}$\;
$\rho_k\gets\rho_k+(a_{i_{k,j}}^Tx_{k,j}-b_{i_{k,j}})^2/\|a_{i_{k,j}}\|^2$\;
$x_{k,j+1}\gets P_{i_{k,j}}(x_{k,j})$\;
}
}
$d_k\gets x_{k,m}-x_k$\;
$\delta_k\gets\|d_k\|^2$\;
$\gamma_k\gets\frac12(\rho_k+\delta_k)$\;
$j_k\gets\max\{k-\ell+1,0\}$\;
$V_k\gets(x_{j_k}-x_k,\ldots,x_{k-1}-x_k)\in\R^{n\times(k-{j_k})}$\;
$p_k\gets V_k^Td_k\in\R^{k-{j_k}}$\;
$C_k\gets C(\gamma_{j_k}\underline{s_{j_k}},\ldots,\gamma_{k-1}\underline{s_{k-1}})
\in\R^{(k-{j_k})\times(k-{j_k})}$\;
$q_k\gets C_kp_k
\in\R^{k-{j_k}}$\;
$\underline{s_k}\gets\frac{\gamma_k}{p_k^Tq_k-\delta_k}\in\R$\;
$\overline{s_k}\gets-\underline{s_k}q_k\in\R^{k-{j_k}}$\;
$x_{k+1}\gets x_k+V_k\overline{s_k}+\underline{s_k}d_k$\;
}
\end{algorithm}
The random Kaczmarz method is known to converge in expectation
when $A$ has full rank and $x^*:=A^{-1}b$ is unique, see Theorem 2 in \cite{Strohmer}.
The final statement of the induction step in its proof shows that
\begin{equation}\label{stro}
\mathbbm{E}\|x_{k+1}-x^*\|^2\le(1-\kappa(A)^{-2})^m\mathbbm{E}\|x_k-x^*\|^2
\quad\forall\,k\in\N,
\end{equation}
where $\kappa(A)>0$ is a specific condition number, see Section 1 of \cite{Strohmer}
for details.
Statement \eqref{stro} reveals that an acceleration of the sequence $(x_k)_k$
that reduces the error $\|x_k-x^*\|^2$ maintains the convergence properties of the
sequence as well as the error estimate.
This motivates us to transfer the acceleration techniques from the previous
sections to the random Kaczmarz method, which gives Algorithm \ref{RK:accel}.
\begin{remark}\label{rK:remark}
Algorithm \ref{RK:accel} is justified by the following reasoning:
The $k$-th epoch of the random Kaczmarz method can be regarded as one cycle
\[\tilde{P}^{(k)}(x):=(P_{i_{k,m-1}}\circ\ldots\circ P_{i_{k,0}})(x)\]
of the deterministic Kaczmarz method applied to the matrix
$\tilde{A}^{(k)}\in\R^{m\times n}$
and a vector $\tilde{b}^{(k)}\in\R^m$ given by
\[\tilde{A}^{(k)}:=(a_{i_{k,0}},\ldots,a_{i_{k,m-1}})^T,\quad
\tilde{b}^{(k)}:=(b_{i_{k,0}},\ldots,b_{i_{k,m-1}})^T,\]
which gives rise to the residual
\[\tilde{r}^{(k)}(x):=\begin{pmatrix}
(a_{i_{k,0}}^Tx-b_{i_{k,0}})/\|a_{i_{k,0}}\|\\
(a_{i_{k,1}}^TP_{i_{k,0}}(x)-b_{i_{k,1}})/\|a_{i_{k,1}}\|\\
\vdots\\
(a_{i_{k,m-1}}^TP_{i_{k,m-2}}\circ\ldots\circ P_{i_{k,0}}(x)-b_{i_{k,m-1}})/\|a_{i_{k,m-1}}\|
\end{pmatrix}.\]
The solution $x^*=A^{-1}b$ also solves $\tilde{A}^{(k)}x^*=\tilde{b}^{(k)}$,
and all statements on errors given in the previous sections remain valid.
\medskip
However, it is no longer true that $\tilde{P}^{(k)}(x_k)\in\aff(x_{j_k},\ldots,x_k)$
if and only if $Ax_k=b$, because $\tilde{A}^{(k)}$
and $\tilde{b}^{(k)}$ are only subsamples of $A$ and $b$, which invalidates
the stopping criteria we previously used.
On the other hand, the inclusion $\tilde{P}^{(k)}(x_k)\in\aff(x_{j_k},\ldots,x_k)$
still implies that $\tilde{P}^{(k)}(x_k)=x_k$ and $\tilde{r}^{(k)}(x_k)=0$.
Hence, in this situation, we can ignore the last epoch in the acceleration scheme
(see line 10 of Algorithm \ref{RK:accel}), and accelerate only when the random
Kaczmarz method made progress.
This guarantees that whenever Theorem \ref{affine:search:thm} is invoked,
its assumptions are satisfied.
\end{remark}
\begin{figure}[t]\begin{center}
\includegraphics[scale=0.8]{updated_10.eps}\hfill
\includegraphics[scale=0.8]{updated_20.eps}\\
\includegraphics[scale=0.8]{updated_40.eps}
\caption{Algorithm \ref{greedy:affine:search} in CT example
with $N\times N$ pixels, $N=10,20,40$.
Notation:
K - Kaczmarz method,
K1 - Kaczmarz method with line-search,
K$\ell$ - Kaczmarz method with at most $\ell$-dimensional affine search space.
\label{uK}}
\end{center}\end{figure}
\begin{figure}[t]\begin{center}
\includegraphics[scale=0.8]{random_10.eps}\hfill
\includegraphics[scale=0.8]{random_20.eps}\\
\includegraphics[scale=0.8]{random_40.eps}
\caption{Algorithm \ref{RK:accel} in CT example
with $N\times N$ pixels, $N=10,20,40$.
Notation:
rK - random Kaczmarz method,
K1 - random Kaczmarz method with line-search,
K$\ell$ - random Kaczmarz method with at most $\ell$-dimensional affine search space.\label{rK}}
\end{center}\end{figure}
\section{Numerical results}
We test the algorithms presented in this paper in the context of the
computerized tomography problem, which is one of the most important
applications of the Kaczmarz method.
To generate benchmark problems, we apply the \texttt{paralleltomo} function
of the AIR Tools library \cite{Hansen} with default parameters to the
Shepp-Logan medical phantom.
\medskip
In all simulations, we choose the initial guess $x_0=0$ and apply an initial
random shuffling to the rows of $A$, because the canonical ordering tends to
induce very slow and hence atypical convergence.
To see how the method behaves under scaling, we investigate the following
scenarios, where
\[\mathrm{oncost}(\ell):=
\frac{\mathrm{cost}(\mathrm{acceleration}(\ell))}
{\mathrm{cost}(\mathrm{Kaczmarz\ cycle})}
\approx\frac{(3+5\ell)n+2m+5\ell}{4\nnz(A)+m}\]
measures the cost of an acceleration step relative in terms of the cost
of a Kaczmarz cycle:
\begin{itemize}
\item [a)] object resolution 10x10,
process matrix $A\in\R^{2296\times 100}$,
number of nonzero elements $\mathrm{nnz}(A)=22820$,
sparsity $\frac{\mathrm{nnz}(A)}{\#\mathrm{entries}(A)}\approx 0.1$,
condition number $\mathrm{cond}(A)\approx 62$,
$\mathrm{oncost}(\ell)\approx 0.052+0.005\ell$.
\item [b)] object resolution 20x20,
process matrix $A\in\R^{4584\times 400}$,
number of nonzero elements $\mathrm{nnz}(A)=91608$,
sparsity $\frac{\mathrm{nnz}(A)}{\#\mathrm{entries}(A)}=0.05$,
condition number $\mathrm{cond}(A)\approx 114$,
$\mathrm{oncost}(\ell)\approx 0.028+0.005\ell$.
\item [c)] object resolution 40x40,
process matrix $A\in\R^{9178\times 1600}$,
number of nonzero elements $\mathrm{nnz}(A)=366496$,
sparsity $\frac{\mathrm{nnz}(A)}{\#\mathrm{entries}(A)}=0.025$,
condition number $\mathrm{cond}(A)\approx 480$,
$\mathrm{oncost}(\ell)\approx 0.016+0.005\ell$.
\end{itemize}
We see that in all three scenarios, the cost of an acceleration step relative
to the cost of a Kaczmarz cycle is small, though the matrices are moderately
sparse.
\medskip
It is well-known that the normal equations \eqref{lgs} are prone to become
ill-conditioned.
We observed unstable behavior of Algorithm \ref{basic:affine:search} e.g.\
in scenario a) at $\ell=20$.
Though Algorithm \ref{greedy:affine:search} essentially solves the same problem,
it remained stable, which is probably a benefit of applying an explicitly known
inverse over solving the linear system numerically.
Only when the approximation error $\|x_k-x^*\|$ was very small (roughly $10^{-13}$),
we observed instability in the form of oscillating errors.
Algorithm \ref{greedy:affine:search} always performs better than Algorithm
\ref{basic:affine:search}, and the outperformance increases with the problem dimension
$n$ and the dimension $\ell$ of the affine search space, see Remark \ref{rem:affine}b
and Remark \ref{discuss:enhanced}.
As it is also more stable, we only display the numerical errors of Algorithm
\ref{greedy:affine:search} in Figure \ref{uK}.
\medskip
The wobble that is most pronounced in the error plot of $K_1$ for $N=40$ is neither
caused by an unstable algorithm nor an artefact.
It is typical across a range of acceleration schemes (not presented in this
paper, but investigated by the author numerically) and seems to be caused
by going back and forth between the Euclidean geometry and the geometry of the
Kaczmarz map $P$.
Roughly speaking, the error curves of the accelerated Kaczmarz methods
cluster at the error curve of $K_\infty$, which is the variant of
Algorithm \ref{greedy:affine:search} that spans the affine search space using
all previously computed iterates in every step.
This seems to suggest that the benefit of working with many or all previous iterates
outweighs the additional cost incurred by processing them.
\medskip
Figure \ref{rK} shows the performance of Algorithm \ref{RK:accel}, and it
demonstrates that our acceleration technique can be successfully applied
to the random Kaczmarz method.
Comparing figures \ref{uK} and \ref{rK} in terms of absolute values is not
meaningful, because the performance of the deterministic Kaczmarz method depends
on the chosen order of the rows of $A$, and the performance of the random
Kaczmarz method depends to some degree on the particular random numbers drawn.
However, there seems to be a slight qualitative difference in the performance
of the accelerated methods between the deterministic and the random setting.
In the random setting, a larger $\ell$ seems to be needed to achieve a similar
level of outperformance of the plain Kaczmarz method as in the deterministic setting, which is particularly noticeable in the error plot of the methods $K_1$ and $rK_1$
for $N=40$.
On the other hand, in both settings, the method $rK_\infty$ always clearly outperforms
the Kaczmarz method.
\bibliographystyle{plain}
\bibliography{gGKa}
\end{document}
| 68,208
|
Bonanza HS Las Vegas, NV Alumni List
Last names starting with "N"
Select letter of last name to filter the alumni list
Registered Alumni
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- Unknown Grad Year
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All Alumni
- Class of 1985 Adren N/a (now Dye)
- Tamara Nance
- Crystal Nash
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- Class of 1986 Carolyn Nygard.
| 297,624
|
TITLE: Evaluate $\lim _{x \to 0} [2x^{-3}(\sin^{-1}x - \tan^{-1}x )]^{2x^{-2}}$=?
QUESTION [1 upvotes]: $$\lim _{x \to 0} \left[\frac{2}{x^3}(\sin^{-1}x - \tan^{-1}x )\right]^{\frac{2}{x^2}}$$
How to find this limit?
My Try: I tried to evaluate this $$\lim _{x \to 0} \left[\frac{2}{x^3}(\sin^{-1}x - \tan^{-1}x )\right]$$ to understand the nature of the problem. I used L'Hopital. But it became too tedious to calculate.
Can anyone please give me suggestion to solve it?
Edit: I used the hint given By lab bhattacharjee. I expand the inverse functions and I got $$\lim _{x \to 0} \left[\frac{2}{x^3}(\sin^{-1}x - \tan^{-1}x )\right] =1$$.
Now I think it remains to find the value of $\lim _{x \to 0}e^{f(x).g(x)}$ where $f(x) = \left[\frac{2}{x^3}(\sin^{-1}x - \tan^{-1}x )\right]$ and $g(x) = 2/x^2$
REPLY [3 votes]: By series expansion of the arcsine and arctangent,
we get
$$\arcsin x-\arctan x = \frac12x^3 - \frac18 x^5 + o(x^5) $$
So you're looking for
$$ \begin{align}\lim_{x\to 0} [1-\tfrac14x^2+o(x^2)]^{2/x^2} &= \lim_{x\to 0} \exp\left(\frac{2}{x^2}\log(1-\tfrac14x^2+o(x^2))\right) \\&=
\lim_{x\to 0} \exp\left(\frac{2}{x^2}\bigl(-\tfrac14x^2+o(x^2)\bigr)\right) = \cdots \end{align} $$
REPLY [1 votes]: Recall that $$\lim_{x \to \infty} \left( 1 + \frac{k}{x}\right)^x = e^k.$$
We can add $o(x^{-1})$ terms to the parentheses without changing this. If we want to evaluate, for example, $$\lim_{x \to \infty} \left( 1 + \frac{k}{x} + \frac{k_2}{x^{1 + \eta}}\right)$$ where $\eta > 0$, then note that for any $\epsilon$, we can choose some large $X$ such that $x > X$ implies $$\left|\frac{k_2}{x^\eta}\right| < \epsilon$$ and thus $$e^{k-\epsilon} = \lim_{x \to \infty} \left( 1 + \frac{k-\epsilon}{x}\right) \leq \lim_{x \to \infty} \left( 1 + \frac{k}{x} + \frac{k_2}{x^{1 + \eta}}\right) \leq \lim_{x \to \infty} \left( 1 + \frac{k+\epsilon}{x}\right) = e^{k+\epsilon}.$$ But since $\epsilon$ was arbitrary, this shows $$\lim_{x \to \infty} \left( 1 + \frac{k}{x} + \frac{k_2}{x^{1 + \eta}} \right) = e^k.$$
Now by Taylor expansions, for $x > 0$, $$\frac{x^3}{2} - \frac{x^5}{8} < \arcsin x - \arctan x < \frac{x^3}{2} - \frac{x^5}{8} + \frac{3x^7}{16} $$ so $$\lim_{x \to 0^+} \left(1 - \frac{x^2}{4} \right)^\frac{2}{x^2} \leq \lim_{x \to 0^+} \left[ \frac{2}{x^3} \left( \arcsin x - \arctan x\right) \right]^\frac{2}{x^2} \leq \lim_{x \to 0^+} \left(1 - \frac{x^2}{4} + \frac{3x^4}{16} \right)^\frac{2}{x^2}.$$ By substituting $y = 2/x^2$, the outer limits are made into $$\lim_{y \to \infty} \left( 1 - \frac{1}{2y} \right)^y$$ and $$\lim_{y \to \infty} \left( 1 - \frac{1}{2y} + \frac{3}{4y^2} \right)^y$$ both of which equal $\boxed{e^{-1/2}}$.
The $x \to 0^-$ limit can be treated very similarly, the only caveat being that the inequalities are reversed.
| 421
|
Serial arsonists could be at work
Lives could be at risk, warn police, who fear serial arsonists could be setting fires in homes they think are empty.
Live updates
Serial arsonists fear
Serial arsonists could be targeting empty flats in Eastleigh after a number of fires broke out in buildings across the town. Police are now warning lives could be at risk.
Detective Constable Claire Reynolds said: "We are concerned people are entering and starting fires in what they believe to be vacant or derelict buildings, when, in fact, people still are living in some of the flats."
The latest fire broke out around 7.30pm Monday, February 11, when police were called to a burning block of 15 flats in Ovington Road, Eastleigh. All but one had been vacated because of redevelopment plans. Fortunately, the tenants of the occupied property were out at the time of the fire.
| 104,385
|
Featured artist: Artist Nadeena Dixon
You are here:
Fantastic open space area to play, learn and relax in. This Park has something for everyone.
Corner Orpington Street and Parramatta Road, Ashfield
Thanks for your feedback. We will use this data to improve the content of this page.
Page last updated: 13 Dec 2018
| 50,675
|
President Obama Visits Baton Rouge After Criticism That He's Late
President Obama tours Louisiana's communities affected by the flood that damaged over 60,000 homes and killed 13 people.
HIP HOP AWARDS
REAL HUSBANDS OF HOLLYWOOD
HOW TO ROCK: DENIM
For a while it seemed like the Republican Party planned to write off African-Americans in the 2012 election cycle, but it’s beginning to look like the GOP is down with the swirl. The Republican National Committee is planning to launch an initiative to win over Black voters, or at least give it a try, CNN reports.
In the next two weeks, the RNC will roll out a new website with testimonials from some of the nation’s best-known African-Americans, including Reps. Allen West (Florida) and Tim Scott (South Carolina), and Florida Lt. Gov. Jennifer Carroll. The goal is to make Black voters aware that the party does indeed have prominent, Black elected officials, “conservative leaders who stand for their principles,” RNC co-chair Sharon Day told the network.
"We need to explain our values to them," she added. "To be honest with you, their values and our values are more similar than dissimilar. They are not Democratic values, they are Republican values."
Day says she’s aiming to develop long-term relationships between the RNC and African-American communities and is not focusing shortsightedly on a specific election cycle. Contributions made on the site will be used for Black voter outreach and candidate recruitment.
"It's not a feel good political campaign site; it is a legacy site, a relationship site, so they have a voice,” Day told CNN.
BET Politics - Your source for the latest news, photos and videos illuminating key issues and personalities in African-American political life, plus commentary from some of our liveliest voices. Click here to subscribe to our newsletter
(Photos: REUTERS/Kevin Lamarque; Orlando Sentinel/MCT.
| 376,826
|
\begin{document}
\maketitle
\begin{abstract}
Given the possibility of communication systems failing
catastrophically, we investigate limits to communicating
over channels that fail at random times. These channels are
finite-state semi-Markov channels. We show that communication with
arbitrarily small probability of error is not possible.
Making use of results in finite blocklength channel coding,
we determine sequences of blocklengths that optimize transmission
volume communicated at fixed maximum message error probabilities.
We provide a partial ordering of communication channels.
A dynamic programming formulation is used to show the structural result
that channel state feedback does not improve performance.
\end{abstract}
\vspace{2mm}
{\small ``a communication channel{\ldots} might be inoperative because of an amplifier failure,
a broken or cut telephone wire, \ldots''} \\
\hspace*{\fill} {\small\emph{--- I.~M.~Jacobs \cite{Jacobs1959}}} \\
\section{Introduction}
Physical systems have a tendency to fail at random times \cite{Davis1952}.
This is true whether considering communication systems embedded in sensor networks that may run
out of energy \cite{DietrichD2009}, synthetic communication systems embedded in
biological cells that may die \cite{CantonLE2008},\footnote{We sidestep teleological
discussions of natural biology \cite{Pfeifer2006,Varshney2006} by considering
synthetic biology \cite{Endy2005}.}
communication systems embedded in spacecraft that may enter black holes \cite{Bekenstein2001},
or communication systems embedded in oceans with undersea cables that may be cut \cite{Headrick1991}.
In these scenarios and beyond, failure of the communication system may be modeled as
communication channel death.
As such, it is of interest to study information-theoretic limits on communicating
over channels that die at random times. This paper gives results on the fundamental
limits of what is possible and what is impossible when communicating over channels that die.
Communication with arbitrarily small probability of error (\emph{Shannon reliability}) is not possible
for any positive communication volume, however a suitably defined notion of $\eta$-reliability is possible.
Schemes that optimize communication volume for a given level of $\eta$-reliability are developed herein.
The central trade-off in communicating over channels that die is in the lengths
of codeword blocks. Longer blocks improve communication performance as
classically known, whereas shorter blocks have a smaller probability of
being prematurely terminated due to channel death. In several settings, a simple greedy
algorithm for determining the sequence of blocklengths yields a certifiably optimal solution.
We also develop a dynamic
programming formulation to optimize the ordered integer partition that determines
the sequence of blocklengths. Besides algorithmic utility, solving the dynamic program demonstrates
the structural result that channel state feedback does not improve performance.
The optimization of codeword blocklengths is reminiscent of
frame size control in wireless networks \cite{HaraOAOM1996,Modiano1999,LettieriS1998,CiSN2005},
however such techniques are used in conjunction with automatic repeat request protocols
and are motivated by amortizing protocol information.
Moreover, the results demonstrate the benefit of adapting to either channel state or
decision feedback. Contrarily, we show that adaptation to channel state provides
no benefit for channels that die.
Limits on channel coding with finite blocklength \cite{Slepian1963,ChangHM1962,MacMullenC1998,Laneman2006,PolyanskiyPV2008,BuckinghamV2008,WiechmanS2008}
are central to our development.
Indeed, channels that die bring the notion of finite blocklength to the fore and provide
a concrete physical reason to step back from infinity.\footnote{The phrase ``back from infinity'' is borrowed
from J.\ Ziv's 1997 Shannon Lecture.} Notions of outage in wireless communication \cite{OzarowSW1994,Goldsmith2005}
and lost letters in postal channels \cite{WolfWZ1970}
are similar to channel death, except that neither outage nor lost letters are permanent conditions.
Therefore blocklength asymptotics are useful to study those channel models but are not
useful for channels that die. Recent work that has similar motivations as this paper
provides the outage capacity of a wireless channel \cite{ZengZC2008}.
The remainder of the paper is organized as follows. Section~\ref{sec:model}
defines discrete memoryless channels that die and shows that these channels have
zero Shannon capacity. Section~\ref{sec:systemmodel} states the communication
system model and also fixes our novel performance criteria. Section~\ref{sec:limcomm}
shows that our notion of Shannon reliability is not achievable, strengthening the result of zero Shannon capacity
and then provides a communication scheme and determines its performance. Section~\ref{sec:optim}
optimizes performance for several death distributions using either a greedy algorithm or
a dynamic programming algorithm. Optimization demonstrates
that channel state feedback does not improve performance.
Section~\ref{sec:ordering} discusses the partial ordering of channels.
Section~\ref{sec:conc} suggests several extensions to this work.
\section{Channel Model}
\label{sec:model}
Consider a channel with finite input alphabet $\mathcal{X}$
and finite output alphabet $\mathcal{Y}$. It has an \emph{alive} state $s = a$ when it acts like a
noisy discrete memoryless channel (DMC) and a \emph{dead} state $s = d$ when it erases the input.\footnote{Our results can be extended to
cover cases where the channel acts like other channels \cite{PolyanskiyPV2009,PolyanskiyPV2009b}
in the alive state.}
Assume throughout the paper that the DMC from the alive state has zero error capacity \cite{Shannon1956}
equal to zero.\footnote{If the channel is noiseless in the alive state, the problem is
similar to settings where fountain codes \cite{Sanghavi2007} are used in the point-to-point case and
growth codes \cite{KamraMFR2006} are used in the network case.}
For example, if the channel acts like a binary symmetric channel (BSC) with crossover
probability $0 < \varepsilon < 1$ in the alive state, with $\mathcal{X} = \{0,1\}$, and
$\mathcal{Y} = \{0,1,?\}$, then the transmission matrix in the alive state is
\begin{equation}
\label{eq:pa}
p(y|x,s=a) = p_a(y|x) = \begin{bmatrix} 1 - \varepsilon & \varepsilon & 0 \\
\varepsilon & 1 - \varepsilon & 0 \end{bmatrix} \mbox{,}
\end{equation}
and the transmission matrix in the dead state is
\begin{equation}
\label{eq:pd}
p(y|x,s=d) = p_d(y|x) = \begin{bmatrix} 0 & 0 & 1 \\
0 & 0 & 1 \end{bmatrix} \mbox{.}
\end{equation}
The channel starts in state $s=a$ and then transitions to $s=d$ at some random time $T$,
where it remains for all time thereafter. That is, the channel is in state $a$
for times $n = 1,2,\ldots,T$ and in state $d$ for times $n = T+1,T+2,\ldots$.
The death time distribution is denoted $p_T(t)$. Note that there is always
a finite $t^{\dagger}$ such that $p_T(t^{\dagger}) > 0$.
\subsection{Finite-State Semi-Markov Channel}
Channels that die can be classified as finite-state channels (FSCs) \cite[Sec.~4.6]{Gallager1968}.
\begin{prop}
\label{prop:FSC}
A channel that dies $(\mathcal{X},p_a(y|x),p_d(y|x),p_T(t),\mathcal{Y})$
is a finite-state channel.
\end{prop}
\begin{IEEEproof}
Follows by definition, since the channel has two states.
\end{IEEEproof}
Channels that die have semi-Markovian \cite[Sec.~4.8]{Ross1996}, \cite[Sec.~5.7]{Gallager1996} properties.
\begin{defin}
A semi-Markov process changes state according to a Markov chain
but takes a random amount of time between changes. More specifically,
it is a stochastic process with states from a discrete alphabet $\mathcal{S}$, such that
whenever it enters state $s$, $s \in \mathcal{S}$:
\begin{itemize}
\item The next state it will enter is state $r$ with probability that depends only on $s,r \in \mathcal{S}$.
\item Given that the next state to be entered is state $r$, the time until the transition from $s$ to $r$
occurs has distribution that depends only on $s,r \in \mathcal{S}$.
\end{itemize}
\end{defin}
\begin{defin}
The Markovian sequence of states of a semi-Markov process is
called the embedded Markov chain of the semi-Markov process.
\end{defin}
\begin{defin}
A semi-Markov process is irreducible if its embedded Markov chain is irreducible.
\end{defin}
\begin{prop}
\label{prop:sM}
A channel that dies $(\mathcal{X},p_a(y|x),p_d(y|x),p_T(t),\mathcal{Y})$ has a channel state sequence that
is a non-irreducible semi-Markov process.
\end{prop}
\begin{proof}
When in state $a$, the next state is $d$ with probability $1$ and given that
the next state is to be $d$, the time until the transition from $a$ to $d$ has distribution $p_T(t)$.
When in state $d$, the next state is $d$ with probability $1$.
Thus, the channel state sequence is a semi-Markov process.
The semi-Markov state process is not
irreducible because the $a$ state of the embedded Markov chain is transient.
\end{proof}
\noindent Note that when $T$ is a geometric random variable, the channel state
process forms a Markov chain, with transient state $a$ and recurrent, absorbing state
$d$.
There are further special classes of FSCs.
\begin{defin}
An FSC is a \emph{finite-state semi-Markov channel} (FSSMC) if its
state sequence forms a semi-Markov process.
\end{defin}
\begin{defin}
An FSC is a \emph{finite-state Markov channel} (FSMC) if its
state sequence forms a Markov chain.
\end{defin}
\begin{prop}
A channel that dies $(\mathcal{X},p_a(y|x),p_d(y|x),p_T(t),\mathcal{Y})$
is an FSSMC and is an FSMC when $T$ is geometric.
\end{prop}
\begin{IEEEproof}
Follows from Props.~\ref{prop:FSC} and \ref{prop:sM}.
\end{IEEEproof}
FSMCs have been widely studied in the literature \cite{Gallager1968,Gallager1972,TatikondaM2009}, particularly the
panic button/child's toy channel of Gallager \cite[p.~26]{Gallager1972}, \cite[p.~103]{Gallager1968} and the
Gilbert-Elliott channel and its extensions \cite{MushkinB1989,GoldsmithV1996}.
Contrarily, FSSMCs seem to not have
been specifically studied in information theory. There are a few works \cite{BratenT2002,WangCA2008,WangP2010} that
give semi-Markov channel models for wireless communications systems but do not
provide information-theoretic characterizations.
\subsection{Capacity is Zero}
A channel that dies has Shannon capacity equal to zero. To show this, first
notice that if the initial state of a channel that dies were not fixed, then it would
be an indecomposable FSC \cite[Sec.~4.6]{Gallager1968}, where the effect of the initial state dies
away.
\begin{prop}
\label{prop:indec}
If the initial state of a channel that dies $(\mathcal{X},p_a(y|x),p_d(y|x),p_T(t),\mathcal{Y})$
is not fixed, then it is an indecomposable FSC.
\end{prop}
\begin{IEEEproof}
The embedded Markov chain for a channel that dies has a unique absorbing state
$d$.
\end{IEEEproof}
Indecomposable FSCs have the property that the upper capacity, defined in \cite[(4.6.6)]{Gallager1968},
and lower capacity, defined in \cite[(4.6.3)]{Gallager1968},
are identical \cite[Thm.~4.6.4]{Gallager1968}. This can be used to show
that the capacity of a channel that dies is zero.
\begin{prop}
\label{prop:cap_zero}
The Shannon capacity, $C$, of a channel that dies $(\mathcal{X},p_a(y|x),p_d(y|x),p_T(t),\mathcal{Y})$
is zero.
\end{prop}
\begin{IEEEproof}
Although the initial state is $s_1=a$ here, temporarily suppose
that $s_1$ may be either $a$ or $d$. Then the channel is
indecomposable by Prop.~\ref{prop:indec}.
The lower capacity $\underline{C}$ equals the upper capacity $\overline{C}$,
for indecomposable channels by \cite[Thm.~4.6.4]{Gallager1968}.
The information rate of a memoryless $p_d(y|x)$ `dead' channel is clearly zero
for any input distribution, so the lower capacity $\underline{C} = 0$.
Thus the Shannon capacity for a channel that dies with initial alive state is $C = \overline{C} = 0$.
\end{IEEEproof}
\section{Communication System}
\label{sec:systemmodel}
In order to information theoretically characterize a channel that dies, a communication system
that contains the channel is described.
We have an information stream (like i.i.d.\ equiprobable bits), which can be grouped into a sequence of
$k$ messages, $(W_1,W_2,\ldots,W_k)$.
Each message $W_i$ is drawn from a message set $\mathcal{W}_i = \{1,2,\ldots,M_i\}$.
Each message $W_i$ is encoded into a channel input codeword $X_1^{n_i}(W_i)$ and these codewords
$(X_1^{n_1}(W_1),X_1^{n_2}(W_2),\ldots,X_1^{n_k}(W_k))$ are transmitted in sequence over the channel. A noisy version
of this codeword sequence is received, $Y_1^{n_1+n_2+\cdots+n_k}(W_1,W_2,\ldots,W_k)$. The receiver then
guesses the sequence of messages using an appropriate decoding rule $g$,
to produce $(\hat{W}_1,\hat{W}_2,\ldots,\hat{W}_k) = g(Y_1^{n_1+n_2+\cdots+n_k})$.
The $\hat{W}_i$s are drawn from alphabets $\mathcal{W}_i^{\ominus} = \mathcal{W}_i \cup \ominus$,
where the $\ominus$ message indicates the decoder declaring an erasure. The receiver
makes an error on message $i$ if $\hat{W}_i \neq W_i$ and $\hat{W}_i \neq \ominus$.
Block coding results are typically expressed with the concern of
sending one message rather than $k$ messages as here.\footnote{Tree
codes are beyond the scope of this paper, since we desire to communicate messages. A reformulation of
communicating over channels that die using tree codes \cite[Ch.~10]{Jelinek1968b} with early termination \cite{Forney1974a}
would, however, be interesting. In fact, communicating over channels that die using convolutional
codes with sequential decoding would be very natural, but would require performance criteria different from the ones
developed herein.}
System definitions can be formalized as follows.
\begin{defin}
An $(M_i,n_i)$ \emph{individual message code} for a channel that dies $(\mathcal{X},p_a(y|x),p_d(y|x),p_T(t),\mathcal{Y})$
consists of:
\begin{enumerate}
\item An individual message index set $\{1,2,\ldots,M_i\}$, and
\item An individual message encoding function $f_{i}: \{1,2,\ldots,M_i\} \mapsto \mathcal{X}^{n_i}$.
\end{enumerate}
The individual message index set $\{1,2,\ldots,M_i\}$ is denoted $\mathcal{W}_i$, and the
set of individual message codewords $\{f_{i}(1),f_{i}(2),\ldots,f_{i}(M_i)\}$ is called the
\emph{individual message codebook}.
\end{defin}
\begin{defin}
An $(M_i,n_i)_{i=1}^k$ \emph{code} for a channel that dies $(\mathcal{X},p_a(y|x),p_d(y|x),p_T(t),\mathcal{Y})$
is a sequence of $k$ individual message codes, $(M_i,n_i)_{i=1}^k$, in the sense of comprising:
\begin{enumerate}
\item A sequence of individual message index sets $\mathcal{W}_1, \mathcal{W}_2, \ldots, \mathcal{W}_k$,
\item A sequence of individual message encoding functions $f = (f_1,f_2,\ldots,f_k)$, and
\item A decoding function $g: \mathcal{Y}^{\sum_{i=1}^k n_i} \mapsto \mathcal{W}_1^{\ominus} \times \mathcal{W}_2^{\ominus} \times \cdots \times \mathcal{W}_k^{\ominus}$.
\end{enumerate}
\end{defin}
There is no essential loss of generality
by assuming that the decoding function $g$ is decomposed into a sequence of
individual message decoding functions $g = (g_1,g_2,\ldots,g_n)$ where $g_{i}: \mathcal{Y}^{n_i} \mapsto \mathcal{W}_i^{\ominus}$
when individual messages are chosen independently,
due to this independence and the conditional memorylessness of the channel.
To define performance measures, we assume that the decoder operates
on an individual message basis. That is, when applying the communication system,
let $\hat{W}_1 = g_1(Y_1^{n_1})$, $\hat{W}_2 = g_2(Y_{n_1 + 1}^{n_1 + n_2})$, and so on.
For the sequel, we make a further assumption on the operation of the decoder.
\begin{assum}
\label{assum:erase}
If all $n_i$ channel output symbols used by individual message decoder $g_i$
are not $?$, then the range of $g_i$ is $\mathcal{W}_i$. If any of
the $n_i$ channel output symbols used by individual message decoder $g_i$
are $?$, then $g_i$ maps to $\ominus$.
\end{assum}
This assumption corresponds to the physical properties of a communication system
where the decoder fails catastrophically. Once the decoder fails, it cannot perform
any decoding operations, and so the $?$ symbols in the channel model of system failure
must be ignored.
\subsection{Performance Measures}
We formally write the notion of error for the communication system as follows.
\begin{defin}
For all $1 \le w \le M_i$, let
\[
\lambda_{w}(i) = \Pr[\hat{W}_i \neq w|W_i = w, \hat{W}_i \neq \ominus]
\]
be the conditional message probability of error given that the $i$th individual message is $w$.
\end{defin}
\begin{defin}
The maximal probability of error for an $(M_i,n_i)$ individual message code
is
\[
\lambda_{\max}(i) = \max_{w \in \mathcal{W}_i} \lambda_{w}(i) \mbox{.}
\]
\end{defin}
\begin{defin}
The maximal probability of error for an $(M_i,n_i)_{i=1}^k$ code is
\[
\lambda_{\max} = \max_{i \in \{1,\ldots,k\}} \lambda_{\max}(i) \mbox{.}
\]
\end{defin}
Performance criteria weaker than traditional in information theory are defined,
since the Shannon capacity of a channel that dies is zero (Prop.~\ref{prop:cap_zero}).
In particular, we define formal notions of how
much information is transmitted using a code and how long it takes.
\begin{defin}
The transmission time of an $(M_i,n_i)_{i=1}^k$ code is
$N = \sum_{i=1}^k n_i$.
\end{defin}
\begin{defin}
The expected transmission volume of an $(M_i,n_i)_{i=1}^k$ code is
\[
V = \E_{T} \left\{\sum_{i \in \{ 1,\ldots,k | \hat{W}_i \neq \ominus \}} \log M_i \right\}\mbox{.}
\]
\end{defin}
Notice that although declared erasures do not lead to errors, they do not contribute
transmission volume either.
The several performance criteria for a code may be combined together.
\begin{defin}
Given $0\le \eta < 1$, a pair of numbers $(N_0,V_0)$ (where $N_0$ is a positive integer and $V_0$ is non-negative)
is said to be an \emph{achievable transmission time-volume at $\eta$-reliability}
if there exists, for some $k$, an $(M_i,n_i)_{i=1}^k$ code for the channel that dies $(\mathcal{X},p_a(y|x),p_d(y|x),p_T(t),\mathcal{Y})$
such that
\begin{align}
\lambda_{\max} &\le \eta \mbox{, }\\
N &\le N_0\mbox{, and }\\
V &\ge V_0\mbox{.}
\end{align}
Moreover, $(N_0,V_0)$ is said to be an \emph{achievable transmission time-volume
at Shannon reliability} if it is an achievable transmission time-volume
at $\eta$-reliability for all $0 < \eta < 1$.
\end{defin}
\section{Limits on Communication}
\label{sec:limcomm}
Having defined the notion of achievable transmission time-volume at various levels of
reliability, the goal of this work is to demarcate what is achievable.
\subsection{Shannon Reliability is Not Achievable}
\label{sec:noShannonRel}
Not only is the Shannon capacity of a channel that dies zero,
but also there is no $V > 0$ such that $(N,V)$ is an achievable transmission time-volume
at Shannon reliability. A coding scheme that always declares erasures would achieve
zero error probability (and therefore Shannon reliability) but would not provide
positive transmission volume; this is also not allowed under Assumption~\ref{assum:erase}.
Lemmas are stated and proved after the proof of the main proposition. For
brevity, the proof is limited to the alive-BSC case, but can be extended to general
alive-DMCs by choosing the two most distant letters in $\mathcal{Y}$ for constructing the repetition
code, among other things.
\begin{prop}
\label{prop:noShaRel}
For a channel that dies $(\mathcal{X},p_a(y|x),p_d(y|x),p_T(t),\mathcal{Y})$,
there is no $V > 0$ such that $(N,V)$ is an achievable transmission time-volume
at Shannon reliability.
\end{prop}
\begin{IEEEproof}
From the error probability viewpoint, transmitting longer codes
is not harder than transmitting shorter codes (Lem.~\ref{lem:longerbetter}) and
transmitting smaller codes is not harder than transmitting larger codes (Lem.~\ref{lem:smallerbetter}).
Hence, the desired result follows from showing that even the longest and smallest code that has positive expected transmission
volume cannot achieve Shannon reliability.
Clearly the longest and smallest code uses a single individual message code
of length $n_1 \to \infty$ and size $M_1 = 2$. Among such codes,
transmitting the binary repetition code is not harder than transmitting
any other code (Lem.~\ref{lem:repetitionbetter}). Hence showing that the binary
repetition code cannot achieve Shannon reliability yields the desired result.
Consider transmitting a single $(M_1 = 2, n_1)$ individual message code that is simply a
binary repetition code over a channel that dies $(\mathcal{X},p_a(y|x),p_d(y|x),p_T(t),\mathcal{Y})$.
Let $\mathcal{W}_1 = \{00000\ldots, 11111\ldots \}$,
where the two codewords are of length $n_1$.
Assume that the all-zeros codeword and the
all-ones codeword are each transmitted with probability $1/2$ and measure average probability
of error, since average error probability lower bounds $\lambda_{\max}(1)$ \cite[Problem 5.32]{Gallager1968}.
The transmission time $N = n_1$ and let $N \to \infty$.
The expected transmission volume is $\log 2 > 0$.
Under equiprobable signaling over a BSC, the minimum error probability decoder is
the maximum likelihood decoder, which in turn is the minimum distance decoder \cite[Problem 2.13]{McEliece2002}.
The scenario corresponds to binary hypothesis testing over a
BSC($\varepsilon$) with $T$ observations (since after the channel dies,
the output symbols do not help with hypothesis testing). Since there is a
finite $t^{\dagger}$ such that $p_T(t^{\dagger}) > 0$, there is a fixed constant $K$
such that $\lambda_{\max} > K > 0$ for any realization $T = t$.
Thus Shannon reliability is not achievable.
\end{IEEEproof}
\begin{lemma}
\label{lem:longerbetter}
When transmitting over the alive state's memoryless channel $p_a(y|x)$,
let the maximal probability of error $\lambda_{\max}(i)$ for an optimal $(M_i, n_i)$ individual
message code and minimum probability of error individual decoder $g_i$ be $\lambda_{\max}(i; n_i)$.
Then $\lambda_{\max}(i; n_i + 1) \le \lambda_{\max}(i; n_i)$.
\end{lemma}
\begin{IEEEproof}
Consider the optimal block-length-$n_i$ individual message code/decoder, which achieves $\lambda_{\max}(i; n_i)$.
Use it to construct an $n_i + 1$ individual message code that appends a dummy symbol
to each codeword and an associated decoder that operates by ignoring this last symbol.
The error performance of this (suboptimal) code/decoder is clearly $\lambda_{\max}(i; n_i)$, and so
the optimal performance can only be better: $\lambda_{\max}(i; n_i + 1) \le \lambda_{\max}(i; n_i)$.
\end{IEEEproof}
\begin{lemma}
\label{lem:smallerbetter}
When transmitting over the alive state's memoryless channel $p_a(y|x)$,
let the maximal probability of error $P_e^{\rm max}(i)$ for an optimal $(M_i, n_i)$ individual
message code and minimum probability of error individual decoder $f_D^{(i)}$ be $P_e^{\rm max}(i; M_i)$.
Then $P_e^{\rm max}(i; M_i) \le P_e^{\rm max}(i; M_i + 1)$.
\end{lemma}
\begin{IEEEproof}
Follows from sphere-packing principles.
\end{IEEEproof}
\begin{lemma}
\label{lem:repetitionbetter}
When transmitting over the alive state's memoryless channel $p_a(y|x)$,
the optimal $(M_i = 2, n_i)$ individual message code can be taken as a binary repetition code.
\end{lemma}
\begin{IEEEproof}
Under minimum distance decoding (which yields the minimum error probability \cite[Problem 2.13]{McEliece2002})
for a code transmitted over a BSC, increasing the distance between codewords can only
reduce error probability. The repetition code has maximum Hamming distance between codewords.
\end{IEEEproof}
Notice that Prop.~\ref{prop:noShaRel} also directly implies Prop.~\ref{prop:cap_zero},
providing an alternate proof.
\begin{cor}
The Shannon capacity of a channel that dies $(\mathcal{X},p_a(y|x),p_d(y|x),p_T(t),\mathcal{Y})$
is zero.
\end{cor}
\subsection{Finite Blocklength Channel Coding}
Before developing an optimal scheme for $\eta$-reliable communication
over a channel that dies, finite block length channel coding is reviewed.
Under our definitions, traditional channel coding results
\cite{Slepian1963,MacMullenC1998,Laneman2006,PolyanskiyPV2008,BuckinghamV2008,WiechmanS2008}
provide information about individual message codes, determining the achievable trios
$(n_i,M_i,\lambda_{\max}(i))$. In particular, the largest possible $M_i$ for a
given $n_i$ and $\lambda_{\max}(i)$ is denoted $M^{*}(n_i,\lambda_{\max}(i))$.
The purpose of this work is not to improve upper and lower bounds on finite
block length channel coding, but to use existing results to study channels that die.
In fact, for the sequel, simply assume that the function $M^{*}(n_i,\lambda_{\max}(i))$ is known,
as are codes/decoders that
achieve this value. In principle, optimal individual message codes
may be found through exhaustive search \cite{MacMullenC1998,KaskiO2006}.
Although algebraic notions of code quality do not directly imply
error probability quality \cite{BargM2005}, perfect codes
such as the Hamming or Golay codes may also be optimal in certain limited
cases.
Recent results comparing upper and lower bounds around Strassen's normal approximation
to $\log M^{*}(n_i,\lambda_{\max}(i))$ \cite{Strassen1962}
have demonstrated that the approximation is quite good \cite{PolyanskiyPV2008}.
\begin{remark}
\label{rem:ass}
We assume that optimal $M^{*}(n_i,\eta)$-achieving individual message codes are known.
Exact upper and lower bounds to $\log M^{*}(n_i,\eta)$ can be substituted to make our results precise.
For numerical demonstrations, we will further assume that optimal codes have performance
given by Strassen's approximation.
\end{remark}
The following expression for $\log M^{*}(n_i,\eta)$ that first appeared in \cite{Strassen1962}
is also given as \cite[Thm.~6]{PolyanskiyPV2008}.
\begin{lemma}
\label{lemma:finiteblock}
Let $M^{*}(n_i,\eta)$ be the largest size
of an individual message code with block length $n_i$ and maximal error probability
upper bounded by $\lambda_{\max}(i) < \eta$. Then, for any DMC with
capacity $C$ and $0 < \eta \le 1/2$,
\[
\log M^{*}(n_i,\eta) = n_iC - \sqrt{n_i\rho} Q^{-1}(\eta) + O(\log n_i) \mbox{,}
\]
where
\[
Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^{\infty} e^{-t^2/2} \, dt\mbox{,}
\]
\[
\rho = \min_{X: C = I(X;Y)} \var\left[ \log \frac{p_{Y|X}(y|x)}{p_Y(y)}\right] \mbox{,}
\]
and standard asymptotic notation \cite{Knuth1976} is used.
\end{lemma}
For the BSC($\varepsilon$), the approximation (ignoring the $O(\log n_i)$ term above) is:
\begin{equation}
\label{eq:PPV_BSC}
\log M^{*} \approx n_i(1 - h_2(\varepsilon)) - \sqrt{n_i\varepsilon(1-\varepsilon)} Q^{-1}(\eta)\log_2 \tfrac{\varepsilon}{1-\varepsilon}\mbox{,}
\end{equation}
where $h_2(\cdot)$ is the binary entropy function.
This BSC expression first appeared in \cite{Weiss1960}.
For intuition, we plot the approximate $\log M^{*}(n_i,\eta)$ function
for a BSC($\varepsilon$) in Fig.~\ref{fig:block_vol}.
Notice that $\log M^{*}$ is zero for small $n_i$ since no code can achieve the target
error probability $\eta$. Also notice that $\log M^{*}$ is a monotonically increasing
function of $n_i$. Moreover, notice in Fig.~\ref{fig:block_rate} that even when
normalized, $(\log M^{*})/n_i$, is a monotonically increasing function of $n_i$.
Therefore longer blocks provide more `bang for the buck.' The curve in
Fig.~\ref{fig:block_rate} asymptotically approaches capacity.
\begin{figure}[ht]
\centering
\subfigure[]{
\includegraphics[width=3.5in]{block_vol.eps}
\label{fig:block_vol}
}
\subfigure[]{
\includegraphics[width=3.5in]{block_rate.eps}
\label{fig:block_rate}
}
\caption{\subref{fig:block_vol}. The expression \eqref{eq:PPV_BSC} for $\varepsilon = 0.01$ and $\eta = 0.001$.
\subref{fig:block_rate}. Normalized version, $(\log M^{*}(n_i,\eta))/n_i$, for $\varepsilon = 0.01$ and $\eta = 0.001$.
The capacity of a BSC($\varepsilon$) is $1-h_2(\varepsilon) = 0.92$.
}
\end{figure}
\subsection{$\eta$-reliable Communication}
\label{sec:scheme}
We now describe a coding scheme that achieves positive expected transmission volume at
$\eta$-reliability. Survival probability of the channel plays a key role in measuring
performance.
\begin{defin}
The \emph{survival function} of a channel that dies
$(\mathcal{X},p_a(y|x),p_d(y|x),p_T(t),\mathcal{Y})$
is $\Pr[T > t]$, is denoted $R_T(t)$, and satisfies
\[
R_T(t) = \Pr[T > t] = 1 - \sum_{\tau = 1}^t p_T(\tau) = 1 - F_T(t) \mbox{,}
\]
where $F_T$ is the cumulative distribution function.
\end{defin}
$R_T(t)$ is a non-increasing function.
\begin{prop}
\label{prop:achievability}
The transmission time-volume
\[
\left(N = \sum_{i=1}^k n_i, V = \sum_{i = 1}^k R_T(e_i) \log M^{*}(n_i,\eta) \right)
\]
is achievable at $\eta$-reliability for any sequence $(n_i)_{i=1}^k$ of individual message
codeword lengths, where $e_0 = 0, e_1 = n_1, e_2 = n_1 + n_2,\ldots, e_k = \sum_{i=1}^k n_i$.
\end{prop}
\begin{IEEEproof}
\emph{Code Design:}
A target error probability $\eta$ and a sequence $(n_i)_{i=1}^k$ of individual message
codeword lengths are fixed. Construct a length-$k$ sequence of $(M_i,n_i)$
individual message codes and individual decoding functions
$(\mathcal{W}_i,f_i,g_i)$ that achieve optimal performance.
The size of $\mathcal{W}_i$ is $|\mathcal{W}_i| = \log M^{*}(n_i,\eta)$. Note that
individual decoding functions $g_i$ have range $\mathcal{W}_i$ rather than
$\mathcal{W}_i^{\ominus}$.
\emph{Encoding:}
A codeword $W_1 = w_1$ is selected uniformly at random from the codebook
$\mathcal{W}_1$. The mapping of this codeword into $n_1$ channel input letters,
$X_{e_0 + 1}^{e_1} = f_1(w_1)$, is transmitted in channel usage times $n = e_0 + 1, e_0 + 2,\ldots,e_1$.
Then a codeword $W_2 = w_2$ is selected uniformly at random from the codebook
$\mathcal{W}_2$. The mapping of this codeword into $n_2$ channel input letters,
$X_{e_1 + 1}^{e_2} = f_2(w_2)$, is transmitted in channel usage times $n = e_1+1,e_1+2,\ldots,e_2$.
This procedure continues until the last individual message code in the code is transmitted.
That is, a codeword $W_k = w_k$ is selected uniformly at random from the codebook
$\mathcal{W}_k$. The mapping of this codeword into $n_k$ channel input letters,
$X_{e_{k-1} + 1}^{e_k} = f_k(w_k)$, is transmitted in channel usage times
$n = e_{k-1}+ 1, e_{k-1}+ 2,\ldots,e_k$.
We refer to channel usage times $n \in \{e_{i-1} + 1,e_{i-1} + 2,\ldots,e_i\}$ as the $i$th transmission epoch.
\emph{Decoding:}
For decoding, the channel output symbols for each epoch are processed separately.
If any of the channel output symbols in an epoch are erasure symbols $?$,
then a decoding erasure $\ominus$ is declared for the message in that epoch,
i.e.\ $\hat{W}_i = \ominus$. Otherwise, the individual
message decoding function $g_i: \mathcal{Y}^{n_i} \to \mathcal{W}_i$ is applied
to obtain $\hat{W}_i = g_i(Y_{e_{i-1} + 1}^{e_i})$.
\emph{Performance Analysis:}
Having defined the communication scheme, we measure the error probability, transmission time,
and expected transmission volume.
The decoder will either produce an erasure $\ominus$ or use an individual
message decoder $g_i$. When $g_i$ is used, the maximal error
probability of individual message code error is bounded as
$\lambda_{\max}(i) < \eta$
by construction. Since declared erasures $\ominus$ do not lead to error, and since all
$\lambda_{\max}(i) < \eta$, it follows that
\[
\lambda_{\max} < \eta \mbox{.}
\]
The transmission time is simply $N = \sum n_i$.
Recall the definition of expected transmission volume:
\[
\E \left\{\sum_{i \in \{ 1,\ldots,k | \hat{W}_i \neq \ominus \}} \log M_i \right\} = \sum_{i \in \{ 1,\ldots,k | \hat{W}_i \neq \ominus \}} \E\left\{ \log M_i \right\}
\]
and the fact that the channel produces the erasure symbol $?$ for all channel usage
times after death, $n > T$, but not before. Combining this with the length of an optimal code, $\log M^{*}(n_i,\eta)$,
leads to the expression
\[
\sum_{i = 1}^k \Pr[T > e_i] \log M^{*}(n_i,\eta) \mbox{,}
\]
since all individual message codewords that are received in their entirety
before the channel dies are decoded using $g_i$ whereas any
individual message codewords that are even partially cut off are declared $\ominus$.
Recalling the definition of the survival function,
the expected transmission volume of the communication scheme is
\[
\sum_{i = 1}^k R_T(e_i) \log M^{*}(n_i,\eta)
\]
as desired.
\end{IEEEproof}
Prop.~\ref{prop:achievability} is valid for any choice of $(n_i)_{i=1}^k$.
Since $(\log M^{*})/n_i$ is monotonically increasing, it is better
to use individual message codes that are as long as possible. With longer individual message
codes, however, there is a greater chance of many channel usages being wasted if
the channel dies in the middle of transmission. The basic trade-off is captured in picking
the set of values $\{n_1,n_2,\ldots,n_k\}$. For fixed and finite $N$, this involves
picking an ordered integer partition $n_1 + n_2 + \cdots + n_k = N$.
We optimize this choice in Section~\ref{sec:optim}.
\subsection{Converse Arguments}
\label{sec:converseargs}
Since we simply have operational expressions and no informational expressions
in our development, as per Remark~\ref{rem:ass}, and since optimal individual message codes and individual message
decoders are assumed to be used, it may seem as though converse arguments are not required.
This would indeed follow, if the following two things were true, which follow
from Assumption~\ref{assum:erase}. First, that there is no benefit
in trying to decode the last partially erased message block. Second, that
there is no benefit to errors-and-erasures decoding \cite{Forney1968} by the $g_i$ for codewords
that are received before channel death.
Under Assumption~\ref{assum:erase}, Prop.~\ref{prop:achievability}
gives the best performance possible.
One might wonder whether Assumption~\ref{assum:erase} is needed.
That there would be no benefit in trying to decode the last partially erased block
follows from the conjecture that an optimal individual message code would have no
latent redundancy that could be exploited to achieve a $\lambda_{\max}(i = \rm{last}) < \eta$,
but this is a property of the actual optimal code.
Understanding the possibility of errors-and-erasures decoding \cite{Forney1968}
by the individual message decoders also requires knowing properties of actual optimal codes.
It is unclear how the choice of threshold in errors-and-erasures decoding
would affect the expected transmission volume
\[
\sum_{i = 1}^k (1-\xi_i) R_T(e_i) \log M^{*}(n_i,\xi_i,\eta) \mbox{,}
\]
where $\xi_i$ would be the specified erasure probability for individual message $i$,
and $M^{*}(n_i,\xi_i,\eta)$ would be the maximum individual message codebook size
under erasure probability $\xi_i$ and maximum error probability $\eta$.
What we can say, however, is that at the level of Strassen's approximation (up to the $\log n$ term),
$\log M^{*}(n_i,\xi_i,\eta)$ and $\log M^{*}(n_i,\eta)$ are the same \cite[Thm.~47]{Polyanskiy2010}.
\section{Optimizing the Communication Scheme}
\label{sec:optim}
In Section~\ref{sec:scheme}, we had not optimized the lengths of the
individual message codes; we do so here. For fixed $\eta$ and $N$, we maximize the expected
transmission volume $V$ over the choice of the ordered integer partition $n_1 + n_2 + \cdots + n_k = N$:
\begin{equation}
\label{eq:optim}
\max_{(n_i)_{i=1}^k: \sum n_i = N} \sum_{i = 1}^k R_T(e_i) \log M^{*}(n_i,\eta) \mbox{.}
\end{equation}
For finite $N$, this optimization can be carried out by an exhaustive search
over all $2^{N-1}$ ordered integer partitions. If the death distribution $p_T(t)$
has finite support, there is no loss of generality in considering only
finite $N$. Since exhaustive search has exponential complexity, however, there
is value in trying to use a simplified algorithm. A dynamic programming
formulation for the finite horizon case is developed in Section~\ref{sec:DP}.
The next subsection develops a greedy algorithm which is applicable
to both the finite and infinite horizon cases and yields the optimal
solution for certain problems.
\subsection{A Greedy Algorithm}
To try to solve the optimization problem \eqref{eq:optim}, we propose a
greedy algorithm that optimizes blocklengths $n_i$ one by one.
\begin{algorithm}
\label{algo:greed}
\quad
\begin{enumerate}
\item Maximize $R_T(n_1) \log M^{*}(n_1,\eta)$ through the choice of $n_1$ independently of any other $n_i$.
\item Maximize $R_T(e_2) \log M^{*}(n_2,\eta)$ after fixing $e_1 = n_1$, but independently of later $n_i$.
\item Maximize $R_T(e_3) \log M^{*}(n_3,\eta)$ after fixing $e_2$, but independently of later $n_i$.
\item Continue in the same manner for all subsequent $n_i$.
\end{enumerate}
\end{algorithm}
Sometimes the algorithm produces the correct solution.
\begin{prop}
\label{prop:localopt}
The solution produced by the greedy algorithm, $(n_i)$, is locally optimal if
\begin{equation}
\label{eq:star}
\frac{R_T(e_i)\log M^{*}(n_i,\eta) - R_T(e_i-1)\log M^{*}(n_i - 1,\eta)}{R_T(e_{i+1})\left[\log M^{*}(n_{i+1} + 1,\eta) - \log M^{*}(n_{i+1},\eta)\right]} \ge 1
\end{equation}
for each $i$.
\end{prop}
\begin{IEEEproof}
The solution of the greedy algorithm partitions time using a set of epoch boundaries $(e_i)$.
The proof proceeds by testing whether local perturbation of an arbitrary
epoch boundary can improve performance. There are two possible perturbations: a shift to the left
or a shift to the right.
First consider shifting an arbitrary epoch boundary $e_i$ to the right by one. This makes the
left epoch longer and the right epoch shorter. Lengthening the left epoch does not improve performance
due to the greedy optimization of the algorithm. Shortening the right epoch does not improve
performance since $R_T(e_i)$ remains unchanged whereas $\log M^{*}(n_i,\eta)$ does not increase since
$\log M^{*}$ is a non-decreasing function of $n_i$.
Now consider shifting an arbitrary epoch boundary $e_i$ to the left by one. This
makes the left epoch shorter and the right epoch longer. Reducing the left epoch will
not improve performance due to greediness, but enlarging the right epoch might improve performance,
so the gain and loss must be balanced.
The loss in performance (a positive quantity) for the left epoch is
\[
\Delta_l = R_T(e_i)\log M^{*}(n_i,\eta) - R_T(e_i-1)\log M^{*}(n_i - 1,\eta)
\]
whereas the gain in performance (a positive quantity) for the right epoch is
\[
\Delta_r = R_T(e_{i+1})\left[\log M^{*}(n_{i+1} + 1,\eta) - \log M^{*}(n_{i+1},\eta)\right] \mbox{.}
\]
If $\Delta_l \ge \Delta_r$, then perturbation will not improve performance.
The condition may be rearranged as
\[
\frac{R_T(e_i)\log M^{*}(n_i,\eta) - R_T(e_i-1)\log M^{*}(n_i - 1,\eta)}{R_T(e_{i+1})\left[\log M^{*}(n_{i+1} + 1,\eta) - \log M^{*}(n_{i+1},\eta)\right]} \ge 1
\]
This is the condition \eqref{eq:star}, so the left-perturbation does not improve
performance.
Hence, the solution produced by the greedy algorithm is locally optimal.
\end{IEEEproof}
\begin{prop}
The solution produced by the greedy algorithm, $(n_i)$, is globally optimal if
\begin{equation}
\label{eq:starK}
\frac{R_T(e_i)\log M^{*}(n_i,\eta) - R_T(e_i-K_i)\log M^{*}(n_i - K_i,\eta)}{R_T(e_{i+1})\left[\log M^{*}(n_{i+1} + K_i,\eta) - \log M^{*}(n_{i+1},\eta)\right]} \ge 1
\end{equation}
for each $i$, and any non-negative integers $K_i \le n_i$.
\end{prop}
\begin{IEEEproof}
The result follows by repeating the argument for local optimality in Prop.~\ref{prop:localopt}
for shifts of any admissible size $K_i$.
\end{IEEEproof}
There is an easily checked special case of global optimality condition \eqref{eq:starK} under
the Strassen approximation, given in the forthcoming Prop.~\ref{prop:epochsizeorder}.
\begin{lemma}
\label{lem:nondecr}
The function $\log M^{*}_{S}(z,\eta) - \log M^{*}_{S}(z-K,\eta)$ is a non-decreasing
function of $z$ for any $K$, where
\begin{equation}
\label{eq:Strass}
\log M^{*}_{S}(z,\eta) = zC - \sqrt{z\rho} Q^{-1}(\eta)
\end{equation}
is Strassen's approximation.
\end{lemma}
\begin{IEEEproof}
Essentially follows from the fact that $\sqrt{z}$ is a concave $\cap$
function in $z$. More specifically $\sqrt{z}$ satisfies
\[
-\sqrt{z} + \sqrt{z - K} \le -\sqrt{z+1} + \sqrt{z+1-K}
\]
for $K \le z$. This implies:
\[
- \sqrt{z}\sqrt{\rho} Q^{-1}(\eta) + \sqrt{z-K}\sqrt{\rho} Q^{-1}(\eta) \le - \sqrt{z+1}\sqrt{\rho} Q^{-1}(\eta) + \sqrt{z+1-K}\sqrt{\rho} Q^{-1}(\eta)\mbox{.}
\]
Adding the positive constant $KC$ to both sides, in the form $zC - zC + KC$ on the left and in the form $(z+1)C - (z+1)C + KC$ on the right
yields
\begin{align*}
&zC - \sqrt{z\rho} Q^{-1}(\eta) - (z-K)C + \sqrt{z-K}\sqrt{\rho} Q^{-1}(\eta) \\ \notag
&\quad \le (z+1)C - \sqrt{z+1}\sqrt{\rho} Q^{-1}(\eta) - (z+1-K)C + \sqrt{z+1-K}\sqrt{\rho} Q^{-1}(\eta)
\end{align*}
and so
\[
\left[\log M^{*}_{S}(z,\eta) - \log M^{*}_{S}(z-K,\eta)\right] \le \left[\log M^{*}_{S}(z+1,\eta) - \log M^{*}_{S}(z+1-K,\eta)\right]\mbox{.}
\]
\end{IEEEproof}
\begin{prop}
\label{prop:epochsizeorder}
If the solution produced by the greedy algorithm using Strassen's approximation
\eqref{eq:Strass} satisfies $n_1 \ge n_2 \ge \cdots \ge n_k$,
then condition \eqref{eq:starK} for global optimality is satisfied.
\end{prop}
\begin{IEEEproof}
Since $R_T(\cdot)$ is a non-increasing survival function,
\begin{equation}
\label{eq:survival}
R_T(e_i - K) \ge R_T(e_{i+1})
\end{equation}
for the non-negative integer $K$.
Since the function $\left[\log M^{*}_{S}(z,\eta) - \log M^{*}_{S}(z-K,\eta)\right]$ is a non-decreasing
function of $z$ by Lem.~\ref{lem:nondecr}, and since the $n_i$ are in non-increasing order,
\begin{equation}
\label{eq:monot}
\log M^{*}_{S}(n_i,\eta) - \log M^{*}_{S}(n_i - K,\eta) \ge \log M^{*}_{S}(n_{i+1} + K,\eta) - \log M^{*}_{S}(n_{i+1},\eta)\mbox{.}
\end{equation}
Taking products of \eqref{eq:survival} and \eqref{eq:monot} and
rearranging yields the condition:
\[
\frac{R_T(e_i - K)\left[ \log M^{*}_{S}(n_i,\eta) - \log M^{*}_{S}(n_i - K,\eta) \right]}{R_T(e_{i+1})\left[ \log M^{*}_{S}(n_{i+1} + K,\eta) - \log M^{*}_{S}(n_{i+1},\eta)\right]} \ge 1 \mbox{.}
\]
Since $R_T(\cdot)$ is a non-increasing survival function,
\[
R_T(e_i - K) \ge R_T(e_i) \ge R_T(e_{i+1}) \mbox{.}
\]
Therefore the global optimality condition \eqref{eq:starK} is also satisfied, by substituting $R_T(e_i)$ for $R_T(e_i - K)$
in one place.
\end{IEEEproof}
\subsection{Geometric Death Distribution}
A common failure mode for systems that do not age
is a geometric death time $T$ \cite{Davis1952}:
\[
p_T(t) = \alpha(1-\alpha)^{(t-1)} \mbox{,}
\]
and
\[
R_T(t) = (1-\alpha)^t \mbox{,}
\]
where $\alpha$ is the death time parameter.
\begin{prop}
\label{prop:equalsize}
When $T$ is geometric, then the solution to \eqref{eq:optim}
under Strassen's approximation yields equal epoch sizes.
This optimal size is given by
\[
\argmax_{\nu} R_T(\nu) \log M^{*}(\nu,\eta) \mbox{.}
\]
\end{prop}
\begin{IEEEproof}
Begin by showing that Algorithm~\ref{algo:greed} will produce a solution with equal epoch sizes.
Recall that the survival function of a geometric random variable with parameter $0 < \alpha \le 1$ is
$R_T(t) = (1-\alpha)^t$. Therefore the first step of the algorithm will choose $n_1$ as
\[
n_1 = \argmax_{\nu} (1-\alpha)^{\nu} \log M^{*}(\nu,\eta) \mbox{.}
\]
The second step of the algorithm will choose
\begin{align*}
n_2 &= \argmax_{\nu} (1-\alpha)^{n_1}(1-\alpha)^{\nu} \log M^{*}(\nu,\eta) \\ \notag
&= \argmax_{\nu} (1-\alpha)^{\nu} \log M^{*}(\nu,\eta) \mbox{,}
\end{align*}
which is the same as $n_1$. In general,
\begin{align*}
n_i &= \argmax_{\nu} (1-\alpha)^{e_{i-1}}(1-\alpha)^{\nu} \log M^{*}(\nu,\eta) \\ \notag
&= \argmax_{\nu} (1-\alpha)^{\nu} \log M^{*}(\nu,\eta) \mbox{,}
\end{align*}
so $n_1 = n_2 = \cdots$.
Such a solution satisfies $n_1 \ge n_2 \ge \cdots$
and so it is optimal by Prop.~\ref{prop:epochsizeorder}.
\end{IEEEproof}
The optimal epoch size for geometric death under Strassen's approximation can
be found analytically, \cite[Sec.~6.4.2]{Varshney2010}. Consider
the setting when the alive state corresponds to a BSC($\varepsilon$).
For fixed crossover probability $\varepsilon$ and target error probability $\eta$, the
optimal epoch size is plotted as a function of $\alpha$ in Fig.~\ref{fig:blocksizes}. The less likely the channel is to
die early, the longer the optimal epoch length.
\begin{figure}
\centering
\includegraphics[width=3.5in]{blocksizes.eps}
\caption{Optimal epoch lengths under Strassen's approximation
for an $(\varepsilon,\alpha)$ BSC-geometric channel that dies for $\varepsilon = 0.01$ and $\eta = 0.001$.}
\label{fig:blocksizes}
\end{figure}
Alternatively, rather than fixing $\eta$, one might fix the number of bits to be communicated
and find the best level of reliability that is possible. Fig.~\ref{fig:bscgeom} shows the best $\lambda_{\max} = \eta$
that is possible when communicating $5$ bits over a BSC($\varepsilon$)-geometric($\alpha$) channel that dies.
\begin{figure}
\centering
\includegraphics[width=3.5in]{bscgeom.eps}
\caption{Achievable $\eta$-reliability in sending $5$ bits
over $(\varepsilon,\alpha)$ BSC-geometric channel that dies.}
\label{fig:bscgeom}
\end{figure}
Notice that the geometric death time distribution
forms a boundary case for Prop.~\ref{prop:epochsizeorder}.
One can consider discrete Weibull death time distributions \cite{KhanKA1989}
to see what happens with heavier tails:
\[
p_T(t) = (1-\alpha)^{(t-1)^{\beta}} - (1-\alpha)^{t^{\beta}} \mbox{,}
\]
and
\[
R_T(t) = (1-\alpha)^{t^{\beta}} \mbox{,}
\]
where $\beta$ is the shape parameter. When $\beta > 1$, the tail is
lighter than geometric and when $\beta < 1$, the tail is heavier than
geometric.
With heavy-tailed death distributions, the greedy algorithm gives
epoch sizes that are non-increasing: $n_1 \ge n_2 \ge \cdots$, and
therefore optimal;
it is better to send long blocks first and then send shorter ones.
\subsection{Dynamic Programming}
\label{sec:DP}
The greedy algorithm of the previous section solves \eqref{eq:optim}
under certain conditions. For finite $N$, a dynamic program (DP)
may be used to solve \eqref{eq:optim} under any conditions.
To develop the DP formulation \cite{Bertsekas2005}, we assume that
channel state feedback (whether the channel output is $?$ or whether it is some other symbol)
is available to the transmitter, however solving the DP will show that channel
state feedback is not required.
\emph{System Dynamics:}
\begin{equation}
\label{eq:dynamics}
\begin{bmatrix} \zeta_n \\ \omega_n \end{bmatrix} = \begin{bmatrix} (\zeta_{n-1} + 1)\hat{s}_{n-1} \\ \omega_{n-1}\kappa_{n-1} \end{bmatrix} \mbox{,}
\end{equation}
for $n = 1,2,\ldots,N+1$. The following state variables, disturbances, and controls are used:
\begin{itemize}
\item $\zeta_n \in \mathbb{Z}^{*}$ is a state variable that counts the location in the current transmission epoch,
\item $\omega_n \in \{0,1\}$ is a state variable that indicates whether the channel is alive ($1$) or dead ($0$),
\item $\kappa_n \in \{0,1\} \sim \Bern\left( R_T(n)\right)$ is a disturbance that kills ($0$) or revives ($1$) the channel in the next time step, and
\item $\hat{s}_n \in \{0,1\}$ is a control input that starts ($0$) or continues ($1$) a transmission epoch in the next time step.
\end{itemize}
\emph{Initial State:}
Since the channel starts alive (note that $R_T(1) = 1$) and since the first transmission epoch starts at the beginning of time,
\begin{equation}
\label{eq:initialstate}
\begin{bmatrix} \zeta_1 \\ \omega_1\end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \mbox{.}
\end{equation}
\emph{Additive Cost:} Transmission volume $\log M^{*}(\zeta_{n} + 1,\eta)$ is credited if the channel is alive (i.e.\ $\omega_n = 1$)
and the transmission epoch is to be restarted in the next time step (i.e.\ $1-\hat{s}_n = 1$). This implies a cost function
\begin{equation}
\label{eq:additivecost}
c_n(\zeta_n,\omega_n,\hat{s}_n) = -(1-\hat{s}_n)\omega_n \log M^{*}(\zeta_n + 1,\eta) \mbox{.}
\end{equation}
This is negative so that smaller is better.
\emph{Terminal Cost:} There is no terminal cost: $c_{N+1} = 0$.
\emph{Cost-to-go:} From time $n$ to time $N+1$ is:
\[
\E_{\vec{\kappa}}\left\{ \sum_{i=n}^N c_i(\zeta_i,\omega_i,\hat{s}_i)\right\}
= -\E_{\vec{\kappa}}\left\{ \sum_{i=n}^N (1-\hat{s}_i)\omega_i \log M^{*}(\zeta_i + 1,\eta)\right\} \mbox{.}
\]
Notice that the state variable $\zeta_n$ which counts epoch time is known to the transmitter and
is determinable by the receiver through transmitter simulation. The state variable $\omega_n$ indicates
the channel state and is known to the receiver by observing the channel output. It may be communicated
to the transmitter through the channel state feedback. The following result follows directly.
\begin{prop}
A communication scheme that follows the dynamics \eqref{eq:dynamics} and additive cost \eqref{eq:additivecost}
achieves the transmission time-volume
\[
\left(N, V = -\E\left[ \sum_{n = 1}^N c_n \right]\right)
\]
at $\eta$-reliability.
\end{prop}
DP may be used to find the optimal control policy $(\hat{s}_n)$.
\begin{prop}
\label{prop:DP}
The optimal $-V$ for the initial state \eqref{eq:initialstate}, dynamics \eqref{eq:dynamics},
additive cost \eqref{eq:additivecost}, and no terminal cost
is equal to the cost of the solution produced by the dynamic programming algorithm.
\end{prop}
\begin{IEEEproof}
The system described by initial state \eqref{eq:initialstate}, dynamics \eqref{eq:dynamics},
and additive cost \eqref{eq:additivecost} is in the form of the \emph{basic problem} of
dynamic programming \cite[Sec.~1.2]{Bertsekas2005}.
Thus the result follows from \cite[Prop.~1.3.1]{Bertsekas2005}
\end{IEEEproof}
The DP optimization computations are now carried out; standard $J$ notation is
used for cost \cite{Bertsekas2005}.
The base case at time $N+1$ is
\[
J_{N+1}(\zeta_{N+1},\omega_{N+1}) = c_{N+1} = 0 \mbox{.}
\]
In proceeding backwards from time $N$ to time $1$:
\[
J_n(\zeta_n,\omega_n) = \min_{\hat{s}_n\in\{0,1\}} \E_{\kappa_n}\left\{ c_n(\zeta_n,\omega_n,\hat{s}_n) + J_{n+1}\left(f_n(\zeta_n,\omega_n,\hat{s}_n,\kappa_n)\right) \right\} \mbox{,}
\]
for $n = 1,2,\ldots,N$, where
\begin{align*}
f_n(\zeta_n,\omega_n,\hat{s}_n,\kappa_n) &= \begin{bmatrix}\zeta_{n+1}&\omega_{n+1}\end{bmatrix}^T \\
&= \begin{bmatrix}(\zeta_n + 1)\hat{s}_n&\omega_n \kappa_n\end{bmatrix}^T\mbox{.}
\end{align*}
Substituting our additive cost function yields:
\begin{align}
\label{eq:dropout}
J_n(\zeta_n,\omega_n) &= \min_{\hat{s}_n\in\{0,1\}} -\E_{\kappa_n}\left\{ (1-\hat{s}_n)\omega_n\log M^{*}(\zeta_n + 1,\eta)\right\} + \E_{\kappa_n}\{J_{n+1}\} \\ \notag
&= \min_{\hat{s}_n\in\{0,1\}} -(1-\hat{s}_n)R_T(n)\log M^{*}(\zeta_n + 1,\eta) + \E_{\kappa_n}\{J_{n+1}\} \mbox{.}
\end{align}
Notice that the state variable $\omega_n$ dropped out of the first term when we took the expectation
with respect to the disturbance $\kappa_n$. This is true for each stage in the DP.
\begin{prop}
For a channel that dies $(\mathcal{X},p_a(y|x),p_d(y|x),p_T(t),\mathcal{Y})$, channel state feedback does not improve performance.
\end{prop}
\begin{IEEEproof}
By repeating the expectation calculation in \eqref{eq:dropout} for each stage
$n$ in the stage-by-stage DP algorithm, it is verified that state variable $\omega$ does not enter into the
stage optimization problem. Hence the transmitter does not require channel state feedback to
determine the optimal signaling strategy.
\end{IEEEproof}
\subsection{A Dynamic Programming Example}
\label{sec:example}
To provide some intuition on the choice of epoch lengths, we present a short example.
Consider the channel that dies with $\mathcal{X} = \{0,1\}$, $\mathcal{Y} = \{0,1,?\}$,
$p_a(y|x)$ given by \eqref{eq:pa} with $\varepsilon = 0.01$, $p_d(y|x)$ given by \eqref{eq:pd},
and $p_T(t)$ that is uniform over a finite horizon of length $40$ (disallowing death
in the first time step):
\[
p_T(t) = \begin{cases} 1/39, & t = 2,\ldots,40, \\ 0 & \mbox{otherwise.}\end{cases}
\]
Our goal is to communicate with $\eta$-reliability, $\eta = 0.001$.
Since the death distribution has finite support, there is no benefit to transmitting
after death is guaranteed. Suppose some sequence of $n_i$s is chosen arbitrarily:
$(n_1 = 13, n_2 = 13, n_3 = 13, n_4 = 1)$. This has expected transmission volume (under the Strassen approximation)
\begin{align*}
V &= \sum_{i=1}^4 R_T(e_i) \log M^{*}(n_i,\eta) \\ \notag
&\stackrel{(a)}{=} \log M^{*}(13,0.001) \sum_{i=1}^3 R_T(e_i) \\ \notag
&= \log M^{*}(13,0.001)[R_T(13) + R_T(26)+ R_T(39)] \\ \notag
&= 4.600[9/13 + 14/39 + 1/39] = 4.954 \mbox{ bits.}
\end{align*}
where (a) removes the fourth epoch since uncoded transmission cannot achieve $\eta$-reliability.
If we run the DP algorithm to optimize the ordered integer partition, we
get the result $(n_1 = 20, n_2 = 12, n_3 = 6, n_4 = 2)$.\footnote{Equivalently
$(n_1 = 20, n_2 = 12, n_3 = 6, n_4 = 1, n_5 = 1)$, since the last two channel usages
are wasted (see Fig.~\ref{fig:block_vol}) to hedge against channel death.}
Notice that since the solution is in order, the greedy algorithm would also
have succeeded. The expected transmission volume for this strategy (under the Strassen approximation) is
\begin{align*}
V &= R_T(20)\log M^{*}(20,0.001) + R_T(32)\log M^{*}(12,0.001) + R_T(38)\log M^{*}(6,0.001) \\ \notag
&= (20/39)\cdot 9.2683 + (8/39)\cdot 3.9694 + (2/39)\cdot 0.5223 \\ \notag
&= 5.594 \mbox{ bits.}
\end{align*}
\subsection{A Precise Solution}
It has been assumed that optimal finite block length codes
are known and used. Moreover, the Strassen approximation has been used for certain
computations. It is, however, also of interest to determine precisely which code should be
used over a channel that dies. This subsection gives an example where a sequence of length-$23$ binary Golay
codes \cite{Golay1949} are optimal. Similar examples may be developed for other perfect codes;
a perfect code is one for which there are equal-radius spheres centered at the
codewords that are disjoint and that completely fill $\mathcal{X}^{n_i}$.
Before presenting the example, the sphere-packing upper bound on $\log M^{*}(n_i,\eta)$ for a
BSC($\varepsilon$) is derived. Recall the notion of decoding radius \cite{Blahut1983}
and let $\rho(\varepsilon,\eta)$ be the largest integer such that
\[
\sum_{s=0}^{\rho} \binom{n_i}{s} \varepsilon^{s} (1-\varepsilon)^{n_i - s} \le 1 - \eta \mbox{.}
\]
The sphere-packing bound follows from counting how many decoding regions
of radius $\rho$ could conceivably fit in the Hamming space $2^{n_i}$ disjointly.
Let $D_{s,m}$ be the number of channel output sequences that are decoded into message $w_m$
and have distance $s$ from the $m$th codeword. By the nature of Hamming space,
\[
D_{s,m} \le \binom{n_i}{s}
\]
and due to the volume constraint,
\[
\sum_{m=1}^M \sum_{s = 0}^{\rho} D_{s,m} \le 2^{n_i} \mbox{.}
\]
Hence, the maximal codebook size $M^{*}(n_i,\eta)$ is upper-bounded as
\begin{align*}
M^{*}(n_i,\eta) &\le \frac{2^{n_i}}{\sum_{s=0}^{\rho} D_{s,m} } \\ \notag
&\le \frac{2^{n_i}}{\sum_{s=0}^{\rho(\varepsilon,\eta)} \binom{n_i}{s} } \mbox{.}
\end{align*}
Thus the sphere-packing upper bound on $\log M^{*}(n_i,\eta)$ is
\[
\log M^{*}(n_i,\eta) \le n_i - \log\left[ \sum_{s=0}^{\rho(\varepsilon,\eta)} \binom{n_i}{s} \right] \triangleq \log M_{sp}(n_i,\eta) \mbox{.}
\]
Perfect codes such as the binary Golay code
of length $23$ can sometimes achieve the sphere-packing bound with equality.
Consider an $(\varepsilon,\alpha)$ BSC-geometric channel that dies, with $\varepsilon = 0.01$ and $\alpha = 0.05$.
The target error probability is fixed at $\eta = 2.9 \times 10^{-6}$. For these values of $\varepsilon$ and $\eta$,
the decoding radius $\rho(\varepsilon,\eta) = 1$ for $2 \le n_i \le 3$. It is $\rho(\varepsilon,\eta) = 2$ for $4 \le n_i \le 10$;
$\rho(\varepsilon,\eta) = 3$ for $11 \le n_i \le 23$; $\rho(\varepsilon,\eta) = 4$ for $24 \le n_i \le 40$; and so on.
Moreover, one can note that the $(n = 23, M = 4096)$ binary Golay code has a decoding radius of $3$; thus it meets the
BSC sphere-packing bound
\[
M_{sp}(23,2.9 \times 10^{-6}) = \frac{2^{23}}{1 + 23 + 253 + 1771} = 4096
\]
with equality.
Now to bring channel death into the picture. If one proceeds greedily, following Algorithm~\ref{algo:greed},
but using the sphere-packing bound $\log M_{sp}(n_i,\eta)$ rather than the optimal $\log M^{*}(n_i,\eta)$,
\begin{align*}
&n_1(\varepsilon = 0.01,\alpha = 0.05, \eta = 2.9 \times 10^{-6}) \\
&\quad = \argmax_{\nu} \bar{\alpha}^{\nu} \log_2 \frac{2^{\nu}}{\sum_{s=0}^{\rho(\varepsilon,\eta)}} = 23 \mbox{.}
\end{align*}
By the memorylessness argument of Prop.~\ref{prop:equalsize}, it follows that
running Algorithm~\ref{algo:greed} with the sphere-packing bound will yield $23 = n_1 = n_2 = \cdots$.
It remains to show that Algorithm~\ref{algo:greed} actually gives the true solution. Had Strassen's
approximation been used rather than the sphere-packing bound, the result would follow directly from Prop.~\ref{prop:equalsize}.
Instead, the global optimality condition \eqref{eq:starK} can be verified exhaustively for
all $23$ possible shift sizes $K$ for the first epoch:
\[
\frac{\bar{\alpha}^{23} \log M_{sp}(23,\eta) - \bar{\alpha}^{23-K}\log M_{sp}(23-K,\eta)}{\bar{\alpha}^{46}\log M_{sp}(23+K) - \bar{\alpha}^{46}\log M_{sp}(23,\eta)} \ge 1 \mbox{.}
\]
Then the same exhaustive verification is performed
for all $23$ possible shifts for the second epoch:
\begin{align*}
\frac{ \bar{\alpha}^{46} \log M_{sp}(23,\eta) - \bar{\alpha}^{46-K}\log M_{sp}(23-K,\eta) }{ \bar{\alpha}^{69}\log M_{sp}(23+K) - \bar{\alpha}^{69}\log M_{sp}(23,\eta)} &\ge 1 \\ \notag
\frac{ \bar{\alpha}^{23} \left[\bar{\alpha}^{23} \log M_{sp}(23,\eta) - \bar{\alpha}^{23-K}\log M_{sp}(23-K,\eta)\right] }{ \bar{\alpha}^{23} \left[\bar{\alpha}^{46}\log M_{sp}(23+K) - \bar{\alpha}^{46}\log M_{sp}(23,\eta)\right]} &\ge 1 \\ \notag
\frac{ \bar{\alpha}^{23} \log M_{sp}(23,\eta) - \bar{\alpha}^{23-K}\log M_{sp}(23-K,\eta) }{ \bar{\alpha}^{46}\log M_{sp}(23+K) - \bar{\alpha}^{46}\log M_{sp}(23,\eta)} &\ge 1 \mbox{.}
\end{align*}
The exhaustive verification can be carried out indefinitely to show that using the length-$23$
binary Golay code for every epoch is optimal.
\subsection{Practical Codes and Empirical Death Distributions}
It should be noted that the algorithms developed for optimizing communication schemes over channels that die
work with arbitrary death distributions, even empirically measured ones, e.g.\ the experimentally
characterized death properties of a synthetic biology communication system \cite[Fig.~3: Reliability]{CantonLE2008}.
Further, rather than considering the $\log M^{*}(n_i, \eta)$ function for optimal finite block length
codes, the code optimization procedures would work just as well if a collection of finite block length codes was
provided. Such a limited set of codes might be selected for decoding complexity or
other practical reasons. As an example, consider the collection $\mathcal{C}$ of $9191$ binary minimum distance codes of lengths
between $6$ and $16$ given in \cite[DVD supplement]{KaskiO2006}. We run the optimization
over the example in Sec.~\ref{sec:example} but restricting to $\mathcal{C}$.
The result obtained for epoch sizes is $(n_1 = 15, n_2 = 15, n_3 = 9, n_4 = 1)$. Under the
Strassen approximation, this set of epoch sizes gives $5.344$ bits, as compared to $5.594$ bits under
the optimal epoch sizes under the Strassen approximation. However the Strassen approximation is not
correct and the actual number of bits achieved with the optimized epoch sizes for $\mathcal{C}$ is
$7.246$ bits. The two minimum distance codes used are the $(n = 15, M = 256, d = 5)$ code and
the $(n = 9, M = 6, d = 3)$ code. It remains to be seen whether the restriction to the collection
of minimum distance codes is actually suboptimal.
\section{Partial Ordering of Channels}
\label{sec:ordering}
It is of interest to order channels that die by quality. The partial ordering
of DMCs was studied by Shannon \cite{Shannon1958}, and as a first step, we
can slightly extend his result to order channels that die having common death distributions.
\begin{defin}
Let $p(i,j)$ be the transition probabilities for a DMC $C_1$ and let $q(k,l)$ be the
transition probabilities for a DMC $C_2$. Then $C_1$ is said to include $C_2$, $C_1 \supseteq C_2$,
if there exist two sets of valid transition probabilities $r_{\gamma}(k,i)$ and $t_{\gamma}(j,l)$, and
there exists a vector $g$: $g_{\gamma} \ge 0$ and $\sum_{\gamma} g_{\gamma} = 1$, such that
\[
\sum_{\gamma,i,j} g_{\gamma} r_{\gamma}(k,i) p(i,j) t_{\gamma}(j,l) = q(k,l) \mbox{.}
\]
\end{defin}
\begin{prop}
Consider two channels that die with identical death distributions:
$(\mathcal{X}_1,p_a,p_d,p_T(t),\mathcal{Y}_1)$ and $(\mathcal{X}_2,q_a,q_d,p_T(t),\mathcal{Y}_2)$.
Let DMC $C_1$ correspond to $p_a$ and let DMC $C_2$ correspond to $q_a$ and moreover suppose
that $C_1 \supseteq C_2$. Fix a transmission time $N$ and an expected transmission volume $V$.
Let $\eta_1$ be the best level of reliability for the first channel and $\eta_2$ be the
best level of reliability for the second channel, under $(N,V)$.
Then $\eta_1 \le \eta_2$.
\end{prop}
\begin{IEEEproof}
The main theorem of \cite{Shannon1958} proves that the average error probability
when transmitting an individual message code over $C_1$ is less than or equal to
the average error probability when transmitting the same individual message code over $C_2$.
Shannon's proof \cite{Shannon1958} holds \emph{mutatis mutandis} for maximum error probability, replacing ``average error
probability'' by ``maximum error probability.''
The desired result follows by concatenating individual message codes into a code.
\end{IEEEproof}
We can also order channels that die having common alive state transition probabilities.
\begin{defin}
Consider two random variables $T$ and $U$ with survival functions $R_T(\cdot)$ and $R_U(\cdot)$ respectively.
Then $U$ is said to stochastically dominate $T$, $U \ge_{{\rm st}} T$, if $R_T(t) \le R_U(t)$ for all $t$.
\end{defin}
\begin{prop}
Consider two channels that die with identical state properties:
$(\mathcal{X},p_a(y|x),p_d(y|x),p_T,\mathcal{Y})$ and $(\mathcal{X},p_a(y|x),p_d(y|x),q_U,\mathcal{Y})$.
Let death random variable $T$ correspond to $p_T$ and let death random variable $U$ correspond to $q_U$
and moreover suppose that $U \ge_{{\rm st}} T$. Fix a transmission time $N$ and a level of
reliability $\eta$. Let $V_1$ be the best expected transmission volume for the first channel and
$V_2$ be the best expected transmission volume for the second channel, under $(N,\eta)$.
Then $V_2 \ge V_1$.
\end{prop}
\begin{IEEEproof}
Recall the expected transmission volume expression \eqref{eq:optim} for the first channel:
\[
\max_{(n_i): \sum n_i = N} \sum_{i} R_T(e_i) \log M^{*}(n_i,\eta)
\]
and for the second channel:
\[
\max_{(\nu_i): \sum \nu_i = N} \sum_{i} R_U(\iota_i) \log M^{*}(\nu_i,\eta) \mbox{.}
\]
Since $R_T(t) \le R_U(t)$ for all $t$, the result follows directly.
\end{IEEEproof}
These two results give individual ordering principles in the two dimensions
essentially depicted in Fig.~\ref{fig:bscgeom}. Putting them together provides a
partial order on all channels that die: if one channel is better than another
channel in both dimensions, than it is better overall.
\begin{prop}
Consider two channels that die: $(\mathcal{X}_1,p_a,p_d,p_T,\mathcal{Y}_1)$ and $(\mathcal{X}_2,q_a,q_d,q_U,\mathcal{Y}_2)$.
Let DMC $C_1$ correspond to $p_a$ and let DMC $C_2$ correspond to $q_a$ and moreover suppose
that $C_2 \supseteq C_1$. Let death random variable $T$ correspond to $p_T$ and let death random variable $U$ correspond to
$q_U$ and moreover suppose that $U \ge_{{\rm st}} T$. Fix a transmission time $N$ and a level of
reliability $\eta$. Let $V_1$ be the best expected transmission volume for the first channel and
$V_2$ be the best expected transmission volume for the second channel, under $(N,\eta)$.
Then $V_2 \ge V_1$.
\end{prop}
\section{Conclusion and Future Work}
\label{sec:conc}
We have formulated the problem of communication
over channels that die and have shown how to maximize expected
transmission volume at a given level of error probability reliability.
There are several extensions to the basic formulation studied in this work
that one might consider; we list a few:
\begin{itemize}
\item Inspired by synthetic biology \cite{CantonLE2008}, rather than thinking of
death time as independent of the signaling scheme $X_1^n$, one might consider
channels that die because they lose fitness as a consequence of operation: $T$ would
be dependent on $X_1^n$. This would be similar to Gallager's panic button/child's toy channel,
and would have intersymbol interference \cite{Gallager1968,Gallager1972}.
There would also be strong connections to channels that heat up \cite{KochLS2009} and
communication with a dynamic cost \cite[Ch.~3]{Eswaran2009}.
\item In the emerging attention economy \cite{DavenportB2001}, agents faced with information overload
\cite{VanZandt2004} may permanently stop listening to certain communication media received over
noisy channels. This setting is exactly modeled by channels that die. The impact of communication
over channels that die on the productivity and efficiency of human organizations may be determined
by building on the results herein.
\item Since channel death is indicated by the symbol $?$, the receiver unequivocally
knows death time. Other channel models might not have a distinct output letter
for death and would need to detect death, perhaps using the theory
of estimating stopping times \cite{NiesenT2009}.
\item Inspired by communication terminals that randomly lie within communication
range, e.g. in vehicular communication, one might also consider a channel that is born at a
random time and then dies at a random time. One would suspect that channel state feedback would be
beneficial. Networks of birth-death channels
are also of interest and would have connections to percolation-style work \cite{Jacobs1959}.
\item This work has simply considered the channel coding problem, however there are several
formulations of end-to-end information transmission problems over channels that die, which
are of interest in many application areas. There is no reason to suspect a separation principle.
\end{itemize}
Randomly stepping back from infinity leads to some new understanding of the fundamental
limits of communication in the presence of noise and unreliability.
\section*{Acknowledgment}
We thank Barry Canton (Ginkgo BioWorks) and Drew Endy (Stanford University)
for discussions on synthetic biology that initially inspired this work.
Discussions with Baris Nakiboglu and Yury Polyanskiy (Princeton University)
are also appreciated.
\bibliographystyle{IEEEtran}
\bibliography{abrv,conf_abrv,lrv_lib}
\end{document}
| 143,985
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For the tour overview: an introduction to Walter Bonatti and his climbing career; the trip concept; locations of all three chapters (cimas); our bikes; and our bikepacking kit list >> head over to our Bonatti Bikepacking Tour Overview…
55km
2000m+
Hut
(Dolomites)
An example of Walter Bonatti’s first expeditions, 1952 & 1953…
After spending our last few days weaving through the mountains amongst either derelict, or very-much-out-of-season chairlift infrastructures and establishments, Cortina d’Ampezzo made a nice change – a perennial tourist town and Outdoor hub with several cable car options, gravel routes and world-famous passes to explore by bike.
Jordan, Dan and I parked the van up pronto and after stocking up on some key Italian Ferrero snacks, rode to the Faloria cable car that shot us up into the heartland of the UNESCO Dolomites, loaded bikes in tow. The change in look and feel of our new Alpine surroundings was remarkable; grainy and true to their ‘Pale Alp’ name compared with the gigantic granite of Courmayeur. No doubt a rock-type technical climbing change that Bonatti had to get used to, when he first arrived to climb here in the early 1950s. Our aim for the day was to reach and stay at one of the rifugios up around the Tre Cima di Lavaredo – the three huge vertical stone fingers, known as the Cima Piccola (Small), Cima Grande (Large) and Cima Ovest (West) that were “the ‘last great problems’ of the thirties until climbed in 1937, and then in winter 1953 by Bonatti.”
Unlike Bonatti, who arrived by train into (the now defunct) Carbonin station and walked/skied up with all his expedition kit in deep snow, the three of us would enjoy the gravel mountain bike and ski-service tracks, before joining the Tre Croci pass, and eventually climbing the somewhat stunning Tre Cime pass up to the famous peaks, and stunning jagged limestone moonscape…
Military Service in the Alpini had meant Bonatti spent fifteen months in the mountains, and it was during this time that the idea for climbing the north face of the Lavaredo came to him – “one day I would pit myself against these giants in the way most appropriate to this austere place—in midwinter, when the north faces are hidden from even the weakest ray of sunshine and the frost and solitude hold sway.” He trained, and returned in winter 1953 when poor Bonatti and climbing companion were so slow going up the trail to the Longeres (now Auronzo) Hut they had to bivvy on the snowy trail before it got dark, to then wake and find it closed and uninhabited anyway. An extra night out in the cold did nothing to hamper their climb and “a little after five in the afternoon on February 27 1953 [they] reached the summit of the Cima Grande, which was still lit by the last rays of the sun.” This ascent of the Lavaredo put his name well and truly on the climbing map; dwarfing our efforts on the bike, of course.
We shared the Lavaredo Hut predominantly with climbers, and spent the evening eating and sharing out the Negroni I’d mixed at the bottom of the mountain, carried up and garnished with fresh orange, under the Dolomite starlight.
Early the next morning as we packed bikes for the last time on the trip, climbers on the table opposite filled their packs whilst checking all their ropes and gear for tackling Via Ferrata and some of Bonatti’s routes in the 1950s. Funnily enough, there was an Austrian guy who’d been in Sheffield climbing at Stanage the week before, as we left for here in the van. Folk always shout about Stanage’s climbing reputation, and that kind of bolted it home for me. Anyway … our last leg, on the ace dedicated (ex-railway) cycle route between Carbonin and Cortina, was the first time we actually saw any other cyclists on either of the three mini-tours. We were overtaken by day-tripping mountain bikers and passed plenty of fully-loaded trad cycle-tourers – one entertaining group of which was towing a guy all the way back to Cortina given a rear hub and wheel failure.
It was a shame to be leaving the mountains, but a week chasing the tracks of Walter Bonatti was definitely enough of an escape on the bikes, and importantly a fun, challenging trip. As Bonatti says “Mountaineering is only one of a thousand ways of living and getting to know yourself. Climbing mountains should signify nothing more than this search for identity. It should never be mere escapism, because sooner or later we must return to our own personalities and feelings … the mountains should prepare us to go further.” I find that theming a trip really helps give focus, and reason; even more so than a simple point-to-point tour…
.”
“….”
A final bivvy night on the Col de la Madeleine, and then morning swim in Lake Annecy on the drive back home were enough to keep our escape alive.
For one more day anyway.
Onto the next trip…
WORDS & ILLUSTRATION
Stefan Amato
PHOTOGRAPHY
Jordan Gibbons
RIDERS / ADDITIONAL PHOTOS
Stefan Amato
Dan Easton
The Mountains Of My Life – Walter Bonatti
Walter Bonatti: The Vertical Dream – Angelo Ponta
Cortina d’Ampezzo (no.55) 1:50,000 – Kompass Maps
| 196,228
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To identify and critically appraise the content, readability, accessibility and usability of websites providing information on cognitive rehabilitation for the families of adults with traumatic brain injury (TBI).
Source: Cognitive Rehabilitation Information for the Families of Individuals with Traumatic Brain Injury: A Review of Online Resources – Archives of Physical Medicine and Rehabilitation
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| 167,124
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Audrey Hepburn, Alfred Hitchcock, Brad Pitt and Mick Jagger have all rested a weary head in Claridge's at one time or another. And we'll bet our bottom dollar that the bar went silent when they propped it up. But did their mere presence whip the entire hotel (and the rest of London) into a frenzy? Maybe. Madonna definitely did last Sunday, at her swanky pre-party for new film W.E. Laurent-Perrier was served to steady beating hearts. The canapés were delectable (particularly the cornets of smoked salmon muscovite) but, ever the consummate host, Madonna spent the evening chatting to guests rather than drinking champagne. Andrea Riseborough looked very much the Hollywood pin-up with her newly blonde hair (guess working with Madonna left an impression), and judging by the crowd's rapturous applause when the credits started to roll, the film will be a big hit when it comes to cinemas in January next year.
| 61,014
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- Purpose(Empty Backbone) Mammalian cell expression plasmid; drives constitutive, low-level expression from retroviral LTR; retroviral vector
- Depositing Lab
-
- Sequence Information
Ordering
This material is available to academics and nonprofits only.
Backbone
- Vector backbonemscv2.2
- Vector typeMammalian Expression, Retroviral
- Promoter LTR
Growth in Bacteria
- Bacterial Resistance(s)Ampicillin
- Growth Temperature37°C
- Growth Strain(s)DH5alpha
- Copy numberUnknown2.2 was a gift from Russell Vance (Addgene plasmid # 60206 ; ; RRID:Addgene_60206)
For your References section:Innate immune recognition of bacterial ligands by NAIPs determines inflammasome specificity. Kofoed EM, Vance RE. Nature. 2011 Aug 28;477(7366):592-5. doi: 10.1038/nature10394. 10.1038/nature10394 PubMed 21874021
| 191,013
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Joined: 31 October 2007
Posts: 9559
The following 2 member(s) liked the above post:
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UK based student tattooed his arm dedicating it to television actress ...
Mansi Sharma, who has been actively doing television shows such as ...
| 397,879
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\begin{document}
\maketitle
\begin{abstract}
We give an analysis and generalizations of some long-established constructive completeness results in terms of categorical logic and pre-sheaf and sheaf semantics.
The purpose is in no small part conceptual and organizational: from a few basic ingredients arises a more unified picture connecting constructive completeness with respect to Tarski semantics, to the extent that it is available, with various completeness theorems in terms of presheaf and sheaf semantics (and thus with Kripke and Beth semantics). From this picture are obtained both (``reverse mathematical'') equivalence results and new constructive completeness theorems; in particular, the basic set-up is flexible enough to obtain strong constructive completeness results for languages of arbitrary size and languages for which equality between the elements of the signature is not decidable.
MSC2010 classification: 03F99, 03G30.
\end{abstract}
\section{Introduction}
Starting with the G\"odel-Kreisel theorem, it has long been well known that the classically ``standard'' semantics---Tarski structures for classical first-order logic (FOL), and Kripke or Beth structures for intuitionistic FOL---are insufficient in a constructive metatheory. For instance, the assumption of strong completeness for intuitionistic FOL with respect to Tarski, Kripke, or Beth semantics in a metatheory such as \textbf{IZF}, \textbf{HAS}, or \textbf{HAA}\footnote{Intuitionistic Zermelo Fraenkel set theory; full intuitionistic second-order arithmetic; intuitionistic second-order arithmetic with arithmetic comprehension} implies the law of excluded middle (LEM), while weak completeness implies Markov's principle (MP) (see \cite{mccarty:08}, \cite{kreisel:62}, \cite{mccarty:94}). On the other hand, constructive completeness theorems exist e.g.\ with respect to formal space valued models, and in sheaf toposes more generally (see e.g.\ \cite{MP}, \cite{coquandsmith:95}, \cite{elephant1}). In a sense intermediate between sheaf semantics and the standard semantics of Tarski and Kripke, completeness was also shown (albeit assuming the Fan Theorem) to hold for ``countable'' intuitionistic first-order theories with respect to fallible Kripke and Beth semantics by Veldman \cite{veldman:76} and by de Swart \cite{deswart:76}.
The insufficiency of the standard semantics can be taken to suggest that constructive model theory should instead be carried out with respect to sheaf models, or some variant thereof. On the other hand, the theorems of Veldman and de Swart indicate that ordinary Tarski semantics can have an important role to play. Thus the underlying conceptual and motivating question of this paper is
the role of Tarski semantics in constructive model theory, or how much ``mileage'' one can get out of ordinary Tarski-models in a constructive setting.
Developments of logic in a constructive setting usually assume that the signatures are, if not in some sense countable, then at least discrete; i.e.\ that decidability (LEM) holds for equality of the basic function and relation symbols. This precludes various classical constructions, such as adding the elements of an arbitrary domain as constants to the language. Or similarly in categorical logic, constructing the internal language of an arbitrary category. Here, we drop this restriction.
In a classical metatheory (with choice), the link between Tarski completeness for classical first-order logic (or its so-called coherent fragment) and Kripke completeness for intuitionistic first-order logic is put to light in a theorem attributed to A.\ Joyal in \cite{makkaireyes}.
The theorem involves, among other things, the technique of considering a theory in terms of an ``approximation'' in a weaker fragment, known sometimes, or in some cases, as \emph{Morleyization}.
However, the theorem is not immediately applicable in a constructive setting as it relies on the assumption of Tarski completeness for the coherent fragment of FOL. The proof (\cite[Thm. 6.3.5]{makkaireyes}) also uses classical techniques.
Nevertheless, while Tarski completeness for the coherent fragment fails to hold in a constructive meta-theory, constructive completeness results for less expressive fragments of FOL exist. By giving a constructive formulation and proof of Joyal's theorem, and using the same ``approximation'' technique, these can be exploited to give completeness results for stronger fragments of FOL, and for FOL itself, in suitable pre-sheaf and sheaf toposes. For instance, the aforementioned completeness theorems for fallible Kripke and Beth semantics can be recovered (and given new proofs) in this way. The purpose and aim of this paper is in no small part to ``tell this story''; that is, to give a unified and conceptual account connecting constructive Tarski completeness, to the extent that it exists, with completeness results in terms of traditionally studied Kripke and Beth-style models. And, furthermore, to give this account using essentially only the two basic ingredients of Joyal's theorem and of Morleyization to fragments for which Tarski completeness holds. In addition to displaying the connections between already established results, such a clearer conceptual picture can also serve to suggest further, and new, ones. As an instance of this, we extend constructive completeness results to languages and theories that are not enumerable, and for which equality of non-logical symbols in the language can not be decided. (We also do not, in general, assume that the sentence $\fins{x}x=x$ is valid). In particular, we give a strong completeness theorem for the disjunctive-free fragment of FOL, over such languages, with respect to fallible Kripke models, and a strong completeness theorem for full FOL with respect to fallible, ``generalized'' Beth models. We also draw attention to a conceptual link between Beth semantics and a ``least coverage forcing the correct interpretation of disjunctions'', and give a constructive version and proof of the completeness theorem of \cite{gabbay:77} with respect to Beth models in which only the forcing clause for disjunction is used.
The paper is structured into the following parts. Section \ref{section: mod tarsk completeness} contains preliminaries and notes on notation and terminology, and recapitulates Tarski completeness for the regular fragment and enumerable coherent theories. The regular completeness theorem for arbitrary theories and signatures is more fully presented in \cite{forsselllumsdaine}. The completeness theorem for enumerable positive coherent theories is known, but is included for conceptual self-containement. (We also note the equivalence between this theorem and the Fan theorem,
and draw a corollary concerning completeness for classical first-order theories.)
Section \ref{section: joyals theorem} gives a constructive reformulation and proof of Joyal's theorem.
Section \ref{section: sheaf completeness} introduces and analyses certain coverages on the category of models, and gives a covering lemma which allows Joyal's theorem to be stated with respect to a poset of structures and homomorphic inclusions. This then linked with the instances of Tarski completeness in Section \ref{section: mod tarsk completeness} to give the aforementioned Kripke and Beth completeness results
theories over non-discrete languages. The specialization to the constructive version of the theorem of \cite{gabbay:77} marks the end.
\section{Constructive Tarski completeness}\label{section: mod tarsk completeness}
\subsection{Preliminiaries}\label{subsection: preliminaries}
\subsubsection{Theories, models and diagrams}
\label{subsection: models and diagrams}
\label{subsubsection: models and diagrams}
In what follows we shall fix our metatheory to be \textbf{IZF} and simply use ``constructive'' to mean that we are working in this setting. Correspondingly, ``classically'' means in the metatheory \alg{ZFC}.
We fix the following terminology, conventions, and notation.
By ``finite'' and ``countable'' we mean cardinal finite and isomorphic to \thry{N}, respectively. A ``list'' is a finite list. \Image{\alg{a}} denotes the set of elements of the list \alg{a}, and $l(\alg{a})$ its length. When deemed safe, we shorten $\alg{a}\in A^{{l(\alg{a})}}$ to $\alg{a}\in A$.
A subset of a set $A$ is decidable if it is given in terms of a function $f\! :\!A\to<100>2$ as \cterm{a\in A}{f(a)=1} and semi-decidable if it is given in terms of a function $f\! :\!A\to<100>2^{\thry{N}}$ as \cterm{a\in A}{\fins{n}(f(a))(n)=1}. By an \emph{enumerable}\ set we mean a semi-decidable subset of a (perhaps implicit) countable set.
Let $\Sigma$ be a first-order signature. We generally assume that $\Sigma$ is \emph{relational} in the sense that it has no function symbols. Thus all functions and constants are taken to be represented by relations and appropriate axioms over $\Sigma$.
Furthermore, we assume, that $\Sigma$ is single-sorted.
Thus $\Sigma$ is simply a set of relation symbols with associated (finite) arities. We do not assume, unless otherwise stated, that $\Sigma$ is \emph{discrete}---i.e.\ that LEM holds for equality betweeen the elements of $\Sigma$. (As usual, however, the logical symbols, including variables, are discrete, and disjoint from $\Sigma$).
Following \cite[D1]{elephant1}, we consider theories in FOL and fragments of FOL formulated in terms of sequents of the form $\phi\vdash_{\mathbf{x}}\psi$. These can be read as $\alle{\alg{x}}\phi\rightarrow \psi$.
The list of variables \alg{x} is required to be a \emph{context} for both $\phi$ and $\psi$ in the sense that it is a list of distinct variables containing (at least) the free variables of the formula (see \cite[D1.1.4]{elephant1}). We write \FIC{\alg{x}}{\phi} for a \emph{formula-in-context}. We write \syntob{\alg{x}}{\phi} for a formula-in-context identified up to $\alpha$-equivalence. (That is, up to renaming of bound variables and variables in the context, as in \cite[D1.4]{elephant1}.) The context, in both cases, is \emph{canonical} if it contains only the free variables of the formula, listed in order of first-appearance.
The logic is ``free'', in the sense that the sequent $\top\vdash \fins{x}x=x$ is, in general, not derivable.
The main fragments we shall be referring to are: the \emph{Horn} fragment\footnote{To prevent confusion with otherwise standard usage of ``Horn clause'' and ``Horn formula'', note the usage here (following \cite{elephant1}) of ``Horn formula'' as simply a formula which is a conjunction of atomic formulas, and ``Horn sequent'' as simply a sequent with such Horn formulas as antecedent and consequent. }, consisting of sequents with formulas over $\Sigma$ involving only the logical constants $\top$ and $\wedge$; the \emph{regular} fragment $\top, \wedge, \exists$; the $\mathit{regular_{\bot}}$ fragment $\top, \wedge, \exists$, and $\bot$; the \emph{positive coherent} fragment $\top, \wedge, \vee$, and $\exists$; the $\mathit{coherent}$ fragment $\top, \wedge, \vee, \exists$, and $\bot$; and, of course, full FOL.
Deduction rules and further details can be found in \cite[D1.3]{elephant1}. The distinguished relation symbol $=$ of equality is included in all languages under consideration. If a theory \theory\ proves a sequent $\phi\vdash_{\mathbf{x}}\psi$ we sometimes write $\phi\vdash_{\mathbf{x}}^{\theory}\psi$ instead of $\theory \vdash (\phi\vdash_{\mathbf{x}}\psi)$. Provable in the empty theory is then written $\phi\vdash_{\mathbf{x}}^{\emptyset}\psi$.
One would usually say that a theory is regular, for instance, if it is axiomatizable by regular sequents. Thus a Horn theory would also be a regular theory etc. For brevity, however, we also mean to indicate what fragment we are considering when we say that a theory is this or that. Thus when we say e.g.\ that \theory\ is a coherent theory and $\phi$ is a formula of \theory, we mean, in particular, that $\phi$ a formula in the coherent fragment over the signature of \theory. In the same vein, if we say that a theory is discrete, we mean that it is over a discrete signature. If we say that a theory is enumerable\ we mean both that it is over a enumerable\ signature and that the set of axioms is enumerable
We say that a coherent sequent is on \emph{canonical form} if it is on the form $\phi\vdash_{\mathbf{x}}\exists{\mathbf{y}_0}\psi_0\vee\ldots\vee\exists{\mathbf{y}_n}\psi_n$ where $\phi$ and all $\psi_i$ are Horn formulas, or on the form $\phi\vdash_{\mathbf{x}}\bot$ where $\phi$ is Horn. A regular sequent is on canonical form if it is so as a coherent sequent. Every coherent ($\mathrm{regular}_{\bot}$, regular) theory can be axiomatized by coherent ($\mathrm{regular}_{\bot}$, regular) sequents on canonical form (see \emph{ibid.}), and we assume that they are.
By \emph{ Tarski structure} for a signature, and \emph{Tarski model} for a theory, we mean the usual notion of a domain set with interpretations of the relation symbols in terms of subsets, and the interpretations of the connectives $\bot$, $\top$, $\wedge$, $\vee$, and $\exists$ by the usual set-theoretic interpretations (as well as $\rightarrow$ and $\forall$, but we do not actually consider Tarski models for anything above the coherent fragment, with the exception of Corollary \ref{corollary: completeness of classical fo theories}).
The domain need not be inhabited or non-empty.
As for the interpretation of the equality relation, we reserve ``structure'' and ``model'' for the case where equality is interpreted as the identity relation, and use \emph{diagram} and \emph{model diagram} for the case where equality is interpreted as a congruence relation\footnote{We avoid using ``diagram'' when extending a language with a structure, talking instead of the language and theory of the structure.}. That is to say, a diagram for a relational signature $\Sigma$ consists of a $\Sigma$-structure \alg{M} together with an equivalence relation $E$ on $|\alg{M}|$ which respects the relations interpreting the symbols of $\Sigma$. The interpretation $\csem{\alg{x}}{\phi}^{\alg{M}}$ of a formula-in-context \FIC{\alg{x}}{\phi} is then defined in the usual way, but interpreting $=$ as $E$.
In part to notationally distinguish diagrams, in this sense, from structures, we consider and write a diagram \alg{M} as a pair $(D,F)$ where $D=|\alg{M}|$ is the domain and $F=\cterm{\pair{\syntob{\alg{x}}{\phi},\alg{d}}}{\phi\ \textnormal{is Horn, and}\ \alg{d}\in\csem{\alg{x}}{\phi}^{\alg{M}} }$, where $\csem{\alg{x}}{\phi}^{\alg{M}}$ is the extension of \FIC{\alg{x}}{\phi} in \alg{M}.
We refer to an element of $F$ as a \emph{fact}.
For a signature $\Sigma$ and theory \theory\ we write \strcat{\Sigma} and \modcat{\theory} for the category of structures and homomorphisms and models and homomorphisms, respectively. We write \strdiag{\Sigma} and \moddiag{\theory} for the category of diagrams and homomorphisms and model diagrams and homomorphisms, respectively, where a homomorphism $h:(D_1, F_1)\to<150>(D_2, F_2)$ between diagrams is a left-total relation $h\subseteq D_1\times D_2$ such that
\begin{enumerate}
\item $h(d_1,d_2)\wedge\pair{\syntob{x,y}{x=y},d_2,d'_2}\in F_2\rightarrow h(d_1,d'_2)$; and
\item $\pair{\syntob{\alg{x}}{\phi},\alg{d}_1}\in F_1)\wedge h(\alg{d}_1,\alg{d}_2)\rightarrow (\pair{\syntob{\alg{x}}{\phi},\alg{d}_2}\in F_2)$, for all (atomic) $\mathrm{Horn}$ formulas $\phi$ over $\Sigma$ and all $\alg{d}_1\in D_1$ and $\alg{d}_2\in D_2$.
\end{enumerate}
(Here $h(\alg{d}_1,\alg{d}_2)$ stands for the expected conjunction. As further notational shortcuts, we allow ourselves to use function notation for homomorphisms between diagrams when no confusion threatens. For instance we might write $\phi[h(d_1)/x]$ instead of $\alle{d_2}h(d_1,d_2) \rightarrow \phi[d_2/x]$. We sometimes write $\phi[\alg{d}/\alg{x}]\in F$, or simply $\phi[\alg{d}]\in F$, instead of $\pair{\alg{d},\syntob{\alg{x}}{\phi}}\in F$, for brevity.)
Then there is an adjoint equivalence
\[\bfig \morphism/{@{>}@/^1em/}/<1000,0>[\strcat{\Sigma}`\strdiag{\Sigma};i]
\morphism|b|/{@{<-}@/_1em/}/<1000,0>[\strcat{\Sigma}`\strdiag{\Sigma};q]
\place(500,0)[\simeq] \efig\]
where $i$ is the inclusion and $q$ is by taking quotients, in the expected way. The unit \funktor{k}{(D,F)}{q(D,F)} preserves and reflects the intepretation of formulas and the truth of sequents. Thus, in particular, the equivalence restricts to models, $\modcat{\theory}\simeq \moddiag{\theory}$.
The diagram notation may seem cumbersome at first, but it is convenient for working with presentations of structures. Let a \emph{presentation} of a diagram, or \emph{pre-diagram}, be a pair $(D,F)$ where $D$ is a set and $F$ is set of facts over $\Sigma$ and $D$, in the sense above; that is to say, $F$ is a set of pairs \pair{\syntob{\alg{x}}{\phi},\alg{d}} with $\phi$ Horn and $\alg{d}\in D^{l(\alg{x})}$. For a pre-diagram $(D,F)$ the least diagram containing it is the diagram \emph{generated by} $(D,F)$. A homomorphism of pre-diagrams is a left-total relation that is a homomorphism of the generated diagrams. The satisfaction relation $\vDash$ for pre-diagrams is defined as satisfaction in the generated diagram.
For $(D,F)$ is a pre-diagram, the theory $\thry{D}_{(D,F)}$ of $(D,F)$ is defined as expected by extending $\Sigma$ with $D$ as constants (or unary predicates, see below) and letting \cterm{\top\vdash \phi[\alg{d}/\alg{x}]}{\pair{\syntob{\alg{x}}{\phi}, \alg{d}}\in F} be axioms. The diagram generated by $(D,F)$ can then be defined as $(D,\cterm{\pair{\syntob{\alg{x}}{\phi}, \alg{d}}}{\top\vdash^{\thry{D}_{(D,F)}} \phi[\alg{d}/\alg{x}]})$. If \theory\ is a theory over $\Sigma$ we write $\theory_{(D,F)}$ for the union of \theory\ and $\thry{D}_{(D,F)}$.
When constants are not discrete, replacing constants with variables in proofs becomes more problematic; that is, one cannot in general replace the same constant with the same variable throughout a formula. This makes the interplay between a theory and the theory of one of its diagrams a little more intricate. The following lemmas, which are completely straightforward for discrete signatures, are shown also to hold for non-discrete signatures in \cite{forsselllumsdaine} and stated here for reference.
\begin{lemma}\label{lemma: replacing constants with variables}
Let \theory\ be a
theory over $\Sigma$. Let $C$ be a set of constants disjoint from $\Sigma$, and write $\Sigma^C=\Sigma\cup C$. Suppose $\phi\vdash_{\alg{x}}\psi$ is a first-order sequent over $\Sigma^C$ which is provable from axioms in \theory. Then there exists a sequent $\phi'\vdash_{\alg{x},\alg{y}}\psi'$ over $\Sigma$ and a ``valuation'' function $\funktor{f}{\alg{y}}{C}$ such that;
\begin{enumerate}[(i)]
\item $\phi'\vdash_{\alg{x},\alg{y}}\psi'$ is provable from \textup{(}the same\textup{)} axioms in \theory; and
\item $\phi'[f]=\phi$, and $\psi'[f]=\psi$.
\end{enumerate}
\end{lemma}
\begin{lemma}\label{lemma: getting rid of constants from the model}
Let \theory\ be a
theory over $\Sigma$ and $(D,F)$ be a \textup{(}pre-\textup{)}diagram. Let $\FIC{\alg{x}}{\psi}$ and $\FIC{\alg{x},\alg{y}}{\phi}$ be first-order $\Sigma$-formulas-in-context. Let \alg{c} be a tuple of elements in $D$. Suppose $\theory_{D,F}$ proves the sequent $\phi[\alg{c}/\alg{y}]\vdash_{\alg{x}}\psi$. Then there is a regular formula $\xi$ in context \alg{y} in $\Sigma$ such that $(D,F)\vDash\xi[\alg{c}/\alg{x}]$ and \theory\ proves the sequent $\xi\wedge\phi\vdash_{\alg{x},\alg{y}}\psi$.
\end{lemma}
As mentioned in the beginning, we generally assume that signatures are relational and thus that any language with function symbols has been translated into an equivalent one without them (cf.\ \cite{bell-machover:1977}). This is in principle so also when extending a signature with the language of one of its diagrams, as above. In practice this becomes burdensome, however, and we leave the translation implicit and swept under the rug.
Finally, we say that a diagram $(D,F)$ is a \emph{subdiagram} of $(D',F')$ if $D\subseteq D'$ and $F\subseteq F'$. We write $(D,F)\subseteq (D',F')$. Note that the inclusion $D\subseteq D'$ induces a homomorphism $\funktor{i}{(D,F)}{(D',F')}$ by $i(d,d')\Leftrightarrow (d=d')\in F'$. (This homomorphism need not be a monomorphism.)
A diagram is \emph{finite} if the domain is finite and the interpretations of $=$ and all relation symbols are finite. For enumerable\ signature $\Sigma$, a diagram is \emph{enumerable}\ if the domain is enumerable\ and the interpretations of $=$ and all relation symbols are enumerable. For discrete signature $\Sigma$, a diagram is \emph{discrete} if the domain is discrete.
We define a \emph{bounded diagram} to be a triple $(D,F,n)$ where $(D,F)$ is a diagram, $n\in \thry{N}$, and the elements of the domain $D$ are pairs where the second component is a natural number less than or equal to $n$.
Clearly, any diagram is canonically isomorphic to a bounded one (with bound $0$, say).
We shall in fact mostly restrict to bounded diagrams, but leave the bound $n$ notationally implicit. Let $\operatorname{Diag}_b(\Sigma)$ be the category of bounded diagrams and diagram homomorphisms.
\subsubsection{Syntactic categories, Morleyization, and exploding models}\label{subsubsection: morleyization}
Recall from e.g.\ \cite[D1.4]{elephant1} the \emph{syntactic category} \synt{C}{\theory} of a theory \theory, consisting of formulas-in-context of the language of the theory. For coherent \theory, the category \synt{C}{\theory} is a coherent category and models of \theory\ in a coherent category \cat{D} can be considered as coherent functors $\synt{C}{\theory}\to<125>\cat{D}$. Similarly for e.g.\ regular and first-order theories, see \emph{loc.\ cit.} for precise statements and details.
The functor $\synt{C}{\theory}\to<125>\cat{D}$ is \emph{conservative} (see e.g.\ \cite{elephant1}) if and only if the corresponding model is conservative, or complete; that is, if only provable sequents are true.
We refer to the rewriting of a theory into a theory of a less expressive fragment as ``Morleyizing'' the theory, after the rewriting of a classical first-order theory as an equivalent coherent theory (as in e.g.\ \cite[D1.5.13]{elephant1}). In categorical terms, the syntactic category \synt{C}{\theory} of, say, an intuitionistic first-order theory \theory\ is a Heyting category, and thus also a coherent and regular category. The Heyting category \synt{C}{\theory} therefore has an internal coherent theory $\theory^{coh}_{\cat{C}_{\theory}}$ and an internal regular one $\theory^{reg}_{\cat{C}_{\theory}}$. These theories are equivalent in the sense that their syntactic categories are equivalent.
\[\synt{C}{\theory}\simeq \synt{C}{\theory^{coh}_{\cat{C}_{\theory}}}\simeq \synt{C}{\theory^{reg}_{\cat{C}_{\theory}}}\]
The categories of models of these theories will in general not be the same (unless we are considering classical theories and their coherent Morleyzations, see loc.cit.). Nevertheless, considering the category of models of the Morleyized theory can be fruitful, not least when the more expressive theory has ``too few models'' (see \cite{makkai:95} for another example).
There is some leeway concerning what precisely one takes the internal language and theory of a category to be. In our case we also start from a given theory and not from a given category. We therefore write down explicitly what we shall take the \emph{regular} and \emph{coherent Morleyizations} of a first-order theory \theory\ over a signature $\Sigma$ to be. Other fragments are similar.
Let $\Sigma^m$ be the signature extending $\Sigma$ with, for each first-order formula ${\phi}$ over $\Sigma$, in canonical context \alg{x}, say, a relation symbol \textbf{P}$_{\phi}$ with arity the length of $\alg{x}$. We write $P_{\phi}$ for the atomic formula over $\Sigma^m$ obtained by assigning $\alg{x}$ to the arity of \textbf{P}$_{\phi}$.
Consider the following axioms.
\begin{itemize}
\item[(\emph{Thry})] \hfill \\ For every sequent $\phi\vdash_{\alg{x}}\psi$ provable in \theory, the axiom
\[P_{\phi}\vdash_{\alg{x}}P_{\psi}\]
\item[(\emph{Atom})] \hfill \\ For every atomic formula $\phi$ over $\Sigma$ in canonical context \alg{x}, \[P_{\phi}\dashv\vdash_{\alg{x}}\phi\]
\item[(\emph{True})] \hfill \\ \[P_{\top}\dashv\vdash_{}\top\]
\item[(\emph{Conj})] \hfill \\ For every conjunction $\theta=\phi\wedge\psi$ over $\Sigma$ in canonical context \alg{x}
\[P_{\theta}\dashv\vdash_{\alg{x}}P_{\phi}\wedge P_{\psi}\]
\item[(\emph{Exist})] \hfill \\ For every existentially quantified formula $\theta=\fins{y}\phi$ over $\Sigma$ in canonical context \alg{x}
\[P_{\theta}\dashv\vdash_{\alg{x}}\fins{y}P_{\phi}\]
\item[(\emph{Disj})] \hfill \\ For every disjunction $\theta=\phi\vee\psi$ over $\Sigma$ in canonical context \alg{x}
\[P_{\theta}\dashv\vdash_{\alg{x}}P_{\phi}\vee P_{\psi}\]
\item[(\emph{False})] \hfill \\ \[P_{\bot}\dashv\vdash_{}\bot\]
\end{itemize}
These axioms define the coherent Morleyization of \theory. The regular Morleyization is obtained by omitting the Disjunction axiom schema and the False axiom. Notice that in, say, the regular Morleyization of a first-order theory, every regular formula is provably equivalent to an atomic formula; and that the sequent $\phi\vdash_{\alg{x}}\psi$ is provable in \theory\ if and only if $P_{\phi}\vdash_{\alg{x}}P_{\psi}$ is provable in $\theory^m$. Notice further that if we Morleyize, say, a regular$_{\bot}$ theory \theory\ to a regular theory, then $\theory^m$ will prove $P_{\bot}\vdash_{\alg{x}}\phi$ for all regular \FIC{\alg{x}}{\phi} over $\Sigma^m$. A model diagram $(D,F)$ such that $P_{\bot}\in F$ will thus have all possible facts in $F$. The corresponding quotient will consist of a single point of which everything is true (it must be inhabited since $\theory^m$ proves $P_{\bot}\vdash \fins{x}\top$). Thus, or in that sense, it is an \emph{exploding} model diagram of \theory.
Finally we note that the notion of enumerability for theories is not affected by Morleyization. We display this for reference.
\begin{lemma}
If \theory\ is enumerable\ then so is $\theory^m$.
\end{lemma}
\subsection{Tarski completeness}\label{section: tarski completeness}
The dynamical method of \cite{costelombardiroy:01}, the chase algorithm of \cite{abiteboulhullvianu:95}, and similar methods\footnote{C.f.\ also \cite{adamekandrosicky:94} and \cite{makkai:90})} can be seen as simultaneous proof searches and (at least partial) completions of structures to models. In essence, one proceeds by repeatedly applying the axioms of the theory to the structure and adding the result; thus if, for instance, $\phi[a/x]$ is true in the structure and $\phi\vdash_x \fins{y}\psi$ is an axiom, one extends the domain with a fresh element $b$ and the interpretation in the least way such that $\psi[a/x,b/y]$ is true. It is in several cases known, or at least folklore, that such methods can be used constructively to obtain completeness results for fragments of FOL. Although the object theories tend to be assumed countable or at least discrete. We summarize in Section \ref{Subsection: Regularbot theories} the relevant results from \cite{forsselllumsdaine} concerning the construction of a functorial ``simultaneous chase'' to the case of regular theories with no size or discreteness constraints. Section \ref{Subsection: Countable coherent theories and fan} displays
the equivalence between the Fan theorem and completeness for enumerable\ positive coherent theories with respect to enumerable\ model diagrams---equivalently of enumerable\ coherent theories with respect to ``possibly exploding'' or ``fallible'' enumerable\ model diagrams.
This equivalence can to a large extent be derived from the literature.
In particular, Veldman's proof \cite{veldman:76} (which relies on the Fan Theorem) of fallible Kripke completeness for first-order (enumerable) theories implies also the Tarski-completeness of positive coherent (enumerable) theories. Nevertheless, since we are, conceptually, regarding first-order fallible Kripke completeness as flowing from the Tarski completeness of positive coherent theories, we supply a direct proof of the latter using the Fan Theorem. The converse, that this completeness theorem implies the Fan Theorem, is rather immediate, and we include a very short and simple proof. This should be compared with the (equally short) proof in \cite{loeb:05} that the contrapositive model existence theorem for decidable (and countable) classical propositional theories is equivalent to the Fan Theorem. (Note that the direction completeness $\Rightarrow$ Fan of Proposition \ref{theorem: fan equivalence} can be carried out in much weaker meta-theories then IZF.
\subsubsection{Chase-complete sets of models for regular theories} \label{Subsection: Regularbot theories}
Let $\Sigma$ be a single-sorted relational signature. That is, $\Sigma$ is a arbitrary set of relation symbols (with assigned arities), not assumed to be of a particular size nor discrete.
\begin{definition}\label{definition: chase functor}
Let \theory\ be a theory over $\Sigma$, and \cat{S} a class of $\Sigma$-diagrams. We say that a functor \funktor{\operatorname{Ch}}{\cat{S}}{\cat{S}} is a \emph{chase functor} if for all diagrams $(D,F)\in \cat{S}$
\begin{enumerate}
\item $(D,F)\subseteq \operatorname{Ch}(D,F)$, naturally in $(D,F)$;
\item $\operatorname{Ch}(D,F)\vDash \theory$; and
\item
for any regular formula $\FIC{\alg{x}}{\psi}$ and $\alg{d}\in D^{l(\alg{x})}$ such that $ \operatorname{Ch}(D,F)\vDash \psi[\alg{d}/\alg{x}]$, there exists a regular formula $\FIC{\alg{x}}{\phi}$ such that $(D,F)\vDash \phi[\alg{d}/\alg{x}]$ and $\phi\vdash^{\theory}_{\alg{x}}\psi$.
\end{enumerate}
\end{definition}
\begin{proposition}\label{theorem: chase completeness for regular theories} \label{proposition: chase completeness for regular theories}
Let \theory\ be a regular theory. Then there exists a chase functor \funktor{\operatorname{Ch}}{{\operatorname{Diag}_b(\Sigma)}}{\operatorname{Diag}_b(\Sigma)}.
\begin{proof}
A straightforward modification of the construction in \cite{forsselllumsdaine}. (The modification being that, working with (bounded) diagrams rather than structures, we can have the components $c_{(D,F)}:(D,F)\to<125>\operatorname{Ch}(D,F)$ of the natural transformation $c:1_{\operatorname{Diag}_b(\Sigma)}\to \operatorname{Ch}$ be inclusions of diagrams, rather than homomorphisms of structures).
\end{proof}
\end{proposition}
In general, we say that a collection of diagrams is \emph{closed under chase} if there is some chase functor \mo{Ch} under which it is closed. The construction of \cite{forsselllumsdaine} shows e.g.\ that if $\Sigma$ is a discrete signature, then discrete diagrams are closed under chase. In a classical meta-theory the stronger property holds that for a regular theory the category of models is weakly reflective in the category of structures (cf. e.g.\ \cite{adamekandrosicky:94}, \cite{makkai:90}). We return to this and the case of enumerable\ signatures and theories in Section \ref{Subsection: Countable coherent theories and fan}.
For purposes of Section \ref{section: joyals theorem} we would, for given signature $\Sigma$ and regular theory \theory, like to restrict to a \emph{set} \cat{S} of diagrams which is nevertheless ``rich enough'' for our purposes. Mainly, this involves being closed under chase, but we add some further conditions. Say that a diagram is
a \emph{finitary extension} of a diagram $(D,F)$ if it is generated by a pre-diagram of the form $(D\cup \operatorname{Im}(\alg{c}), F\cup \pair{\syntob{\alg{x},\alg{y}}{\phi}, \alg{d},\alg{c}})$, where $\phi$ is a Horn formula, \alg{d} a (possibly empty) list of elements of $D$, and \alg{c} is a (possibly empty) list of elements such that \Image{\alg{c}} is finite and disjoint from $D$. Say that a collection \cat{S} of diagrams is \emph{chase-complete}
if: the empty diagram is in \cat{S}; \cat{S} is closed under finitary extensions (up to isomorphism); \cat{S} is closed under chase; and finally we add that, for any finite list of diagrams in $ \cat{S}$ there exists mutually disjoint isomorphic copies in \cat{S} of those diagrams. We say that a collection \cat{M} of model diagrams is \emph{chase-complete} if \cat{M} is the collection of model diagrams in a chase-complete category of diagrams. By Proposition \ref{theorem: chase completeness for regular theories}, $\operatorname{MDiag}_b(\Sigma)$ is chase-complete.
With reference to the chase functor construction of \cite{forsselllumsdaine}, the restriction to a small and chase-complete subcategory of diagrams can be done e.g.\ by building a set \cat{U} based on the natural numbers and the syntax of the theory closed under finite lists; and then consider the bounded diagrams whose domains are subsets of $\cat{U}$. We leave the details.
For reference, we state, then:
\begin{theorem}\label{theorem: Reg theories have replete sets of models}\label{corollary: completeness for regular}
Every regular theory \theory\ has a chase-complete category \cat{M} of model diagrams. If the signature is discrete, then the theory has a chase-complete set of discrete model diagrams.
\end{theorem}
We say that \cat{M} is \emph{conservative} for a class \cat{K} of sequents if for every sequent $\sigma$ in \cat{K}, if $\sigma$ is true in all diagrams in \cat{M} then $\theory\vdash\sigma$. (If \cat{K} is left implicit it is understood to be all sequents of the fragment of the theory). We say that \cat{M} is \emph{strongly conservative} for \cat{K} if it is conservative for \cat{K}, and, moreover, for every $(D,F)\in \cat{M}$, the set of $\theory_{(D,F)}$-model diagrams the reducts of which are in \cat{M} is conservative for $\theory_{(D,F)}$.
\begin{lemma}\label{lemma: regular strong completeness}
Let \theory\ be a regular theory and \cat{M} a chase-complete set of model diagrams. Then \cat{M} is strongly conservative.
\begin{proof}
Let \theory\ and $(D,F)$ be given, and let $\phi\vdash_{\alg{x}}\fins{\alg{y}}\psi$ be a normal form regular sequent over $\Sigma$ extended with $D$ as constants. Assume this sequent to be true in all $\theory_{(D,F)}$-model diagrams the reducts of which are in \cat{M}. Replacing every occurrence of a constant from $D$ in $\phi$ with a fresh variable $z$, write $\phi=\phi'[\alg{d}/\alg{z}]$. Let \alg{s} be a list of fresh constants, disjoint from $D$, of the same length as \alg{x} such that $\operatorname{Im}(\alg{s})$ is finite. Then $\operatorname{Ch}(D\cup\operatorname{Im}(\alg{s}),F\cup\{\pair{\syntob{\alg{x},\alg{z}}{\phi'}, \alg{s}, \alg{d}}\})\vDash \theory_{D,F}$ and $\operatorname{Ch}(D\cup\operatorname{Im}(\alg{s}),F\cup\{\pair{\syntob{\alg{x},\alg{z}}{\phi'}, \alg{s}, \alg{d}}\}\vDash \phi[\alg{s}/\alg{x}]$, so $\operatorname{Ch}(D\cup\operatorname{Im}(\alg{s}),F\cup\{\pair{\syntob{\alg{x},\alg{z}}{\phi'}, \alg{s}, \alg{d}}\}\vDash \fins{\alg{y}}\psi[\alg{s}/\alg{x}]$. Whence $\theory_{(D\cup\operatorname{Im}(\alg{s}),F\cup\{\pair{\syntob{\alg{x},\alg{z}}{\phi'}, \alg{s}, \alg{d}}\}}$ proves the sequent $\top\vdash \fins{\alg{y}}\psi[\alg{s}/\alg{x}]$. Then $\theory_{(D,F)}$ proves the sequent $\phi[\alg{s}/\alg{x}]\vdash \fins{\alg{y}}\psi[\alg{s}/\alg{x}]$. With $\operatorname{Im}(\alg{s})$ being a finite set, we can conclude that $\theory_{(D,F)}$ proves the sequent $\phi\vdash_{\alg{x}} \fins{\alg{y}}\psi$.
\end{proof}
\end{lemma}
A similar argument shows also that a chase-complete \cat{M} is conservative for geometric sequents over the signature of \theory. We state this for reference.
\begin{lemma}\label{lemma: chasecomplete gives geometric conservative}
Let \theory\ be a regular theory and \cat{M} a chase-complete set of model diagrams. Then \cat{M} is conservative for geometric sequents over the signature of \theory.
\begin{proof}
See \cite{forsselllumsdaine}.
\end{proof}
\end{lemma}
Finally, for the statement of Joyal's theorem in Section \ref{subsection: joyals theorem} we transfer the relevant results above to the case of structures for not necessarily purely relational signatures. This is a straightforward application of using the adjoint equivalence between diagrams and structures and of translating between signatures with function symbols and signatures without them, and we display it for reference. For theory \theory\ and model \alg{M}, the theory $\theory_{\alg{M}}$ of \alg{M} is defined as usual, so that $\modcat{\theory_{\alg{M}}}\simeq (\alg{M}\downarrow \modcat{\theory})$.
\begin{corollary}\label{corollary: strongly complete set of models}
Let \theory\ be a regular theory over an arbitrary signature $\Sigma$ (not necessarily purely relational). Then there exists a strongly complete set of models for \theory.
\end{corollary}
\subsubsection{Enumerable coherent theories and Fan}\label{Subsection: Countable coherent theories and fan}
The construction of the functor \mo{Ch} of \cite{forsselllumsdaine} relied upon in the previous section involves applying all axioms of the theory simultaneously at each step. (In that sense it could be said to be a ``simultaneous chase''.)
In the enumerable setting one can, instead, apply a single axiom in each step.
With disjunctions allowed in the axioms, this produces a finitely branching tree of structures, instead of a sequence of structures. Passing from regular to coherent theories, we therefore need the Fan theorem (with decidable bar\footnote{For statement and basic equivalents of the Fan theorem see e.g.\ \cite[Sect.\ 7]{troelstra:73}, where the relevant principle is named FAN$_\textnormal{D}$.}) to prove completeness.
The construction in this case is akin to e.g.\ \cite{costelombardiroy:01} (and, as mentioned, the resulting proposition known), and we only outline it. Recall that by positive coherent we mean the coherent fragment without the logical constant $\bot$.
\begin{proposition}[Fan]\label{lemma: completeness for sd coherent}\label{proposition: completeness for sd coherent}
Enumerable positive coherent theories are complete with respect to enumerable\ model diagrams.
\begin{proof}
Let \theory\ be a enumerable\ positive coherent theory over a relational signature $\Sigma$, assumed to be axiomatized by sequents on normal form.
An \emph{application} of such an axiom
\[\theta\vdash_{\alg{x}}\bigvee_{1\leq i\leq n} \fins{\alg{y_i}}\psi_i\]
to a diagram $(D,F)$ is a function $f:\alg{x}\rightarrow D$ such that $\pair{\syntob{\alg{x}}{\theta},f(\alg{x})}\in F$. Such an application induces $n$ children $(D_i,F_i)$, where $(D_i,F_i)$ is the least extension of $(D,F)$ containing a list $\alg{y'}_i$ of distinct fresh elements of the same length as $\alg{y}_i$ and such that $\psi_i[f(\alg{x})/\alg{x},\alg{y'}_i/\alg{y}_i]$ is true.
Given a finite diagram $(D,F)$, build a finitely branching tree \cat{T} of finite diagrams with root $(D,F)$ e.g.\ as follows. With \theory\ enumerable\ we can find finite subtheories $\theory_n$ such that $\theory_0\subseteq \ldots \theory_n \subseteq \ldots \bigcup_{n\in \thry{N}}\theory_n=\theory$. Build a sequence of finite trees $\cat{T}_n$ with $\cat{T}_0$ consisting just of $(D,F)$ and $\cat{T}_n$ an initial subtree of $\cat{T}_{n+1}$ as follows. For each leaf $(D',F')$ of the tree $\cat{T}_n$, list all possible applications $a_1,\ldots,a_m$ of $\theory_n$ to that leaf. Apply $a_1$ to $(D',F')$. This induces a finite number of children which are extensions of $(D',F')$. Apply $a_2$ to those, $a_3$ to the children induced by that again, and proceed until the list of applications runs out. This produces a finite tree $\cat{T'}_{(D',F')}$ with $(D',F')$ as its root. Then $\cat{T}_{n+1}$ is obtained by appending $\cat{T'}_{(D',F')}$ to each leaf $(D',F')$ of $\cat{T}_n$. Let \cat{T} be the union of the $\cat{T}_n$. Notice that: 1) $\cat{T}_n$ constitutes (or translates to) a dynamical cover (c.f.\ \cite{costelombardiroy:01}) of $(D,F)$ with respect to $\theory$; and 2) the union of the diagrams along a path of \cat{T} is a (enumerable) \theory-model diagram.
Now, let \[\phi\vdash_{\alg{x}}\bigvee_{ i\leq n} \fins{\alg{y_i}}\psi_i\] be a positive coherent sequent (on normal form, without loss of generality). Assume it is true in all \theory-model diagrams. Let $(D,F)$ be the finite diagram presented by the formula-in-context \FIC{\alg{x}}{\phi}; that is the least diagram for which the domain consists of a finite set bijective with $\mo{Im}(\alg{x})$, we can write $\bar{\alg{x}}$, and $\phi[\bar{\alg{x}}/\alg{x}]$ is true. Construct the tree \cat{T} on $(D,F)$ as above. Define the subset of nodes
\[B=\cterm{(D',F')}{\fins{i\leq n}\fins{\alg{d}\in D'}\psi_i[\bar{\alg{x}}/\alg{x},\alg{d}/\alg{y}_i]\in F'}\]
This is a decidable subset, and since the sequent is true in all models, and thus in all paths, it is a bar. Thus by the Fan theorem, it is a universal bar. Accordingly, there is an $n$ such that every leaf node in $\cat{T}_n$ is in $B$. And since $\cat{T}_n$ is a dynamical cover, the sequent is provable in $\theory$.
\end{proof}
\end{proposition}
The fallible Kripke semantics of e.g.\ \cite{veldman:76} allows for \emph{exploding} nodes, in the form of nodes that force $\bot$. Such nodes force all other formulas as well. Similarly we could define an exploding structure as one that interprets $\bot$ as true. We prefer to look at this through the lense of Morleyization; a structure for $\theory^m$ yields a structure for \theory\ in which some of the logical constants are interpreted non-standardly. In particular, if \theory\ is a coherent theory and $\theory^m$ its positive coherent Morleyization, then a $\theory^m$-model diagram $(D,F)$ induces an interpretation of the formulas of \theory, by letting the extension of a formula $\phi$ be the extension of the corresponding predicate $P_{\phi}$. We say that this interpretation is \emph{exploding} if $P_{\bot}\in F$. And we refer to the interpretation of \theory\ in terms of the models of $\theory^m$ as possibly exploding or \emph{fallible} Tarski semantics. We will consider further modifications in the next section.
We now have the following corollary of Proposition \ref{lemma: completeness for sd coherent}.
\begin{corollary}[Fan]\label{corollary: Coherent modified tarski completeness}
Enumerable coherent theories are complete with respect to fallible Tarski semantics.
\end{corollary}
If \theory\ is an enumerable classical first-order theory, then its coherent Morleyization $\theory^m$ is an enumerable\ coherent theory. A model for $\theory^m$ can be regarded as an ordinary Tarski model for \theory\ satisfying LEM; that is to say, every set that is the extension of a first-order formula of the language of the theory \theory\ must be complemented. Thus regarding a classical first-order theory as a first-order theory containing the LEM axiom scheme, we have as a consequence:
\begin{corollary}[Fan]\label{corollary: completeness of classical fo theories}
Classical enumerable\ first-order theories are complete with respect to fallible Tarski semantics.
\end{corollary}
\begin{remark}\label{Remark: on the completeness of cfol}
Corollary \ref{corollary: completeness of classical fo theories} provides an intuitionistic completeness theorem for classical logic provided the notion of model is relaxed to allow exploding models. Such theorems have been derived before, notably by Krivine (see \cite{krivine:96} and also \cite{berardiandvalentini:04}).
In fact, Krivine proves the model existence theorem for consistent classical first-order theories (which is intuitionistically stronger than the completeness theorem),
but with respect to models in which disjunction is non-standardly interpreted. In comparison, Corollary \ref{corollary: completeness of classical fo theories} retains the standard semantics for all connectives except $\bot$.
\end{remark}
Finally, we show that the use of the Fan theorem in Proposition \ref{lemma: completeness for sd coherent} is essential, and conclude:
\begin{theorem}\label{theorem: fan equivalence}
The completeness of enumerable\ positive coherent theories with respect to enumerable\ model diagrams is equivalent to the Fan theorem.
\begin{proof}
Let $F$ be a fan containing a decidable bar $\alg{B}$. We follow the notation in \cite[4.1]{troelstravandalen:88}. Consider the theory $\theory$ over the signature consisting of a propositional variable $P_{\alg{n}}$ for each element $\alg{n} \in F$ and a propositional variable $B$, and whose axioms are:
\begin{enumerate}
\item $\top \vdash P_{\langle \rangle}$, where $\langle \rangle$ is the root of the Fan;
\item $P_{\alg{n}} \vdash \bigvee_{\alg{n}\ast m \in F} P_{\alg{n}\ast m}$;
\item $P_{\alg{n}} \vdash B$ for each $\alg{n} \in \alg{B}$.
\end{enumerate}
For each branch $\alpha$ in $F$ let $S_{\alpha}$ be the set of sentences
\[S_{\alpha}=\cterm{P_{\alg{n}}}{\fins{x}\bar{\alpha}x=\alg{n}}\]
Let $\alg{M}=(D,F)$ be a enumerable\ \theory -model. Since it is enumerable\ we can find a branch $\alpha$ such that $S_{\alpha}\subseteq F$.
Since \alg{B} is a bar, we then have that $B\in F$. Thus $\alg{M}\vDash (\top\vdash B)$. By completeness, there is a proof of $\top\vdash B$ in \theory, with finitely many axioms, whence \alg{B} must be uniform.
\end{proof}
\end{theorem}
Since classical first-order logic is conservative over coherent logic (see e.g.\ \cite{negri:03}), we could have added the fallible Tarski-completeness of classical FOL as a third equivalent statement in \ref{theorem: fan equivalence}. We proceed now to the theorem of Joyal by which one can add the fallible Kripke completeness of FOL as a fourth.
Before doing so, however, we note, for use in Section \ref{section: sheaf completeness}, that if \theory\ is a regular theory then the construction of Proposition \ref{lemma: completeness for sd coherent} yields a sequence of diagrams, the union of which is a model of \theory. That this construction, extended to general enumerable\ diagrams, can be used to show that enumerable\ \theory\ models are weakly reflective in enumerable\ diagrams, as in the classical case, is rather expected and straightforward.
We therefore state the following for reference and without proof.
\begin{proposition}\label{proposition: sd regular chase}
\noindent Let \theory\ be an enumerable\ regular theory. There exists a chase-complete set \cat{S} of enumerable\ diagrams with a chase functor $\funktor{\operatorname{Ch}}{\cat{S}}{\cat{M}}$---where \cat{M} is the subcategory of model diagrams---which is moreover a weak reflection. That is,
for any homomorphism \funktor{h}{(D,F)}{(D',F')} in \cat{S} where $(D',F')\in \cat{M}$ there exists a homomorphism $\funktor{\hat{h}}{\operatorname{Ch}(D,F)}{(D',F') }$ such that
\[\bfig
\ptriangle/>`<-`<-/<600,400>[\operatorname{Ch}(D,F)`(D',F')`(D,F);\hat{h}`c_{(D,F)}`h]
\efig\]
commutes.
\end{proposition}
\section{Joyal's theorem}\label{section: joyals theorem}\label{subsection: joyals theorem}
Let \theory\ be a coherent (or regular) theory. A coherent (regular) formula-in-context \syntob{\alg{x}}{\phi} induces an evaluation functor
\[\funktor{Ev_{\syntob{\alg{x}}{\phi}}}{\modcat{\theory}}{\Sets}\]
by $\alg{M}\mapsto \csem{\alg{x}}{\phi}^{\alg{M}}$.
Mapping a formula-in-context to its corresponding evaluation functor defines (by soundness) a functor $\operatorname{Ev}:\synt{C}{\theory}\to<125>\Sets^{\modcat{\theory}}$, which we also call the evaluation functor, trusting that context will prevent confusion.
Since the coherent structure in a presheaf category is computed pointwise, the following is immediate and stated only for emphasis and reference.
\begin{lemma}\label{lemma: ev functor is coherent}
Let \theory\ be a coherent \textup{(}regular\textup{)} theory and \synt{C}{\theory} its coherent \textup{(}regular\textup{)} syntactic category. The functor
\[\operatorname{Ev}\ :\ \synt{C}{\theory}\to \Sets^{\modcat{\theory}}\]
which sends a formula to its corresponding evaluation functor is coherent \textup{(}regular\textup{)}.
\end{lemma}
We can now give the ``constructive content'' of the Kripke completeness theorem of A.\ Joyal---cf.\ \cite[Thm 6.3.5]{makkaireyes}---first in the form of the following theorem for regular theories. Since the purpose is to give a constructive restatement of this classical theorem, we state it first in terms of arbitrary signatures and ordinary Tarski models. The proof is not in essence dissimilar from the one in \cite{makkaireyes}. Recall that by Corollary \ref{corollary: strongly complete set of models} there are strongly complete sets of models for regular theories.
\begin{theorem}
\label{theorem: joyals theorem}
Let $\Sigma$ be a single sorted theory, not restricted in size, nor necessarily discrete (and possibly containing function symbols). Let \theory\ be a regular theory over $\Sigma$, and let $\cat{M}$ be a full subcategory of $\modcat{\theory}$ such that \cat{M} is strongly conservative.
Then the functor
\[\operatorname{Ev}\ :\ \synt{C}{\theory}\to \Sets^{\cat{M}}\]
is a\textup{)} conservative and b\textup{)} whenever the pullback functor $\funktor{f^*}{\mathrm{Sub}_{\synt{C}{\theory}}(B)}{\mathrm{Sub}_{\synt{C}{\theory}}(A)}$ induced by a morphism $f\! :\! A\to<125>B$ in \synt{C}{\theory} has a right adjoint $\forall_f$ we have for all $S\in\sublat{\synt{C}{\theory}}{A}$ that $Ev(\forall_f(S))=\forall_{Ev(f)}(Ev(S))$.
\begin{proof} a) For formulas \syntob{\alg{x}}{\phi} and \syntob{\alg{x}}{\psi}, if $\operatorname{Ev_{\syntob{\alg{x}}{\phi}}}(\alg{M})\subseteq \operatorname{Ev_{\syntob{\alg{x}}{\psi}}}(\alg{M})$ for all $\alg{M}\in\cat{M}$ then $\alg{M}\vDash (\phi\vdash_{\alg{x}}\psi)$ for all $\alg{M}\in\cat{M}$, whence $\phi\vdash^{\theory}_{\alg{x}}\psi$ by completeness.
b) The non-trivial direction is $Ev(\forall_f(S))\supseteq\forall_{Ev(f)}(Ev(S))$. It suffices to consider a situation
\[\bfig
\square/`^{(}->`^{(}->`>/<2000,300>[S=\syntob{\alg{x}}{\theta}`\forall_f(S)=\syntob{\alg{y}}{\gamma}`A=\syntob{\alg{x}}{\phi}`B=\syntob{\alg{y}}{\psi};```f=|\syntob{\alg{x},\alg{y}}{\lambda}|]
\efig\]
in \synt{C}{\theory}, where $\theta\vdash^{\theory}_{\alg{x}}\phi$ and $\gamma\vdash^{\theory}_{\alg{y}}\psi$.
Applying the functor Ev and evaluating at a model $\alg{M}$ we have
\[\bfig
\square/`^{(}->`^{(}->`>/<2000,300>[\cterm{\alg{d}}{\alg{M}\vDash\theta(\alg{d})}`\cterm{\alg{c}}{\alg{M}\vDash\gamma(\alg{c})}`\cterm{\alg{d}}{\alg{M}\vDash\phi(\alg{d})}`\cterm{\alg{c}}{\alg{M}\vDash\psi(\alg{c})};```\operatorname{Ev}_f(\alg{M})=\cterm{\alg{d},\alg{c}}{\alg{M}\vDash\lambda(\alg{d},\alg{c})}]
\efig\]
Let $\alg{c}\in \forall_{\operatorname{Ev}_{f}}(\operatorname{Ev}_{\syntob{\alg{x}}{\theta}})(\alg{M})\subseteq \operatorname{Ev}_{\syntob{\alg{y}}{\psi}}(\alg{M})$. Accordingly, for all $g\! :\! \alg{M}\to<125>\alg{N}$ in $\cat{M}$ we have:
\begin{equation}\label{Eq: Joyals-forall eq}\evalmod{f}{(\alg{N})}^{-1}(\operatorname{Ev}_{\syntob{\alg{y}}{\psi}}(g)(\alg{c}))\subseteq \operatorname{Ev}_{\syntob{\alg{x}}{\theta}}(\alg{N}).\end{equation}
We show $\alg{M}\vDash\gamma(\alg{c})$. Let $g\! :\! \alg{M}\to<125>\alg{N}$ be a morphism in $\cat{M}$, with $\alg{N}'$ the corresponding $\theory_{\alg{M}}$-model.
By (\ref{Eq: Joyals-forall eq}) we have
\begin{equation}\label{Eq: Joyals-forall seq}
\alg{N}'\vDash (\lambda[\alg{c}/\alg{y}]\vdash_{\alg{x}}\theta)
\end{equation}
Thus, the sequent (\ref{Eq: Joyals-forall seq}) is true in all $\theory_{\alg{M}}$-models corresponding to homomorphisms from $\alg{M}$ in \cat{M},
and therefore provable in $\theory_{\alg{M}}$, by the assumption of strong completeness. By Lemma \ref{lemma: getting rid of constants from the model}, there is a regular formula $\xi$ in context \alg{y} such that $\alg{M}\vDash\xi(\alg{c})$ and $\theory$ proves the sequent $(\xi\wedge\lambda\vdash_{\alg{x},\alg{y}}\theta)$. But then, since
$\forall_f(\syntob{\alg{x}}{\theta})=\syntob{\alg{y}}{\gamma}$, we have that \theory\ proves the sequent $(\xi\wedge \psi\vdash_{\alg{y}}\gamma)$. Whence $\alg{M}\vDash \gamma(\alg{c})$.
\end{proof}
\end{theorem}
It is convenient to have a name for the property proved in Theorem \ref{theorem: joyals theorem}. Following e.g.\ \cite{butz:02} (at least for the first notion):
\begin{definition}\label{definition: subheyting}
We say that a functor $\funktor{F}{\cat{C}}{\cat{D}}$ from a coherent category to a Heyting category is \emph{conditionally Heyting} if it is coherent and preserves any right adjoints to pullback functors that might exist in \cat{C}. If $F$ and \cat{C} are regular, we say $F$ is \emph{conditionally sub-Heyting} if it is regular and preserves any right adjoints to pullback functors that might exist in \cat{C}.
\end{definition}
The equivalence $\modcat{\theory}\simeq \moddiag{\theory}$ induces an equivalence $\Sets^{\modcat{\theory}} \simeq \Sets^{\moddiag{\theory}} $. Accordingly, if \cat{M} is a strongly complete set of model diagrams, the composite
\[ \synt{C}{\theory}\to^{\operatorname{Ev}} \Sets^{q(\cat{M})} \simeq \Sets^{\cat{M}}\]
is conservative and conditionally sub-Heyting. When returning to working with diagrams in the sequel, we shall consider this functor, also under the name $\operatorname{Ev}$.
As a corollary Theorem \ref{theorem: joyals theorem} we have the following version for coherent theories. Strongly complete sets of models do not in general exist for coherent theories. By Proposition\ref{lemma: completeness for sd coherent}, however, they do for enumerable\ positive coherent theories under the assumption of the Fan theorem.
\begin{corollary}\label{corollary: classical joyals}
Let \theory\ be a coherent theory, with \synt{C}{\theory} its coherent syntactic category, and suppose \cat{M} is a strongly conservative category of \theory-model diagrams.
Then the evaluation functor
\[\operatorname{Ev}\ :\ \synt{C}{\theory}\to \Sets^{\cat{M}}\]
is a conservative and conditionally Heyting functor.
\begin{proof}The proof of \ref{theorem: joyals theorem} can be repeated for this case. Alternatively, consider the regular Morleyization $\theory^m$ of the coherent theory \theory. Notice, first, that \cat{M} can then be considered as a full subcategory of $\operatorname{Mod}(\theory^m)$, and as such it is then strongly conservative for $\theory^m$. Then notice that evaluation restricted to \cat{M} yields a coherent functor from $\synt{C}{\theory}\simeq \synt{C}{\theory^m}$ to $\Sets^{\cat{M}}$.
\end{proof}
\end{corollary}
\section{Sheaf completeness}\label{section: sheaf completeness}
\subsection{Modified completeness}\label{subsection: modified completeness}
Loosely and informally, let us say that a model is \emph{modified} if some connectives are interpreted in a non-standard way, and \emph{standard} otherwise. We say that it is \emph{fallible} if the only connective treated non-standardly is $\bot$.
It is a corollary of Theorem \ref{corollary: completeness for regular} and Lemma \ref{lemma: regular strong completeness} that regular$_{\bot}$ theories are complete with respect to fallible Tarski semantics.
Now, if \theory\ is, say, a regular$_{\bot}$ theory, $\theory^m$ it's regular Morleyization, and \cat{M} is a small, full, and strongly conservative subcategory of $\operatorname{MDiag}(\theory^m)$, we have that the evaluation functor
\[\operatorname{Ev}\ :\ \synt{C}{\theory}\simeq \synt{C}{\theory^m}\to \Sets^{\cat{M} }\]
is regular (in particular). But it does not preserve the initial object, as $\evalmod{\syntob{}{P_{\bot}}}{-}$ is not the constant empty functor $0$. Thus it can be viewed as a conservative fallible presheaf model of \theory. We shall obtain a conservative standard sheaf model by taking sheaves with respect to the least coverage (on \op{\cat{M}}) so that $\evalmod{\syntob{}{P_{\bot}}}{-}$ is identified with $0$. Accordingly, we obtain a model of \theory\ in a closed subtopos (in the sense of \cite[A4.5.3]{elephant1}) of $\Sets^{\cat{M}}$. Similarly, if \theory\ is a coherent theory and $\theory^m$ its regular Morleyization, $\operatorname{Ev}\ :\ \synt{C}{\theory}\to \Sets^{\cat{M} }$ does not preserve finite disjunctions. A conservative standard model will be obtained by sheafifying with respect to the least coverage such that finite disjunctions are preserved. A conservative model will also be given by a slightly stronger coverage given (in part) in terms of binary trees and which is, in that sense, akin to a (fallible) Beth model. Classically, or in a enumerable setting, the latter two coverages are equivalent. In a countable setting, they also give rise to a Beth-completeness theorem, establishing a link between the least coverage forcing a standard interpretation and Beth semantics.
The coverages are given in terms of sieves on \op{\cat{M}}, and thus cosieves on \cat{M}.
Explicitly, then, let \theory\ be a theory in a fragment with $\bot$, $\theory^m$ its regular Morleyization, and \cat{M} a small full subcategory of \moddiag{\theory^m}.
Let the \emph{exploding coverage} ${E}$ be the coverage which assigns to each $(D,F)\in \cat{M}$ the set of cosieves $E(D,F)=\cterm{\emptyset}{P_{\bot}\in F}$.
This is a coverage (in the sense of \cite[A2.1.9, C2.1.1]{elephant1}) since if $S\in E(D_1,F_1)$ and $f:(D_1,F_1)\to<150> (D_2,F_2)$ is a homomorphism, then $S=\emptyset$ and $P_{\bot}\in F_1$, whence $P_{\bot}\in F_2$, since it is preserved by $f$, so $\emptyset\in E(D_2,F_2)$.
We then have the following addition to Theorem \ref{theorem: joyals theorem}.
\begin{proposition}\label{proposition: modified joyals theorem for regular}
Let \theory\ be an \textup{(}at least\textup{)} regular$_{\bot}$ theory, $\theory^m$ its regular Morleyization, \cat{M} be a strongly conservative, small, full subcategory of $\operatorname{MDiag}(\theory^m)$ and $E$ be the exploding coverage. Then the evaluation functor factors through sheaves
\[\bfig
\qtriangle/>`>`<-^{)}/<500,300>[\synt{C}{\theory}\simeq\synt{C}{\theory^m}`\Sets^{\cat{M}}`\Sh{\op{\cat{M}},E};\operatorname{Ev}``]
\efig\]
So that $\operatorname{Ev}:\synt{C}{\theory}\to<150>\Sh{\op{\cat{M}},E}$ is conservative, conditionally sub-Heyting, and preserves the initial object.
\begin{proof} By Theorem \ref{theorem: joyals theorem} it remains only to show that $\operatorname{Ev}$ factors through \Sh{\op{\cat{M}},E} and that $\evalmod{\syntob{}{{\bot}}}{-}$ is terminal in \Sh{\op{\cat{M}},E}. First, $\evalmod{\syntob{\alg{x}}{\phi}}{-}$ is a sheaf since if
$S \in E(D,F)$, then $S$ is empty and $(D,F)$ is exploding, which means that $\evalmod{\syntob{\alg{x}}{\phi}}{D,F}=\{\ast\}$.
Second, $\evalmod{\syntob{}{{\bot}}}{D,F}=\cterm{\ast}{P_{\bot}\in F}$ which is the initial sheaf in \Sh{\op{\cat{M}},E}.
\end{proof}
\end{proposition}
Note that classically the standard (i.e.\ non-exploding) models in \cat{M} are dense (in the sense of \cite{elephant1}), so that then $\Sh{\op{\cat{M}},E}\simeq \Sets^{\cat{M}^s}$ where $\cat{M}^s$ is the full subcategory of standard models.
Next, let \theory\ be a theory in a fragment with $\vee$ and $\bot$, with $\theory^m$ its regular Morleyization and \cat{M} a small full subcategory of \moddiag{\theory^m}. Again,
\[\operatorname{Ev}\ :\ \synt{C}{\theory}\simeq\synt{C}{\theory^m}\to \Sets^{\cat{M}}\]
is regular and conservative, but fails to preserve $\vee$ as well as $\bot$. Again we make explicit the least coverage $B$ forcing a standard interpretation. That is, the least coverage such that the initial object $0$ is dense in $\evalmod{\syntob{}{P_{\bot}}}{-}$ and, for all disjunctions $\syntob{\alg{x}}{\phi\vee\psi}$ of \theory, $\evalmod{\syntob{\alg{x}}{P_{\phi}}}{-}\vee\evalmod{\syntob{\alg{x}}{P_{\psi}}}{-}$ is dense in $\evalmod{\syntob{\alg{x}}{P_{\phi\vee\psi}}}{-}$. First, for all disjunctions(-in-context) $\syntob{\alg{x}}{\phi\vee\psi}$ of \theory, model diagrams $(D,F)$ in \cat{M}, and lists of elements $\alg{d}\in D^{l(\alg{x})}$, let $S_{\pair{\syntob{\alg{x}}{P_{\phi\vee\psi}},\alg{d}}}$ be the following cosieve on $(D,F)$:
\begin{align*}S_{\pair{\syntob{\alg{x}}{P_{\phi\vee\psi}},\alg{d}}}=&\cterm{\funktor{h}{(D,F)}{(D',F')}}{(D',F')\vDash P_{\phi}[ h(\alg{d})/\alg{x}]\vee P_{\psi}[ h(\alg{d})/\alg{x}]}
\end{align*}
Then let $B$ be specified by
\[B(D,F)=E(D,F)\cup\cterm{S_{\pair{\syntob{\alg{x}}{P_{\phi\vee\psi}}},\alg{d}}}{\pair{\syntob{\alg{x}}{P_{\phi\vee\psi}},\alg{d}}\in F}\]
Again, $B$ is a coverage. We refer to it as the \emph{minimal coverage}. A connection to Beth semantics will be displayed in Section \ref{subsubsection: beth completeness}. The proof of the following is similar to that of Theorem \ref{proposition: modified joyals theorem for coherent with C coverage}, and a corollary of it if \cat{M} is chase-complete, and is therefore omitted.
\begin{proposition}\label{proposition: modified joyals theorem for coherent}
Let \theory\ be an \textup{(}at least\textup{)} coherent theory, $\theory^m$ its regular Morleyization, \cat{M} a strongly conservative, small, full subcategory of $\moddiag{\theory^m}$, and $B$ be the minimal coverage on \op{\cat{M}}. Then the evaluation functor $\mo{Ev}$ composed with the sheafification functor $\funktor{a}{\Sets^{\cat{M}}}{\Sh{\op{\cat{M}},B}}$
\[\bfig
\qtriangle/>`>`@{<-^{)}}@<5pt>/<500,300>[\synt{C}{\theory}`\Sets^{\cat{M}}`\Sh{\op{\cat{M}},B};\operatorname{Ev}`a\circ\mo{Ev}`]
\qtriangle|abl|/>`>`{@{>}@<-5pt>}/<500,300>[\synt{C}{\theory}`\Sets^{\cat{M}}`\Sh{\op{\cat{M}},B};\operatorname{Ev}`a\circ\mo{Ev}`a]
\efig\]
is conservative, coherent, and conditionally Heyting.
\end{proposition}
If \cat{M} is chase-complete, then the minimal coverage on $\op{\cat{M}}$ can be strengthened while still yielding a conservative \theory-model as follows.
Let $(D,F)\in \cat{M}$, let \syntob{\alg{x}}{\phi\vee \psi} be a disjunction of \theory, and let $\alg{d}\in D$ such that $\pair{\syntob{\alg{x}}{P_{\phi\vee\psi}},\alg{d}}\in F$. Then we have,
\[\bfig
\Vtriangle/`<-`<-/<500,300>[\operatorname{Ch}(D,F\cup\{P_{\phi}{[}\alg{d}{]}\})`\operatorname{Ch}(D,F\cup\{P_{\psi}{[}\alg{d}{]}\})`(D,F);`c_0`c_1]
\efig\]
with $c_0$ the homomorphism induced by $(D,F)\subseteq (D,F\cup\{P_{\phi}{[}\alg{d}{]}\}) \subseteq \operatorname{Ch}(D,F\cup\{P_{\phi}{[}\alg{d}{]}\})$, and similarly for $c_1$.
We refer to such a pair, given by a fact of the form $\pair{\syntob{\alg{x}}{P_{\phi\vee\psi}},\alg{d}}$ in $F$, as a \emph{chase pair}
over $(D,F)$. Since $\operatorname{Ch}$ is a functor, assigning to each $(D,F)\in\cat{M}$ the set of chase pairs over it is a coverage. To this we also add the coverage $E$, so that the family of covering families over $(D,F)$ is the union of the set of chase pairs and the set \cterm{\emptyset}{(D,F)\vDash P_\bot}. Denote the resulting coverage by $C$, and the least Grothendieck coverage containing $C$ by $\overline{C}$. Similarly, write $\overline{B}$ for the least Grothendieck coverage containing $B$. We refer to $C$ as the \emph{disjunctive} coverage.
The two coverages compare as follows.
\begin{lemma}\label{lemma: B and C same if choice}
Let \theory\ be a coherent theory, $\theory^m$ its regular Morleyization, and \cat{M} a chase-complete category of model diagrams for $\theory^m$. Let $\overline{B}$ and $\overline{C}$ be the least Grothendieck coverages containing the minimal and disjunctive coverages on \op{\cat{M}}, respectively. Then $\overline{B}\subseteq \overline{C}$. Moreover,
\begin{enumerate}
\item If $\operatorname{Ch}$ is a weak reflection
then also $\overline{C}\subseteq\overline{B}$.
\item The statement that $\overline{C}= \overline{B}$ for arbitrary \theory\ is equivalent to the Axiom of Choice.
\end{enumerate}
\begin{proof}
(1) and that $\overline{C}\supseteq \overline{B}$ is clear. Since (2) can be considered more of a remark that will play no further role for us here, we only outline the proof:
Consider the \emph{coherent} theory \theory\ with unary predicate symbols $B$ and $\bar{B}$, one binary relation symbol $R$, and the axioms
\begin{align*}
B(x)\wedge \bar{B}(x)\vdash_x \bot \quad \quad
B(x)\vdash_x \fins{y}R(x,y)
\end{align*}
A surjection $e:Y\to<125> X$ can be considered as a \theory-model, and therefore a $\theory^m$-model \alg{E} by, briefly, letting $|\alg{E}|$ be $X+Y$, $X$ be the extension of $B$, and the extension of $R$ be the inverse of the graph of $e$.
Write $\beta^{\alg{x}}:= \bigwedge_{x\in\alg{x}}(B(x)\vee\bar{B}(x))$.
Let $(D,F)$ be the $\Sigma^m$-diagram presented by $D=\{\ast\}$ and\\ $F=\cterm{\pair{\syntob{\alg{x}}{P_{\phi}},\vec{\ast}}}{\syntob{\alg{x}}{\phi} \textnormal{ coherent in canonical context, and } \beta^{\alg{x}}\vdash^{\theory}_{\alg{x}}\phi}$.\\
Then $(D,F)$ is a (finite) $\theory^m$-model diagram.
Now, if $\overline{C}= \overline{B}$ then
the family $S_{\pair{\syntob{x}{P_{B(x)\vee\bar{B}(x)}},{\ast}}}$ is the sieve generated by the chase pair given by \pair{\syntob{x}{P_{B(x)\vee\bar{B}(x)}},\ast}. Then, for a surjection $e:Y\to<125> X$ , elements in $X$ induces homomorphisms $(D,F)\to<125>\alg{E}$, and the lifting through $\operatorname{Ch}(D,F\cup\{P_{B(x)}{[}\ast{]}\})$ gives a splitting of $e$ (see the proof of Theorem 5.16 in \cite{forsselllumsdaine}).
\end{proof}
\end{lemma}
\begin{theorem}\label{proposition: modified joyals theorem for coherent with C coverage}\label{theorem: modified joyals theorem for coherent with C coverage}
Let \theory\ be an at least coherent theory, $\theory^m$ its regular Morleyization, \cat{M} a full, chase-complete subcategory of $\moddiag{\theory^m}$, and $C$ be the disjunctive coverage on \op{\cat{M}}. Then the evaluation functor $\mo{Ev}$ composed with the sheafification functor $\funktor{a}{\Sets^{\cat{M}}}{\Sh{\op{\cat{M}},C}}$
\[\bfig
\qtriangle/>`>`@{<-^{)}}@<5pt>/<500,300>[\synt{C}{\theory}`\Sets^{\cat{M}}`\Sh{\op{\cat{M}},C};\operatorname{Ev}`a\circ\mo{Ev}`]
\qtriangle|abl|/>`>`{@{>}@<-5pt>}/<500,300>[\synt{C}{\theory}`\Sets^{\cat{M}}`\Sh{\op{\cat{M}},C};\operatorname{Ev}`a\circ\mo{Ev}`a]
\efig\]
is conservative, coherent, and conditionally Heyting.
\begin{proof} Let $U^n=\evalmod{\syntob{\alg{x}}{\top}}{-}$ where the length of \alg{x} is $n$. Recall from e.g.\ \cite{maclane92} that the (coherent) sheafification functor $a$ restricts to (one half of) a poset isomorphism between closed subobjects of $U^n$ and subobjects of $a(U^n$),
\[\operatorname{ClSub}_{}(U^n)\cong\operatorname{Sub}_{}(a(U^n))\]
It follows that it is sufficient to show that, for any formula $\syntob{\alg{x}}{\theta}$ of \theory, the functor $\evalmod{\syntob{\alg{x}}{P_{\theta}}}{-}$ is $C$-closed in the subobject lattice of $U^{n}$. Let $(D,F)\in \cat{M}$, let \syntob{\alg{y}}{\phi\vee\psi} be a disjunction of \theory, and let $\alg{d}\in D^{\operatorname{length}(\alg{y})}$ such that $(D,F)\vDash P_{\phi\vee\psi}[\alg{d}/\alg{y}]$. Let $(D_1,F_1)= \operatorname{Ch}(D,F\cup\{P_{\phi}[\alg{d}]\}))$ and $(D_2,F_2)= \operatorname{Ch}(D,F\cup\{P_{\psi}[\alg{d}]\}))$. Let $\alg{c}\in D^n$ and assume $(D_1,F_1),(D_2,F_2)\vDash P_{\theta}[\alg{c}/\alg{x}]$. Then $\theory^m_{(D,F)}$ proves the sequents $P_{\phi}[\alg{d}/\alg{y}]\vdash P_{\theta}[\alg{c}/\alg{x}]$ and $P_{\psi}[\alg{d}/\alg{y}]\vdash P_{\theta}[\alg{c}/\alg{x}]$. Thus there exists proofs with premisses in $\theory^m$ of $\chi\wedge P_{\phi}[\alg{d}/\alg{y}]\vdash P_{\theta}[\alg{c}/\alg{x}]$ and $\xi\wedge P_{\psi}[\alg{d}/\alg{y}]\vdash P_{\theta}[\alg{c}/\alg{x}]$ where $\chi$ and $\xi$ are conjunctions of atomic sentences over $\Sigma\cup D$ which are true in $(D,F)$. By Lemma \ref{lemma: replacing constants with variables}, there are proofs with the same premisses of sequents $\chi'\wedge P'_{\phi}\vdash_{\alg{w}} P'_{\theta}$ and $\xi'\wedge P''_{\psi}\vdash_{\alg{v}} P''_{\theta}$ over $\Sigma$, where we assume $\alg{w}$ and $\alg{v}$ disjoint, and a function \funktor{f}{\alg{w},\alg{v}}{D} such that $\chi'[f]=\chi$ and $\xi'[f]=\xi$ and $P'_{\phi}[f]=P_{\phi}[\alg{d}/\alg{y}]$ and $P''_{\psi}[f]=P_{\psi}[\alg{d}/\alg{y}]$ and $P'_{\theta}[f]=P''_{\theta}[f]=P_{\theta}[\alg{c}/\alg{x}]$.
Form the finite set $E$ of equalities with variables from the (assumed disjoint) lists \alg{w},\alg{v}, \alg{x}, and \alg{y} as follows: for each argument slot in $\alg{P}_{\phi}$, with that slot occupied by, say, $y$ in $P_{\phi}$, $w$ in $P'_{\phi}$, and $v$ in $P''_{\phi}$, add $y=w$ and $y=v$ to $E$; for each argument slot in $\alg{P}_{\psi}$, with that slot occupied by, say, $y$ in $P_{\psi}$, $w$ in $P'_{\psi}$, and $v$ in $P''_{\psi}$, add $y=w$ and $y=v$ to $E$; and for each argument slot in $\alg{P}_{\theta}$, with that slot occupied by, say, $x$ in $P_{\theta}$, $w$ in $P'_{\theta}$, and $v$ in $P''_{\theta}$, add $x=w$ and $x=v$ to $E$. Let $\rho$ be the conjunction of the equalities in $E$. Extend $f$ by: if $x$ occurs in an equality in $\rho$, say $x=w$, then add $x\mapsto f(w)$; and if $y$ occurs in an equality in $\rho$, say $y=w$, then add $y\mapsto f(w)$. Notice that this is well defined and $P_{\phi}[f]=P_{\phi}[\alg{d}/\alg{y}]$, $P_{\psi}[f]=P_{\psi}[\alg{d}/\alg{y}]$, and $P_{\theta}[f]=P_{\theta}[\alg{c}/\alg{x}]$. Then: 1) $\rho[f]$ is true in $(D,F)$ and
2) the sequents
$\fins{\alg{w},\alg{v}}\rho \wedge \chi' \wedge \xi' \wedge P_{\phi}\vdash_{\alg{x},\alg{y}}P_{\theta}$
and
$\fins{\alg{w},\alg{v}}\rho \wedge \chi' \wedge \xi' \wedge P_{\psi}\vdash_{\alg{x},\alg{y}}P_{\theta}$
are provable in $\theory^m$. We can replace the regular formula $\fins{\alg{w},\alg{v}}\rho \wedge \chi' \wedge \xi' $ by an atomic formula in $\Sigma^m$; say $P_{\gamma}$. We then have that \theory\ proves the sequents
$\gamma \wedge {\phi}\vdash_{\alg{x},\alg{y}}{\theta}$
and
$\gamma \wedge {\psi}\vdash_{\alg{x},\alg{y}}{\theta}$. Thus \theory\ proves $\gamma \wedge (\phi\vee \psi)\vdash_{\alg{x},\alg{y}}{\theta}$, whence $\theory^m$ proves $P_{\gamma} \wedge P_{\phi\vee \psi}\vdash_{\alg{x},\alg{y}}P_{\theta}$. Since substituting \alg{d} for \alg{y} and \alg{c} for \alg{x} makes the antecedent true in the $\theory^m$-model $(D,F)$, $P_{\theta}[\alg{c}/\alg{x}]$ must also be true in $(D,F)$.
\end{proof}
\end{theorem}
\subsubsection{Kripke and generalized Beth models}\label{subsection: models on posets}
The models of \ref{proposition: modified joyals theorem for regular} and \ref{proposition: modified joyals theorem for coherent with C coverage} can be translated to models in presheaves and sheaves on posets using e.g.\ the Diaconescu cover (see e.g.\ \cite{maclane92} for a description of the Diaconescu cover). However, in our current setting, we can use, more directly, the poset of model diagrams and inclusions.
We state this also as covering lemma, the technical heart of which is the following. Write $\operatorname{MDiag}^{\subseteq}(\theory)$ for the poset of of model diagrams and inclusions. Write $\funktor{\pi}{\operatorname{MDiag}^{\subseteq}(\theory)}{\moddiag{\theory}}$ for the functor that sends an inclusion to the homomorphism it induces.
\begin{lemma}\label{lemma: open cover}
Let \moddiag{\theory} be the category of model diagrams for some (regular or coherent) theory \theory.
For any homomorphism $\funktor{h}{(D_0,F_0)}{(D_1,F_1)}$ where $D_0$ and $D_1$ are disjoint there exists an extension $a\!:\!(D_0,F_0)\subseteq (D_2,F_2)$ and homomorphisms $r$ and $i$ as in the following diagram
\[\bfig
\dtriangle|arb|/<-`@{>}@/^5pt/`>/<750,500>[(D_1,F_1)`(D_0,F_0)`(D_2,F_2);h`i`\pi(a)]
\dtriangle|alb|/<-`@{<-}@/_5pt/`>/<750,500>[(D_1,F_1)`(D_0,F_0)`(D_2,F_2);`r`\pi(a)]
\efig\]
such that the outer triangle commutes and $r\circ i$ is the identity on $(D_1,F_1)$.
\begin{proof} Let $D_2=D_0 \cup D_1$ and let $F_2$ be the diagram generated by $F_0\cup F_1\cup \cterm{d=d'}{h(d,d')}$. We have
$a:(D_0,F_0)\subseteq (D_2,F_2)$.
Let $i$ be the homomorphism induced by the inclusion $(D_1,F_1)\subseteq (D_2,F_2)$. Then
\[\pi(a)(d,d') \Leftrightarrow \fins{c\in D_1} h(d,c)\wedge i(c,d')\]
so the outer triangle commutes. Let $r(d',d)\Leftrightarrow (d'=d)\in F_1 \vee h(d',d)$. It is then straightforward that $r$ is a well-defined homomorphism, as well as both a left and right inverse to $i$, from which it also follows that $(D_2,F_2)$ is a \theory-model diagram.
\end{proof}
\end{lemma}
Clearly, the assumption that the domain and codomain of $h$ are disjoint can be done without loss if the codomain can be replaced by an isomorphic copy disjoint from both it and the domain.
If \cat{M} is a chase-complete set of model diagrams for a regular theory \theory\ and $\cat{M}^{\subseteq}$ is the poset of model diagrams in \cat{M} and inclusions, then Lemma \ref{lemma: open cover} implies (by e.g.\ \cite[C3.1.2]{elephant1}) that the restriction functor
\[\Sets^{\cat{M}_{}}\to^{\pi^*}\Sets^{\cat{M}_{}^{\subseteq}}\]
is Heyting and conservative. By Theorem \ref{theorem: joyals theorem} we then obtain the following new version of Joyal's theorem:
\begin{theorem}\label{Theorem: joyals with inclusions}
Let \theory\ be a regular theory and $\cat{M}_{}^{\subseteq}$ a chase-complete set of model diagrams ordered by inclusions. Then the evaluation functor
\[\synt{C}{\theory}\to \Sets^{\cat{M}^{\subseteq}_{}}\]
is conservative and conditionally sub-Heyting.
\end{theorem}
We have, as corollaries to Theorem \ref{theorem: joyals theorem} (or Proposition \ref{proposition: modified joyals theorem for regular}), (fallible) Kripke completeness results for theories in certain fragments of first-order logic.
By \ref{lemma: open cover},
the underlying poset of the Kripke models can be taken to be a set of model diagrams for the regular Morleyization of \theory\ ordered by inclusion.
Similarly, as a corollary of Theorem \ref{proposition: modified joyals theorem for coherent with C coverage} we have a completeness theorem for first-order theories with respect to a generalized version of Beth semantics.
These are fairly straighforward cases of translating models on presheaves and sheaves on posets to Kripke and Beth-style presentations. For explicitness, we give some further details, also making clear
what notions of Kripke and ``generalized'' Beth models we have in mind:
Let $\Sigma$ be a relational signature. Write $\topo{S}$ for the partially ordered class of $\Sigma$-diagrams and (homomorphic) inclusions. Thus an object $S$ in $\topo{S}$ is a Tarski structure for $\Sigma$ with a congruence relation $\sem{=}^S$ interpreting $=$. Write $\topo{F}$ for the partially ordered class of \emph{fallible} $\Sigma$-diagrams and diagram inclusions: an object $S$ in $\topo{F}$ is an inhabited diagram for $\Sigma$ with a subset $\sem{\bot}^S\subseteq 1$ of the terminal set interpreting $\bot$, and such that $S$ satisfies the axioms $\bot \vdash_{\alg{x}} \phi$ for all atomic formulas $\phi$ in canonical context \alg{x} over $\Sigma$. The inclusions in \topo{F} must preserve $\sem{\bot}$, i.e.\ $S\subseteq S' \Rightarrow \sem{\bot}^S\subseteq \sem{\bot}^{S'}$.
\begin{definition}\label{definition: generalized beth model}
(I)
Let $\Sigma$ be a relational signature.
By a \emph{generalized \textup{(}fallible\textup{)} Beth structure} for $\Sigma$ we mean a triple \pair{P,D,T} where $P$ is a poset; $\mathfrak{D}$ is a functor from $P$ to \topo{S} (\topo{F}); and $T$ is an assignment of inhabited sets of subposets of $P$ to nodes of $P$ such that:
\begin{enumerate}[(i)]
\item all elements of $T(p)$ are finite, binary trees with root $p$, and $T(p)$ is closed under initial binary subtrees;
\item if $t\in T(p)$ with leaf nodes $q_1,\ldots,q_n$ and $t_1\in T(q_1),\ldots, q_n\in T(q_n)$ then the tree obtained by extending $t$ with the $t_i$'s is in $T(p)$;
and for all $p\in P$, $t\in T(p)$, and $q\in t$, $t\cap \uparrow q \in T(q)$;
and
\item for all $p\leq p'$ in $P$ and $t\in T(p)$ there exists $t'\in T(p')$ such that for all leaf nodes $q'$ of $t'$ there exists a leaf node $q$ of $t$ such that $q\leq q'$.
\end{enumerate}
The clauses of the forcing relation $p\Vdash \phi[\alg{d}/\alg{x}]$ between $p\in P$, first-order formulas-in-context \FIC{\alg{x}}{\phi}, and $\alg{d}\in \mathfrak{D}(p)^{l(\alg{x})}$ are then:
\begin{enumerate}[(a)]
\item for $\phi$ atomic or equal $\bot$ or $\top$, $p\Vdash \phi[\alg{d}/\alg{x}]$ if there exists $t\in T(p)$ such that for all leaf nodes $q\in t$ it is the case that $\mathfrak{D}(q)\vDash \phi[\alg{d}/\alg{x}]$;
\item for $\phi=\psi\wedge\theta$, $p\Vdash \phi[\alg{d}/\alg{x}]$ if $p\Vdash \psi[\alg{d}/\alg{x}]$ and $p\Vdash \theta[\alg{d}/\alg{x}]$;
\item for $\phi=\psi\vee\theta$, $p\Vdash \phi[\alg{d}/\alg{x}]$ if there exists $t\in T(p)$ such that for all leaf nodes $q$ in $t$ it is the case that $q\Vdash \psi[\alg{d}/\alg{x}]$ or $q\Vdash \theta[\alg{d}/\alg{x}]$;
\item for $\phi=\psi\rightarrow\theta$, $p\Vdash \phi[\alg{d}/\alg{x}]$ if for all $p'\geq p$ it is the case that if $p'\Vdash \psi[\alg{d}/\alg{x}]$ then $p'\Vdash \theta[\alg{d}/\alg{x}]$;
\item for $\phi=\fins{y}\psi$, $p\Vdash \phi[\alg{d}/\alg{x}]$ if there exists $t\in T(p)$ such that for all leaf nodes $q$ in $t$ there exists $c\in \mathfrak{D}(q)$ such that $q\Vdash \psi[c/y,\alg{d}/\alg{x}]$; and
\item for $\phi=\alle{y}\psi$, $p\Vdash \phi[\alg{d}/\alg{x}]$ if for all $p'\geq p$ and all $c\in \mathfrak{D}(p')$ it is the case that $q\Vdash \psi[c/y,\alg{d}/\alg{x}]$.
\end{enumerate}
(II) By a \emph{ \textup{(}fallible\textup{)} Kripke structure} we mean a generalized (fallible) Beth structure where $T(p)$ contains only the one node tree on $p$.
(III) By (\emph{fallible}) \emph{Beth structure} we mean a generalized (fallible) Beth structure where $P$ is a binary tree
and $T(p)$ is the set of initial binary subtrees of $\uparrow(p)$. (This notion of Beth structure is, then, with respect to the strong rather than the weak notion of forcing, cf.\ \cite[Ch.13 1.8]{troelstravandalen:88ii}.)
(IV) By a (\emph{generalized, fallible}) \emph{Beth$^{\star}$ structure} we mean a (generalized, fallible) Beth structure where ``covers are only relevant for disjunctions'', i.e.\ one satisfying the following additional conditions:
\begin{enumerate}[(1)]
\item for $\phi$ atomic or $\bot$, it is the case that $p\Vdash\phi[\alg{d}]$ iff $p\vDash\phi[\alg{d}]$, and
\item for all formulas of the form $\fins{x}\phi\in \cat{L}$, it is the case that $p\Vdash\fins{x}\phi[\alg{d}]$ iff there exists $a\in p$ such that $p\Vdash\phi[a,\alg{d}]$.
\end{enumerate}
\end{definition}
We state corollaries of Theorem \ref{Theorem: joyals with inclusions}, Proposition \ref{proposition: modified joyals theorem for regular}, and Theorem \ref{proposition: modified joyals theorem for coherent with C coverage} in terms of Definition \ref{definition: generalized beth model}. Let the \emph{$\vee$-free} fragment of FOL be the fragment consisting of sequents not mentioning the connective $\vee$, and the \emph{$\bot$,$\vee$-free} fragment be the one not mentioning $\bot$ or $\vee$.
\begin{corollary}\label{corollary: Kripke comp for bot and disj free}
Let \theory\ be a theory in the $\bot$,$\vee$-free fragment over the signature $\Sigma$. Then there exists a Kripke model for \theory\ which is conservative (with respect to the $\bot$,$\vee$-free fragment).
\begin{proof}
Let $\theory^m$ over $\Sigma^m$ be the regular Morleyization of \theory. By Theorem \ref{theorem: Reg theories have replete sets of models} there exists a chase-complete category \cat{M} of model diagrams for $\theory^m$. By Theorem \ref{Theorem: joyals with inclusions}, the evaluation functor $\operatorname{Ev}:\synt{C}{\theory}\simeq\synt{C}{\theory^m}\to \Sets^{\cat{M}^{\subseteq}_{}}$ is conditonally sub-Heyting, thus giving a conservative (w.r.t.\ the $\bot$,$\vee$-free fragment) model of \theory. Define a Kripke structure $K$ by letting the poset $P$ be $\cat{M}^{\subseteq}$ and the functor $\funktor{\mathfrak{D}}{\cat{M}^{\subseteq}}{\topo{S}}$ be the forgetfull functor. Let \FIC{\alg{x}}{\phi} be $\bot$,$\vee$-free over $\Sigma$, $S\in \cat{M}^{\subseteq}$, and $\alg{d}$ a list of the same length as \alg{x} of elements in the domain of $S$ . Using that $S\vDash P_{\phi}[\alg{d}/\alg{x}]\Leftrightarrow [\alg{d}]\in \evalmod{\FIC{\alg{x}}{\phi}}{S}$ and that $\operatorname{Ev}$ is conditionally sub-Heyting, a straightforward induction argument on $\FIC{\alg{x}}{\phi}$ shows that $S\Vdash^{K} \phi[\alg{d}/\alg{x}]\Leftrightarrow S\vDash P_{\phi}[\alg{d}/\alg{x}] $.
\end{proof}
\end{corollary}
\begin{corollary}\label{corollary: Mod Kripke comp for disj free}
Let \theory\ be a $\vee$-free theory. Then there exists a fallible Kripke model for \theory\ which is conservative with respect to the \emph{$\vee$-free} fragment.
\begin{proof}
From Proposition \ref{proposition: modified joyals theorem for regular} and Theorem \ref{Theorem: joyals with inclusions} (similarly to \ref{corollary: Kripke comp for bot and disj free}).
\end{proof}
\end{corollary}
\begin{remark}\label{Remark: completeness for disjunction free}
The restrictions are essential.
The existence of a conservative Kripke model (that is, a non-fallible one) for $\vee$-free theories implies LEM (see \cite{mccarty:08})\footnote{In fact, it is equivalent to it, since with LEM we can distinguish between exploding and non-exploding models.}.
A Kripke completeness theorem for $\bot$-free theories, or a fallible Kripke completeness theorem for full FOL, would imply e.g.\ that the Boolean Prime Ideal theorem is provable in ZF.
For theories whose axioms do not mention $\bot$ or $\vee$, such as the empty theory, the existence of a Kripke model which is conservative with respect to all first-order sequents implies MP (see \cite{mccarty:08}).
\end{remark}
As an example application of Theorem \ref{Theorem: joyals with inclusions}, we give a short, semantic proof of the disjunction property for (arbitrary) $\vee$-free theories (cf.\ \cite{troelstraschwichtenberg:96}) by reducing it to the disjunction property for regular theories (see e.g.\ \cite{elephant1}; note that the disjunction property for regular theories also directly follows from Proposition \ref{theorem: chase completeness for regular theories}).
\begin{corollary}\label{corollary: disjunction property}
Let \theory\ be a first-order theory the axioms of which are $\vee$-free, and let $\phi$, $\psi$, and $\theta$ be $\vee$-free formulas. If \theory\ proves the sequent $\phi\vdash_{\alg{x}}\psi\vee \theta$, then \theory\ proves either $\phi\vdash_{\alg{x}}\psi$ or $\phi\vdash_{\alg{x}} \theta$.
\begin{proof}
Consider the fallible Kripke model
of Corollary \ref{corollary: Mod Kripke comp for disj free}. (As in \ref{corollary: Kripke comp for bot and disj free}) the nodes are models of $\theory^m$ and for all $\vee$-free formulas $\xi$ we have $(D,F)\vDash P_{\xi}[\alg{d}/\alg{x}]$ $\Leftrightarrow$ $(D,F)\Vdash \xi[\alg{d}/\alg{x}]$. Then
for any $(D,F)$ in $\cat{M}^{\subseteq}$ and $\alg{d}\in D^{l(\alg{x})}$ we have: $(D,F)\vDash P_{\phi}[\alg{d}/\alg{x}]$ $\Leftrightarrow$ $(D,F)\Vdash \phi[\alg{d}/\alg{x}]$ $\Rightarrow$ $(D,F)\Vdash (\psi\vee \theta)[\alg{d}/\alg{x}]$ $\Rightarrow$ $(D,F)\Vdash \psi[\alg{d}/\alg{x}]$ or $(D,F)\Vdash \theta[\alg{d}/\alg{x}]$ $\Leftrightarrow$ $(D,F)\vDash P_{\psi}[\alg{d}/\alg{x}]$ or $(D,F)\vDash P_{\theta}[\alg{d}/\alg{x}]$ $\Leftrightarrow$ $(D,F)\vDash (P_{\psi}\vee P_{\theta})[\alg{d}/\alg{x}]$. Thus by Lemma \ref{lemma: chasecomplete gives geometric conservative}, $\theory^m$ proves the sequent
$P_{\phi}\vdash_{\alg{x}} P_{\psi}\vee P_{\theta}$. Therefore, $\theory^m$ proves the sequent $P_{\phi}\vdash_{\alg{x}} P_{\psi}$ or the sequent $P_{\phi}\vdash_{\alg{x}} P_{\theta}$, whence \theory\ proves $\phi\vdash_{\alg{x}}\psi$ or $\phi\vdash_{\alg{x}} \theta$.
\end{proof}
\end{corollary}
Finally, for full first-order logic we have:
\begin{corollary}\label{corollary: generalized beth model}
Let \theory\ be a first-order theory. Then \theory\ has a conservative generalized fallible Beth$^{\star}$ model.
\begin{proof}
Let $\theory^m$ be the regular Morleyization of \theory\ over extended signature $\Sigma^m$, with $\Sigma$ the signature of \theory. By Theorem \ref{theorem: Reg theories have replete sets of models} there exists a chase-complete category \cat{M} of model diagrams for $\theory^m$. By Theorem \ref{proposition: modified joyals theorem for coherent with C coverage} the functor $a\circ \operatorname{Ev}:\synt{C}{\theory}\simeq\synt{C}{\theory^m}\to \Sh{\op{\cat{M}}, C}$ is conservative and Heyting. From Lemma \ref{lemma: open cover}, by \cite[C2.3.18--19(i)]{elephant1} and \cite[C3.1.23]{elephant1}, the (right) top functor of the following commutative (up to isomorphism) diagram
\[\bfig
\square(1000,0)/>`<-`<-`>/<1000,500>[\Sh{\op{\cat{M}}, C}`\Sh{\op{\cat{M}^{\subseteq}},C}`\Sets^{\cat{M}}`\Sets^{\cat{M}^{\subseteq}};a\circ\pi^{\ast}\circ i`a`a`\pi^{\ast}]
\dtriangle/<-`<-`>/<1000,500>[\Sh{\op{\cat{M}}, C}`\synt{C}{\theory}\simeq\synt{C}{\theory^m}`\Sets^{\cat{M}};a\circ \operatorname{Ev}``\operatorname{Ev}]
\efig\]
is Heyting and conservative. Hence so is the composite top functor. As in the proof of \ref{proposition: modified joyals theorem for coherent with C coverage}, the subpresheaves of the form $\pi^{\ast}\circ\operatorname{Ev}_{\syntob{\alg{x}}{\phi}}\to/^{(}->/<125> \pi^{\ast}\circ\operatorname{Ev}_{\syntob{\alg{x}}{\top}}$ are $C$-closed.
Define a generalized fallible Beth structure $B$ as follows. Let $P$ be $\cat{M}^{\subseteq}$, and let the functor $\funktor{\mathfrak{D}}{\cat{M}^{\subseteq}}{\topo{F}}$ send a $\Sigma^m$-diagram $S$ to its $\Sigma$ reduct extended with $\sem{\bot}^S:=\sem{P_{\bot}}^S$.
For $S\in\cat{M}^{\subseteq}$ let $T(S)$ be the set of finite binary trees with nodes in $\cat{M}^{\subseteq}$, root $S$, and such that the children of any node $S'$ form a chase pair (as given in the paragraph following \ref{proposition: modified joyals theorem for coherent}) over $S'$.
Let \FIC{\alg{x}}{\phi} be first-order over $\Sigma$, $S\in \cat{M}^{\subseteq}$, and $\alg{d}$ a list of the same length as \alg{x} of elements in the domain of $S$ .
We show by induction on $\phi$ that $S\Vdash^{B} \phi[\alg{d}/\alg{x}]\Leftrightarrow S\vDash P_{\phi}[\alg{d}/\alg{x}] $.
And, simultaneously, that $B$ satisfies the conditions for being a Beth$^{\star}$ structure.
Let $\phi$ be atomic or $\phi = \bot$. Suppose $S\Vdash^{B} \phi[\alg{d}/\alg{x}]$. Then there exists a tree $t\in T(S)$ such that for all leaves $S'$ it is the case that $\mathfrak{D}(S')\vDash \phi[\alg{d}]$. Hence $S'\vDash P_{\phi}[\alg{d}/\alg{x}] $. Now, the inclusions $S\subseteq S'$ define a $C$-cover, so $S\vDash P_{\phi}[\alg{d}/\alg{x}] $, and thus $\mathfrak{D}(S)\vDash \phi[\alg{d}/\alg{x}] $. The converse is immediate. The case for the existential quantifier is similar, and the the case for conjuction is immediate.
Let $\phi = \psi \vee \theta $. Suppose $S\Vdash^{B} \phi[\alg{d}/\alg{x}]$. Then there exists a tree $t\in T(S)$ such that for all leaves $S'$ it is the case that $S'\Vdash^{B} \psi[\alg{d}]$ or $S'\Vdash^{B} \psi[\alg{d}]$. By induction hypothesis $S'\vDash P_{\psi}[\alg{d}]$ or $S'\vDash P_{\theta}[\alg{d}]$, so $S'\vDash P_{\phi}[\alg{d}]$, and since $t$ defines a $C$-cover on $S$, $S\vDash P_{\phi}[\alg{d}]$. Conversely, suppose $S\vDash P_{\phi}[\alg{d}/\alg{x}]$. Then \pair{\syntob{\alg{x}}{\phi},\alg{d}} defines a chase pair on $S$, yielding a tree in $T(S)$ with two leaves $S'$ and $S''$ such that, by the induction hypothesis, $S'\Vdash^B {\psi}[\alg{d}]$ and $S''\Vdash^B {\phi}[\alg{d}]$.
Let $\phi = \alle{y}\psi$. Suppose $S\Vdash^{B} \phi[\alg{d}/\alg{x}]$. Then for all $S\subseteq S'$ we have $\mathfrak{D}(S')\Vdash^{B} \psi[\alg{d}/\alg{x},c/y]$ for all $c$ in the domain of $S'$, thus by induction hypothesis $S'\vDash P_{\psi}[\alg{d},c]$. Hence, since $\funktor{\pi^{\ast}\circ \operatorname{Ev}}{\synt{C}{\theory}\simeq \synt{C}{\theory^m}}{\Sets^{\cat{M}^{\subseteq}}}$ is Heyting, $S\vDash P_{\phi}[\alg{d}/\alg{x}] $. The converse follows from $P_{\alle{y}\psi}\vdash_{\alg{x},y}^{\theory^m}P_{\psi}$. And $\rightarrow$ is similar.
\end{proof}
\end{corollary}
\subsubsection{Beth completeness}\label{subsubsection: beth completeness}
For enumerable first-order \theory\ we specialize \ref{proposition: modified joyals theorem for coherent with C coverage}/\ref{corollary: generalized beth model} to the effect that for every $\theory^m$-model diagram $(D,F)$ in suitable \cat{M} there is a Beth model \alg{B} with root domain $D$ such that $\alg{B}\Vdash \phi[\alg{d}/\alg{x}]\Leftrightarrow (D,F)\vDash P_{\phi}[\alg{d}/\alg{x}]$ for all first-order $\phi$. This yields a Beth$^{\star}$ completeness theorem for \theory.
Specifically, let \theory\ be a enumerable\ first-order theory over a signature $\Sigma$ and $\theory^m$ its regular Morleyization. In this section, we assume that the sequent $\top\vdash\fins{x} x=x$ is an axiom of \theory. We refer to theories having this sequent as an axiom as \emph{habitative} Let \cat{M} be the subcategory of $\operatorname{MDiag}_b(\theory^m)$ consisting of diagrams $(D,F)$ of the following form. The domain $D$ is a semi-decidable subset of \thry{N}, coding a bounded relation from \thry{N} to \thry{N}. $D$ comes equipped with the least upper bound. Denote by $f_D$ the function $\funktor{f_D}{\thry{N}}{2^{\thry{N}}}$ such that $D=\cterm{n\in \thry{N}}{\fins{m\in\thry{N}}f(n)(m)=1}$. The set of facts $F$ is, similarly, (coded as) a semi-decidable subset of \thry{N} (with function $f_F$). Then, straightforwardly, \cat{M} is chase-complete (for $\theory^m$) with a weakly reflective chase functor. We can assume that if $(D,F)$ is a diagram and $(D',F')=\operatorname{Ch}(D,F)$ then $f_D\leq f_{D'}$ and $f_F\leq f_{F'}$. We fix an enumeration $g$ of (codes of) all possible facts of the form \pair{\syntob{\alg{x}}{P_{\phi\vee\psi}},\alg{d}}, where every such fact is revisited an infinite number of times.
For any $(D,F)\in\cat{M}$ we define a binary tree $\cat{T}_{D,F}$ over $(D,F)$, where $\cat{T}_{D,F}$ occurs as a subcategory of \cat{M}, as follows. The diagram $(D,F)$ is the root. At node $(D',F')$ at level $n$, if $g(n)$ is, say, $\pair{\syntob{\alg{x}}{P_{\phi\vee\psi}},\alg{d}}$ and $\fins{m\leq n}f_{F'}(g(n),m)=1$ then $(D',F')$ has the children $\operatorname{Ch}(D',F'\cup\{P_{\phi}[\alg{d}]\})$ and $\operatorname{Ch}(D',F'\cup\{P_{\psi}[\alg{d}]\})$. Else the children of $(D',F')$ are both $(D',F')$ itself. In the former case, we say that $\pair{\syntob{\alg{x}}{P_{\phi\vee\psi}},\alg{d}}$ \emph{is chased}.
$\cat{T}_{D,F}$ becomes a fallible Beth structure for $\Sigma$ by setting $P=\cat{T}_{D,F}$, and letting the functor $\funktor{\mathfrak{D}}{\cat{T}_{D,F}}{\topo{F}}$ send a $\Sigma^m$-diagram $S$ to its $\Sigma$ reduct extended with $\sem{\bot}^S:=\sem{P_{\bot}}^S$.
It is clear that for any node $q$ of $\cat{T}_{D,F}$, any level of $\cat{T}_{D,F}$ above it can be seen as a $\overline{C}$-cover of $q$ (and therefore also a $\overline{B}$-cover, since $\overline{B}=\overline{C}$ in this setting). We refer to it therefore also as ``a cover of $q$''. By the proof of Theorem \ref{proposition: modified joyals theorem for coherent with C coverage}, we have, therefore, that for any atomic formula $\FIC{\alg{x}}{P_{\phi}}$ of $\Sigma^m$, any node $q$ in $\cat{T}_{D,F}$, and any elements $\alg{d}$ of $q$, if $P_{\phi}[\alg{d}/\alg{x}]$ is true on a cover of $q$ then $q\vDash P_{\phi}[\alg{d}/\alg{x}]$.
\begin{lemma}\label{lemma: nodes force the interpretation} Let $(D,F)\in \cat{M}$ and consider the Beth-structure $\cat{T}_{D,F}$. Let $p$ be a node in $\cat{T}_{D,F}$,
let $\FIC{\alg{x}}{\phi}$ be a first-order formula over $\Sigma$, with $P_{\phi}$ the corresponding atomic formula in $\Sigma^m$, and let $\alg{d}\in p$. Then
\[p\Vdash \phi[\alg{d}/\alg{x}]\Leftrightarrow p\vDash P_{\phi}[\alg{d}/\alg{x}]\]
\begin{proof}
By induction on $\phi$, as follows.
$\phi$ atomic or $\phi = \bot$: by the remark immediately preceding this lemma.
$\phi\equiv \varphi\wedge \psi$ or $\phi=\top$: immediate.
$\phi= \varphi\vee \psi$: Assume $p\Vdash (\varphi\vee \psi)[\alg{d}]$. Then there exists a cover of $p$ such that for all $q$ in the cover either $q\Vdash \varphi[\alg{d}]$ or $q\Vdash \psi[\alg{d}]$. By induction hypothesis, then, either $q\vDash P_{\varphi}[\alg{d}]$ or $q\vDash P_{\psi}[\alg{d}]$. As $\theory^m$ proves e.g.\ $P_{\varphi}\vdash_{\alg{x}}P_{\varphi\vee\psi}$ therefore $q\vDash P_{\varphi\vee\psi}[\alg{d}]$. Whence $p\vDash P_{\varphi\vee\psi}[\alg{d}]$.
\noindent Conversely, assume that $p\vDash P_{\varphi\vee\psi}[\alg{d}]$. Then for all $q\geq p$, we have $q\vDash P_{\varphi\vee\psi}[\alg{d}]$. Therefore, there exists a cover of $p$, say at level $n$, such that on that cover $P_{\varphi\vee\psi}[\alg{d}]$ is chased. Then for all $q\geq p$ at level $n+1$ we have that either $q\vDash P_{\varphi}[\alg{d}]$ or $q\vDash P_{\psi}[\alg{d}]$. Whence, by induction hypothesis, either $q\Vdash {\varphi}[\alg{d}]$ or $q\Vdash {\psi}[\alg{d}]$, and so $p\Vdash (\varphi\vee \psi)[\alg{d}]$.
$\phi=\fins{y}\varphi$: We have that $p\Vdash \fins{y}\varphi[\alg{d}]$ iff $q\Vdash \varphi[a_q,\alg{d}]$ on a cover iff $q\vDash P_{\varphi}[a_q,\alg{d}]$ on a cover iff $q\vDash \fins{y}P_{\varphi}[\alg{d}]$ on a cover iff $q\vDash P_{\fins{x}\varphi}[\alg{d}]$ on a cover iff $p\vDash P_{\fins{y}\varphi}[\alg{d}]$ .
$\phi= \varphi\rightarrow \psi$: Assume that $p\vDash P_{\varphi\rightarrow \psi}[\alg{d}]$. Let $q\geq p$ and assume that $q\Vdash \varphi[\alg{d}]$. By induction hypothesis $q\vDash P_{\varphi}[\alg{d}]$. Now, $\theory^m$ proves the sequent $P_{\varphi}\wedge P_{\varphi\rightarrow \psi}\vdash_{\alg{x}}P_{\psi}$. Whence $q\Vdash P_{\psi}[\alg{d}]$. And so applying the induction hypothesis again, $q\Vdash {\psi}[\alg{d}]$. Hence $p\Vdash (\varphi\rightarrow \psi)[\alg{d}]$.
\noindent For the converse direction, observe first that for $(D,F)\in \cat{M}$ we have that $(D,F)\vDash P_{\varphi\rightarrow \psi}[\alg{d}]$ iff $\operatorname{Ch}(D,F\cup \{P_{\varphi}[\alg{d}]\})\vDash P_{\psi}[\alg{d}]$: for the right-to-left direction, the right hand side implies that there is a $\theory^m$-provable sequent $P_{\chi}\wedge P_{\phi}\vdash_{\alg{x}} P_{\psi}$ such that $(D,F)\vDash P_{\chi}[\alg{d}/\alg{x}]$. Whence \theory\ proves ${\chi}\wedge {\phi}\vdash_{\alg{x}} {\psi}$ and therefore $\chi\vdash_{\alg{x}} \phi\rightarrow \psi$, so that $\theory^m$ proves $P_{\chi}\vdash_{\alg{x}} P_{\phi\rightarrow \psi}$.
Now, assume that $p\Vdash (\varphi\rightarrow \psi)[\alg{d}]$. We have $p\vDash P_{\top\vee \varphi}[\alg{d}]$, so there exists a level $n$ where it is chased. For a node $q=(D',F')$ on level $n$, the right child is therefore $q'=\operatorname{Ch}(D,F\cup \{P_{\varphi}(\alg{d})\})$. By induction hypothesis, $q'\vDash P_{\varphi}[\alg{d}]$ implies that $q'\Vdash {\varphi}[\alg{d}]$. Thus since $q'\geq p$ we have by assumption that $q'\Vdash {\psi}[\alg{d}]$, so $q'\vDash P_{\psi}[\alg{d}]$. So, by the observation, $q\vDash P_{\varphi\rightarrow\psi}[\alg{d}]$. With $P_{\varphi\rightarrow\psi}[\alg{d}]$ true on a cover of $p$ we have, then, that $p\vDash P_{\varphi\rightarrow\psi}[\alg{d}]$.
$\phi=\alle{y}\varphi$: Assume that $p\vDash P_{\alle{y}\varphi}[\alg{d}]$. Then for all $q\geq p$ and $a\in q$ we have that $q\vDash P_{\varphi}[a,\alg{d}]$, by applying the axiom $P_{\alle{y}\varphi}\vdash_{y,\alg{x}}P_{\varphi}$ of $\theory^m$. So, by induction hypothesis, $q\Vdash {\varphi}[a,\alg{d}]$. Thus $p\Vdash {\alle{x}\varphi}[\alg{d}]$.
\noindent For the converse direction, observe first, similar to the case of the conditional above, that that for $(D,F)\in \cat{M}$ we have that $(D,F)\vDash P_{\alle{x}\varphi}[\alg{d}]$ iff $\operatorname{Ch}(D+1,F)\vDash P_{\varphi}[e,\alg{d}]$, for all elements $e$ of $\operatorname{Ch}(D+1,F)$ (specifically the new element of $D+1$). Since \theory\ is habitative, this implies that $(D,F)\vDash P_{\alle{x}\varphi}[\alg{d}]$ iff $\operatorname{Ch}(D,F)\vDash P_{\varphi}[e,\alg{d}]$, for all elements $e$ of $\operatorname{Ch}(D,F)$ (specifically any fresh element $e$ added by an application of the axiom $\top\vdash\fins{x}x=x$). Assume, then, that $p\Vdash {\alle{x}\varphi}[\alg{d}]$. We have $p\vDash P_{\top\vee\top}$, so there exist a level $n$ on which $P_{\top\vee\top}$ is chased. For $(D',F')=q\geq p$ on level $n$, either child is $\operatorname{Ch}(D',F')$ and $\operatorname{Ch}(D',F')\Vdash \varphi[e,\alg{d}]$, for all elements $e$ in $(D',F')$. By induction hypothesis, $\operatorname{Ch}(D',F')\vDash P_{\varphi}[e,\alg{d}]$, so $q=(D',F')\vDash P_{\alle{x}\varphi}[\alg{d}]$. With $ P_{\alle{x}\varphi}[\alg{d}]$ thus true on a cover, we conclude $p\vDash P_{\alle{x}\varphi}[\alg{d}]$.
\end{proof}
\end{lemma}
It is clear that given a Beth$^{\star}$ model of a first order \theory, the nodes, and the root in particular, model $\theory^m$. We can now state the converse
\begin{theorem}
Let \theory\ be a habitative enumerable first-order theory. Let $\theory^m$ be its regular Morleyization. For every enumerable Tarski model \alg{M} of $\theory^m$ there exists a fallible Beth$^{\star}$ model \alg{B} of \theory\ such that the domain of the root $r$ is the domain of $\alg{M}$, and such that for all $\alg{m}\in \alg{M}^{l(\alg{m})}$ and \FIC{\alg{x}}{\phi} first-order,
\[r\Vdash \phi[\alg{m}/\alg{x}]\Leftrightarrow \alg{M}\vDash P_{\phi}[\alg{m}/\alg{x}]\]
\end{theorem}
The following can then be seen as a constructive version of the completeness theorem of \cite{gabbay:77}.
\begin{corollary}\label{theorem: Beth completeness for s.d.} Let \theory\ be a habitative enumerable\ first-order theory. Then \theory\ is complete with respect to fallible Beth$^{\star}$ models.
\end{corollary}
\section*{Acknowledgements}
We wish to particularly thank H\aa{}kon R. Gylterud, Hugo Herbelin, Panagis Karazeris, Peter L.\ Lumsdaine, and, especially, Erik Palmgren, for discussions, pertinent observations, and feedback.
\bibliographystyle{plain}
\bibliography{bibliografiEv}
\newpage
\end{document}
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Clare Nuttall in Chisinau -
Moldova has no intention of turning its back on plans to sign an EU free trade and association agreement later this month, Prime Minister Iurie Leanca told an investment conference on June 12. Both Leanca and European Commission President Jose Manuel Barroso sent out a strong message to “enemies” of change, after Moldova has come under months of pressure from Russia to back away from closer EU integration.
Both Moldova and Georgia are due to sign an Association Agreement and Deep and Comprehensive Free Trade Agreement (DCFTA) with the EU on June 27. The deal is an important step on Moldova’s road to integration with the EU and its ultimate goal of EU accession. While the agreements cover primarily economic and technical issues, politically the signing is a decisive move by Chisniau towards Europe and away from Russia’s sphere of influence, and is seen by Leanca’s government as a pivotal moment for the country. “We are certain [signing the agreement] is for the good of the country, the economy and the people,” Leanca told the conference. “Europe is in our history, our culture, our language and our values – but most importantly it is where we see our future.”
Barroso also emphasised the importance of the move for Moldova’s future. “It is our conclusion and our determination that Europe is the best future for Moldova. It is the best way to guarantee a united, modern and prosperous Moldova. There is no doubt that Moldova is a European country,” he said.
Pressure
Moldova and Georgia defied heavy pressure from Russia to initial their EU association agreements at the Vilnius summit in November 2013. The decision by now-ousted Ukrainian president Viktor Yanukovych not to sign a similar agreement sparked mass protests in Kyiv and several months of violence, which is still continuing in parts of the country. Yanukovych was ousted in February, and the new government under President Petro Poroshenko had hoped to sign the deal on June 27, though on June 11 this was suspended on technical grounds. Meanwhile in Armenia, two months before the Vilnius summit, President Serzh Sargsyan abruptly announced that the country would join Vladimir Putin’s rival Customs Union – effectively ruling out the signing of an EU deal.
With less than two weeks to go, it is unlikely now that Moldova will be deterred from its course towards EU integration. However, both Leanca and Barroso alluded to the pressures Chisinau is still facing. “We combat resistance to change both here and abroad. Change has many enemies,” said Leanca. “This is not just a choice between different visions. It is a choice between the forces of modernisation and those who want to take us back to stagnation, isolation and authoritarian politics. We will not draw back from EU integration, we will take the country forward.”
Barroso noted “spurious and... irrational objections” to Moldova’s ambitions to sign the agreement. “We say to Russia that the association agreement is compatible with Moldova’s free trade agreement with the [Commonwealth of Independent States]. A stable and prosperous Moldova will benefit Russian exporters and investors. We call on Russia not to take further measures around the signature of the agreement as there is no economic reason or legal justification for this.”
Russia has considerable scope to put economic pressure on Moldova. Immediately before the Vilnius summit, Russia banned imports of Moldovan wine, citing several batches of contaminated brandy. Russia’s Gazprom is also Moldova’s sole gas supplier, though construction of the Iasi-Ungheni gas interconnector pipeline will allow Moldova to start imports from Romania when it comes online.
Politically, Moscow wields leverage through its financial and military support for Transnistria (also known as Transdniestria), a tiny self-declared republic between the east bank of the Dniester river and Moldova’s border with Ukraine. Encouraged by Russia's annexation of Crimea in April, Transnistria has stepped up its efforts to secede from Moldova. Its parliament voted on April 16 to appeal to Moscow for official recognition, followed by entry to the Russian Federation.
New markets
However, Moldova, Europe’s poorest country, is keenly aware of the economic benefits of integration with the EU, as well as the political factors. The International Monetary Fund (IMF) expects Moldova’s economy to grow by 3.5% in 2014, but its future growth is highly dependent on growth within the EU, its main trading partner, and on regional stability.
Signing the DCFTA is expected to benefit Moldova by opening up EU markets and to encourage investment by EU firms. Over half of Moldova’s investments already go to the EU, and Chisinau is eager to increase this total. Barroso told the conference that studies have shown Moldova’s GDP will grow by an additional 5.4% in the medium term after signing the agreement. On April 29, Moldovans were also granted visa-free travel to most countries in the EU.
Complying with the DCFTA will, however, require considerable effort on the part of the Moldovan authorities and private companies to bring production, quality checks and other activities into line with EU standards.
Currently, Moldova’s most pressing problems include a lack of job opportunities, especially for young people, which has resulted in a high level of emigration. In 2012, remittances from migrant workers abroad accounted for 25% of GDP, putting the country in joint third place worldwide in terms of the contribution of remittances to GDP, according to a World Bank report.
Leanca told the conference that “even more important than access to the EU market is creating the legal institutions and economic environment to attract more investment”. “FDI is critical to our economy, and the DCTFA is imperative for Moldovan policy – first for job creation and second for innovation,” he said. “We have massive emigration because there are not local employment opportunities, and therefore job creation is a key priority. More investments are a pre-condition for more jobs here and bringing back migrants.”
After the deal is signed on June 27, Chisinau is expected to continue to pursue its long-term goal of EU membership. Speaking on April 30, the day after Moldovans were allowed to travel visa-free to most EU countries, Leanca said that Moldova has set itself a target of entry to the EU within five years and will do "everything possible" to achieve.
| 274,481
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FAQs
By nature, the experiment involves spreading the solution with centrifugal force. This will give
ripples when either the solution quantity is in excess or the speed/acceleration is lower than
required. (Few other factors might also result in ripple formation)
- Low maintenance
- Highly efficient
- Long life span
- High torque to weight ratio
All Spin Coaters are not necessarily Glove Box friendly. In order to satisfy this criterion, the
instrument must be compact and simple. To achieve this, it is recommended that the device be
standalone device (without the need of a vacuum pump).
This helps the researchers to quickly recall and replay a program from the past. This helps in judging
the repeatability and its parameters. NT12000 stores not only the program data, but also the details
of the experiment which will equip the researcher with the data quickly.
Though the spin coating process has its own way to judge the resultant thickness, researchers often
encounter a problem with narrowing down to the required parameters. This equation will help them
perform experiments without and delay in calculation so that they arrive at the perfect mix of
parameters quickly.
This maybe required for several reasons, few of the important ones being coating highly viscous
solutions and to achieve extremely thin films. With high acceleration, the outward force increases
exponentially, possibly even removing ripples formed on the film.
- Can switch between vacuum and vacuum-less chucks at ease.
- Makes more space for cleaning the chamber and makes it comfortable to reach to the
edges.
- If the chuck is accidentally damaged, replacement is quick and effortless.
Vibrations in the horizontal plane will cause differences in centrifugal force in different points of the
substrate which will result in more dispersion along few radially outward lines and less among the
rest whereas vibrations in the vertical plane will result in ripples, deformation and undulations in the
film.
- This allows the instrument to be compact and standalone. With this, it can be fitted into any
space. This becomes an important point because the spaces in labs are at a premium and
should be used wisely.
- In addition to it, higher RPMs can be achieved which is not possible in the case of vacuum-
based Spin Coaters due to the limited vacuum pressure that can be applied on the
substrates.
- Reduces the time between two successive experiments without the need to switch the
vacuum pump.
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uBiome convinced Silicon Valley that testing poop was worth $600 million. Then the FBI came knocking. Here's the inside story.
uBiome; Yutong Yuan/Business Insider
uBiome founders Zac Apte and Jessica Richman.
- Silicon Valley health startup uBiome is in hot water on the heels of an FBI raid in April.
- Founded in 2012, uBiome raised $105 million from investors on the promise of exploring the microbiome, a 'forgotten organ.'
- As uBiome advanced from a citizen science project to a clinical testing company, it overstated the medical value of its tests and prioritized growth over patient care, according to insiders, lawyers, and government officials.
- "Some of my uBiome results remind me of astrology," one ex-employee said.
- uBiome may also have run afoul of federal and state regulations while running some of its tests, according to the experts, insiders, and documents.
- This is a preview of the full inside story on uBiome, which is available exclusively to BI Prime subscribers.
When the three founders of the "microbial genomics" startup uBiome began collecting human poop, they kept it in an erstwhile storage closet, inside second-hand freezers from a discount lab supply website. It was a far cry from a state-of-the-art facility.
But the setup matched uBiome's image at the time as a crowd-funded citizen science initiative that sought to seize knowledge about our bowels from lab scientists and place it in the hands of regular people.
It wasn't until roughly three years later, at the end of 2014, that the company established a lab space professional enough to be certified by government regulators. That certification was part of uBiome's attempt to transform itself from a fun, collaborative science project to a bonafide medical-testing outfit - one that could justify investments from high-powered VC firms like Andreessen Horowitz and 8VC and ultimately garner a $600 million valuation.
Read the full inside story on uBiome, exclusively on BI Prime.
That transformation hasn't gone very well: At the end of April, FBI agents busted through the door of uBiome's San Francisco headquarters and executed a search warrant, collecting information from employees' computers and hauling away cardboard boxes full of evidence, CNBC reported. The warrant was reportedly part of an investigation into the company's billing practices.
But the problems at uBiome extended far beyond billing issues, according to interviews with 11 former employees across its billing, operations, marketing, and science, departments, as well as with customers, lawyers, and medical experts.
Read Business Insider's full story on uBiome, available exclusively to BI Prime subscribers.
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View original post 30 more words
| 311,003
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TITLE: Constructing blocks from $n$ square blocks of size $1\times 1$
QUESTION [1 upvotes]: Given $n$ square blocks of size $1\times 1$, how many blocks up to rotation and reflection are possible to construct if we paste all of them together side to side $($ let's call it $\, B(n) \, )$ . For example, $B(1)=B(2)=1$, $B(3)=2$ and $B(4)=5$.
The problem is simple for small values of $n$, but for higher values the complexity increases drastically, I believe that there is some kind of relation between $B(n)$ and the partitions of an integer $n$. It took me some time to calculate $B(5)$, as there are several possibilities. I am not sure whether the figure below contains all possible blocks for $n=5$. I don't see any pattern here. How do we find $B(n)$ for any natural number $n$?
REPLY [1 votes]: For $n=5$ their number is $12$. You missed $2$ of them:
For more details see free polyominoes. There is no known formula for the number of distinct polyominoes of a given number of squares.
| 127,952
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Stories inspire me. And if you’re anything like me, your daily cup of coffee is also your getaway to a world of stories that are either inspired or a product of your imagination. I believe we are all living our stories, and mine’s on its 26th chapter.
Everyday, I feel the need to share some of the experiences/learnings of my 26-year old self. It could be the love that I have for myself, or the unverified fact that my short life has given me learnings of a lifetime. So i’m shaping into words my great learnings and how a 26 year old could use these to keep the mid-life crisis at bay.
Work Matters : When they were designing the cycle of life, they put a very large part into this aspect called ‘earning’. People who are still struggling with what they want to do in life scare me. At 26, you ought to be, you must be sorted with what you want to do. Your dream job may be working as a wine-taster, or even better, becoming a pizza-roaster, but you need to have a serious career. What’s more important – try to work with your passion. Understand, put time, take advice, search over and over until you find what you’re really looking for. It’s never too late to start doing what you actually love doing since what you wouldn’t achieve with time, you certainly would with self-satisfaction. Also, do not restrict yourself to job titles. Do not ever restrict yourself when it comes to work. Come on time, leave on time. Learn as much as possible, it will always be of help. That said, always put your dignity first, no job will be as important as your self-respect, ever.
Experiencing beyond breathing : As Indians, we’re illogically over protected. We don’t want to cross the lines drawn in the influence of ‘Log kya khenge’ syndrome. We’re better at breathing than at experiencing. Plus, you need to have a collection of memoirs of life to share with your kids. Wouldn’t it be cool to tell your kids, “when I was 25, I spent all that I had and solo-tripped across Europe”. While in college, I freelanced across industries to understand my interest areas. 26 is a good age to break them barriers. Go beyond your imagination – learn, laugh, live to your heart’s content. And come back a changed person.
Staying Alive : As a 26 year old, this is most important. The urban lifestyle is killing us with every slice of that cheese-burst pizza. At times, I feel lucky to have access to fresh, (hopefully) organic food, as compared to the genetically modified supplements of the west. Your body will talk to you if you listen. I have had my own share of spicy pav-bhajis and greasy chola-bhaturas, but I can say this – one day, you’ll be sorry. It’s harsh but it’s the truth. So, my advice, listen to your body, and not your heart, when it comes to food. Add a lot of fresh fruits and vegetables in your diet and eat dinner before 8. Thank me later.
Love is not a game : It really is not. Modern dating makes me want to punch myself in the throat. Swap left, swap right. Really? I may be old-school, but love should be as heart warming as an old book. I wouldn’t say that 26 is when you should’ve found real love, but it surely is the age to start looking. My experience says that a good partner helps you become a better person, every single day. The smiles that we lose in the stress and monotony of life, that one person will make up for it. Plus who likes to life their life without a pinch of drama?
Learn to live alone : Our lives are supported by so many instruments – living and non-living. Just as that hardworking gardener works on keeping your grass green, your refrigerator keeps your coffee cold. If you’re as privileged as me, your food gets cooked, your bed gets made, your cars get cleaned, your floors are wiped, your dinner is served with little or no interventions. You have really nice neighbours to help you out when you lose your keys or good friends who get drunk with you when your heart gets broken. But, I believe, it is extremely important for every 26 year old to learn how to live alone. With personal experience, I can say that an encounter with yourself in the alone is incredibly interesting. Try going to a cafe and sitting by yourself. It would be awkward at first but it will really help you in getting in touch with yourself, along with a reality check on your confidence. Knowing how to do things is an asset that every young person should equip him/herself with. Learn to cook, clean, wash and be independent – you don’t have to like it. Plus, that easy peasy recipe of spaghetti Arrabbiata could really come in handy when you are trying to impress that special one.
Find a goto thing : It could be anything but cigarettes and alcohol. I believe that once a person finds out his/her passion spot, they become wiser, they make better decisions for themselves. The trick is to do a lot of things and then discovering what really makes you happy. For me it is food, or travelling, or travelling with food. Good food is my ultimate source of joy, and so is experiencing a new place. For you, it could be music or books or coins or sleeping – anything that gives you complete happiness, rejuvenates your soul – makes you feel more alive and kicking than ever.
Learn to smile, and forgive : Forgiveness is a powerful virtue. Smiling is the best weapon. You feel as unworthy as a tiny speck of sand after your boss gives you an earful, come out the door smiling like you’ve conquered the world. It would be hard, but equally worthwhile when your colleagues would assume/tell you “maybe he’s got a promotion” or “what a lucky ass!”. Same goes with forgiveness. It is something you, and only you possess. Try it sometime, will leave you feeling lighter and nicer. Plus good karma.
Mostly, don’t let anyone ever tell you how you should be. I know, it is ironical, given this article. What I mean is – listen, understand but do what YOU want. One life, right?
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The craft brewing scene is exploding in Minnesota as it is across the country: Two-thirds of the breweries in the state have opened just since 2010, according to licensing records.
When Ryan Petz signed his lease in 2010 for a new brewery in the shadow of Minneapolis' Target Field, he never dreamed they'd outgrow the 6,000-square-foot building in just a few years.
Fulton Beer expects to pump out 7,000 barrels — that's about 1.7 million pints — of beer this year, more than double their production in 2012. They'll have to cram in two more brewing tanks to meet demand for their brews, like the Guns N' Roses-inspired India Pale Ale called Sweet Child of Vine. There is barely enough room for one.
The craft brewing scene is exploding in Minnesota as it is across the country: Two-thirds of the breweries in the state have opened just since 2010, according to licensing records. Twenty of those started in the last year alone.
Much of that growth has been driven on the local level by city officials, who see a new craft brewery in their town as a cultural and tourism magnet, a job creator and a growing economic engine.
Nowhere is that more clear than in Minneapolis. Mayor R.T. Rybak crashes brewery openings, and the City Council has shown its willingness to tweak zoning and other city ordinances to help out individual breweries.
"That has been one of the biggest difference makers," said Petz, the president of Fulton. "If the city of Minneapolis hadn't been willing to ... actually be proactive in helping us, the industry would be in a different place."
Minneapolis is home to eight of the 35 craft breweries and brewpubs that have opened since 2010. Duluth, another hot spot for growth, has four.
"We have done everything we could to make it clear that we're strongly behind the new beer culture," Rybak said.
In November 2011, the City Council changed a longstanding ordinance that separated churches and establishments that serve alcohol so that Dangerous Man Brewing could open a taproom in northeast Minneapolis. Dangerous Man finally opened its doors earlier this year.
And last spring, food trucks parked in Fulton's lot to feed hungry customers until Petz got a notice that he had to send the trucks away — the city's zoning ended three blocks east of their building. The city fixed the issue in less than a week, which Petz called "light speed."
"It could have been a protracted thing. We could not even have food trucks now, a year later," he said.
On the banks of the St. Croix River — outside the cities where the industry has flourished most — Stillwater has been nothing but supportive of Lift Bridge Brewing, said CEO Dan Schwarz.
"They understand that it's a part of the culture, it's a part of the community," Schwarz said.
For the industry to keep growing in Minnesota, craft brewers may need more help from the state.
The so-called Surly bill (named for a Brooklyn Center brewery), which allowed brewers to open taprooms and sell pints onsite, was a huge win for the industry when it passed in 2011. But other victories have been harder to score at the Capitol, where lawmakers and a powerful liquor lobby are wary of easing alcohol restrictions too fast.
Rep. Joe Atkins, an Inver Grove Heights Democrat who has gained a reputation for handling alcohol issues at the Legislature, said it makes sense for cities to approach craft breweries as an opportunity for economic development. But the state's role is a regulatory one, he said, which means changes often come slow.
In just the past five years, Atkins said he and others at the Legislature "have gone from perceiving the craft brewer as something of a novelty to something that makes a great deal of sense to embrace."
This year, a group of small breweries is pushing to revise state law so that growing brewers can keep selling 64-ounce growlers of beer.."
For Petz, it's just a matter of time before the Legislature embraces the industry the way Minneapolis has. Minnesota brewers want to catch up to other states like California, Colorado and Oregon, where craft beer is ingrained in the culture.
"People are just getting that idea now," he said.
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Know more about this business than we do? Cool! Please submit any corrections or missing details you may have.Help us make it right
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Univar USA Inc can be found at Paper Mill Rd 2500. The following is offered: Chemicals & Allied Products. The entry is present with us since Sep 8, 2010 and was last updated on Nov 12, 2013. In Mobile there are 15 other Chemicals & Allied Products. An overview can be found here.
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TUNAP Automotive stands for over forty years of success and outstanding quality. We follow a single goal: we want to make your garage standards even better with innovative applications and constant development.
We develop tailored solutions for improved mobility together with partners at home and abroad. The product range includes chemical maintenance, repair and troubleshooting products for use in all motor vehicle fields from engines and brakes via bodywork to maintenance and care of air conditioning systems and interiors that is not harmful to humans. TUNAP offers sustainable goods and services with added value and thus creates win-win situations for makers, garages and motorists.
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\begin{document}
\title{On Schur 2-groups}
\author{Mikhail Muzychuk}
\address{Netanya Academic College, Netanya, Israel}
\email{muzy@netanya.ac.il}
\author{Ilya Ponomarenko}
\address{Steklov Institute of Mathematics at St. Petersburg, Russia}
\email{inp@pdmi.ras.ru}
\thanks{The work of the second author was partially supported by the RFBR Grant 14-01-00156 А}
\date{}
\begin{abstract}
A finite group $G$ is called a Schur group, if any Schur ring over~$G$ is the transitivity module of a point stabilizer
in a subgroup of $\sym(G)$ that contains all right translations. We complete a classification of abelian $2$-groups by proving that the
group $\mZ_2\times\mZ_{2^n}$ is Schur. We also prove that any non-abelian Schur $2$-group of order larger than $32$ is dihedral
(the Schur $2$-groups of smaller orders are known). Finally, in the dihedral case, we study Schur rings of rank at most $5$, and show that
the unique obstacle here is a hypothetical S-ring of rank $5$ associated with a divisible difference set.
\end{abstract}
\maketitle
\section{Introduction}
Following R.P\"oschel \cite{Poe74}, a finite group $G$ is called a {\it Schur group}, if any S-ring over~$G$ is the transitivity module
of a point stabilizer in a subgroup of $\sym(G)$ that contains all right translations (for the exact definitions, we refer to
Section~\ref{120813a}). He proved there that if $p\ge 5$ is a prime, then a finite $p$-group is Schur if and only if
it is cyclic. For $p=2$ or $3$, a cyclic $p$-group is still Schur, but P\"oschel's
theorem is not true: a straightforward computation shows that an elementary abelian group
of order $4$ or $9$ is Schur. In this paper, we are interested in Schur 2-groups.\medskip
Recently in \cite{EKP14}, it was proved that every finite abelian Schur group belongs to one
of several explicitly given families. In particular, from Lemma~5.1 of that paper it
follows that all abelian Schur $2$-groups are known except for the groups
$\mZ_{2^{}}\times \mZ_{2^n}$, where $n\ge 5$. We prove that all these groups
are Schur (Theorem~\ref{090514b}). As a by-product we can complete the classification of
abelian Schur $2$-groups.
\thrml{170814a}
An abelian $2$-group $G$ is Schur if and only if $G$ is cyclic, or elementary abelian of order at most $32$, or
is isomorphic to $\mZ_{2^{}}\times \mZ_{2^n}$ for some~$n\ge 1$.
\ethrm
Non-abelian Schur groups have been studied in~\cite{PV} where it was proved that they are
metabelian. In particular, from Theorem~4.2 of that paper
it follows that non-abelian Schur $2$-groups are known except for dihedral groups and groups
\qtnl{170914a}
M_{2^n}=\grp{a,b:\ a^{2^{n-1}}=b^2=1,\ bab=a^{1+2^{n-2}}},
\eqtn
where $n\ge 4$. In this paper we prove that the latter groups are not Schur (Theorem~\ref{071113a}).
As a by-product we obtain the following statement.
\thrml{170814b}
A non-abelian Schur $2$-group of order at least $32$ is dihedral.
\ethrm
We do not know whether or not a dihedral $2$-group of order more than $32$ is Schur. A standard technique based
on Wielandt's paper~\cite{W49} enables us to describe S-rings of rank at most~$5$ as follows (see Subsection~\ref{231014w} for a connection between S-rings and divisible difference sets).
\thrml{170814c}
Let be $\cA$ an S-ring over a dihedral $2$-group. Suppose that
$\rk(\cA)\le 5$. Then one of the following statements is true:
\nmrt
\tm{1} $\cA$ is isomorphic to an S-ring over $\mZ_{2^{}}\times \mZ_{2^n}$,
\tm{2} $\cA$ is a proper dot or wreath product,
\tm{3} $\rk(\cA)=5$ and $\cA$ is associated with a divisible difference set in $\mZ_{2^n}$.
\enmrt
\ethrm
S-rings in statement (1) of this theorem are schurian by Theorem~\ref{170814a}. By
induction, this implies that all S-rings in statement~(2) are also schurian. Thus, by
Theorem~\ref{081014w} we obtain the following corollary.
\crllrl{031014a}
Under the hypothesis of Theorem~\ref{170814c}, the S-ring $\cA$ is
not schurian only if $\cA$ is associated with a divisible difference set in a cyclic $2$-group.
\ecrllr
In fact, we do not know whether there exists a non-trivial divisible difference set in a cyclic $2$-group that produces an S-ring $\cA$
in part~(3) of Theorem~\ref{170814c}. If such a set does exist, then the corresponding dihedral $2$-group is not Schur. On the other hand,
in Subsection~\ref{231014w}, we show that using a relative difference set (which is a special case of a divisible one), one can construct
an S-ring of rank~$6$ (over a dihedral $2$-group). These difference sets, and, therefore, S-rings, do exist, but are relatively rare. The only known example is the classical $(q+1,2,q,(q-1)/2)$-difference set
where $q$ is a Mersenne prime. Thus, the question whether a dihedral $2$-group is Schur,
remains open.\medskip
The paper consists of fourteen sections. In Sections~\ref{120813a}, \ref{111014a}
and \ref{231014v}, we give a background of S-rings, Cayley schemes\footnote{Here, we assume
some knowledge of association scheme theory.} and cite
some basic facts on S-rings over cyclic and dihedral groups. In Sections~\ref{231014a1}--
\ref{231014a3}, we develop a theory of S-rings over $\mZ_2\times\mZ_{2^n}$ that is
culminated in Theorem~\ref{090514b} stating that this group is Schur.
In Section~\ref{231014a4}, we show that the group $M_{2^n}$ is not Schur
for all $n\ge 4$ (Theorem~\ref{071113a}).
In Sections~\ref{231014a6}--\ref{231014a7}, we study S-rings over a dihedral $2$-group:
here, we start with constructions based on cyclic divisible difference sets, and then
complete the proof of Theorem~\ref{170814c}.\medskip
{\bf Notation.}
As usual, by $\mZ$ we denote the ring of rational integers.
The identity of a group $D$ is denoted by $e$; the set of non-identity elements in $D$ is denoted by $D^\#$.
Let $X\subseteq D$. The subgroup of $D$ generated by $X$ is denoted by $\grp{X}$;
we also set $\rad(X)=\{g\in D:\ gX=Xg=X\}$. The element $\sum_{x\in X}x$ of the group
ring $\mZ D$ is denoted by $\und{X}$. The set $X$ is called {\it regular} if the order
$|x|$ of an element $x\in X$ does not depend on the choice of~$x$.
For a group $H\trianglelefteq D$, the quotient epimorphism from $D$ onto $D/H$ is denoted by $\pi_{D/H}$.
The group of all permutations of $D$ is denoted by $\sym(D)$. The set of orbits of a group
$G\le\sym(D)$ is denoted by $\orb(G)=\orb(G,D)$. We write $G\twoe G'$ if the groups $G,G'\le\sym(D)$
are {\it $2$-equivalent}, i.e. have the same orbits in the coordinate-wise action on~$D\times D$.
Given two subgroups $L\trianglelefteq U\leq D$, the quotient group $U/L$ is called the {\it section} of $D$. For a
set $\Delta\subseteq\sym(D)$ and a section $S = U/L$ of $D$ we set
$$
\Delta^S=\{f^S:\ f\in \Delta,\ S^f=S\},
$$
where $S^f = S$ means that $f$ permutes the right $L$-cosets in $U$ and $f^S$ denotes the bijection of
$S$ induced by~$f$.
The cyclic group of order $n$ is denoted by $\mZ_n$.
\section{A background on S-ring theory}\label{120813a}
In what follows, we use the notation and terminology of~\cite{EP12}.\medskip
Let $D$ be a finite group. A subring~$\cA$ of the group ring~$\mZ D$ is called a {\it Schur
ring} ({\it S-ring}, for short) over~$D$ if there exists a partition $\cS=\cS(\cA)$ of~$D$
such that
\nmrt
\tm{S1} $\{e\}\in\cS$,
\tm{S2} $X\in\cS\ \Rightarrow\ X^{-1}\in\cS$,
\tm{S3} $\cA=\Span\{\und{X}:\ X\in\cS\}$.
\enmrt
When $\cS=\orb(K,D)$ where $K\le\aut(D)$, the S-ring $\cA$ is called {\it cyclotomic} and
denoted by $\cyc(K,D)$. A group isomorphism $f:D\to D'$ is called a {\it Cayley isomorphism} from an S-ring $\cA$ over
$D$ to an S-ring $\cA'$ over $D'$ if $\cS(\cA)^f=\cS(\cA')$.\medskip
It follows from (S3) that given $X,Y\in\cS(\cA)$ there exist non-negative integers
$c_{X Y}^Z$, $Z\in\cS(\cA)$, such that
$$
\und{X}\,\und{Y}=\sum_{Z\in\cS(\cA)}c_{X Y}^Z\und{Z}.
$$
One can see that $c_{XY}^Z$ equals the number of different representations $z=xy$ with
$(x,y)\in X\times Y$ for a fixed (and hence for all) $z\in Z$. It is a well-known fact that
$$
c_{Y^{-1}X^{-1}}^{Z^{-1}}=c_{X^{}Y^{}}^{Z^{}}\qaq |Z|c_{XY}^{Z^{-1}}=|X|c_{YZ}^{X^{-1}}=|Y|c_{ZX}^{Y^{-1}}
$$
for all $X,Y,Z$. A ring isomorphism $\varphi:\cA\to \cA'$ is said to be {\it algebraic} if for any $X\in\cS(\cA)$
there exists $X'\in\cS(\cA')$ such that $\varphi(\und{X})=\und{X'}$.\medskip
The classes of the partition $\cS$ and the number $\rk(\cA)=|\cS|$ are called the {\it basic
sets} and the {\it rank} of the S-ring~$\cA$, respectively. Any union of basic sets
is called an {\it $\cA$-subset of~$D$} or {\it $\cA$-set}. The set of all of them
is closed with respect to taking inverse and product. Given an $\cA$-set $X$, we denote by $\cA_X$
the submodule of~$\cA$ spanned by the elements $\und{Y}$, where $Y$ belongs to the set
$$
\cS(\cA)_X=\{Y\in\cS(\cA):\ Y\subseteq X\}.
$$
Any subgroup of $D$ that is an $\cA$-set, is called an {\it $\cA$-subgroup} of~$D$ or
{\it $\cA$-group}. With each $\cA$-set $X$, one can naturally associate two $\cA$-groups,
namely $\grp{X}$ and $\rad(X)$ (see Notation). The following useful lemma was proved
in \cite[p.21]{EvdP09}.\footnote{Apparently, for the first time this lemma was proved in~\cite[Proposition~4.5]{M87}.}
\lmml{090608a}
Let $\cA$ be an S-ring over a group $D$ and $H\le D$ an $\cA$-group. Then
given $X\in\cS(\cA)$, the cardinality of the set $X\cap xH$ does not depend on $x\in X$.
\elmm
A section $S=U/L$ of the group $D$ is called an {\it $\cA$-section}, if both $U$ and
$L$ are $\cA$-groups. In this case, the module
$$
\cA_S=\Span \{\pi_S(X):\ X\in\cS(\cA)_U\}
$$
is an S-ring over the group~$S$, the basic sets of which are exactly the sets $\pi_S(X)$ from the
right-hand side of the formula.\medskip
The S-ring $\cA$ is called {\it primitive} if the only $\cA$-groups are $e$ and $D$,
otherwise this ring is called {\it imprimitive}. One can see that if $H$
is a minimal $\cA$-group, then the S-ring $\cA_H$ is primitive. The classical results
on primitive S-rings over abelian and dihedral groups were obtained in papers~\cite{S33,K37,W49}.
A careful analysis of the proofs shows that the schurity assumption there
was superfluous. Therefore, in the first part of the following statement, we formulate the
corresponding results in slightly more general form.
\thrml{030414a}
Let $D$ be a $2$-group which is cyclic, dihedral, or isomorphic to the group $\mZ_2\times\mZ_{2^n}$. Then any primitive S-ring over $D$
is of rank~$2$. In particular, if $\cA$ is an S-ring over $D$ and $H\le D$ is a minimal $\cA$-group,
then $H^\#$ is a basic set of~$\cA$.
\ethrm
Let $S=U/L$ be an $\cA$-section of the group $D$. The S-ring $\cA$ is called the {\it generalized $S$-wreath product}
\footnote{In \cite{LM98}, the term {\it wedge product} was used.} if the group $L$ is
normal in~$D$ and $L\le\rad(X)$ for all basic sets $X$ outside~$U$; in this case we write
\qtnl{050813a}
\cA=\cA_U\wr_S\cA_{D/L},
\eqtn
and omit $S$ if $U=L$. When the explicit indication of~$S$ is not important,
we use the term {\it generalized wreath product}. The generalized $S$-wreath product is
{\it proper} if $L\ne e$ and $U\ne D$. When $U=L$, the generalized $S$-wreath product coincides with the ordinary
wreath product.\medskip
Let $D=D_1D_2$, where $D_1$ and $D_2$ are trivially intersecting subgroups of $D$.
If $\cA_1$ and $\cA_2$ are S-rings over the groups $D_1$ and $D_2$ respectively, then the
module
$$
\cA=\Span\{\und{X_1\cdot X_2}: X_1\in\cS(\cA_1),\ X_2\in\cS(\cA_2)\}
$$
is an S-ring over the group $D$ whenever $\cA_1$ and $\cA_2$ are commute with each other.
In this case, $\cA$ is called the {\it dot product} of $\cA_1$ and
$\cA_2$, and denoted by $\cA_1\cdot\cA_2$ \cite{LM98}. When $D=D_1\times D_2$, the
dot product coincides with the {\it tensor product} $\cA_1\otimes \cA_2$. The following
statement was proved in \cite{EKP14}.
\lmml{050813b}
Let $\cA$ be an S-ring over an abelian group $D=D_1\times D_2$. Suppose that $D_1$ and $D_2$
are $\cA$-groups. Then $\cA=\cA_{D_1}\otimes \cA_{D_2}$ whenever $\cA_{D_1}=\mZ D_1$
or $\cA_{D_2}=\mZ D_2$.
\elmm
The following two important theorems go back to Schur and Wielandt (see \cite[Ch.~IV]{Wie64}).
The first of them is known as the Schur theorem on multipliers, see~\cite{EvdP09}.
\thrml{261009b}
Let $\cA$ be an S-ring over an abelian group $D$. Then given an integer~$m$ coprime
to $|D|$, the mapping $X\mapsto X^{(m)}$, $X\in\cS(\cA)$, where
\qtnl{141014a}
X^{(m)}=\{x^m:\ x\in X\},
\eqtn
is a bijection. Moreover, $x\mapsto x^m$, $x\in D$, is a Cayley automorphism of~$\cA$.
\ethrm
Given a subset $X$ of an abelian group $D$, denote by $\tr(X)$ the {\it trace} of $X$, i.e. the union
of all~$X^{(m)}$ over the integers $m$ coprime to~$|D|$. We say that
$X$ is {\it rational} if $X=\tr(X)$. When $\tr(X)=\tr(Y)$ for some $Y\subseteq D$, the sets
$X$ and $Y$ are called {\it rationally conjugate}. For an S-ring $\cA$ over $D$, the module
$$
\tr(\cA)=\Span\{\tr(X):\ X\in\cS(\cA\}
$$
is also an S-ring; it is called the {\it rational closure} of~$\cA$. Finally, the S-ring is
{\it rational} if it coincides with its rational closure, or equivalently, if each of its
basic sets is rational.\medskip
In general, Theorem~\ref{261009b} is not true when $m$ is not coprime to the order of $D$. However, the following
weaker statement holds.
\thrml{261009w}
Let $\cA$ be an S-ring over an abelian group $D$. Then given a prime divisor~$p$ of~$|D|$,
the mapping $X\mapsto X^{[p]}$, $X\in 2^ D$, where
\qtnl{030713a}
X^{[p]}=\{x^p:\ x\in X,\ \nmmod{|X\cap Hx|}{0}{p}\}
\eqtn
with $H=\{g\in D:\ g^p=1\}$, takes an $\cA$-set to an $\cA$-set.
\ethrm
We complete the section by the theorem on separating subgroup that was proved
in~\cite{EP05}.
\thrml{t100703}
Let $\cA$ be an S-ring over a group $D$. Suppose that $X\in\cS(\cA)$ and $H\le D$ are such that
$$
X\cap H\ne\emptyset\qaq X\setminus H\ne\emptyset\qaq
\grp{X\cap H}\le\rad(X\setminus H).
$$
Then $X=\grp{X}\setminus\rad(X)$ and $\rad(X)\le H\le\grp{X}$.
\ethrm
\section{S-rings and Cayley schemes}\label{111014a}
In this section, we freely use the language of association scheme theory; in our exposition,
we follow \cite{EP09,MP09}.
\sbsnt{The 1-1 correspondence.}
For a group $D$, denote by $R(D)$ the set of all binary relations on $D$ that are
invariant with respect to the group $D_{right}$ (consisting of the permutations of the set~$D$ induced
by the right multiplications in the group~$D$). Then the mapping
\qtnl{171014a}
2^D\to R(D),\quad X\mapsto R_D(X)
\eqtn
where $R_D(X)=\{(g,xg):\ g\in D,x\in X\}$, is a bijection. If $\cA$ is an S-ring
over the group $D$, then the pair
\qtnl{210215a}
\cX=(D,S),
\eqtn
where $S=R_D(\cS(\cA))$, is
an association scheme. Moreover, it is a {\it Cayley scheme} over $D$, i.e.
$D_{right}\le\aut(\cX)$. Each basis relation $s\in S$ of this scheme, is a Cayley digraph
over~$D$ the connection set of which is equal to $es=\{g\in D:\ (e,g)\in s\}$. Conversely,
given a Cayley scheme~\eqref{210215a} the module
$$
\cA=\Span\{\und{es}:\ s\in S\}
$$
is an S-ring over~$D$.
\thrml{100909a}{\rm \cite{K85}}
The mappings $\cA\mapsto\cX$, $\cX\mapsto\cA$ form a 1-1 correspondence
between the S-rings and Cayley schemes over the group~$D$.
\ethrm
It should be mentioned that the above correspondence preserves the inclusion. Moreover,
the mapping~\eqref{171014a} induces a ring isomorphism from $\cA$ onto the adjacency algebra
of the Cayley scheme $\cX$ associated with $\cA$. It follows that
$c_{XY}^Z=c_{rs}^t$ for all $X,Y,Z\in\cS(\cA)$, where $r=R_D(X)$, $s=R_D(Y)$
and $t=R_D(Z)$. In particular, the number $|X|$ is equal to the valency $n_r$ of the relation $r$, and
the S-ring~$\cA$ is commutative if and only if so is the Cayley scheme~$\cX$.
\sbsnt{Isomorphisms and schurity.}
We say that S-rings $\cA$ and $\cA'$ are (combinatorial) {\it isomorphic} if
the Cayley schemes associated with $\cA$ and $\cA'$ are isomorphic. Any isomorphism
between these schemes is called the {\it isomorphism} of $\cA$ and $\cA'$. The group
$\iso(\cA)$ of all isomorphisms from $\cA$ to itself has a normal subgroup
$$
\aut(\cA)=\{f\in\iso(\cA):\ R_D(X)^f=R_D(X)\ \text{for all}\; X\in\cS(\cA)\};
$$
any such $f$ is called a (combinatorial) {\it automorphism} of the S-ring~$\cA$.
In particular, if $\cA=\mZ D$ (resp. $\rk(\cA)=2$), then $\aut(\cA)=D_{right}$
(resp. $\aut(\cA)=\sym(D)$).\medskip
The S-ring $\cA$ is called {\it schurian} (resp. {\it normal}) if so is the Cayley scheme
associated with $\cA$. Thus, $\cA$ is schurian if and only if $\cS(\cA)=\orb(\aut(\cA)_e,D)$,
and normal if and only if $D_{right}\trianglelefteq\aut(\cA)$.\medskip
From our definitions, it follows that $\cA=\cA_1\wr\cA_2$ if and only if $\cX=\cX_1\wr\cX_2$
where $\cX$, $\cX_1$ and $\cX_2$ are the Cayley schemes associated with the S-rings~$\cA$, $\cA_1$ and
$\cA_2$ respectively. Similarly, $\cA=\cA_1\cdot\cA_2$ if and only if $\cX=\cX_1\otimes\cX_2$.
On the other hand, the tensor and wreath product of association schemes (and permutation groups) are special cases
of the crested product introduced and studied in~\cite{BC05}. Thus, Theorem~\ref{211014a} below immediately follows from
Remark~23 of that paper and Theorems~21 and~22 proved there.
\thrml{211014a}
Let $\cA=\cA_1\ast\cA_2$, where $\ast\in\{\wr,\cdot\}$. Then $\cA$ is schurian if and
only if so are $\cA_1$ and $\cA_2$. Moreover,
$$
\aut(\cA_1\wr\cA_2)=\aut(\cA_1)\wr\aut(\cA_2)\qaq
\aut(\cA_1\cdot\cA_2)=\aut(\cA_1)\times\aut(\cA_2).
$$
\ethrm
The following simple statement is an obvious consequence of the definition of wreath product
and Theorem~\ref{211014a}.
\crllrl{021014a}
Let $\cA$ be an S-ring over a group $D$ and $H$ an $\cA$-group such that
$\rk(\cA)=\rk(\cA_H)+1$. Then $\cA$ is isomorphic to the wreath product of $\cA_H$
by an S-ring of rank $2$ over the group $\mZ_{[D:H]}$.
Moreover, $\cA$ is schurian if and only if so is $\cA_H$.
\ecrllr
\sbsnt{Quasi-thin S-rings.} An S-ring $\cA$ is called {\it quasi-thin} if any of its basic sets consists of at most
two elements. Thus, $\cA$ is quasi-thin if and only if
the Cayley scheme associated with $\cA$ is quasi-thin (the latter means that the valency
of any its basic relation is at most~$2$).
\lmml{100914a}
Let $\cA$ be an S-ring over an abelian group $D$. Suppose that $X\in\cS(\cA)$ is such
that $|X|=2$ and $\grp{X}=D$. Then $\cA$ is quasi-thin.
\elmm
\proof The Cayley scheme~$\cX$ associated with $\cA$ is commutative,
because the group $D$ is abelian. Moreover, the relation $r=R_D(X)$ corresponding
to the set $X$, is of valency $|X|=2$. Thus, the equality $\grp{X}=D$ implies that $\cX$ is a $2$-cyclic
scheme generated by the tightly attached relation $r$ in the sense of~\cite{HM}. So, by Proposition~3.11 of that paper,
$\cX$ is a quasi-thin scheme. Therefore, the S-ring~$\cA$ is also quasi-thin.\bull
Following the theory of quasi-thin schemes in \cite{MP}, we say that a basic set $X\ne\{e\}$
of a quasi-thin S-ring~$\cA$ is an {\it orthogonal} if $X\subseteq Y\, Y^{-1}$ for some
$Y\in\cS(\cA)$.
\lmml{100914y}
Any commutative quasi-thin S-ring $\cA$ is schurian. Moreover, if
it has at least two orthogonals, then the group $\aut(\cA)_e$ has a faithful regular orbit.
\elmm
\proof The first statement immediately follows from ~\cite[Theorem~1.2]{MP}. To prove
the second one, denote by $\cX$ the Cayley scheme associated with the S-ring~$\cA$. Then
from \cite[Corollary~6.4]{MP} it follows that the group $\aut(\cX)_{e,x}$ is trivial for
some $x\in D$. This means that $x^{\aut(\cX)_e}$ is a faithful regular orbit of the group $\aut(\cA)_e$.\bull
\section{S-rings over cyclic and dihedral groups}\label{231014v}
\sbsnt{Cyclic groups.}\label{141114b}
Let $C$ be a cyclic group of order $2^n$, $n\ge 1$. Then the group $\aut(C)$
consists of permutations $\sigma_m:x\mapsto x^m$, $x\in C$, where $m$ is an odd integer. In what
follows, $c_1$ denotes the unique involution in $C$.
\lmml{211113a}
Let $X\in\orb(K,C)$, where $K\le\aut(C)$. Then
\nmrt
\tm{1} $\rad(X)=e$ if and only if $X$ is a singleton, or $n\ge 3$ and $X=\{x,\varepsilon x^{-1}\}$
where $x\in X$ and $\varepsilon\in \{e,c_1\}$,
\tm{2} if $K\ge\{\sigma_m:\ \mmod{m}{1}{2^{n-k}}\}$, then $2^k$ divides $|\rad(X)|$.
\enmrt
\elmm
\proof Statement (1) follows from \cite[Lemma~5.1]{EP02}, whereas statement~(2) is straightforward.\bull
Let $\cA$ be an S-ring over the group $C$. By the Schur theorem on multipliers,
the group $\rad(X)$ does not depend on a set $X\in\cS(\cA)$ that contains a generator of $C$.
This group is called the {\it radical} of $\cA$ and denoted by $\rad(\cA)$. Since $C$ is a $2$-group, from~\cite[Lemma~6.4]{EP02}
it follows that if $\rad(\cA)=e$, then either $n\ge 2$ and $\rk(\cA)=2$, or $\cA=\cyc(K,C)$, where $K\le\aut(C)$ is the group
generated by the automorphism taking a generator $x$ of $C$ to an element in $\{x,x^{-1},c_1x^{-1}\}$
(see also statement~(1) of Lemma~\ref{211113a}). In any case, $\cA$ is, obviously, schurian. In fact, the latter statement holds
for any S-ring over a cyclic $p$-group \cite{EKP}.\medskip
For any basic set $X$ of the S-ring~$\cA$, one can form an $\cA$-section $S=\grp{X}/\rad(X)$. Then the radical
of the S-ring~$\cA_S$ is trivial. Since in our case, $|S|$ is a $2$-group, the result in previous paragraph shows
that either the S-ring $\cA_S$ is cyclotomic or $|S|$ is a composite number and $\rk(\cA_S)=2$. In the
former case, $X$ is an orbit of an automorphism group of $C$, whereas in the latter case, $X=\grp{X}\setminus\rad(X)$. Thus, any
basic set of $\cA$ is either regular or equals the set difference of two distinct $\cA$-groups.
\lmml{021109a}
Let $\cA$ be a cyclotomic S-ring over a cyclic $2$-group.
Suppose that $\rad(\cA)=e$. Then $\rad(\cA_S)=e$ for any $\cA$-section $S$ such that $|S|\ne 4$.
\elmm
\proof Follows from \cite[Theorem~7.3]{EKP}.\bull
The following auxiliary lemma will be used in Section~\ref{140914b}.
\lmml{150414a}
Let $C$ be a cyclic $2$-group, and let $X$ and $Y$ be orbits of some subgroups of $\aut(C)$. Suppose
that $\grp{X}\ne\grp{Y}$ and $\rad(X)=\rad(Y)=e$. Then the product $X\,Y$ contains
no coset of $\grp{c_1}$.
\elmm
\proof Without loss of generality, we can assume that $\grp{Y}$ is a proper subgroup of~$\grp{X}$.
Then from statement (1) of Lemma~\ref{211113a}, it follows that
$X=\{x\}$ or $\{x,\varepsilon x^{-1}\}$, and $Y=\{y\}$ or $\{y,\varepsilon' y^{-1}\}$
where $|x|>|y|$. Therefore, the required statement trivially holds whenever $X$ or $Y$ is a singleton. Thus,
we can assume that $|X|=|Y|=2$, and hence $|x|>|y|\ge 8$. Furthermore,
$$
X\,Y=\{xy,\quad \varepsilon'xy^{-1},\quad \varepsilon x^{-1}y,\quad \varepsilon''x^{-1}y^{-1}\},
$$
where $\varepsilon''=\varepsilon\varepsilon'$. Suppose on the contrary, that this product contains a coset
of $\grp{c_1}$. Then, obviously, $c_1xy=\varepsilon'xy^{-1}$ or
$c_1\varepsilon x^{-1}y=\varepsilon''x^{-1}y^{-1}$.
In any case, $y^2\in\{e,c_1\}$ and hence $|y|\le 4$. Contradiction.\bull
\sbsnt{Dihedral groups.}\label{011014b}
Throughout this subsection, $D$ is a dihedral group and $C$ is the cyclic subgroup of $D$ such that all the
elements in $D\setminus C$ are involutions. A set $X\subseteq D$ is called {\it mixed} if
the sets $X_0=X\cap C$ and $X\setminus X_0$ are not empty. For an
element $s\in D\setminus C$, we denote by $X_1=X_{1,s}$ the subset of
$C$ for which $X=X_0\,\cup\,X_1s$. The following statement (in the other notation) was proved in~\cite{W49}.
\lmml{240913a}
Let $\cA$ be an S-ring over the dihedral group $D$ and $X$ a mixed basic set of $\cA$. Then
\nmrt
\tm{1} the sets $X_0$, $X_1s$ and $X$ are symmetric, and $X_0$ commutes with $X_1s$,
\tm{2} given an integer $m$ coprime to $|D|$, there exists a unique $Y\in\cS(\cA)$ such that $(X_0)^{(m)}=Y_0$.
\enmrt
\elmm
When it does not lead to confusion, the set $Y$ from statement~(2) of Lemma~\ref{240913a} will be also denoted by $X^{(m)}$;
for $X\subseteq C$, this notation is consistent with~\eqref{141014a}. For any $\cA$-set $X$ such that $X_0\ne\emptyset$, one can define
its trace $\tr(X)$ to be the union of sets $X^{(m)}$, where $m$ runs over all integers coprime to $|D|$. The following statement was
also proved in~\cite{W49}.
\lmml{290514b}
Let $\cA$ be an S-ring over the dihedral group $D$. Suppose that $X_0\ne\emptyset$ for all $X\in\cS(\cA)$. Then
given an integer $m$ coprime to $|D|$, the mapping $X\mapsto X^{(m)}$ induces an algebraic isomorphism of~$\cA$; in particular,
$|X^{(m)}|=|X|$.
\elmm
The algebraic fusion of the S-ring $\cA$ from Lemma~\ref{290514b} with respect to the group of all algebraic isomorphisms
defined in this lemma, is an S-ring any basic set of which is of the form $\tr(X)$ where $X\in\cS(\cA)$. This S-ring is called the
{\it rational closure} of $\cA$ and denoted by $\tr(\cA)$. It should be stressed that this notation has sense only
if the hypothesis of Lemma~\ref{290514b} is satisfied.
\section{S-rings over $D=\mZ_2\times\mZ_{2^n}$: basic sets containing involutions}\label{231014a1}
In what follows, $C\le D$ is a cyclic group of order $2^n$ and $E$ is the Klein subgroup of~$D$. The non-identity elements of
this subgroup are the involution $c_1\in C$ and the other two involutions $s\in D\setminus C$ and $c_1s$.
For an S-ring over~$D$ and an element $t\in E$, we denote by $X_t$ the basic set that contains~$t$. Then
by the Schur theorem on multipliers (Theorem~\ref{261009b}), this set is rational. In this section we completely
describe the sets~$X_t$'s.
\thrml{250814a}
Let $\cA$ be an S-ring over the group $D$. Then the set $H=\bigcup_{t\in E}X_t$
is an $\cA$-group and for a suitable choice of $s$ one of the following statements holds
with $U=\grp{X_{c_1}}$:
\nmrt
\tm{1} $X_{c_1}=X_s=X_{sc_1}=U\setminus e$,
\tm{2} $X_{c_1}=U\setminus e$ and $X_s=X_{sc_1}=H\setminus U$,
\tm{3} $X_{c_1}=X_{sc_1}=U\setminus\grp{s}$ and $X_s=\{s\}$,
\tm{4} $X_{c_1}=U\setminus e$, $X_s=\{s\}$ and $X_{sc_1}=sX_{c_1}$.
\enmrt
\ethrm
The proof of Theorem~\ref{250814a} will be given later. The following
auxiliary statement is, in fact, a consequence of the Schur theorem on multipliers. Below, we fix an S-ring~$\cA$ over
the group~$D$.
\lmml{290714c}
Suppose that $X\in\cS(\cA)$ contains two elements $x$ and $y$ such that $|x| > |y|\geq 2$. Then $x\{e,c_1\}\subseteq X$.
\elmm
\proof Set $m=1 + |x|/2$. Then by Schur's theorem on multipliers, $Y:=X^{(m)}$ is a basic set of $\cA$. On the other hand,
since $|x| > |y|$, we have $y^m=y$. Thus, $y^m\in X$. This implies that $X=Y$, and hence $x^m\in X$. Since $|x|>2$ and
$x^m=xc_1$, we conclude that $x\{e,c_1\}\subseteq X$ as required.\bull\medskip
In the following lemma, we keep the notation of Theorem~\ref{250814a}.
\lmml{290714b}
Either $X_{c_1}=U\setminus e$, or $H$ is an $\cA$-group and statement~(3) of Theorem~\ref{250814a} holds.
\elmm
\proof
The statement is trivial if the set $X:=X_{c_1}$ is contained in~$E$. So, we can assume that $X$ contains at least one element
of order greater than two. Then $x\{e,c_1\}\subseteq X$ for each $x\in X$ with $|x|>2$ (Lemma~\ref{290714c}).
Thus, $c_1\in\rad(X\setminus E)$. We observe that $X\cap E$ is equal to one of the following sets:
$$
\{c_1\}\qoq \{c_1,s,sc_1\}\qoq \{c_1,t\},
$$
where $t\in\{s,sc_1\}$.
In the first two cases, $c_1\in\rad(X\setminus\{c_1\})$. So by Theorem~\ref{t100703} with $H=\grp{c_1}$, we
conclude that $X=\grp{X}\setminus\rad(X)$ and $c_1\not\in\rad(X)$. However, the only non-trivial subgroups of $D$
not containing $c_1$, are $\grp{s}$ and $\grp{sc_1}$. Since none of them equals $\rad(X)$,
we conclude that $\rad(X)=e$ and $X=U\setminus e$.\medskip
In the remaining case, $X\cap E=\{c_1,t\}$, and hence $|c_1 X\cap X|=|X|-2$. Since $c_1\in X$, this implies that the latter number
equals~$c_{XX}^X$ (see Section~\ref{120813a}).
Therefore, $|x^{-1}X\cap X|=|X|-2$ for each $x\in X$. On the other hand, the set $\{c_1,t\}t=\{c_1t,e\}$ does not
intersect $X$. Thus, $t(X\setminus E)=X\setminus E$, and hence
$$
\rad(X\setminus E)=E.
$$
By Theorem~\ref{t100703} with
$H=E$, we conclude that $X=\grp{X}\setminus\rad(X)$ and $E\setminus\rad(X)$ is contained in~$X$. Therefore, $\rad(X)=\grp{t'}$,
where $t'$ is the element of $\{s,sc_1\}$ other than $t$. Thus, $X=U\setminus\grp{t'}$, $H=U$ is an $\cA$-group and we are done.\bull\medskip
{\bf Proof of Theorem~\ref{250814a}.} By Lemma~\ref{290714b}, we can assume that $X:=X_{c_1}=U\setminus e$.
If, in addition, $X$ contains $s$ or $sc_1$, then $H=U$ is an $\cA$-group and statement~(1) of the theorem holds. Thus, we can
also assume that
$X\ne X_t$ for each $t\in \{s,sc_1\}$. Suppose first that $X_s=X_{sc_1}$; denote this set by $Y$. Then $Y\cap E=\{s,sc_1\}$, and hence
$c_1\in\rad(Y)$ (Lemma~\ref{290714c}). Since $X=U\setminus e$, this implies that $U\le\rad(Y)$.
Thus, $H=\grp{Y}$ is an $\cA$-group, $X_s=X_{sc_1}=Y=H\setminus U$ and statement~(2) holds.\medskip
Let now $X_s\ne X_{sc_1}$ and $t\in \{s,sc_1\}$. Then $Y\cap E=\{t\}$ where $Y=X_t$. It follows that $|c_1 Y\cap Y|=|Y|-1$, and hence
\qtnl{280814a}
\und{Y}^2 = |Y|e+(|Y|-1) \und{X} + \cdots,
\eqtn
where the omitted terms in the right-hand side contain neither $e$ nor elements of $X$ with non-zero coefficients. However,
$|X|c_{YY}^X=|Y|c_{YX}^Y$ because $X=X^{-1}$ and $Y=Y^{-1}$. Since $c_{YY}^X=|Y|-1$, this implies
that $|X|$ is divided by $|Y|$. If, in addition, $|Y|=1$, then we have $\{X_s,X_{sc_1}\}=\{\{t\},tX_{c_1}\}$ and $H=U\cup tU$ is an $\cA$-group. So statement~(4) holds. If $|Y|\ne 1$, equality~\eqref{280814a}
implies that $|Y|=|X|=|U|-1$ and $\und{Y}^2 = |Y|e+(|Y|-1) \und{X}$. It follows that $c_{YX}^Y=|X|-1$. Therefore,
$$
|tU\cap Y|=|X|-1=|U|-2.
$$
This is true for $t=c_1$ and $t=sc_1$. On the other hand, $sU=sc_1U$ and the sets $X_s$ and $X_{sc_1}$ are disjoint. Thus,
$$
|U|=|tU|\ge |sU\cap X_s|+|sc_1U\cap X_{sc_1}|=2(|U|-2).
$$
It follows that $|U|=2$ or $|U|=4$. In the former case, $H=E$ and statement~(4) trivially holds. In the
latter one, $|X|=|X_s|=3$ and $|sU\cap Y|=2$. Therefore, there exists a unique $x\in X_s$ outside $sU$. It
follows that $|xU\cap X_s|=1$ which is impossible by Lemma~\ref{090608a}.\bull\medskip
\section{S-rings over $D=\mZ_2\times\mZ_{2^n}$: non-regular case}\label{140914a}
A set $X\subseteq D$ is said to be {\it highest} (in $D$) if it contains an element of order $2^n$.
Given an S-ring $\cA$ over $D$, denote by $\rad(\cA)$ the group generated by
the groups $\rad(X)$, where $X$ runs over the highest basic sets of $\cA$.
Clearly, $\rad(\cA)$ is an $\cA$-group, and it is equal to $e$ if and only if each highest basic set of $\cA$ has trivial radical.
In what follows, we say that $\cA$ is {\it regular}, if each highest basic set of $\cA$ is regular. Now, the
main result of this section can be formulated as follows.
\thrml{170414a}
Let $\cA$ be an S-ring over the group $D$. Suppose that $\rad(\cA)=e$. Then $\cA$ is either regular
or rational. Moreover, in the latter case, $\cA=\cA_H\otimes\cA_L$, where $\rk(\cA_H)=2$ and
$|L|\le 2\le |H|$; in particular, $\cA$ is schurian.
\ethrm
The proof of Theorem~\ref{170414a} will be given in the end of the section. The key point of the proof is the
following statement.
\thrml{290714a}
Let $\cA$ be an S-ring over the group $D$. Then any non-regular basic set of $\cA$ either intersects $E$,
or has a non-trivial radical.
\ethrm
\proof
Let $X$ be a non-regular basic set of $\cA$ that does not intersect $E$. Then the minimal order of an element in $X$
equals $2^m$ for some $m\ge 2$. Denote by $X_m$ the set of all elements in~$X$ of order $2^m$. Clearly,
each of the sets $X\setminus X_m$ and $X_m$ is non-empty. Suppose, towards a contradiction, that $\rad(X)=e$. Then $c_1\not\in\rad(X)$, and $c_1\in\rad(X\setminus X_m)$ (Lemma~\ref{290714c}). It follows that
\qtnl{300814u}
c_1\not\in\cA,
\eqtn
because otherwise $c_1X$ is a basic set other than $X$ that intersects $X$.\medskip
Denote by $K$ the setwise stabilizer of $X$ in the group $G\cong\mZ_{2^n}^*$ of all permutations $x\mapsto x^m$,
$x\in D$, where $m$ is an odd integer. Then
by Schur's theorem on multipliers, $X_m$ is the union of at most two $K$-orbits (one inside $C$ and the other outside). The
radicals of these orbits must be trivial, because
$\rad(X)=e$ and $c_1\in\rad(X\setminus X_m)$. Thus, by statement~(1) of Lemma~\ref{211113a}, we have
\qtnl{300814a}
X_m=\{x\}\qoq \{x,x^{-1}\varepsilon\}\qoq\{x,ys\}\qoq\{x,x^{-1}\varepsilon,ys,y^{-1}\varepsilon s\},
\eqtn
where $x,y\in X_m$ are such that $\grp{x}=\grp{y}$, and $\varepsilon\in\{e,c_1\}$. It should be mentioned that $x\ne \varepsilon y$,
for otherwise $\varepsilon s\in\rad(X)$.\medskip
Let us define $\cA$-groups $U$ and $H$ as in Theorem~\ref{250814a}. Then
$X\subseteq D\setminus H$, because $X$ does not intersect $E$. By the definition of $H$, this implies that it does not contain elements
of order $2^m$. Since $U\le H$, we conclude that
$xU\cap X\subseteq X_m$ for each $x\in X_m$. Therefore, $X_m$ is a disjoint union
of some sets $xU\cap X$ with such $x$. However, by Lemma~\ref{090608a} the number $\lambda:=|xU\cap X|$ doesn't depend
on a choice of $x\in X$. Thus, $\lambda$ divides $|X_m|$.
By~\eqref{300814a} this implies that
$\lambda\in\{1,2,4\}$. Moreover, setting $Y$ to be the basic set
containing $c_1$, we have
\qtnl{141114a}
c_{X^{} Y^{}}^X=\lambda-1,
\eqtn
because $U=Y\setminus e$ or $U=Y\setminus \grp{\varepsilon s}$,
and $x\ne \varepsilon y$.\medskip
Denote by $\alpha$ the number of $z\in X_m$ for which $c_1z\not\in X_m$.
If $\alpha=1$, then from Theorem~\ref{261009w}, it follows that
$\cS(\cA)$ contains $X^{[2]}=\{z^2\}$ for an appropriate $z\in X_m$.
Since $z\not\in E$, this implies that $c_1\in\cA$ in contrast
to~\eqref{300814u}. Thus, $\alpha\ne 1$. Therefore, $\alpha$ is an
even number less or equal than $|X_m|\le 4$ (see~\eqref{300814a}).
Moreover, it is not zero, because otherwise $c_1\in\rad(X)$. Besides,
from~\eqref{141114a} it follows that
\qtnl{310714a}
|X|\,(\lambda - 1)=|X|c_{X^{} Y^{}}^X =
|Y|\,c_{X^{} X^{-1}}^Y = |Y|\,|c_1X\cap X| = |Y|\,(|X|-\alpha).
\eqtn
Since $|X|>|X_m|\ge\alpha$, this implies that the right-hand side of
the equality is not zero. Thus, $\lambda\ne 1$, and finally
\qtnl{010914a}
\lambda,\alpha\in\{2,4\}.
\eqtn
\lmml{310714b}
In the above notation $|X|\geq 2|X_m|$, and the equality holds only if
\nmrt
\tm{1} $X_m$ is a union of two $K$-orbits and $X\setminus X_m$ is a $K$-orbit,
\tm{2} any element in $X\setminus X_m$ is of order $2^{m+1}$.
\enmrt
\elmm
\proof By the Schur theorem on multipliers, the stabilizers of
an element $x\in X$ in the groups $K$ and $G$, coincide. However,
the stabilizer in $G$ consists of raising to power $1+i|x|$, where
$i=0,1,\ldots,2^n/|x|-1$. Therefore, $|K_x|=2^n/|x|$. For $x\in X_m$ and
$y\in X\setminus X_m$, this implies that
$|K_x|\ge 2|K_y|$, and hence
$$
|x^K|=\frac{|G|}{|K_x|}\le \frac{|G|}{2|K_y|}=\frac{|y^K|}{2}.
$$
Taking into account that $X_m$ is a disjoint union of
at most two $K$-orbits, we obtain that
$$
|X|-|X_m|\geq |y^K|\geq 2|x^K|\geq |X_m|
$$
as required. Since the equality holds only if the second and third
inequalities in the above formula are equalities, we are done.\bull\medskip
We observe that $|Y|\ne\lambda-1$: indeed, for $\lambda=2$, this follows from~\eqref{300814u} whereas
for $\lambda=4$, the assumption $|Y|=\lambda-1$ implies by \eqref{310714a} an impossible equality $|X|=|X|-\alpha$.
Thus, by~\eqref{310714a} and Lemma~\ref{310714b}
we have
$$
\frac{\alpha\,|Y|}{|Y|-(\lambda-1)}=|X|\ge 2|X_m|\ge 2\alpha.
$$
Furthermore, if $\lambda=2$, then $|Y|=2$ and $|X|=2\alpha$. On the other hand, if $\lambda=4$, then $\lambda-1<|Y|\le 6$ and
$|Y|\in \{2^a-1,2^a-2\}$ for some~$a$ (Theorem~\ref {250814a}); but then $|Y|=6$ and $|T|=2\alpha$.
Thus, by~\eqref{010914a} there are exactly four possibilities:
\nmrt
\tm{1} $\alpha =2, \lambda=2, |Y|=2, |X|=4, |X_m|=2$,
\tm{2} $\alpha =4, \lambda=2, |Y|=2, |X|=8, |X_m|=4$,
\tm{3} $\alpha =2, \lambda=4, |Y|=6, |X|=4, |X_m|=2$,
\tm{4} $\alpha =4, \lambda=4, |Y|=6, |X|=8, |X_m|=4$.
\enmrt
In all cases, $|X|=2|X_m|$. Therefore, by Lemma~\ref{310714b} and \eqref{300814a}, we conclude that
$X_m=\{x,ys\}$ in cases (1) and~(3), and $X_m = \{x,x^{-1}\varepsilon,sy,sy^{-1}\varepsilon\}$
in cases (2) and~(4). Moreover, since the number $|Y|$ is even, $U\setminus Y$ is a group of order two. Without loss
of generality, we assume that it is $\grp{s}$.\medskip
Let $\pi$ be the quotient epimorphism from $D$ to $D'=D/U$. Then the group $D'$ is cyclic, the S-ring $\cA'=\cA_{D'}$ is
circulant\footnote{Any S-ring over a cyclic group is called a circulant one.} and $X'=\pi(X)$ is a non-regular basic set of it (the elements in $X'_m=\pi(X_m)$ and in $X_{}'\setminus X'_m$ have
different orders). However, any non-regular basic set of a circulant S-ring over a $2$-group is a set difference of two its subgroups
(see Subsection~\ref{141114b}). Therefore,
$$
X'=\grp{X'}\setminus\rad(X').
$$
Since $X_{}'\ne X'_m$, this implies that $|X'|\ge 3$. On the other hand, $|X'|=|X|/\lambda$ by the definition of $\lambda$. Thus,
we can exclude cases (1), (3) and (4). In case (2) let $|\rad(X')|=2^i$ for some $i\ge 0$. Then $4=|X'|=2^{i+2}-2^i=3\,2^i$.
Contradiction.\bull\medskip
Any basic set $X$ of an S-ring $\cA$ over $D$ that intersects the
group~$E$, must contain an involution. Therefore, such $X$ is rational.
By Theorem~\ref{290714a}, this proves the following statement.
\crllrl{011213d}
Let $X$ be a basic set of an S-ring over $D$. Suppose that $\rad(X)=e$. Then $X$
is either regular or rational.\bull
\ecrllr
{\bf Proof of Theorem~\ref{170414a}.}\ Suppose that $\cA$ is not regular. Then there exists a highest set $X\in\cS(\cA)$
that is not regular.
Since $\rad(X)=e$, we conclude by Theorem~\ref{290714a} that the set $X\cap E$ is not empty. Therefore,
$X$ is contained in the $\cA$-group~$H\ge E$ defined in Theorem~\ref{250814a}. But then $H=D$, because
the set $X$ is highest. Now, the first statement follows, because the
S-ring $\cA_H=\cA$ is, obviously, rational. Moreover, statements (2) and (3) of Theorem~\ref{250814a} do not
hold, because $\rad(\cA_H)=\rad(\cA)=e$. Thus, the second statement of our theorem is true for $L=e$
(resp. $L=\grp{s}$) if statement~(1) (resp. statement~(4)) of Theorem~\ref{250814a} holds.\bull\medskip
From the proof of Theorem~\ref{170414a}, it follows that if one of the highest basic sets of $\cA$ is not regular,
then all highest basic sets are rational. This implies the following statement.
\crllrl{180514a}
Let $\cA$ be an S-ring over the group $D$. Suppose that $\rad(\cA)=e$. Then either every highest basic set of
$\cA$ is regular, or every highest basic set of $\cA$ is rational.\bull
\ecrllr
\section{S-rings over $D=\mZ_2\times\mZ_{2^n}$: regular case}\label{100414a}
Throughout this section, $C=C_n$ is a cyclic subgroup
of~$D=D_n$ that is isomorphic to $\mZ_{2^n}$.
We denote by $c_1$, $c_2$ and $s$, respectively, the unique involution in $C$, one of the two elements of $C$ of order~$4$, one of the two
involutions in $D\setminus C$. The main result is given by the following theorem.
\thrml{230214a}
Let $\cA$ be a regular S-ring over the group $D$. Suppose that $\rad(\cA)=e$. Then $\cA$ is a cyclotomic S-ring.
More precisely, $\cA=\cyc(K,D)$, where $K\le\aut(D)$ is one of the groups listed in Table~\ref{280414b}.
\ethrm
\begin{table}
\centering
\begin{tabular}[tc]{|c|l|c|c|c|}
\hline
$K$ & generators & $|K|$ & $n$ & comment \\
\hline
$K_1$ & $(x,s)\mapsto (x,s)$ & 1 & $n\ge 2$ & $X_1=\emptyset$\\
\hline
$K_2$ & $(x,s)\mapsto (x^{-1},s)$ & 2& $n\ge 3$ & $X_1=\emptyset$ \\
\hline
$K_3$ & $(x,s)\mapsto (c_1x^{-1},s)$ & 2& $n\ge 3$ & $X_1=\emptyset$ \\
\hline
$K_4$ & $(x,s)\mapsto (x^{-1},sc_1)$ & 2 & $n\ge 3$ & $X_1=\emptyset$ \\
\hline
$K_5$ & $(x,s)\mapsto (c_1x^{-1},sc_1)$ & 2 & $n\ge 3$ & $X_1=\emptyset$ \\
\hline
$K_6$ & $(x,s)\mapsto (sc_2x,sc_1)$,\ $(x,s)\mapsto (x^{-1},s)$ & 4 & $n\ge 4$ & $X_a\ne\emptyset$\\
\hline
$K_7$ & $(x,s)\mapsto (sc_2x,sc_1)$,\ $(x,s)\mapsto (c_1x^{-1},s)$ & 4 & $n\ge 4$ & $X_a\ne\emptyset$\\
\hline
$K_8$ & $(x,s)\mapsto (sx^{-1},s)$ & 2 &$n\ge 4$ & $X_a\ne\emptyset$\\
\hline
$K_9$ & $(x,s)\mapsto (sc_1x^{-1},s)$ & 2& $n\ge 4$ & $X_a\ne\emptyset$\\
\hline
$K_{10}$ & $(x,s)\mapsto (sc_2x,sc_1)$ & 2& $n\ge 3$ & $X_a\ne\emptyset$\\
\hline
$K_{11}$ & $(x,s)\mapsto (sc_2x^{-1},sc_1)$ & 2& $n\ge 4$ & $X_a\ne\emptyset$\\
\hline
\end{tabular}\\[2mm]
\caption{The groups of cyclotomic rings with trivial radical}
\label{280414b}
\end{table}
\crllrl{100514d}
Under the assumptions of Theorem~\ref{230214a}, let $K=K_i$, where $i=1,\ldots,11$. Then the following
statements hold:
\nmrt
\tm{1} $\grp{c_1}$ is an $\cA$-group,
\tm{2} $C$ is an $\cA$-group if and only if $i\le 5$,
\tm{3} $\grp{\varepsilon s}$ with $\varepsilon\in\{e,c_1\}$, is an $\cA$-group if and only if
$i\in\{1,2,3,8,9\}$.\bull
\enmrt
\ecrllr
In what follows, given a basic set $X\in\cS(\cA)$, we denote by $X_0$ and $X_1$ the uniquely determined subsets of $C$ for which
$X=X_0\cup sX_1$.\medskip
{\bf Proof of Theorem~\ref{230214a}.} Let $X$ be a highest basic set
of the S-ring~$\cA$. Then it is regular by the theorem hypothesis.
By the Schur theorem on multipliers, this implies that
if the set $X_a$ is not empty for some $a\in\{0,1\}$, then $X_a$ is an orbit of
the group $\aut(C)_{\{X_a\}}$. Therefore, $X_a$ is of the form given in
statement~(1) of Lemma~\ref{211113a}. The rest of the proof consists of Lemmas~\ref{040914a},
\ref{040914b} and \ref{040914c} below: in the first one, $X_0$ or $X_1$ is empty, and in the
other two, both $X_0$ and $X_1$ are not empty and $|X_0|=2$ or $1$,
respectively.
\lmml{040914a}
Let $X_0=\emptyset$ or $X_1=\emptyset$. Then $\cA=\cyc(K_i,D)$ with $i\in\{1,2,3,4,5\}$.
\elmm
\proof
Without loss of generality, we can assume that $n\ge 3$ and $X_1=\emptyset$. Then $X=X_0$ generates $C$. Therefore, $C$ is
an $\cA$-group and $X$ is a
highest basic set of a circulant S-ring $\cA_C$. Since $\rad(X)=e$, this implies that $\rad(\cA_C)=e$. If, in addition,
$\grp{s}$ is an $\cA$-group, then $\cA=\cA_C\otimes\cA_{\grp{s}}$
by Lemma~\ref{050813b}, and hence $\cA=\cyc(K_i,D)$ with $i=1,2,3$. Thus, we can assume that
\qtnl{290414a}
s\not\in\cA.
\eqtn
Let us prove by induction on $n$ that $\cA=\cyc(K_i,D)$ with $i=4$ or $5$. For $n=3$ this statement can be
verified by a computer computation. Let $n>3$. Denote by $X'$ the basic set of $\cA$ that contains $x'=xs$,
where $x\in X$ is a generator of~$C$. Then by the theorem hypothesis,
$X'$ is a regular set with trivial radical. It follows that $C'=\grp{X'}$ is the order $2^n$ cyclic subgroup of $D$
other than $C$. In particular,
$$
\rad(\cA_{C'})=\rad(X')=e.
$$
Besides, since $C^2=(C')^2$, the S-rings $\cA_C$ and $\cA_{C'}$ have the same basic sets inside the group $C^2$; in particular,
$|X|=|X'|$. Moreover, these S-rings are not Cayley isomorphic. Indeed, otherwise $X'=sY$, where $Y=X^{(m)}$ for some
odd~$m$. Then $s$ is the only element that appears in the product $\und{Y}^{-1}\und{X'}$ with multiplicity~$|X|$. However,
in this case $s\in\cA$, contrary to~\eqref{290414a}. Thus,
\qtnl{040914f}
X=\{x,\varepsilon x^{-1}\}\qaq X'=\{sx,sc_1\varepsilon x^{-1}\}.
\eqtn
Set $i=4$ or $5$ depending on $\varepsilon=e$ or $c_1$, respectively. Then
from~\eqref{040914f} and the Schur theorem on multipliers, it follows that the S-rings $\cA$ and $\cyc(K_i,D)$ have the same
highest basic sets. We also observe that $D_{n-1}$ is an $\cA$-group.\medskip
Since the S-ring $\cA_C$ is cyclotomic, it follows from~\eqref{040914f} that $Y=\{x^2,x^{-2}\}$
is a basic set of $\cA$. Denote by $Y'$ the basic set containing $sx^2$. Then, obviously,
$$
Y'\subseteq X\,X'=\{sx^2,sc_1x^{-2},s,sc_1\}.
$$
However, $|sx^2|\ge 8$, because $n\ge 4$. Thus, $Y'\subseteq\{sx^2,sc_1x^{-2}\} $: otherwise $Y\cap E$ is
not empty and from Theorem~\ref{250814a} it follows that $|Y'|>4$. Therefore,
$Y'$ is regular and $\rad(Y')=e$. Since also $\rad(Y)=e$
and $Y,Y'$ are highest basic sets of the S-ring $\cA_{D_{n-1}}$, the latter
satisfies the hypothesis of Lemma~\ref{040914a}. By the induction, we conclude that
$\cA_{D_{n-1}}=\cyc(K_4,D_{n-1})$. Thus, $\cA=\cyc(K_i,D)$ as required.\bull
\lmml{040914b}
Let $|X_0|=2$ and $X_1\ne\emptyset$. Then $\cA=\cyc(K_i,D)$ with $i\in\{6,7\}$.
\elmm
\proof By statement~(1) of Lemma~\ref{211113a}, we have $X_0=\{x,\varepsilon x^{-1}\}$. Since $X$ is regular, this implies that
\qtnl{010514a}
X=\{x,\varepsilon x^{-1},sy,s\varepsilon y^{-1}\}
\eqtn
for some generator $y$ of the group $C$. For $n=3$, we tested in the computer that no S-ring over $D$
has a highest basic set $X$ such that $X_0=\{x,\varepsilon x^{-1}\}$ and $X_1\ne\emptyset$.
Suppose that $n\ge 4$. Let us prove that the lemma statement holds for $i=6$ or $i=7$ depending on
$\varepsilon=e$ or $\varepsilon=c_1$, respectively. For $n=4$, we tested this statement in the computer. Thus,
in what follows, we can assume that $n\ge 5$.
\lmml{010514b}
In the above notations, the following statements hold:
\nmrt
\tm{1} $C_{n-1}$ is an $\cA$-group whereas $\grp{s}$ and $\grp{sc_1}$ are not,
\tm{2} $Y_x=\{x^{\pm 2},c_1x^{\pm 2}\}$ and $Z_x=\{s x^{\pm 2}\}$ are $\cA$-sets,
\tm{3} $y=xc_2$ for a suitable choice of $y$ and $c_2$.\footnote{One can interchange $y$ and $y^{-1}$,
and $c_2$ and $c_2^{-1}$.}
\enmrt
\elmm
\proof Since $n\ge 5$, we have $x^2\ne x^{-2}$ and $y^2\ne y^{-2}$. Besides, since neither $s$
nor $sc_1$ belongs to $\rad(X)$, we have $x^2\ne y^{\pm 2}\ne x^{-2}$. Thus,
\qtnl{030514a}
|\{x^2, x^{-2}, y^2, y^{-2}\}|=4.
\eqtn
However, $X^{[2]}=\{x^2, x^{-2}, y^2, y^{-2}\}$ and $X^{[2]}$ is an $\cA$-set by Theorem~\ref{261009w}.
Thus, the first part of statement~(1) holds, because $C_{n-1}=\grp{X^{[2]}}$. To prove the second part of
statement~(1), suppose on the contrary that $L:=\grp{s\varepsilon'}$ is an $\cA$-group
for some $\varepsilon'\in\{e,c_1\}$. Then the circulant S-ring $\cA_{D/L}$ has a basic set
$$
\pi(X)=\{\pi(x),\pi(\varepsilon x^{-1}),\pi(y),\pi(\varepsilon y^{-1})\},
$$
where $\pi:D\to D/L$ is the quotient epimorphism. However, from~\eqref{030514a} it easily follows that
$|\rad(\pi(X))|\ge 2$. Therefore, one of the quotients $\pi(x)/\pi(\varepsilon x^{-1})$, $\pi(x)/\pi(y)$ or
$\pi(x)/\pi(\varepsilon y^{-1})$ has order~$2$. This implies, respectively, that the order of $\pi(x)$ is $8$,
$\pi(x)=\pi(c_1)\pi(y)$, and $\pi(x)=\pi(c_1)\pi(\varepsilon y^{-1})$. The former case is impossible, because $n\ge 5$,
whereas in the other two, we have $x\in c_1yL$ and $x\in c_1\varepsilon y^{-1}L$ which is impossible due
to~\eqref{030514a}.\medskip
To prove the second part of statement~(2), we observe that $C_{n-1}\cup\tr(X)$ is an $\cA$-set. But the complement to it
in~$D$ coincides with $sC_{n-1}$. Thus, it is also an $\cA$-set. Besides,
\qtnl{110214a}
\und{X}^2\,\circ\,\und{sC_{n-1}}=2\,\und{sX'},
\eqtn
where $X'=\{(xy)^{\pm 1},\varepsilon(xy^{-1})^{\pm 1}\}$. Therefore, $sX'$ is an $\cA$-set.
However, it is easily seen that $|xy|\ne|\varepsilon xy^{-1}|$. Moreover, $|xy|=2^{n-1}$ or
$|\varepsilon xy^{-1}|=2^{n-1}$, because
$x$ and $y$ are generators of the group $C$ and $n\ge 3$. Thus, the
elements $sxy$ and $s\varepsilon xy^{-1}$ can not belong to the same basic set of $\cA$.
Indeed, otherwise assuming $|xy|=2^{n-1}$, we conclude by Lemma~\ref{290714c} that this basic set contains $sxyc_1$.
Then $xyc_1\in X'$, and hence $xyc_1\in\{(xy)^{-1},\varepsilon x^{-1}y\}$. Consequently, $(xy)^2=c_1$ or $x^2=c_1\varepsilon$. In any
case, $n-1\le 2$. Contradiction. A similar argument leads to a contradiction when $|\varepsilon xy^{-1}|=2^{n-1}$.
In the same way, one can verify that no two elements, one in $\{s(xy)^{\pm 1}\}$ and the other one in $\{s\varepsilon (xy^{-1})^{\pm 1}\}$,
cannot belong to the same basic set of $\cA$. Thus, $sX'$ is a disjoint union
of two $\cA$-sets of the form $\{sz^{\pm 1}\}$ with $z\in C$, and one of them consists of elements
of order $2^{n-1}$. This implies that $Z_x$ is an $\cA$-set, as required.\medskip
To prove statement~(3), suppose on the contrary that $x^4\ne y^{\pm 4}$. Then since $Y:=X^{[2]}$, $Z_x$, and $Z_y$
are $\cA$-sets, the S-ring $\cA$ contains the element
$$
(\und{Y_{}}\,\und{Z_x})\,\circ\, (\und{Y_{}}\,\und{Z_y})
=2s(2e+x^{\pm 2}y^{\pm 2}).
$$
Since $n\ge 3$, this implies that only $s$ appears in the right-hand
side with multiplicity~$4$. By the Schur-Wielandt principle, this implies that $s\in\cA$. However, this contradicts the second part of statement~(1).\medskip
To complete the proof, we note that by statement~(3), we have $Y_x=X^{[2]}$. Since $X^{[2]}$ is an $\cA$-set,
the first part of statement~(2) follows.\bull
Let us continue the proof of Lemma~\ref{040914b}.
Denote by $\cA_i$ the minimal S-ring over $D$ that contains $X$ as a basic set; we recall that
$i=6$ or $i=7$ depending on $\varepsilon=e$ or $\varepsilon=c_1$. We claim that
\qtnl{040514c}
\cA_i=\cyc(K_i,D).
\eqtn
Indeed, from statements~(2) and~(3) of Lemma~\ref{010514b}, it follows that the sets
$X$, $Y_x$, and $Z_x$ are orbits of the group $K_i$ (see Table~\ref{tblsg1}, where the generic orbits of the group $K_i$
contained inside $C_{n-1}$ and $sC_{n-1}$ are given).
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|}
\hline
$C_{n-1}$ & $sC_{n-1}$ \\
\hline
$x^{\pm 2},c_1x^{\pm 2}$ & $s\varepsilon x^{\pm 2}$ \\
\hline
$x^4,x^{-4}$ & $sx^{\pm 4},sc_1x^{\pm 4}$ \\
\hline
$\cdots$ & $\cdots$ \\
\hline
$c_2^{},c_2^{-1}$ & $sc_2^{},sc_2^{-1}$ \\
\hline
$c_1$ & $s,sc_1$ \\
\hline
$e $ & \\
\hline
\end{tabular}\\[2mm]
\caption{The orbits of $K_6$ and $K_7$}\label{tblsg1}
\end{center}
\end{table}
Therefore, by the Schur theorem on multipliers, we have
\qtnl{040514a}
(\cA_i)_{D\setminus D_{n-2}}=\cyc(K_i,D)_{D\setminus D_{n-2}}.
\eqtn
Next, $C':=\grp{Z_x}$ is a cyclic $\cA$-group of order $2^{n-1}$ other than $C_{n-1}$. Moreover, $Z_x$ is a highest
basic set of the circulant S-ring $(\cA_i)_{C'}$. Therefore, this S-ring has trivial radical. From the results discussed
just after Lemma~\ref{211113a}, it follows that it is the cyclotomic S-ring $\cyc(K',C')$, where
$K'$ is the subgroup of $\aut(C')$ that has $Z_x$ as an orbit. Since $\orb(K_i,C')=\orb(K',C')$,
we conclude that
\qtnl{040514b}
(\cA_i)_{C_{n-2}}=\cyc(K_i,D)_{C_{n-2}}.
\eqtn
Let us complete the proof of~\eqref{040514c}. To do this, taking into account that $\varepsilon(e+c_1)=e+c_1$, we find that
\qtnl{040514u}
\und{Y_x}\,\und{Z_x}=sx^{\pm 4}(e+c_1)+2s(e+c_1).
\eqtn
Moreover, since $n\ge 5$, the elements $x^{\pm 4},sx^{\pm 4}c_1$
appear in the right-hand side with coefficient~$1$. By the Schur-Wielandt principle, this implies that $\{s,sc_1\}x^{\pm 4}$
and $\{s,sc_1\}$ are $\cA$-sets. Thus,
\qtnl{040514e}
(\cA_i)_{sC_m\setminus sC_{m-1}}=\cyc(K_i,D)_{sC_m\setminus sC_{m-1}}\qaq
(\cA_i)_{sC_1}=\cyc(K_i,D)_{sC_1},
\eqtn
where $m=n-2$. For all $m=n-3,\ldots,2$ the first equality is proved in a similar way by the induction on~$m$; the sets $Y_x$ and
$Z_x$ in equation~\eqref{040514u} are replaced by the $\cA$-sets $\{x^{\pm 2^{m+1}}\}$ and $(e+c_1)\{x^{\pm 2^{m+1}}\}$, respectively.
Thus, the claim follows from~\eqref{040514a}, \eqref{040514b} and~\eqref{040514e}.\medskip
Let us continue the proof of Lemma~\ref{040914b}.
Now, since, obviously, $\cA\ge\cA_i$, we conclude by \eqref{040514c} that $\cA\ge\cyc(K_i,D)$. To verify
the converse inclusion, we have to prove that every $K_i$-orbit $Z'$ belongs to~$\cS(\cA)$. Suppose
on the contrary that some $Z'$ properly contains a set $Z\in\cS(\cA)$. Then
\qtnl{010314a}
Z\subseteq D_{n-1}\setminus D_1.
\eqtn
Indeed, from~\eqref{010514a} and statement~(3) of Lemma~\ref{010514b}, it follows that the $K_i$-orbits
outside $D_{n-1}$ are the basic sets of~$\cA$. Besides, the orbits $\{e\}$ and $\{c_1\}$ are also basic sets,
because $\cA\ge\cyc(K_i,D)$. Finally, the orbit $\{s,c_1s\}$ belongs to $\cS(\cA)$ by the second
part of statement~(1) of Lemma~\ref{010514b}.\medskip
From \eqref{010314a} and the first part of statement~(1) of Lemma~\ref{010514b}, it follows that the set $Z$ is regular.
So it is an orbit of an automorphism group of $C$. This implies that $Z$ has cardinality $1$, $2$, or $4$.
The latter case is impossible, because otherwise $Z=Z'$. We claim that the first case is also impossible. Indeed,
otherwise $Z=\{z\}$ for some $z\in D_{n-1}\setminus D_1$. Then
$zX$ is a highest basic set of~$\cA$. However, $(zX)_0=\{zx,z\varepsilon x^{-1}\}$.
Therefore, $\varepsilon (zx)^{-1}=z\varepsilon x^{-1}$, and hence $z=z^{-1}$. Thus, $z\in D_1$. Contradiction.\medskip
To complete the proof of Lemma~\ref{040914b}, let $|Z|=2$. Without loss of generality, we
can assume that the order of an element in $Z$
is minimal possible. Clearly, $|Z'|=4$, and hence
$$
Z'=\{z^{\pm 1},c_1z^{\pm 1}\},
$$
where either $z=x^2$, or $z=sx^{2^m}$ with $m\in\{3,\ldots,n-3\}$ (see Table~\ref{tblsg1}). Choose $z\in Z$ so that
$$
Z=\{z,c_1z\}\qoq Z=\{z,\varepsilon' z^{-1}\},
$$
where $\varepsilon'\in\{e,c_1\}$. In the first case, the singleton $Z^2=\{z^2\}$ is a basic set of $\cA$. By the above
argument, this implies that $z^2\in D_1$. Thus, $z\in D_2$. Contradiction.
In the second case, $\cA$ contains the element
$$
(sz+sz^{-1})(z+\varepsilon' z^{-1})=sz^2+s\varepsilon' z^{-2}+s+s\varepsilon'.
$$
Since $s\not\in\cA$, this implies that $\varepsilon'=c_1$. Therefore, $\{sz^2,s\varepsilon' z^{-2}\}$ cannot be a basic
set of $\cA$, and hence $m\ne 3$. But then, the minimality of $|z|$ implies that
$\{sz^2,s\varepsilon' z^{-2},s,s\varepsilon'\}\in\cS(\cA)$ that is impossible, because $s+sc_1\in\cA$.\bull
\lmml{040914c}
Let $|X_0|=1$ and $X_1\ne\emptyset$. Then $\cA=\cyc(K_i,D)$
with $i\in\{8,9,10,11\}$.
\elmm
\proof In this case $X=\{x,ys\}$, where $\grp{y}=C$. It follows that $X$ generates $D$. Moreover, $n\ge 3$, because
$c_1\not\in\rad(X)$. Therefore, there are two orthogonals in the S-ring~$\cA$: one of them is in
$(X\,X^{-1})\cap Cs$, and another one is in $(X\,X^{-1})\cap C$. Thus, $\cA$ is quasi-thin
by Lemma~\ref{100914a}, and hence schurian by Lemma~\ref{100914y}. The latter implies also that
the stabilizer $K$ of the point $e$ in the group $\aut(\cA)$, has a faithful regular orbit. Therefore, the index of $D$ in $\aut(\cA)$
is equal to~$2$. But then $D\trianglelefteq \aut(\cA)$, and hence $K\le\aut(D)$.
Consequently, $\cA=\cyc(K,D)$ and the group $K$ is generated by an involution
$\sigma\in\aut(D)$. This involution interchanges $x$ and $ys$. Therefore, the automorphism $\sigma$
is uniquely determined. We leave the reader to verify that $\sigma$ is one of automorphisms that are listed
in the rows 8, 9, 10, 11 of Table~\ref {280414b}.\bull
\section{S-rings over $D=\mZ_2\times\mZ_{2^n}$: automorphism groups in regular case}\label{231014a2}
In this section we find the automorphism group of a regular S-ring over $D$ with trivial radical. For this purpose, we
need the following concept introduced in~\cite{KK}: a permutation group is called {\it $2$-isolated} if no other
group is $2$-equivalent to it. The following statement is the main result of this section;
it shows, in particular, that a regular S-ring over $D$ with trivial radical is normal.
\thrml{090514a}
Let $\cA$ be a regular S-ring over the group $D$. Suppose that $\rad(\cA)=e$. Then for any
$\cA$-group $L$ of order at most $2$, the group $\aut(\cA_{D/L})$ is $2$-isolated. In
particular, if $\cA=\cyc(K,D)$ for some $K\le\aut(D)$, then
$\aut(\cA)=DK$.
\ethrm
The proof will be given in the end of the section. The following statement provides a sufficient condition
for a permutation group to be $2$-isolated.
\lmml{160514a}
Let $\cA$ be an S-ring and $G=\aut(\cA)$. Suppose that the point stabilizer of $G$ has a faithful regular
orbit. Then the group $G$ is $2$-isolated.
\elmm
\proof It was proved in \cite[Theorem~$3.5'$]{KK} that $G$ is
$2$-isolated whenever it is $2$-closed and a two-point stabilizer
of $G$ is trivial. However, the latter exactly means that a
point stabilizer of~$G$ has a faithful regular orbit.\bull\medskip
To apply Lemma~\ref{160514a}, we need the following auxiliary statement giving a sufficient condition
providing the existence of a faithful regular orbit of a point stabilizer in the automorphism group
of an S-ring.
\lmml{150514d}
Let $\cA$ be an S-ring over an abelian group $H$. Suppose that a set $X\in\cS(\cA)$ satisfies the
following conditions:
\nmrt
\tm{1} $\grp{\tr(X)}=H$,
\tm{2} $c_{XY}^Z=1$ for each $Z\in\cS(\cA)_{\tr(X)}$ and some $Y\in\cS(\cA)$,
\tm{3} $c_{XY}^X=1$ for each $Y\in\cS(\cA)_{XX^{-1}}$.
\enmrt
Then $X$ contains a faithful regular orbit of the group $\aut(\cA)_e$.
\elmm
\proof Denote by $\cX$ the Cayley scheme over $H$ associated with the S-ring~$\cA$. Then the
relation $r=R_H(\tr(X))$ is a union of basic relations of $\cX$. Clearly, $r$ is symmetric, and
connected (condition~(1)). Moreover, conditions~(2)
and~(3) imply that the coherent configuration $(\cX_e)_{er}$ is semiregular. Thus, by \cite[Theorem~3.3]{P13}
given $x\in X$, the two-element set $\{e,x\}$ is a base of the scheme $\cX$, and hence of the group $\aut(\cX)$.
This implies that $\{x\}$ is a base of the group $K=\aut(\cX)_e$. Thus, $x^K\subseteq X$ is a faithful regular
orbit of~$K$.\bull
{\bf Proof of Theorem~\ref{090514a}.}\ From Theorem~\ref{230214a}, it follows that $\cA=\cyc(K,D)$, where $K=K_i$ is one of the groups
listed in Table~\ref{280414b}, $1\le i \le 11$. Therefore, taking into account that the groups $DK$ and $\aut(\cA)$ are $2$-equivalent,
we conclude that the second part of the theorem statement immediately follows from the first one. To prove the latter, without loss of generality, we assume that $i\ge 2$ and $n\ge 4$. \medskip
Let $L\le D$ be an $\cA$-group. In what follows, we set $H=D/L$ and $\pi=\pi_L$. To prove that the group $\aut(\cA_H)$ is $2$-isolated,
it suffices to verify that its point stabilizer has a faithful regular orbit (Lemma~\ref{160514a}). The remaining part of the proof
is divided into three cases.\medskip
{\bf Case 1:} $L=\grp{\varepsilon s}$, where $\varepsilon\in\{e,c_1\}$. Here, $H$ is a cyclic group and the S-ring $\cA_H=\cyc(\pi(K),H)$
is cyclotomic. Moreover, $i\in\{2,3,8,9\}$ by statement~(3) of Corollary~\ref{100514d}. Therefore, the order of the group $\pi(K)\le\aut(H)$
is at most~$2$. By the implication $(3)\Rightarrow(2)$ of~\cite[Theorem~6.1]{EP02}, this implies
that the group $\aut(\cA_H)_e$ has a faithful regular orbit. Thus, the group $\aut(\cA_H)$ is $2$-isolated by Lemma~\ref{160514a}.\medskip
{\bf Case 2:} $e\le L \le\grp{c_1}$ and $|K|=2$. Here, $i\not\in\{1,6,7\}$ and each basic set of $\cA$ is of cardinality at most~$2$.
Since $\cA$ is commutative, the latter is also true for the basic sets of~$\cA_H$. Therefore, this S-ring is quasi-thin. So by the
second part of
Lemma~\ref{100914y} it suffices to prove that $\cA_H$ has at least two orthogonals.
To do this let $X$ be a basic set of $\cA$ that contains a generator of~$C$. Since
the S-ring $\cA$ is cyclotomic, $\pi(X^{(2)})$ and $\pi(X^{(4)})$ are basic sets of $\cA_H$.
Moreover, they are distinct, because $n\ge 4$. Finally, they are orthogonals,
because $\pi(X^{(2)})\subseteq \pi(X)\,\pi(X^{-1})$ and
$\pi(X)^{(4)}\subseteq \pi(X^{(2)})\,\pi(X^{(-2)})$.\medskip
{\bf Case 3:} $e\le L \le\grp{c_1}$ and $|K|=4$. Here $i=6$ or $7$. It suffices to verify that the hypothesis of
Lemma~\ref{150514d} is satisfied for a highest basic set $X$ of the S-ring~$\cA_H$. To do this we first observe
that the sets $X_0$ and $X_1$ are not empty. Therefore,
$\tr(X)=D\setminus D_{n-1}$, and condition~(1) is, obviously, satisfied.\medskip
To verify conditions (2) and (3), suppose first that $L=e$. Then for $x\in X_0$, we have
$$
X=\{x,\varepsilon x^{-1},sc_2x,sc_2^{-1}\varepsilon x^{-1}\},
$$
where $\varepsilon\in\{e,c_1\}$. Since $n\ge 4$, the elements $xy^{-1}$ with $y\in X$
belong to distinct $K$-orbits of cardinalities $1,2,2$ and $4$. A straightforward check
shows that if $Y$ is one of these orbits, then
$$
|Y|c_{XX^{-1}}^Y=4.
$$
Therefore, $4=|Y|c_{XX^{-1}}^Y=|X|c_{X^{}Y^{-1}}^X$, and condition~(3) is satisfied,
because $|X|=4$. Let now $Z\in\cS(\cA)_{\tr(X)}$. Then
$$
Z=\{xy,\varepsilon (xy)^{-1},sc_2xy,sc_2^{-1}\varepsilon (xy)^{-1}\}
$$
for some $y\in C_{n-1}$. Let $Y$ be the set $\{y^{\pm 1},c_1y^{\pm 1}\}$ ,
$\{y^{\pm 1}\}$, or $\{y\}$ depending on whether $y$ belongs to $C_{n-1}\setminus C_{n-2}$,
$C_{n-2}\setminus C_1$, or $C_1$, respectively. Then $Y$ is a basic set of $\cA$. Moreover,
a straightforward computation shows that in any case, $c_{YZ}^X=1$.
Since $c_{X^{}Y^{}}^{Z^{}}=c_{Y^{-1}Z^{-1}}^{X^{-1}}=c_{Y^{}Z^{}}^{X^{}}$,
condition~(2) is also satisfied.\medskip
Let now $L=\grp{c_1}$. To simplify notations, we identify the group $H=D/L$ with
$D_{n-1}$, write $\cA$ instead of $\cA_H$,\footnote{In our case, the S-ring $\cA_H$ does
not depend on the choice of $i\in\{5,6\}$.} and use the notation $x$ and $s$ for the
$\pi$-images of $x$ and $sc_1$, respectively. Thus, $\cA$ is a cyclotomic
S-ring over $D_{n-1}$ and
$$
X=\{x,x^{-1},sx,sx^{-1}\}
$$
is a highest basic set of $\cA$. It follows that $C_{n-2}$ is an $\cA$-group and any basic set inside $C_{n-2}$ is of
the form $\{z^{\pm 1}\}$ for a suitable $z\in C_{n-2}$. Since $sC_{n-2}$ is an $\cA$-set, the elements $xy^{-1}$
with $y\in X$ belong to distinct basic sets of~$\cA$. Therefore, the set $X^{}\,X^{-1}$
consists of basic sets $Y$ for which $c_{XY}^X=1$. Thus, condition~(3) is satisfied.
Let now $Z\in\cS(\cA)_{\tr(X)}$. Then
$$
Z=\{xy,(xy)^{-1},sxy,s (xy)^{-1}\}
$$
for some $y\in C_{n-2}$. Taking $Y$ to be the basic set $\{y^{\pm 1}\}$, we find that
$c_{YZ}^X=1$. Thus, condition~(2) is also satisfied, and we are done.\bull
\section{S-rings over $D=\mZ_2\times\mZ_{2^n}$: non-trivial radical case}\label{140914b}
In Theorems~\ref{170414a} and~\ref{230214a}, we completely described the structure of an S-ring over~$D$ that has
trivial radical. In this section, we study the remaining S-rings.
\thrml{060414a}
Let $\cA$ be an S-ring over a group $D$. Suppose that $\rad(\cA)\ne e$. Then $\cA$ is a proper generalized
$S$-wreath product, where the section $S=U/L$ is such that
\qtnl{171114a}
\cA_S=\mZ S\qoq |S|=4\qoq \rad(\cA_U)=e\ \text{and}\ |L|=2.
\eqtn
\ethrm
\proof Denote by $U$ the subgroup of~$D$ that is generated by all $X\in\cS(\cA)$ such that $\rad(X)=e$.
\lmml{170414b}
$U$ is an $\cA$-group and $\rad(\cA_U)=e$.
\elmm
\proof The first statement is clear. To prove the second one, without loss of generality, we
can assume that $U=D$. Then there exists a highest set $X\in\cS(\cA)$ such that
$\rad(X)=e$. Suppose first that $X$ is not regular. Then $X\cap E\ne\emptyset$ by
Theorem~\ref{290714a}. Therefore, $X$ is one of basic sets $X_{c_1}$, $X_s$, or $X_{sc_1}$
from Theorem~\ref{250814a}. Since $\rad(X)=e$, we have $X=X_{c_1}$ and $X=X_s$
in statements~(2) and~(3) of this theorem, respectively. Moreover, since $X$ is highest, $D=\grp{X}$
in statements~(1), (2), (3). Therefore, in these three cases $\rk(\cA)=2$, and hence $\rad(\cA)=e$. In the
remaining case (statement~(4) of Theorem~\ref{250814a}), we have $D=H$, and hence
$\cA=\cA_U\otimes\cA_{\grp{s}}$. Since each of the factors is of rank~$2$, this
implies that again $\rad(\cA)=e$.\medskip
From now on, we can assume that any highest basic set of~$\cA$ with trivial radical,
is regular (Corollary~\ref{011213d}). Moreover, if $X$ is one of them and both $X_0$ and
$X_1$ are not empty, then all highest basic sets are pairwise rationally conjugate and, hence
$\rad(\cA)=\rad(X)=e$. Thus, we can also assume that $\grp{X}$ is a cyclic group $C$
of order at least~$4$ that has index $2$ in $D$.\medskip
Since $\rad(\cA_C)=\rad(X)=e$, the circulant S-ring $\cA_C$ is cyclotomic (see
Subsection~\ref{141114b}). Together with $|C|\ge 4$, this shows that $c_1\in\cA$.
We claim that any $Y\in\cS(\cA)$ such that $\rad(Y)=e$ and $D=\grp{X,Y}$,
is regular. Indeed, otherwise by Theorem~\ref{290714a}, we have $Y=X_h$,
where $h$ is a non-identity element of the group~$E$. However, $h\ne c_1$: otherwise $Y=\{c_1\}$ by above,
and $\grp{X,Y}=C$, in contrast to the assumption. Since $X_{c_1}=\{c_1\}$ and $\rad(Y)=e$,
only statement~(4) of Theorem~\ref{250814a} can hold. But then, $Y$ is a singleton
in $E$, and hence it is regular. Contradiction.\medskip
To complete the proof, let $Y$ be a regular basic set with trivial radical. We can assume
that $Y\subseteq D\setminus C$, for otherwise $D=C$ and $\rad(\cA)=\rad(X)=e$.
If $Y$ is not highest, then any basic set $Z\subseteq XY$ is highest, and $\grp{X,Z}=D$.
Moreover, $\rad(Z)=e$ by Lemma~\ref{150414a}. Thus, we can also assume that $Y$ is highest.
Then any highest basic set of $\cA$ is rationally conjugate to either $X$ or $Y$. Thus,
$\rad(\cA)=e$, as required.\bull
By the theorem hypothesis and Lemma~\ref{170414b}, we have $U\ne D$. We observe also that
by Theorem~\ref{030414a}, the group $U$ contains every minimal $\cA$-group.
\lmml{130914a}
Suppose that there is a unique minimal $\cA$-group, or $c_1\in\rad(X)$ for all $X\in\cS(\cA)_{D\setminus U}$. Then the statement of Theorem~\ref{060414a} holds.
\elmm
\proof Let $L$ be a unique minimal $\cA$-group. Then the definition of $U$ implies that
$\cA$ is a proper generalized $S$-wreath
product, where $S=U/L$. If, in addition, $|L|\le 2$, then the third statement in~\eqref{171114a} follows
from Lemma~\ref{170414b} and we are done. Let now $|L|>2$. Then $\grp{c_1}$ is not
an $\cA$-group. By statement~(1) of Corollary~\ref{100514d}, this implies that the S-ring
$\cA_U$ is not regular. So, by the first part of Theorem~\ref{170414a}, it is rational.
Now, by the second part of this theorem, the uniqueness of~$L$ implies that $U=L$. Thus,
$|S|=1$ and the first statement in~\eqref{171114a} holds.\medskip
To complete the proof, suppose that there are at least two minimal $\cA$-groups. Then, obviously,
one of them, say $H$, contains $c_1$. Therefore, $c_1\in H\le U$. On the other hand, by the
lemma hypothesis, $c_1\in\rad(X)$ for all $X\in\cS(\cA)_{D\setminus U}$. Thus, $\cA$ is a proper generalized
$S$-wreath product, where $S=U/H$. Without loss of generality, we can assume that $|H|>2$.
If the S-ring $\cA_U$ is rational, then from the second part of
Theorem~\ref{170414a}, it follows that there is another minimal $\cA$-group $L$ of order $2$
and such that
$$
\cA_U=\cA_H\otimes\cA_L.
$$
Thus, $|S|=|L|=2$, and the first statement in~\eqref{171114a} holds. In the remaining
case, $\cA_U$ is a regular S-ring by the first part of Theorem~\ref{170414a}. By statement~(1)
of Corollary~\ref{100514d}, this implies that $H=\grp{c_1}$. Thus, $|H|=2$, and
the third statement in~\eqref{171114a} holds.\bull
Denote by $V$ the union of all sets $X\in\cS(\cA)$ such that $\rad(X)=e$ or $c_1\in\rad(X)$.
Then, obviously, $U\subseteq V$ and $V$ is an $\cA$-set. By Lemma~\ref{130914a}, we can
assume that $V\ne D$, and that $U$ contains two distinct minimal $\cA$-groups. It is easily seen that in this case
$E\subseteq V$.
\lmml{130914c}
In the above assumptions let $X\in\cS(\cA)_{D\setminus V}$. Then
\nmrt
\tm{1} $\rad(X)=\grp{s}$ or $\grp{sc_1}$,
\tm{2} $X$ is a regular set such that both $X_0$ and $X_1$ are not empty.
\enmrt
\elmm
\proof Since $X\not\subseteq U$, we have $\rad(X)\ne e$. Besides, $c_1\not\in\rad(X)$ by the
definition of $V$. Thus, statement (1) holds, because $\grp{s}$ and $\grp{sc_1}$ are the
only subgroups of $D$ that do not contain $c_1$. To prove statement (2), set
$$
L=\rad(X)\qaq\pi=\pi_{D/L}.
$$
Then $\rad(\pi(X))=e$. However, $D/L$ is a cyclic $2$-group by statement~(1). Therefore,
$\pi(X)$ is the basic set of a circulant S-ring $\cA_{D/L}$. From the description
of basic sets of such an S-ring given in Subsection~\ref{141114b}, it follows
that $\pi(X)$ is regular or is of the form
$$
\pi(X)=\pi(H)^\#
$$
for some $\cA$-group $H\ge D_1$ such that $|H/L|\ge 4$. In the latter case,
$X=H\setminus L$, and hence $L$ is a unique minimal $\cA$-group in contrast to
our assumption on $U$. Thus, the set $\pi(X)$ is regular. This implies that the
set $X$ is also regular. Finally, the fact that $X_0$ and $X_1$ are not empty,
immediately follows from statement~(1).\bull
By statement~(2) of Lemma~\ref{130914c}, the union of $\tr(X)$, where $X$ runs over the set
$\cS(\cA)_{D\setminus V}$, is of the form $D\setminus D_k$ for some $k\ge 1$. However,
the set $V$ coincides with the complement to this union. Thus, $V=D_k$ is an $\cA$-group.
A similar argument shows that $D_m$ is an $\cA$-group for all $m\ge k$.
\lmml{130914u}
Let $m=\max\{2,k\}$. Then the group $L:=\rad(X)$ does not depend on the choice of
$X\in\cS(\cA)_{D\setminus D_m}$.
\elmm
\proof Suppose on the contrary that there exist basic sets $X$ and $Y$ outside $D_m$ such that
$\grp{Y}\subsetneq\grp{X}$ and $\rad(X)\ne\rad(Y)$. Then by statement (1) of Lemma~\ref{130914c},
without loss of generality, we can assume that
\qtnl{181114a}
\rad(X)=\grp{s}\qaq \rad(Y)=\grp{sc_1}.
\eqtn
By statement~(2) of that lemma, $X_0$ is a regular non-empty set. Therefore, it
is an orbit of a subgroup of $\aut(C)$. Moreover, from the first equality in \eqref{181114a},
it follows that $\rad(X_0)=e$. Let now $\pi:D\to D/\grp{s}$ be the quotient epimorphism. Then
$\pi(X)=X_0$ is a basic set of a circulant S-ring $\cA'=\pi(\cA)$. It follows that
$\cA'_{\grp{X_0}}$ is a cyclotomic S-ring with trivial radical. Moreover, $Y'=\pi(Y)$ is a
basic set of this S-ring and $|\grp{Y'}|\ge 2^{m+1}\ge 8$. Therefore, by Lemma~\ref{021109a} applied
for $\cA=\cA'_{\grp{X_0}}$ and $S=\grp{Y'}/e$, we obtain that
$$
\rad(\cA'_{\grp{Y'}})=e.
$$
This implies that $\rad(Y')=e$. On the other hand, by the second equality in~\eqref{181114a},
we have $\rad(Y')=\grp{c_1}\ne e$. Contradiction.\bull
By Lemma~\ref{130914u}, the S-ring $\cA$ is the generalized $S$-wreath product,
where $S=D_2/L$ if $k=1$, and $S=V/L$ if $k\ge 2$. The only case when this generalized
product is not proper, is $D=D_2$ and $k=1$. However, in this case $\cA$ is, obviously, a proper
$E/L$-wreath product and $|E/L|=2$. Thus, if $k\le 2$, then the first or the second statement
in~\eqref{171114a} holds, and we are done. To complete the proof of Theorem~\ref{060414a},
it suffices to verify that the third statement in~\eqref{171114a} holds whenever $k\ge 3$.
But this immediately follows from the lemma below.
\lmml{220414u}
If $k\ge 3$, then $\rad(\cA_V)=e$. In particular, $V=U$.
\elmm
\proof The second statement follows from the first one and statement~(1) of Lemma~\ref{130914c}.
To prove that $\rad(\cA_V)=e$, let $X$ be a highest basic set of the S-ring $\cA_V$. Since $V\ne D$, there exists
a set $Y\in\cS(\cA)_{D\setminus V}$ such that $X\subseteq Y^2$. By Lemma~\ref{130914c},
we have $Y=LY_0$, where $Y_0$ is an orbit of a subgroup of $\aut(C)$ such that $\rad(Y_0)=e$.
However, $Y_0=\{y\}$ or $Y_0=\{y,\varepsilon y^{-1}\}$, where $\varepsilon\in\{e,c_1\}$
(statement~(1) of Lemma~\ref{211113a}). Therefore,
\qtnl{240414b}
Y^2=LY_0^2=L\times\css
\{y^2\}, &\text{if $Y_0=\{y\}$},\\
\{\varepsilon,y^{\pm 2}\}, &\text{if $Y_0=\{y,\varepsilon y^{-1}\}$}.\\
\ecss
\eqtn
On the other hand, since $X\subset V$, the definition of $V$ implies that $\rad(X)=e$ or $c_1\in\rad(X)$. In
the former case, $\rad(\cA_V)=e$, and we are done. Suppose that $c_1X=X$. Then the set $Y^2$ contains
a $\grp{c_1}$-coset $\{x,c_1x\}$ for all $x\in X$. By~\eqref{240414b}, this implies that
$$
\{x,c_1x\}\subset \{\varepsilon,y^{\pm 2}\}
$$
which is impossible, because $|x|=2^k\ge 8$.\bull
\section{S-rings over $D=\mZ_2\times\mZ_{2^n}$: schurity}\label{231014a3}
In this section, based on the results obtained in Sections~\ref{140914a}--\ref{140914b}, we prove the following main
theorem.
\thrml{090514b}
For any integer $n\ge 1$, every S-ring over the group $D=\mZ_2\times\mZ_{2^n}$ is schurian. In particular,
$D$ is a Schur group.
\ethrm
\proof
The induction on $n$. An exhaustive computer search of all S-rings over small groups shows that
$D$ is a Schur group for $n\le 4$. Let $n\ge 5$. We have to verify that any S-ring $\cA$
over~$D$ is schurian. However, if $\rad(\cA)=e$, then this is true by Theorems~\ref{170414a}
and~\ref{230214a}. For the rest of the proof, we need the following result from~\cite{EP12}
giving a sufficient condition for a generalized wreath product of S-rings to be schurian.
\thrml{140914f}{\rm \cite[Corollary~5.7]{EP12}}
Let $\cA$ be an S-ring over an abelian group~$D$. Suppose that $\cA$ is the generalized $S$-wreath
product of schurian S-rings $\cA_{D/L}$ and $\cA_U$, where~$S=U/L$. Then $\cA$ is schurian if and only if there exist two groups
$\Delta_0\ge(D/L)_{right}$ and $\Delta_1\ge U_{right}$, such that
\qtnl{140914r}
\Delta_0\twoe\aut(\cA_{D/L})\qaq
\Delta_1\twoe\aut(\cA_U)\qaq
(\Delta_0)^{U/L}=(\Delta_1)^{U/L}.
\eqtn
\ethrm
\crllrl{110514a}
Under the hypothesis of Theorem~\ref{140914f}, the S-ring $\cA$ is schurian whenever the
group $\aut(\cA_S)$ is $2$-isolated.
\ecrllr
\proof Set $\Delta_0=\aut(\cA_{D/L})$ and $\Delta_1=\aut(\cA_U)$. Then the first two
equalities in~\eqref{140914r} hold, because the S-rings $\cA_{D/L}$
and $\cA_U$ are schurian. Since the group $\aut(\cA_S)$ is $2$-isolated, we have
$
(\Delta_0)^S=\aut(\cA_S)=(\Delta_1)^S,
$
which proves the third equality in~\eqref{140914r}. Thus, $\cA$ is schurian
by Theorem~\ref{140914f}.\bull
Let us turn to the proof of Theorem~\ref{090514b}. Now, we can assume that
$\rad(\cA)\ne e$. Then by Theorem~\ref{060414a}, the S-ring~$\cA$ is a proper generalized $S$-wreath
product, where the section $S=U/L$ is such that formula~\eqref{171114a} holds. Besides, by induction,
the S-rings $\cA_{D/L}$ and $\cA_U$ are schurian. Suppose that $\cA_S=\mZ S$, or $|S|=4$,
or $|L|=2$ and $\cA_U$ is a regular S-ring with trivial radical. Then the group $\aut(\cA_S)$
is $2$-isolated: this is obvious in the first two cases and follows from Theorem~\ref{090514a}
(applied for $\cA=\cA_U$) in the third one. Thus, $\cA$ is schurian
by Corollary~\ref{110514a}.\medskip
To complete the proof, we can assume that $|S|=|2^m|$, where $m\ge 3$, and that
$\cA_U$ is a non-regular S-ring with trivial radical. Then
$\cA_U=\cA_H\otimes\cA_L$, where $|H|\ge 4$ and $\rk(\cA_H)=2$ (Theorem~\ref{170414a}).
Therefore,
\qtnl{140914w}
\aut(\cA_U)^S=(\sym(H)\times\sym(L))^{U/L}=\sym(S).
\eqtn
On the other hand, $L=\grp{s}$ or $L=\grp{sc_1}$, because $c_1\in H$. Therefore,
the S-ring $\cA_{D/L}$ is circulant. Besides, $S$ is an $\cA_{D/L}$-section of
composite order. By \cite[Theorem~4.6]{EP12}, this implies that
\qtnl{150914a}
\aut(\cA_{D/L})^S=\sym(S).
\eqtn
By \eqref{140914w} and \eqref{150914a}, relations \eqref{140914r} are true
for the groups $\Delta_0:=\aut(\cA_{D/L})$ and $\Delta_1:=\aut(\cA_U)$. Thus, the
S-ring~$\cA$ is schurian by Theorem~\ref{140914f}.\bull
\section{A non-schurian S-ring over $M_{2^n}$}\label{231014a4}
The main result of this section is the following theorem in the proof of which
we construct a non-schurian S-ring over the group $M_{2^n}$
defined in~\eqref{170914a}.
\thrml{071113a}
For any $n\ge 4$, the group $M_{2^n}$ is not Schur.
\ethrm
\proof The group $M_{16}$ is not Schur \cite[Lemma~3.1]{PV}. Suppose that $n\ge 5$.
Denote by $e$ the identity of the group~$G=M_{2^n}$, and
by $H$ the
normal subgroup of $G$ that is generated by the elements $c=a^{2^{n-3}}$ and $b$. Then
$H\simeq \mZ_4\times \mZ_2$ and
\qtnl{110413b}
H=Z_0\,\cup\, Z_1\,\cup\, Z_2,
\eqtn
where the sets $Z_0=\{e\}$, $Z_1=\{c^2\}$, and $Z_2=H\setminus\grp{c^2}$ are mutually disjoint. Next, let us
fix two other decompositions of $H$ into a disjoint union of subsets:
$$
H=\underbrace{B\,\cup\,Bc^3}_{X_1}\ \cup\ \underbrace{Bc^2\,\cup\,Bc}_{Y_1}
\quad=\quad
\underbrace{B'\,\cup\,B'c}_{X_2}\quad\cup\quad\underbrace{B'c^2\,\cup\,B'c^3}_{Y_2},
$$
where $B$ and $B'$ are the groups of order~$2$ generated by the involutions~$b$ and $b':=c^2b$.
Then a straightforward computation shows that
\qtnl{110413a}
Ha\cup Ha^{-1}=\underbrace{X_1a\cup X_2a^{-1}}_{Z_3}\ \cup\ \underbrace{Y_1a\cup Y_2a^{-1}}_{Z_4},
\eqtn
in particular, the sets $Z_3$ and $Z_4$ are disjoint. Moreover, $Z_3c^2=Z_4$, because
$Y_1=X_1c^2$ and $Y_2=X_2c^2$. Finally, there are exactly $m'=2^{n-3}-3$ cosets of $H$ in $G$,
other than $H$, $Ha$, and $Ha^{-1}$. Let us combine them in pairs as follows
\qtnl{110413ae}
Z_{i+3}:=Ha^i\cup Ha^{-i},\quad i=2,3,\ldots m,
\eqtn
where $m=(m'-1)/2+1$ and $Z_{m+1}=Ha^{2^{n-2}}$.
Then the sets $Z_0,Z_1,\ldots,Z_{r-1}$ with $r=m+5$ form
a partition of the group~$G$; denote it by~$\cS$. The submodule of $\mZ G$ spanned
by the elements $\und{Z_i}$, $i=0,\ldots,r-1$, is denoted by $\cA$.
\lmml{170914s}
The module $\cA$ is an S-ring over $G$. Moreover, $\cS(\cA)=\cS$.
\elmm
\proof
From the above definitions, it follows that $Z_i^{-1}=Z_i^{}$ for all $i$. Thus, it suffices
to verify that given $i$ and $j$, the product $\und{Z_i}\,\und{Z_j}$ is a linear combination
of $\und{Z_k}$, $k=0,\ldots,r-1$. However, it is easily seen that
$$
\und{Ha^i}\,\und{Ha^j}=\und{Ha^j}\,\und{Ha^i}=\und{Ha^{i+j}}=\und{a^{i+j}H}
$$
for all $i,j,k$. Therefore, the required statement holds whenever $i,j\not\in\{3,4\}$.
To complete the proof, assume that $i=3$ (the case $i=4$ is considered analogously). Then
a straightforward check shows that
\nmrt
\item[$\bullet$] $\und{Z_3}\,\und{Z_1}=\und{Z_4}$,\quad $\und{Z_3}\,\und{Z_2}=\und{Z_4}$,\quad $\und{Z_3}\,\und{Z_{r-1}}=4\und{Z_{r-2}}$,
\item[$\bullet$] $\und{Z_3}\,\und{Z_3}=8\,\und{Z_0}+2\,\und{Z_5}+4\,\und{Z_2}$,
\item[$\bullet$] $\und{Z_3}\,\und{Z_4}=8\,\und{Z_1}+2\,\und{Z_5}+4\,\und{Z_2}$,
\item[$\bullet$] $\und{Z_3}\,\und{Z_5}=4\und{Z_3}+4\und{Z_4}+4\und{Z_6}$,
\item[$\bullet$] $\und{Z_3}\,\und{Z_i}=4\und{Z_{i-1}}+4\und{Z_{i+1}}$, $i=6,\ldots r-2$.
\enmrt
Since $\und{Z_3}$ commutes with $\und{Z_j}$ for all $j$, we are done.
\bull
By Lemma~\ref{170914s}, the statement of Theorem~\ref{071113a} immediately
follows from the lemma below.
\lmml{231112a}
The S-ring $\cA$ is not schurian.
\elmm
\proof Suppose on the contrary that $\cA$ is schurian. Then it
is the S-ring associated with the group
$\Gamma=\aut(\cA)$. It follows that the basic set $Z_2$ is an orbit of the one-point
stabilizer of~$\Gamma$. Since $|Z_2|=6$, there exists an element
$\gamma\in\Gamma$ such that
\qtnl{231112b}
|\gamma^{Z_2}|=3.
\eqtn
On the other hand, due to~\eqref{110413a}, the quotient S-ring $\cA_{G/H}$ is isomorphic
to the S-ring associated with the dihedral group of order $2^{n-2}$ in its natural permutation representation of degree $2^{n-3}$.
Therefore, $\aut(\cA_{G/H})$ is a $2$-group. It contains a subgroup $\Gamma^{G/H}$,
and hence the element $\gamma^{G/H}$. So by~\eqref{231112b}, the
permutation~$\gamma$ leaves each $H$-coset fixed (as a set). Therefore,
\qtnl{281112a}
\gamma^{H\cup Ha}\in\aut(C),
\eqtn
where $C$ is the bipartite graph with vertex set $H\cup Ha$ and the edges $(h,hax)$ with $x\in X_1$. However,
the graph $C$ is isomorphic to the lexicographic product of the empty graph with~$2$ vertices and the
undirected cycle of length~$8$. Therefore, $\aut(C)$ is a $2$-group. By~\eqref{281112a}, this implies
that $|\gamma^H|$ is a power of~$2$. But this contradicts to~\eqref{231112b}, because $Z_2\subset H$.\bull
\section{S-rings over $D=D_{2n}$: divisible difference sets}\label{231014a6}
\sbsnt{Preliminaries.} In the rest of the paper, we deal with S-rings over a
dihedral $2$-group $D=D_{2n}$ of order $2n$. Interesting examples of such rings arise from difference sets. To
construct them, let us recall some definitions from~\cite{P95}.\medskip
Let $T$ be a $k$-subset of a
group $G$ of order $mn$ such that every element outside a subgroup $N$ of order
$n$ has exactly $\lambda_2$ representations as a quotient $gh^{-1}$ with elements
$g,h\in G$, and elements in $N$ different from the identity have exactly
$\lambda_1$ such representations,
\qtnl{161014a}
\underline{T}\cdot\underline{T}^{-1}=
k\cdot e + \lambda_1\underline{N\setminus e}+\lambda_2\underline{G\setminus N}.
\eqtn
Then $T$ is called an {\it $(m,n,k,\lambda_1,\lambda_2)$-divisible difference set} in $G$
relative to $N$. If $\lambda_1 =0$ (resp. $n = 1$), then we say that $T$
is a {\it relative difference set} or {\it relative $(m,n,k,\lambda_2)$-difference set}
(resp. difference set). A difference set $T$
is {\it trivial} if it equals $G$, $\{x\}$ or $G\setminus\{x\}$, where $x\in G$.
\thrml{231014a}
Let $C$ be a cyclic $2$-group. Then
\nmrt
\tm{1} any difference set in $C$ is trivial,
\tm{2} there is no relative $(2^a,2,2^a,2^{a-1})$-difference set in $C$.
\enmrt
\ethrm
\proof Statement (1) follows from \cite[Theorem~II.3.17]{BJL} and \cite[Theorem~1.2]{DS94}.
Statement~(2) follows from \cite[Theorems~4.1.4,4.1.5]{P95}.\bull
\sbsnt{Constructions.}\label{231014w} Let $D$ be a dihedral group of order $2n$, $C$ the cyclic subgroup
of $D$ of order $n$ and $H$ a subgroup of $C$. Let $T$ be a non-empty subset of~$C$
such that $|T\cap xH|$ does not depend on $x\in T$ ({\it the intersection condition},
cf. Lemma~\ref{090608a}). Set
$$
\cS:=\{e,\ H\setminus e,\ C\setminus H,\ Ts,\ T's\},
$$
where $T'=C\setminus T$. Clearly, $\cS$ is a partition of $D$ such that condition (S1)
is satisfied. Since all the elements of $\cS$ are symmetric, condition (S2) is also
satisfied. Set
\qtnl{161014c}
\cA:=\cA(T,C)=\Span\{\und{X}:\ X\in\cS\}.
\eqtn
\thrml{p0}
In the above notation, $\cA$ is an S-ring over $D$ with $\cS(\cA)=\cS$ if and only if $T$
is a divisible difference set in $C$ relative to $H$.
\ethrm
\proof To prove the ``only if'' part, suppose that $\cA$ is an S-ring with $\cS(\cA)=\cS$.
Then $Ts$ is a basic set of $\cA$ and $Ts\,Ts=T\,T^{-1}$ is a subset of~$C$.
Therefore,
\qtnl{191014a}
\underline{T}\cdot\underline{T}^{-1}=
\underline{Ts}^2=
|T|e+\lambda_1\underline{H\setminus e} + \lambda_2\,\underline{C\setminus H},
\eqtn
where $\lambda_1=c_{Ts\,Ts}^{H\setminus e}$ and
$\lambda_2=c_{Ts\,Ts}^{C\setminus H}$. Thus, $T$ is a divisible difference
set in $C$ relative to~$H$.\medskip
To prove the ``if'' part, suppose that $T$ is a divisible difference set in $C$ relative
to~$H$. It suffices to verify that $\cA\cdot\cA\subseteq \cA$. To do this, denote by $\cA'$
the module spanned by $e$, $\underline{H}$, $\underline{C}$, and $\underline{D}$.
Then, obviously, $\cA'\cdot\cA'\subseteq\cA'$ and $\cA =\Span\{\cA',\underline{sT}\}$.
Thus, we have to check that
$$
\underline{sT}^2\in\cA\qaq \cA'\cdot\underline{sT}\,\subseteq\, \cA.
$$
The first inclusion follows from~\eqref{161014a}, because $T$ is a divisible difference set.
Routine calculations show that the second inclusion is equivalent to the inclusion
$\und{T}\cdot\und{H}\in\cA$. However, this easily follows from the intersection condition.\bull
We do not know any divisible difference set over a cyclic $2$-group that satisfies the intersection condition.
However, we can slightly modify the construction by taking the set $T$ to satisfy the intersection condition, but
this time inside $C\setminus H$. Then using the same argument, one can
construct an S-ring of rank $6$ that coincides with $\cA$ on $C$ and has three
basic sets outside $C$: $Ts$, $T's$, and $Hs$, where $T'=C\setminus (T\cup H)$.\medskip
The S-rings of this form do exist. It suffices to take a classical relative
$(q+1,2,q,(q-1)/2)$-difference set $T$ defined as follows (see~\cite[Theorem~2.2.13]{P95}).
Take an affine line $L$ in a $2$-dimensional linear space over finite field $\mF_q$ that does not contain
the origin. Then $L$ is a relative $(q+1,q-1,q,1)$-difference set in the
multiplicative group of $\mF_{q^2}$. Let $\pi$ be a quotient epimorphism from this group
onto $C$ such that $|\ker(\pi)|=m$ for some divisor $m$ of $q-1$. Then $\pi(L)$ is a
cyclic relative $(q+1,(q-1)/m,q,m)$-difference set in~$C$. When $C$ is a $2$-group, the
number $q+1$ is a $2$-power, and so, $q$ is a Mersenne number. If, in addition,
$m=(q-1)/2$, then we come to the required set $T$.
\crllrl{120614a}
Let $\cA$ be an S-ring over a dihedral $2$-group $D$. Suppose that $C$ is an $\cA$-group. Then
\nmrt
\tm{1} if $\rk(\cA_C)=2$, then $\cA\cong\cA_C\ast\cB$, where $\ast\in\{\wr,\cdot\}$
and $\cB=\mZ\mZ_2$,
\tm{2} if $\rk(\cA_C)=3$ and $\rk(\cA)=5$, then $\cA=\cA(T,C)$, where $T$ is
a divisible difference set in $C$.
\enmrt
\ecrllr
\proof To prove statement~(1), suppose that $\rk(\cA_C)=2$. Let $X$ be a basic set outside $C$.
Then $X=Ts$ for some set $T\subseteq C$. From~\eqref{191014a} with $H=e$, it follows that $T$ is
a difference set in~$C$. By statement~(1) of Theorem~\ref{231014a}, this implies that either
$T=C$, or $T$ or $C\setminus T$ is a singleton. It
is easily seen, that $\cA$ is isomorphic to $\cA_C\wr\cB$ in the former case, and to $\cA_C\otimes \cB$
in the other two.\medskip
To prove statement (2), suppose that $\rk(\cA_C)=3$ and $\rk(\cA)=5$. Then
$$
\cA_C=\Span\{e,\und{H},\und{C}\}
$$
for some $\cA$-group $H<C$. Besides, any $X\in\cS(\cA)_{D\setminus C}$ is of the form
$X=Ts$ for some set $T\subseteq C$ satisfying the intersection condition (Lemma~~\ref{090608a}).
Thus, $\cA=\cA(T,C)$. So, $T$ is a divisible difference set in $C$ by Theorem~\ref{p0}.\bull
\sbsnt{Schurity.}
The main goal of this subsection is to prove the following theorem showing that the first
construction given in the previous subsection, produces mainly non-schurian S-rings.
\thrml{081014w}
Let $T$ be a divisible difference set in a cyclic $2$-group~$C$ relative to
a group $H\le C$. Suppose that the intersection condition holds and
$HT\ne T$. Then the S-ring $\cA(T,C)$ defined in~\eqref{161014c} is not schurian.
\ethrm
We will deduce this theorem in the end of this subsection from a general statement on
schurian S-rings over a dihedral $2$-group. This statement shows that if $T$ is a
divisible difference set in a cyclic $2$-group relative to a subgroup $H$, and
the S-ring of rank $6$ associated with $T$, is schurian but not a proper generalized
wreath product, then $H$ is of order~$2$.
\thrml{231014s}
Let $\cA$ be a schurian S-ring over a dihedral $2$-group $D$ and $H<C$ a minimal
$\cA$-group. Suppose that $\cA$ is not a proper generalized wreath product. Then $|H|=2$.
\ethrm
\proof By the hypothesis, $\cS(\cA)=\orb(G_e,D)$ for some group $G\leq\sym(D)$ containing $D_{right}$.
Since $H$ is an $\cA$-group, the partition
$D/H$ of the group $D$ into the right $H$-cosets, forms an imprimitivity system for~$G$. Denote by $N$
the stabilizer of this partition in $G$,
$$
N=\{g\in G:\ (Hx)^g=Hx\ \,\text{for all}\ \,x\in D\}.
$$
Since $H_{right}\leq N$, the group $N^X$ is transitive for each block $X\in D/H$.
The following statement can also be deduced from~\cite[Lemma 2.1]{KMM11}.
\lmml{p1}
For each block $X\in D/H$, the group $N^X$ is $2$-transitive.
\elmm
\proof Let $X\in D/H$. Then the group $G^X$ is $2$-transitive, because $\rk(\cA_H)=2$. Moreover, it contains a regular cyclic
subgroup isomorphic to $H_{right}$. Since $H$ is a $2$-group,
the classification of primitive groups having a regular cyclic subgroup~\cite{Jo02},
implies that $G^X$ contains
a unique minimal normal subgroup $K$ that is also $2$-transitive. However, $N^X$ is a non-trivial normal subgroup
of $G^X$. Therefore, $N^X$ contains $K$, and hence is $2$-transitive.\bull\medskip
Let us define an equivalence relation $\sim$ on the $H$-cosets by setting $X\sim Y$ if and
only if the actions of $N$ on~$X$ and $Y$ have the same permutation character. Then by
Lemma~\ref{p1} and a remark in~\cite[p.2]{C72}, the group $N_e$ acts transitively on each
$X$ not equivalent to $H$. Denote by $U$ the class of $\sim$ that
contains $H$. Then each orbit of $G_e$ outside $U$ is a union of $N$-orbits. Thus,
$\cA$ is the generalized $U/H$-wreath product. By the theorem hypothesis, this implies that $U=D$.
Therefore, all classes of the equivalence relation $\sim$ are singletons. So by Lemma~\ref{p1},
we have
$$
|\orb(N,X\times Y)|=2\quad\text{for all}\ X,Y\in D/H.
$$
This yields
us two symmetric block-designs between $X$ and $Y$ which are complementary to each other.
Since $N$ contains a cyclic subgroup $H$ which acts regularly on $X$ and $Y$, these
block-designs are circulant, and so correspond to cyclic difference sets.
By statement~(1) of Lemma~\ref{231014a}, they are trivial. Thus, the group
$N_x$ with $x\in X$, has two orbits on $Y$ of cardinalities $1$ and $|Y|-1$.\medskip
To complete the proof, suppose on the contrary that $|H|>2$. Then, obviously, $|Y| > 2$. Therefore,
the group $N_e$ fixes exactly one point in each $H$-coset~$Y$. This implies that the set $F$ of
all fixed points of $N_e$, is of cardinality $[D:H]$. On the other hand, by
\cite[Proposition~5.2]{KMM11}, the set $F$ is a block of $G$. Therefore, $F$ is a subgroup of~$D$.
Moreover, since $F\cap H=e$, it is a complement for $H$ in $D$. Thus, $H=C$. Contradiction.\bull
{\bf Proof of Theorem~\ref{081014w}.} Suppose on the contrary that the S-ring $\cA=\cA(T,C)$
is schurian. Then the hypothesis of Theorem~\ref{231014s} is satisfied, because
$H$ is a minimal $\cA$-group and $HT\ne T$.
Thus, $|H|=2$. Denote by $x$ the element of order $2$ in~$H$. Then $x\in\cA$, and hence
$x(Ts)=T's$. This implies that $|Ts|=|T's|=m$, where $m=|C|/2$, and that $x$ appears
neither in $\und{Ts}^2$, nor in $\und{T's}^2$. Therefore, $T$ has parameters
$(m,2,m,0,m/2)$. It follows that $T$ is a relative $(m,2,m,m/2)$-difference set in $C$.
However, this contradicts part~(2) of Theorem~\ref{231014a}.\bull
\section{S-rings over $D=D_{2^{n+1}}$: a unique minimal $\cA$-group not in $C$}
In this section we deal with S-rings over a dihedral group $D=D_{2^{n+1}}$ of
order $2^{n+1}$
and keep the notation of Subsection~\ref{011014b}. The main result here is given
by the following statement.
\thrml{011014a}
Let $\cA$ be an S-ring over the dihedral group $D$. Suppose that there
is a unique minimal $\cA$-group $H$, and that $H\not\le C$.
Then $\cA$ is isomorphic to an S-ring over $\mZ_{2^{}}\times\mZ_{2^n}$.
\ethrm
\proof The hypothesis on $H$ implies that every basic set $X$ outside
$H$ is mixed, for otherwise $\grp{X}$ contains a non-identity
$\cA$-subgroup of $C$. Moreover, either $H=\grp{s}$ for some $s\in D\setminus C$,
or $H$ is a dihedral group. Let us consider these two cases
separately.\medskip
{\bf Case 1: $H=\grp{s}$} for some $s$. In this case, all basic sets except
for $\{e\}$ and $\{s\}$ are mixed. By statement~(1) of Lemma~\ref{240913a}, this
implies that $X_0^{-1}=X_0^{}$ for all $X\in\cS(\cA)$. Besides,
$Xs\in\cS(\cA)$, because $s\in\cA$, and
$(Xs)_0=X_1$ and $(Xs)_1=X_0$.
Thus, $X_1^{-1}=X_1^{}$ also for all $X$.\medskip
Denote by $\sigma$ the automorphism of $D$ that takes $(c,s)$ to $(c^{-1},s)$,
where $c$ is a generator of $C$. Then by the above paragraph, we have
$$
X^\sigma=(X_0\,\cup\,X_1s)^\sigma=X_0^{-1}\,\cup\,X_1^{-1}=
X_0\,\cup\,X_1s=X
$$
for all $X\in\cS(\cA)$. Therefore, the semidirect product
$D\rtimes\grp{\sigma}\le\sym(D)$ is an automorphism group of~$\cA$.
The element $s\sigma$ of this group has order two and commutes with~$c$. Therefore, the group $D'=\grp{s\sigma,c}$
is isomorphic to $\mZ_2\times\mZ_{2^n}$. On the other hand, $D'$ is a regular subgroup
in $\sym(D)$. Thus, the Cayley scheme over $D$ associated with $\cA$ is
isomorphic to a Cayley scheme over $D'$. Consequently, $\cA$ is isomorphic to an
S-ring over $\mZ_{2^{}}\times\mZ_{2^n}$.\medskip
{\bf Case 2: $H$ is dihedral}. In this case all basic sets of $\cA$
other than $\{e\}$ are mixed. Moreover, the S-ring $\cA_H$ is primitive
by the minimality of $H$. Therefore, $\rk(\cA_H)=2$ by
Theorem~\ref{030414a}. In particular, $\aut(\cA_H)=\sym(H)$. Below,
we will prove that
\qtnl{011014c}
H\le\rad(X)\quad\text{for all}\ X\in\cS(\cA)_{D\setminus H}.
\eqtn
Then the Cayley scheme associated with $\cA$ is isomorphic to
the wreath product of the scheme associated with $\cA_H$ and a
circulant scheme on the right $H$-cosets. Therefore, the group $\aut(\cA)$
contains a subgroup isomorphic to $\sym(H)\wr\mZ_m$, where $m=[D:H]$.
Since the latter group contains a regular subgroup isomorphic to
$\mZ_{2^{}}\times\mZ_{2^n}$, we are done.\medskip
To complete the proof, we will check statement~\eqref{011014c} in two
steps: first for rational S-rings, and then in general.
\lmml{280813a}
Statement~\eqref{011014c} holds whenever the S-ring $\cA$ is rational.
\elmm
\proof The rationality of $\cA$ implies that it is symmetric, and
hence commutative. Toward to a contradiction,
suppose that $HX\ne X$ for some basic set $X$ contained in $D\setminus H$. Then
the product $HX$ is a union of $m>1$ basic sets $X,Y,\ldots$.
Without loss of generality, we may assume that
$|X|\leq |Y|\leq\cdots$.\medskip
Since $H$ is an $\cA$-group, it follows from Lemma~\ref{090608a} that
the number $\lambda=|X\cap xH|$ does not depend on the choice of $x\in X$. Therefore,
each $x$ appears $\lambda$ times in the product $\und{H_{}}\,\und{X_{}}$,
i.e.
\qtnl{300514a}
\und{H_{}}\,\und{X_{}} = \lambda(\und{X}+\und{Y}+\cdots).
\eqtn
By the minimality of $X$, this implies that
$|H|\,|X|\geq\lambda m |X|$, and hence $|H|\geq \lambda m$. On the
other hand, $(H\cap C) X_0 = X_0$ by the rationality of $X$. Therefore,
the element $\und{X}$ appears in the product $\und{H}\,\und{X}$
at least $|H\cap C|=|H|/2$ times. Thus, $\lambda\geq |H|/2$,
and
$$
|H|\geq \lambda m\ge m|H|/2.
$$
Due to $m>1$, we have $m=2$ and $\lambda=|H|/2$. Consequently, $H_0=H_1=H\cap C$, because
the group $H$ is dihedral. Therefore,
$$
\und{H_{}}\,\und{X_{}} = \und{H_0}(e+s)(\und{X_0}+s\und{X_1}) =
$$
$$
\und{H_0}\,\und{X_0} +\und{H_0}\,\und{sX_0} +\und{H_0}\,\und{X_1} +\und{H_0}\,\und{sX_1} =
|H_0|\und{X_0} +\und{H_0}\,\und{X_1}+\cdots
$$
By~\eqref{300514a}, all coefficients in the last expression are equal
to $\lambda=|H_0|$. Therefore, the set $H_0X_1\cap X_0$ must be empty.
It follows that $\und{H_0}\,\und{X_1} = |H_0|\und{H_0X_1}$. However,
this means that $H_0\le\rad(X_1)$. Since also $H_0\le\rad(X_0)$,
we conclude that $H_0\leq\rad(X)$. But $H_0\ne e$, because $H$ is
dihedral. Thus, $\rad(X)$ is non-trivial, and so contains the minimal
$\cA$-group~$H$. But then, $HX = X$. Contradiction.\bull
To complete the proof of~\eqref{011014c}, take a basic set $X$ outside $H$. Then $Y:=\tr(X)$ is also
outside $H$. So by Lemma~\ref{280813a}, we have
$$
\und{Y_0}+\und{sY_1}=\frac{1}{|H|}\und{H}\,\und{Y}=
\frac{1}{|H|}(e+s)\,(\und{H_0Y_0}+\und{H_0Y_1}).
$$
Therefore, $|Y_0|=|Y_1|$. On the other hand, for every integer $m$ coprime to $|D|$, we have $|(X^{(m)})_0|=|X_0|$. By
Lemma~\ref{290514b}, this implies that $|(X^{(m)})_1|=|X_1|$. Thus,
$$
k|X_0|=|Y_0|=|Y_1|=k|X_1|,
$$
where $k$ is the number of all distinct sets $X^{(m)}$'s. It follows that $|X_0|=|X_1|$ for each
basic set $X$ outside $H$.\medskip
Denote by $\rho$ the restriction to $\cA$ of the one-dimensional representation of $D$ that takes $s$ and $c$ to $-1$ and
$1$, respectively. Then $\rho$ is an irreducible representation of $\cA$ such that $\rho(e)=1$ and $\rho(\und{H^\#})=-1$. Moreover,
for any basic set $X$ outside $H$, we obtain by above that $\rho(\und{X}) =-|X_1|+|X_0| = 0$. In particular,
$\rho(\und{X^{-1}})=0$. Therefore,
$$
0=\rho(\und{X^{}}\,\und{X}^{-1})=\sum_{Y\in\cS(\cA)} c_{X^{}X^{-1}}^Y\rho(\und{Y})=
$$
$$
c_{X^{}X^{-1}}^e\rho(e)+c_{X^{}X^{-1}}^{H^\#}\rho(H^\#)=
c_{X^{}X^{-1}}^e-c_{X^{}X^{-1}}^{H^\#}.
$$
It follows that $|X|=c_{X^{}X^{-1}}^e=c_{X^{}X^{-1}}^{H^\#}$. Therefore, $HX = X$ for all basis sets
outside $H$, and we are done.\bull
\section{Proof of Theorem~\ref{170814c}}\label{231014a7}
Let $D$ be a dihedral $2$-group and $C$ its cyclic subgroup of index $2$.
Let $\cA$ be an S-ring over the group $D$. Suppose that $r:=\rk(\cA)$ is
at most $5$. For $r=2$, part (1) of Theorem~\ref{170814c} holds trivially. Let $r\ge 3$.
Then from Theorem~\ref{030414a}, it follows that the S-ring $\cA$ is imprimitive; denote
by $H$ a minimal non-trivial $\cA$-group. Then $\rk(\cA_H)=2$. Now,
if $r=3$, then $\cA$ is a proper wreath product by Corollary~\ref{021014a}.
Thus, we can assume that $r=4$ or $r=5$.
\lmml{300514e}
If there is a minimal $\cA$-group $L\ne H$, then statement~(2) holds.
\elmm
\proof By the minimality of the groups $H$ and $L$, we have $H\cap L=e$. Therefore, at least one of them intersects~$C$ trivially. Moreover,
if $H\cap C=L\cap C=e$, then $\grp{HL}$ is an $\cA$-group contained in~$C$, and we replace $H$ by a minimal $\cA$-subgroup in
$\grp{HL}$. Thus, without loss of generality, we can assume that
$$
H\cap C\ne e\qaq L\cap C=e.
$$
Then $L=\grp{s}$ for some involution
$s\in D\setminus C$. Moreover, $sHs=H$ by the minimality of $H$.
Thus, $HL$ is an $\cA$-group and the set $\cS(\cA_{HL})$ contains $4$ elements: $\{e\}$, $H^\#$, $\{s\}$,
and $sH^\#$. Therefore,
$$
\cA_{HL}=\cA_H\cdot\cA_L.
$$
This implies the required statement for $r=4$, and by Corollary~\ref{021014a}
also for $r=5$.\bull
By Lemma~\ref{300514e}, we can assume that $H$ is a unique minimal
$\cA$-group. If it is not contained in $C$, then statement (1) holds
by Theorem~\ref{011014a}. Thus, from now, on we also assume that $H\le C$.
Denote by $F$ the union of all basic sets of $\cA$ that are not $C$-mixed.
Clearly, $H\subseteq F$.
\lmml{240913c}
$F$ is an $\cA$-group. Moreover, if $r=r(\cA_F)+1$, then
$\cA$ is a proper wreath product.
\elmm
\proof The second part of our statement follows from the first one and
Corollary~\ref{021014a}. To prove the first statement, denote by $U$
and $V$ the unions of all basic sets of $\cA$ contained
in $C$ and $D\setminus C$, respectively. We have to prove
that $U\cup V$ is a group. Since $U$ is, obviously, an $\cA$-group,
without loss of generality, we may assume that $V$ is not empty. Then
$V=U's$, where $s\in D\setminus C$ is such that $U\cap Us$ is a
subgroup of $D$. It follows that
$$
UU'\subseteq U'\qaq U'U'\subseteq U.
$$
Since also $U'\subseteq UU'$, the first inclusion implies that $U'=UU'$.
Therefore, $U'$ is a union of some $U$-cosets contained in $C$. Since
the group $C$ is a cyclic $2$-group, this together with the
second inclusion implies $U'=U$. Thus, the set
$U\cup V=U\cup Us$ is a group.\bull
Suppose first that $F_0=C$. Then from the definition of $F$, it follows that $C$ is an
$\cA$-group. By Corollary~\ref{021014a}, we can assume that $r_C=\rk(\cA_C)$
is not equal to $r-1$. Since, obviously, $r_C\ge 2$,
we have
$$
(r,r_C)=(4,2),\ (5,2)\ \text{or}\ (5,3).
$$
In the former two cases, we are done by statement~(1) of Corollary~\ref{120614a},
whereas in the third one by statement~(2). Thus, in
what follows, we can assume that
$$
F_0<C\qaq H=F\ \text{or}\ r_F=3,
$$
where $r_F=\rk(\cA_F)$. In particular, there are two
or three basic sets outside $F$ (notice, that they are $C$-mixed).
\lmml{041014a}
Let $X\in\cS(\cA)_{D\setminus F}$. Suppose that $X$ is rational or
$[V:H]\ge 4$, where $V=\grp{X_0}$. Then $H\le\rad(X)$.
\elmm
\proof It suffices to verify that $H\le\rad(X_0)$. Indeed, then
the coefficient at $\und{X}$ in $\und{H}\,\und{X}$ is at least $|H|$.
Since it can not be larger than $|H|$, we are done.\medskip
Suppose first that $X$ is rational. Then by statement~(2) of Lemma~\ref{240913a} the
set $X_0$ is rational. Since $X_0\subseteq C\setminus H$, we conclude that
$H\le\rad(X_0)$, as required.\medskip
Let now $[V:H]\ge 4$. Then $V\cong\mZ_{2^k}$ for some $k\ge 2$. Since $r\le 5$, there are at most two basic
rationally conjugate to $X$. Therefore, the stabilizer of $X_0$ in the group $(\mZ_{2^k})^*$,
has index at most $2$ in it. It follows that this stabilizer
contains the subgroup of all elements $x\mapsto x^{1+4m}$, $x\in \mZ_{2^k}$,
with $m\in \mZ_{2^k}$. By statement~(2) of Lemma~\ref{211113a}, this implies that
$\rad(X_0)\ge V^4\ge H$.\bull
From Lemma~\ref{041014a}, it follows that if all basic sets outside $F$ are
rational, then $\cA$ is a proper wreath product. Indeed, this is obvious when $H=F$.
If $H\ne F$, this is also true, because then $F\setminus H$ is a basic set, the radical
of which equals $H$. Thus, we can assume that two of basic sets outside $F$, say $X$ and $Y$,
are rationally conjugate, and the third one (if exists) is rational. The rest of the proof is divided
into four cases below.\medskip
{\bf Case 1: $F=H$ and $r=4$.}
Using the computer package COCO, \cite{PRZ} we found exactly five and three
S-rings of rank~$4$ over the groups $D_8$ and $D_{16}$, respectively.
In both cases, only two of them have a unique minimal $\cA$-group
contained in $C$ and they are proper wreath products. Thus, in what follows, we assume
that $|D|\ge 32$.\medskip
In our case, the non-trivial basic sets of $\cA$ are $X$, $Y$ and $Z=H^\#$. It is easily seen that
the hypothesis of Lemma~\ref{290514b} is satisfied. Since $X$ and $Y$
are rationally conjugate, there is an algebraic isomorphism of $\cA$ that takes $X$
to $Y$, and $Y$ to $X$. Therefore,
\qtnl{081014a}
\und{H}\,\und{X}=a(\und{X}+\und{Y}),
\eqtn
where $a=|H|/2$. Consequently, $c_{ZX}^X=a-1$. On the other hand, since $X$ and $Z$ are
symmetric, we have $C_{XX}^Z=\frac{|X|}{|Z|}C_{ZX}^X$. It follows that $|Z|=2a-1$ divides
$$
|X|c_{ZX}^X=\frac{(d-2a)(a-1)}{2},
$$
where $d=|D|$. However, by Lemma~\ref{041014a}, without loss of generality,
we can assume that $|C:H|=2$. Therefore, $2a=|H|=d/4$. It follows that
$2a-1$ divides $3a(a-1)$. But $a$ being a power of $2$, must be coprime to $2a-1$.
Consequently, $2a-1$ divides $3a-3$. Since this is possible only for $a\le 2$, i.e.
when $d\le 16$, we are done.\medskip
{\bf Case 2: $F=H$ and $r=5$.} Denote by $Z$ the basic set in $\cS(\cA)_{D\setminus H}$
other than $X$ and $Y$. It is easily seen that the hypothesis of Lemma~\ref{290514b}
is satisfied. Since $X$ and $Y$ are rationally conjugate, there is an algebraic isomorphism of $\cA$
that takes $X$ to $Y$, $Y$ to $X$ and leaves $Z$ fixed. Therefore, the rational closure of
$\cA$ is of rank~$4$. So by Lemma~\ref{041014a}, it is the
wreath product $\cA_H\wr\cB$, where $\cB$ is isomorphic to the rational closure of $\cA_{D/H}$.
Therefore, $\rk(\cB)=3$, and hence there exists a non-trivial $\cB$-group. Denote by $U$
its preimage in $\cA$. Then, obviously, $H<U<D$.\medskip
Since $X$ and $Y$ are rationally conjugate, we have $X\cup Y\subseteq U$ or
$X\cup Y\subseteq D\setminus U$. However, $\rk(\cA)=r=5$. Therefore, $Z=D\setminus U$ in the former
case, and $Z=U\setminus H$ in the latter one. In any case, $H\le\rad Z$. By Lemma~\ref{041014a},
this implies that if $Z=U\setminus H$, then
\qtnl{140215a}
\rad(X)=\rad(Y)\ge H,
\eqtn
and $\cA$ is a proper wreath product. Let now, $Z=D\setminus U$. Then $\rk(\cA_U)=4$, and $\cA_U$ is
the wreath product $\cA_H\wr\cA_{U/H}$. Therefore, again \eqref{140215a} holds, and $\cA$ is a proper wreath product.\medskip
{\bf Case 3: $C\not\ge F>H$.} In this case, $F=H\cup Hs$. So by Lemma~\ref{041014a},
the sets $X_0$ and $Y_0$ are orbits of an index $2$ subgroup of $\aut(C)$, unless
$\cA$ is a proper wreath product.
This implies that the group $\rad(X_0)=\rad(Y_0)$ has index $2$ in $H$. Therefore,
equality~\eqref{081014a} holds with $a=|H|/2$. Exactly as in Case~2, we conclude
that $2a-1$ divides
$$
|X|c_{ZX}^X=\frac{(d-4a)(a-1)}{2},
$$
where $Z=H^\#$ and $d=|D|=4|H|=8a$. Thus, $2a-1$ divides $2a(a-1)$. Contradiction.\medskip
{\bf Case 4: $C\ge F>H$.} In this case, $F=F_0<C$ by the above assumption. Therefore,
$X_0$ (and also $Y_0$) contains a generator of $C$. It follows that
$$
[\grp{X_0}:H]=[C:H]\ge [C:F][F:H]\ge 4.
$$
By Lemma~\ref{041014a}, this implies that \eqref{140215a} holds. Since also
$F\setminus H$ is the basic set and $H=\rad(F\setminus H)$, the group $H$ is contained
in the radical of every basic set outside~$H$. Thus, $\cA$ is a proper wreath product.\bull
| 101,833
|
TITLE: Closed form of recurrence relation $a_{n}=a_{n-1}(a_{n-2}^{2}-a_{0})-a_{1}$
QUESTION [1 upvotes]: Given the following sequence:
$$\begin{cases}a_{0}=2\\a_{1}=\frac{5}{2}\\a_{n}=a_{n-1}(a_{n-2}^{2}-a_{0})-a_{1}\end{cases}$$
Prove that
$$a_{n}=2^{\frac{2^{n}-(-1)^{n}}{3}}+2^{\frac{(-1)^{n}-2^{n}}{3}}$$
For all natural numbers $n$.
My try: (using induction)
Let $b_{n}=\frac{2^{n}-(-1)^{n}}{3}$
For $n=0$: correct $\checkmark$
Now for $n+1$:
$$
\begin{aligned}
a_{n+1} &= a_{n}(a_{n-1}^{2}-a_{0})-a_{1})=(2^{b_{n}}+2^{-b_{n}})(2^{2b_{n}}+2^{-2b_{n}}+2) \\
&=2^{3b_{n}}+2^{-3b_{n}}+2^{b_{n}}+2^{-b_{n}}+2^{b_{n}+1}+2^{-b_{n}+1}
\end{aligned}
$$
How do I proceed from here?
REPLY [3 votes]: You want to proof that $a_n = f(n)$ for any $n\inℕ$ with
$$ f(n) := 2^{\frac{2^{n}-(-1)^{n}}{3}}+2^{\frac{(-1)^{n}-2^{n}}{3}}.$$
Check of two (!) initial conditions:
$f(0) = 2 = a_0 \checkmark$
$f(1) = \frac52 = a_1 \checkmark$
Inductive step:
$$\begin{aligned}
a_{n+1} &= a_n (a_{n-1}^2 - a_0) - a_1\\
&= f(n) (f(n-1)^2 - f(0)) - f(1)\\
&= (2^{b_n} + 2^{-b_n}) \cdot ((2^{b_{n-1}} + 2^{-b_{n-1}})^2 - 2) - \frac52\\
&= (2^{b_n} + 2^{-b_n}) \cdot (2^{2b_{n-1}} + 2^{-2b_{n-1}}) - \frac52 \\
&= 2^{b_n+2b_{n-1}} + 2^{b_n-2b_{n-1}} + 2^{-b_n+2b_{n-1}} + 2^{-b_n-2b_{n-1}} - \frac52 \end{aligned}$$
Simple computations show that:
$$\begin{aligned}
b_n+2b_{n-1} &= \frac13(2^{n+1} + (-1)^n) = b_{n+1}\\
b_n-2b_{n-1} &= (-1)^{n+1}.
\end{aligned}$$
Thus
$$a_{n+1} = f(n+1) + 2^{(-1)^{n+1}} + 2^{(-1)^n} - \frac52$$
With the equality (for $n\in\mathbb N$)
$$2^{(-1)^{n+1}} + 2^{(-1)^n} = \frac52$$
we can finish the proof:
$$a_{n+1} = f(n+1).$$
| 60,114
|
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