text:
\begin{document}
\title{The Borel-Ritt problem in Beurling ultraholomorphic classes}

\author[A. Debrouwere]{Andreas Debrouwere}
\thanks{A. Debrouwere was supported by  FWO-Vlaanderen through the postdoctoral grant 12T0519N}
\address{A. Debrouwere, Department of Mathematics: Analysis, Logic and Discrete Mathematics\\ Ghent University\\ Krijgslaan 281\\ 9000 Gent\\ Belgium}
\email{andreas.debrouwere@UGent.be}

\subjclass[2010]{\emph{Primary.} 30E05, 30D60. \emph{Secondary.} 46E10, 46M18, 46A63}
\keywords{Borel-Ritt problem; Beurling ultraholomorphic classes; homological algebra methods in functional analysis}

\begin{abstract}
We give a complete solution to the Borel-Ritt problem in non-uniform spaces $\mathscr{A}^-_{(M)}(S)$ of ultraholomorphic functions of Beurling type, where $S$ is  an unbounded sector  of the Riemann surface of the logarithm and $M$ is a strongly regular weight sequence. Namely, we characterize the surjectivity and the existence of a continuous linear right inverse of the asymptotic Borel map on $\mathscr{A}^-_{(M)}(S)$ in terms of the aperture of the sector $S$ and the weight sequence $M$. Our work improves previous results by  Thilliez \cite{Thilliez} and Schmets and Valdivia \cite{S-V}.
\end{abstract}
\maketitle

\section{Introduction}
The Borel-Ritt problem for spaces of ultraholomorphic functions goes back to the seminal work of Ramis \cite{Ramis} and plays an important role  in the study of formal power series solutions to various kinds of equations \cite{Balser}; see \cite{Debrouwere-Stieltjes,JG-S-S, JG-S-S-2, S-V,Thilliez} for an account of known results and open problems. In the present article, we consider this problem in the setting of non-uniform Beurling ultraholomorphic classes.

Let $M = (M_{p})_{p \in \N}$ be a sequence of positive real numbers and fix a sector of the Riemann surface $\Sigma$ of the logarithm
$$
S_\gamma = \{ z \in \Sigma \, | \, | \operatorname{Arg} z | < \frac{\gamma \pi}{2} \}, \qquad \gamma > 0.
$$
We define $\mathscr{A}_{(M)}(S_\gamma)$ as the Fr\'echet space consisting of all functions $f$ holomorphic in $S_\gamma$ such that
$$
\sup_{p \in \N} \sup_{z \in S_\gamma} \frac{n^p |f^{(p)}(z)|}{p!M_p} < \infty, \qquad \forall n \in \N,
$$
and set
$$
\mathscr{A}^-_{(M)}(S_\gamma) := \varprojlim_{\delta \to \gamma^-} \mathscr{A}_{(M)}(S_\delta).
$$
We call $\mathscr{A}_{(M)}(S_\gamma)$ and $\mathscr{A}^-_{(M)}(S_\gamma)$ \emph{the uniform and non-uniform space of ultraholomoprhic functions of class $(M)$ (of Beurling type) in $S_\gamma$}, respectively.  The corresponding sequence  space $\Lambda_{(M)}(\N)$ is defined as the Fr\'echet space consisting of all $(c_p)_{p \in \N} \in \C^\N$ such that
$$
\sup_{p \in \N}  \frac{n^p |c_p|}{p!M_p} < \infty,  \qquad \forall n \in \N.
$$
Consider the \emph{asymptotic Borel map}
\begin{equation}
B: \mathscr{A}_{(M)}(S_\gamma) \rightarrow \Lambda_{(M)}(\N), \, f \mapsto (f^{(p)}(0))_{p \in \N},
\label{borel-mapping}
\end{equation}
where $f^{(p)}(0) := \lim_{\substack{z \to 0^+ \\ z \in S_{\gamma}}} f^{(p)}(z)$. We also define
\begin{equation}
B: \mathscr{A}^-_{(M)}(S_\gamma) \rightarrow \Lambda_{(M)}(\N), \, f \mapsto (f^{(p)}(0))_{p \in \N},
\label{borel-mapping-nu}
\end{equation}
where now $f^{(p)}(0) = \lim_{\substack{z \to 0^+ \\ z \in S_{\delta}}} f^{(p)}(z)$ for each $\delta < \gamma$.
Obviously, these maps are well-defined and continuous. In this context, the Borel-Ritt problem consists in finding conditions on $\gamma$  and $M$ which ensure that the maps  \eqref{borel-mapping} and \eqref{borel-mapping-nu} are surjective.  If this is the case, it is natural to ask in addition whether these maps have a continuous linear right inverse.

In 2003 Thilliez showed the following fundamental result; see Section \ref{sect-prelim} for unexplained notions about weight sequences.
\begin{theorem} \label{theorem-T} \cite[Corollary 3.4.1]{Thilliez} Let $M$ be a strongly regular weight sequence and let $0 < \gamma < \gamma(M)$. Then, the map \eqref{borel-mapping} is surjective.
\end{theorem}
Schmets and Valdivia \cite{S-V}  already proved in 2000 the surjectivity of the map \eqref{borel-mapping} under the following more restrictive condition: $\gamma < n < \gamma(M)$ for some $n \in \Z_+$. But, in contrast to Thilliez, they also showed that the map \eqref{borel-mapping} has a continuous linear right inverse. Theorem \ref{theorem-T} is  believed to be optimal in the sense that the condition $\gamma < \gamma(M)$ is also necessary for the map \eqref{borel-mapping} to be surjective. However, for general strongly regular weight sequences, it is only known that the surjectivity of the map \eqref{borel-mapping} implies $\gamma \leq \gamma(M)$ (cf.\ \cite{JG-S-S})\footnote{This result is shown in \cite[Corollary 4.19]{JG-S-S} in the Roumieu case but, as we shall explain in Section \ref{sect-main}, the proof provided there can be adapted to the Beurling case.}.

In this article we give a complete solution to the Borel-Ritt problem in the non-uniform spaces $\mathscr{A}^-_{(M)}(S_\gamma)$. More precisely, we show the following result.
\begin{theorem} \label{intro}  Let $M$ be a strongly regular weight sequence and let $\gamma > 0$. Then, the following statements are equivalent:
\begin{itemize}
\item[$(i)$] $\gamma \leq \gamma(M)$.
\item[$(ii)$] $B: \mathscr{A}^-_{(M)}(S_\gamma) \rightarrow \Lambda_{(M)}(\N)$ is surjective.
\item[$(iii)$] $B: \mathscr{A}^-_{(M)}(S_\gamma) \rightarrow \Lambda_{(M)}(\N)$ has a continuous linear right inverse.
\end{itemize}
\end{theorem}
The implication $(ii) \Rightarrow (i)$ follows  from the analogous statement for the uniform spaces $\mathscr{A}_{(M)}(S_\gamma)$ mentioned above, whereas the implication $(i) \Rightarrow (ii)$ for $\gamma < \gamma(M)$ is a consequence of Theorem \ref{theorem-T}. The main novelty of Theorem \ref{intro}  comprises of  the following two results: The map \eqref{borel-mapping-nu} is surjective in the limit case $\gamma = \gamma(M)$; the map \eqref{borel-mapping-nu} is surjective if and only if it has a continuous linear right inverse. Furthermore,
Theorem \ref{intro} yields the following corollary for the uniform spaces $\mathscr{A}_{(M)}(S_\gamma)$; it simultaneously improves the results of Thiliez \cite{Thilliez} (Theorem \ref{theorem-T})  and Schmets and Valdivia \cite{S-V} mentioned above.
\begin{corollary}Let $M$ be a strongly regular weight sequence and let $0 < \gamma < \gamma(M)$. Then, the map \eqref{borel-mapping} has a continuous linear right inverse.
\end{corollary}
Our proof of Theorem \ref{intro} is based on two classical results from the general theory of Fr\'echet spaces, namely, the abstract Mittag-Leffler Lemma \cite{Wengenroth} and the $(DN)$-$(\Omega)$ splitting theorem \cite{Vogt}.  We explain these results in Section \ref{sect-aux1}. Moreover, we make use of certain linear topological properties of spaces of holomorphic functions with rapid decay on strips, recently obtained by the author in  \cite{Debrouwere}. These spaces and their properties are reviewed in Section \ref{sect-aux2}. We also show there how these spaces  are connected to the Borel-Ritt problem for Beurling ultraholomorphic classes. Finally, the proof of our main result Theorem \ref{intro}  is given in Section \ref{sect-main}. We would like  to point out that the technical core of the proof of Theorem \ref{intro} is contained in the results we use from \cite{Debrouwere}.

\section{Beurling ultraholomorphic classes}\label{sect-prelim}
In this preliminary section we introduce various spaces of ultraholomorphic functions of Beurling type that are used throughout this paper. Furthermore, we state our main result.

We start by discussing weight sequences. A sequence $M = (M_{p})_{p \in \N}$ of positive real  numbers is called a \emph{weight sequence} if $\lim_{p \to \infty} M_{p}^{1/p} =  \infty$. We set $m_p = M_p/M_{p-1}$ for $p \in \Z_+$.
We consider the following conditions on a weight sequence $M$:
\begin{itemize}
\item[$(M.1)$] \emph{(log-convexity)}  $M^2_{p} \leq M_{p-1}M_{p+1}$, $p \in \Z_+$.
\item[$(M.2)$]  \emph{(moderate growth)} $M_{p+q} \leq CL^{p+q} M_{p} M_{q}$, $p,q\in \N$, for some $C,L>0$.
\item[$(M.3)$]   \emph{(strong non-quasianalyticity)}$\displaystyle  \sum_{q = p+1}^\infty \frac{1}{m_q} \leq  \frac{Cp}{m_{p+1}}$, $p \in \Z_+$, for some $C > 0$.
\end{itemize}
We refer to  \cite{Komatsu} for the meaning of these conditions. Following Thilliez \cite{Thilliez}, we call a weight sequence $M$ \emph{strongly regular} if $M$ satisfies $(M.1)$ and $(M.2)$, and $(p!M_p)_{p \in \N}$ satisfies $(M.3)$.  The most important examples of strongly regular weight sequences are the \emph{Gevrey sequences} $p!^{\alpha}$, $\alpha>0$.

Let $M$ be a weight sequence and let $\gamma > 0$. We say that $M$ satisfies property $(P_\gamma)$ if there exists a sequence $(n_p)_{p \in \Z_+}$ of positive real numbers such that $(n_p/p^\gamma)_{p \in \Z_+}$ is non-decreasing and there is $C > 0$ such that $C^{-1}n_p \leq m_p \leq Cn_p$ for all $p \in \Z_+$. We define the \emph{growth index} $\gamma(M)$ of $M$ as \cite[Definition 1.3.5]{Thilliez}
$$
\gamma(M) := \sup \{\gamma > 0 \, | \, M \mbox{ satisfies } (P_\gamma) \}.
$$
If $M$ is strongly regular, then $0 <\gamma(M) < \infty $ \cite[Lemma 1.3.2]{Thilliez}.

The \emph{associated function} of a weight sequence $M$ is defined as
$$
\omega_M(t) := \sup_{p\in\N}\log \frac{t^pM_0}{M_p},\qquad t \geq 0.
$$
Suppose that $M$ satisfies $(M.1)$. Then, $M$ satisfies $(M.2)$ if and only if  \cite[Proposition 3.6]{Komatsu}
\begin{equation}
2\omega_M(t) \leq \omega_M(Lt) +  C, \qquad t \geq 0,
\label{assM2}
\end{equation}
for some $C,L >0$, whereas $\gamma(M) > 0$ implies that \cite[Corollary 2.14 and Corollary 4.6$(i)$]{JG-S-S-ind}
\begin{equation}
\omega_M(2t) \leq L\omega_M(t) + C, \qquad t \geq 0,
\label{assgamma}
\end{equation}
for some $C,L > 0$.

The relation $M \prec N$ between two weight sequences means that  for all $L > 0$ there is $C > 0$ such that
$M_p\leq CL^{-p}N_{p}$ for all $p\in\N$. By \cite[Lemma 3.10]{Komatsu}, we have that $M \prec N$ if and only if for all
$$
\omega_N(Lt) \leq \omega_M(t) +  C, \qquad t \geq 0,
$$
for all $L > 0$ and suitable $C = C_L > 0$.

Let $M$ be a weight sequence and let $s > 0$. We define $M^s = (M^s_p)_{p \in \N}$. Note that $\omega_{M^s}(t) = s \omega_M(t^{1/s})$ for $t \geq 0$ and $\gamma(M^s) = s \gamma(M)$. Furthermore, $M^s$ is strongly regular if $M$ is so \cite[Lemma 1.3.4]{Thilliez}.

We  need the following characterization of $\gamma(M)$  \cite{JG-S-S-ind}.
\begin{lemma} \label{gammaM3}
Let $M$ be a strongly regular weight sequence  and let $\gamma > 0$. Then, the following statements are equivalent:
\begin{itemize}
\item[$(i)$] $\gamma < \gamma(M)$.
\item[$(ii)$] $M^{1/\gamma}$ satisfies $(M.3)$.
\item[$(iii)$] There are $C,L > 0$ such that
$$
\int_1^\infty \frac{\omega_M(ts)}{s^{1+ (1/\gamma)}} ds \leq L \omega_M(t) + C, \qquad t \geq 0.
$$
\end{itemize}
\end{lemma}
\begin{proof}
$(i) \Leftrightarrow (ii)$ This is shown in \cite[Corollary 3.12$(iii)$]{JG-S-S-ind}.

$(i) \Leftrightarrow (iii)$ This follows from \cite[Theorem 2.11 and Corollary 4.6$(iii)$]{JG-S-S-ind}.
\end{proof}
Next, we introduce various spaces of ultraholomorphic functions of Beurling type. We denote by $\Sigma$ the Riemann surface of the logarithm. Given an open subset $\Omega$ of $\Sigma$ (or  of the complex plane $\C$), we write $\mathscr{O}(\Omega)$ for the space of holomorphic functions in $\Omega$.  We define the sectors
$$
S_\gamma := \{ z \in \Sigma \, | \, | \operatorname{Arg} z | < \frac{\gamma \pi}{2} \}, \qquad \gamma > 0.
$$

Let $M$ be a weight sequence and let $\gamma >0$. We define $\mathscr{A}_{(M)}(S_\gamma)$ as the Fr\'echet space consisting of all $f \in \mathscr{O}(S_\gamma)$ such that
$$
\sup_{p \in \N} \sup_{z \in S_\gamma} \frac{n^p |f^{(p)}(z)|}{p!M_p} < \infty, \qquad \forall n \in \N,
$$
and $\mathscr{S}_{(M)}(S_\gamma)$ as the Fr\'echet space consisting of all $f \in \mathscr{O}(S_\gamma)$ such that
$$
\sup_{p \in \N} \sup_{z \in S_\gamma} \frac{n^p |f^{(p)}(z)|e^{\omega_M(n|z|)}}{p!M_p} < \infty, \qquad \forall n \in \N.
$$
We are mainly interested in the following non-uniform spaces
$$
\mathscr{A}^{-}_{(M)}(S_\gamma) := \varprojlim_{\delta \rightarrow \gamma^-} \mathscr{A}_{(M)}(S_\delta), \qquad \mathscr{S}^{-}_{(M)}(S_\gamma) := \varprojlim_{\delta \rightarrow \gamma^-} \mathscr{S}_{(M)}(S_\delta).
$$
Note that $\mathscr{S}_{(M)}(S_\gamma) \subset \mathscr{A}_{(M)}(S_\gamma)$ and $\mathscr{S}^{-}_{(M)}(S_\gamma) \subset \mathscr{A}^{-}_{(M)}(S_\gamma)$  continuously.
We define $\Lambda_{(M)}(\N)$ as the Fr\'echet space consisting of all sequences $(c_p)_{p \in \N} \in \C^\N$ such that
$$
 \sup_{p \in \N} \frac{n^p |c_p|}{p!M_p} < \infty, \qquad \forall n \in \N.
$$
The \emph{asymptotic Borel map} is defined as
$$
B: \mathscr{A}^{-}_{(M)}(S_\gamma) \rightarrow \Lambda_{(M)}(\N), \, f \mapsto  (f^{(p)}(0))_{p \in \N},
$$
where
$$
f^{(p)}(0) := \lim_{\substack{z \to 0^+ \\ z \in S_{\delta}}} f^{(p)}(z) \in \C, \qquad p \in \N,
$$
exist for each $\delta < \gamma$. Obviously, this map is well-defined and continuous. We  also use the notation $B$ for the restriction of the asymptotic Borel map to various subspaces of $\mathscr{A}^{-}_{(M)}(S_\gamma)$. To avoid confusion, we always clearly state the domain and range of these maps.

 We are ready to formulate the main result of this article.

\begin{theorem}\label{main}
Let $M$ be a strongly regular weight sequence and let $\gamma > 0$. The following statements are equivalent:
\begin{itemize}
\item[$(i)$] $\gamma \leq \gamma(M)$.
\item[$(ii)$] $B:\mathscr{A}^{-}_{(M)}(S_\gamma) \rightarrow \Lambda_{(M)}(\N)$ is surjective.
\item[$(iii)$] $B: \mathscr{A}^{-}_{(M)}(S_\gamma) \rightarrow \Lambda_{(M)}(\N)$ has a continuous linear right inverse.
 \item[$(iv)$] $B: \mathscr{S}^{-}_{(M)}(S_\gamma) \rightarrow \Lambda_{(M)}(\N)$ is surjective.
\item[$(v)$] $B: \mathscr{S}^{-}_{(M)}(S_\gamma) \rightarrow \Lambda_{(M)}(\N)$ has a continuous linear right inverse.
\end{itemize}
\end{theorem}
 \begin{remark}
 We are mainly interested in the equivalences of $(i)$--$(iii)$ in Theorem \ref{main} (cf.\ Theorem \ref{intro}). However,  the spaces $\mathscr{S}^{-}_{(M)}(S_\gamma)$ play an essential role in our proof of Theorem \ref{main}; see  Remark \ref{explanation} below for more details.
 \end{remark}
  The proof of Theorem \ref{main} is given in Section \ref{sect-main}.  In preparation for it, we  present various auxiliary results in the next two sections.

\section{Two functional analytic tools}\label{sect-aux1}
\subsection{The Mittag-Leffler procedure}\label{sect-ML} We explain the abstract Mittag-Leffler lemma for Fr\'echet spaces and state it in terms of the derived projective limit functor; we refer to the book \cite{Wengenroth} for more information on this subject.

A \emph{projective spectrum}  $\mathscr{X} = (X_n, \varrho_{n+1}^n)_{n \in \N}$ consists of vector spaces $X_n$  and linear
maps $\varrho^n_{n+1}: X_{n+1} \rightarrow X_n$ (called the spectral maps). We define $\varrho^n_n  = \operatorname{id}_{X_n}$ and $\varrho^n_{m} = \varrho_{n+1}^n \circ \cdots \circ \varrho^{m-1}_{m} : X_m \rightarrow X_n$ for $n,m \in \N$ with $m > n$. Given two projective spectra $\mathscr{X} = (X_n, \varrho_{n+1}^n)_{n \in \N}$  and $\mathscr{Y} = (Y_n, \sigma_{n+1}^n)_{n \in \N}$, a \emph{morphism} $T = (T_n)_{n \in \N}: \mathscr{X} \rightarrow \mathscr{Y}
$ consists of linear maps $T_n : X_n \rightarrow Y_n$ such that $T_n \circ \varrho_{n+1}^n = \sigma_{n+1}^n \circ T_{n+1}$ for all $n \in \N$. We set
$$
\operatorname{Proj} \mathscr{X} = \{(x_n)_{n \in \N} \in \prod_{n \in \N} X_n \, | \,  x_n = \varrho^n_{n+1}(x_{n+1}),  \forall n \in \N\}
$$
and
$$
\operatorname{Proj} T : \operatorname{Proj} \mathscr{X} \rightarrow \operatorname{Proj} \mathscr{Y}, \, (x_n)_{n \in \N} \mapsto (T_n(x_n))_{n \in \N}.
$$
For each $n \in \N$ we also write $\varrho^n : \operatorname{Proj} \mathscr{X} \rightarrow X_n, \, (x_j)_{j \in \N} \mapsto x_n$. Furthermore, we define
$$
\operatorname{Proj}^1 \mathscr{X} =   \prod_{n \in \N}X_n / B(\mathscr{X}),
$$
where
$$B(\mathscr{X}) = \{(x_n)_{n \in \N} \in \prod_{n \in \N} X_n \, | \, \exists \, (u_n)_{n \in \N} \in \prod_{n \in \N} X_n \, : \,  x_n = u_n - \varrho^n_{n+1}(u_{n+1}), \forall n \in \N\}.
$$

Let
 \begin{center}
\begin{tikzpicture}
  \matrix (m) [matrix of math nodes, row sep=2em, column sep=2.5em]
    {0 & \mathscr{X} & \mathscr{Y} & \mathscr{Z} & 0 \\
   };
  { [start chain] \chainin (m-1-1);
\chainin (m-1-2);
\chainin (m-1-3) [join={node[above,labeled] {S}}];
\chainin (m-1-4)[join={node[above,labeled] {T}}];
\chainin (m-1-5)[join={node[above,labeled] {}}];
}
\end{tikzpicture}
\end{center}
be an exact sequence of projective spectra, i.e., a commutative diagram
\begin{center}
\begin{tikzpicture}
  \matrix (m) [matrix of math nodes, row sep=2em, column sep=2.5em]
    {
        0 &  X_1 & Y_1  & Z_1 & 0 \\
        0 &  X_{2} & Y_{2}  & Z_{2} & 0 \\
         & \vdots & \vdots  & \vdots &  \\
          };

  { [start chain] \chainin (m-1-1);
  \chainin (m-1-2);
\chainin (m-1-3)[join={node[above,labeled] {S_1}}];
\chainin (m-1-4)[join={node[above,labeled] {T_1}}];
\chainin (m-1-5);
; }
  { [start chain] \chainin (m-2-1);
  \chainin (m-2-2);
\chainin (m-2-3)[join={node[above,labeled] {S_{2}}}];
\chainin (m-2-4) [join={node[above,labeled] {T_{2}}}];
\chainin (m-2-5);
; }
  { [start chain] \chainin (m-3-2);
  \chainin (m-2-2);
    \chainin (m-1-2);

; }
  { [start chain] \chainin (m-3-3);
  \chainin (m-2-3);
    \chainin (m-1-3);

; }
  { [start chain] \chainin (m-3-4);
  \chainin (m-2-4);
    \chainin (m-1-4);

; }

\end{tikzpicture}
\end{center}
where the vertical arrows stand for the spectral maps and each row is a short exact sequence of vector spaces. If $\operatorname{Proj}^1 \mathscr{X} = 0$, then
\begin{center}
\begin{tikzpicture}
  \matrix (m) [matrix of math nodes, row sep=2em, column sep=2.5em]
    {0 & \operatorname{Proj}\mathscr{X} & \operatorname{Proj}\mathscr{Y} & \operatorname{Proj}\mathscr{Z} & 0 \\
   };
  { [start chain] \chainin (m-1-1);
\chainin (m-1-2);
\chainin (m-1-3) [join={node[above,labeled] {\operatorname{Proj} T}}];
\chainin (m-1-4)[join={node[above,labeled] {\operatorname{Proj} S }}];
\chainin (m-1-5)[join={node[above,labeled] {}}];
}

\end{tikzpicture}
\end{center}is again exact \cite[Cor.\ 3.1.5]{Wengenroth}.

Two projective spectra  $\mathscr{X} = (X_n, \varrho_{n+1}^n)_{n \in \N}$ and  $\mathscr{Y} = (Y_n, \sigma_{n+1}^n)_{n \in \N}$ are said to be \emph{equivalent} if there are increasing sequences $(k_n)_{n \in \N}$ and $(l_n)_{n \in \N}$ of natural numbers with $n \leq l_n \leq k_n \leq l_{n+1}$ and  linear maps $T_n : X_{k_n} \rightarrow Y_{l_n}$ and $S_n : Y_{l_{n+1}} \rightarrow X_{k_n}$ such that
$S_n \circ T_{n+1} = \varrho^{k_n}_{k_{n+1}}$ and $T_n \circ S_{n} = \sigma^{l_n}_{l_{n+1}}$ for all $n \in \N$. In such a case, we have that  $\operatorname{Proj}^1 \mathscr{X} \cong \operatorname{Proj}^1 \mathscr{Y}$ \cite[Proposition 3.1.7]{Wengenroth}.

 A \emph{projective spectrum of Fr\'echet spaces}  $\mathscr{X} = (X_n, \varrho_{n+1}^n)_{n \in \N}$ is a projective spectrum consisting of Fr\'echet spaces $X_n$  and continuous
 linear spectral maps $\varrho^n_{n+1}: X_{n+1} \rightarrow X_n$. Given a Fr\'echet space $X$, we denote by $\mathscr{U}_0(X)$ its collection of neighbourhoods of $0$.
 For projective spectra of Fr\'echet spaces  the vanishing of the derived projective limit functor may be characterized as follows (this result is known as the abstract Mittag-Leffler lemma).
\begin{theorem}\label{M-L} \cite[Theorem 3.2.8]{Wengenroth} Let $\mathscr{X} = (X_n, \varrho_{n+1}^n)_{n \in \N}$ be a projective spectrum of Fr\'echet spaces. Then, the following statements are equivalent:
\begin{itemize}
\item[$(i)$] $\operatorname{Proj}^1 \mathscr{X} = 0$.
\item[$(ii)$] $\forall n \in \N, \, U \in \mathscr{U}_0(X_n) \, \exists m \geq n \, \forall k \geq m \, : \, \varrho^n_m(X_m) \subset \varrho^n_k(X_k) + U$.
\item[$(iii)$] $\forall n \in \N, \, U \in \mathscr{U}_0(X_n) \, \exists m \geq n  \, : \, \varrho^n_m(X_m) \subset \varrho^n(\operatorname{Proj} \mathscr{X}) + U$.
\end{itemize}
\end{theorem}
In what follows we shall not write explicitly the spectral maps of projective spectra as they will be clear from the context.
\subsection{A splitting theorem} We present a particular version of the $(DN)$-$(\Omega)$ splitting theorem for Fr\'echet spaces; see \cite{Vogt} for more information.

A Fr\'echet space $X$ with a fundamental increasing system $(U_n)_{n \in \N}$ of neighborhoods of zero is said to satisfy property $(\Omega)$ if
$$
\forall n \in \N \, \exists m \geq n  \, \forall k \geq m  \, \exists C,L > 0 \, \forall \varepsilon > 0 \, : U_m \subset  C\varepsilon^{-L}U_k + \varepsilon U_n.
$$
Let $\alpha = (\alpha_j)_{j \in \N}$ be an increasing sequence of positive real numbers with $\lim_{n \to \infty}\alpha_n =  \infty$.  We define the \emph{power series space $\Lambda_\infty(\alpha)$ of infinite type} as the Fr\'echet space consisting of all $(c_j)_{j \in \N} \in \C^\N$ such that
$$
\sum_{j = 0}^\infty |c_j| e^{n \alpha_j} < \infty, \qquad \forall n \in \N.
$$
We then have:
\begin{theorem} \cite[Theorem 5.1]{Vogt}\label{splitting} Let
\begin{center}
\begin{tikzpicture}
  \matrix (m) [matrix of math nodes, row sep=2em, column sep=2.5em]
    {0 & X & Y & Z & 0 \\
   };
  { [start chain] \chainin (m-1-1);
\chainin (m-1-2);
\chainin (m-1-3) [join={node[above,labeled]{ S}}];
\chainin (m-1-4)[join={node[above,labeled] { T}}];
\chainin (m-1-5)[join={node[above,labeled] {}}];
}
\end{tikzpicture}
\end{center}
be a short exact sequence of Fr\'echet spaces with continuous linear maps. If $X$ satisfies $(\Omega)$ and $Z$ is isomorphic to a power series space of infinite type, then $T$ has a continuous linear right inverse.
\end{theorem}

\section{Spaces of holomorphic functions with rapid decay on strips}\label{sect-aux2}
In this section we introduce a class of weighted Fr\'echet spaces of  holomorphic functions with rapid decay on strips and give two properties of these spaces, which were essentially shown in \cite{Debrouwere}. Moreover,  we establish a connection between these spaces and the spaces $\mathscr{S}^{-}_{(M)}(S_\gamma)$. We point out that the notation employed here is different from the one used in \cite{Debrouwere}.

A continuous increasing function $\nu: [0,\infty) \rightarrow [0,\infty)$ is called a \emph{weight function} if $\nu(0) = 0$ and $\log t = o(\nu(t))$.
Let $h > 0$. A weight function $\nu$ is said to satisfy condition $(Q_h)$ if there are $C,L >0$ such that
$$
\int^\infty_0\nu(t+s)  e^{-s/ h }ds \leq L\nu(t) + C, \qquad t \geq 0.
$$
We set
$$
h(\nu) := \sup \{ h > 0 \, | \, \  \nu \mbox{ satisfies } (Q_h) \}.
$$

We define the horizontal strips
$$
V_h := \{ z \in \C \, | \,  | \operatorname{Im} z | < \frac{ h \pi}{2} \}, \qquad h > 0.
$$
 Let $\nu$ be a weight function. For $h > 0$ we define $\mathscr{U}_{(\nu)}(V_h)$ as the Fr\'echet space consisting of all $f \in \mathscr{O}(V_h)$  such that
$$
\sup_{z \in V_h} |f(z)|e^{n\nu(| \operatorname{Re} z |)} < \infty, \qquad \forall n \in \N.
$$
The corresponding non-uniform spaces are defined as
$$
\mathscr{U}^{-}_{(\nu)}(V_h) := \varprojlim_{k \to h^{-}}\mathscr{U}_{(\nu)}(V_k).
$$
We  need the following two facts about these spaces.
\begin{theorem}\label{thmstrips}
Let $\nu$ be a weight function and let $0 <h \leq h(\nu)$. Then,
\begin{itemize}
\item[$(i)$] Let $(h_n)_{n \in \N}$ be an increasing sequence of positive real numbers with $h_n < h$ such that $\lim_{n \to \infty} h_n = h$. Consider the projective spectrum $\mathscr{X} = ( \mathscr{U}_{(\nu)}(V_{h_n}))_{n \in \N}$ (with restriction as spectral maps). Then, $\operatorname{Proj}^1 \mathscr{X} = 0$.
\item[$(ii)$] $\mathscr{U}^-_{(\nu)}(V_{h})$ satisfies $(\Omega)$.
\end{itemize}
\end{theorem}
\begin{proof}
$(i)$ We show that $\mathscr{X}$ satisfies condition $(iii)$ from Theorem \ref{M-L}.  For $l > 0$ and $n \in \N$ we define $\mathscr{U}_{\nu,n}(V_l)$ as the Banach space consisting of all $f \in \mathscr{O}(V_l)$  such that
$$
\sup_{z \in V_l} |f(z)|e^{n\nu(| \operatorname{Re} z |)} < \infty.
$$
Consider the projective spectrum $\mathscr{Y} = ( \mathscr{U}_{\nu,n}(V_{h_n}))_{n \in \N}$ (with restriction as spectral maps). Note that $\operatorname{Proj} \mathscr{Y} =  \operatorname{Proj} \mathscr{X} = \mathscr{U}^-_{(\nu)}(V_{h})$. The proof of \cite[Proposition 3.1]{Debrouwere} (see the statement at the beginning of this proof) implies  that $\mathscr{Y}$ satisfies condition $(ii)$ from Theorem \ref{M-L}. Hence,  $\mathscr{Y}$ also satisfies condition $(iii)$ from this result, i.e.,
$$
\forall n \in \N, \, U \in \mathscr{U}_0(\mathscr{U}_{\nu,n}(V_{h_n})) \, \exists m \geq n \, : \, \mathscr{U}_{\nu,m}(V_{h_m}) \subset \mathscr{U}^-_{(\nu)}(V_{h}) + U.
$$
This implies that
$$
\forall n \in \N, \, U \in \mathscr{U}_0(\mathscr{U}_{(\nu)}(V_{h_n})) \, \exists m \geq n \, : \, \mathscr{U}_{(\nu)}(V_{h_m}) \subset \mathscr{U}^-_{(\nu)}(V_{h}) + U,
$$
which means that  $\mathscr{X}$ satisfies condition $(iii)$ from Theorem \ref{M-L}.

$(ii)$ This is shown in \cite[Proposition 3.1]{Debrouwere}.
\end{proof}
Next, we show how the spaces $\mathscr{S}^-_{(M)}(S_\gamma)$ and $\mathscr{U}^-_{(\nu)}(V_h)$ are related to each other. Let $M$ be  a weight sequence and let $\gamma > 0$. We define $\mathscr{S}^{0}_{(M)}(S_\gamma)$ and $\mathscr{S}^{-,0}_{(M)}(S_\gamma)$ as the kernel of the map $B: \mathscr{S}_{(M)}(S_\gamma) \rightarrow  \Lambda_{(M)}(\N)$ and $B: \mathscr{S}^-_{(M)}(S_\gamma) \rightarrow  \Lambda_{(M)}(\N)$, respectively.
\begin{lemma}\label{kernel}
Let $M$ be a strongly regular weight sequence and let $\gamma > 0$. Then, $f \in \mathscr{O}(S_\gamma)$ belongs to  $\mathscr{S}^{-,0}_{(M)}(S_\gamma)$ if and only if
$$
\| f \|_{n,\delta} = \sup_{z \in S_\delta} |f(z)| e^{n(\omega_M(1/|z|) + \omega_M(|z|))} < \infty, \qquad \forall n \in \N, \delta < \gamma.
$$
Moreover, the topology of $\mathscr{S}^{-,0}_{(M)}(S_\gamma)$ is generated by the family of seminorms $\{ \| \, \cdot \, \|_{n,\delta} \, | \, n \in \N, \delta < \gamma \}$.
\end{lemma}
\begin{proof}
Taylor's theorem and the Cauchy estimates imply that  $f \in \mathscr{O}(S_\gamma)$ belongs to  $\mathscr{S}^{-,0}_{(M)}(S_\gamma)$ if and only if
$$
| f |_{n, \delta} = \sup_{z \in S_\delta} |f(z)| e^{\omega_M(n/|z|) + \omega_M(n|z|)} < \infty, \qquad \forall n \in \N, \delta < \gamma,
$$
and that the topology of $\mathscr{S}^{-,0}_{(M)}(S_\gamma)$ is generated by the family of seminorms $\{ | \, \cdot \, |_{n,\delta} \, | \, n \in \N, \delta < \gamma \}$.
The result therefore follows from \eqref{assM2} and \eqref{assgamma}.
\end{proof}
Consider the biholomorphic map $\operatorname{Log}: \Sigma \rightarrow \C$ and note that $\operatorname{Log}(S_\gamma) =  V_\gamma$ for all $\gamma > 0$.
\begin{lemma}\label{equivalence}
Let $M$ be a strongly regular weight sequence  and set $\nu_M(t) = \omega_M(e^t)$ for $t \geq 0$. Then,
\begin{itemize}
\item[$(i)$] $\gamma(M) = h(\nu_M)$.
\item[$(ii)$] For each $\gamma > 0$ the map
$$
\mathscr{U}^-_{(\nu_M)}(V_\gamma) \rightarrow \mathscr{S}^{-,0}_{(M)}(S_\gamma), \, f \mapsto f \circ \operatorname{Log}
$$
is a topological isomorphism.
\end{itemize}
\end{lemma}
\begin{proof}$(i)$ This follows from Lemma \ref{gammaM3}.

$(ii)$ This follows from Lemma \ref{kernel}.
\end{proof}
Lemma \ref{equivalence} enables us to give an analogue of Theorem \ref{thmstrips} for the spaces $\mathscr{S}^{0}_{(M)}(S_\gamma)$ and $\mathscr{S}^{-,0}_{(M)}(S_\gamma)$.
\begin{corollary}\label{proj-omega}
Let $M$ be a strongly regular weight sequence and let $0 < \gamma \leq \gamma(M)$. Then,
\begin{itemize}
\item[$(i)$] Let $(\gamma_n)_{n \in \N}$ be an increasing sequence of positive real numbers with $\gamma_n < \gamma$ such that $\lim_{n \to \infty} \gamma_n = \gamma$. Consider the projective spectrum $\mathscr{X} = (\mathscr{S}^{0}_{(M)}(S_{\gamma_n}))_{n \in \N}$. Then, $\operatorname{Proj}^1 \mathscr{X} = 0$.
\item[$(ii)$] $\mathscr{S}^{-,0}_{(M)}(S_\gamma)$ satisfies $(\Omega)$.
\end{itemize}
\end{corollary}
\begin{proof}
$(i)$ Consider the projective spectra
$$\mathscr{Y} = (\mathscr{S}^{-,0}_{(M)}(S_{\gamma_n}))_{n \in \N}, \qquad  \mathscr{Z} = (\mathscr{U}^{0}_{(\nu_M)}(V_{\gamma_n}))_{n \in \N}.
$$
 It is clear that $\mathscr{X}$ and $\mathscr{Y}$ are equivalent, while Lemma \ref{equivalence}$(ii)$ implies that $\mathscr{Y}$ and $\mathscr{Z}$ are equivalent. Hence, by Theorem \ref{thmstrips}$(i)$ and Lemma \ref{equivalence}$(i)$, $\operatorname{Proj}^1\mathscr{X} \cong \operatorname{Proj}^1\mathscr{Y} \cong \operatorname{Proj}^1\mathscr{Z} = 0$ (cf.\ Subsection \ref{sect-ML}).

$(ii)$ This follows from Theorem \ref{thmstrips}$(ii)$ and Lemma \ref{equivalence}.
\end{proof}

\begin{remark}\label{explanation}
Corollary \ref{proj-omega} is the reason why we also consider the spaces $\mathscr{S}^{-}_{(M)}(S_\gamma)$ in Theorem \ref{main}. We do not know whether a result similar to Corollary \ref{proj-omega} holds for the spaces $\mathscr{A}^{-}_{(M)}(S_\gamma)$.
\end{remark}
\section{Proof of the main result}\label{sect-main}
This section is devoted to the proof of Theorem \ref{main}. We start by introducing an auxiliary space of ultraholomorphic functions. Let $M$ be a weight sequence and let $\gamma > 0$. A function $f \in \mathscr{O}(S_\gamma)$ is said to admit the formal power series $\sum_{p= 0}^\infty a_p z^p$, $a_p \in \C$, as its \emph{uniform (M)-asymptotic expansion in $S_\gamma$} if for all $L > 0$
$$
 \sup_{p \in \N} \sup_{z \in S_\gamma} \frac{L^p}{M_p|z|^p} \left | f(z) - \sum_{q=0}^{p-1} a_q z^q \right| < \infty.
$$
We denote by $\widetilde{\mathscr{A}}_{(M)}(S_\gamma)$  the space consisting of all $f \in \mathscr{O}(S_\gamma)$ that admit a uniform $(M)$-asymptotic expansion  in $S_\gamma$.
Taylor's theorem and the Cauchy estimates imply that
$$
\mathscr{A}_{(M)}(S_\gamma) \subset \widetilde{\mathscr{A}}_{(M)}(S_\gamma) \subset \mathscr{A}^-_{(M)}(S_{\gamma})
$$
and that $a_p =f^{(p)}(0)/p!$, $p \in \N$, for all $f \in \widetilde{\mathscr{A}}_{(M)}(S_\gamma)$, where $\sum_{p= 0}^\infty a_p z^p$ is a uniform (M)-asymptotic expansion of $f$ in $S_\gamma$. In particular, the coefficients $a_p$ are unique.   Our interest in the spaces  $\widetilde{\mathscr{A}}_{(M)}(S_\gamma)$ stems from the following observation (cf.\ the proof of \cite[Theorem 4.17]{JG-S-S}).
\begin{lemma} \label{ramification}
Let $M$ be a strongly regular weight sequence, let $n \in \Z_+$, and let $\gamma > 0$. Suppose that $g \in \widetilde{\mathscr{A}}_{(M^{1/n})}(S_{\gamma/n})$ is such that $g^{(p)}(0) = 0$ for all $p \in \N \backslash n \N$. Define $f(z) = g(z^{1/n})$ for $z \in S_\gamma$. Then, $f \in \widetilde{\mathscr{A}}_{(M)}(S_{\gamma})$ and $f^{(p)}(0)/p! = g^{(pn)}(0)/(pn)!$ for all $p \in \N$.
\end{lemma}
\begin{proof}
It is clear that $f \in \mathscr{O}(S_\gamma)$. The assumptions on $g$ imply that  \begin{align*}
&\sup_{z \in S_\gamma}\left | f(z) - \sum_{q=0}^{p-1}  \frac{g^{(qn)}(0)}{(qn)!} z^q \right| =  \sup_{w \in S_{\gamma/n}}\left | g(w) - \sum_{q=0}^{p-1}  \frac{g^{(qn)}(0)}{(qn)!} z^{nq} \right| \\
&=  \sup_{w \in S_{\gamma/n}}\left | g(w) - \sum_{j=0}^{pn-1}  \frac{g^{(j)}(0)}{(j)!} z^{j} \right| \leq CL^{-p}M^{1/n}_{pn}, \qquad p \in \N,
\end{align*}
for all $L > 0$ and suitable $C = C_L >0$.  The result now follows from the fact that, by $(M.2)$,
$$
M^{1/n}_{pn} \leq CH^pM_p, \qquad p \in \N,
$$
for some $C,H > 0$.
\end{proof}

We need the following improvement of  Theorem \ref{theorem-T}.
\begin{proposition}\label{surjective-S}
Let $M$ be a strongly regular weight sequence and let $0 < \gamma < \gamma(M)$. Then, $B: \mathscr{S}_{(M)}(S_\gamma) \rightarrow \Lambda_{(M)}(\N)$ is surjective.
\end{proposition}

\begin{proof}
We distinguish two cases (cf.\ the proof of \cite[Theorem 3.2.1]{Thilliez}).

CASE I: $\gamma < 2$. In this case, we may view $S_\gamma$ as a subset of the complex plane $\C$. Choose $\gamma < \delta < \gamma(M)$ and set $N = M^{\delta/\gamma(M)}$. Then, $\gamma(N) = \delta$ and $N \prec M$. By Proposition \cite[Theorem 2.3.1]{Thilliez} (applied to $N$), there exists $G \in \mathscr{O}(S_\gamma)$ such that $G$ does not vanish on $S_\gamma$ and
$$
|G(z)| \leq e^{-\omega_N(\kappa/|z|)}, \qquad z \in S_\gamma,
$$
for some $\kappa > 0$. Since $N \prec M$, we have that
$$
|G(z)| \leq Ce^{-\omega_M(L/|z|)}, \qquad z \in S_\gamma,
$$
for all $L > 0$ and suitable $C = C_L > 0$.  Set $R_\gamma = \{ z \in \C \, | \, z+1 \in S_\gamma \}$ and define $P(z) = G(1/(z+1))$ for $z \in R_\gamma$. Then, $P$ does not vanish on $R_\gamma$ and
\begin{equation}
|P(z)| \leq Ce^{-\omega_M(L|z|)}, \qquad z \in R_\gamma,
\label{bounds-H}
\end{equation}
for all $L > 0$ and suitable $C = C_L > 0$. Choose $\varepsilon > 0$ such that for  each $z \in S_\gamma$ the closed disc with center $z$ and radius $\varepsilon$ belongs to $R_\gamma$. The Cauchy estimates for $P$ and \eqref{bounds-H} imply that
\begin{equation}
|P^{(p)}(z)| \leq C\varepsilon^{-p}p! e^{-\omega_M(L|z|)}, \qquad p\in \N,z \in S_\gamma,
\label{bounds-Hder}
\end{equation}
for all $L > 0$ and suitable $C =C_L > 0$.
As $P$ does not vanish on $R_\gamma$, the Cauchy estimates for $1/P$ yield that
\begin{equation}
|(1/P)^{(p)}(0)| \leq C\varepsilon^{-p}p!, \qquad p\in \N,
\label{inverse}
\end{equation}
for some $C > 0$. Now let $a =(a_p)_{p \in \N} \in \Lambda_{(M)}(\N)$ be arbitrary. For $p \in \N$ we define
$$
b_p =  \sum_{j=0}^p \binom{p}{j} a_j (1/P)^{(p-j)}(0).
$$
Inequality \eqref{inverse} implies that $b= (b_p)_{p \in \N}   \in  \Lambda_{(M)}(\N)$. By  Theorem \ref{theorem-T}, there exists $g \in \mathscr{A}_{(M)}(S_\gamma)$ such that $B(g) = b$. Set $f = gP$ and note that, by \eqref{bounds-Hder}, $f \in \mathscr{S}_{(M)}(S_\gamma)$. Finally, it is clear that $B(f) = a$.

CASE II: $\gamma \geq 2$. Choose $\gamma < \delta < \gamma(M)$ and $n \in \Z_+$ such that $\delta/n < 2$. Let $a = (a_p)_{p \in \N} \in \Lambda_{(M)}(\N)$ be arbitrary.  Consider $b = (b_p)_{p \in \N} \in \C^\N$, where
	\[b_p = \begin{cases}
	\frac{(nj)!}{j!}a_j,&  p = nj \mbox{ for some $j \in \N$}, \\
	0, & \mbox{otherwise}.\\
	\end{cases}\]
Condition $(M.1)$ implies that $b \in 	 \Lambda_{(M^{1/n})}(\N)$. Since $\delta/n < 2$ and $\delta/n < \gamma(M)/n = \gamma(M^{1/n})$, CASE I (applied to $M^{1/n}$) implies that there exists $g \in \mathscr{S}_{(M^{1/n})}(S_{\delta/n})$ such that $B(g) = b$. Define $f(z) = g(z^{1/n})$ for $z \in S_{\delta} $. By Lemma \ref{ramification}, we have that $f \in \widetilde{\mathscr{A}}_{(M)}(S_{\delta})$ and $B(f) = a$. We now show that $f \in \mathscr{S}_{(M)}(S_{\gamma})$. We have that $f \in \widetilde{\mathscr{A}}_{(M)}(S_{\delta}) \subset \mathscr{A}^-_{(M)}(S_{\delta})$. Since $g \in \mathscr{S}_{(M^{1/n})}(S_{\delta/n})$, the equality $\omega_{M^{1/n}}(t) = \omega_M(t^n)/n$ for $t \geq 0$ and \eqref{assM2} yield that

$$
|f(z)| = |g(z^{1/n})| \leq Ce^{-\omega_M(L|z|)}, \qquad z \in S_\delta,
$$

for all $L > 0$ and suitable $C = C_L > 0$. Hence,  the Cauchy estimates imply that $f \in \mathscr{S}_{(M)}(S_{\gamma})$.

\end{proof}

The next result will be used to prove the implication $(ii) \Rightarrow (i)$ in Theorem \ref{main}.
\begin{proposition} \label{necc}
Let $n \in \Z_+$ and suppose that $B: \widetilde{\mathscr{A}}_{(M)}(S_n) \rightarrow \Lambda_{(M)}(\N)$ is surjective. Then, $\gamma(M) > n$.
\end{proposition}
Proposition \ref{necc} is essentially shown in \cite{S-V} (see also \cite[Theorem 4.14$(i)$]{JG-S-S} for the Roumieu variant of this result). We briefly indicate the main idea of the proof for the reader's convenience. Let $M$ be a weight sequence and let $n \in \Z_+$. We denote by $\mathscr{N}_{(M),n}([0,\infty))$ the space consisting of all $f \in C^\infty([0,\infty))$ such that $f^{(p)}(0) = 0$ for all $p \in \N \backslash n \N$ and
$$
\sup_{p \in \N} \sup_{x \geq 0} \frac{L^p|f^{(np)}(x)|}{p!M_p} < \infty
$$
for all $L > 0$. Note that  this definition differs from the one used in  \cite{S-V}.
\begin{proof}[Proof of Proposition \ref{necc}] We define $M^* = (M_p/p!)_{p \in \N}$. In \cite[Proposition 4.3]{S-V} it is shown that the surjectivity of the map
\begin{equation}
\mathscr{N}_{(M^*),n}([0,\infty))  \rightarrow \Lambda_{(M^*)}(\N), \, f \mapsto (f^{(np)}(0))_{p \in \N}
\label{borel}
\end{equation}
implies that  $M^{1/n}$ satisfies $(M.3)$. Hence, by Lemma \ref{gammaM3}, it suffices to show that  the map  \eqref{borel} is surjective if the map $B: \widetilde{\mathscr{A}}_{(M)}(S_n) \rightarrow \Lambda_{(M)}(\N)$ is so. In the Roumieu case, this is done in \cite[Theorem 4.14$(i)$]{JG-S-S} and the same proof applies to the Beurling case (see also the proof of \cite[Theorem 4.6]{S-V}). We point out that the asymptotic Borel map and the sequence space associated to $M$ employed here are different from the ones used in  \cite{JG-S-S}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{main}]
We  show the chain of implications $(i) \Rightarrow (iv) \Rightarrow (v) \Rightarrow (iii) \Rightarrow (ii) \Rightarrow (i)$. The implications $(v) \Rightarrow (iii)$ and $(iii) \Rightarrow (ii)$ are trivial. We now prove the three other ones.

$(i) \Rightarrow (iv)$ Let $(\gamma_n)_{n \in \N}$ be an increasing sequence of positive real numbers with $\gamma_n < \gamma$ such that $\lim_{n \to \infty} \gamma_n = \gamma$. We set $B_n = B:
\mathscr{S}_{(M)}(S_{\gamma_n}) \rightarrow \Lambda_{(M)}(\N)$ for $n \in \N$. Consider the projective spectra
$$
\mathscr{X} = (\mathscr{S}^{0}_{(M)}(S_{\gamma_n}))_{n \in \N}, \qquad \mathscr{Y} = (\mathscr{S}_{(M)}(S_{\gamma_n}))_{n \in \N}, \qquad \mathscr{Z} = (\Lambda_{(M)}(\N))_{n \in \N},
$$
and note that $(B_n)_{n \in \N} : \mathscr{Y} \rightarrow \mathscr{Z}$ is a morphism. We need to show that
$$
B = \operatorname{Proj} (B_n)_{n \in \N} : \mathscr{S}^{-}_{(M)}(S_{\gamma}) =  \operatorname{Proj} \mathscr{Y}  \rightarrow  \Lambda_{(M)}(\N) =  \operatorname{Proj} \mathscr{Z}
$$
is surjective. Proposition \ref{surjective-S} yields that
\begin{center}
\begin{tikzpicture}
  \matrix (m) [matrix of math nodes, row sep=2em, column sep=3.5em]
    {0 & \mathscr{X}  & \mathscr{Y}  & \mathscr{Z}  & 0 \\
   };
  { [start chain] \chainin (m-1-1);
\chainin (m-1-2);
\chainin (m-1-3) [join={node[above,labeled]{ }}];
\chainin (m-1-4)[join={node[above,labeled] { (B_n)_{n \in \N}}}];
\chainin (m-1-5)[join={node[above,labeled] {}}];
}
\end{tikzpicture}
\end{center}
is a short exact sequence of projective spectra. As explained in Subsection \ref{sect-ML}, the result therefore follows from the fact that $\operatorname{Proj}^1 \mathscr{X} = 0$ (Corollary \ref{proj-omega}$(i)$).

$(i) \Rightarrow (iv)$ The assumption yields that
\begin{center}
\begin{tikzpicture}
  \matrix (m) [matrix of math nodes, row sep=2em, column sep=2.5em]
    {0 & \mathscr{S}^{-,0}_{(M)}(S_{\gamma})   & \mathscr{S}^{-}_{(M)}(S_{\gamma})   & \Lambda_{(M)}(\N)  & 0 \\
   };
  { [start chain] \chainin (m-1-1);
\chainin (m-1-2);
\chainin (m-1-3) [join={node[above,labeled]{ }}];
\chainin (m-1-4)[join={node[above,labeled] { B}}];
\chainin (m-1-5)[join={node[above,labeled] {}}];
}
\end{tikzpicture}
\end{center}
is a short exact sequence of Fr\'echet spaces  with continuous linear maps. Since $\mathscr{S}^{-,0}_{(M)}(S_{\gamma})$ satisfies $(\Omega)$ (Corollary \ref{proj-omega}$(ii)$) and $\Lambda_{(M)}(\N) \cong \Lambda_\infty(j)$, the result is a consequence of Theorem \ref{splitting}.

$(ii) \Rightarrow (i)$ It suffices to show that for all $\delta \in \mathbb{Q}$ with $0 < \delta < \gamma$ it holds that $\delta < \gamma(M)$. Write $ \delta= n/m$ with $n,m \in \Z_+$.  We claim that $B :  \widetilde{\mathscr{A}}_{(M^m)}(S_{n}) \rightarrow  \Lambda_{(M^m)}(\N)$ is surjective. Then, by Proposition \ref{necc}, we would have that $\gamma(M^m) > n$ and, thus, $\gamma(M) > n/m =  \delta$. We now show the claim. Let $a = (a_p)_{p \in \N} \in \Lambda_{(M^m)}(\N)$ be arbitrary.  For $p \in \N$ we define
	\[b_p = \begin{cases}
	\frac{(mj)!}{j!}a_j,&  p = mj \mbox{ for some $j \in \N$}, \\
	0, & \mbox{otherwise}.\\
	\end{cases}\]
Condition $(M.1)$ implies that $b  = (b_p)_{p \in \N} \in  \Lambda_{(M)}(\N)$. The assumption yields that there exist $g \in \mathscr{A}^-_{(M)}(S_{\gamma}) \widetilde{\mathscr{A}}_{(M)}(S_{\delta})$ such that $B(g) = b$. Define $f(z) = g(z^{1/m})$ for $z \in S_{\delta m} = S_n$. By Lemma \ref{ramification}, we have that $f \in \widetilde{\mathscr{A}}_{(M^m)}(S_{n})$ and $B(f) = a$.
\end{proof}


TITLE: What is the domain of $f(x)=\sqrt{2x-8}$?
QUESTION [1 upvotes]: Is the domain of $f(x)$ simply any number, but in this Khan Academy video, it said the domain is $x \geqslant 4$.

REPLY [1 votes]: Since $f$ is a real-valued function of a real variable, unless otherwise specified, its domain is understood to be the set of all real numbers $x$ such that $f(x)$ is real-valued. For a square root function, this means the radicand, the term inside the radical, must be nonnegative since no negative number has a real-valued square root.  Hence, for $x$ to be in the domain of $f$,
\begin{align*}
2x - 8 & \geq 0\\
2x & \geq 8\\
x & \geq 4
\end{align*}
Thus, the domain of $f$ is $\text{Dom}_f = [4, \infty)$.


TITLE: contact surgery diagram on Brieskorn manifolds
QUESTION [4 upvotes]: For the Brieskorn manifold $\Sigma(p,q,r)=\{z_1^p+z_2^q+z_3^r=0\} \cap S^5 \subset C^3$, replacing zero with $\epsilon$ in the above, realizes $\Sigma$ as boundary of a Stein domain which induces a contact structure on the Brieskorn manifold. I want to know how to get a contact surgery diagram of this contact structure. I guess this should have been written down but I could not find any reference.

REPLY [2 votes]: Here's a partial answer that works when $p=2$. If $\Sigma (a_0,a_1, \dots ,a_n)$ is defined as the link of the the singularity $\sum z_i ^{a_i}$, the map $\pi_0:\Bbb C^{n+1}\to \Bbb C^n$ with $\pi_0(z_0,\dots,z_n)=(z_1,\dots, z_n) $ restricts to an $a_0$-fold branched contact covering $\Sigma(a_0,a_1,\dots,a_n)\to S^{2n-1}$ with branch set $\Sigma(a_1,\dots, a_n)$ which is transverse. This is a theorem of Ozturk and Niederkruger (Thm 6 in the paper here: http://arxiv.org/abs/math/0509660).
In the case of $\Sigma (2,q,r)$, we have $\Sigma (2,q,r)$ is the branched double cover of $S^3$ over $\Sigma (q,r)$ (better known as the $(q,r)$ torus knot).  In this paper: http://projecteuclid.org/euclid.jsg/1175790955, Plamenevskaya gives an algorithm to compute the Legendrian surgery diagram of the contact double branched cover over a transverse knot (using a braid diagram of the transverse knot). Of course the knot here is just a torus knot which is already in braid position so you don't have to go through the trouble of finding a transverse representative.
EDIT: It has come to my attention that not to long after the above paper of Plamenevskaya, Harvey-Kawamuro-Plamenevskaya extended this algorithm to give a Legendrian surgery diagram for the $p$-fold contact cover of a transverse link in http://arxiv.org/abs/0712.1557 , so the whole problem is solved. These pictures can get quite complicated.


TITLE: Ballot problem with tie
QUESTION [0 upvotes]: We have two candidates $c_1$ and $c_2$. At the end of a ballot each candidate receives $p$-many votes.
What is the probability that $c_2$ has never been ahead of $c_1$ the whole process of vote counting? (assume that each path of votes has the same probability)

My approach:
We are allowed to use the ballot theorem: $c_1$ wins with $a$-many votes and $c_2$ loses with $b$-many votes ($a>b$). The number of paths where $c_1$ leads is given by ${a+b\choose a}\frac{a-b}{a+b}$.
First we define the sample space by $\Omega:=\left\{\omega\in\{-1,1\}^{2p}\right\}$ where $1$ rerpresents a vote for $c_1$ and $-1$ represents a vote for $c_2$. We define the set $A:=\{\omega\in\Omega\mid \sum\limits_{i=1}^{n}\omega_i\geq 0\text{ for each } 1\leq n\leq 2p\}$ which denotes the set of paths where $c_2$ never leads/$c_1$ always leads or has the same amounts of votes as $c_2$.
Now we consider a second ballot where $c_1$ got $p+1$-many votes and $c_2$ only $p$- many votes. The respective sample space is $\Omega':=\left\{\omega'\in\{-1,1\}^{2p+1}\right\}$. We know by ballot theorem that all paths where $c_1$ leads can be counted by ${2p+1\choose p}\frac{1}{2p+1}$. Let's denote this set of paths by $B$ where $B:=\{\omega'\in\Omega'\mid \sum\limits_{i=1}^{n}\omega_i\geq 1\text{ for each } 1\leq n\leq 2p+1\}$.
We construct a bijection between $A$ and $B$. Let be $\omega'\in B$ if we remove the first vote of $\omega'$ then we get a new path $\omega$ which lies in $A$. On the other hand, if we consider a path $\omega\in A$ and add a vote for $c_1$ at the beginning then we get a new path $\omega'$ which lies in $B$. Moreover, it is obviously that this mapping is injective. Hence, $|A|=|B|={2p+1\choose p}\frac{1}{2p+1}\implies P(A)=\frac{|A|}{|\Omega|}=\frac{1}{p+1}$.

Is this approach correct?

REPLY [0 votes]: While I can't comment on your approach, your result is correct.
The thing that confused me in the comments is that probability of $c_1$ is always leading or equal to $c_2$ within $2p$ votes distributed evenly is actually equal to $c_1$ is always leading or equal to $c_2$ within $2p-1$ votes with $c_1$ getting $p$ of them. Thus, removing a single envelope does not change the result.
In other words, we know that $c_1$ and $c_2$ will have a draw at the end of counting. So, for $c_1$ never to fall behind, the last envelope has to be for $c_2$ anyway. Thus, removing an envelope does not alter the probability.
Now we can use Bertrand's Ballot theorem directly for (p, p-1) votes, giving us $\frac{1}{p+1}$.
I have the simulation results and the simulation code here.

def hitting_simulation(m,p):
    hit=0
    for j in range(m):
        votes=np.ones(2*p)
        while not sum(votes)==0:
            a=random.randint(0,2*p-1)
            votes[a]=-1
        voting=np.cumsum(votes)
        #print(voting)
        if any(voting<0):
            hit+=1
    return m-hit


TITLE: To construct a graph with desired eccentricity.
QUESTION [2 upvotes]: I am trying to solve a problem.
$G$ is a graph where eccentricity of vertices $x$ and $y$ is $8$,
and rest of the vertices have eccentricity $7$. Also $d(x,y) = 8$. I am trying to make eccentricities of $x$ and $y$ also $7$ by adding a vertex (or two vertices). The old graph must be induced in new graph. I proposed a construction:

Let $w$ be any vertex in $G$ such that $d(y,w) = 3$. Add a new vertex $z$ and make it adjacent to $y$ and $w$.

Is this true that every vertex in this new graph will have eccentricity $7$? For a few examples, I got the desired result. Or is there any other way to construct this(by adding one more vertex)?
PS: As asked by @bof, I tried for a smaller value. Like in the following figure, two vertices have eccentricity three and rest have eccentricity two, I tried to make eccentricity of every as two by adding a new vertex. As required the old graph is induced in new graph.

REPLY [2 votes]: It is not necessarily true that the eccentricity of every vertex in the new graph is exactly $7$.
In the graph below, the eccentricity of $z$ is $6$.


text:
\section{Stable homology of configuration spaces with labels in a fibre bundle}\label{appendix2}
In this appendix we prove Theorem \ref{thm:McDufflabelssection4}. Another proof can be obtained adapting step by step the proof in \cite{McDuff} for trivial labels, as pointed out in the introduction to that paper. We give here a sketch of this proof with some shortcuts, taking advantage of knowing the homology stability theorem with labels (Proposition \ref{pImprovedRangeTwisted}) in the spirit of \cite{GMTW}.

\begin{df} Let $M$ be an open manifold, let $c\colon D^{n-1}\times (0,1]$ be a proper embedding and let $M_1 = M\cup_c (\mathring{D}^{n-1}\times (-1,1])$. Define $\psi^{\delta,\gamma}(M;\theta)$ to be the space whose underlying set is $C^{\delta,\gamma}(M;\theta):= \coprod_k{C_k^{\delta,\gamma}(M;\theta)}$ with the following topology: Consider the quotient $Y$ of $C^{\delta,\gamma}(M_1;\theta)$ under the relation $\sim$ where $(\q,\epsilon,\{f_q\}_{q\in\q})\sim(\q',\epsilon',\{f'_{q'}\}_{q'\in\q'})$ if and only if
\begin{enumerate}
\item $\q\cap M = \q'\cap M$,
\item if the above intersection is non-empty, then $\epsilon=\epsilon'$,
\item if the above intersection is non-empty, then $f_q = f'_q$ for all $q\in \q\cap M$.
\end{enumerate}
The natural inclusion $C^{\delta,\gamma}(M;\theta)\to C^{\delta,\gamma}(M_1;\theta)$ induces a bijection $C^{\delta,\gamma}(M;\theta)\cong Y$, which we use to endow $C^{\delta,\gamma}(M;\theta)$ with a new topology.
\end{df}
Recall that we defined the non-linear scanning map with labels
\[\ssnl{\theta,\gamma}\colon C_k^{\delta,\gamma}(M;\theta)\lra \Gamma_c(\psi^\delta(T^{1}M;\theta))\]
in \S\ref{sUpToDim}. Following the same recipe we can define a scanning map
\[\ssnl{\theta,\gamma}\colon \psi^{\delta,\gamma}(M;\theta)\lra \Gamma(\psi^\delta(T^{1}M;\theta))\]
whose target is now the whole space of sections.

\begin{lemma}[{\cite[\S 2.3]{Hesselholt:derivative}}]\label{lemma:hesselholt}  If $M$ is connected, then the scanning map is a homotopy equivalence.
\end{lemma}
In the paper, Hesselholt considers $E$ to be a fibre bundle of based spaces. This lemma is a particular case of his theorem, taking a disjoint basepoint in each fibre and the submanifold $N$ in his theorem to be connected.

Let $\pi_1\colon M\dasharrow I$ be the partially defined function that sends a point in the image of $D^{n-1}\times I$ to the second coordinate. Define $\psi^{\delta,\gamma}(M;\theta)_\bullet$ to be the semi-simplicial space whose space of $i$-simplices is the space of tuples $(\q,\epsilon,\{f_q\}_{q\in\q},a_0,\ldots,a_i)$, where $(\q,\epsilon,\{f_q\}_{q\in\q})\in \psi^{\delta,\gamma}(M;\theta)$ and $(a_0,\ldots,a_i)\in I^{i+1}$ and $\pi_1(\q)\cap \{a_0,\ldots,a_i\}=\emptyset$. The $j$th face map forgets $a_j$, and there is an augmentation to $\psi^{\delta,\gamma}(M;\theta)$ that forgets all the $a_j$'s.
\begin{lemma} The realization of the augmentation
\[\|\psi^{\delta,\gamma}(M;\theta)_\bullet\|\to \psi^{\delta,\gamma}(M;\theta)\]
 is a weak homotoy equivalence.
\end{lemma}
\begin{proof} This is an augmented topological flag complex \cite{GR-W2} satisfying the conditions of Theorem 6.2 in that paper, hence a weak homotopy equivalence.
\end{proof}

\begin{proposition} If $\theta\colon E\to M$ has path-connected fibres, then the restriction of the scanning map
\[\ssnl{\gamma,\theta}\colon C_k^{\delta,\gamma}(M\setminus c;\theta) \lra \Gamma_c(\Psi^{\delta}(T^1M\setminus c;\theta))\]
is a homology isomorphism in the range in which the stabilisation map of Proposition \ref{pImprovedRangeTwisted} is a homology isomorphism. Since $M\setminus c\cong M$, the same holds for $M$.
\end{proposition}
\begin{proof}
We have constructed the following commutative diagram:
\begin{equation}\label{eq:488}
\begin{split}
\xymatrix@C-1.5ex{
\|\psi^{\delta,\gamma}(M;\theta)_\bullet\|\ar[r]\ar[d] & \psi^{\delta,\gamma}(M;\theta) \ar[r]\ar[d]& \Gamma(\Psi^\delta(T^1M;\theta)) \ar[d] \\
\|\psi^{\delta,\gamma}(D^{n-1}\times I;\theta)_\bullet\|\ar[r] & \psi^{\delta,\gamma}(D^{n-1}\times I;\theta) \ar[r]& \Gamma(\psi^{\delta}(T^1(D^{n-1}\times I);\theta))
}
\end{split}
\end{equation}
All the horizontal maps are homotopy equivalences, by the previous two lemmas. The rightmost vertical map is a fibration. We now choose another properly embedded ray $D^{n-1}\times (0,1]\cong L\subset \partial M\setminus c$, and take the colimit
\[P(M;\theta)_\bullet := \colim \left(\psi^{\delta,\gamma}(M;\theta)_\bullet\overset{s}{\lra} \psi^{\delta,\gamma}(M;\theta)_\bullet\lra \ldots\right)\]
with respect to the stabilisation maps $s$ that push the configurations outside $L$ and adds a point in $L$ with some prescribed label. The scanning of this operation $\ssnl{\gamma,\theta}(s)$ gives also a sequence of maps between spaces of sections, whose colimit we denote by
\[G(M;\theta) := \colim \left(\Gamma(\psi^{\delta}(T^1M;\theta))\overset{\ssnl{\theta,\gamma}(s)}{\lra} \Gamma(\psi^{\delta}(T^1M;\theta)) \lra \ldots\right)\]
Observe that the maps $\ssnl{\gamma,\theta}(s)$ increase the degree by $1$. We can consider instead the maps that push the source of the scanning map away from $L$ and glue there the \emph{reflection} of the scanning of some point in $L$, together with some prescribed label. This latter map is a homotopy inverse of $\ssnl{\gamma,\theta}(s)$, hence the maps $\ssnl{\gamma,\theta}(s)$ are homotopy equivalences.

By Proposition \ref{pImprovedRangeTwisted}, it follows that the semi-simplicial map
\[P(M;\theta)_\bullet\lra \psi^{\delta,\gamma}(D^{n-1}\times I;\theta)_\bullet\]
satisfies the hypotheses of \cite[Proposition~4]{McDuff-Segal}, so the realization
\[\|P(M;\theta)_\bullet\|\lra \|\psi^{\delta,\gamma}(D^{n-1}\times I;\theta)_\bullet\|\]
 is a homology fibration. Its fibre over any point is the colimit of the space $C^{\delta,\gamma}(M;\theta)$ with respect to the stabilisation map $s$. The map
\[G(M;\theta)\lra \Gamma(\psi^{\delta,\gamma}(T^1M;\theta))\]
is a Serre fibration (it is a union of Serre fibrations). Its fibre over any point is the colimit of $\Gamma_c(\psi^\delta(T^1M);\theta)$ with respect to the map obtained by scanning $s$:
\begin{equation}\label{eq:489}\begin{gathered}
\xymatrix{
\colim C^{\delta,\gamma}(M;\theta)\ar[rr]\ar[d] && \colim \Gamma_c(\psi^\delta(T^1M);\theta)\ar[d] \\
\|P(M;\theta)_\bullet\|\ar[rr]^\simeq\ar[d]^{a} & & G(M;\theta) \ar[d]^b \\
\|\psi^{\delta,\gamma}(D^{n-1}\times I;\theta)_\bullet\|\ar[rr]^\simeq & & \Gamma(\psi^{\delta}(T^1M;\theta))
}
\end{gathered}
\end{equation}

The fibres of \eqref{eq:489} together with the maps to the fibres of \eqref{eq:489} give the following commutative diagram
\[\xymatrix{
C^{\delta,\gamma}(M;\theta)\ar[rr]^{\ssnl{\gamma,\theta}}\ar[d] && \Gamma_c(\psi^\delta(T^1M);\theta) \ar[d] \\
\colim C^{\delta,\gamma}(M;\theta) \ar[rr] && \colim \Gamma_c(\psi^{\delta}(T^1M);\theta).
}\]
The bottom map is a homology equivalence because the horizontal maps in diagram \eqref{eq:488} are homotopy equivalences. The left vertical map is a homology equivalence in the stable range of Proposition \ref{pImprovedRangeTwisted}. The right vertical map is a homotopy equivalence. As a consequence, the upper horizontal map is a homology equivalence in the range provided by Proposition \ref{pImprovedRangeTwisted}
\end{proof}
The following is proved in the same way as Theorem 1.1 at the bottom of page 34 in \cite{McDuff}.
\begin{corollary}\label{cor:blabla} If $M$ is a manifold with empty boundary, then the non-linear scanning map
\[\ssnl{\theta,\gamma}\colon C^{\delta,\gamma}(M;\theta) \lra \Gamma_c(\psi^\delta(T^1M;\theta))\]
is a homology isomorphism in the range in which the stabilisation map is a homology isomorphism.
\end{corollary}
We showed in Section \S\ref{sUpToDim} that the triangle
\begin{equation}
\centering
\begin{split}
\begin{tikzpicture}
[x=1mm,y=1mm]
\node (tl) at (0,15) {$C_k^{\delta,\gamma}(M;\theta)$};
\node (tr) at (40,15) {$\Gamma_c(\psi^\delta (T^1 M;\theta))$};
\node (br) at (40,0) {$\Gamma_c(\dot{T}^{1,\theta} M).$};
\draw[->] (tl) to node[above,font=\small]{$\ssnl{\theta,\gamma}$} (tr);
\draw[->] (tl) to node[anchor=north east,font=\small]{$\ssl{}$} (br);
\draw[->] (tr) to[out=300,in=60] node[right,font=\small]{$h$} (br);
\draw[->] (br) to[out=120,in=240] node[left,font=\small]{$i$} (tr);
\node at (40,7.5) {$\simeq$};
\end{tikzpicture}
\end{split}
\end{equation}
commutes, hence from Corollary \ref{cor:blabla} it follows that:
\begin{theorem}[McDuff's Theorem with labels]\label{thm:McDufflabels} The linear scanning map with labels
\[\ssl{\delta,\theta}\colon C_k(M;\theta)\lra \Gamma_c(\dot{T}^\theta(M))_k\]
induces an isomorphism on homology groups in the stable range provided by Proposition \ref{pImprovedRangeTwisted}.
\end{theorem}

config: arxiv


TITLE: prove $f(x)$ and $g(x)$ have the same number of roots in $\mathbb{F}_{p}$
QUESTION [1 upvotes]: Suppose $p$ is a prime,and $f(x)=\sum\limits_{k=0}^{p-1}k!x^{k}$, $g(x)=\sum\limits_{k=0}^{p-1}\frac{1}{k!}x^{k}$,prove $f,g$ have the same number of roots in $\mathbb{F}_{p}$
I tried to consider the relation between $f(x)$ and $g(1/x)$, but I do not know what should I do next.Can anyone help me? Thanks a lot.

REPLY [2 votes]: This is an application of Wilson's Theorem and Fermat's Little Theorem. The first states that $p$ is prime if and only if
$$(p-1)!\equiv -1 (\text{mod}\ p)$$
This gives us that for $0\leq k<p$ we have
$$(p-1)!=k!(p-1)(p-2)...(k+1)\equiv -1(\text{mod}\ p)$$
$$\Rightarrow k!\equiv -[(p-1)(p-2)...(k+1)]^{-1}(\text{mod}\ p)$$
$$\Rightarrow (k!)^{-1}\equiv -(p-1)(p-2)...(k+1)$$
$$=-(-1)(-2)...(-p+k+1)=(-1)^{k+1}(p-1-k)!(\text{mod}\ p)$$
This implies that if $r\equiv x^{-1}(\text{mod}\ p)$ then
$$g(-r)=\sum_{k=0}^{p-1}(k!)^{-1}(-r)^{k}\equiv-\sum_{k=0}^{p-1}(p-1-k)!r^k(\text{mod}\ p)$$
Then using Fermat's Little Theorem we know
$$r^k\equiv x^{-k}\equiv x^{p-1-k}(\text{mod}\ p)$$
This implies
$$-\sum_{k=0}^{p-1}(p-1-k)!r^k\equiv-\sum_{k=0}^{p-1}(p-1-k)!x^{p-1-k}=-\sum_{k=0}^{p-1}k!x^k=-f(x)(\text{mod}\ p)$$
Thus $f(x)\equiv-g(r)\ (\text{mod }p)$. Since $f(0)\neq 0\neq g(0)$ we are assured that the roots inverses exist.


TITLE: Prove that if A is symmetric and invertible, then$ (A^{-1})^t = (A^t)^{-1}$.
QUESTION [2 upvotes]: I have a problem. Prove that if A is symmetric and invertible, then $(A^{-1})^t = (A^t)^{-1}$.
This is what I have done, please tell me if it is correct:
Say $C=A^{-1}$. Then, $C^t = ((C^{-1})^t)^{-1} = ((C^{-1})^{-1})^t = C^t$.

REPLY [6 votes]: If $A$ is an invertible matrix (symmetric or not symmetric) then
$$AA^{-1}=I\Rightarrow \left(AA^{-1}\right)^T=\left(A^{-1}\right)^TA^T=I$$
so the matrix $A^T$ is invertible and
$$\left(A^T\right)^{-1}=\left(A^{-1}\right)^T$$


TITLE: Create a formula that makes x approach 100 as j approaches 0
QUESTION [0 upvotes]: I'm normally a computer programmer but right now I'm trying to create a math formula for my computer program.
I have a variable i = 70, and another variable J = 15.
As J approaches 0, X should approach 100.  I'm stumped what formula I should use.
It should be written something like this:
X = i + ( put operations that make x approach 100 as J approaches 0, J is inside these parenthesis )
And also, I can change 70 to a number between 0 to 100 and the operations above would still make x approach 100 as J approaches 0.
So with my example, if i is 70,
when J = 7.5, X = 85
when J = 5, X = 90
when J = 2.5, X = 95
And so on... but i is a parameter that gets input by the user from 0 to 100
Please ask any other questions if I'm not explaining well enough.

REPLY [1 votes]: In your example, you have the points  (7.5,85), (5,90) and (2.5,95). Those can be written as (7.5, 100-7.5*2),(5,100-5*2),(100-2.5*2), so that suggests the formula X = 100-2J. More generally, a function that linearly approaches the point (0,100) will have the formula X = 100+mJ. If you want the function to also go through the point (15,i), then you can solve for m in terms of i: plugging in J = i and X = 15 gives you i = 100+15m, so m=(i-100)/15, giving you X=(i-100)J/15+100. You can also write it as X = 100-(100-i)J/15; this is a bit more transparent what's going on. Suppose X is height, and J is time. You start out at height 100, and as time goes on you go down from there. (Note that you've worded it as J starts at 15 and goes to 0, which makes things a bit confusing. I've worded it as starting at 0 and going to 15.)
final_J_value = 15 #your parameter i represents your height at time 15
initial_value = 100 #you start at 100
final_distance = initial_value-i #calculate how far away from the goal you are at the final time
speed = final_distance/final_J_value #the speed your height is decreasing is the total amount you decreased divided by how long it took to decrease that amount
current_distance = speed*J #your distance from the initial height is proportional to your speed and how long you have to go been going
X = initial_value-current_distance #current_distance is how far away X is from 100, so X is 100 minus that value


TITLE: Let $H = \{\alpha^2 : \alpha \in S_4\}$. Show that $H = A_4$.
QUESTION [0 upvotes]: Let $H = \{\alpha^2 : \alpha \in S_4\}$. Show that $H = A_4$.

Suppose $x \in H$. Then $x = \alpha^2$, where $\alpha \in S_4$. Since elements of $S_4$ takes on the form of 2-cycles, 3-cycles, product of two 2-cycles, and 4-cycles, we look at their individual squares. If $\alpha = (a b)$, then $\alpha ^2 = \epsilon \in A_4$ (identity). If $\alpha = (a b c)$, then $\alpha ^2 = (a c b) = (c b )(a b) \in A_4$. If $\alpha = (ab)(cd)$ or $(ac)(bd)$ or $(ad)(bc)$, then $\alpha^2 = \epsilon \in A_4$. If $\alpha = (abcd)$, then $\alpha ^2 = (ac)(bc) \in A_4$. Hence, $ H \subseteq A_4$.

Is this direction correct? Also, how can I prove the other inclusion? Do I look at each element $y \in A_4$ and try to find some element $x \in S_4$ such that $x^2 = y$? Is there another way to do this?

REPLY [2 votes]: Your proof is correct. For the reverse inclusion, I will show that for each $y\in A_4$, there exists $x\in S_4$ such that $y=x^2$. This can be done similarly by considering the cycle structure of elements in $A_4$.
(i) If $y=\epsilon$, then $y=(12)^2$.
(ii) If $y=(ab)(cd)$, then $y=(acbd)^2$.
(iii) If $y=(abc)$, then $y=(acb)^2$.


text:
\section{Proof sketch}
In this section we sketch the proof for two of our main results, namely \cref{fedac:a1}(a) and \ref{fedacii:a2}.
We focus on the proof of \cref{fedac:a1}(a) to outline our proof framework, and then illustrate the difference in the proof of \cref{fedacii:a2}.
\subsection{Proof sketch of \cref{fedac:a1}(a): \fedaci under \cref{asm1} }
\label{sec:proof:sketch:1}
Our proof framework consists of the following four steps.

\paragraph{Step 1: potential-based perturbed iterate analysis.}
The first step is to study the difference between \fedac and its fully synchronized idealization, namely the case of $K=1$ (recall $K$ denotes the number of local steps).
To this end, we extend the perturbed iterate analysis \citep{Mania.Pan.ea-SIOPT17} to potential-based setting to analyze accelerated convergence.
For \fedaci, we study the \emph{decentralized} potential $\Psi_t := \frac{1}{M} \sum_{m=1}^M F( {w_{t}^{\mathrm{ag}, m}})  - F^* + \frac{1}{2}\mu \|\overline{w_{t}} - w^*\|^2$ and establish the following lemma.
$\Psi_t$ is adapted from the common potential for acceleration analysis \citep{Bansal.Gupta-19}.
\begin{lemma}[Simplified version of \cref{fedaci:conv:main}, Potential-based perturbed iterate analysis for \fedaci]
  \label{fedaci:conv:sketch}
  In the same settings of \cref{fedac:a1}(a), the following inequality holds
    \begin{align}
       \expt \left[\Psi_T \right]    &      \leq
      \exp \left( - \gamma \mu T \right)  \Psi_0 + \frac{\eta^2 L \sigma^2}{2\gamma \mu} + \frac{\gamma\sigma^2}{2M} \tag{\text{Convergence rate in the case of $K=1$}}
      \\
      &+ \underbrace{ L \cdot \max_{0 \leq t < T}  \expt
      \left[\frac{1}{M} \sum_{m=1}^M \left\| \overline{w_t^{\mathrm{md}}} - w_t^{\mathrm{md}, m}  \right\|
      \left\| \frac{1}{1 + \gamma \mu} (\overline{w_t} - w_t^m) + \frac{\gamma \mu}{1 + \gamma \mu}   (\overline{w_t^{\mathrm{ag}}} - w_t^{\mathrm{ag},m})  \right\|  \right]}_{\text{Discrepancy overhead}}.
      \label{eq:fedaci:conv:sketch}
    \end{align}
\end{lemma}
We refer to the last term of \eqref{eq:fedaci:conv:sketch} as ``discrepancy overhead'' since it characterizes the dissimilarities among workers due to infrequent synchronization. The proof of \cref{fedaci:conv:sketch} is deferred to \cref{sec:fedaci:conv:main}.

\paragraph{Step 2: bounding discrepancy overhead.}
The second step is to bound the discrepancy overhead in \eqref{eq:fedaci:conv:sketch} via stability analysis.
Before we look into \fedac, let us first review the intuition for \fedavg.
There are two forces governing the growth of discrepancy of \fedavg, namely the (negative) gradient and stochasticity.
Thanks to the convexity, the gradient only makes the discrepancy lower.
The stochasticity incurs $\mathcal{O}(\eta^2 \sigma^2)$ variance per step, so the discrepancy
$\expt [\frac{1}{M} \sum_{m=1}^M \|\overline{w_t} - w_t^m\|^2 ]$
grows at rate $\mathcal{O}(\eta^2 K \sigma^2)$ linear in $K$.
The detailed proof can be found in \citep{Khaled.Mishchenko.ea-AISTATS20,Woodworth.Patel.ea-ICML20}.

For \fedac, the discrepancy analysis is subtler since acceleration and stability are at odds --- the momentum may amplify the discrepancy accumulated from previous steps.
Indeed, we establish the following \cref{instability:sketch}, which shows that the \emph{standard deterministic} Accelerated GD (\agd) may \emph{not} be initial-value stable even for strongly convex and smooth objectives, in the sense that initial infinitesimal difference may grow exponentially fast.
We defer the formal setup and the proof of \cref{instability:sketch} to \cref{sec:instability} in Appendix.
\begin{theorem}[Initial-value instability of deterministic standard \agd]
  \label{instability:sketch}
  For any $L, \mu > 0$ such that $\nicefrac{L}{\mu} \geq 25$, and for any $K \geq 1$, there exists a 1D objective $F$ that is $L$-smooth and $\mu$-strongly-convex, and an $\varepsilon_0 > 0$, such that for any positive $\varepsilon < \varepsilon_0$, there exists initialization $w_0, u_0, w_0^{\mathrm{ag}}, u_0^{\mathrm{ag}}$ such that $|w_0 - u_0| \leq \varepsilon$, $|w_0^{\mathrm{ag}} - u_0^{\mathrm{ag}}| \leq \varepsilon$, but
  the trajectories $\{w_t^{\mathrm{ag}}, w_t^{\mathrm{md}}, w_t\}_{t=0}^{3K}$, $\{u_t^{\mathrm{ag}}, u_t^{\mathrm{md}}, u_t\}_{t=0}^{3K}$ generated by applying deterministic \agd with initialization $(w_0, w_0^{\mathrm{ag}})$ and $(u_0, u_0^{\mathrm{ag}})$ satisfies
  \begin{equation}
      |w_{3K} - u_{3K}| \geq \frac{1}{2} \varepsilon (1.02)^K,
      \qquad
      |w^{\mathrm{ag}}_{3K} - u^{\mathrm{ag}}_{3K}| \geq \varepsilon (1.02)^K.
  \end{equation}
\end{theorem}
\begin{remark}
  It is worth mentioning that the instability theorem \textbf{does not contradicts the convergence} of \agd \citep{Nesterov-18}. The convergence of \agd suggests that $w_{t}^{\mathrm{ag}}$, $w_t$, $u_t^{\mathrm{ag}}$, and $u_t$ will all converge to the same point $w^*$ as $t \to \infty$, which implies $\lim_{t \to \infty} \|w_t^{\mathrm{ag}} - u_t^{\mathrm{ag}}\| = \|w_t - u_t\| = 0$. However, the convergence theorem does not imply the stability with respect to the initialization --- it does not exclude the possibility that the difference between two instances (possibly with very close initialization) first expand and only shrink until they both approach $w^*$. Our \cref{instability:sketch} suggests this possibility: for any finite steps, no matter how small the (positive) initial difference is, it is possible that the difference will grow exponentially fast.
This is fundamentally different from the Gradient Descent (for convex objectives), for which the difference between two instances does not expand for standard choice of learning rate $\eta = \frac{1}{L}$ (where $L$ is the smoothness).
\end{remark}

Fortunately, we can show that the discrepancy can grow at a slower exponential rate via less aggressive acceleration, see \cref{fedaci:stab:sketch}. As we will discuss shortly, we adjust $\gamma$ according to $K$ to restrain the growth of discrepancy within the linear regime. The proof of \cref{fedaci:stab:sketch} is deferred to \cref{sec:fedaci:stab:main}.
\begin{lemma}[Simplified version of \cref{fedaci:stab:main}, Discrepancy overhead bounds for \fedaci]
  \label{fedaci:stab:sketch}
  In the same setting of \cref{fedac:a1}(a), the following inequality holds
    \begin{equation}
      \text{``Discrepancy overhead'' in \cref{eq:fedaci:conv:sketch}} \leq \begin{cases}
        7 \eta \gamma L K \sigma^2 \left(1 + \frac{2\gamma^2\mu}{\eta}\right)^{2K}
         & \text{if~} \gamma \in (\eta, \sqrt{\frac{\eta}{\mu}} ],
        \\
        7 \eta^2 L K \sigma^2
         &
        \text{if~} \gamma = \eta.
      \end{cases}
      \label{eq:fedaci:stab:sketch}
    \end{equation}
\end{lemma}
\paragraph{Step 3: trading-off acceleration and discrepancy.}
Combining \cref{fedaci:conv:sketch,fedaci:stab:sketch} gives
  \begin{equation}
    \label{fedaci:combine:sketch}
    \expt \left[\Psi_T \right]
    \leq \underbrace{\exp \left( - \gamma \mu T \right)  \Psi_0}_{\mathrm{(I)}}
    + \frac{\eta^2 L \sigma^2}{2\gamma \mu}
    + \frac{\gamma\sigma^2}{2M}
    +
    \underbrace{
      \begin{cases}
        7 \eta \gamma L K \sigma^2 \left(1 + \frac{2\gamma^2\mu}{\eta}\right)^{2K}
         & \text{if~} \gamma \in (\eta, \sqrt{ \frac{\eta}{\mu}} ],
        \\
        7 \eta^2 L K \sigma^2
         &
        \text{if~} \gamma = \eta.
      \end{cases}
    }_{\mathrm{(II)}}
  \end{equation}

The value of $\gamma \in [\eta, \sqrt{\eta/\mu}]$ controls the magnitude of acceleration in (I) and discrepancy growth in (II).
The upper bound choice $\sqrt{\eta/\mu}$ gives full acceleration in (I) but makes (II) grow exponentially in $K$.
On the other hand, the lower bound choice $\eta$ makes (II) linear in $K$ but loses all acceleration.
We wish to attain as much acceleration in (I) as possible while keeping the discrepancy (II) grow moderately. \textbf{Our balanced solution} is to pick $\gamma = \max \{ \sqrt{{\eta}/{(\mu K)}}, \eta\}$. One can verify that the discrepancy grows (at most) linearly in $K$.
Substituting this choice of $\gamma$ to \cref{fedaci:combine:sketch} leads to
  \begin{equation}
    \expt \left[ \Psi_T \right]
    \leq
    \underbrace{\exp \left(  \min \left\{ -\eta \mu T, - \frac{\eta^{\frac{1}{2}} \mu^{\frac{1}{2}} T}{K^{\frac{1}{2}}} \right\} \right) \Psi_0}_{\text{Monotonically decreasing } \varphi_{\downarrow}(\eta)}
    +
    \underbrace{\mathcal{O} \left( \frac{\eta^{\frac{1}{2}} \sigma^2}{ \mu^{\frac{1}{2}} M K^{\frac{1}{2}} }
      + \frac{\eta \sigma^2}{M}
      + \frac{\eta^{\frac{3}{2}} L K^{\frac{1}{2}} \sigma^2} {\mu^{\frac{1}{2}}}
      + \eta^2 L K \sigma^2\right)}_{\text{Monotonically increasing } \varphi_{\uparrow}(\eta)}.
    \label{eq:fedaci:eta:sketch}
  \end{equation}
\paragraph{Step 4: finding $\eta$ to optimize the RHS of \cref{eq:fedaci:eta:sketch}.}
It remains to show that \eqref{eq:fedaci:eta:sketch} gives the desired bound with our choice of $\eta = \min \{ \frac{1}{L}, \tilde{\Theta} (\frac{K}{\mu T^2} )\}$.
The increasing $\varphi_{\uparrow}(\eta)$ in \eqref{eq:fedaci:eta:sketch} is bounded by
\(
\tildeo ( \frac{\sigma^2}{\mu M T} +  \frac{LK^2 \sigma^2}{\mu^2 T^3} ).
\)
The decreasing term $\varphi_{\downarrow}(\eta)$ in \eqref{eq:fedaci:eta:sketch} is bounded by $\varphi_{\downarrow}({\frac{1}{L}}) + \varphi_{\downarrow} (\tilde{\Theta} (  \frac{K}{\mu T^2} ) )$, where $    \varphi_{\downarrow} ({\frac{1}{L}} )
  = \exp ( \min \{ - \frac{\mu T}{L}, - \frac{\mu^{\frac{1}{2}}T}{L^{\frac{1}{2}}K^{\frac{1}{2}} }\} )$,
and
$\varphi_{\downarrow} (\tilde{\Theta} (  \frac{K}{\mu T^2} ) )
\leq
\exp \left( - \frac{\mu^{\frac{1}{2}} T}{K^{\frac{1}{2}}} \cdot \sqrt{\tilde{\Theta} \left(  \frac{K}{\mu T^2} \right)} \right)$
can be controlled by the bound of $\varphi_{\uparrow}(\eta)$ provided $\tilde{\Theta}$ has appropriate $\polylog$ factors.
Plugging the bounds to \eqref{eq:fedaci:eta:sketch} and
replacing $K$ with $\nicefrac{T}{R}$ completes the proof of \cref{fedac:a1}(a).
We defer the details to \cref{sec:fedaci}.
\subsection{Proof sketch of \cref{fedacii:a2}: \fedacii under \cref{asm2}}
\label{sec:proof:sketch:2}
In this section, we outline the proof of \cref{fedacii:a2} by explaining the differences with the proof in \cref{sec:proof:sketch:1}.
The first difference is that for \fedacii we study an alternative \emph{centralized potential} $\Phi_t = F(\overline{w_t^{\mathrm{ag}}}) - F^* + \frac{1}{6}\mu \| \overline{w_t} - w^*\|^2$, which leads to an alternative version of \cref{fedaci:conv:sketch} as follows.
  \begin{equation}
    \expt [\Phi_T] \leq \exp \left( - \frac{\gamma \mu T}{3} \right) \Phi_0
    + \frac{3\eta^2 L \sigma^2 }{2 \gamma \mu M}
    + \frac{\gamma \sigma^2}{2 M}
    + \frac{3}{\mu} \max_{0 \leq t < T}   \expt \left\| \frac{1}{M} \sum_{m=1}^M \nabla F (w_t^{\mathrm{md}, m}) - \nabla F(\overline{w_t^{\mathrm{md}}})    \right\|^2 .
    \label{eq:fedacii:conv:sketch}
  \end{equation}
The second difference is that the particular discrepancy in \eqref{eq:fedacii:conv:sketch} can be bounded via \nth{3}-order smoothness $Q$ since
(we omit ``md'' and $t$ to simplify the notations)
\begin{align}
  & \left\|  \frac{1}{M} \sum_{m=1}^M \nabla F(w^m) - \nabla F (\overline{w})    \right\|^2 = \left\|  \frac{1}{M} \sum_{m=1}^M \left( \nabla F(w^m) - \nabla F (\overline{w}) - \nabla^2 F (\overline{w})(w^m - \overline{w_t}) \right)   \right\|^2
\\
\leq & \frac{1}{M} \sum_{m=1}^M\left\|   \nabla F(w^m) - \nabla F (\overline{w}) - \nabla^2 F (\overline{w})(w^m - \overline{w}) \right\|^2   \leq \frac{Q^2}{4M}  \sum_{m=1}^M\left\| w^m - \overline{w} \right\|^4.
\end{align}
The proof then follows by analyzing the \nth{4}-order stability of \fedac. We defer the details to \cref{sec:fedacii}.


TITLE: What is the correct term for a module but with the scalars being elements of a monoid?
QUESTION [0 upvotes]: From my understanding, a module is similar to a vector space over a field, except the scalars come from a ring. What about the cases when the scalars come from a monoid? Is there a specific term used when the scalars are from a monoid instead?

REPLY [1 votes]: Some people call them monoid actions, by analogy with group actions. There's also the term "monoid act," but I don't know who uses it.