text:
\begin{document}

\maketitle

\section{Introduction}
Consider a surface $S_{g,n}$ with genus $g$ and $n$ punctures with negative Euler characteristic.  The group of self homeomorphisms of the surface up to isotopy is called the \textit{mapping class group}, $\mbox{Mod}^+(S_{g,n})$.  By the Nielsen-Thurston classification \cite{FLP} an element of the mapping class group is either periodic, reducible(i.e. there exists a non-trivial invariant curve), or pseudo-Anosov.  A pseudo-Anosov element is one which fixes a pair of transverse measured singular foliations,

$$\phi((\mathcal{F}^\pm,\mu^{\pm}))=(\mathcal{F}^{\pm},\lambda^{\pm 1}\mu^{\pm})$$

\noindent
up to scaling the measures $\mu^{\pm 1}$ by a constants $\lambda^{\pm 1}$ where $\lambda>1$.  The number $\lambda$ is called the \textit{dilatation} of $\phi$.  The set of dilatations is discrete and therefore bounded away from $1$ \cite{AY} \cite{Iv}.  Furthermore for fixed $g$ and $n$ the minimum, $\delta_{g,n}$, is achieved by some mapping class $\phi\in \mbox{Mod}^+(S_{g,n})$.  One open question about the spectrum of dilataitons is the following one.

\begin{quest}
What is the value of $\delta_{g,n}$ given $(g,n)$ defining a surface of negative Euler characteristic?
\end{quest}

This question is only answered for a handful of cases of small $g$ and $n$, see \cite{Ko-Los-Song}, \cite{Ham-Song}, \cite{Cho-Ham}.  More is known about the general behavior of these numbers.  Penner explored the behavior for closed surfaces and proved the following theorems.

\begin{thm}\cite{Pen91}\label{lowerbound-thm}
\begin{eqnarray}\label{penner-geq}
\log(\delta_{g,n})\geq\frac{\log(2)}{12g-12+4n}
\end{eqnarray}
\end{thm}

\begin{thm}\cite{Pen91}\label{penner-asymp-thm}
\begin{eqnarray}\label{Penner-eqn}
\log (\delta_{g,0}) \asymp \frac{1}{g}.
\end{eqnarray}
\end{thm}

Tsai \cite{Tsai}(cf \cite{H-K}) continued this investigation for punctured surfaces and showed that for $g=0$ or $g=1$ and $n$ even the behavior is:

$$\log{\delta_{g,n}}\asymp\frac{1}{n}.$$

\noindent
However for surfaces of genus $g>1$ the minimal dilatations behave like,

$$\log(\delta_{g,n})\asymp\frac{\log(n)}{n}.$$

\noindent
This lead to the following question.

\begin{quest}\cite{Tsai}
What is the behavior for the minimal dilatations for different sequences of $(g,n)$?
\end{quest}

In this paper we provide a partial answer to Tsai's question.

\begin{thm}\label{main-thm}
Given any rational number $r$ the asymptotic behavior along the ray defined by $g=rn$ is,
$$\log(\delta_{g,n}) \asymp \frac{1}{|\chi(S_{g,n})|},$$
where $\chi(S_{g,n})$ is the Euler characteristic of the surface $S_{g,n}$.
\end{thm}

The proof follows Penner's proof of Theorem \ref{penner-asymp-thm}.  In \cite{Pen91} Penner proves a general lower bound and defines a sequence of pseudo-Anosov mapping classes $\phi_g:S_{g,0}\rightarrow S_{g,0}$ such that $\lambda((\phi_g)^g)$ is bounded by some constant.  We use Penner's lower bound \cite{Pen91} and generalize his examples.  This generalization allows us to construct sequences with dilatation bounded by some constant multiple of $\frac{1}{\mid\chi(S_{g,n})\mid}$.  Given certain choices we can find bounding examples for the upper bound in Theorem \ref{main-thm}.

In Section 2 we recall some known results about pseudo-Anosov mapping classes and train tracks and some techniques of Penner's used in providing the upper bound for closed surfaces.  In Section 3 we define generalized Penner sequences and begin to prove that such a sequence $\phi_n$ has $\log(\lambda(\phi_n))\asymp\frac{1}{\mid\chi(S_{g,n})\mid}$.  In Section 4 we apply our construction and its behavior to the proof of Theorem \ref{main-thm}.

\noindent
\textbf{Acknowledgments:}  I would like to thank Eriko Hironaka for suggesting the problem solved in this paper and for many helpful conversations.  I would also like to thank Beson Farb for reading an earlier version and for his helpful comments.

\section{Background}

In this section we recall some facts about pseudo-Anosov mapping classes and train tracks that will be used later in this paper.

Dehn showed in \cite{Dehn} that the mapping class group is generated by finitely many Dehn twists $d_x$ where $x$ is a simple closed curve in the surface.  The following theorem of Penner's gives a partial answer to the question of which words of Dehn twists define pseudo-Ansov mapping classes.

\begin{thm}[Penner's Semigroup Criterion]\label{Penner-semigroup-thm} \cite{Pen88}
Suppose $\mathcal{C}$ and $\mathcal{D}$ are each disjointly embedded collections of simple closed curves in an oriented surface $S$.  Suppose $\mathcal{C}$ interesects $\mathcal{D}$ minimally and $\mathcal{C}\cup\mathcal{D}$ fills $S$(i.e. the connected components of $S\slash C\cup D$ have non-negative Euler characteristic).  Let $R(\mathcal{C}^+,\mathcal{D}^-)$ be the semigroup generated by $\{d_c\mid c\in\mathcal{C}\}\cup\{d_d^{-1}\mid d\in\mathcal{D}\}$.  If $\omega\in R(\mathcal{C}^+,\mathcal{D}^-)$, then $\omega$ is pseudo-Anosov if each $d_c$ and $d_d^{-1}$ appear in $\omega$.
\end{thm}

Every pseudo-Anosov mapping class $\phi:S_{g,n}\rightarrow S_{g,n}$ has an associated train track $\tau\subset S_{g,n}$ such that $\phi(\tau)$ is smoothly homotopic into $\tau$, or $\phi(\tau)$ is \textit{carried} by $\tau$.  This homotopy defines a transition matrix $T[a_{i,j}]$ on the edges of $\tau$ where the entry $a_{i,j}$ is the incidence of the edge $i$ with the edge $j$ after applying $\phi$ followed by the homotopy.  This matrix defines a linear action on the vector space of admissable measures of $\tau$.  Each admissable measure of $\tau$ defines a measured singular foliation of $S_{g,n}$ and so finding the real eigenvectors of $T$ is equivalent to finding the invariant foliations where the real eigenvalue is the dilatation.  For a further discussion of train tracks see \cite{P-H} \cite{Pen88}.

Recall that a non-negative matrix $M$ such that $M^n$ is positive for some $n>0$ is said to be Perron-Frobenious.  Such a matrix has a unique real eigenvector whose eigenvalue is the spectral radius of the matrix.  We will use the following lemma to bound the spectral radii of these matrices.  The result is well known and we include the proof for the convenience ofthe reader.

\begin{lem}\label{columnsum-lem}
The spectral radius of a Perron-Frobenius matrix is bounded by the largest column sum.
\end{lem}

\noindent\textbf{Proof:}  Let $M$ be a Perron-Frobenius matrix with spectral radius $\lambda$ and corresponding eigenvector $v$ with norm $1$.  This eigenvector is real and positive.
$$\mid vM\mid=\lambda=\frac{\sum_{i=1}^nv_im_{ij}}{v_j}\mbox{ for all $j=1...n$}.$$

Let $v_j$ be the largest component of $v$. Then we have:
$$\mid vM\mid=\lambda=\sum_{i=1}^n\frac{v_i}{v_j}m_{ij}.$$
But each term $\frac{v_i}{v_j}\leq 1$ and each term $m_{ij}\geq 0$ and so we have the inequality:
$$\lambda\leq \sum_{i=1}^nm_{ij}.$$
Thus $\lambda$ is bounded by the largest column sum.$\Box$

We will construct pseudo-Anosov maps and corresponding Perron-Frobenius matrices.

As we have already mentioned a pseudo-Anosov mapping class defines a pair of transverse measured singular foliations $(\mathcal{F}^{\pm},\mu^{\pm})$.  The following lemma tells us when a pseudo-Anosov mapping class extends under the forgetful map to another with the same dilatation, see \cite{H-K}.

\begin{lem}\label{filling-lem}
If $\phi$ is a pseudo-Anosov mapping class on the surface $S_{g,n}$, some subset of the punctures $I$ is fixed setwise, and if none of the points in $I$ are 1-pronged then the punctures may be filled in and the induced mapping class $\tilde{\phi}$ is pseudo-Anosov with $\lambda(\phi)=\lambda(\tilde{\phi})$.
\end{lem}

An \textit{m-gon} is a possibly punctured disc in $S_{g,n}\backslash\tau$ with $m$ cusps.  We will be able to apply Lemma \ref{filling-lem} with the use of the following Lemma, see \cite{P-H}.

\begin{lem}\label{transitionmatrix-lem} Let $\phi$ be a pseudo-Anosov mapping class and $\tau$ a compatible train track.  Then we have the following:
\begin{enumerate}
\item The singularities and punctures of $S$ defined by the stable foliation of
$\phi$ are in one-to-one correspondence with the the $m$-gons of $\tau$, and
\item  a singularity or puncture is $m$-pronged if and only if it is contained in an $m$-gon of $\tau$.
\end{enumerate}
\end{lem}

We will use this information in order to construct examples, find transition matrices that bound their dilatation and lastly extend these examples to ones suitable for the $gn$-rays we are interested in.

\section{Penner Sequences}

In this section we will define Penner sequences and give the stepping stones to prove the following theorem.

\begin{thm}\label{sequence-thm}
Given a Penner sequence of mapping classes $\phi_m : F_m \rightarrow F_m$ there is a constant $P$ such that
$$\log(\lambda(\phi_m))\asymp\frac{P}{\mid\chi(F_m)\mid}$$
where $\chi(F_m)$ is the Euler characteristic of the surface $F_m$.
\end{thm}

First we contstruct the surfaces the mapping classes are defined on.  Consider an oriented surface with boundary and punctures, $S_{g,n,b}$, with two sets of disjoint arcs on the boundary components $a^-$ and $a^+$ such that

$$a^-\cap a^+=\emptyset$$

\noindent
and an orientation reversing homeomorphism,

$$\iota:a^+\rightarrow a^-.$$

\noindent
Let $\Sigma_i$ be homeomorphic copies of $S_{g,n,b}$ and let $h_i:S_{g,n,b}\rightarrow \Sigma_i$ be a homeomorphism for each $i\in\mathbb{Z}$.  Set

$$F_{\infty}=\bigcup_{i \in \mathbb{Z}} \Sigma_i/\sim,$$

\noindent
where $y_i \sim y_j$ if, for some $x \in a^+$ and $k\in\mathbb{Z}$,

$$(y_i,y_j)=(h_k(x),h_{k+1}(\iota(x))).$$

The action

$$\rho=h_{i+1}h^{-1}_i$$

\noindent
acts properly discontinuously on $F_{\infty}$.  Then we define

$$F_m=F_\infty\slash\rho^m.$$

We then pick two sets of multicurves $C$ and $D$ on $\Sigma_1$ satisfying Theorem \ref{Penner-semigroup-thm}.  A \textit{connecting curve} is a curve, $\gamma$, on $F_{\infty}$ such that $\gamma\subset\Sigma_1\cup\Sigma_2$, $C\cup\rho(C)\cup\gamma$ is a multicurve, $\gamma$ intersects $D\cup\rho(D)$ minimally, and the set of curves,

$$\displaystyle{J=\{\rho^i(C\cup D\cup\gamma)\}_{-\infty}^{\infty}}$$

\noindent
fills the surface $F_{\infty}$.

\begin{defn}
A sequence of mapping classes $\phi_m:F_m\rightarrow F_m$ is called a Penner sequence if for some $(C,D)$ as in Theorem \ref{Penner-semigroup-thm},
$$\phi_m=\rho d_{\gamma}\omega,$$
where $\omega\in R(C^+,D^-)$ is pseudo-Anosov on $\Sigma_1$.
\end{defn}

Next we want to show that these mapping classes are pseudo-Anosov.  We start with the following lemma about train tracks.  Here we allow our train tracks to have bigons.

\begin{lem}\label{track-lem}
Given a Penner sequence $\phi_m$ there exists invariant train tracks on each surface $F_m$ such that:
\begin{enumerate}
\item  The curves in $C$ and $D$ and the connecting curve are carried on the train track.
\item The images of the curves used to define $\omega$ and the image of the connecting curve under the map $\phi_m$ are carried on the train track.
\end{enumerate}
\end{lem}

\noindent\textbf{Proof:}  We construct a train track on the surface $F_{\infty}$ and then project it to $F_{m}$.  Consider the set of curves $J$ in the definition of $F_{\infty}$.  Then assign positive orientation to all curves,

$$\displaystyle{\{\rho^i(C\cup\gamma)\}_{-\infty}^{\infty}},$$

\noindent
and negative orientation to all curves,

$$\displaystyle{\{\rho^i(D)\}_{-\infty}^{\infty}}.$$

\noindent
We then smooth the intersections according to Figure \ref{smooth-fig}.

\begin{figure}[h]\label{smooth-fig}
\centering
\includegraphics[scale=.5]{gentrack.png}
\caption{Smoothing}
\end{figure}

This provides a train track, $\tau$, which may be projected to a track $\tau_m$ on $F_m$.  It is easy to see that all the curves in $J$ are carried by the train track.  If we consider a curve $x$ that is carried by $\tau_m$ and perform a Dehn twist by an element $y\in J$ then the resulting curve $d_y(x)$ is carried by the edges carrying $x$ and the edges that carry the element $y$.  Since the train track is symmetric with respect to the rotation map we are done.$\Box$

The next lemma allows us to compute the entries of a transition matrix on this train track.

\begin{lem}\label{measure-lem}
Consider $x,y\in J$ where,

$$J=\displaystyle{\cup_{i=1...m}\rho^i(C\cup D\cup \gamma)}.$$

If $\mu_x$ and $\mu_y$ are the elements of $W_{\tau_m}$ induced by the curves $x$ and $y$ and $d_x^{\star}$ is the map on $W_{\tau_m}$ induced by $d_x$ then
$$d_x^{\star}(\mu_y)=\mu_y+i(x,y)\mu_x$$
\end{lem}

\noindent\textbf{Proof:}  Collapsing a curve is invariant up to homotopy so the edges that elements of $J$ collapse to on the train track are well defined.  Therefore we can consider a curve as a free homotopy class.  If two curves $x,y\in J$ are in minimal position then performing a Dehn twist on $y$ about $x$ will send $y$ to $d_x(y)=y+i(x,y)x$.  We collapse the homology elements to $\tau_m$ and obtain the desired result.$\Box$

\begin{lem}\label{PF-lem}
The matrix $T_{\phi_m}$ is Perron-Frobenius for each $m \geq 2$.
\end{lem}

\noindent\textbf{Proof:}  This follows from the previous lemma and the fact that given the mapping class $\phi_m^m$ where we perform Dehn twists about all the curves in,

$$J_m=\displaystyle{\cup_{i=1...m}\rho^i(C\cup D\cup \gamma)},$$

\noindent
each curve is connected to another through at most $m(r+s+1)$ curves.  Therefore $T_{\phi_m}^{m^2(r+s+1)}$ is strictly positive where $r=\sharp C$ and $s=\sharp D$.$\Box$

\bigskip

\noindent\textbf{Remark:}  An admissable measure on a bigon track defines a measured foliation up to an equivalence of the admissable measures \cite{P-H}.  We only consider a subset of the admissable measures when computing transition matrices.  Since we have show that the transition matrix on these measures is Perron-Frobenius the eigenvector is positive, defining an invariant foliation for the mapping class.  We will prove that these mapping classes are pseudo-Anosov in the next section, therefore the invariant expanding foliation is unique and we need not worry about the equivalent measures or measures outside the considered subset.

\section{Asymptotic Behavior}

\noindent\textbf{Proof of Theorem \ref{sequence-thm}:}  The mapping classes $(\phi_m)^m$ are pseudo-Anosov by Penner's semigroup criteria and so the mapping classes $\phi_m$ are as well.  Lemma \ref{track-lem} gives an invariant bigon train track.

As stated in Theorem \ref{columnsum-lem} the spectral radius of a Perron-Frobenius matrix is bounded by the largest column sum of the matrix.  So now we would like to compute the matrix defining the action on the transverse measures.  This matrix will be Perron-Frobenius by Lemma \ref{PF-lem} and can be computed using Lemma \ref{measure-lem}.

The map $\rho$ permutes the curves of $J_m=\displaystyle{\cup_{i=1...m}\rho^i(C\cup D\cup \gamma)}$ and so the induced map on the space of weights spaned by $\mu_1...\mu_{m(r+s+1)}$, where $r=\sharp C$ and $s=\sharp D$, is defined by a block permutation matrix.

$$M_{\rho}=\left(
\begin{array}{ccccc}
0&I&0&.&0\\
0&0&I&.&0\\
.&.&.&.&.\\
I&0&0&.&0
\end{array}\right)$$

\noindent
The Dehn twist about the connecting curve gives the map with transition matrix defined by,

$$M_{d_{\gamma}}=\left(
\begin{array}{ccccc}
U&0&0&.&0\\
V&I&0&.&0\\
.&.&.&.&.\\
0&0&0&.&I
\end{array}\right).$$

\noindent
Last the transition matrix for the map induced by the word $\omega$ is given below.

$$M_{\omega}=\left(
\begin{array}{ccccc}
W&0&0&.&0\\
0&I&0&.&0\\
0&0&I&.&0\\
.&.&.&.&.\\
X&0&0&.&I
\end{array}\right)$$

\noindent
Then the matrix for the map $\phi_m$ is given below by matrix multiplication after making the identifications $WU=Y$ and $XK=Z$.

$$M_{\phi_m}=\left(\begin{array}{ccccccc}
0&Y&0&0&.&.&0\\
0&V&I&0&.&.&0\\
0&0&0&I&.&.&0\\
.&.&.&.&.&.&.\\
.&.&.&.&.&.&.\\
0&0&0&0&.&.&I\\
I&Z&0&0&.&.&0
\end{array}\right)$$

The matrix $M$ is an $m\times m$ block matrix of $r+s+1\times r+s+1$ blocks.  The matrix $Y$ depends on the word $\omega$.  The matrix $V$ may have non-zero entries in the last column except that the last row must be zero since $\gamma\cap\rho(\gamma)=\emptyset$.  The observation $V^2=0$ will be important later.

$$V=\left(\begin{array}{cccccc}
0&.&.&.&0&v_1\\
0&.&.&.&0&v_2\\
.&.&.&.&.&.\\
.&.&.&.&.&.\\
0&.&.&.&0&v_{rs}\\
0&.&.&.&0&0
\end{array}\right)$$

\noindent
The matrix $Z$ will depend on $\omega$ as well but only has non-zero entries in the last row.  Now we want to consider the matrix $M^m$.  Inductively we see that for $1<k<m$ the matrix $M^k$ is given by the following matrix.  Here we use the fact that $V^2$ is the zero matrix.

$$M_{\phi_m}^k=\left(
\begin{array}{cccccccccc}
0 &0&    0&    .& YV&   Y& 0& 0& .&0\\
.& .&    .&    .& 0&    V& I& 0& .&.\\
.& .&    .&    .& .&    0& 0& I& .&0\\
0& 0&    .&    .& .&    .& .& 0& .&I\\
I& Z&    0&    .& .&    .& .& .& .&0\\
0& Y+ZV& Z&    .& .&    .& .& .& .&.\\
.& YV&   Y+ZV& .& .&    .& .& .& .&.\\
.& 0&    YV&   .& .&    .& .& .& .&.\\
.& .&    0&    .& 0&    .& .& .& .&.\\
.& .&    .&    .& C&    0& .& .& .&.\\
0& 0&    0&    .& Y+ZV& Z& 0& 0& .&0
\end{array}\right)$$

\noindent
Then we can find the transition matrix for the $m$th iterate.

$$M^m=\left(
\begin{array}{ccccccc}
Y& YZ  &   0   &    .& .& 0   &    YV\\
V& Y+ZV&   Z   &    .& .& .   &    0\\
0& YV  &   Y+ZV&    .& .& .   &    .\\
.& 0   &   YV  &    .& .& .   &    .\\
.& .   &   0   &    .& .& 0   &    .\\
.& .   &   .   &    .& .& Z   &    0\\
0& .   &   .   &    .& .& Y+ZV&    Z\\
C& 0   &   0   &    .& .& YV  &   Y+ZV
\end{array}\right)$$

Then this matrix is Perron-Frobenius by Lemma \ref{PF-lem} and by Theorem \ref{columnsum-lem} the spectral radius is bounded by the largest column sum.  A block column sum is either equal to a column sum of $YV+Z+ZV$, $YZ+Y+ZV+YV$, or $Y+V+Z$.  Therefore the dilatation of the $m$th iterate is bounded by a constant, say $P$.  This tells us that

$$\log(\lambda(\phi_m))\leq \frac{P}{m}.$$

\noindent
Then Theorem \ref{lowerbound-thm} with the upper bound just given finishes the proof.$\Box$

\medskip

\noindent
Next we use this to prove Theorem \ref{main-thm}

\medskip

\noindent\textbf{Proof of Theorem \ref{main-thm}:}  With Penner's lower bound we only need the upper bound to prove the asymptotic behavior.  Suppose a ray has slope $\frac{p}{q}$ with $(p,q)=1$ then let $S_{g,n,1}$ have $g=p$, and $n=q$ and create any Penner sequence with $a^+$ being an arc on the boundary component and $a^-$ another disjoint arc on the same boundary component.  This sequence of mapping classes then has the required upper bound on dilatation.  This gives sequences on $gn$-rays through $(2,0)$.  Further if we choose our curves as in Figure \ref{proof-fig1}, which is shown with a chosen connecting curve as well, then we can find a train track for $\phi_m$ given in Figure \ref{proof-fig2}.

\begin{figure}\label{proof-fig1}
\centering
\includegraphics[scale=.5]{mapsmapsmaps.png}
\caption{Choice for curves}
\end{figure}

\begin{figure}\label{proof-fig2}
\centering
\includegraphics[scale=.35]{pqtrack.png}
\caption{Train track}
\end{figure}

From this we can see by Theorem \ref{transitionmatrix-lem} that the two fixed punctures are not 1-pronged.  Filling in both fixed punctures we obtain sequences of mapping classes with the same dilatation and two fewer punctures, the sequences for the lines passing through the origin.$\Box$

\bibliography{mybib}
\bibliographystyle{alpha}
\end{document}


TITLE: Is there any such way (analytically or computationally) to solve this system?
QUESTION [0 upvotes]: I have to fit some data to a nonlinear least squares fit. In doing so I need an initial guess for the function $y(\sigma)= x_1(x_2\sigma)^{x_3}$ . With some the data I get the following set of equations:
$x_1(x_2)^{x_3} = 3.29$
$x_1(2x_2)^{x_3} = 4.3$
$x_1(4x_2)^{x_3} = 7.17$
Is there any way of getting an initial guess for these values using analytical or numerical methods?

REPLY [2 votes]: Note that $x_2$ is extraneous.  You can write $y(\sigma)=x_1x_2^{x_3}\sigma^{x_3}$ and define $x_1'=x_1x_2^{x_3}$ to get down to two parameters.  Now you have a standard exponential fit.  If you take the log of the equation it becomes linear:
$$\log y(\sigma)=\log (x_1')+x_3 \log(\sigma)$$
Now you have an analytic solution if you are happy with least squares in the logs.


TITLE: What is a quasar flux ratio anomaly?
QUESTION [4 upvotes]: In the context of strongly lensed, multiply-imaged quasar observations, I sometimes hear about "flux ratio anomalies". What are these? Why are they important?

REPLY [4 votes]: A massive object (such as a galaxy) along the line of sight to a distant bright source (such as a quasar) bends the light along its path. If the "lensing" object is massive enough and the geometry is right, the background object can be seen as multiple sources. For instance, here is a galaxy (central point) and four images of a single quasar:

For a well-behaved, nicely symmetric lens the positions of the images yield a strong prediction for the relative flux (brightness) of each image. The images lie on a ring (with a radius called the "Einstein radius"). For example, if the top and bottom images appeared closer to the left-hand image in the picture above, the right-hand image would be relatively fainter. Systems that do not obey the predicted flux ratios are said to exhibit a flux ratio anomaly.
A flux ratio could have a number of causes:

Dust or other faint/diffuse structure along the line of sight to one of the images (but not the others) causing one of the images to dim.
If the background source (quasar) is time-varying, the flux ratios may be different because of the different light travel times for the different images. In other words, the same background quasar is seen simultaneously at different times.
Some asymmetric structure in the lens, or additional structure along the line of sight, can break the symmetry that yielded the flux ratio predictions in the first place.

This last possibility is of considerable interest, especially if the "substructure" perturbing the lens (or along the line of sight) is a low-mass dark matter halo ($\lesssim 10^{7}\,{\rm M}_\odot$). The existence of these halos is a fundamental prediction of the standard $\Lambda{\rm CDM}$ theory of cosmology. Conversely, such low mass halos are not expected to exist at all in $7\,{\rm keV}$ sterile neutrino warm dark matter models, the currently preferred "flavor" of warm dark matter, except perhaps for systems undergoing stripping by a more massive nearby structure. A detection, or strongly constrained non-detection, of such a low mass dark matter halo is therefore a strong test of cosmology.


text:
\begin{document}
\begin{abstract}
We study the distribution of the cokernels of adjacency matrices (the Smith groups) of certain models of random $r$-regular graphs and directed graphs, using recent mixing results of M\'esz\'aros.  We explain how convergence of such distributions to a limiting probability distribution implies asymptotic nonsingularity of the matrices, giving another perspective on recent results of Huang and M\'esz\'aros on asymptotic nonsingularity of adjacency matrices of random regular directed and undirected graphs, respectively.
We also remark on the new distributions on finite abelian groups that arise, in particular in the $p$-group aspect when $p\mid r$.
\end{abstract}

\maketitle

\section{Introduction}

The singularity problem in combinatorial random matrix theory states that if a square matrix $A_n$ of size $n$ is ``sufficiently
random'', then $A_n$ is non-singular asymptotically almost surely as
$n$ tends to infinity, in other words $p_n$, the probability of $A_n$
being singular, tends to 0. This problem has a rich history, for which
we now mention briefly. In the early 60s Koml\'os \cite{Komlos67}
showed that if the entries of $A_n$ take values $\{0,1\}$
independently with probability 1/2 then $p_n=O(n^{-1/2})$. This bound
was significantly improved by Kahn, Koml\'os and Szemer\'edi
\cite{KKSz95} to $p_n \le 0.999^n$ for random $\pm 1$ matrices, by Tao
and Vu \cite{TV07} to $p_n \le (\frac{3}{4}+o(1))^n$, by Rudelson and
Vershynin \cite{RV08}, and by Bourgain, Vu and Wood \cite{BVW10}
where it was shown that $p_n= (\frac{1}{\sqrt{2}}+o(1))^n$ for random
$\pm 1$ matrices. The methods of these results also give exponential
bounds for other more general iid ensembles. Since then, there have been
subsequent papers addressing the sparse cases, such as those by Wood
\cite{PWood12}, and by Basak and Rudelson \cite{AR17} where in the
later result the entries of $A_n$ can take values 0 with probability as large as
$1-O(\frac{\log n}{n})$. We refer the reader to \cite{TV10, PWood12, AR17} and
the references therein to various extension and application of the
singularity problem for the iid models.

This singularity problem views the $A_n$ as matrices over $\R$, but since the entries are integers they could also be viewed as elements of the field $\Z/p\Z$ for any prime $p$.  A matrix is singular mod $p$ exactly when its determinant is $0$ mod $p$, and so heuristically, one expects this to happen about $1/p$ of the time instead of $0\%$ of the time.  From this point of view, it is natural to ask more refined questions, including the limiting distribution of ranks of the matrices mod $p$, or the distribution of the cokernels $\Z^n/A_n(\Z^n)$ of the matrices, i.e. the failure of the matrices to be surjective as maps $A_n: \Z^n\ra \Z^n$.   The second author  determined the limiting distribution for the cokernels of random matrices, including the $A_n$ considered above \cite{Wood2015a}.  For square matrices,  these distributions have non-zero limiting behavior if one considers the Sylow $p$-subgroups of the cokernels for only finitely many primes at once.  However, for non-square matrices, recent work of the authors \cite{Nguyen2018a} finds the distribution of the entire cokernel.  The distributions of cokernels that arise are the Cohen-Lenstra distributions on finite abelian groups \cite{Cohen1984}.

In another direction, there
have been results studying the singularity problem for matrices with various dependency conditions on the entries. For
instance in \cite{Nguyen13} the first author studied random (dense) matrices of given
row sums, or in  \cite{AChW16} Adamczak, Chafai and Wolff studied
random matrices
with exchangeable entries.
Cook \cite{Cook2017} studied the singularity of adjacency matrices of random $r$-regular digraphs where he showed that $p_n =r^{-\Omega(1)}$
as long as $\min(r,n-r) \geq C \log^2 n$ for some absolute constant
$C$. A similar result was also established by Basak, Cook and
Zeitouni \cite{BCZ18} for sum of $r$ random permutation
matrices as long as $r \ge \log^{12-o(1)} n$. While these results are
highly non-trivial, the random matrices are still relatively
dense and so one might still believe that the matrices are still non-singular
with high probability. However, it has been conjectured that the
phenomenon continue to hold for extremely sparse matrices with some row and
column constraints. Let us mention here one such conjecture. For a positive integer $r$, let
$A_{n,r}$ be a uniformly distributed matrix on the set of all
$\{0,1\}$ square
matrices of size $n$ with $r$ ones in every row and column (i.e. $A_{n,r}$ is the
adjacency matrix of a random $r$-regular directed graph on $n$
vertices where loops are allowed.) Motivated by
Conjecture \ref{conj:sym} (to be mentioned below), the following has
been asked by Cook  \cite[Conjecture 2]{Cook2017}.
\footnote{It was originally stated as $p_n=O(n^{-c})$ for some absolute
    positive constant $c$.}

\begin{conjecture}\label{conj:non-sym} For any $3\le r\le n-3$ we have
  $p_n=o(1)$.
\end{conjecture}
For this model, the recent work by Litvak, Lytova, Tikhomirov,
Tomczak-Jaegermann and Youssef in \cite{Litvak2017} shows that $p_n \leq \frac{C\log^3r}{\sqrt{r}}$ as long as $C\leq r \leq
cn/\ln^2 n$ for some constants $c,C$. As a consequence, this bound implies that $p_n \to
\infty$ if $r \to \infty$. By a
more involved study of the structure of the eigenvectors of $A_{n,r}$, it has been
shown by the same goup of authors in \cite{Litvak2018a} that
asymptotically almost surely the rank of $A_{n,r}$ is at least $n-1$
as long as  $r>C$ for sufficiently large constant $C$. Finally, very
recently Huang \cite{Huang2018} confirmed Conjecture \ref{conj:non-sym}
for this regime of $r$ and proved the following.

\begin{theorem}\label{T:D} Conjecture \ref{conj:non-sym} is true for fixed $r$.
\end{theorem}

In this note we will give another proof of Theorem~\ref{T:D} (see Theorem~\ref{T:main2}), via determination of the distribution of the cokernels of related matrices. Specifically we show the following.

\begin{theorem}\label{T:sandpile2}
For an integer $r\geq 3$, and an integer $n,$ let $D_{n}$ be the sum of $r$ independent uniform random $n\times n$ permutation matrices.
For  a finite set $P$ of  primes not dividing $r$, and a finite abelian group $V$ such that $|V|$ is a product of powers of primes in $P$, we have
\begin{equation}\label{E:probs2}
\lim_{\substack{n\ra\infty}} \P( \cok D_n\tensor\prod_{p\in P} \Z_p \isom V)=
 \frac{1}{|\Aut(V)|}\prod_{p\in P} \prod_{k\geq 1}(1-p^{-k}).
\end{equation}

\end{theorem}

Our method to prove Theorems~\ref{T:sandpile2} (and the symmetric version, Theorem~\ref{T:sandpile} below) uses recent powerful mixing results of M\'esz\'aros \cite{Meszaros2018} to find the moments of this distribution and results of the second author  \cite{Wood2017} on the moment problem for random finite abelian groups.  We note that M\'esz\'aros \cite{Meszaros2018} proved his mixing results as a step towards determining the distribution of the cokernels of the Laplacians of the random graphs from Theorems~\ref{T:sandpile2} and Theorem~\ref{T:sandpile}, and so it is not surprising they are also useful for the closely related adjacency matrices.  The distribution in Theorem~\ref{T:sandpile2} is the Cohen-Lenstra distribution, as appears in the  case of independent entries \cite{Wood2015a} and in the Laplacian version of the underlying graphs \cite{Meszaros2018}.  However, unlike in these previous cases, the entire cokernels of $D_n$ are not distributed in the Cohen-Lenstra distribution, but rather only the prime-to-$r$ part of $\cok D_n$ follows the
Cohen-Lenstra distribution.  In Section~\ref{S:newd}, we find the limiting moments for the distribution of the entire cokernel and remark on some new and interesting behavior of these distributions at primes dividing $r$.

Now we discuss symmetric random matrices.
 Answering a question of Weiss, for symmetric random matrices $A_n$,  Costello, Tao and Vu
\cite{Costello06} showed in 2006 that if the upper diagonal entries of $A_n$
take values $\{0,1\}$ independently with probability 1/2 then
$p_n=O(n^{-\Omega(1)})$. This bound has been improved by the first author
in \cite{Nguyen2012} to $p_n =O(n^{-\omega(1)})$ and then to
$p_n=O(\exp(-n^c))$ in parallel by Vershynin \cite{Vershynin14}. For
sparse matrices,  it was shown by Costello and Vu \cite{CV08} that as
long as $A_n$ is the adjacency matrix of the Erd\H{o}s-R\'enyi graph
$G(n,p)$ with $\frac{(1+\epsilon)\log n}{n} \le p \le 1/2$, we have
$p_n=o(1)$. Here we note that the threshold $\frac{\log n}{n}$ is
optimal.
For Laplacians of Erd\H{o}s--R\'{e}nyi
random graphs, Clancy, Leake, and Payne \cite{Clancy2015a}, conjectured a limiting distribution for their cokernels, which was proven by the second author \cite{Wood2017}.

For symmetric matrices with further constraints, a popular
conjecture, first appeared in \cite[Question
10.1]{CV08} by Costello and Vu, and then subsequently in \cite[Conjecture
8.4]{Vu2008}, \cite[Conjecture 5.8]{Vu2014}, and \cite[Section 9,
Problem 7]{Frieze2014}, says that the adjacency matrix of random
regular graphs are non-singular asymptotically almost surely.

\begin{conjecture}\label{conj:sym} Let $3\le r\le n$. Let $G(n,r)$ be a uniform random
  $r$-regular simple graph on the vertex set $\{1,\dots,n\}$ and let
  $A_{n,r}$ be the adjacency matrix of $G(n,r)$, then $p_n=o(1)$.
\end{conjecture}

Compared to the non-symmetric model, this problem is much less studied. There is recent work by Landon, Sosoe and Yau
\cite{Landon2016} where it can be deduced that $p_n=o(1)$ as long as $r \ge n^\varepsilon$
for any given $\varepsilon>0$. It seems plausible that the method
there can be extended all the way to $r = \omega(\log n)$, but it
seems to the current authors that the singularity problem for smaller
$r$ requires significantly new ideas.
M\'esz\'aros \cite{Meszaros2018} has recently shown that $p_n\ra 0$ for the adjacency matrices of the random multi-graphs on an even number of vertices that are given by the union of $r$ independent perfect matchings, as well as determining the cokernel distribution of the Laplacians of theses graphs.  In this note we prove the following result on the distribution of cokernels of the adjacency matrices, using results from \cite{Meszaros2018}  and \cite{Wood2017} as discussed above.
\begin{theorem}\label{T:sandpile}
For an integer $r\geq 3$, and an even integer $n,$ let $C_n$ be the adjacency matrix of the multigraph given by taking the union of $r$ independent uniform random perfect matchings on $n$ labelled vertices.
For  a finite set $P$ of odd primes not dividing $r$, and a finite abelian group $V$ such that $|V|$ is a product of powers of primes in $P$, we have
\begin{equation}\label{E:probs}
\lim_{\substack{n\ra\infty\\ n \textrm{ even}}} \P( \cok C_n\tensor\prod_{p\in P} \Z_p \isom V)=
 \frac{\#\{\textrm{symmetric, bilinear, perfect } \phi:V\times V \ra \C^* \}}{|V||\Aut(V)|}\prod_{p\in P} \prod_{k\geq 0}(1-p^{-2k-1}).
\end{equation}
\end{theorem}
The distribution in Theorem~\ref{T:sandpile} is the same as appears in the  cases of independent entries, Laplacians of Erd\H{o}s--R\'{e}nyi
random graphs \cite{Wood2017}, and in the Laplacian version of the underlying graphs in Theorem~\ref{T:sandpile} \cite{Meszaros2018}.
We also explain in Section~\ref{S:ns} how the limiting distribution of cokernels of these matrices can be used to give a different proof of M\'esz\'aros's result on the asymptotic nonsingularity of the $C_n$, and point out that M\'esz\'aros's result \cite[Proposition 5]{Meszaros2018} has the following simple corollary for the uniform $r$-regular graphs.
\begin{theorem}\label{T:main}
Given a fixed $r\geq 3$, and $p_n$ as in Conjecture~\ref{conj:sym}, we have
$$
\lim_{\substack{n\ra\infty\\ n \textrm{ even}}} p_n=0.
$$
\end{theorem}

\subsection{Notation}
For a prime $p$, we write $\Z_p$ for the $p$-adic integers.
For a finite abelian group $V$ and a prime $p$, we write $V_p$ for $V\tensor \Z_p$.
If $V$ is a finite abelian group, then $V_p$ is the Sylow $p$-subgroup of $V$.
If $V=\Z^k \times T$, for a finite abelian group $T$, then $V_p=\Z_p^k \times T_p$.
For a set of primes $P$, we let $V_P=\prod_{p\in P}V_p$.

For two groups $G,H$, we write $\Hom(G,H)$ for the set of group homomorphisms from $G$ to $H$ and $\Sur(G,H)$ for the set of surjective group homomorphisms from $G$ to $H$.  We write $\Aut(G)$ for the set of group automorphisms of $G$.  We use $\isom$ to denote an isomorphism of groups.

For an $n\times n$ integer matrix $M$, we write $\cok M$ for $\Z^n/M(\Z^n)$, i.e. the cokernel of the map $M:\Z^n\ra \Z^n$.

We use $\P$ for probability and $\E$ for expectation.
\section{Cokernel distributions for non-symmetric matrices: proof of Theorem~\ref{T:sandpile}}

In this section we give the proof of Theorem~\ref{T:sandpile}.
A.~M\'esz\'aros \cite{Meszaros2018} has recently studied the sandpile group of the graphs in the theorem, and we first explain a key result of his that we will use.

To start, we let $V$ be any abelian group. We consider vectors $q\in V^n=\Hom(\Z^n, V)$, with coordinates $q_i\in V$ that give the image of the $i$th standard basis vector.
For a vector $q\in V^n$, let $\operatorname{MinCos}_q$ be the minimal coset of $V$ containing all the entries $q_i$ of $q$.  If $\operatorname{MinCos}_q=\gamma +H$ for $\gamma\in V$ and $H$ a subgroup of $V$, we write $r\cdot \operatorname{MinCos}_q$ for $r\gamma +H$.
For $s\in V^n$, we define $\langle q,s \rangle\in V\tesnor V$ to be $\sum_{i=1}^n q_i\tensor s_i$. By definition of $C_n$, we have that
$\langle q,qC_n \rangle$ is a sum of elements of the form $\langle q,q' \rangle$ where $q'$ is obtained from $q$ by performing an involution with no fixed points on the coordinates.  Thus $\langle q,qC_n \rangle$ is in the subgroup $I_2(V)$ of $V\tensor V$ generated by elements of the form $a\tensor b+b \tensor a$.
Let $R^S(q,r)=\{s\in (r\cdot \operatorname{MinCos}_q)^n | \langle q,s \rangle \in I_2(V) \textrm{ and } \sum_{i=1}^n s_i = r \sum_{i=1}^n q_i \}$.

Note that $q C_n \in R^S(q,r)$.
M\'esz\'aros \cite[Theorem 4]{Meszaros2018} proved that
 $$
\lim_{\substack{n\ra\infty\\ n \textrm{ even}}}  \sum_{\substack{q\in V^n
}} \max_{s\in R^S(q,r)} |\P(q C_n =s)-|R^S(q,r)|^{-1} |=0.
 $$
M\'esz\'aros used this result to determine the moments of the cokernel of the graph Laplacian of $\hat{G}(n,r)$, and we can similarly use it to determine the moments of the cokernel $\cok C_n$ of the adjacency matrix.
 Note that $q\in \Hom(\Z^n, V)$ is surjective if and only if the coordinates $q_i$ of $q$ generate $V$, and that $q$ descends to a homomorphism from $\cok C_n$ if and only if $q C_n=0$.  So $\E(|\Sur(\cok C_n,V)|)$  is exactly the expected number of $q\in V^n$ such that $q C_n=0$ and the $q_i$ generate $V$ (as in \cite[Section 3]{Wood2017} or \cite[Proposition 34]{Meszaros2018}).
We conclude
\begin{align*}
\lim_{\substack{n\ra\infty\\ n \textrm{ even}}} \E( |\Sur(\cok C_n,V)|)&= \lim_{\substack{n\ra\infty\\ n \textrm{ even}}} \sum_{\substack{q\in \Sur(\Z^n,V)
\\ 0\in R^S(q,r)
}}
|\P(q C_n=0)|\\
&= \lim_{\substack{n\ra\infty\\ n \textrm{ even}}} \sum_{\substack{q\in \Sur(\Z^n,V)
\\ 0\in R^S(q,r)
}}
|R^S(q,r)|^{-1}. \\
\end{align*}

Now we suppose that $|V|$ is odd, and we will show that for $q\in \Sur(\Z^n,V)$ with $\operatorname{MinCos}_q=V$, we have
$|R^S(q,r)|=|V|^{n-1}/|\wedge^2 V|,$ where $\wedge^2 V$ is the abelian group that is the quotient of $V\tensor V$ by the subgroup generated by elements of the form $a\tensor a$.  Note that since $|V|$ is odd, we have that $I_2(V)$ is the subgroup generated by elements of the form $a\tensor a$, and so $V/I_2(V)=\wedge^2 V$.  Our claim about the size of $|R^S(q,r)|$ will then follow from the fact that the map
\begin{align*}
V^n &\ra \wedge^2 V \times V\\
s &\mapsto \left( \sum_{i=1}^n q_i \wedge s_i , \sum_{i=1}^n s_i \right)
\end{align*}
is surjective.  The surjectivity to the first factor follows, because for $v,w\in V$, if $v=\sum_{i=1}^n a_iq_i$ for $a_i\in \Z$ (which we have from the surjectivity of $q$), then $s$ with $s_i= a_iw$ maps to $v\wedge w$ in the first factor.
Note if $s \mapsto (b,c)$ then if $s'$ has $s'_i=s_i$ for $i\neq j$ and $s_j=s_j+ q_j$, then
$s'\mapsto (b,c+q_j)$, and it follows the map above is surjective.

For the rest of this proof, we assume that $V$ is as in the theorem statement.
We will count the number of $q\in \Sur(\Z^n,V)$
such that $0\in\R^S(q,r)$.  Note that $0\in\R^S(q,r)$ is equivalent to $0\in r\cdot\operatorname{MinCos}_q$ and  $r \sum_{i=1}^n q_i=0$.
Since $r$ is relatively prime to $|V|$, that is equivalent to $\operatorname{MinCos}_q$ being a subgroup and $\sum_{i=1}^n q_i=0$.
Note that $q\in \Sur(\Z^n,V)$ implies that the $q_i$ generate $V$, so any subgroup that all the $q_i$  belong to in fact must be $V$.
Conversely, if $q\in\Hom(\Z^n,V)$ has $\operatorname{MinCos}_q=V$, then $q\in \Sur(\Z^n,V)$.
So
$$
\#\{ q\in \Sur(\Z^n,V) | 0\in\R^S(q,r) \}\leq  \#\{ q\in \Hom(\Z^n,V) | \sum_{i=1}^n q_i=0 \}=|V|^{n-1}
$$
and
$$
\#\{ q\in \Sur(\Z^n,V) | 0\in\R^S(q,r) \}\geq \#\{ q\in \Hom(\Z^n,V) | \sum_{i=1}^n q_i=0 \} - \sum_{\substack{\gamma+H\\ \textrm{ proper coset of $V$}}}
\#\{ q\in  (\gamma+H)^n \}.
$$
Note that
$$\sum_{\substack{\gamma+H\\ \textrm{ proper coset of $V$}}} |H|^n=o(|V|^n)$$ (where the constant in the little $o$ notation may depend on $|V|$).
So
$$
\#\{ q\in \Sur(\Z^n,V) | 0\in\R^S(q,r) \}=|V|^{n-1}+o(|V|^n).
$$

 Putting together the above, for any $V$ as in the theorem statement, we conclude that
\begin{align*}
\lim_{\substack{n\ra\infty\\ n \textrm{ even}}} \E( |\Sur(\cok C_n,V)|)&=
  |\wedge^2 V|.
\end{align*}
  Let $H_n$ be a random $n\times n$ symmetric matrix with entries on and above the diagonal independently drawn from Haar measure in $\prod_{p\in P} \Z_p$.
 By \cite[Corollary 9.2]{Wood2017}, we have
 \begin{align*}
\lim_{\substack{n\ra\infty}} \P( \cok H_n\isom V)&=
 \frac{\#\{\textrm{symmetric, bilinear, perfect } \phi:V\times V \ra \C^* \}}{|V||\Aut(V)|}\prod_{p\in P}\prod_{k\geq 0}(1-p^{-2k-1}).
\end{align*}
By \cite[Theorem 11]{Clancy2015} (or see \cite[Theorem 6.1]{Wood2017}) we have
 \begin{align*}
\lim_{\substack{n\ra\infty}} \E( |\Sur(\cok H_n,V)|)&=|\wedge^2 V|.
\end{align*}
Thus by \cite[Theorem 8.3]{Wood2017}, which says that these limiting moments determine a unique limiting distribution, we conclude for every integer $a$ that is a product of powers of the primes in $P$,
 \begin{align*}
\lim_{\substack{n\ra\infty\\ n \textrm{ even}}} \P( \cok C_n \tensor \Z/a\Z \isom V)=\lim_{\substack{n\ra\infty}} \P( \cok H_n\tensor \Z/a\Z \isom V).
\end{align*}
Note that if $W$ is a finitely generated abelian group and $V$ is a finite abelian group  as  in the theorem statement, and $V$ has exponent $a=\prod_{p\in P } p^{e_p}$, then $V\tensor \Z/a'\Z\isom V$, and further, we have $W_P \isom V$ if and only if
$W \tensor \Z/a'\Z \isom V$, where $a'=\prod_{p\in P } p^{e_p+1}$. From this we conclude the theorem.

\section{Directed graph (non-symmetric matrix) analog: proof of Theorem~\ref{T:sandpile2}}\label{S:directed}

In this section, we give a proof of Theorem~\ref{T:sandpile2}, which is  simpler than our proof of Theorem~\ref{T:sandpile} above.

We start with a result of M\'esz\'aros \cite{Meszaros2018}, which he proved in order to determine the asymptotic distribution of the sandpile groups with sink of the directed graphs associated to $D_{n}$.
Let $V$ be any abelian group. We consider vectors $q\in V^n=\Hom(\Z^n, V)$, with coordinates $q_i\in V$ that give the image of the $i$th standard basis vector.
Recall the notation $\operatorname{MinCos}_q$ from the proof of Theorem~\ref{T:main}.
Let $R(q,r)=\{s\in (r\cdot \operatorname{MinCos}_q)^n |  \sum_{i=1}^n s_i = r \sum_{i=1}^n q_i \}$.

Note that $q D_n \in R(q,r)$.
M\'esz\'aros \cite[Theorem 3]{Meszaros2018} proved that
\begin{equation}\label{E:M}
\lim_{\substack{n\ra\infty}}  \sum_{\substack{q\in V^n \\ q_i \textrm{ generate $V$}
}} \max_{s\in R(q,r)} |\P(q C_n =s)-|R(q,r)|^{-1} |=0.
\end{equation}

As above in the undirected case and in \cite{Meszaros2018} for the graph Laplacian, we use this to determine the moments of $\cok D_n$.
 As in the proof of Theorem~\ref{T:sandpile}, we  conclude
\begin{align*}
\lim_{\substack{n\ra\infty}} \E( |\Sur(\cok D_n,V)|)&= \lim_{\substack{n\ra\infty}} \sum_{\substack{q\in \Sur(\Z^n,V)
\\ 0\in R(q,r)
}}
|\P(q D_n=0)|\\
&= \lim_{\substack{n\ra\infty}} \sum_{\substack{q\in \Sur(\Z^n,V)
\\ 0\in R(q,r)
}}
|R(q,r)|^{-1}. \\
\end{align*}

For $q\in \Sur(\Z^n,V)$ with $\operatorname{MinCos}_q=V$, we have
$|R(q,r)|=|V|^{n-1}$.
For the rest of this proof, we assume that $r$ and $|V|$ are relatively prime.
The proof of Theorem~\ref{T:main} shows that for
$q\in \Sur(\Z^n,V)$, and $r$ relatively prime to $|V|$, and $0\in\R^S(q,r)$, we have $\operatorname{MinCos}_q=V,$ and further shows that
$$
\#\{ q\in \Sur(\Z^n,V) | 0\in\R(q,r) \}=|V|^{n-1}+o(|V|^n).
$$

For $V$ as in the theorem statement, we have concluded above that
\begin{align*}
\lim_{\substack{n\ra\infty}} \E( |\Sur(\cok D_n,V)|)&=
1.
\end{align*}

Thus by \cite[Theorem 3.1 and Proof of Corollary 3.4]{Wood2015a} (see also \cite[Proposition 8.3]{Ellenberg2016} for the case $P=\{p\}$), we conclude the theorem.

\section{Nonsingularity over $\R$}\label{S:ns}
In this section, we show the relationship between limiting cokernel distributions and non-singularity.  The key is that singular matrices can be detected by their infinite cokernels, even after tensoring with $\Z_p$.

\begin{lemma}\label{L:Fatou}
If $G_n$ for $n\geq 0$ is a sequence of random abelian groups, and for each finite abelian group $W$, the limit
$\lim_{n\ra\infty} \P(G_n\isom W)$ exists, and $\mu$ defined by $\mu(W):= \lim_{n\ra\infty} \P(G_n\isom W)$ for each finite abelian group $W$ is a probability measure on the set finite abelian groups, then $\lim_{n\ra\infty} \P(G_n \textrm{ is infinite})=0$.
\end{lemma}

\begin{proof}
We have
\begin{align*}
\lim_{\substack{n\ra\infty}}  \P(G_n \textrm{ is finite})
&=\lim_{\substack{n\ra\infty}}  \sum_{W \textrm{ fin. ab. group}} \P(G_n\isom W)\\
&\geq \sum_{W \textrm{ fin. ab. group}}\lim_{\substack{n\ra\infty}}  \P(G_n\isom W)\\
&= \sum_{W \textrm{ fin. ab. group}} \mu(W)\\
&=1,
\end{align*}
where the inequality is given by Fatou's Lemma.
\end{proof}

However, it does not suffice to apply Lemma~\ref{L:Fatou} to the cokernels of our sequences of matrices, because for all the sequences $A_n$ of matrices considered in this paper we have $\lim_{n\ra\infty} \P(\cok A_n\isom W)=0$ for each finite abelian group $W$ (since $\prod_{p \textrm{ prime}} (1-p^{-1}))=0$, see the argument for \cite[Corollary 9.3]{Wood2017}).  However, if we tensor our cokernels with $\Z_p$ we can use the following.

\begin{corollary}\label{C:getsing}
If $p$ is a prime, $A_n$ is a sequence of random integral matrices, and $\mu$ a probability measure on finite abelian $p$-groups such that $\lim_{n\ra\infty} \P((\cok A_n)_p\tensor W)=\mu(W)$ for every finite abelian $p$-group, then
$\lim_{n\ra\infty} \P(\det A_n=0)=0$.
\end{corollary}
\begin{proof}
If $M$ is an $n\times n$ integral matrix with $\det M=0$, then $\cok M\isom \Z^k \oplus T$ for some $k\geq 1$ and finite abelian group $T$, and in particular
$(\cok C_n)_P=(\cok C_n)\tensor \Z_p$ is infinite, and so the theorem follows.
\end{proof}

Next we will see how the above results can be used to give proofs of Theorem \ref{T:main} and the directed analog.  We only further need the facts that the limiting distribution is a probability distribution, and arguments of contiguity and conditioning to move between different models of random matrices.  We remark that the proofs of Huang \cite{Huang2018} and M\'esz\'aros \cite{Meszaros2018} only use the $\Z/p\Z$-moments of the cokernels distributions for infinitely many primes $p$ to establish asymptotic nonsingularity.  We have used all of the moments of the cokernel distributions to determine the entire distribution of the cokernels above, but for the nonsingularity argument only need the part that follows from the $W$-moments (i.e. $\E(|\Sur(-,W)|$) for each finite abelian $p$-group $W$ for a single prime $p$.

\begin{proof}[Proof of Theorem~\ref{T:main}]
We first use work of Bollob\'as \cite{Bollobas1980}, to replace $G(n,r)$ with a random multi-graph $G^*(n,r)$ given as follows (see \cite[Corollary 2.18]{Bollobas2001}).  We take $nr$ half-edges
labelled $(i,j)$ for $1\leq i\leq n$ and $1\leq j\leq r$, and then choose a uniform random perfect matching on these $nr$-half edges.  Then we create the random $r$-regular multi-graph $G^*(n,r)$ on vertex set $\{1,\dots,n\}$ by making an edge $ab$ for each pair $(a,j),(b,k)$ in the matching.  If we condition on $G^*(n,r)$ having no loops or multiple edges, then we get exactly $G(n,r)$.  Let $G'(n,r)$ be given by $G^*(n,r)$ conditioned on having no loops.  Since, for fixed $r$, the probability that
$G^*(n,r)$ has a loop or multiple edge is bounded away from $1$ (see \cite[Proof of Theorem 2.16]{Bollobas2001}), it  suffices to prove the theorem with $G(n,r)$ replaced by $G^*(n,r)$ or $G'(n,r)$.

By work of Janson, $G'(n,r)$ is contiguous with the random multi-graph  $\hat{G}(n,r)$, which is given
by taking the union of $r$ independent uniform random perfect matchings on the vertex set $\{1,\dots,n\}$ (\cite[Theorem 11]{Janson1995}, see also \cite[Theorem 3]{Molloy1997}).  To say that two sequences of random graphs are \emph{contiguous} means that a property that holds asymptotically almost surely for one sequence holds asymptotically almost surely for the other sequence.  Thus is suffices to replace $G'(n,r)$ with $\hat{G}(n,r)$.  Note that the adjacency matrix of $\hat{G}(n,r)$ is the matrix $C_n$ in Theorem~\ref{T:sandpile}.  (M\'esz\'aros \cite[Proposition 5]{Meszaros2018} gives asymptotic nonsingularity of these matrices, and thus Theorem~\ref{T:main}
follows in this way, though we provide a different approach below.)

For a fixed odd prime $p$, not dividing $r$, let $\mu(V)$ denote the right-hand side of Equation~\eqref{E:probs} when $P=\{p\}$.  We will now show that $\mu$ gives a probability measure on finite abelian $p$-groups.
Let $\mathcal{A}$ denote the set of (isomorphism classes of) pairs $(V,\delta)$ where $V$ is a finite abelian $p$-group and $\delta$ is a
symmetric, bilinear, perfect pairing $\delta: V \times V \ra \C^*$.
By \cite[Proposition 7]{Clancy2015}, we have
$$
\sum_{(W,\delta)\in \mathcal{A}} \frac{1}{|W||\Aut(W,\delta)|}=\prod_{k\geq 0}(1-p^{-2k-1})^{-1}.
$$
We have that $\Aut(W)$ acts naturally on the set $P_W$ of symmetric, bilinear, perfect pairings on $V$, with stabilizer of $\delta$ being $\Aut(W,\delta)$,
and its orbits are in bijection with isomorphism classes $(W,\delta)\in \mathcal{A}$ such that $W\isom V$.
By the orbit-stabilizer theorem, for each finite abelian $p$-group $V$, we have
$$
\sum_{\substack{(W,\delta)\in \mathcal{A}\\ W\isom V}} \frac{1}{|W||\Aut(W,\delta)|}=
\sum_{\delta\in P_W} \frac{1}{|W||\Aut(W)|},
$$
and so, summing over $W$, we have
$$
\prod_{k\geq 0}(1-p^{-2k-1})^{-1}=\sum_{W \textrm{ fin. ab. $p$-group}} \frac{\#\{\textrm{symmetric, bilinear, perfect } \phi:W\times W \ra \C^* \}}{|W||\Aut(W)|}.
$$
Thus, we conclude $\mu$ is a  probability measure on finite abelian $p$-groups, and the theorem follows from Corollary~\ref{C:getsing}.
\end{proof}

In the nonsymmetric matrix case we can conclude the following.
\begin{theorem}\label{T:main2}
Given an integer $r\geq 3$ and an integer $n$, let $D_{n}$ be the sum of $r$ independent uniform random $n\times n$ permutation matrices.
We have
$$
\lim_{\substack{n\ra\infty}} \P(\det D_n=0)=0.
$$
The theorem is also true if we replace $D_n$ by the adjacency matrix of a uniform random $r$-regular (in-degree $r$ and out-degree $r$) random directed graph on $n$ labeled vertices.
\end{theorem}

A version of Theorem~\ref{T:main2} for the configuration model of random regular directed graphs was  proven by Huang \cite[Theorem 1.1]{Huang2018} (and other models including Theorem~\ref{T:main2} can follow from his work by the same types of contiguity and conditioning arguments that allow us to move between models).

\begin{proof}[Proof of Theorem~\ref{T:main2}]
For a finite abelian $p$-group $V$, let $\mu'(V)$ denote the right-hand side of Equation~\eqref{E:probs2} when $P=\{p\}$.
The measure $\mu'$ is  the well known Cohen-Lenstra probability measure on finite abelian $p$-groups (see e.g. \cite{Hall1938} for a proof that is is a probability measure).
So the first statement of the theorem follows from Corollary~\ref{C:getsing}.

Let $\Gamma(n,r)$ be a uniform random $r$-regular (in-degree $r$ and out-degree $r$) random directed graph on $n$ labeled vertices.
Let $\bar{\Gamma}(n,r)$ be the random multi-graph given by a union of $r$ independent uniform 1-regular (no loops or multiple edges) directed graphs on $n$ labeled vertices.
Let  $\tilde{\Gamma}(n,r)$ be $\bar{\Gamma}(n,r)$ conditioned on no multiple edges.
By \cite[Section 4]{Molloy1997}, we have that $\Gamma(n,r)$ is contiguous with $\tilde{\Gamma}(n,r)$ (see also \cite[Section 4]{Janson1995}), so it suffices to prove the theorem for adjacency matrices of $\tilde{\Gamma}(n,r)$.
  Since the probability that $\bar{\Gamma}(n,r)$ has multiple edges is bounded away from  $1$ (e.g. see \cite[Theorem 7]{Janson1995}),
  it suffices to prove the theorem for adjacency matrices of $\bar{\Gamma}(n,r)$.
  If we let $\Gamma^*(n,r)$ be the random graph whose adjacency matrix is the sum of $r$ independent uniform random $n\times n$ permutation matrices,
  then $\bar{\Gamma}(n,r)$ is $\Gamma^*(n,r)$ conditioned on no loops.  Since the probability that $\Gamma^*(n,r)$ has loops is bounded away from $1$ (e.g. see \cite[Theorem 7]{Janson1995}),
  it suffices to prove the theorem for adjacency matrices of $\Gamma^*(n,r)$, which is what we have already done.
\end{proof}

\section{New distributions}\label{S:newd}
The matrices $D_n$ and $C_n$ from Theorems~\ref{T:sandpile2} and \ref{T:sandpile} have each row and column summing to $r$.
So when $p$ is a prime such that $p\mid r$, these matrices are always singular mod $p$, and $(\cok D_n)_p$ and $(\cok C_n)_p$ are never trivial, and in particular these cokernels  are not distributed in the familiar distributions given in Theorems~\ref{T:sandpile2} and \ref{T:sandpile}.  In this section we give some remarks on the distribution in the non-symmetric matrix case, and find the limiting moments of this distribution.

For integers $r,m$, we define an \emph{$(r,m)$-pair} to be a pair $(G,C)$, where $G$ is an abelian group $G$, and $C$ is a coset of $G$, such that the following conditions hold: if we write $C=\gamma+H$, where $\gamma\in G$ and $H$ is a subgroup of $G$, then (1) $r(G/H)=0$, (2) $\gamma$ generates $G/H$, and (3) $mr\gamma\in rH$.  Note that if $G$ is finite and $r$ is relatively prime to $|G|$, then condition (1) implies $H=G$ and thus $C=G$.

We note that $\cok D_n=\Z^n/D_n(\Z^n)$ is not just an abelian group, but naturally has the structure of an $(r,n)$-pair with coset $e_1+E$, where $e_i$ are the images of the standard generators of $\Z^n$, and $E$ is the subgroup generated by the elements $e_i-e_j$ for $1\leq i <j \leq n$.  Condition (2) would be satisfied for any matrix.  If we let $d_{i,j}$ be the entries of $D_n$, then for each $j$, we have $\sum_i d_{i,j}=r$.  Since in $\cok D_n$, we have $\sum_i d_{i,j} e_i=0$, in $(\cok D_n)/E$, we have $re_1=0$,  and thus condition (1) is satisfied.  Since $\sum_{i,j} d_{i,j} e_i=0$, we have $0=nre_1+\sum_{i,j} d_{i,j} (e_i-e_1)=nre_1+\sum_{i} r(e_i-e_1)$, which gives condition (3).

For abelian groups $G,G'$ with cosets $C,C'$ respectively, we let $\Sur((G,C),(G',C'))$ be the set of surjective group homomorphisms from $G$ to $G'$ in which $C$ has image $C'$.  For a random $(r,m)$-pair $(G,C)$, we define its moments to be indexed by fixed $(r,m)$-pairs $(A,B)$, and for the $(A,B)$-moment to be $\E(|\Sur((G,C),(A,B))|).$
For a group $G$, we let $G[r]$ denote the subgroup of elements $g\in G$ such that $rg=0$.

\begin{theorem}\label{T:sandpile3}
Let $r$ and $m$ be positive integers. Let $D_n$ be as in Theorem~\ref{T:sandpile2}, and $e_1$ and $E$ as above.  Let $(V,B)$ be an $(r,m)$-pair such that $B$ is a coset of the subgroup $H$ of $V$.
Then
$$
\lim_{\substack{n\ra\infty\\ m|n}} \E(|\Sur((\cok D_n,e_1+E),(V,B))|)= |H[r]|.
$$
\end{theorem}

\begin{proof}
Let $\tilde{e}_i$ be the standard generators of $\Z^n$ and $\tilde{E}$ the subgroup generated by $\tilde{E}$.
Note that a surjection from $(\cok D_n,e_1+E)$ to $(V,B)$ is exactly given by surjection $q:(\Z^n,\tilde{e}_1+\tilde{E})\ra (V,B)$  (we write $q\in V^n$ as in the proof of Theorem~\ref{T:sandpile2}) such that $q D_n=0$.
A surjection $q:(\Z^n,\tilde{e}_1+\tilde{E})\ra (V,B)$ is exactly a surjection $q: \Z^n \ra V$ such that $\operatorname{MinCos}_q=B$.  (Recall the notation from the proof of Theorem~\ref{T:sandpile}, and note that the minimal coset containing elements $q_i$ is exactly $q_1+H$, where $H$ is the subgroup generated by all the $q_i-q_j$.)

We write $B=\gamma+H$, for some $\gamma\in V$.
For $q\in \Sur(\Z^n, V)$ with $\operatorname{MinCos}_q=B$, we next determine $|R(q,r)|$.
(Recall the notation from the proof of Theorem~\ref{T:sandpile2}.)
 By condition (1), we have $r\cdot B=H$.  We have that $R(q,r)$ is the set of $s\in H^n$ such that $\sum_{i=1}^n s_i = r\sum_{i=1}^n q_i$.  By condition (1), we have $rq_i\in H$, and so   $|R(q,r)|=|H|^{n-1}$.

Next we determine the number of $q\in \Sur(\Z^n, V)$ with $\operatorname{MinCos}_q=B$ such that $0\in R(q,r)$.  Note that $0\in R(q,r)$ is equivalent to $0\in r\cdot \operatorname{MinCos}_q$ and $r\sum_{i=1}^n q_i=0$.  For $q$ with $\operatorname{MinCos}_q=B$, we have that $0\in r\cdot \operatorname{MinCos}_q$ is implied by condition (1) on $B$.  If we let $q_1,\dots, q_{n-1}$ be any choices of elements in $B$, then we will count the number of  $q_n\in B$ such that
$r\sum_{i=1}^n q_i=0$.  This is the same as the number of $h$ such that $r(\sum_{i=1}^{n-1} q_i +\gamma + h)=0$, which is the same as the number of $h'$ such that
$r(n\gamma +h')=0$.  By condition (3) and $m\mid n$, we have $rn\gamma\in rH$, and thus there are $H[r]$ choices of $q_n$ satisfying $r\sum_{i=1}^n q_i=0$.
We conclude there are $|H|^{n-1}|H[r]|$ choices of $q\in B^n$ such that $r\sum_{i=1}^n q_i=0$.
  By condition (2), we have $q\in V^n$ is a surjection if and only if the $q_i-q_j$ generate $H$.  Also, for $q\in B^n$, we have $\operatorname{MinCos}_q=B$ if and only if the $q_i-q_j$ generate $H$.
If we choose $q_1$ and then $q_i-q_i$ for each $i$, we see that the number of $q\in B^n$ such that the $q_i-q_j$ do not generate $H$ is a most
$$
|H| \sum_{\substack{H' \textrm{proper}\\ \textrm{subgroup of $H$}}} |S|^{n-1}=o(|H|^n),
$$
where the constant in the little $o$ notation depends on $H$.  So we conclude that the number of $q\in \Sur(\Z^n, V)$ with $\operatorname{MinCos}_q=B$ such that $0\in R(q,r)$ is $|H|^{n-1}|H[r]|+o(|H|^n)$.

Thus, using Equation~\eqref{E:M}, we conclude
\begin{align*}
\E(|\Sur((\cok D_n,e_1+E),(V,B))|)&=  \sum_{\substack{q\in \Sur(\Z^n,V)\\\operatorname{MinCos}_q=B
\\ 0\in R(q,r)
}}
|\P(q D_n=0)|\\
&=  \sum_{\substack{q\in \Sur(\Z^n,V)\\\operatorname{MinCos}_q=B
\\ 0\in R(q,r)
}}
|R(q,r)|^{-1} \\
&= |H[r]|+o(1).
\end{align*}
\end{proof}

It is natural to combine these moments into the usual moments of finite abelian groups.
For a finite set of primes $P$ and an integer $n$, we define $n_P$ to be the largest divisor of $n$ that is a product of powers of primes in $P$.

\begin{corollary}\label{C:mom}
Let $r$ and $m$ be positive integers. Let $D_n$ be as in Theorem~\ref{T:sandpile2}.  Let $P$ be a finite set of primes and $V$ be a finite abelian group whose order is a product of powers of primes in $P$.
Then
$$
\lim_{\substack{n\ra\infty\\ n_P=m}} \E(|\Sur(\cok D_n,V)|) =\sum_{\substack{\gamma+H \textrm{coset of $V$}\\
(V,\gamma+H) \textrm{ an $(r,m)$-pair}
}} |H[r]|.
$$
\end{corollary}

\begin{proof}
It suffices to show that every surjection $\phi:\cok D_n \ra V$ has $(V,\phi(e_1+E))$ an $(r,m)$-pair, so each surjection is a surjection to some $(r,m)$-pair.  Since $r((\cok D_n)/E)=0$, we have
$r(V/\phi(E))=0$.  Since $e_1$ generates $(\cok D_n)/E$, we have that $\phi(e_1)$ generates $V/E$.  Since $nre_1 \in rE$, we have
$nr\phi(e_1) \in r\phi(E)$, and since $n_P=m$ that implies $mr\phi(e_1) \in r\phi(E).$
\end{proof}

We notice two interesting features of the moments in Theorem~\ref{T:sandpile3}.
If $r$ is relatively prime to the $p\in P$, then we noted above that $(V,V)$ is the only $(r,m)$-pair, and the moments are all $1$.
The first interesting feature, as expected above, is that the moments will be bigger when $r$ and $V$ are not relatively prime.  For example, if $V=\Z/p\Z$, and $p\mid r$,
then each non-zero element $\gamma\in V$ gives an $(r,m)$-pair with trivial $H$, and we also have the $(r,m)$-pair $(V,V)$ with $|V[r]|=p$, and so we have total limiting $\Z/p\Z$-moment $2p-1$.

We also note that for $r$ and $V$ fixed, the moments also can depend on $n_P$, i.e. there is not necessarily a limit as $n\ra\infty$ if we include all $n$.  This is why we sort $n$ by the values of $n_P$ in Corollary~\ref{C:mom} above.  For example, let $r=p$ and $V=\Z/p^2\Z$.  The $(r,p)$-pairs have cosets $B=V$ and $B=\gamma+p\Z/p^2\Z$ for $\gamma\not\in p\Z/p^2\Z$ (there are $p-1$ choices of the latter cosets).  However, the only $(r,1)$-pair is with coset $B=V$, because the cosets $B=\gamma+p\Z/p^2\Z$ do not satisfy condition (3), as $p\gamma$ is not trivial but $p (p\Z/p^2\Z)$ is trivial. So for $n$ with $n_p=1$, the $\Z/p^2\Z$-moment approaches $p$, but for
$n$ such that $p\mid n$, the $\Z/p^2\Z$-moment approaches $p^2$.

In conclusion, for each $r$ and $m$ we see new distributions on finite abelian groups, and it would be interesting to better understand these distributions.

\subsection*{Acknowledgements}
The authors thank Nick Cook for helpful comments on an earlier version of this manuscript.
During this work, the first author was partially supported by National
Science Foundation grants DMS-1600782 and DMS-1752345.
During this work, the second author was supported by a Packard Fellowship for Science and Engineering, a Sloan Research Fellowship,  National Science Foundation grants DMS-1652116 and DMS-1301690, and a Vilas Early Career Investigator Award.

\newcommand{\etalchar}[1]{$^{#1}$}

config: arxiv


TITLE: Limit of the integrals of a $\left\{f_{k}\right\}_{k \in \mathbb{N}}$ decreasing succession of integrable measurable functions,
QUESTION [0 upvotes]: Let $\left(X,M,\mu\right)$ be  a measurable space and $f_{k}:X\rightarrow \left[0,\infty\right]$  with $k \in \mathbb{N}$  a  decreasing succession of measurable functions with $\int_{X}f_{k}d\mu<\infty$ for all $k \in \mathbb{N}$.
Show that $$\lim_{k\rightarrow\infty}\int_{X}f_{k}d\mu=\int_{X}\lim_{k\rightarrow\infty}f_{k}d\mu$$
Remark: I have tried the following:
We know that if $f_{k}\geq f_{k+1}$ then $\int_{X}f_{k}d\mu \geq \int_{X}f_{k+1}d\mu$. Therefore:
$$\liminf_{k\rightarrow\infty}f_{k}=\lim_{k\rightarrow\infty}f_{k}\; \;\mbox{ y }\; \; \liminf_{k\rightarrow\infty}\int_{X}f_{k}d\mu=\lim_{k\rightarrow\infty}\int_{X}f_{k}d\mu$$
Hence, by Fatou's Lemma we have $$\int_{X}\lim_{k\rightarrow\infty}f_{k}d\mu\leq\lim_{k\rightarrow\infty}\int_{X}f_{k}d\mu$$
The problem is to prove the other inequality.

REPLY [0 votes]: The other direction is the "easy" direction: it follows directly from definitions.
Write out the definition of the integral as the supremum of the integrals over simple functions below $f$. Now what can you say about the integral of each $f_k$ compared to the integral of $f$?
This is a common feature in Lebesgue integration arguments. Equalities are proved using two inequalities, and one inequality is usually free from the definitions.
p.s. Don't forget you need to address measurability of $\lim_{k\rightarrow\infty} f_k$ before you can do anything.


TITLE: It would be nice if someone has some idea! (A Diophantine system associated with a network flow)
QUESTION [1 upvotes]: Assume that we are given a connected network flow with n nodes, $\{1, ..., n\}$, and m arcs. For each arc, say $x_{ij}$ from node i to node j, there is a maximum capacity level given as $M_{ij}$. Assume node 1 as the source node and node n as the sink node. Assume that $x_{ij}$ denotes the amount of flow from node i to node j. I am wondering if there is any algorithm to solve the following system.
1) $\sum_{j} x_{1j}=k$    (i.e., the outgoing flow of the source node is equal to k)
2) $\sum_{j} x_{jn}=k$   (i.e., the incoming flow of the sink node is equal to k)
3) $\sum_{j} x_{ij} = \sum_{l} x_{li}$ , for i=2, ..., n-1. (i.e., the outgoing flow from each node is equal to the incoming flow to that node).
4) $x_{ij} \leq M_{ij}$
Please note that I do not like to use the network characteristics to solve the system. In fact, I need an algorithm to solve the above mentioned system as a Diophantine system to find all the solutions as m-tuple vectors. I just need to have the integer system's solutions.
I appreciate you all in advance.

REPLY [1 votes]: First way You can first solve the related equations to the source and sink nodes, and then complete the solutions by going through the related equations to the other nodes.
Second way You may consider all the possible solutions according to $x_{ij} \leq M_{ij}$, and then check each solution for the flow-conversation law.


TITLE: Exercise 1.64 of Megginson's "An Introduction to Banach Space Theory".
QUESTION [3 upvotes]: Could someone verify whether my solution to the following exercise is correct? The reason I am a bit in doubt is because the chapter of which this Exercise is part consists of the Banach-Steinhaus Theorem and Closed Graph Theorem, so I would expect I have to use those theorems. However, I guess my direct approach works.

Let $X$ be a Banach space and let $T : X \to \ell^{\infty} (\mathbb{N})$ be a linear operator. For each $n \in \mathbb{N}$, let $(Tx)_n$ denote the $n^{th}$ entry of $Tx$ and let $f_n$ be a linear functional on $X$ given by $x \mapsto (Tx)_n$. Prove that $T$ is bounded if and only if each $f_n$ is bounded.

My solution is the following:

Suppose $T$ is bounded. Then there exists a $M > 0$ such that $\| Tx \|_{\infty} \leq M \| x \|_X$ whenever $x \in X$. Therefore, it holds for all $x \in X$ that
    $$ |f_n(x)| = |(Tx)_n| \leq \sup_{n \in \mathbb{N}} |(Tx)_n| \leq M \|x\|_X, $$
    and hence $f_n$ is bounded.
Suppose $f_n$ is bounded. Then, for each $n \in \mathbb{N}$, there exists an $M_n > 0$ such that $|f_n(x)| \leq M_n \| x \|_X$. Let
    $$ K := \sup_{n \in \mathbb{N}} \frac{|f_n (x)|}{\|x\|_X} $$
    Then, for all $x \in X$,
    $$ \| Tx\|_{\infty} = \sup_{n \in \mathbb{N}} |f_n (x)| \leq K \| x \|_X ,$$
    and hence $T$ is bounded.

The next exercise is the same as the one above, but with $\ell^{\infty}$ replaced with $\ell^{1}$. I guess a similar approach works there?

REPLY [2 votes]: What you did for the first part is correct.
For the converse, however, there is a problem. You have to prove that the sequence $\left(M_n\right)_{n\geqslant 1}$ is bounded. Your $K$ depends on $X$, and the task is to prove that $K$ is finite.
You can solve the question using the uniform boundedness principle: since $Tx$ belongs to $\ell^\infty$ for each $x\in X$, we have that $\sup_{n\in\mathbb N}\left|f_n(x)\right|\lt+\infty$.


TITLE: The square root of $z^2$ is analytic?
QUESTION [1 upvotes]: Let us consider a disk $|z|<\epsilon$. Clearly, the square root of $z$ is not analytic on this disk. In order to show it, I compute the intergral over a path $\gamma$ containing $z=0$ in $Int(\gamma)$. But, what can I say of the principal branch of the square root of $z^2$? Is it holomorphic in the disk $|z|<\epsilon$?

REPLY [1 votes]: The proposed function is not continuous on the imaginary axis, negating any holomorphy claims on this basic level. Indeed, the square of $z=\pm\delta+ib$ wth $0<δ\ll|b|$ is
$$
z^2=δ^2-b^2\pm i2bδ.
$$
which lie on opposite sides of the negative real ray, the branch cut for the usual complex square root.
The sign the main branch of the square root takes values on the half-plane with positive real values, which would be
$$
+δ+ib\mapsto δ+ib\\
-δ+ib\mapsto δ-ib
$$
which gives a jump in the sign while crossing the imaginary axis.


TITLE: Ambiguous intersection notation: $\mathcal G(\mathbb R^{[0,1]})\cap C([0,1])$
QUESTION [0 upvotes]: I am attempting to understand a theorem from my lecture's notes, however it seems to me that the notation used there is ambiguous and therefore unclear.
The full statement and proof:

Where $\mathcal G(X)$ stands for open sets in the space $X$, $C[0,1]$ is the space of continuous functions on $[0,1]$, $\mathcal U(y,\epsilon| C([0,1]))$ is the $\epsilon$-neighborhood of the point $y$ with respect to the space $C([0,1])$
Now $\mathcal G(\mathbb R^{[0,1]})\cap C([0,1])$ is confusing as one is a set system and the other is a set of functions, i.e. is it $\{A\cap C([0,1]): A\in \mathcal G(\mathbb R^{[0,1]})\}$ ?
Thank you.

REPLY [2 votes]: Your thinking is correct. The point of the theorem is this: Given the (unmentioned) topologies for the space of functions from $[0,1]$ to $\mathbb{R}$ and the space of continuous functions from $[0,1]$ to itself (I suspect you've established some sort of natural choice for these topologies previously in the notes). If you take any open set in the space of functions with codomain $\mathbb{R}$ and intersect that with the (smaller) space of continuous functions the result is an open set in the topology for the latter space.


text:
\begin{document}
\normalsize

\title{\Large\bf Archimedeanization of ordered vector spaces}
\author{\bf Eduard Yu. Emelyanov\footnote{Middle East Technical University, Ankara, Turkey}}
\maketitle

{\bf Abstract:} {\normalsize
In the case of an ordered vector space (briefly, OVS) with an order unit, the Archimedeanization
method has been developed recently by Paulsen and Tomforde \cite{PT}. We present a general version of the
Archimedeanization which covers arbitrary OVS.}

{\bf MSC:} {\normalsize 46A40}

{\bf Keywords:} {\normalsize Ordered vector space, Pre-ordered vector space, Archimedean, Archimedean element,
almost Archimedean, Archimedeanization, Linear extension.}

\section{Preliminaries}

In the present paper, we always exclude 0 from natural numbers ($\nn:=\{1,2,...\}$)
and consider vector spaces only over reals.\\

A subset $W$ of a vector space $V$ is called a {\it wedge} if it satisfies
$$
  W+W\subseteq W
$$
and
$$
  rW\subseteq W\ \ \ \ (\forall\ r\ge 0).
  \eqno(1)
$$
A wedge $K$ is called a {\it cone} if
$$
  K\cap(-K)=\{0\}.
$$
A cone $K$ is said to be {\it generating} if
$$
  K-K=V.
  \eqno(2)
$$
Given a cone (resp., a wedge) $V_+$ in $V$, we say that $(V,V_+)$ is an {\it ordered vector space}
(briefly an OVS) (resp., a {\it pre-ordered vector space} or a POVS). The partial ordering (resp., partial pre-ordering)
$\le$ on $V$ is defined by
$$
  x\le y \ \ \text{\rm whenever} \ \ y-x\in V_+\,.
$$
$(V,V_+)$ is also denoted by $(V,\le)$ or simply by $V$, if the partial ordering $\le$ is well understood.
We say that $e\in V$ is an {\it order unit} if for each $x\in V$ there is some $n\in\nn$ such that
$x\le ne$. Clearly, $V_+$ is generating if $V$ possesses an order unit.

For every $x,y$ in a POVS $(V,\le)$, the (possibly empty, if $x\not\le y$) set
$$
  [x,y]:=\{z:x\le z\le y\}
$$
is called an {\it order interval}. A subset $A\subseteq V$ is said to be {\it order convex}
if $[a,b]\subseteq A$ for all $a,b\in A$. Any order convex vector subspace of $V$ is called an {\it order ideal}.
If $Y$ is an order ideal in a POVS $V$, the quotient space $V/Y$ is partially ordered by:
$$
  [0]\le [f] \ \ \text{\rm if} \ \ \exists q\in Y \ \text{\rm with} \ \  0\le f+q\ .
$$
The order convexity of $Y$ is needed for the property
$$
  [f]\le [g] \le [f] \ \Rightarrow [f]=[g]\,,
$$
which guarantees that $[V_+]_Y:=V_++Y$ is a cone in  $V/Y$.

Let $x_n\in V$ and $u\in V_+$. The sequence $(x_n)_n$ is said to be {\it $u$-uniformly convergent}
to a vector $x\in V$, in symbols $x_n\stackrel{(u)}{\to}x$, if
$$
  k_n(x_n-x)\in[-u,u]
$$
for some sequence $k_n\uparrow\infty$ in $\nn$. Clearly, $x_n\stackrel{(u)}{\to}x$ iff
$$
  x_n-x\in[-\varepsilon_nu,\varepsilon_nu]
$$
for some sequence $\varepsilon_n$ of reals such that $\varepsilon_n\downarrow 0$. A subset $A\subseteq V$ is said to be
{\it uniformly closed} in $V$ if it contains all limits of all $u$-uniformly convergent sequences $(x_n)_n$ in $A$ for all $u\in V_+$.

\begin{definition}\label{D1}
A wedge $K$ in a vector space $V$ is said to be {\rm Archi\-me\-dean} $($also we say that the POVS $(V,K)$ is {\rm Archi\-medean}$)$, whenever
$$
  (\forall x,y\in V)\bigg[[(\forall n\in\nn) \ x-ny\in K]\Rightarrow [y\in -K]\bigg]\,.
  \eqno(3)
$$
\end{definition}
\noindent
Clearly, $(3)$ in Definition \ref{D1} can be replaced by
$$
  (\forall x\in V_+,y\in V)\bigg[[(\forall n\in\nn) \ x-ny\in K]\Rightarrow [y\in -K]\bigg]\,.
  \eqno(3')
$$
It is obvious that any subspace of an Archimedean POVS is Archi\-medean.
It is also worth to notice that if $V$ admits a linear topology $\tau$ for which
a wedge $K\subseteq V$ is closed, then $K$ is Archi\-medean (cf. \cite[Lem.2.3]{AT}).
In the case when $\tau$ is Hausdorff and a wedge $K\subseteq V$ has nonempty $\tau$-interior,
$K$ is Archi\-medean if and only if it is $\tau$-closed (cf. \cite[Lem.2.4]{AT}).

\begin{asser}\label{A1}
An OVS $(V,V_+)$ is Archimedean if and only if
$$
  (\forall y\in V_+)\inf_{n\ge 1}\frac{1}{n}y=0\,.
$$
\end{asser}

Clearly, one may define the Archimedean property by saying that, whenever $x\in V_+$
and $y\in V$ with $0\le x+ny$ for all $n\in\nn$, then $0\le y$. This observation
leads to the following definition, that is motivated by the definition of an {\it Archimedean order unit}
in \cite[Def. 2.7]{PT}.

\begin{definition}\label{D2}
If $(V,V_+)$ is a POVS and $x\in V_+$, we say that $x$ is an {\rm Archimedean element} if,
whenever $y\in V$ with $x+ny\in V_+$ for all $n\in\nn$, then $y\in V_+$.
\end{definition}
\noindent
Notice that $x=0$ is always an Archimedean element. Clearly, $V$ is Archimedean
if and only if every element $x\in V_+$ is Archimedean. Furthermore, if $0\le z\le x$ and $x$ is Archimedean, then $z$
is Archi\-me\-dean. In particular, in any POVS $V$ with an Archimedean order unit, all elements of $V_+$ are Archimedean.

\begin{definition}\label{D3}
A wedge $W$ in a vector space $V$ is said to be {\rm almost Archime\-dean}
$($also we say that the POVS $(V,W)$ is {\rm almost Archi\-medean}$)$ if
$$
  (\forall x,y\in V)\bigg[[(\forall n\in\zz) \ x-ny\in W]\Rightarrow [y=0]\bigg]\,.
  \eqno(4)
$$
\end{definition}
\noindent
Clearly, any $x\in V$ satisfying $[(\forall n\in\zz) \ x-ny\in W]$ in (4) is indeed an element of $W$.
OVS which are not almost Archimedean given in the following example.

\begin{exam}\label{E0}
Let $V=\rr^2$ be ordered by the cones
$$
  K_1:=\{(x,y)|x>0\}\cup\{0\}\,, \ \ K_2:=\{(x,y)|x>0\}\cup\{(0,y)|y\ge 0\}\,.
$$
The OVS $(V,K_2)$ is denoted by $(\rr^2,\le_{lex})$ and called the {\em Euclidean plane with
the lexicographic ordering}.
\end{exam}
\noindent

Given a POVS $(V,V_+)$, denote
$$
  N_V:=\{x\in V|(\exists y\in V)(\forall n\in\nn)[-y\le nx\le y]\}
  \eqno(5)
$$
the set of all {\em infinitely small elements} of $V$. The wedge $V_+$ is almost Archi\-medean iff $N_V=\{0\}$.
Since $V_+\cap (-V_+)\subseteq N_V$, we see that every almost Archi\-medean POVS is an OVS. Although, the wedge $V$
in a vector space $V\ne\{0\}$ is Archimedean but not almost Archi\-medean, any Archi\-medean OVS is clearly almost
Archi\-medean. The converse is not true even if $\dim(V)=2$ (cf. also Example \ref{E2}).

\begin{exam}\label{E1}
Let $\Gamma$ be a set containing at least two elements, and $V$ be the space of all bounded real functions
on $\Gamma$, partially ordered by
$$
  f\le g \ \ \text{\it if} \ \ \text{\it either} \ f=g \ \  \text{\it or} \ \ \inf_{t\in\Gamma}[g(t)-f(t)]>0\,.
$$
The OVS $(V,\le)$ is almost Archi\-me\-dean but not Archi\-me\-dean. It can be shown that
$\inf_{n\ge 1}\frac{1}{n}f$ does not exist for any $0\ne f\in V_+$.
\end{exam}

\begin{asser}\label{A2}
A wedge $K$ in a vector space $V$ is almost Archime\-dean if and only if it does not contain
a straight line, say $x+ty$ with $x,y\in V$, $y\ne 0$. Furthermore, a POVS $(V,\le)$
is almost Archimedean if and only if
$$
  \bigcap_{n\ge 1}\Bigl[-\frac{1}{n}x,\frac{1}{n}x\Bigr]=\{0\}
$$
for every $x\in V_+$.
\end{asser}
\noindent
The following assertion is immediate.
\begin{asser}\label{A3}
Any subcone of an almost Archimedean cone is almost Archi\-medean.
\end{asser}

It is worth to notice, and it follows directly from Definitions \ref{D1}, \ref{D3}, that the intersection of
any nonempty family of Archimedean (resp., almost Archimedean) wedges in a vector space is an Archimedean
(resp., almost Archimedean) wedge.

If $A$ is an order ideal in a POVS $(V,V_+)$ such that the POVS $(V/A,V_++A)$ is almost Archimedean,
then $A$ is uniformly closed (cf. the part (a) of Proposition \ref{P2}). When $A=\{0\}$, the converse
is true by the part (b) of Proposition \ref{P2}. Thus $V$ is almost Archimedean if and only if $\{0\}$
is uniformly closed. In general, the question, whether or not $V/A$ is Archimedean assuming an order
ideal $A$ to be uniformly closed, is rather nontrivial (it has a positive answer due to Veksler \cite{V}
in the vector lattice setting). An OVS $V$ is said to be a {\it vector lattice} (or a {\it Riesz space})
if every nonempty finite subset of $V$ has a least upper bound. It is well known that any almost
Archimedean vector lattice is Archimedean. In Example \ref{E1}, $V=V/\{0\}$ is not Archimedean although $\{0\}$
is uniformly closed. Similarly to Definition \ref{D3}, we introduce almost Archimedean elements.

\begin{definition}\label{D4}
If $(V,V_+)$ is an POVS and $x\in V_+$, we say that $x$ is an {\rm almost Archimedean element} if,
whenever $y\in V$ with $x+ny\in V_+$ for all $n\in\zz$, then $y=0$.
\end{definition}

We shall use also the following definition.

\begin{definition}\label{D5}
Let $V=(V,V_+)$ be an OVS. Consider the category $\mathcal{C}_{Arch}(V)$ whose objects are pairs $\left\langle U,\phi\right\rangle$,
where $U=(U,U_+)$ is an Archimedean OVS and $\phi:V\to U$ is a positive linear map $($that is: $\phi(V_+)\subseteq U_+$$)$, and morphisms
$\left\langle U_1,\phi_1\right\rangle\to\left\langle U_2,\phi_2\right\rangle$ are positive linear maps $q_{12}: U_1\to U_2$ such that
$q_{12}\circ \phi_1=\phi_2$. If $\mathcal{C}_{Arch}(V)$ possesses an initial object $\left\langle U_0,\phi_0\right\rangle$, then
$U_0$ is said to be an {\em Archimedeanization} of $V$.
\end{definition}

It should be noted that in Definitions \ref{D1}-\ref{D5} we use only the fact that $V$ is a {\em commutative ordered group},
where we replace the condition (1) in the definition of a wedge by
$$
  W\subseteq nW\ \ \ \ (\forall n\in\nn),
  \eqno(1')
$$
and positive linear maps in Definition \ref{D5} by group homomorphisms.

For further information on ordered vector spaces we refer to \cite{AT}.

\section{The Archimedeanization of an ordered vector space}

The following theorem is the main result of the present paper.

\begin{thm}\label{T1}
Any ordered vector space possesses a unique, up to an order isomorphism, Archime\-deanization.
\end{thm}

The proof is presented in the end of this section. First of all we introduce and discuss some
notions which may be considered as a generalization of the Archimedeanization method of
Paulsen and Tomforde \cite{PT}. Given a POVS $(V,V_+)$, denote
$$
  D_V:=\{x\in V|(\exists \xi\in V_+)(\forall  n\in\nn)\ nx+\xi\in V_+\}\,,
$$
and let $I_V$ be the intersection of all uniformly closed order ideals in $(V,V_+)$.

It is straightforward to verify, that $D_V\cap(-D_V)=N_V$, where $N_V$ was defined in (5),
$N_V$ is an order ideal in $(V,V_+)$, $D_V$ is a wedge, and $V_+\subseteq D_V$.
Consider the sets $V_++N_V=[V_+]_{N_V}$ and $D_V+N_V=[D_V]_{N_V}$ in the quotient space $V/N_V$.
Since $V_+$, $D_V$, and $N_V$ are wedges, then $V_++N_V$ and $D_V+N_V$ are wedges in $V/N_V$.
However, in Example \ref{E0}, $D_V$ is not a cone. Clearly,
$$
  (D_V+N_V)\cap -(D_V+N_V)=(D_V+N_V)\cap(-D_V+N_V)=
$$
$$
  D_V\cap(-D_V)=N_V.
$$
So, $D_V+N_V$ is a cone in $V/N_V$, and $V_++N_V\subseteq D_V+N_V$ is a cone as well.
Let us collect some further elementary properties.

\begin{prop}\label{P1}
Given an POVS $(V,V_+)$. Then

{\em (a)}\ $N_V$ is an order ideal in the POVS $(V,D_V)$$;$

{\em (b)}\ $N_V\subseteq I_V$$;$

{\em (c)}\ if $V$ possesses an order unit then $N_{V/A}=I_{V/A}$ for every ideal $A\subseteq V$$;$

{\em (d)}\ if $(V,V_+)$ is a Riesz space then $(V/N_V,D_V+N_V)$ is a Riesz space.
\end{prop}

{\bf Proof:}\
(a)\ Let $u,v\in N_V$, $w\in V$, with $u\le w\le v$ in $(V,D_V)$. Then
$$
  -x\le nu\le x\,, \ \ -y\le nv\le y \ \ \ (\forall n\in\nn)
  \eqno(6)
$$
for some $x,y\in V_+$, and
$$
  0\le n(w-u)+\xi_1\,, \ \ 0\le n(v-w)+\xi_2 \ \ \ (\forall n\in\nn)
  \eqno(7)
$$
for some $\xi_1,\xi_2\in V_+$. It follows from (6) and (7) that
$$
  0\le nw+x+\xi_1\,, \ \ 0\le -nw+y+\xi_2 \ \ \ (\forall n\in\nn).
$$
Then
$$
  -x+\xi_1\le nw\le y+\xi_2 \ \ \ (\forall n\in\nn)\,,
$$
and therefore
$$
  -(x+y+\xi_1+\xi_2)\le nw\le (x+y+\xi_1+\xi_2) \ \ \ (\forall n\in\nn)\,,
$$
which means that $w\in N_V$.

(b)\ Take an $f\in N_V$, then for some $g\in V$
$$
  -g\le nf\le g \ \ (\forall n\in\nn)
$$
or equivalently
$$
  -\frac{1}{n}g\le 0-f\le \frac{1}{n}g \ \ (\forall n\in\nn).
$$
Let $A\subseteq V$ be a uniformly closed order ideal. The sequence $a_n:=0\in A$ converges
$g$-uniformly to $f$. Thus, $f\in A$. Since $A$ is arbitrary, then $f\in N_V$.

(c)\ Denote by $e$ an order unit in $V$. Firstly, assume that $A=\{0\}$. Given $f\in N_V$,
then for some $g\in V$
$$
  -g\le nf\le g \ \ (\forall n\in\nn)
$$
and, since $g\le ce$ for some $c\in\rr$, one gets
$$
  -e\le nf\le e \ \ (\forall n\in\nn).
$$
We have to show that $N_V$ is uniformly closed. Suppose $f_n\stackrel{(u)}{\to}z$, then $f_n\stackrel{(e)}{\to}z$.
That is
$$
  -e \le k_n(f_n-z)\le e\ \ (\forall k\in\nn)
$$
for some $\nn\ni k_n\uparrow\infty$. But also
$$
  -e \le k_nf_n\le e\ \ (\forall k\in\nn).
$$
Hence
$$
  -2e \le k_nz\le 2e\ \ (\forall k\in\nn).
$$
Since $k_n\uparrow\infty$, then $z\in N_V$.

Now, let $A\subseteq V$ be an arbitrary ideal. Since $e$ is an order unit in $V$, then $[e]$ is an order unit in $V/A$.
So, $N_{V/A}=I_{V/A}$ as it was shown above.

(d)\ Denote by $u\vee v=\sup(u,v)$ and by $u\wedge v=\inf(u,v)$ the supremun and infimum of $u,v\in V$ in $(V,V_+)$.
Take $u,v\in V$. Since $u\vee v -u, u\vee v -v \in V_+$ and $V_+\subseteq D_V$, then $u\vee v -u\in D_V$ and $u\vee v -v\in D_V$.

Let $w-u\,, w-v\in D_V$, then
$$
  0\le n(w-u)+\xi_1\,, \ \ 0\le n(w-v)+\xi_2 \ \ \ (\forall n\in\nn)
$$
for some $\xi_1,\xi_2\in V_+$. Denoting $\xi=\xi_1+\xi_2$, we obtain
$$
  0\le(n(w-u)+\xi)\wedge(n(w-v)+\xi)=n((w-u)\wedge(w-v))+\xi=
$$
$$
  n(w+(-u)\wedge(-v))+\xi=n(w-u\vee v)+\xi\ \ \ (\forall n\in\nn)\,,
$$
which means that $w-u\vee v\in D_V$. As $N_V=D_V\cap(-D_V)$, the supremum of $[u],[v]$ exists, and is equal to $[u\vee v]$ in
the quotient OVS $(V/N_V,D_V+N_V)$. Since $u,v\in V$ are taken arbitrary, $(V/N_V,D_V+N_V)$ is a Riesz space.~$\blacksquare$

\begin{prop}\label{P2}
Given a POVS $(V,V_+)$. The following assertions hold true.

{\em (a)}\ Let $A\subseteq V$ be an order ideal. Then
$$
  V/A\ \ \text{\it is} \ \ \text{\rm almost}\ \ \text{\it Archimedean} \ \Rightarrow \ A \ \text{\it is} \ \
  \text{\it uniformly} \ \ \text{\it closed}\,.
$$

{\em (b)}\ \ $V \text{\it is} \ \ \text{\it almost}\ \ \text{\it Archimedean} \ \Leftrightarrow \ \ N_V=\{0\} \ \ \Leftrightarrow I_V=\{0\}$.

{\em (c)}\ If $(V,V_+)$ possesses an order unit then $(V/N_V,D_V+N_V)$ is an Archimedean OVS.

{\em (d)}\ If $(V/N_V, D_V+N_V)$ is Archi\-me\-dean then $(V/N_V, V_++N_V)$ is almost Archi\-me\-dean. In particular,
if $(V,V_+)$ possesses an order unit then $(V/N_V, V_++N_V)$ is almost Archimedean.
\end{prop}

{\bf Proof:}\
(a)\ Take a sequence $(y_n)\subseteq A$ such that $y_n\stackrel{(u)}{\to}x$ for some $u\in V_+$, and $x\in V$. We may assume that
$$
  n(y_n-x)\in [-u,u] \ \ \ (\forall n\in\nn).
$$
Thus
$$
  n[-x]=n[y_n-x]\in [-[u],[u]] \ \ \ (\forall n\in\nn).
$$
Since $V/A$ is almost Archimedean, $[-x]=[0]$, and hence $x\in A$.

(b)\ The implication ``$V$ is almost Archimedean $\Rightarrow\ \{0\}$ is uniformly closed'' follows from $(a)$.

The implication ``$\{0\}$ is uniformly closed $\Rightarrow\ I_V=\{0\}$'' is true due to the definition of $I_V$.

The implication ``$I_V=\{0\}\ \Rightarrow\ N_V=\{0\}$'' is true by the part $(b)$ of Proposition \ref{P1}.

The implication ``$N_V=\{0\}\ \Rightarrow\ V \ \text{\rm almost}\ \text{\rm Archimedean}$'' follows from Definition \ref{D3}.

(c)\ It has been proved in \cite[Thm.2.35]{PT}.

(d)\ It follows from Assertion \ref{A3}, as $V_++N_V$ is a subcone of an almost Archimedean cone $D_V+N_V$ in $V/N_V$.~$\blacksquare$\\

The following example shows that, for the cone $D_V+N_V$ in $V/N_V$ to be Archi\-me\-dean, existence of an order unit in $(V,V_+)$
is not essential.

\begin{exam}\label{E2}
Let $V$ be the vector space $\mathcal{P}(\rr)$ of all real polynomials on $\rr$ with the positive cone
$$
  V_+:=\{p\in V|(\forall t\in\rr) p(t)>0\}\cup\{p\ |p(t)\equiv 0\}.
$$
The OVS $(V,V_+)$ has no order unit, $N_V=\{0\}$, and $D_V:=\{p\in V|(\forall t\in\rr) p(t)\ge0\}$.
Take $x\equiv 1$, $y=-t^2$ in $V$. Then
$$
   x-ny\in V_+\ \ \ \ (\forall n\in\nn)
$$
but $y\not\in -V_+$. Thus $(V,V_+)$ is not Archime\-dean, however $N_V=\{0\}$.
The OVS $(V/N_V,D_V+N_V)=(V,D_V)$ is clearly Archi\-medean.

Notice that if we take $V=\mathcal{P}[0,1]$ with the same ordering, then $V$ becomes a non-Archime\-dean but almost
Archimedean OVS with an order unit and uniformly clo\-sed order ideal $N_V=\{0\}$. Moreover, it is well known that
this OVS satisfies the {\em Riesz decomposition property}, however is not a Riesz space.
\end{exam}

Now, consider the following example due to T. Naka\-yama (see, for instance, \cite[p.436]{LZ}), in which
the cone $D_V+N_V$ in $V/N_V$ is not even almost Archi\-me\-dean.

\begin{exam}[T. Nakayama]\label{E3}
Let
$$
  V=\{a=(a^1_k,a^2_k)_k|\ (a^1_k,a^2_k)\in(\rr^2,\le_{lex})\,,
  a^1_k\ne 0   \ \text{\it for}\ \text{\it finitely}\ \text{\it many}\ k\}\,.
$$
Then $V$ is a Riesz space with respect to the pointwise ordering and operations. Denote
$$
  A=\{a\in V|\ a^1_k=0 \ \text{\it for}\ \text{\it all}\ k\,, a^2_k\ne 0 \ \text{\it for} \ \text{\it finitely}\ \text{\it many}\ k\}\,,
$$
$$
  \ \text{\it and}\ \ B:=\{a\in V|\ a^1_k=0 \ \text{\it for}\ \text{\it all}\ k\}\,.
$$
Clearly $A\subseteq B$, $A\ne B$, and $A\subseteq N_V$. Furthermore, $B=I_V$ and $A=N_V$. Indeed,
take an $a\in B$, then
$$
  -u \le n(a-a_n)\le u\ \ (\forall n\in\nn)
$$
for
$$
  a_n:=\{(0,a^2_1),(0,a^2_2),\ldots,(0,a^n_1),(0,0,),(0,0,),\ldots\}\in A
$$
and $u:=(0,k|a^2_k|)_{k\in\nn}\in V$. So, $a_n\stackrel{(u)}{\to}a$ and $a\in I_V$, since $a_n\in N_V\subset I_V$.
Therefore, $B\subseteq I_V$. Denote $M_k=\{a\in V|\ a^1_k=0\}$ for $k\in\nn$.
Clearly, any $M_k$ is a uniformly closed ideal in $V$, and hence
$$
  I_V\subseteq\bigcap_{k\in\nn}M_k=B\,.
$$
So, $B=I_V$. In particular, $N_V\subseteq B$. But if $a\in N_V$ then
$$
  -b \le na\le b\in V\ \ \ (\forall n\in\nn)\,,
$$
and $b=(b^1_k,b^2_k)_{k\in\nn}$ with only finitely many $b^1_k\ne 0$. We may assume that if\ \ $b^1_k=0$
then also $b^2_k=0$. Hence $a=(a^1_k,a^2_k)_{k\in\nn}$ has all $a^1_k=0$ and only finitely many $a^2_k\ne 0$
Therefore $a\in A$ and, finally, $A=N_V$. Notice that
$$
  N_V=\{a\in V|\ (\forall k) a^1_k=0 \ \text{\it and}\ a^2_k\ne 0 \ \text{\it for}\ \text{\it finitely}\ \text{\it many}\ k\}.
$$
$$
  D_V=\{a\in V|\ (\forall k) a^1_k\ge 0 \ \text{\it and}\ a^2_k<0 \ \text{\it for}\ \text{\it finitely}\ \text{\it many}\ k\}.
$$
Thus, $(V/N_V,D_V+N_V)$ is not necessarily almost Archimedean even if $V$ is a Riesz space. To see this, consider $c_n\in V$ with
$$
  \bigg[c_n\bigg]_k=
  \begin{cases}
  (0,-\frac{1}{2^k}) & 1 \le k \le n \\
  (0,0)      & k>n
  \end{cases}.
$$
Then $c_n\in N_V$, and $c_n$ converges uniformly to $d\in V$, where
$$
  d=\bigg((0,-\frac{1}{2^k})\bigg)_{k=1}^{\infty}\not\in N_V.
$$
Thus, $N_V$ is not uniformly closed in $(V,V_+)$. Moreover, $c_n$ converges uniformly to $d$ in $(V,D_V)$,
however $d\not\in D_V$. In view of part $(2)$ of Proposition $\ref{P1}$, $(V/N_V,D_V+N_V)$ is a Riesz space.
So, by Veksler's result {\em \cite{V}}, the Riesz space $(V/N_V,D_V+N_V)$ is not Archime\-dean and therefore
not almost Archimedean.
\end{exam}

In connection with Example \ref{E2}, a question arises, whether or not any almost Archimedean
OVS $(V,V_+)$ satisfies the property that $(V,D_V)$ is Archimedean. Fortunately, it has a positive
answer as the following proposition shows.

\begin{prop}\label{P3}
Let $(V,V_+)$ be an almost Archimedean OVS. Then the OVS $(V,D_V)$ is Archimedean.
\end{prop}

{\bf Proof:}\
Assume in contrary that the cone $D_V$ is not Archimedean. Then for some $x,y\in V$, $x\in D_V$,
$$
  (\forall r>0)\,\   rx+y\in D_V \ \ \text{\rm but}\ \ y\not\in D_V\,.
$$
Since $x\in D_V$, then for some $v\in V_+$
$$
  (\forall \alpha>0)\, \ x+\alpha v\in V_+\,.
$$
Take
$$
  K=\spa (x,v,y)\cap V_+\,.
$$
Denote by $cl(C)$ the closure in Euclidean topology on $\spa(x,v,y)$ of $C\subseteq\spa(x,v,y)$.
Then $x\in\cl(K)$ and $rx+y\in\cl(K)$ for all $r>0$ and hence $y\in \cl(K)\subseteq D_V$
which is impossible. Therefore, $D_V$ is Archimedean.~$\blacksquare$

\begin{coro}\label{C1}
Any almost Archimedean cone can be embedded into an Archime\-dean cone.
Moreover, given a POVS $(V,V_+)$, then the OVS $(V/N_V,D_V+N_V)$ is Archimedean if and
only if the OVS $(V/N_V,V_++N_V)$ is almost Archimedean.
\end{coro}

{\bf Proof:}\ The first part is immediate by Proposition \ref{P3}. For the second part, denote $N=N_V$ and $D=D_V$.

The necessity follows from Assertion \ref{A3}, since $V_++N$ is a subcone of the Archimedean cone $D+N$ in $V/N$.

For the sufficiency, assume that $(V/N,V_++N)$ is almost Archi\-medean.
Then the cone $D_{V/N}$ is Archimedean in $V/N$, by Proposition \ref{P3}. Since
$$
  D+N=\bigg[D\bigg]_N=\bigg[\{u\in V|(\exists\ \xi\in V_+)(\forall n\in\nn)\ nu+\xi\in V_+\}\bigg]_N\subseteq
$$
$$
  \{[u]\in V/N|(\exists\ [\xi]\in V_++N)(\forall n\in\nn)[nu+\xi]\in V_++N\}=D_{V/N},
$$
the cone $D+N$ is almost Archimedean in $V/N_V=V/N$.~$\blacksquare$

Although, the first part Corollary \ref{C1} gives already an Archimedeanization of any almost Archimedean space $(V,V_+)$
(one only needs to replace $V_+$ by the intersection of all Archimedean cones containing $V_+$), our main result in this paper,
Theorem \ref{T1} do not require any restrictions on the cone $V_+$ and it covers also such OVS as $V$ in Example \ref{E3}
where $V/N$ still has nonzero infinitely small elements.

{\bf Proof of Theorem \ref{T1}:}\ Let $V=(V,V_+)$ be an OVS. Denote $N_0=\{0\}$,
$$
  N_1=N_V=\{x\in V|[x]_{N_0}=x\ \text{\rm is}\ \text{\rm infinitely}\ \text{\rm small}\ \text{\rm in}\ V/N_0=V\},
$$
$$
  N_{n+1}=\{x\in V|[x]_{N_n}\ \text{\rm is}\ \text{\rm infinitely}\ \text{\rm small}\ \text{\rm in}\ V/N_n\},
$$
and more generally
$$
  N_{\alpha}=N_{\alpha}(V)=\{x\in V|[x]_{\cup_{\beta<\alpha}N_{\beta}}\ \text{\rm is}\ \text{\rm infinitely}\
  \text{\rm small}\ \text{\rm in}\ V/_{\cup_{\beta<\alpha}N_{\beta}}\}
$$
for an arbitrary ordinal $\alpha>0$. It follows directly from the definition of $N_{\alpha}$ that
$$
  N_{\alpha_1}\subseteq N_{\alpha_2} \ \ \ (\forall \alpha_1\le\alpha_2).
$$
Take the first ordinal, say $\lambda_V$, such that $N_{\lambda_V+1}=N_{\lambda_V}$. Then the OVS\\
$(V/N_{\lambda_V},[V_+]_{N_{\lambda_V}})$ has no nonzero infinitely small elements and hence
is almost Archimedean. By Corollary \ref{C1}, the OVS
$$
  (V/N_{\lambda_V},[D_{V/N_{\lambda_V}}]_{N_{\lambda_V}})=(V/N_{\lambda_V},D_{V/N_{\lambda_V}})
$$
is Archimedean. Denote by $p_V$ the quotient map $V\to V/N_{\lambda_V}$, then $p_V$ is positive and linear.
For any other pair $\left\langle U,\phi \right\rangle$, where $U=(U,U_+)$ is an Archimedean OVS and
$\phi:V\to U$ is a positive linear map, we have that $\phi(x)\in N_U$ for every $x\in N_{\alpha}$,
where $\alpha$ is an arbitrary ordinal. Since every Archimedean OVS is almost Archimedean, $N_U=\{0\}$ and therefore
$N_{\lambda_V}\subseteq\ker(\phi)$. So, the map $\tilde{\phi}:V/N_{\lambda_V}\to U$ is well defined
by $\tilde{\phi}([x]_{N_{\lambda_V}})=\phi(x)$ and satisfies $\tilde{\phi}\circ p_V=\phi$.
Moreover, if $[x]_{N_{\lambda_V}}\in D_{V/N_{\lambda_V}}$, then
$$
  n[x]_{N_{\lambda_V}}+[\xi]_{N_{\lambda_V}}\in [V_+]_{N_{\lambda_V}} \ \ \ (\forall n\in\nn)
$$
for some $[\xi]_{N_{\lambda_V}}\in [V_+]_{N_{\lambda_V}}$. Applying $\phi$ gives
$$
  n\phi(x)+\phi(\xi)\in U_+ \ \ \ (\forall n\in\nn),
$$
and, since $(U,U_+)$ is Archimedean, it follows that $\phi(x)\in U_+$. Thus
$$
  \tilde{\phi}([x]_{N_{\lambda_V}})=\phi(x)\in U_+\,,
$$
and $\tilde{\phi}$ is a positive linear map. To show that $\tilde{\phi}$ is unique, take any $\psi:V/N_{\lambda_V}\to U$, that
satisfies $\psi\circ p_V=\phi$. Then
$$
  \psi([y]_{N_{\lambda_V}})=\psi(p_V(y))=\phi(y)=\tilde{\phi}(p_V(y))=\tilde{\phi}([y]_{N_{\lambda_V}})\ \ \ \ (\forall y\in V)\,,
$$
and hence $\psi=\tilde{\phi}$. Thus, $\left\langle(V/N_{\lambda_V},D_{V/N_{\lambda_V}}),p_V\right\rangle$
is an initial object of the category $\mathcal{C}_{Arch}(V)$. Hence, the OVS $(V/N_{\lambda_V},D_{V/N_{\lambda_V}})$
is an {\em Archimedeanization} of the OVS $(V,V_+)$.
~$\blacksquare$

In connection with the proof of Theorem \ref{T1}, the following question arises naturally.
Whether or not for any ordinal $\alpha$ there is an OVS $(V,V_+)$ for which $\alpha$
is the first ordinal such that $N_{\alpha+1}(V)=N_{\alpha}(V)$?


TITLE: Show that if $T(x)=\int_0^\infty e^{-t}t^{x-1}dt $, then $T(1)=1$
QUESTION [1 upvotes]: I need to calculate the following
Given:
$ T(x)=\int_0^\infty e^{-t}t^{x-1}dt $
I need to show that $ T(1)=1 $
Solution:
My logic was to plug $1$ in for $x$ before I integrated, but I am not sure if I can do that
So,
$$\int_0^\infty e^{-t}t^{1-1}dt=-e^{-t}\vert_0^\infty,$$ which gives me a final answer of 1

REPLY [2 votes]: What you've done is correct, indeed, in general if we have $T(x) = \int_{a}^b f(x, t) \, \mathrm{d}t$ then $T(1)$ is given by $\int_a^b f(1, t) \, \mathrm{d}t$.


TITLE: How the amount that is wagered influences the way bankroll goes?
QUESTION [0 upvotes]: I found the following question on the internet:

A player sits down at a gambling table to play a game with 1000 dollars. He bets 1 dollar at a time with probability of winning being equal to 49.3212% with a 1:1 payout. What is the probability that the player's bankroll reaches 1200 before he goes bust?

I think the question is pretty interesting. Here's the answer:
$$\frac{\left (\frac{1-p}{p}  \right ) ^{1000} - 1}{\left (\frac{1-p}{p}  \right ) ^{1200} - 1}= \frac{\left (\frac{0.506788}{0.493212}  \right ) ^{1000} - 1}{\left (\frac{0.506788}{0.493212}  \right ) ^{1200} - 1} \approx .004378132$$
Now the answer also states that if instead of betting 1 dollar each time the player decides to bet the following amounts, the probability of reaching 1200 before losing the initial 1000 dollars also changes as follows:

5: 0.336507
10: 0.564184
25: 0.731927
50: 0.785049
100: 0.809914

How are those numbers found? The initial formula doesn't include the amount wagered as a variable. So how are those different probabilities are found?

REPLY [1 votes]: The value 1000 in the formulas is \$1000 (the original amount) divided by \$1 (the wager). If you increase the bets to \$25, you have \$1000/\$25 = 40 bets' worth of money. I trust you can take it from here?


TITLE: Limits of Baire Spaces
QUESTION [3 upvotes]: A Baire space is a topological space in which intersection of open dense sets is dense.
Two of the ways in which we could obtain a Baire space is to have a complete metric space or a locally compact Hausdorff space.
Completeness does not seem to pass through limits, for example, $\{x\in L^{\infty}: ||x||\geq 1/n\}$ does not remain complete in the limit $n\to\infty$.
Question: Is it true that (countable) union of Baire spaces is Baire?
I cannot appeal to completeness because it might not hold under countable unions.

REPLY [4 votes]: This is not true, since $$\mathbb{Q} =\bigcup_{q\in\mathbb{Q}} \{q\} $$ and every $\{q\} $ is a Baire space.