TITLE: Corollary of Vitali's theorem from measure theory
QUESTION [0 upvotes]: Definition: Suppose $E\subset \mathbb{R}$ and and let $K=\{I_{\alpha}\}_{\alpha\in \Omega}$ be a family of closed nonempty intervals in $\mathbb{R}$. We say that $K$ is Vitali covering of $E$ if for any $x\in E$ and for any $\epsilon>0$ there is $I=I_{x,\epsilon}$ such that $x\in I$ and $\mu(I)<\epsilon$, where by $\mu$ I mean Lebesgue measure on the real line.

Vitali's theorem: Let $E$ be a bounded subset of $\mathbb{R}$ and let $K$ is a Vitali covering of $E$. Then there is countable family of
  pairwise disjoint $\{I_n\}_{n=1}^{\infty}\subset K$ such that
  $$\mu^*(E- \sqcup _{n=1}^{\infty}I_n)=0$$

Corollary: Under the assumptions of the above theorem one can find finitely many pairwise disjoint intervals $\{I_k\}_{k=1}^{n}\subset K$ such that $$\sum \limits_{k=1}^n\mu(I_k)>\frac{1}{2}\mu^*(E).$$
Can anyone explain to me how the corollary follows the theorem, please? It is not so clear to me .
EDIT: I was able to derive the following inequality: $$\mu^*(E)\leq \sum \limits_{k=1}^{\infty}\mu(I_k)$$

REPLY [1 votes]: Since $I=\cup_{n=1}^\infty I_n$ is measurable
$\mu^* E = \mu^*(E \setminus I) + \mu^*(E \cap I) \le \mu^* I = \mu I$.
Since $\lim_{N \to \infty} \cup_{n=1}^N \mu I_n = \mu I$ we can find some $N$ such that
$\cup_{n=1}^N I_n  > { 1\over 2} \mu I \ge {1 \over 2} \mu^* E$.


TITLE: How to solve $\int \:\frac{3x^2+1}{x^3+2}dx$?
QUESTION [0 upvotes]: I am trying to solve the following integral:
$$\int \:\frac{3x^2+1}{x^3+2}dx$$
I am not sure how to do it. I tried using partial fractions but you can not factor $x^3+2$, and changing the variable with $u$ does not seem to work either. Anyone has any ideas?

REPLY [0 votes]: $$I=\int \frac{3x^2+1}{x^3+2}dx=\int \frac{3x^2}{x^3+2}dx+\int \frac{dx}{x^3+2}$$
$$I=\log(x^3+2)+\int \frac{dx}{x^3+2}$$ For the remaining integral, let
$$x=\sqrt[3]2\, t\implies dx=\sqrt[3]2\,dt$$
$$J=\int \frac{dx}{x^3+2}=\frac{\sqrt[3]2}2 \int \frac{dt}{t^3+1}$$
Now
$$t^3+1=(t-1)(t-a)(t-b)$$ Partial fraction decomposition
$$\frac 1 {t^3+1}=\frac{1}{(a-1) (a-b) (t-a)}+\frac{1}{(b-1) (b-a) (t-b)}+\frac{1}{(a-1) (b-1) (t-1)}$$ Integrate to get a weighted sum of logarithms with complex values (the weights are complex too) and recombine them to get the answer already given.


TITLE: Joint Distribution applied to quadratic equations.
QUESTION [1 upvotes]: If i pick random number $b$ and $c$ from $[0,1]$ and then define $p(x) = x^2+bx+c$, what is the probability that p has two real roots?.
I've been thinking that it would be enough to know $P\{4c < b^2\}$, but i'm stuck there, i've been thinking about defining two random variables with uniform distribution for all the possible values of $b$ and $c$ but it doesn't seem to do any good.

REPLY [2 votes]: Denoting by $f_b$ the pdf of $b$, you can write
$$
P(c < b^2/4) = \int_0^1P(c < x^2/4\ |b = x)f_b(x)dx.
$$
Now, assuming that $b$ and $c$ are uniform distributed,  $f_b(x) = x \mathbb 1_{x\in (0,1)}$ and $P(c < x) = x$, so
$$
\int_0^1P(c < x^2/4\ |b = x) x  dx = \int_0^1 \frac{x^2}{4} x  dx = \frac 1{16}.
$$


text:
\begin{document}

\title{Morse-Smale flow, Milnor metric, and  dynamical zeta function}
\author{Shu Shen}
\author{Jianqing Yu}
\address{Institut de Math\'ematiques de Jussieu-Paris Rive Gauche,
Sorbonne Universit\'e,  4 place Jussieu, 75252 Paris Cedex 05, France.}

\email{shu.shen@imj-prg.fr
}
\address{School of Mathematical Sciences, University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui 230026, P. R. China.}
\email{jianqing@ustc.edu.cn}
\thanks {}

\date{\today}

\dedicatory{}

\begin{abstract}
	With the help of  interactions  between the fixed points and the closed orbits of a Morse-Smale	flow, we introduce  a
	Milnor metric on the determinant  line of  the cohomology of the
	underlying  closed manifold with coefficients in a flat vector
	bundle.
	This allows us to generalise the notion of the absolute value at zero
	point of the
	Ruelle dynamical zeta function, even in the case where  this
	value is not well
	defined in the classical  sense. 	We give a
	formula relating the Milnor metric and the Ray-Singer metric. An
	essential ingredient of our proof is  Bismut-Zhang's Theorem.
\end{abstract}

\maketitle
\tableofcontents
\settocdepth{section}
\section*{Introduction}
The study of the relation between the combinatorial/analytic torsion  of
a flat vector bundle and the Morse-Smale flow is initiated by Fried \cite{Friedconj} and
S\'anchez-Morgado \cite{MorgadoMorseSmale}.
In this paper, we give a formula relating
\begin{itemize}
	\item  a spectral invariant: the Ray-Singer metric associated
	to  a  flat  vector bundle with a Hermitian metric on a closed Riemannian
	manifold;
	\item  a dynamical  invariant: the Milnor metric which reflects
	the interactions 	between the fixed points and the
	closed orbits of the Morse-Smale flow, and  generalises the
	absolute value at zero point of the Ruelle dynamical zeta function;

	\item a transgressed Euler class:

the Mathai-Quillen current.
\end{itemize}

 \subsection{Background}
 Let $X$ be a connected closed   smooth   manifold. Let $F$ be a complex
 flat vector bundle of rank $r$ on $X$, which amounts to specifying a
representation
$\rho: \pi_1(X)\rightarrow \GL_r(\bC)$ of the
fundamental group $\pi_1(X)$.
Denote by $H^\cdot(X,F)$ the cohomology of
the sheaf of locally constant sections of $F$, and by $\lambda$ the
determinant line of $H^{\cdot}(X,F)$.

{Assume that  $H^{\cdot}(X,F)=0$ and that $F$ is equipped
with a flat metric, which is equivalent to say that  its holonomy
representation $\rho$ is unitary.
The Reidemeister torsion  \cite{ReidemeisterTorsion,FranzTorsion,dRTorsion}  is a positive real number
defined with the  help of a triangulation on $X$. However, it does not depend on the
triangulation and becomes a topological invariant.  It is the first
invariant that could distinguish closed manifolds such as lens spaces
which are homotopy equivalent but not homeomorphic.}

The analytic torsion was introduced by Ray and Singer
\cite{RSTorsion} as an analytic counterpart of the Reidemeister torsion.
In order to define the analytic torsion one has to choose a
Riemannian metric on $X$. The analytic torsion is a certain
weighted alternating product of regularized determinants of the Hodge
Laplacians  acting on the space of differential forms  with values in
$F$.

The celebrated Cheeger-M\"uller Theorem
\cite{Ch79,Muller78} tells us that the Ray-Singer analytic torsion
coincides with the Reidemeister combinatorial torsion. Bismut-Zhang \cite{BZ92} and M\"uller \cite{Muller2} simultaneously
considered generalisations of this result.
M\"uller \cite{Muller2} extended his result to the case where $F$ is
unimodular, i.e., $|\det
\rho(\gamma)|=1$ for all $\gamma\in \pi_{1}(X)$. Bismut and Zhang \cite[Theorem 0.2]{BZ92}  generalised the original Cheeger-M\"uller
Theorem to arbitrary flat vector bundles with arbitrary Hermitian
metrics.  Subsequently Bismut-Zhang's theorem was generalised to the
equivariant case  by the same authors  \cite{BZ94}, to the family case by
Bismut-Goette  \cite{BG01}, and to
manifolds with arbitrary boundaries by Br\"uning-Ma \cite{BruningMa_2012}.

Let us explain Bismut-Zhang's theorem \cite[Theorem 0.2]{BZ92} in more detail. Indeed, to
formulate their result in the case  where the flat vector bundle is not
necessarily  acyclic or unitarily flat, Bismut and Zhang introduced the so-called Ray-Singer metric, which  is a
metric on $\lambda$  defined as the product of
the analytic torsion with an $L^2$-metric on $\lambda$.
Also they introduced the Milnor metric on $\lambda$ which is  a combinatorial
metric associated to a  Morse-Smale gradient  flow. It generalises the
Reidemeister torsion to  the case where $F$ is neither  acyclic
nor unitarily flat. In this way,
they were able to extend the Cheeger-M\"uller Theorem to a comparison theorem of two metrics on $\lambda$, one is analytic and the other
one is combinatorial.

The study of the relation between the combinatorial/analytic torsion  and the dynamical system
can be traced  back to Milnor \cite{MilnorZcover}.  Fried
\cite{FriedRealtorsion}  showed that on hyperbolic manifolds the analytic torsion of an acyclic unitarily flat vector
bundle is equal to the  value at zero point   of the Ruelle dynamical zeta
function of the geodesic flow.  He conjectured
\cite[p.\,66~Conjecture]{Friedconj} that similar results should hold true for  more general
flows. In \cite{Shfried}, following previous contributions
by
Moscovici-Stanton \cite{MStorsion}, using Bismut's orbital integral formula
\cite{B09}, the  author  affirmed  the Fried
conjecture for  geodesic flows on  closed locally symmetric
manifolds. In \cite{Shen_Yu}, the authors made a further
generalisation to closed locally symmetric orbifolds.

Besides the gradient flow, Morse-Smale flow is the simplest structurally stable dynamical system
which has only two  types of  recurrent behaviors: closed orbits and
fixed points.  Fried \cite[Theorem 3.1]{Friedconj}
proved  his conjecture for the Morse-Smale flows without fixed points.
When compared with Bismut-Zhang's Theorem  \cite[Theorem 0.2]{BZ92}, it seems  nature to ask
whether there is a relation between the torsion invariant (or more generally the
Ray-Singer metric for non acyclic and non unitarily flat vector bundle) and a general Morse-Smale flow which has both
fixed points and closed orbits.

This is one of the motivations  of S\'anchez-Morgado's work
\cite{MorgadoMorseSmale}. He showed that the
heteroclinic orbits have a non trivial contribution in the torsion
invariant, and in this way he constructed  a counterexample to Fried's
conjecture on Seifert manifolds.

In this
paper, we introduce a new Milnor metric,
which indeed contains the heteroclinic  contributions and
generalises  the absolute  value at zero point of the Ruelle dynamical zeta function, and we
give a comparison theorem for the Milnor and Ray-Singer metrics on
$\lambda$. We believe
that in this way we  give  a complete answer in the affirmative to the above  question.

Let us  mention that there is another interpretation of the Ruelle dynamical zeta
function  provided by Dang-Rivi\`ere \cite{DR_MSR}. See also
\cite{DR_MSG, DR_MSI, DR_MSII, DR_MSW} for related works.

 \subsection{A new Milnor metric}
  A vector field $V$ is called Morse-Smale if $V$ generates a flow
  whose  nonwandering set is the union of a finite set $A$ of
  hyperbolic fixed points and a finite set $B$ of hyperbolic prime
  orbits, and if the stable and unstable manifolds of the critical
  elements in $A\coprod B$ intersect transversally.

  Let us take a Hermitian metric $g^{F}$ on $F$. In Section \ref{sSF},  we construct on $\lambda$   a   Milnor metric
$\|\cdot\|^{M,2}_{\lambda, V}$ using  long exact
sequences associated to a Smale
filtration of the Morse-Smale flow.  Note that
the long exact
sequences  encode the information about the interactions
between the critical elements in $A\coprod B$.

 If $V$ is a gradient,
then our Milnor metric is just the classical one as defined in
\cite{MilnorWhiteheadTor} and \cite[Definition 1.9]{BZ92}.

Our first result says that  the  Milnor metric
$\|\cdot\|^{M,2}_{\lambda, V}$ is a generalisation of  the absolute value at
zero point of
the Ruelle dynamical zeta function. For a closed orbit $\gamma\in B$,
let $\ell_{\gamma}\in \bR^{*}_{+}$ be its minimal period, and let
$\ind(\gamma)\in \mathbf{N}$  be its index
(see \eqref{indexg}). Take  $\Delta(\gamma)$ to be $1$ if
$\gamma$ is untwist and $-1$ in the contrary  case (see \eqref{Delta}). The  Ruelle dynamical zeta function
 is given by
\begin{align}\label{intoR}
	R_{\rho}(s)=\prod_{\gamma\in
	B}\det\(1-\Delta(\gamma)\rho(\gamma)^{-1}e^{-s
	\ell_\gamma}\)^{(-1)^{\ind(\gamma)}}.
\end{align}
Note that the above definition of the dynamical zeta function
corresponds to  Fried's definition  \cite{Friedconj} associated to
the holonomy of the dual bundle $F^{*}$.

\begin{prop}\label{prop}
	If $V$ does not have any fixed points, and if none of $\Delta(\gamma)$ is an eigenvalue of $\rho(\gamma)$, then
$H^{\cdot}(X,F)=0$, and the norm of the canonical section
$\mathbf{1}\in \bC=\lambda$ is given by
\begin{align}
	\|\mathbf{1}\|^{M}_{\lambda,V}=|R_{\rho}(0)|^{-1}.
\end{align}
\end{prop}

\subsection{The main result of the paper}
Let $g^{TX}$ be a metric on $TX$. Let
$\psi(TX,\nabla^{TX})$ be the Mathai-Quillen current  associated to the
Levi-Civita connection $\nabla^{TX}$ (see Section \ref{sMQ}). It is a
current of degree $n-1$ defined
on the total space of the tangent bundle $TX$, which takes values in $o(TX)$, the orientation
bundle of $TX$. Let $\|\cdot\|^{RS,2}_{\lambda}$ be the Ray-Singer metric
on $\lambda$ associated to $(g^{TX},g^{F})$ (see Section \ref{sRS}).
Let $
\theta(F,g^{F})=\Tr\[g^{F,-1}\nabla^{F}g^{F}\]\in \Omega^{1}(X)$.
Our main result  is the following.

\begin{thm}\label{thm1} We have
	\begin{align}
	\log
	\(\frac{\|\cdot\|^{RS,2}_{\lambda}}{\|\cdot\|^{M,2}_{\lambda,V}}\)=-\int_{X}\theta\(F,g^{F}\)V^{*}\psi\(TX,\nabla^{TX}\).\end{align}
\end{thm}

If $V$ does not have any closed orbits, Theorem \ref{thm1} reduces to
\cite[Theorem 0.2]{BZ92}. If $V$ does not have any fixed  points,
and if $F$ is unitarily flat, by Proposition \ref{prop},  then  our
theorem corresponds to \cite[Theorem 3.1]{Friedconj}.

Our proof of Theorem \ref{thm1} is based on a result of Franks
\cite[Proposition 5.1]{Franks1979}, who
constructed  a gradient flow by destroying the
closed orbits of the Morse-Smale flow.
In Section \ref{sCMM}, we first
establish a comparison formula between
our Milnor metric
associated to the original Morse-Smale flow
and the classical one associated to Franks'
gradient flow. In Section \ref{S2}, to obtain Theorem \ref{thm1}, we apply Bismut-Zhang's formula \cite[Theorem
0.2]{BZ92}, which compares the
Ray-Singer metric with the
Milnor metric for Franks' gradient flow.

Recall that $F$ is said to be  unimodular, if its holonomy representation
$\rho$ is unimodular, i.e., $|\det \rho(\gamma)|= 1$ for all
$\gamma\in \pi_{1}(X)$.  This is
equivalent to the fact that there is a Hermitian metric $g^{F}$
such that $\theta\(F,g^{F}\)=0.$

By Theorem \ref{thm1}, we get
\begin{cor}
	If $(F,g^{F})$ is unimodular, then
	\begin{align}
	\|\cdot\|^{RS,2}_{\lambda}=\|\cdot\|^{M,2}_{\lambda,V}.
\end{align}
\end{cor}

\subsection{Organisation  of the paper}

In Section \ref{Sp1}, we
introduce some conventions on the determinant line, the cohomology
of a circle, and also a long exact sequence associated to three manifolds $Y_{1}\subset Y_{2}\subset Y_{3}$.  In Section \ref{S1M}, we recall some background on Morse-Smale
flows. We also introduce the Milnor metric, and we show Proposition
\ref{prop}. In
Section \ref{S2}, we recall the constructions of Mathai-Quillen
current and Ray-Singer metric. We show our main result.
We use the convention $\mathbf{N}=\{0,1,2,\ldots\}$ and $\mathbf{R}_{+}^{*}=(0,\infty)$.

\subsection{Acknowledgement}
We are indebted to  Xiaolong Han and Xiaonan Ma for reading  a
preliminary version of this paper and for useful suggestions. S.S.
would like to thank Nguyen Viet Dang and Gabriel  Rivi\`ere for
fruitful  discussions  on Morse-Smale flows. This paper was written
while J.Y. was visiting the Institut de
Math\'ematiques de Jussieu-Paris Rive Gauche in Spring 2018.
The hospitality of the Institute and the partial financial support
from NSFC under Grant No.~11771411 are gratefully acknowledged.

\settocdepth{subsection}
\section{Preliminary} \label{Sp1}
This section is organised as follows. In Section \ref{sDL}, we
introduce our convention on the determinant line. In Section
\ref{sCS1}, we give a metric on the determinant line of  the
cohomology of $\mathbb{S}^{1}$. This is  our model case near the
closed orbits of a flow. In Section \ref{Fp}, we explain a long exact
sequence associated to a triple  of manifolds $Y_{1}\subset
Y_{2}\subset Y_{3}$.

\subsection{The determinant line}\label{sDL}
Let $W$ be a complex finite dimensional vector space. We denote by
$W^*$ the dual space. If $\dim W=1$, we write $W^{-1}=W^{*}$.  Take
$\Lambda^\cdot(W)=\oplus_{j=0}^{\dim W}\Lambda^{j}(W)$ to be the exterior
algebra.

Set
\begin{align}
\det W =\Lambda^{\dim W} (W).
\end{align}
Clearly, $\det W$ is a complex line. If $W=0$, then
\begin{align}
	\det
W=\mathbf{\bC}.
\end{align}
Let $E^{\cdot}=\oplus_{i\in \bZ} E^{i}$ be a finite dimensional  $\mathbf{Z}$-graded
space. Put
\begin{align}
	\det E^{\cdot}= \bigotimes_{i\in \bZ} \(\det E^{i}\)^{(-1)^{i}}.
\end{align}

For $m\in \mathbf{N}$, let
\begin{align}
	(C^{\cdot},d): 0\to C^{0}\to C^{1}\to\cdots\to C^{m}\to 0
\end{align}
be a complex of finite vector
spaces. By \cite{KM76} or
\cite[(1.5)]{BGS1}, we
have the canonical isomorphism of lines
\begin{align}\label{detCdetH}
	\det C^{\cdot}\simeq \det H^{\cdot}(C^{\cdot},d).
\end{align}
If $H^{\cdot}(C^{\cdot},d)={0}$, we can choose $s_{j}^{k}\in C^{k}$
such that $\{s_{j}^{k}\}_{j=1}^{k_{j}}$ projects to a basis of
$C^{k}/dC^{k-1}$, then
\begin{align}\label{Maja1}
	\(\wedge_{j} s^{0}_{j}\)\otimes \(\wedge_{j} ds^{0}_{j}\wedge (
	\wedge_{j}s^{1}_{j})\)^{-1} \otimes\cdots \otimes\(\wedge_{j}
	ds^{m-1}_{j}\)^{(-1)^m}\in \det C^{\cdot}
\end{align}
defines a canonical element of $\det C^\cdot$. Under
\eqref{detCdetH}, this element maps to $\mathbf{1}\in \bC=\det
H^{\cdot}(C^{\cdot},d)$.

\subsection{The cohomology of $\bbS^{1}$}\label{sCS1}
Let $\mathbb{S}^{1}=\mathbf{R}/\mathbf{Z}$ be an oriented circle.

Let $F$ be a flat vector bundle of rank $r$ on
$\mathbb{S}^{1}$. Let $A\in \GL_{r}(\bC)$ be
the holonomy of $F$ along the oriented circle.
The canonical triangulation on $\bbS^{1}$ induces a
complex of simplicial cochains with values in $F$,\footnote{The cochain  complex $\(C^{\cdot}(\bbS^{1},F),d\)$
is the dual of the chain complex $\(C_{\cdot}(\bbS^{1},F^{*}),d\)$.
This explains why $A^{-1}$ appears in the complex \eqref{CS1}.}
\begin{align}\label{CS1}
\(C^{\cdot}(\bbS^{1},F),d\):		\xymatrixcolsep{3pc}\xymatrix{
0  \ar[r] &\bC^{r}
\ar[r]^{A^{-1}-1} &\bC^{r} \ar[r]          &0.}
\end{align}

By \eqref{detCdetH}, the canonical element $
\mathbf{1}\in \bC=\det C^{\cdot}(\bbS^{1},F)$ defines  an element
\begin{align}\label{eqsigmaa}
\sigma_{A}\in 	\det H^{\cdot}\(\mathbb{S}^{1},F\).
\end{align}
We equip $\det H^{\cdot}\(\mathbb{S}^{1},F\)$ with a metric
$\|\cdot\|^{2}_{\det H^{\cdot}\(\mathbb{S}^{1},F\)}$ such that
\begin{align}\label{nsa}
	\|\sigma_{A}\|_{\det H^{\cdot}\(\mathbb{S}^{1},F\)}=1
\end{align}

If $1$ is not an eigenvalue of $A$, then $H^{\cdot}(\bbS^{1},F)=0$.
By \eqref{Maja1},  the norm of the canonical element
$\mathbf{1}\in \bC=\det H^{\cdot}(\bbS^{1},F)$ is given by
\begin{align}\label{sa}
	\|\mathbf{1}\|_{\det
	H^{\cdot}\(\mathbb{S}^{1},F\)}=\left|\det\(1-A^{-1}\)\right|^{-1}.
\end{align}

\begin{re}
	Equation \eqref{sa} is just Proposition \ref{prop} for  the 	rotation flow on  	$\mathbb{S}^{1}$.
\end{re}

\begin{re}
	Since the flat vector bundle is not necessarily unimodular, i.e.,
	$|\det(A)|$ is
	not necessarily equal to $1$,  the choice of the
	orientation on $\bbS^{1}$ is very important.
\end{re}

\subsection{A fusion principle}\label{Fp}
Let $Y_{1}\subset Y_{2}\subset Y_{3}$ be three compact smooth
manifolds with boundaries of
the same dimension such that $Y_{1}\subset \mathring{Y_{2}}$ and
$Y_{2}\subset \mathring{Y_{3}}$. Let $F$ be a flat vector bundle on
$Y_{3}$ of rank $r$. Denote by $H^{\cdot}(Y_{3},Y_{2},F)$,
$H^{\cdot}(Y_{3},F)$,\ldots, the corresponding relative or absolute
cohomologies  with coefficients in  $F$.

As in \cite[(0.16)]{BruningMa_2012}, we have a long exact sequence
\begin{align}\label{Les}
\cdots\to 	H^{i}(Y_{3},Y_{2},F)\to H^{i}(Y_{3},Y_{1},F)\to
	H^{i}(Y_{2},Y_{1},F)\to \cdots.
\end{align}
By \eqref{detCdetH} and \eqref{Les}, we get an isomorphism of lines
\begin{align}\label{Maja3}
	f_{21,32	}:\det H^{\cdot}(Y_{2},Y_{1},F)\otimes \det 		H^{\cdot}(Y_{3},Y_{2},F)\simeq \det H^{\cdot}(Y_{3},Y_{1},F).
\end{align}
Using the other triples  $(\varnothing,Y_{1},Y_{2})$ and
$(\varnothing,Y_{1},Y_{3})$, we get similar isomorphisms
\begin{align}
f_{1,21}:	\det H^{\cdot}(Y_{1},F)\otimes \det
	H^{\cdot}(Y_{2},Y_{1},F)\simeq \det H^{\cdot}(Y_{2},F),
\end{align}
and
\begin{align}\label{Maja4}
f_{1,31}:	\det H^{\cdot}(Y_{1},F)\otimes \det
	H^{\cdot}(Y_{3},Y_{1},F)\simeq \det H^{\cdot}(Y_{3},F).
\end{align}

By \eqref{Maja3}-\eqref{Maja4}, we see that
$f_{2,32}\circ\(f_{1,21}\otimes\mathrm{id}\) $ and $f_{1,31}\circ \(\mathrm{id} \otimes f_{21,32}\)$
define  two isomorphisms
\begin{align}\label{FYYY}
	\det H^{\cdot}(Y_{1},F)\otimes \det H^{\cdot}(Y_{2},Y_{1},F)\otimes \det
	H^{\cdot}(Y_{3},Y_{2},F)	\simeq  \det H^{\cdot}(Y_{3},F).
\end{align}

\begin{prop}\label{propFp}
	There is $\mu\in \bC$ with $| \mu |=1$ such that
	\begin{align}\label{mabu}
f_{2,32}	\circ\(f_{1,21}\otimes\mathrm{id}\) = \mu \times
	f_{1,31}\circ \(\mathrm{id} \otimes f_{21,32}\).
	\end{align}
\end{prop}
\begin{proof}
	Let us give a short analytic proof. Take  a Riemannian metric on $Y_{3}$ such that  $\p Y_{1}$ and $\p Y_{2}$ are totally
	geodesic. Take also a Hermitian metric on $F$. By abuse of
	notation denote by
	$\|\cdot\|^{RS,2}$  the Ray-Singer
	metrics\footnote{We will not use the explicit definition  of this metric.
	We refer the reader to \cite{BruningMa_2012} for the  detail. }
	on $\det H^{\cdot}(Y_{1},F)$, $\det
	H^{\cdot}(Y_{2},Y_{1},F),\ldots$, or the induced  metric on the
	lines likes
	\begin{align}
		\Big(\det H^{\cdot}(Y_{1},F)\otimes \det H^{\cdot}(Y_{2},Y_{1},F)\otimes \det
	H^{\cdot}(Y_{3},Y_{2},F)	\Big)^{-1}\otimes  \det
	H^{\cdot}(Y_{3},F).
	\end{align}

	By \cite[Theorem 0.3]{BruningMa_2012},  we have
	\begin{align}\label{f121}
		\|f_{2,32} \circ\(f_{1,21}\otimes\mathrm{id}\)
		\|^{RS,2}
	=\|f_{2,32}\|^{RS,2}\|f_{1,21}\|^{RS,2}=2^{r\chi(\p
	Y_{1})+r\chi\(\p Y_{2}\)},
	\end{align}
	and also
	\begin{align}\label{f131}
		\|f_{1,31}\circ \(\mathrm{id} \otimes f_{21,32}\)\|^{RS,2}
	=\|f_{1,31}\|^{RS,2}\|f_{21,32}\|^{RS,2}=2^{r\chi(\p
	Y_{1})+r\chi\(\p Y_{2}\)},
	\end{align}
	where $r$ is the rank of $F$ and $\chi\(\p Y_{1}\),\chi\(\p
	Y_{2}\)$ are the Euler characteristic numbers of the boundaries  $\p Y_{1}, \p Y_{2}$.
	By \eqref{f121} and \eqref{f131}, we get \eqref{mabu}.
\end{proof}
\begin{re}
	Instead of using the analytic Ray-Singer metric and \cite[Theorem
	0.3]{BruningMa_2012}, 	we can also use the Milnor
	metric \cite[Definition 1.6]{BruningMa_2012} and 	\cite[(3.71)]{BruningMa_2012}.
\end{re}

\section{Milnor metric}\label{S1M}
This section is organised as follows. In Sections \ref{sMS} and
\ref{sDZf}, we recall
the definitions of Morse-Smale flow and the associated Ruelle dynamical zeta
function. In Section \ref{sMSMF}, we recall  some results due to Franks
\cite{Franks1979,Franks1982} on the construction of  a new
gradient  flow  by destroying  the closed orbits of the original Morse-Smale flow.  In Section
\ref{sSF}, using the Smale filtration, we introduce the Milnor
metric. In Section  \ref{sCMM}, we establish a comparison  formula for the two Milnor
metrics,  one is associated to the Morse-Smale flow and the other is
associated to the gradient  flow constructed by Franks.
\subsection{Morse-Smale flow}\label{sMS}
Let $X$ be a connected closed smooth manifold of dimension $m$. Let $V$ be a
 vector field on $X$, and let
$\(\phi_{t}\)_{t\in \bR}$ be the
 flow generated

 by $V$.

\begin{defin}
A fixed point $x\in X$ of the flow $\phi_{\cdot}$ is called hyperbolic if there is a splitting
\begin{align}
	T_{x}X= E^{u}_{x}\oplus E^{s}_{x},
\end{align}
and there exist $C>0,\theta>0$ and a Riemannian metric on $X$
such that for  $v\in E^{u}_{x}$,  $v'\in E^{s}_{x}$, and $t>0$, we have
\begin{align}\label{eqMS}
&\left|\phi_{- t,*}v\right|\l
Ce^{-\theta|t|}\left|v\right|,&\left|\phi_{ t,*}v'\right|\l
Ce^{-\theta|t|}\left|v'\right|.
\end{align}
The unstable and stable manifolds of the  hyperbolic fixed point $x$ are defined by
\begin{align}
	&W^{u}_{x}=\Big\{y\in X: \lim_{t\to -
	\infty}d_{X}(\phi_{t}(y),x)=0\Big\},&W^{s}_{x}= \Big\{y\in X: \lim_{t\to
	\infty}d_X(\phi_{t}(y),x)=0 \Big\},
\end{align}
where $d_{X}$ denotes the Riemannian distance. The index $\ind(x)\in \mathbf{N}$ of $x$ is defined by
\begin{align}
	\ind(x)=\dim E^{u}_{x}.
\end{align}
\end{defin}

\begin{defin}
A  closed orbit $\gamma$ of the flow $\phi_{\cdot}$ is called
hyperbolic,
if there is  a $\phi_{t}$-invariant continuous  splitting
\begin{align}\label{eq:AF1}
TX|_{\gamma}=\mathbf{R}V\oplus E^{u}_{\gamma}\oplus E^{s}_{\gamma},
\end{align}
such that \eqref{eqMS} holds.  The associated unstable and
stable manifolds are defined by
\begin{align}
\begin{aligned}
	W^{u}_{\gamma}=\bigcup_{x\in \gamma} \Big\{y\in X: \lim_{t\to
	-\infty}d_{X}(\phi_{t}(y),\phi_{t}(x))=0 \Big\},\\ W^{s}_{\gamma}=\bigcup_{x\in \gamma} \Big\{y\in X: \lim_{t\to
	+\infty}d_{X}(\phi_{t}(y),\phi_{t}(x))=0 \Big\}.
	\end{aligned}
\end{align}
The index $\ind(\gamma)\in \mathbf{N}$ of $\gamma$ is defined by
\begin{align}\label{indexg}
	\ind(\gamma)=\dim E^{u}_{\gamma}.
	\end{align}
\end{defin}

Denote by $A$ the set of fixed points and by $B$ the set of
prime closed orbits.

\begin{defin}
	A vector field $V$ or a flow $\phi_{\cdot}$ is called Morse-Smale if
	\begin{itemize}
		\item the sets $A$ and $B$ are finite and contain only hyperbolic elements;
       \item the nonwandering set of
$\phi_{\cdot}$ is equal to
$	A\cup \bigcup_{\gamma\in B}\gamma$;
     \item the stable  and unstable manifold of any critical
	 element in $A\coprod B$  intersect transversally.
	\end{itemize}
\end{defin}

In the sequel , we assume
that $V$ is a
Morse-Smale vector field.

\subsection{Ruelle dynamical zeta function}\label{sDZf}
 For $\gamma\in B$, denote by $\ell_{\gamma}\in \bR^{*}_{+}$ its
 minimal period.

 A closed orbit $\gamma\in B$
 is called untwist if $E^{u}_{\gamma}$ is orientable along $\gamma$, and is
 called twist otherwise. Put
 \begin{align}\label{Delta}
	 \Delta(\gamma)=\left\{
	 \begin{array}{cl}
		 1,&\hbox{if } \gamma \hbox{ is untwist},\\
		 -1,&\hbox{if } \gamma \hbox{ is twist}.
	\end{array}
	 \right.
 \end{align}

 Let $\rho:\pi_{1}(X)\to \mathrm{GL}_{r}(\bC)$ be a representation of the fundamental group of $X$. If
 $\gamma\in B$, denote by $\rho(\gamma)$
 the holonomy along $\gamma$. Clearly, $\rho(\gamma)$ is well
 defined up to a conjugation.

\begin{defin} The twist Ruelle dynamical zeta function is a meromorphic
	function on $\bC$ defined by
\begin{align}\label{rrs}
	R_{\rho}(s)=\prod_{\gamma\in
	B}\det\(1-\Delta(\gamma)\rho(\gamma)^{-1}e^{-s
	\ell_\gamma}\)^{(-1)^{\ind(\gamma)}}.
\end{align}
This definition of the Ruelle dynamical zeta function
corresponds to  Fried's definition  \cite{Friedconj} associated to
the  holonomy of the  dual bundle  $F^{*}$.
\end{defin}

\subsection{Frank's Morse function}\label{sMSMF}
 We follow \cite[Section 1]{Franks1979}. Let $\mathbb{D}^{r}$ be the
$r$-dimensional open unit ball of center $0\in
\mathbf{R}^{r}$.
 A fixed point $x\in A$ of index $p$ is said to be
 of standard form if there
 is a system of
 coordinates $(y_{1},\ldots, y_{m})\in \mathbb{D}^{m}$ on a neighborhood
 of $x$ such that $x$ is represented by $0$ and
 \begin{align}
	 V=\sum_{i=1}^{p}y_{i}\frac{\p}{\p
	 y_{i}}-\sum_{i=p+1}^{m}y_{i}\frac{\p}{\p y_{i}}.
\end{align}

For closed orbits we must distinguish the following four cases  in
establishing the standard forms. Assume $\gamma\in B$ such that
$\ind(\gamma)=p$.
\begin{description}
	\item[Case (1)] Suppose that $TM|_{\gamma}$ is orientable and that  $\gamma$ is
	untwist. In this case, $\gamma$ is said to be of standard form, if there
	is a  system of coordinates $(t,y_{1},\ldots,
 y_{m-1})\in \mathbb{S}^{1}\times \mathbb{D}^{m-1}$
 on a tubular neighborhood
	$U_{\gamma}$	of $\gamma$ such that $\gamma$ is represented by
 $(t,0)\in  \mathbb{S}^{1}\times \mathbb{D}^{m-1}$ and
\begin{align}\label{eqV}
	 V=\frac{\p}{\p t}+\sum_{i=1}^{p}y_{i}\frac{\p}{\p
	 y_{i}}-\sum_{i=p+1}^{m-1}y_{i}\frac{\p}{\p y_{i}}.
\end{align}
	\item[Case (2)] Suppose that $TM|_{{\gamma}}$ is orientable and
	that   $\gamma$ is
	twist. In this case, $\gamma$ is said to be of standard form, if
	 $U_{\gamma}$ and $V$ can be obtained
	from Case (1) by  the identification
	\begin{align}\label{enid}
	(t,
	x_{1},\ldots,x_{m-1})\sim 	(t+1/2,
	-x_{1},x_{2},\ldots, x_{p}, -x_{p+1}, x_{p+2},\ldots,x_{m-1}).
	\end{align}
	\item[Case (3)]  Suppose that   $TM|_{{\gamma}}$ is not orientable  and
 that  $\gamma$ is
	untwist. In this case, $\gamma$ is said to be of standard form, if
	 $U_{\gamma}$ and $V$ can be obtained
	from Case (1) by  the identification
	\begin{align}
	(t,
	x_{1},\ldots,x_{m-1})\sim 	(t+1/2,
	x_{1},\cdots,x_{p},-x_{p+1},x_{p+1},\ldots,x_{m-1}).	\end{align}
	\item[Case (4)]  Suppose that  $TM|_{{\gamma}}$ is not orientable
	and that 	 $\gamma$ is
	twist. In this case,  $\gamma$ is said to be of standard form, if
	 $U_{\gamma}$ and $V$ can be obtained
	from Case (1) by  the identification
	\begin{align}
		(t,
	x_{1},\ldots,x_{m-1})\sim (t+1/2,
	-x_{1},x_{2},\cdots,x_{m-1}).
	\end{align}
\end{description}
Note that in \cite[Section 1]{Franks1979} the author assumed that $M$ is orientable, so only the first two cases appear.

The following three propositions are  \cite[Proposition 1.6,
Theorem 2.2, Proposition 5.1]{Franks1979}.
Their proofs can be generalised to the non orientable case
with some evident modifications. We omit the details.

 \begin{prop}\label{propV01}
	 For any Morse-Smale vector field $V$, there is a family of Morse-Smale
	 vector fields $(V_{\ell})_{0\l \ell\l 1}$ such that $V_{0}=V$ and that the critical
	 elements of
	 $V_{1}$ are all of
standard forms and are precisely the same as the critical
	 elements of $V$. Moreover, $V_{0}$ and $V_{1}$ are topologically
	 conjugated, i.e., there is a homeomorphism carrying the  orbits of
	 $V_{0}$ to those
of $V_{1}$ and preserving their orientations.
\end{prop}

The relation between the Morse-Smale flow and the  Morse function is
summarised in the following two propositions. The first one is due to Smale \cite{S61}.

\begin{prop}
	If $V$ is a Morse-Smale vector field whose flow has fixed points
	in standard form and no closed orbits, then $V$ is a  gradient of
a certain Morse function with respect to some Riemannian metric.
\end{prop}

To state the following  proposition, let us introduce some notation. For $x,y\in A$  such that $\ind(y)=\ind(x)-1$, then $W^{u}_{x}\cap W^{u}_{y}$
consists of a finite set $\Gamma(x,y)=\{a_{\cdot}\}$ of integral curves of
$V$ such that $a_{-\infty}=x$ and $a_{\infty}=y$ along
which $W^{u}_{x}$ and $W^{s}_{y}$ intersect transversally. Define
$n(a)=\pm1$ as in  \cite[(1.28)]{BZ92}, whose definition does not
require the manifold to be orientable.

\begin{prop}\label{propfV}
	For some small neighborhood $U=\cup_{\gamma\in B}U_{\gamma}$ of
	closed orbits $\cup_{\gamma\in B}\gamma$, there is
	a 	Morse function $f$ on $M$ whose gradient vector field $\nabla f$
	with respect to a certain Riemannian metric is
	Morse-Smale, such that
	\begin{itemize}
		\item on $M\backslash U$, we have
	\begin{align}\label{eqdf=V}
		\nabla f=V,
	\end{align}
	\item on each $U_{\gamma}$, the Morse function  $f$ has only two  critical points $x_{\gamma},x'_{\gamma}$ of
 index $\ind(\gamma)+1$ and $\ind(\gamma)$ respectively.
	\end{itemize}
	  Also, $\Gamma(x_{\gamma},x'_{\gamma})$ consists of two
 integral curves  $a_{\gamma},a'_{\gamma}$ (see Figure \ref{fig:1})
 such that their composition
 $a_{\gamma}\circ  (a'_{\gamma})^{-1}$ and the
 closed orbit $\gamma$ lie
 in the same freely  homotopy class of
 loops on $X$ and  that
 \begin{align}\label{nana}
	 n(a_{\gamma})n(a'_{\gamma})=-\Delta(\gamma).
\end{align}
\end{prop}

\begin{figure}[htbp]
\centering
\includegraphics[width=3.5in]{MS}
\caption{A closed orbit  and  integral curves }\label{fig:1}
\end{figure}

\begin{re}\label{reK}We recall the essential step in the
	construction of the gradient $\nabla f$ given by Franks \cite[Proposition
	8.5]{Franks1982}. For simplicity, assume that	there is only one
	closed orbit 	$\gamma$ and it is of index $p$ and in standard
	form of Case (1).
For $\e>0$ small enough, let $\rho\in
C^{\infty}_{c}(\mathbb{D}^{m-1},[0,1])$
be a cutoff function, which is equal to
$1$ on $|y|<\e$ and to $0$ on $|y|>2\e$. Put
\begin{align}
		V_{1}=\left\{
		\begin{array}{lr}
		\big((1-\rho)+\rho\sin\(2\pi t\)\big)\frac{\p}{\p t}+\sum_{i=1}^{p}y_{i}\frac{\p}{\p
	 y_{i}}-\sum_{i=p+1}^{m-1}y_{i}\frac{\p}{\p y_{i}}, &\hbox{ in
	 } 	 U_{\gamma},\\
	 V,& \hbox{ outside } U_{\gamma}.
		\end{array}\right.
	\end{align}
It is easy to see that the
	nonwandering set of $V_{1}$ in $U_{\gamma}$ consists of two
	points $(1,0),(\frac{1}{2},0)\in
	\mathbb{S}^{1}\times \mathbb{D}^{m-1}$.
	Then, by a small perturbation on $V_{1}$,  we  get a Morse-Smale gradient vector
	field  $\nabla f$ which
	has the desired transversality and other properties.

We remark that by the above construction, we can find  a	family of vector fields $(V_{\e})_{0\l\e\l1}$ connecting $V$ and
	$\nabla f$ such that near $\{1/4\}\times \mathbb{D}^{m-1}$, for
	any $ \e\in [0,1]$, $V_{\e}$ does not vanish.
	Similar remark holds for $\gamma$ in standard forms of Cases (2)-(4).
	In Section \ref{s33}, we will use this fact to simplify the proof of our main theorem.
\end{re}

\subsection{Smale filtration and Milnor metric}\label{sSF}
 Following \cite[Definition 9.10]{Franks1982},
let
\begin{align}\label{eqSmale}
	\varnothing=X^{0}\subset X^{1}\subset\cdots \subset X^{N}=X
\end{align}
be a Smale filtration on $X$ associated to $V$. Note that each
$X^{p}\subset X$ is a submanifold with boundary, and can be constructed by the sublevel set  of a smooth Lyapunov function. Also, we
have
\begin{itemize}
	\item on each $\p X^{p}$, $V$ does not vanish and points toward the inside of $X^{p}$;
	\item  there is only one critical element $c\in A\coprod
	B$ contained in $X^{p+1}\backslash
	X^{p}$ and

	\begin{align}\{c\}=\bigcap_{t\in \bR}\phi_{t}\(X^{p+1}\backslash
	X^{p}\).
	\end{align}
\end{itemize}

Let $F$ be a flat vector bundle on $X$ induced by the representation
$\rho$.
Let $H^{\cdot}(X,F)$ be the cohomology of the sheaf of locally constant
sections of $F$. Put
\begin{align}\label{eqlambda}
	\lambda=\det H^{\cdot}(X,F).
\end{align}
We use the notation in Section \ref{Fp}. By \eqref{FYYY}, we get an isomorphism
 \begin{align}\label{eqSS}
 	\sigma_{V}:\bigotimes_{p=0}^{N-1}\det
	H^{\cdot}(X^{p+1},X^{p},F)\simeq \lambda.
 \end{align}
By Proposition \ref{propFp}, up to a multiplication  by a unit in $\bC$, the
morphism $\sigma_{V}$ does not depend on the way that the
cohomologies are fused.

By \cite[Theorem 9.11]{Franks1982} (see also \cite[Section 2]{MorgadoMorseSmale}), if the critical element  $c\in
X^{p+1}\backslash X^{p}$ is a fixed
point, then
\begin{align}\label{E1x}
	H^{q}\(X^{p+1},X^{p},F\)=\left\{ \begin{array}{ll}
	F_{c},& q=\ind (c),\\
	0,&\hbox{otherwise},
\end{array}\right.
\end{align}
and if  the critical element  $c\in
X^{p+1}\backslash X^{p}$ is a closed orbit, then
\begin{align}\label{E1g}
	H^{q}\(X^{p+1},X^{p},F\)=\left\{ \begin{array}{ll}
	H^{q-\ind (c)}\big(c,o(E^{u}_{c})\otimes F|_{c}\big),& q-\ind( c)=0
	\hbox{ or }1,\\
	0,&\hbox{otherwise,}
\end{array}\right.
\end{align}
where $o(E_{c}^{u})$ is the orientation line bundle of $E^{u}_{c}$ along the
closed orbit $c$.

We equip $\det
	H^{\cdot}\(\gamma,o(E^{u}_{\gamma})\otimes
	F|_{\gamma}\)$ with the metric $\|\cdot\|^{2}_{\det
	H^{\cdot}\(\gamma,o(E^{u}_{\gamma})\otimes
	F|_{\gamma}\)}$ defined  in \eqref{nsa}.

Let $g^{F}$ be a Hermitian metric on $F$. By \eqref{eqSS}-\eqref{E1g}, the
restriction  $g^{F}|_{A}$
and  the metric $\|\cdot\|^{2}_{\det
	H^{\cdot}\(\gamma,o(E^{u}_{\gamma})\otimes
	F|_{\gamma}\)}$ induce a metric

$\|\cdot\|^{M}_{\lambda,V}$ on
$\lambda$. By Proposition \ref{propFp},
this metric  does not depend on the way that the
cohomologies are fused.

\begin{defin}
	The  metric $\|\cdot\|^{M}_{\lambda,V}$ on
$\lambda$ is called the Milnor metric associated to
$V$.
\end{defin}

\begin{re}
	If $V$ is a Morse-Smale gradient vector field, then
	$\|\cdot\|^{M}_{V}$ coincides with the one constructed by
	Bismut-Zhang \cite[Definition 1.9]{BZ92}. In fact, there is a small difference with Bismut-Zhang's
	construction, where  they used a filtration \cite[(1.37)]{BZ92} induced
	by sublevel sets of a nice 	Morse function.  Using Proposition
	\ref{propFp}, we can deduce  that the two constructions coincide.
\end{re}

\begin{re}
	The Milnor metric for
	general Morse-Smale flow 	does not depend on the Smale
filtration \eqref{eqSmale}. We will not give the direct proof  since it is a consequence of our main Theorem \ref{thm1}.
\end{re}

Let us restate and prove Proposition \ref{prop}.

\begin{prop}\label{Prop}
	If $V$ does not have any fixed points, and
if none of $\Delta(\gamma)$ is an  eigenvalue of $\rho(\gamma)$, then
$H^{\cdot}(X,F)=0$, and the norm of the canonical section
$\mathbf{1}\in \bC=\det H^{\cdot}(X,F)$ is given by
\begin{align}\label{eqReMi}
	\|\mathbf{1}\|^{M}_{\lambda,V}=|R_{\rho}(0)|^{-1}.
\end{align}
\end{prop}
\begin{proof}
	By \eqref{Les}, \eqref{E1x}, and \eqref{E1g}, we can deduce  that  $H^{\cdot}(X,F)=0$. By
	\eqref{sa}, \eqref{rrs}, and \eqref{eqSS}, we get
	\eqref{eqReMi}.
\end{proof}

\subsection{A comparison formula for  Milnor metrics}\label{sCMM}
In this section, we assume that all the critical elements of
$V$ are in standard forms, and that $f$ is chosen as in
Proposition 	\ref{propfV}.

Let $\det \tau({a'_{\gamma}})\in \det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}$ be the canonical element induced by the
	parallel transport  with respect to the flat connection	along the
	integral curve $a_{\gamma}'$ (see Figure \ref{fig:1}). Let $\|\cdot\|^{2}_{\det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}}$ be the metric on $\det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}$ induced by $g^{F}_{x_{\gamma}}$ and
		$g^{F}_{x'_{\gamma}}$.

\begin{prop}\label{propMM}

	The following identity holds,
	\begin{align}\label{eqMM}
		\log\(\frac{\|\cdot\|^{M,2}_{\lambda,\nabla
		f}}{\|\cdot\|^{M,2}_{\lambda,V}}\)=\sum_{\gamma\in
		B}(-1)^{\ind(\gamma)}\log
		\left\|\det\tau(a_{\gamma}')\right\|^{2}_{\det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}}.
	\end{align}

\end{prop}
\begin{proof}
	We refine the filtration \eqref{eqSmale} by the new critical
	points of $f$. By Propositions \ref{propFp} and \ref{propfV}, and by
	\eqref{eqSS}, the following diagram commutes
	 \begin{align}\label{eqDia}
		\begin{aligned}
			\xymatrix{
	\bigotimes_{x\in A}  \big(\!\det F_{x}\big)^{ (-1)^{\ind (x)}}
		\bigotimes_{\gamma\in B} \left\{\det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}\right\}^{(-1)^{\ind (\gamma)}}
		\ar[d]
		\ar[r]^-{\sigma_{\nabla f}} &\lambda\ar[d]
\\
		\bigotimes_{x\in A}\!\big(\!\det
F_{x}\big)^{(-1)^{\ind (x)}} \!
	\bigotimes_{\gamma \in B}\Big\{\!\det
	H^{\cdot}\(\gamma,o(E^{u}_{\gamma})\otimes
	F|_{\gamma}\)\!\Big\}^{(-1)^{\ind
	(\gamma)}}\ar[r]^-{\sigma_V}         &\lambda}
	\end{aligned}
	\end{align}

where the first vertical arrow  is induced by \eqref{eqsigmaa} and
the second vertical arrow is a multiplication  by a unit in $\mathbf{C}$.

	The Milnor metric $\|\cdot\|_{\lambda,\nabla f}^{M,2}$ is
	obtained   from the metric $g^{F}|_{A \cup
	\{x_{\gamma},x'_{\gamma}:\gamma\in B\}}$ via $\sigma_{\nabla f}$.

	By \eqref{eqDia}, we get
	\eqref{eqMM}.
\end{proof}

\section{An extension of  Bismut-Zhang's Theorem to Morse-Smale
flow}\label{S2}
This section is organised as follows. In Sections
\ref{sec:Bere}-\ref{sRS},  following \cite{BZ92}, we recall the
constructions of the Berezin integral, the  Mathai-Quillen current and the Ray-Singer metric.
In Section \ref{s33}, we restate and prove our main theorem.
\subsection{Berezin integral}\label{sec:Bere}
Let $E$ be a real Euclidean space of dimension $n$ with
the scalar product $\<,\>$, and let $W$ be a real vector space of finite dimension.

Suppose temporarily that $E$ is oriented and that $e_1,\ldots, e_n$
is an oriented orthogonal basis of $E$. Let $e^1,\ldots,e^n$ be the corresponding dual basis of $E^*$. We define $\int^B$ to be the linear map from $\Lambda^\cdot(W^*)\widehat{\otimes}\Lambda^\cdot(E^*)$ into $\Lambda^\cdot(W^*)$, such that if $\alpha\in \Lambda^\cdot(W^*), \beta\in \Lambda^\cdot(E^*)$,
\begin{align}\label{eq:defBere}
& \int^B\alpha\beta=0, \hbox{ if } \deg \beta<n,&\int^B\alpha
e^1\wedge\cdots\wedge e^n=\frac{(-1)^{n(n+1)/2}}{\pi^{\frac{n}{2}}}\alpha.
\end{align}
More generally, if $o(E)$ is the orientation line of $E$, then $\int^B$ defines a linear map from $\Lambda^\cdot(W^*)\widehat{\otimes}\Lambda^\cdot(E^*)$ into $\Lambda^\cdot(W^*)\widehat{\otimes} o(E)$, which is called a Berezin integral.

Let $A\in \End^{\mathrm{anti}}(E)$ be an antisymmetric endomorphism of $E$. We identify $A$ with
\begin{align}\label{eq:ALam}
 \dot{A}=\frac{1}{2}\sum_{1\l i,j\l n}\<e_i,Ae_j\>e^i\wedge e^j\in \Lambda^2(E^*).
\end{align}
By definition, the Pfaffian $\mathrm{Pf}[A]\in o(E)$ of $A$ is given by
\begin{align}\label{eq:PfA}
\mathrm{Pf}\[\frac{A}{2\pi}\]=\int^B\exp\(-\frac{\dot{A}}{2}\).
\end{align}
Clearly, $\mathrm{Pf}[A]$ vanishes if $n$ is odd.

\subsection{Mathai-Quillen formalism }\label{sMQ}
Recall that  $X$ is a connected closed smooth    manifold of dimension $m$. Let
$E$ be a Euclidean vector bundle of rank $n$ on  $X$ with a Euclidean
metric $g^{E}$ and a metric connection  $\nabla^{E}$. Let $R^{E}=\(\nabla^{E}\)^{2}\in
\Omega^{2}(X,\End^{\rm anti}(E))$ be its curvature.  Denote by $o(E)$  the orientation line
bundle of $E$.
The Euler form of
$(E,\nabla^{E})$ is given by
\begin{align}
	e\(E,\nabla^{E}\)=\mathrm{Pf}\[\frac{R^{E}}{2\pi}\] \in
	\Omega^{n}(X,o(E)).
\end{align}
Clearly, $e\(E,\nabla^{E}\)=0$,  if $n$ is odd.

Let $\cE$ be the total space of $E$, and let $\pi: \cE\to X$ be the
natural projection. We will use the formalism of the Berezin integral developed in
Section \ref{sec:Bere} with $W = T\cE$. If $\omega$ is a smooth section of
 $\Lambda^{\cdot}(T^{*}\cE)\otimes\pi^{*}\Lambda^{\cdot}(E^{*})$
 over $\cE$, then $\int^{B}\omega$ is a smooth section of $\Lambda^{\cdot}
 (T^{*} \cE)
 \otimes \pi^{*}o(E)$ over $\cE$.

Let $T^{V}\cE\subset T\cE$ be the vertical subbundle of $T\cE$, and
let $T^{H}\cE\subset T\cE$ be the horizontal  subbundle of $T\cE$
with respect to $\nabla^{E}$, so that
\begin{align}\label{splitting}
	T\cE=T^{H}\cE\oplus  T^{V}\cE.
\end{align}
By the identification
$T^{V}\cE\simeq \pi^{*}E$,
the
vertical projection with
respect to the splitting \eqref{splitting} induces a section
$P^{E}\in C^{\infty}(\cE,T^{*}\cE \otimes \pi^{*}E)$. Using the metric $g^{E}$, we identify $P^{E}$ with $\dot{P}^{E}\in C^{\infty}(\cE,T^{*}\cE \otimes \pi^{*}E^{*})$.
Let $Y\in C^{\infty}(\cE,\pi^{*}E)$ be the tautological section.
Write $\widehat{Y}$ the corresponding  section  in
$C^{\infty}(\cE,\pi^{*}E^{*})$ induced by  $g^{E}$.
Recall that $\dot{R}^{E}\in C^{\infty}(X,\Lambda^{2}(T^{*}X)\otimes \Lambda^{2}(E^{*}))$ is
defined in \eqref{eq:ALam}.

\begin{defin}
	For
$T\g 0$, set
\begin{align}
	A_{T}=\frac{1}{2}\pi^{*}\dot{R}^{E}+\sqrt{T}\dot{P}^{E}+T|Y|^{2}\in C^{\infty}\big(\cE,\Lambda^{\cdot}(T^{*}\cE)\otimes \pi^{*}\Lambda^{\cdot}(E^{*})\big).
\end{align}
	Let $(\alpha_{T})_{T\g0}$ and $(\beta_{T})_{T>0}$ be  families  of forms on $\cE$ defined
	by
	\begin{align}
		\begin{aligned}
		&\alpha_{T}=\int^{B}\exp\(-A_{T}\)\in
		\Omega^{n}(\cE,\pi^{*}o(E)),\\
		&\beta_{T}=\int^{B}\frac{\widehat{Y}}{2\sqrt{T}}\exp\(-A_{T}\)\in \Omega^{n-1}(\cE,\pi^{*}o(E)).
		\end{aligned}
	\end{align}
\end{defin}
Clearly,
\begin{align}
	\alpha_{0}=\pi^{*}e\(E,\nabla^{E}\).
\end{align}
Let us recall \cite[Theorems 3.4, 3.5, and 3.7]{BZ92}.
\begin{thm}
For $T\g 0$, the form $\alpha_{T}$ is closed whose  cohomology class
does not depend on $T$.
For $T>0$,  $\alpha_{T}$ represents the Thom
class of $E$ so that $\pi_{*}\alpha_{T}=1$, and  we have
\begin{align}

&\frac{\p \alpha_{T}}{\p T}=-d\beta_{T}.
\end{align}
\end{thm}

We identify $X$ as a submanifold of $\cE$ by the zero section.  The normal bundle
to  $X$ in $\cE$ is exactly $E$ and the conormal bundle is $E^{*}$.
Let $\delta_{X}$ be the current on $\cE$ defined by  the integration  on $X$.
For a current $v$ on $\cE$, denote by $\mathrm{WF}(v)\subset T^{*}\cE$ its wave front set.

\begin{thm}
Let $K\subset
\cE$ be a compact subset of $\cE$. There is $C_{K}>0$ such that  for any $u\in
\Omega^{\cdot}(\cE,\pi^{*}o(TX))$ whose support is contained in $K$
and for any $T\g 1$, we have
\begin{align}\label{eqabT}
&\left|	\int_{\cE}u(\alpha_{T}-\delta_{X})\right|\l
\frac{C_{K}}{\sqrt{T}}\|u\|_{C^{1}},&\left|\int_{\cE}u \beta_{T}\right|\l
\frac{C_{K}}{T^{3/2}}\|u\|_{C^{1}},
\end{align}
where $\|\cdot\|_{C^1}$ denotes the $C^{1}$-norm.
\end{thm}

 In view of \eqref{eqabT},
 \begin{align}
	 \psi\(E,\nabla^{E}\)=\int_{0}^{\infty}\beta_{T}dT,
	 \end{align}
 is a well defined current of degree $n-1$ on $\cE$ with values in $\pi^{*}o(E)$.

\begin{thm}
The current $\psi\(E,\nabla^{E}\)$ is locally
integrable such that
\begin{align}\label{WFphi}
	\mathrm{WF} \big(\psi\(E,\nabla^{E}\)\big)\subset E^{*}.
\end{align}
  The following identity of currents
on $\cE$ holds,
\begin{align}
	d\psi\(E,\nabla^{E}\)=\pi^{*}e\(E,\nabla^{E}\)-\delta_{X}.
\end{align}
\end{thm}

 \begin{defin}
The  current  $\psi\(E,\nabla^{E}\)$ is called the Mathai-Quillen
current.
\end{defin}

\begin{re}
	The restriction of $\psi(E,\nabla^{E})$ to the
	sphere bundle  of $E$
	was first constructed in Mathai-Quillen \cite[Section
	7]{MathaiQuillen}.
	If $E=TX$, this restriction  coincides with the transgressed Euler
	class defined by Chern \cite{Chern2}.
\end{re}

\subsection{Ray-Singer metric}\label{sRS}
We use the notation in Section \ref{S1M}. Recall that $F$ is a flat  vector bundle on $X$.
Let $(\Omega^{\cdot}(X,F),d^{X})$ be the de Rham complex with values in
$F$.  By de Rham's theory, its  cohomology is $H^{\cdot}(X,F)$.

Take metrics $g^{TX}$ and $g^{F}$ on $TX$ and $F$. Let
$\<,\>_{\Lambda^{\cdot}(T^{*}X)\otimes F}$  be the induced metric on
$\Lambda^{\cdot}(T^{*}X)\otimes F$.  Let $dv_{X}\in \Omega^{m}(X,,o(TX))$  be the volume form on $X$.
For $s_{1},s_{2}\in \Omega^{\cdot}(X,F)$, set
\begin{align}\label{eqX}
	\<s_{1},s_{2}\>_{\Omega^{\cdot}(X,F)}=\int_{x\in X}\<s_{1}(x),s_{2}(x)\>_{\Lambda^{\cdot}(T^{*}X)\otimes
	F}dv_{X}.
\end{align}
Then \eqref{eqX} defines an $L^{2}$-metric on $\Omega^{\cdot}(X,F)$.

Let $d^{X*}$  be the formal adjoint of $d^{X}$ with respect to the
$L^{2}$-metric $\<\cdot,\cdot\>_{\Omega^{\cdot}(X,F)}$. Put
\begin{align}
	\Box^{X}=d^{X}d^{X*}+d^{X*}d^{X}.
\end{align}
Then $\Box^{X}$ is a formally self-adjoint  second order elliptic
operator acting on $\Omega^{\cdot}(X,F)$. By Hodge theory, we have
\begin{align}\label{Hodge}
	\ker \Box^{X}\simeq H^{\cdot}(X,F).
\end{align}
By \eqref{eqlambda} and \eqref{Hodge}, the restriction of the $L^{2}$-metric $\<\cdot,\cdot\>_{\Omega^{\cdot}(X,F)}$ to $\ker
\Box^{X}$ induces a metric $|\cdot|^{RS,2}_{\lambda}$ on $\lambda$.

Let $(\ker \Box^X)^{\bot}$ be the orthogonal  space to $\ker \Box^X$
in $\Omega^{\cdot}(X,F)$. Then $\Box^{X}$ acts as an invertible
operator on $\(\ker\Box^{X}\)^{\bot}$. Let $\(\Box^{X}\)^{-1}$ be the
inverse of $\Box^{X}$ acting on $\(\ker\Box^X\)^{\bot}$. Let
$N^{\Lambda^{\cdot}(T^{*}X)}$ be the  number operator on
$\Lambda^{\cdot}(T^{*}X)$, which is multiplication by $p$ on
$\Lambda^{p}(T^{*}X)$. For $s\in \bC$ such that $\Re(s)>\frac{m}{2}$, set
\begin{align}
	\theta(s)=-\Tr\[(-1)^{N^{\Lambda^{\cdot}(T^{*}X)}}N^{\Lambda^{\cdot}(T^{*}X)}\(\Box^{X}\)^{-s}\].
\end{align}
By a result of Seeley \cite{Seeley66} or by \cite[Theorem 7.10]{BZ92}, $\theta(s)$ extends to a meromorphic function
of $s\in \bC$, which is holomorphic at $s=0$.

\begin{defin}
The Ray-Singer metric $\|\cdot\|^{RS,2}_{\lambda}$ on
$\lambda$ is defined  by
\begin{align}
	\|\cdot\|^{RS,2}_{\lambda}=|\cdot|^{RS,2}_{\lambda}\exp\big(\theta'(0)\big).
\end{align}
\end{defin}

Let $\nabla^{TX}$ be the Levi-Civita connection on $(TX,g^{TX})$. Let
 $\psi(TX,\nabla^{TX})$ be the Mathai-Quillen current.

By \eqref{eqMS} and \eqref{WFphi}, for any Morse-Smale vector field
$V$, the pull back $V^{*}\psi(TX,\nabla^{TX})$ is a well
 defined current of degree $(m-1)$ on $X$ with values in $o(TX)$. Set
\begin{align}
	\theta\(F,g^{F}\)=\Tr\[g^{F,-1}\nabla^{F}g^{F}\]\in \Omega^{1}(X).
\end{align}
Then $\theta(F,g^{F})$ is a closed $1$-form.

The main result of Bismut-Zhang \cite[Theorem 0.2]{BZ92} is the following.

\begin{thm}\label{BZ}
	Suppose that  $f$ is a Morse function, whose gradient $\nabla f$ with respect to
some Riemannian metric is Morse-Smale. The following identity holds,
	\begin{align}\label{eqBZ}
	\log
	\(\frac{\|\cdot\|^{RS,2}_{\lambda}}{\|\cdot\|^{M,2}_{\lambda,\nabla f}}\)=-\int_{X}\theta\(F,g^{F}\)\(\nabla f\)^{*}\psi(TX, \nabla ^{TX}).
\end{align}
\end{thm}

\subsection{Proof of the main result}\label{s33}

 We restate our main
result Theorem \ref{thm1}, which is an extension of Theorem \ref{BZ}.
\begin{thm}\label{Thm1}
	Suppose that $V$ is a Morse-Smale flow. The following identity holds,
	\begin{align}
	\log
	\(\frac{\|\cdot\|^{RS,2}_{\lambda}}{\|\cdot\|^{M,2}_{\lambda,V}}\)=-\int_{X}\theta\(F,g^{F}\)V^{*}\psi\(TX,\nabla^{TX}\).
\end{align}
\end{thm}
\begin{proof}
Take 	$\(V_{\ell}\)_{\ell\in [0,1]}$ as in Proposition
\ref{propV01}. By our construction, we have
\begin{align}\label{MVMV1}
	\|\cdot\|^{M,2}_{\lambda,V}=\|\cdot\|^{M,2}_{\lambda,V_{1}}.
\end{align}
Proceeding as in \cite[Proposition 6.1]{BZ92},  we have
\begin{align}\label{XVV1}
	\int_{X}\theta\(F,g^{F}\)V^{*}\psi\(TX,
	\nabla^{TX}\)=\int_{X}\theta\(F,g^{F}\)V^{*}_{1}\psi\(TX, \nabla^{TX}\).
\end{align}
By \eqref{MVMV1} and \eqref{XVV1}, it is enough to show our theorem
for the Morse-Smale vector field $V$ whose  critical elements are all
of standard forms.

Take $f$  as in  Proposition
	\ref{propfV}.

By Proposition \ref{propMM} and Theorem  \ref{BZ}, we have
\begin{multline}\label{LLa}
	\log
	\(\frac{\|\cdot\|^{RS,2}_{\lambda}}{\|\cdot\|^{M,2}_{\lambda,V}}\)=-\int_{X}\theta\(F,g^{F}\)\(\nabla f\)^{*}\psi(TX, \nabla ^{TX})\\
	+\sum_{\gamma\in
		B}(-1)^{\ind(\gamma)}\log
		\left\|\det\tau(a_{\gamma}')\right\|^{2}_{\det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}}.
\end{multline}
By \eqref{LLa}, it remains to show
\begin{multline}\label{eqaaa}
	\int_{X}\theta\(F,g^{F}\)\(\nabla
	f\)^{*}\psi\(TX,
	\nabla^{TX}\)-\int_{X}\theta\(F,g^{F}\)V^{*}\psi\(TX,\nabla^{TX}\)\\
=\sum_{\gamma\in
		B}(-1)^{\ind(\gamma)}\log
		\left\|\det\tau(a_{\gamma}')\right\|^{2}_{\det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}}.
\end{multline}
We assume all the closed orbits are in standard form
of Case (1).  Cases (2)-(4) can be dealt similarly.

Following \cite[Section IV c)]{BZ92},
choose a smooth triangulation $K$ of $X$ such that $A\cap
K^{m-1}=\varnothing$, and such that on $\ol{U}_{\gamma}\simeq \mathbb{S}^{1}\times
\ol{\mathbb{D}}^{m-1}$ the
triangulation is given by the $m$-simplex
$\sigma^{m}_{\gamma}=\(\mathbb{S}^{1}-\{1/4\}\)\times
\ol{\mathbb{D}}^{m-1}$ and $(m-1)$-simplex
$\sigma^{m-1}_{\gamma}=\{1/4\}\times \ol{\mathbb{D}}^{m-1}$.

On each simplex $\sigma\in K^{m}\backslash K^{m-1}$ of maximal degree, choose a
primitive $W_{0,\sigma}\in C^{\infty}(\sigma)$ of $\theta(F,g^{F})$,
such that on $\sigma$ we have
\begin{align}
	dW_{0,\sigma}=\theta\(F,g^{F}\).
\end{align}
Let $W_{0}$ be the locally integrable current  on $X$, such that
for each $\sigma\in  K^{m}\backslash K^{m-1}$ we have
\begin{align}
W_{0}|_{\sigma}=W_{0,\sigma}.
\end{align}
 By our construction of $K$, for $\gamma\in B$, the two points
$x_{\gamma},x_{\gamma}'$, and the integral curve $a_{\gamma}'$ are in the same simplex $\sigma^{m}_{\gamma}\in
K^{m}$, so that
\begin{align}\label{W0W0}
	W_{0}(x_{\gamma}')-W_{0}(x_{\gamma})=\log
		\left\|\det\tau(a_{\gamma}')\right\|^{2}_{\det
		F_{x'_{\gamma}}\otimes \(\det
		F_{x_{\gamma}}\)^{-1}}.
\end{align}

Set
\begin{align}
W_{1}=\theta\(F,g^{F}\)-dW_{0}.
\end{align}
Then $W_{1}$ is  a closed
current  of degree $1$ on $X$ such that $\mathrm{Supp}(W_{1})\subset K^{m-1}$. By \cite[(6.16)]{BZ92} and by \eqref{W0W0}, we see that \eqref{eqaaa} is equivalent to
\begin{align}\label{eqLast}
	\int_{X}W_{1} \wedge (\nabla
	f)^{*}\psi\(TX,\nabla^{TX}\)=\int_{X}W_{1}\wedge
	V^{*}\psi\(TX,\nabla^{TX}\).
\end{align}

By \eqref{eqdf=V}, on any simplex in
$K^{m-1}$ other than $\sigma_{\gamma}^{m-1}$, we have $\nabla f=V$.
By Remark \ref{reK}, near $\sigma_{\gamma}^{m-1}$, $\nabla
	f$ and $V$ can be connected by a family of vector fields without
	zero. Using the fact that $\mathrm{Supp}(W_{1})\subset K^{m-1}$,
	and proceeding as  \cite[Proposition 6.4]{BZ92}, we get \eqref{eqLast}.

The proof of our  theorem is completed.
\end{proof}

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text:
Our proof for the semi-postivity (that is, nefness and pseudo-effectivity) and the positivity (that is, ampleness and bigness) statements are different. Hence, we discuss the corresponding outlines  separately in \autoref{sec:outline_semi_positivity} and in \autoref{sec:outline_positivity}, respectively. Additionally, as it is an indispensable link between semi-positivity and positivity, we present the ideas behind the nefness threshold statement of \autoref{thm:bounding_nef_threshold} in \autoref{sec:outline_nefness_threshold}. For simplicity, we restrict in all cases to the non-logarithmic situation, that is, to statements about $-K_{X/T}$ instead of $-(K_{X/T} + \Delta)$. As all the assumptions and consequences are invariant under base-extension to another algebraically closed field, we may also assume that $k$ is uncountable. In particular, the very general geometric fibers assumed then exist also as closed fibers.

\subsubsection{ Semi-positivity statements.}
\label{sec:outline_semi_positivity}
As nefness and pseudo-effectivity can be checked via non-negative intersection with effective or moving 1-cycles, respectively, points \autoref{itm:semi_positive_boundary:pseff} and \autoref{itm:semi_positive_boundary:nef} of \autoref{thm:semi_positive_boundary} can be reduced to the case of a curve base. Hence, we assume that the base of our fibration $f: X \to T$ is a curve, in which case pseudo-effectivity and nefness are both equal to the degree being at least zero. So, we are supposed to prove that $\deg \lambda_f \geq 0$ or equivalently that $(-K_{X/T})^{n+1} \leq 0$ (see \autoref{eq:CM_line_bundle_log}).

We argue by contradiction, so we assume that $(-K_{X/T})^{n+1}>0$. If we fix a $\bQ$-divisor $H$ on $T$ of small enough positive degree, then by the continuity of the intersection product $(-K_{X/T} - f^*H )^{n+1}>0$ also holds.   As $X$ is normal and fibered over the curve $T$ over which $-K_{X/T}$ is ample, this implies via a Riemann-Roch computation that the $\bQ$-linear system $|-K_{X/T} - f^* H|_{\bQ}$ is non-empty (see \autoref{rem:big}). Our initial idea is to obtain a contradiction from this, as \autoref{prop:cont} tells us that there can exist no $\Gamma \in |-K_{X/T} - f^* H|_{\bQ}$ such that $(X_t,\Gamma_t)$ is klt for general $t \in T$. The only problem is that there are examples where $|-K_{X/T} - f^* H|_{\bQ}$ is non-empty but the above klt condition fails. Indeed, every family with negative CM line bundle  has to be an example like that according to \autoref{prop:cont}. An explicit example is given in \autoref{ex:negative_degree}.

Our second idea is that maybe the $K$-stable assumption leads us to a $\Gamma$ as above that also satisfies the klt condition. According to the delta invariant description of $K$-semi-stability (\autoref{def:K_stable}), if $X_t$ is $K$-semi-stable, then up to a little perturbation one can obtain klt divisors the following way: for $q \gg 0$, let $D_1,\dots,D_l$ be divisors corresponding to any basis of $H^0\left(X_t, -qK_{X_t} \right)$; then the divisor $D := \displaystyle\sum_{i=1}^l \frac{D_i}{ql} \in |-K_{X_t}|_{\bQ}$
is such that $(X_t, D)$ is klt.

Now, we would like to lift such a divisor to $|-K_{X/T} - f^* H|_{\bQ}$. To this end, it is enough to lift for $q \gg 0$, every element of a basis of $H^0\left(X_t, -qK_{X_t} \right)$ to elements of $H^0(X,q(-K_{X/T} - f^* H))$. Using some perturbation argument, one can get by by finding linearly independent sections $s_1,\dots,s_l \in H^0\left(X_t, -qK_{X_t} \right)$ such that $s_i$ lifts, and $\frac{l}{h^0\left(-qK_{X_t} \right)}$ is close enough to $1$.

This in turn would be implied by the following: let $\sE_q$ be the subsheaf of $f_* \sO_X(-qK_{X/T})$ spanned by the global sections. Then we would need to show that
\begin{equation}
\label{eq:outline_goal}
\displaystyle\lim_{q \to \infty} \frac{\rk \sE_q}{\rk f_* \sO_X(q(-K_{X/T}-f^*H))}=1.
\end{equation}
(For the readers more familiar with the language of volumes and restricted volumes, we note that \autoref{eq:outline_goal} is equivalent to showing that the restricted volume of $-K_{X/T}$ over a general fiber is equal to the anti-canonical volume of the fibers.)

Unfortunately, \autoref{eq:outline_goal}  is still not doable. For example, if one takes the isotrivial family
\begin{equation*}
X:=\bP_{T} (\sO_{T}(-n)\oplus \underbrace{\sO_T(1)\oplus \dots \oplus \sO_T(1)}_{\textrm{$n$ times}})
\end{equation*}
of $\bP^n$'s over $T:= \bP^1$ (as in \autoref{ex:negative_degree} for $n=2$), then
\begin{equation*}
f_* \sO_X(-qK_{X/T}) \cong S^{(n+1)q}(\sO_{T}(-n)\oplus \sO_T(1)\oplus \dots \oplus \sO_T(1)).
\end{equation*}
 In this situation $\sE_q$ is the direct sum of the factors with  degree greater than $q \deg H \sim q \varepsilon$ (here $1 \gg \varepsilon>0$). Then one can compute that \autoref{eq:outline_goal} does not hold. For example, in the case of $n=1$,
\begin{equation*}
S^{2q}(\sO_T(-1) \oplus \sO_T(1)) = \sO_T(-q) \oplus \sO_T(-q+1) \oplus \dots \oplus \sO_T(q).
\end{equation*}
 So, we see that the limit of \autoref{eq:outline_goal} is $\frac{1}{2} - \varepsilon$.

The idea that saves the day at this point is the \emph{product trick}, which was pioneered in the case of semi-posivity questions by Viewheg \cite{Viehweg_Weak_positivity}. The precise idea is to replace $X$ by an $m$-times self fiber product $X^{(m)}$ over $T$. Let $f^{(m)} : X^{(m)} \to T$  be the induced morphism (\autoref{sec:product_notation}). Then, one can replace the initial goal with showing that there exists  $\Gamma \in \left| - K_{X^{(m)}/T} - \left( f^{(m)}\right)^* m H \right|_{\bQ}$ such that $\left(X^{(m)}_t, \Gamma_t \right)$ is klt for $t \in T$ general. Running through the previous arguments for $X^{(m)}$ instead of $X$,  this would boil down to showing that
\begin{equation}
\label{eq:outline_goal_2}
\displaystyle\lim_{m \to \infty} \frac{\rk \sE_{q,m}}{\rk f_*^{(m)} \sO_{X^{(m)}} \left(q\left(-K_{X^{(m)}/T}- \left(f^{(m)} \right)^* m H\right)\right) }=1,
\end{equation}
where $\sE_{q,m}$ is a  subsheaf given by certain condition specified below of the subsheaf generated by global sections   of
\begin{equation}
\label{eq:tensor_product_bundle_outline}
f_*^{(m)} \sO_{X^{(m)}} \left(q\left(-K_{X^{(m)}/T}- \left(f^{(m)} \right)^* m H\right)\right) \cong \bigotimes_{\textrm{$m$ times}}f_* \sO_X(q(-K_{X/T}-f^*H)).
\end{equation}
The extra condition in the definition of $\sE_{q,m}$ is due to the need that $\Gamma$ has to be klt on a general fiber. This would be automatic if  the conjecture that products of $K$-semi-stable klt Fanos are $K$-semi-stable was known. Unfortunately this is a surprisingly hard unsolved conjecture in the theory of $K$-stability. Hence, we elude it by considering only bases of $H^0\left( X^{(m)}_t, -q K_{X^{(m)}_t} \right) \cong \displaystyle\bigotimes_{m \textrm{ times}} H^0 \left( X_t, -q K_{X_t}\right)$ that are induced from bases of $H^0 \left( X_t, -q K_{X_t}\right)$. As log canonical thresholds are known to behave well under taking products  (\autoref{prop:prod_basis}),
if the restriction   $\Gamma|_{X_t^{(m)}}$ to a general fiber is  a divisor corresponding to such basis, the $K$-stability of $X_t$ implies that $\left(X_t^{(m)}, \Gamma|_{X_t^{(m)}} \right)$ is klt. Hence, the additional condition in the definition of $\sE_{q,m}$ is that it is the biggest subsheaf as above such that $\left( \sE_{q,m} \right)_t$ is spanned by simple tensors for a basis $t_1,\dots,t_l$ of $\left( f_* \sO_X(q(-K_{X/T}-f^*H)) \right)_t$ to be specified soon.

So, we are left to specify a basis of $\left( f_* \sO_X(q(-K_{X/T}-f^*H)) \right)_t \cong H^0(X_t, -qK_{X_t})$ for which \autoref{eq:outline_goal_2} holds. For that we use the Harder-Narasimhan filtration $0=\sF^0 \subseteq \dots \subseteq \sF^r$ of $f_* \sO_X(q(-K_{X/T}-f^*H))$. Let the basis $v_1,\dots, v_l$ be any basis adapted to the restriction of this filtration over $t$, that is, to $0=\sF^0_t \subseteq \dots \subseteq \sF^r_t$. The lower part of the filtration, until  the graded pieces reach slope $2g$ (where $g$ is the genus of $T$),   is globally generated. Furthermore, there is an induced Harder-Narasimhan filtration on \autoref{eq:tensor_product_bundle_outline}. The slope at least $2g$ part of the latter filtration is globally generated such that its restriction over $t \in T$ is generated by simple tensors in $v_i$ (\autoref{prop:HN_Tensor_product}). Hence, if $\sE_{q,m}'$ is this part of the Harder-Narasimhan filtraton, then it is enough to prove that
\begin{equation}
\label{eq:outline_goal_3}
\displaystyle\lim_{m \to \infty} \frac{\rk \sE_{q,m}'}{\rk f_*^{(m)} \sO_{X^{(m)}} \left(q\left(-K_{X^{(m)}/T}- \left(f^{(m)} \right)^* m H\right)\right) }=1,
\end{equation}
The final trick of the semi-positivity part is then that \autoref{eq:outline_goal_3} can be translated to a probability limit, which then is implied by the central limit theorem of probability theory (\autoref{prop:vectP2}).

We explain here the probability theory argument via the example of
\begin{equation*}
\sF_m:=\displaystyle\bigotimes_{m \textrm{ times}} (\sO_{\bP^1}(-1) \oplus \sO_{\bP^1}(2)).
\end{equation*}
 The claim then is that as $m$ goes to infinity the rank of the non-negative degree part of $\sF_m$ over the rank of $\sF_m$ converges to $1$. It is easy to see that this is the following limit:
\begin{equation*}
\sum_{0 \leq i \leq m, 2i - (m-i) \geq 0 } {m \choose i} \left(\frac{1}{2} \right)^n =

\underbrace{\sum_{0 \leq i \leq m, i \geq \frac{m}{3} } {m \choose i} \left(\frac{1}{2} \right)^n \geq \sum_{0 \leq i \leq m, i \geq \frac{m}{2}- A\frac{\sqrt{m}}{4} } {m \choose i} \left(\frac{1}{2} \right)^n }_{\textrm{for $m$ big enough, where $A>0$ is an arbitrary fixed real number}}
\end{equation*}
The latter is the probability when flipping a coin $m$ times one gets at least $\frac{m}{2} - A\frac{\sqrt{m}}{4}$ heads. Note that for this $m$-times flipping the expected value is $\frac{m}{2}$ and $\sqrt{m}$-times the square deviation is $\frac{\sqrt{m}}{4}$. Hence, the above probability converges to $\int_A^\infty \frac{1}{\sqrt{2 \pi}} e^{\frac{-x^2}{2}} dx$  by the classical De Moivre-Laplace theorem, a special case of the central limit theorem. We obtain \autoref{eq:outline_goal_3} by taking $A \to \infty$ limit, and using that the above integral integrates the density function of the standard Gaussian normal distribution.

\subsubsection{ Nefness threshold.}
\label{sec:outline_nefness_threshold}
This part uses the same ideas as the above semi-positivity part, but in a different logical framework. That is, the argument is not a proof by contradiction. Instead, the starting point is that $\left(-K_{X/T} + \left( f^{(m)} \right)^* \left(\frac{\lambda_f}{v (n+1)}  + H \right) \right)^{n+1}>0$. Hence, again up to a little perturbation and by using the ideas of the previous point, there is an integer $m>0$ such that there exists a  $\Gamma \in \left|-\delta K_{X^{(m)}/T} + \left( f^{(m)} \right)^* m \left(\frac{\delta \lambda_f}{v (n+1)}  + H \right) \right|_{\bQ}$ for which  $\left( X^{(m)}_t,\Gamma_t \right)$ is klt for $t \in T$ general. Then standard semi-positivity argument (\autoref{prop:semi_positivity_engine_downstairs}) shows that
\begin{equation*}
K_{X^{(m)}/T} -\delta K_{X^{(m)}/T} + \left( f^{(m)} \right)^* m \left(\frac{\delta \lambda_f}{v (n+1)}  + H \right) = (1- \delta)  K_{X^{(m)}/T} + \left( f^{(m)} \right)^* m \left(\frac{\delta \lambda_f}{v (n+1)}  + H \right)
\end{equation*}
is nef. Lastly, one divides by $\delta -1$, converges to $0$ with $H$, and lastly by a standard lemma (\autoref{lem:product_nef}) removes the $(\_)^{(m)}$.

\subsubsection{ Positivity.}
\label{sec:outline_positivity}
The rough idea here is to use a twisted version of the ampleness lemma (\cite[3.9 Ampleness Lemma]{Kollar_Projectivity_of_complete_moduli}, with little modifications in \cite[Thm 5.1]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}). We need a twisted version of the ampleness lemma as the techniques developed until this point in the article  do not work directly over higher dimensional bases. The main idea here is that to get bigness of $\lambda_f$ it is enough to show positivity of $\lambda_f$ over a very general element $C$ of each moving family of curves of $T$, in a bounded way. Below we explain how we do this.

The main benefit of proving the nefness threshold result above

is the following:  one can prove, again using standard semi-positivity arguments (\autoref{prop:semi_positivity_engine_downstairs}), that $\sQ:=f_* \sO_X(-r K_{X/T} + \alpha f^* \lambda_f)$ is nef, for some constants $r$ and $\alpha$. Furthermore, these constants $r$ and $\alpha$  can be chosen to be uniform, as $f$ runs through all families obtained by base-changing on a very general element $C$ of a moving family of curves on $T$. Then, the ampleness lemma (\autoref{thm:ampleness}) gives an ample line bundle $B$ on $T$ such that for all curves $C$ as above, $C \cdot B  \leq C \cdot \det \sQ$. Then one can use another  trick from (semi-)positivity theory, which we also learned from Viehweg's works. That is, for $q :=  \rk \sQ$, there is an embedding
\begin{equation*}
\det \sQ \to \bigotimes_{q \textrm{ times}} f_* \sO_X(-r K_{X/T} + \alpha f^* \lambda_f) \cong f_*^{(q)} \sO_X\left(-r K_{X^{(q)}/T} + q \alpha \left( f^{(q)} \right)^* \lambda_f \right),
\end{equation*}
Using the adjunction of $f^{(q)}_*$ and $\left( f^{(q)} \right)^*$ this implies the inequality of divisors
\begin{equation*}

\left(f^{(q)}\right)^* B \leq  \left(f^{(q)}\right)^* \det \sQ \leq - r K_{X^{(q)}/T} + q \alpha \left( f^{(q)} \right)^* \lambda_f,
\end{equation*}
which survives the restriction over $C$ by the genericity assumption in the definition of the latter.
From here, a simple intersection computation shows that $C \cdot B$ bounds $\deg \lambda_f|_C$ from below up to some uniform constants, not depending on the choice of $C$ (see the end of the proof of  point \autoref{itm:semi_positive_no_boundary:big} of \autoref{thm:semi_positive_no_boundary}).


TITLE: Count bulbs in ON state
QUESTION [0 upvotes]: A room has N (1 to N inclusive) bulbs and N switches. N people go in one by one. 1st person goes in and toggles none of the switches. 2nd person goes in and toggles all switches other than the multiples of 2(2, 4, 6...). 3rd person goes in and toggles all switches other than multiples of 3(3, 6, 9...), and so on (Till Nth person toggles all the switches except Nth switch). Once the process is finished, how many bulbs are in 'on' state. (Assume that all bulbs to be in 'off' state initially).
Example : If N=3 then answer is 2 and when N=7 then answer is 5.
How to find it for given N ?

REPLY [2 votes]: Notice that for the switches to be in ON state, they would have to be "touched" an odd number of times. Consider the $k$th switch. It is touched by all except persons whose serial number is a divisor of $k$. Since the number of divisors of $k$ is odd iff the number is a perfect square, we should have all switches ON except switches whose serial number is a perfect square.


TITLE: Where can a mathematician read Hawking's proof for the existence of a black hole's radiation?
QUESTION [3 upvotes]: Can a mathematician with no knowledge in physics understand this proof? Is there a paper that presents in suitable way for me?

REPLY [0 votes]: Where can a mathematician read Hawking's proof for the existence of a black hole's radiation?

IMHO the first port of call is his papers. Hawking’s first paper on Hawking radiation was arguably the four laws of black hole mechanics co-authored with Brandon Carter and Jim Bardeen in 1973. Section 1 talked about gravitational collapse and gave three references to Hawking’s papers followed by three references to Carter’s. Section 2 talked about an integral formula and Killing vectors and covariant derivatives and antisymmetrization. It talked about hypersurfaces and angular momentum measured asymptotically from infinity. It also talked about complex conjugate null vectors and null tetrads, and it gave a great deal of mathematics including expressions like this:

In section 4 the authors said “we shall pursue the analogy between black holes and thermo-dynamics and shall formulate four laws which correspond to and in some ways transcend the four laws of thermodynamics”. They started with the second law wherein “the area A of the event horizon of each black hole does not decrease with time”. They claimed that it establishes an analogy with entropy. Their first law which said “any two neighboring stationary axisymmetric solutions are related by δM = κ/8π δA + ΩHδJH + ∫ΩδdJ + ∫μδdN + ∫ΩδdS”. And that “it can be seen that κ/8π in analogous to temperature”. They also said "it should be emphasized that κ/8π and A are distinct from the temperature and entropy of the black hole" along with "the effective temperature of a black hole is absolute zero”. Their zeroth law said “the surface gravity, κ of a stationary black hole is constant over the event horizon”. Their third law said “it is impossible by any procedure, no matter how idealized, to reduce κ to zero by a finite sequence of operations”.
Hawking’s next significant paper was black hole explosions? which appeared in Nature in 1974. Hawking said “any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π)(ħ/2k) ≈ 10−6 (M /M)K where κ is the surface gravity of the black hole”. He also said the temperature will increase as the hole loses energy and a smaller black hole would end its life in an explosion "equivalent to about 1 million 1 Mton hydrogen bombs". He talked about a massless Hermitian scalar field in an asymptotically flat spacetime. He referred to the Heisenberg operator ϕ with ai and ai+ interpreted as creation and annihilation operators. He talked about outgoing waves and waves crossing the event horizon and positive and negative frequencies. He said the number of particles created and emitted in a gravitational collapse can therefore be determined, and he talked about a wave propagating backward in spacetime from future null infinity to past null infinity.
Hawking’s next paper was his Hawking radiation paper particle creation by black holes dating from 1975. He said “it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies”. In section 1 he set the scene talking about general relativity and quantum mechanics in curved spacetime, saying “one can interpret this as implying that the time dependent metric or gravitational field has caused the creation of a certain number of particles". He said the uncertainty in the local energy can be thought of as corresponding to the local energy density of particles created by the gravitational field. He said "the gravitational field of a black hole will create particles and emit them to infinity at just the rate that one would expect if the black hole were an ordinary body with a temperature in geometric units of κc/2π, where κ is the ‘surface gravity’ of the black hole". Then he said that as the mass of the black hole decreased, the area of the event horizon would have to go down, thus "violating the law that, classically, the area cannot decrease". Hawking also said "this violation must, presumably, be caused by a flux of negative energy". He then said "one might picture this negative energy flux in the following way. Just outside the event horizon there will be virtual pairs of particles, one with negative energy and one with positive energy". He said one of the particles has negative energy relative to infinity, and that the other particle of the pair "having a positive energy, can escape to infinity where it constitutes a part of the thermal emission described above". There's more, but IMHO you have to sit down and read it for yourself.

Can a mathematician with no knowledge in physics understand this proof? Is there a paper that presents in suitable way for me?

It's hard to say. I think you can understand what Hawking is saying even if you don't know too much physics. See of the black hole thermodynamics by Xinyong Fu for more information, along with work by Yvan Leblanc.


TITLE: Are $\Bbb R^2$ and $\Bbb R^3$ homeomorphic?
QUESTION [2 upvotes]: I know $\Bbb R$ and $\Bbb R^2$ are not homeomorphic.but

Are $\Bbb R^2$ and $\Bbb R^3$ homeomorphic?

REPLY [7 votes]: $\mathbb{R^n}$ and $\mathbb{R^m}$ aren't homeomorphic for $m \neq n$. An easy proof using algebraic topology can be given: remove a point from each of the two spaces. Then the spaces become homotopic to $S^n$ and $S^m$ respectively. But $nth$ homology group of $S^m$ is $0$ if $m \neq n$. So, $\mathbb{R^n}$ and $\mathbb{R^m}$ aren't homeomorphic for $m \neq n$.


TITLE: PDA for this CFL $L=\{a^i b^j c^k, k=i*j, i,j=0,1,2...\}$
QUESTION [0 upvotes]: I have troubles constructing pushdown automata for the following language: $L=\{a^i b^j c^k \mid k=i*j; \quad i,j=0,1,2...\}$ .I am not interested in the automata itself, just the idea of handling the product.

REPLY [1 votes]: It is very reasonable that you are having problems constructing an PDA, because the language is not context-free. We can use the pumping lemma to show this. Let $p$ be the pumping constant from this lemma and consider the word $a^pb^pc^{p\cdot p}\in L$.
Any pumping must affect both $a^pb^p$ and $c^{p\cdot p}$, because otherwise we obtain words not in the language. Since $|vwx|\leq p$ this means that $v\in b^+$ and $x\in c^+$.
Even for the minimal case of $v = b$, $x$ would need to be $c^p$ to arrive at $a^pb^{p+1}c^{p\cdot (p+1)}$. But again because of $|vwx|\leq p$ the factor $x$ could at most be $c^{p-1}$.
So there is no pumping factorization for the word $a^pb^pc^{p\cdot p}$ and consequently $L$ is not context-free.


TITLE: Decomposition of Free Module over PID
QUESTION [0 upvotes]: Let $M$ be a free module with rank $2$ over the PID $R$ having basis $B=\{b_1,b_2\}$. I have a few questions regarding the submodule $Rm$, where $m$ is some element of our module $M$. Suppose $m$ is of the form $xb_1+xb_2$ for $x\in R$. How can we show that there is some submodule $T$ such that $M=Rm\oplus T$ if and only if $x$ has an inverse in $R$?
C
an we also show that if $m=xb_1+yb_2$, then there is $T$ s.t. $M=Rm\oplus T \iff$ the ideal generated by $x$ and $y$ gives the whole ring $R$?
I've given this some thought and have ideas to use a decomposition as done here, but I'm not really sure how to approach this. I'm also not sure how $x$ being invertible is relevant. Thanks in advance.

REPLY [0 votes]: This is easiest to see working with $2\times2$-matrices over $R$: Let me show that if $m=xb_1 + yb_2$ then there exists $T\subseteq M$ such that $M = Rm \oplus T$ if and only if $Rx+Ry = R$.
First assume that $Rx+Ry=R$. Then there are $w,z\in R$ such that $wx - zy = 1$. Now define $t := zb_1 + wb_2\in M$ and $A := \begin{pmatrix}x & y\\ z & w\end{pmatrix}\in {\rm Mat}_{2\times 2}(R)$ is invertible (its inverse being $\begin{pmatrix}w & -y\\ -z & x\end{pmatrix}$). Thus, we have
$$
\begin{pmatrix} m\\ t\end{pmatrix} = \begin{pmatrix} x & y\\ z & w\end{pmatrix}\cdot \begin{pmatrix} b_1\\ b_2\end{pmatrix}.
$$
Setting $T:= Rt$, it is clear that $M = Rm\oplus T$. For example linear independence of $m$, $t$: Let $\lambda,\mu\in R$ such that $0 = \lambda m + \mu t = \begin{pmatrix} \lambda & \mu\end{pmatrix}\cdot \begin{pmatrix} m\\ t\end{pmatrix}$. Then
$$
0 = \begin{pmatrix} \lambda & \mu\end{pmatrix}\cdot \begin{pmatrix} m\\ t\end{pmatrix} = \left(\begin{pmatrix} \lambda & \mu\end{pmatrix}\cdot \begin{pmatrix} x & y\\ z & w\end{pmatrix}\right)\cdot \begin{pmatrix} b_1\\ b_2\end{pmatrix}.
$$
Since $b_1, b_2$ is a basis, we have $\begin{pmatrix} \lambda & \mu\end{pmatrix}\cdot \begin{pmatrix} x & y\\ z & w\end{pmatrix} = 0$ and hence
$$
0 = \begin{pmatrix} \lambda & \mu\end{pmatrix}\cdot \begin{pmatrix} x & y\\ z & w\end{pmatrix} \cdot \begin{pmatrix} w & -y\\ -z & x\end{pmatrix} = \begin{pmatrix} \lambda & \mu\end{pmatrix}\cdot \begin{pmatrix} 1 & 0\\ 0 & 1\end{pmatrix} = \begin{pmatrix} \lambda & \mu\end{pmatrix},
$$
i. e. $\lambda = \mu = 0$.
Conversely, if there exists $T\subseteq M$ such that $M = Rm\oplus T$ then $T$ has to be free of rank $1$, generated by, say $t=zb_1 + wb_2\in M$. Since $m, t$ is also a basis of $M$, there exist $a,b,c,d\in R$ such that $b_1 = am + bt$ and $b_2 = cm +dt$. Hence
$$
\begin{pmatrix}m\\t\end{pmatrix} = \begin{pmatrix}x & y\\ z & w\end{pmatrix}\cdot \begin{pmatrix} b_1 \\ b_2\end{pmatrix} = \begin{pmatrix} x & y\\ z & w\end{pmatrix}\cdot \begin{pmatrix} a & b\\ c & d\end{pmatrix} \cdot \begin{pmatrix} m\\ t\end{pmatrix}.
$$
Since $m,t$ are linearly independent, we conclude $\begin{pmatrix} x & y\\ z&w\end{pmatrix}\cdot \begin{pmatrix} a & b\\ c & d\end{pmatrix} = \begin{pmatrix} 1 & 0\\ 0&1\end{pmatrix}$. In particular, $ax + cy = 1\in R^\times$, i. e. $Rx + Ry = R$.


TITLE: Conditional expectation for Kelly strategy/criterion
QUESTION [2 upvotes]: Consider a gambler who at each gamble either wins or loses his bet with
  probabilities $p$ and $1-p$, independent of earlier gambles. When $p > 1/2$
  a popular gambling system  known as the Kelly strategy, is to always bet the fraction $2p-1$ of the current fortune. Compute the expected fortune after $n$ gambles, starting with $x$ units and employing the Kelly strategy.

So, I feel I need to use the rule $$\Bbb E[X] =\Bbb E\left[\Bbb E[X\mid Y]\right]$$ where $X =$ Fortune after $n$ gambles and $Y =$ Fortune after $n-1$ gambles.
Also if I start with a fortune $a$, then I will earn $a(2p-1)$ with probability $p$ or lose $a(2p-1)$ with probability $(1-p)$. So my expected current fortune should be $$ap(2p-1) + a(1-p)(2p-1)$$
I don't really know how to continue after this.

REPLY [6 votes]: Say the gambler currently has fortune $a$, then if he wins, he will earn $a(2p-1)$ and his fortune will become $a + a(2p-1) = 2pa$. If he loses, he will lose $a(2p-1)$ and his fortune will become $a-a(2p-1) = (2-2p)a$.
And we know that all the winning/losing events in the sequence are independent, so the joint probability can be calculated with simple multiplication.
Let's say, out of the $n$ gambles, he wins $i$ times and loses $(n-i)$ times, then after the $i$ wins and $(n-i)$ loss, his fortune will be
$$ a \cdot (2p)^i \cdot (2-2p)^{n-i}$$
Then let's look at the wins and losses of the $n$ gambles. The probability of winning $i$ times (and clearly losing $n-i$ times) is equal to
$$ \binom{n}{i} p^{i}(1-p)^{n-i} $$
because there are $\binom{n}{i}$ ways of winning exactly $i$ times (and losing exactly $(n-i)$ times).
Thus the expectation after $n$ gambles would be
$$ \sum_{i = 0}^{n}\binom{n}{i}p^i(1-p)^{n-i}a(2p)^{i}(2-2p)^{n-i} $$


TITLE: squaring the equality constraints
QUESTION [0 upvotes]: When creating an unconstrained optimization problem from an equality constrained one, the usual way to build the Lagrangian, is by adding a term consisting of a multiplier, multiplied by the equality constraint. Are there problem instances, where it makes better sense to square the equality constraint and then use that in the unconstrained problem?

REPLY [3 votes]: When the equality constraint has the form $g(x)=0$ then replacing it with $h(x):=g^2(x)=0$ is fatal. Lagrange's method detects the points where $df(x)=\lambda dg(x)$ with a finite $\lambda$, and together with $g(x)=0$ you can expect to obtain a finite number of "conditionally stationary points" on the surface $S: \>g(x)=0$. But $dh(x)=0$ at all points of $S$, so the equation $df(x)=\lambda dh(x)$ will not produce a single point that is of interest.
It is another matter with the objective function $f$. When $f$ appears as a square root (e.g., if $f$ represents a distance between certain points) then no harm is done when you replace $f$ by $f^2$.


TITLE: Where can I find rigorous statements about the spectral decomposition of reductive groups?
QUESTION [3 upvotes]: Given a global field $F$ and a reductive group $G$, where can I find the spectral decomposition of
$$ L^2( Z(\mathbb{A}) G(F) \backslash G( \mathbb{A})).$$
I will need the result in this generality, means for a general reductive group and for function and number fields.
I have just seen some instances of such theorems yet and would be happy about a reference. Of course, I expect that the function field and number field case have been treated, but probably in different places.

REPLY [2 votes]: This is the main aim of the book "Spectral decomposition and Eisenstein series" by Moeglin and Waldspurger, which is based on the original work of Langlands but updated to use more modern notation and techniques. This has a very detailed account of the spectral decomposition theorems in chapter VI.


TITLE: Counting probability of disease
QUESTION [1 upvotes]: There are 2 types of people in a big factory: prone to COVID (30% of workers), and nonprone to COVID (70% of workers). For a COVID-prone person, p(COVID) = $0.4$. For a nonprone to COVID person, p(COVID) = $0.2$.
Compute:
a) the probability a random factory worker will have a COVID
b) given a person who has a COVID, what is p(person is COVID-prone).
I'm not sure, should I use binomial probability or usual, or how I should count in general. I suppose in (a) I need to calculate $0.3*0.4 + 0.7*0.2 = 0.26$

REPLY [1 votes]: For a) you are correct. The probability that someone has Covid is the sum of the probability for a covid-prone person to get it, plus the probability for a non-prone person to get it, or
$$0.4\times 0.3 +0.2\times 0.7 = 0.26$$
For b) you want to know the probability for a person who has covid to be covid-prone. The probability to have covid if you are prone is
$$0.4\times 0.3$$and since you know that the person has already covid you divide by the probability of the known event calculated above (see conditional probability)
$$\dfrac{0.4\times 0.3}{0.26} \approx 0.4615$$

REPLY [1 votes]: If we write everything we know, we got :
$P(Covid~\vert~prone~Covid) = 0.4 $
$P(prone~Covid) = 0.3$
$P(non-prone~Covid) = 0.7$
$P(Covid~\vert~non-prone~Covid) = 0.2$
We can deduce : $P(Covid) = P(Covid\bigcap prone~Covid) + P(Covid \bigcap non-prone~ Covid)$
This expression corresponds to the answer you found for a)
For b) : You have to calculate $P(prone~Covid \vert ~Covid)$ using what you found previously


TITLE: Is every quadratic surface in $\mathbb{P}^3$ ruled?
QUESTION [0 upvotes]: Let $\mathbb{P}$ be the projective line over an algebraically closed field $k$.
Is it true that every quadratic surface in $\mathbb{P}^3$ is ruled?
How can one see that this is the case?
This seems rather strange to me, but I was told this by a classmate.

REPLY [2 votes]: Real case
The sphere is an unruled quadric, so the answer is no.
Wikipedia distinguishes several cases, in particular three non-degenerate ones:

The first case is the empty set.
The second case generates the ellipsoid, the elliptic paraboloid or the hyperboloid of two sheets, depending on whether the chosen plane at infinity cuts the quadric in the empty set, in a point, or in a nondegenerate conic respectively. These all have positive Gaussian curvature.
The third case generates the hyperbolic paraboloid or the hyperboloid of one sheet, depending on whether the plane at infinity cuts it in two lines, or in a nondegenerate conic respectively. These are doubly ruled surfaces of negative Gaussian curvature.

So it effectively depends on the sign pattern of the normal form, which is the sign pattern of the eigenvalues of the symmetric matrix representing the quadric.
To see that the statement is true for the last case, it is enough to show it for one representative of that class, since all others are related to that by some projective transformation, and a projective transformation will preserve the ruling.
Algebraically closed field
OK, as Marioano pointed out in a comment, there is a comment to the question indicating that the question is not about real projective space, but instead over an algebraically closed field. In that case, the concept of signs breaks down, and all non-degenerate quadrics are related by projective transformations, and hence equivalent. So if any of them is ruled, all are. Showing that at least one of them is, by giving an explicit ruling, should be fairly simple.
Degenerate conics will factor in a pair of planes, which are obviously ruled as well.


text:
{\bf Problem.} Let $x$ and $y$ be distinct real numbers such that
\[
\begin{vmatrix} 1 & 4 & 9 \\ 3 & x & y \\ 3 & y & x \end{vmatrix}
= 0.\]Find $x + y.$
{\bf Level.} Level 3
{\bf Type.} Precalculus
{\bf Solution.} Expanding the determinant, we obtain
\begin{align*}
\begin{vmatrix} 1 & 4 & 9 \\ 3 & x & y \\ 3 & y & x \end{vmatrix} &= \begin{vmatrix} x & y \\ y & x \end{vmatrix} - 4 \begin{vmatrix} 3 & y \\ 3 & x \end{vmatrix} + 9 \begin{vmatrix} 3 & x \\ 3 & y \end{vmatrix} \\
&= (x^2 - y^2) - 4(3x - 3y) + 9(3y - 3x) \\
&= x^2 - y^2 - 39x + 39y \\
&= (x - y)(x + y) - 39(x - y) \\
&= (x - y)(x + y - 39).
\end{align*}Since this is 0, either $x - y = 0$ or $x + y - 39 = 0.$  But $x$ and $y$ are distinct, so $x + y = \boxed{39}.$

set_name: MATH


TITLE: Two tensions in massless string
QUESTION [0 upvotes]: There should only be one tension in a massless string.
But imagine this situation. Let a massless rope hang from a ceiling and two monkeys of different weights are holding on to the rope at two different heights. Then there should be two different tensions which balance the two weights of the monkeys.
How do you account for this?

REPLY [0 votes]: Then there should be two different tensions which balance the two weights of the monkeys.

If the monkeys are of different weights, the ropes are not balanced at all (unless the friction in the ceiling is great enough). This is the case, when the monkeys (and, of course, the rope with them) move with some acceleration. Because the rope does not extend during the process, the differences between tensions for each little segment of the rope are the same. Thus, the tension is uniform throughout the rope.
P.s.: yes, I do write "each little segment of the rope" by which I mean a segment of mass dm. You might say, that the rope is massless, but in reality, it does not have zero mass, it has a mass, which is negligible by the conditions of current problem. So this interpretation of breaking the rope in little sections is right.