text:
\begin{document}

\begin{frontmatter}

\title{An iterative estimation for disturbances of semi-wavefronts to the delayed Fisher-KPP equation}

\author[a]{Rafael Benguria D.}
\address[a]{Instituto de F\'isica, Pontificia Universidad Cat\'olica de Chile, Casilla 306, Santiago, Chile
\\ {\rm E-mail: rbenguri@fis.uc.cl}}
\author[b]{Abraham Solar}

\address[b]{Instituto de F\'isica, Pontificia Universidad Cat\'olica de Chile, Casilla 306, Santiago, Chile
\\ {\rm E-mail: asolar@fis.uc.cl}}

\begin{abstract} We give an iterative method to estimate the disturbance of semi-wavefronts of  the equation:  $\dot{u}(t,x) = u''(t,x) +u(t,x)(1-u(t-h,x)),$ $x \in \mathbb{R},\ t >0;$ where $h>0.$ As a consequence, we show the exponential stability, with an unbounded weight, of semi-wavefronts with speed $c>2\sqrt{2}$  and $h>0$. Under the same restriction of  $c$ and $h$, the uniqueness of semi-wavefronts is obtained.
\end{abstract}

\begin{keyword}semi-wavefront\sep
stability \sep  uniqueness, delay, reaction-diffusion equations
\MSC[2010] 34K12\sep 35K57\sep 92D25
\end{keyword}

\end{frontmatter}

\section{Introduction}

In this work, we present some results that answer  key questions about bounded solutions of the form $u(t,x):=\phi_c(x+ct)$ to the diffusive Hutchison equation, also called delayed Fisher-KPP equation, with delay $h\geq0$,
\begin{eqnarray}\label{ie}
\dot{u}(t,x) &=& u''(t,x) + u(t,x)(1-u(t-h,x)),\quad t >0,\ x \in \R,
\end{eqnarray}
(here $\ \dot{} \ $ indicates the temporal derivate whereas $\ ' \ $ indicates the spatial derivate) satisfying  $\phi_c(-\infty)=0$ and $\liminf_{z\to+\infty}\phi_c(z)>0$ for some speed $c>0$ and profile $\phi_c:\R\to\R_+$. Such solutions are called {\it semi-wavefronts} which clearly satisfy the ordinary differential equation with delay

\begin{eqnarray}\label{swe}\nonumber
\phi''_c(z)-c\phi'_c(z)+\phi_c(z)(1-\phi_c(z-ch))=0,\quad z\in\R.
\end{eqnarray}
 If $\phi_c(+\infty)=1$  semi-wavefronts are called {\it wavefronts}.

If $h=0$, the questions on existence, uniqueness, geometry and stability of wavefronts have been satisfactory responded (see, e.g. \cite{STG} and \cite{UC} and references therein). In this case the general conclusions are: (i) semi-wavefronts are indeed monotone  wavefronts existing for all $c\geq 2$, (ii) two wavefronts with same speed are unique up to translations and (iii) wavefronts are stable under suitable perturbations.

 However, for $h>0$ it has been only recently  established the existence of semi-wavefronts on the domain $\{(h,c)\in\R^2:  h\geq 0\ \hbox{and} \ c\geq 2\}$ (see \cite{HT} and \cite{NP}). The study of the existence of wavefronts to (\ref{ie}) was initiated by Wu and Zou \cite{WZ} who constructed a monotone integral operator for small $h$, depending on $c$, whose iterative application to adequate sub and super-solutions converges to a monotone wavefront. Next, Hasik and Trofimchuk \cite{HT} demonstrated that such monotone wavefronts are unique up to translations. Moreover, the existence and uniqueness of monotone wavefronts has been completely established \cite{AGT,HT}.
In \cite{HT} and \cite{NP} the authors gave significative information about the geometry of semi-wavefronts, they demonstrated the existence of non-monotone wavefronts in the domain $\{(h,c)\in\R^2: 0\leq h\leq 1\ \hbox{and} \ c\geq 2\}$ and the existence of asymptotically periodic semi-wavefronts whenever $h\geq \pi/2$, in a neighborhood of $\pi/2$, and $c(h)$ sufficiently large (see \cite[Theorem 1.7]{HT} and \cite[Theorem 2.3]{NP}). One of the open problems proposed in \cite{NP} is to prove that the semi-wavefronts are unique, up to translation, in $\{(h,c)\in\R^2:  h\geq 0\ \hbox{and} \ c\geq 2\}.$ In this paper we  obtain the uniqueness of semi-wavefronts in the a region of parameters $(h,c)$ that includes the region $\{(h,c)\in\R^2: h\geq 0\ \hbox{and}\ c>2\sqrt{2}\}$, and we provide a result about the stability of these semi-wavefronts; as much as we know, there are no results concerning the stability of semi-wavefronts of (\ref{ie}) when $h>0$. In our main result, the description of domain the parameters $(h,c)$ to the stability depends on a uniform estimation of semi-wavefronts. Thus, the estimation $\phi_c(z)\leq e^{ch}$, for all $z\in\R$, obtained in \cite{HT},  turns out very profitable.

 The study of the asymptotic behavior of solutions to reaction-diffusion equations with delay normally require some kind of maximum principle. For example, the Mackey-Glass type equations
 \begin{eqnarray}\label{MG}
 \dot{u}(t,x)=u''(t,x)-u(t,x)+g(u(t-h,x)),\quad t>0, x\in\R,
 \end{eqnarray}
 has been widely studied under the KPP condition $g(u)\leq g'(0)u$, $u>0$, for which, initially, the monotonicity of $g$ was assumed (see, e.g.,  \cite{MOZ}). Then, it was shown in \cite{STR1} that the conclusion of stability of semi-wavefronts  to (\ref{MG}) can be obtained only assuming the Lipschitz continuity of $g$ (type monostable) by a suitable application of the maximum principle. However, this technique does not work with (\ref{ie}) since this equation is strongly non monotone'.

 Our technique is closed to the contractive approach used by Travis and Webb \cite{TW} which  establishes the stability and uniqueness of stationary solutions to nonlinear partial functional differential equation $\dot{u}(t)=Au(t)+F(u_t), t>0$, where $u_t(\theta):=u(t+\theta)$ with $\theta\in[-h,0]$ and $u:[-h,+\infty)\to X$ for some Banach space $X$, $F:C([-h,0],X)\to X$ and $A$ is a certain linear operator. The main assumption in \cite{TW} is that  $A$ generates a strongly continuous semigroup with a growth bound greater than the global Lipschitz constant of $F.$ Unfortunately,  the Travis-Webb approach does not apply to (\ref{ie}) because the nonlinearity is not Lipschitz in the usual spaces. However, we show that this technique can be adapted using a variation of parameter formula given in terms of a certain fundamental solution to the linearization of (\ref{ie}) around of a semi-wavefront. Here we follow the theory of fundamental solutions constructed by Friedman \cite{AF}.

 So, in order to pose the Cauchy problem we take an initial datum in the space  $\mathcal{C}:=C([-h,0],C^{1,1}(\R))$ where, as usual,
$$
C^{1,1}(\R)=\{f:\R\to\R \ \hbox{is bounded and Lipschitz continuous} \}
$$

 \noindent with norm
  $$|f|_{C^{1,1}}:=\sup_{x\in\R}|f(x)|+\sup_{x\neq y}\frac{|f(x)-f(y)|}{|x-y|}$$
and the norm in $\mathcal{C}$ is the sup-norm, i.e.,  $|f|_{\mathcal{C}}:=\sup_{s\in[-h,0]}|f(s)|_{C^{1,1}}$. With all this notation, we give the following global existence result to solutions of (\ref{ie}):

\begin{proposition}\label{prop1}
If $u_0\in\mathcal{C}$ then there exists a unique solution $u(t,\cdot)$ to (\ref{ie}) such that $u(t,x)=u_0(t,x)$ for all $(t,x)\in[-h,0]\times\R$ and $u_t\in\mathcal{C}$ for all $t\geq 0$. Also, $\lim_{t\to 0}u(t,x)=u_0(0,x)$ for all $x\in\R$ and  if $u_0\geq 0$ then
$u(t,x)\geq 0$ for all $(t,z)\in[-h,+\infty)\times\R$.
\end{proposition}

\begin{proof}
We denote  the fundamental solution to the heat equation by $\Gamma_0(t,x):=e^{-x^2/4t}/2\sqrt{\pi t}$. Next, from \cite[Lemma 4.1]{UC} the integral equation
\begin{eqnarray} \label{inte}
u(t,x)=\Gamma_0(t,\cdot)\ast u_0(0,\cdot)(x)+\int_0^t\Gamma_0(t-s,\cdot)\ast[1-u(t-h,\cdot)]u(t,\cdot)(x)ds,
\end{eqnarray}
has a unique bounded solution on $[0,h]\times\R$ which is a solution to (\ref{ie}) since $u(t-h,\cdot)$ is Lipschitz continuous, uniformly with respect to $t\in[0,h]$. Then, differentiating (\ref{inte}) we conclude that
\begin{eqnarray}\label{derivate}
|u_x(t,x)|\leq \operatorname{ess}\sup_{y\in\R}|u_x(0,y)|+\frac{\sqrt{t}}{2\sqrt{\pi}}\sup_{(s,y)\in[0,h]\times\R}|u(s,y)[1-u(s-h,y)]|\ \hbox{on} \ [0,h]\times\R
\end{eqnarray}

\noindent so $u(t,\cdot)$ is  Lipschitz continuous, uniformly respect to $t\in[0,h]$. Repeating the process to the intervals $[h,2h],[2h,3h]...$ the first assertion is true. Next, since the parabolic operator $Lu=u''+(1-u_0)u-\dot{u}$ has a fundamental solution on $[0,h]\times\R$( see, e.g., \cite[Chapter 1, Theorem 10]{AF}) we have that $u(t,x)$ tends to $u_0(0,x)$ when $t$ tends to 0, for all $x\in\R.$ Finally, by the Phragm\`en-Lindel\"of principle \cite[Chapter 3, Theorem 10]{PW} applied to the operator $L$ we obtain $u(t,x)\geq 0$, for all $(t,x)\in[0,h]\times\R$, whenever $u_0\geq 0$ and in a similar form we have $u(t,\cdot)\geq 0$ for the intervals $[h,2h],[2h,3h]...$
\end{proof}
\begin{remark}
We note that  if the initial datum is such that $u_0(s,x)=O(x^2)$ the comparison principle works too, however, the boundedness of derivate in (\ref{derivate}) requires the boundedness of $u_0(s, \cdot)$ uniformly in $s\in[0,h]$ and without the uniform boundedness  in time the uniqueness of solution fails  to the step $[0,h]$ (see, e.g.,  \cite{ChKm}). Also, we note that the typical results  to global existence (see, e.g., \cite[Proposition 2.1]{TW} and \cite[Theorem 2.3]{Wu}) do not apply to (\ref{ie}).

\end{remark}

An interesting open problem is to determine whether the solutions of (\ref{ie}), with initial data in $\mathcal{C}$ for example, are globally bounded in time. As much as we know, this issue has been only dealt to the boundary-initial Cauchy problem on a bounded domain (see, e.g., \cite{Fri}).

\section{Main Result}
Now, we denote by $0<\lambda_1(c)\leq \lambda_2(c)$ the zeros of quadratic function
$$
\epsilon_{\lambda}:=\lambda^2-c\lambda+1.
$$

Next, for $\lambda\in\R$ we define
$$
|f|_{\lambda}=\sup_{z\in\R}e^{-\lambda z}|f(z)|=||e^{-\lambda \cdot}f(\cdot)||_{\infty},
$$

and
$$
L_{\lambda}^{\infty}=\{f:\R\to\R, |f|_{\lambda}<\infty \}.
$$

\noindent Also, denoting  by $z=x+ct$ and $v(t,z):=u(t,z-ct)$ we have

\begin{eqnarray}\label{eqc}
\dot{v}(t,z)=v''(t,z)-cv'(t,z)+v(t,z)(1-v(t-h,z-ch)), \quad (t,z)\in\R_+\times\R
\end{eqnarray}

\noindent so, a semi-wavefront $\phi_c$ is a stationary solution of (\ref{eqc}) and we can establish the following result on exponential stability of semi-wavefronts.

\begin{thm}[Stability with unbounded weigth]\label{sthm}
Let $\lambda_c\in(\lambda_1(c),\lambda_2(c))$. Suppose that $\phi_c$ is semi-wavefront  to (\ref{ie}) and $v_0\in\mathcal{C}$ an initial datum to (\ref{eqc}). Then the following assertions are true

\begin{itemize}
\item[(i)] If $v_0$ is non negative such that $v_0(0,\cdot)\in L^{\infty}_{\lambda_c}$ then

\begin{eqnarray}\label{zeroc}\nonumber
\lim_{t\to\infty}v(t,z)=0\quad \hbox{for all} \quad z\in\R.
\end{eqnarray}

\item[(ii)] If
\begin{eqnarray}\label{norm}\nonumber
e^{-\lambda_c ch}|\phi_c|_{0}\leq \epsilon_{\lambda_c}
\end{eqnarray}
and $\delta\leq 0$ is number satisfying
\begin {eqnarray}\label{inq1}
e^{-\lambda_c ch}|\phi_c|_{0}\leq e^{\delta h}(\delta+\epsilon_{\lambda_c})
\end{eqnarray}
then
\begin{eqnarray}\label{inp}
\phi_c(\cdot)-v_0\in C([-h,0],L^{\infty}_{\lambda_c})
\end{eqnarray}
implies
\begin{eqnarray}\label{expst}
|v(t,\cdot)-\phi_c|_{\lambda_c}\leq K_{\lambda_c} e^{\delta t}\quad \hbox{for all}\quad t\geq -h,
\end{eqnarray}
where $K_{\lambda_c}=\max_{s\in[-h,0]}|\phi_c-v_0(s,\cdot)|_{\lambda_c}.$

Finally, if we take $\lambda_c=c/2$ in (\ref{inp}) we obtain (\ref{expst}) on the domain $\{(h,c)\in\R^2: h\geq 0\ \hbox{and}\ c>2\sqrt{2}\}.$
\end{itemize}
\end{thm}

Denoting by
$$
\kappa_h^c:=\frac{c}{2}-\frac{1}{ch}+\frac{1}{2}\sqrt{c^2+\frac{4}{c^2h^2}-4},
$$
the number $\kappa_h^c$ is a critical point of the function $Q:[\lambda_1(c),\lambda_2(c)]\to\R$ defined by
$$
Q(\lambda):=e^{\lambda ch}(-\lambda^2+c\lambda-1),
$$
we can establish the following result for the uniqueness of semi-wavefronts to (\ref{ie}) up to translations.
\begin{corollary}[Uniqueness of semi-wavefronts]\label{coro1}
Let $c>2$. If $\phi_c$ and $\psi_c$ are semi-wavefronts such that
\begin{eqnarray}\label{uniq}
\min\{|\psi_c|_{0},|\phi_c|_{0}\}<Q(\kappa_h^c)
\end{eqnarray}
then $\phi_c(\cdot)=\psi_c(\cdot+z_0)$ for some $z_0\in\R$. In particular, if $c>2\sqrt{2}$ the semi-wavefronts to (\ref{ie}) with speed $c$ are unique up to translations for all $h\geq0$.
\end{corollary}
 \bigskip

\bigskip

Now, for a function $v:[-h,+\infty)\times\R\to\R$ and an initial datum $\eta_0$ we define the following Cauchy problem
\begin{eqnarray}\label{ACP}
\dot{\eta}(t,z)=\eta''(t,z)-c\eta'(t,z)+[1-v(t-h,z-ch)]\eta(t,z)\quad (t,z)\in\R_+\times\R,\\
\eta(0,z)=\eta_0(z) \ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad z\in\R.\label{ACP1}
\end{eqnarray}

\begin{lemma}\label{lem1}
Let $\lambda\in\R$ and $v_t\in\mathcal{C}$ for all $t\geq 0$ be. If   $\eta_0\in L^{\infty}_{\lambda}$  then (\ref{ACP})-(\ref{ACP1}) has a unique solution $\eta(t,\cdot)$ in $L^{\infty}_{\lambda}$ for all $t\geq 0$. Moreover, denoting by $\Gamma_v(t)\eta_0:=\eta(t,\cdot)$  we have the following estimate
\begin{eqnarray}\label{estimate1}
|\Gamma_v(t)\eta_0|_{\lambda}\leq e^{-\epsilon_{\lambda}t}|\eta_0|_{\lambda}\quad\hbox{for all}\quad t\geq 0.
\end{eqnarray}
\end{lemma}
\begin{proof}
Making the change of variable $p(t,z):=e^{-\lambda z}\eta(t,z)$ we have that $p$ satisfies  the Cauchy problem
\begin{eqnarray}\label{ACP2}
\dot{p}(t,z)=p''(t,z)+(2\lambda-c)p'(t,z)+(\epsilon_{\lambda}-v(t-h,z-ch))p(t,z)\quad(t,z)\in\R_+\times\R\\
 p(0,z)=e^{-\lambda z}\eta_0(z).\quad\quad\quad\quad z\in\R \label{ACP3}
\end{eqnarray}

\noindent If we translate the spatial variable by $(2\lambda-c)t \ $  then, using the Lipschitz continuity of the function $\tilde{v}(t,\cdot):=v(t,\cdot-(2\lambda-c)t)$ (uniformly in $t\in[0,T]$, for any $T<\infty$),  by \cite[Lemma 4.1]{UC} we conclude that (\ref{ACP2})-(\ref{ACP3}) has a unique bounded solution (uniformly in $t\in[0,T]$, for any $T<\infty$), so $\eta(t,\cdot)\in L^{\infty}_{\lambda}$ for all $t\geq 0$. Then, by the Phragm\`en-Lindel\"of principle $\eta(t,\cdot)\geq 0$ for all $t\geq 0$ and the estimation (\ref{estimate1}) is obtained from \cite[Lemma 4.1]{UC}.

\end{proof}

\begin{proof}[Theorem \ref{sthm}]
(i)   It follows from Proposition \ref{prop1} and Lemma \ref{lem1} with $\eta=v$ and $\lambda=\lambda_c$.

\noindent (ii) We denote by $U=[0,h]\times\R$, $R=|\phi_c|_{0}e^{-\lambda_c ch}$ and $w_k(t,x):= w(t+(k-1)h,x)$  where $k=0,1,2,...$
If we take $w=v-\phi_c$ then
\begin{eqnarray}\label{eqt}
\dot{w}_k(t,z)=w''_k(t,z)-cw'_k(t,z)+[1-v_{k-1}(t,z-ch)]w_k(t,z)+f_{k-1}(t,z), \ \hbox{on} \ U,
\end{eqnarray}

\noindent where $f_{k-1}(t,\cdot)=-\phi_c(\cdot) \ w_{k-1}(t,\cdot-ch).$

Now, we define the sequence $C_k:=K_{\lambda_c}e^{(k-1)\delta h}$ for $k=0,1,2,...$

\noindent Clearly, $|w_0(t)|_{\lambda_c}\leq C_0e^{\delta t}$ for $t\in[0,h]$. Next, we assume that

\begin{eqnarray}\label{ih}
|w_{l}(t)|_{\lambda_c}\leq C_{l}e^{\delta t}\quad \hbox{for all}\quad t\in[0,h]
\end{eqnarray}

 \noindent and some  $l\geq1.$

 \noindent Next, by Lemma \ref{lem1} there exist $D_l>0$ such that
  \begin{eqnarray}\label{idk}
 |f_l(t,z)|\leq D_le^{cz/2}, \ \hbox{on} \ U.
 \end{eqnarray}
Therefore, using Proposition \ref{prop1},   \cite[Chapter 1,Theorem 12]{AF} implies that the associated Cauchy problem  to (\ref{eqt}) with initial datum $w_{l+1}(0,\cdot)$ has a unique bounded solution which is represented by
\begin{eqnarray}\nonumber
w_{l+1}(t)=\Gamma_{v_l}(t) w_{l+1}(0)+\int_0^t \Gamma_{v_l}(t-\tau) f_{l}(\tau)d\tau.
\end{eqnarray}

\noindent Next, using $w_{l+1}(0)=w_l(h)$ and Lemma \ref{lem1} we have

\begin{eqnarray}\nonumber

|w_{l+1}(t)|_{\lambda_c}&\leq& e^{-\epsilon_{\lambda_c}t}|w_{l}(h)|_{\lambda_c}+ R\int_0^t e^{-\epsilon_{\lambda_c}(t-\tau)}|w_{l}(\tau)|_{\lambda_c}d\tau.
\end{eqnarray}
Because of (\ref{ih}) we have
\begin{eqnarray}\nonumber
|w_{l+1}(t)|_{\lambda_c}&\leq& e^{-\epsilon_{\lambda_c}t}e^{\delta h}C_{l}+Re^{-\epsilon_{\lambda_c}t}C_l[e^{(\epsilon_{\lambda_c}+\delta)t}- 1]/(\epsilon_{\lambda_c}+\delta)\\ \nonumber
&=& e^{\delta t}C_{l}[e^{-(\epsilon_{\lambda_c}+\delta)t+\delta h}+R(1-e^{-(\epsilon_{\lambda_c}+\delta)t})/(\epsilon_{\lambda_c}+\delta)].
\end{eqnarray}
Now, due to (\ref{inq1}), the function $M:[0,h]\to\R$ defined by
$$
M(t)=e^{-(\epsilon_{\lambda_c}+\delta)t+\delta h}+R(1-e^{-(\epsilon_{\lambda_c}+\delta)t})/(\epsilon_{\lambda_c}+\delta)
$$

\noindent is  non increasing. Therefore,
$$
|w_{l+1}(t)|_{\lambda_c}\leq e^{\delta (t+h)}C_l=e^{\delta t}C_{l+1}\quad\hbox{for all}\quad t\in[0,h].
$$

\noindent So, the general conclusion is
\begin{eqnarray}\nonumber
|w(t,\cdot)|_{\lambda_c}\leq K_{\lambda_c}e^{\delta t} \quad \hbox{for all}\quad t\geq -h.
\end{eqnarray}
 Finally, due to the estimate $|\phi_c|_{0}<e^{ch}$ \cite[Theorem 4.3]{HT}, taking $\lambda_c=c/2$ we have that (\ref{norm}) is satisfied for all $c>2\sqrt{2}$.
 \end{proof}

\begin{proof}[Corollary \ref{coro1}]
By a standard argument of ordinary differential equations the asymptotic behavior of non negative solutions of (\ref{swe}) at $-\infty$ is
\begin{eqnarray}\label{asw}
\phi_c(z)=A_{\phi_c}z^{j_c}e^{\lambda_1z}+o(z^je^{\lambda_1z})\\ \nonumber
\psi_c(z)=A_{\psi_c}z^{j_c}e^{\lambda_1z}+o(z^je^{\lambda_1z})
\end{eqnarray}
 where $A_{\phi_c}, A_{\psi_c}>0$, $j_c=0,1$  and $j_c=1$ if and only if $c=2$. So, for some $z_0\in\R$ we have $A_{\psi_c(\cdot+z_0)}=A_{\phi_c(\cdot)}$.

 Otherwise, denoting $\Phi_c(z)=\phi_c(z)-\psi_c(z+z_0)$ we have
\begin{eqnarray*}\label{equn}
\Phi''_c(z)-c\Phi'_c(z)+(1-\psi_c(z))\Phi_c(z)-\phi_c(z)\Phi_c(z-ch)=0\quad z\in\R.
\end{eqnarray*}

 \noindent Next, using (\ref{asw}) and  applying \cite[Proposition 7.2]{MP} we can conclude
\begin{eqnarray}\nonumber
\Phi_c(z)&=&O(e^{\lambda_2z})\quad \hbox{if}\quad c>2.

\end{eqnarray}

Next, we take, for example, the initial datum $v_0(s,z)=\min\{1,A_{\phi_c}e^{\lambda_1z}\}\in\mathcal{C}$. Then, applying Theorem  \ref{sthm} with $\lambda_c=\kappa_h^c$, $w_0^1=v_0-\phi_c$ and $w_0^2=v_0-\psi_c$ we conclude that $\phi_c(\cdot)=\psi_c(\cdot+z_0)$ under the  condition (\ref{uniq}). Similarly, taking $\lambda_c=c/2$ we conclude   that $\phi_c(\cdot)=\psi_c(\cdot+z_0)$ on the domain $\{(h,c)\in\R^2: h\geq 0\ \hbox{and}\ c>2\sqrt{2}\}$.
\end{proof}

 \section*{Acknowledgments} This work was supported by FONDECYT (Chile) through the Postdoctoral  Fondecyt  2016 program with project number 3160473, and FONDECYT project 116--0856.
 The authors are  grateful to Sergei Trofimchuk for his important comments on this work and to Fran\c cois Hamel for some technical advice.

\vspace{5mm}


TITLE: Semisimplicity of Frobenius operation on etale cohomology?
QUESTION [12 upvotes]: Let $X_0$ be a variety defined over a finite field of characteristic $p \neq l$.
Is it true, that the action of the frobenius on the l-adic cohomology $H_l^*(X)$ is semisimple (say for smooth $X_0$)? If not, what would be a counter-example?

REPLY [21 votes]: Let $X$ be a smooth projective variety over a finite field
$\mathbb{F}_{q}$ of caracteristic $p$ and let $l$ be a prime number different from p.
 We consider the following statement :
(A) The action of the Frobenius on the etale cohomology $H^{i}_{et}( X_{\overline{\mathbb{F}}_{q}}, \mathbb{Q}_{l})$ is semisimple.
How to suppress the projective hypothesis is the subject of the mathoverflow question
link text
(A) is true in the following cases :
1) $X=A$ an abelian variety (and so for $X$ a curve via the jacobian).
As mentionned in comment by Emerton, it is a consequence of the Weil's
work on the Riemann hypothesis in this case.
Fix a polarization on $A$. For $x$ an endomorphism of $X$ which gives an endomorphim on $H^{1}_{et}$, we can define an endomorphism $x'$ (' : "Rosati involution") by $x' = *x^{T}* $
where in the middle we have the transposition with respect to the intersection product
and * comes from the duality theory of abelian varieties ( the polarisation gives a identification between $H^{1}_{et}(A)$ and $H^{1}_{et}(\check{A})$).
Weil proved that $Tr(xx')>0$ if $x$ is non-zero. Let $F$ be the (geometric) Frobenius. For
$x = q^{-1/2}F$, we have x'=$x^{-1}$. So $Tr(aa')$ is a definite positive bilinear form
on the $\mathbb{Q}$ algebra generated by x and is preserved by multiplication by $x$ : so
multiplication by $x$ is unitary which shows that $x$ is semi-simple (and eigenvalues
of modulus one gives the Riemann hypothesis).
2) $X$ a K3 surface. As mentionned in comment by shenghao, it is a consequence of the work of Deligne : link text
The result is deduced from the case of abelian varieties
via the Kuga-Satake construction (of course there is a non-trivial thing to do because
Kuga-Satake construction is a priori of transcendental nature but Deligne did it).
For $X$ general, (A) is conjectured. It is a consequence of standard conjectures. More precisely, things should work as in the case of abelian varieties. We can still define
$x \mapsto x'$ at the cohomological level but $Tr(xx')>0$ is conjectural : a standard conjecture
of Lefschetz type implies $x'$ algebraic if $x$ is, which permits to use a trace formula
expressing $Tr(xx')$ as an intersection product. The positivity should then be a consequence of a standard conjecture of Hodge type.
For more details, as mentionned in comment by Damian Rössler, see Kleiman "The standard conjectures" (whose some details depend on Kleiman, "Algebraic cycles and the Weil conjectures").


TITLE: probability of unbalanced dice
QUESTION [0 upvotes]: An unbalanced die (with $6$ faces, numbered from $1$ to $6$) is thrown. The probability that the face value is odd is $90$% of the probability that the face value is even. The probability of getting any even numbered face is the same.
If the probability that the face is even given that it is greater than $3$ is $0.75$, what is the probability that the face valued exceeds $3$?

REPLY [3 votes]: Everything here can be expressed in terms of the probability $p$ that a particular even-numbered face comes up: $\Pr(\text{even})=3p$, and $\Pr(\text{odd})=0.9\cdot\Pr(\text{even})=2.7p$. Of course
$$\Pr(\text{even})+\Pr(\text{odd})=1\;,$$
so you can solve for $p$. Let $A$ be the event that the die comes up $4,5$, or $6$, and let $B$ be the event that it comes up $4$ or $6$. You’re told that $\Pr(B\mid A)=\frac34$, you know that $\Pr(B)=2p$, and you should easily be able to see what $\Pr(A\mid B)$ is just by thinking about the two events. Now use Bayes’ theorem to find $\Pr(A)$.


TITLE: Why $\mathbf Z[\sqrt{-5}]/(2, 1+\sqrt{-5})\simeq \mathbf Z[x]/(2,x+1,x^2+5)$?
QUESTION [4 upvotes]: Why is $(2, 1+\sqrt{-5})$ not principal?
\begin{align*}\mathbf Z[\sqrt{-5}]/(2, 1+\sqrt{-5})&\simeq  \mathbf Z[x]/(2,x+1,x^2+5)\simeq \mathbf Z_2[x]/(x+1,x^2+1)\\
&=\mathbf Z_2[x]/\bigl(x+1,(x+1)^2\bigr)=\mathbf Z_2[x]/(x+1)\simeq\mathbf Z_2.
\end{align*}
It is said to be used that $(R/I)/(J/I)\simeq R/J$, but can you explain in more detail?
And I still don't understand about the other homomorphisms and equations (especially $\mathbf Z_2[x]/(x+1,x^2+1)=\mathbf Z_2[x]/\bigl(x+1,(x+1)^2)$

REPLY [4 votes]: This is why: $\;\mathbf Z[\sqrt{-5}]\simeq\mathbf Z[x]/(x^2+5)$, in which $x$ corresponds to $\sqrt{-5}$, so
\begin{align*}\mathbf Z[\sqrt{-5}]/(2,1+\sqrt{-5})&\simeq\mathbf Z[x]/(x^2+5)\textbf{/}(2,1+x)\mathbf Z[x]/(x^2+5)\\&=\mathbf Z[x]/(x^2+5)\textbf{/}(2,1+x, x^2+5)/(x^2+5)
\end{align*}
Note. – The general situation is this:
Suppose you have a ring $R$ and two ideals $I,J$ in $R$. Then $I\cdot R/J=(I+J)/J$, so that
$$ (R/J)\textbf{/}(I\cdot R/J)\simeq R/(I+J).$$
For your second question, it comes from the fact that, in characteristic $2$, $\;(x+1)^2=x^2+1^2=x^2+1$.


TITLE: What is a contour integral, and how do you calculate one?
QUESTION [0 upvotes]: So, I am currently learning Multivariable Calculus, and I know how to evaluate integrals and derivatives, as well as how to parametrize multivariable functions, and how to evaluate line integrals. I am now wondering how to evaluate contour integrals. I've checked everywhere, including here, but the answers were lengthy and hard to understand, so I didn't grasp any of the information provided. So, I resorted to asking my own question. I would like a plainly stated explanation of this, along with a formula, if possible, along with the derivation of that formula, and an explanation of what's going on when we do the math.
Thanks in advance.

REPLY [2 votes]: Suppose we parameterize a path $\gamma$ in $\Bbb C$ with some real parameter, so $\gamma$ runs from $t=t_i$ to $t=t_f$. The contour integral of $f(z)$ along $\gamma$ is then $\int_{z(t_i)}^{z(t_f)}f(z(t))dz(t)=\int_{t_i}^{t_f}f(z(t))z^\prime(t)dt$. In this sense, it's not fundamentally different from integrals along an interval in $\Bbb R$. A common example is anticlockwise around the circle $|z-a|=r$, giving $\oint_\gamma f(z)dz=\int_0^{2\pi}f(a+re^{it})ire^{it}dt$.
If $f=\frac{1}{z-a}$, this is $\int_0^{2\pi}idt=2\pi i$, which is nonzero even though the path is closed. (Under certain conditions not applicable here, an integral around a closer contour would be $0$, just as $\int_b^bg(x)dx=0$ for "conventional" integrals.) This is the basis for how we typically calculate contour integrals. The basic idea is that a loop around a pole $z=a$ of a function $f$ contributes to the integral around that loop, in proportion to the coefficient of $\frac{1}{z-a}$ in an expansion of $z$. (Other powers don't matter, because $\int_0^{2\pi}e^{int}dt=0$ for any integer $n\ne0$.) Fleshing this out, we can (often) write the full integral around a closed path as a sum of multiples of such coefficients. Often, an integral of interest is something easy minus something else easy (indeed, often something equal to $0$).
For example, let's take the $t=0$ case here, You can prove $\int_{-\infty}^\infty\frac{dx}{1+x^2}=\pi$ by regarding this as the $R\to\infty$ limit of the integral along the diameter of a semicircular contour whose arc is $Re^{i\theta},\,\theta\in[0,\,\pi]$. As $R\to\infty$, the integral along the arc $\to0$ (proof explained in the link). But the integral around the semicircle is$$2\pi i\lim_{z\to i}\frac{z-i}{z^2+1}=\pi$$(once you've understood the theory summarized above); so, subtracting the arc, the original integral is also $\pi$.