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For any smooth random Gaussian function $\Phi$ on $\mathbb{R}^m$ we denote by $Z_N(\Phi)$ the number of critical points of $\Phi$ inside the cube $[0,N]^m$. We prove that for certain isotropic random functions $\Phi$ the ratio $N^{-m}Z_N(\Phi)$ converges a.s. and $L^2$ to a universal explicit constant $C_m(\Phi)$., Comment: 26 pages, added an appendix on stationary random measures, added references

We use the Kac-Rice formula to compute the Mehta integral describing the normalization constant arising in the statistics of the Gaussian Orthogonal Ensemble., Comment: 10 pages

We consider finite element approximations to the optimal constant for the Hardy inequality with exponent $p=2$ in bounded domains of dimension $n=1$ or $n \geq 3$. For finite element spaces of piecewise linear and continuous functions on a mesh of size $h$, we prove that the approximate Hardy constant converges to the optimal Hardy constant at a rate proportional to $1/| \log h |^2$. This result holds in dimension $n=1$, in any dimension $n \geq 3$ if the domain is the unit ball and the finite element discretization exploits the rotational symmetry of the problem, and in dimension $n=3$ for general finite element discretizations of the unit ball. In the first two cases, our estimates show excellent quantitative agreement with values of the discrete Hardy constant obtained computationally., Comment: Review: Significantly improved estimates compared to the original version (23 pages, 6 figures)

We study the existence and uniqueness of source-type solutions to the Cauchy problem for the heat equation with fast convection under certain tail control assumptions. We allow the solutions to change sign, but we will in fact show that they have the same sign as the initial data, which is a multiple of the Dirac delta. As an application, we obtain the large-time behaviour of nonnegative bounded solutions with integrable initial data to heat equations with fast convection, covering the case of several dimensions that remained open since the end of last century., Comment: 15 pages

We study a non-local evolution equation on the hyperbolic space $\mathbb{H}^N$. We first consider a model for particle transport governed by a non-local interaction kernel defined on the tangent bundle and invariant under the geodesic flow. We study the relaxation limit of this model to a local transport problem, as the kernel gets concentrated near the origin of each tangent space. Under some regularity and integrability conditions on the kernel, we prove that the solution of the rescaled non-local problem converges to that of the local transport equation. Then, we construct a large class of interaction kernels that satisfy those conditions. We also consider a non-local, non-linear convection-diffusion equation on $\mathbb{H}^N$ governed by two kernels, one for each of the diffusion and convection parts, and we prove that the solution converges to the solution of a local problem as the kernels get concentrated. We prove and then use in this sense a compactness tool on manifolds inspired by the work of Bourgain-Brezis-Mironescu.

In this paper we study the large time asymptotic behaviour of the heat equation with Hardy inverse-square potential on corner spaces $\mathbb{R}^{N-k}\times (0,\infty)^k$, $k\geq 0$. We first show a new improved Hardy-Poincar\'{e} inequality for the quantum harmonic oscillator with Hardy potential. In view of that, we construct the appropriate functional setting in order to pose the Cauchy problem. Then we obtain optimal polynomial large time decay rates and subsequently the first term in the asymptotic expansion of the solutions in $L^2(\mathbb{R}^{N-k}\times (0,\infty)^k)$. Particularly, we extend and improve the results obtained by V\'{a}zquez and Zuazua (J. Funct. Anal. 2000), which correspond to the case $k=0$, to any $k\geq 0$. We emphasize that the higher the value of $k$ the better time decay rates are. We employ a different and simplified approach than V\'{a}zquez and Zuazua, managing to remove the usage of spherical harmonics decomposition in our analysis.

We study the large-time behaviour of nonnegative solutions to the Cauchy problem for a nonlocal heat equation with a nonlinear convection term. The diffusion operator is the infinitesimal generator of a stable L\'evy process, which may be highly anisotropic. The initial data are assumed to be bounded and integrable. The mass of the solution is conserved along the evolution, and the large-time behaviour is given by the source-type solution with this mass of a limit equation that depends on the relative strength of convection and diffusion. When diffusion is stronger than convection the original equation simplifies asymptotically to the purely diffusive nonlocal heat equation. When convection dominates, it does so only in the direction of convection, and the limit equation is still diffusive in the subspace orthogonal to this direction, with a diffusion operator that is a "projection" of the original one onto the subspace. The determination of this projection is one of the main issues of the paper. When convection and diffusion are of the same order the limit equation coincides with the original one. Most of our results are new even in the isotropic case in which the diffusion operator is the fractional Laplacian. We assume in principle that the convection nonlinearity is locally Lipschitz, but we are also able to cover some situations in which it is only locally H\"older. This possibility has never been considered before in the nonlocal diffusion setting., Comment: 48 pages, 2 figures

We study local (the heat equation) and nonlocal (convolution type problems with an integrable kernel) evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions to both local and nonlocal problems is given by the solution of the heat equation, but on a star shaped graph in which there is only one node and as many infinite edges as in the original graph. In this way we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single point when looking at the asymptotic behaviour of the solutions. For this star shaped limit problem the asymptotic behaviour of the solutions is just given by the solution to the heat equation in a half line with a Neumann boundary condition at $x = 0$ and initial datum $(2M/N)\delta_{x=0}$ where M is the total mass of the initial condition for our original problem and $N$ is the number of edges of infinite length. In addition, we show that solutions to the nonlocal problem converge, when we rescale the kernel, to solutions to the heat equation (the local problem), that is, we find a relaxation limit., Comment: To appear in Annali di Matematica Pura ed Applicata

In this paper we consider the linear Schrodinger equation (LSE) on a regular tree with the last generation of edges of infinite length and analyze some unique continuation properties. The first part of the paper deals with the LSE on the real line with a piece-wise constant coefficient and uses this result in the context of regular trees. The second part treats the case of a LSE with a real potential in the framework of a star-shaped graph., Comment: 30 pages, 1 figure

We consider semidiscrete finite differences schemes for the periodic Scr\"odinger equation in dimension one. We analyze whether the space-time integrability properties observed by Bourgain in the continuous case are satisfied at the numerical level uniformly with respect to the mesh size. For the simplest finite differences scheme we show that, as mesh size tends to zero, the blow-up in the $L^4$ time-space norm occurs, a phenomenon due to the presence of numerical spurious high frequencies. To recover the uniformity of this property we introduce two methods: a spectral filtering of initial data and a viscous scheme. For both of them we prove a $L^4$ time-space estimate, uniform with respect to the mesh size. {Warning 2019}: This paper was submitted to M2AN in 2007 and it was assigned the number 2007-29. It passed a first review round ( three reviews :-) ) without decision ("It is only after the consideration of a thoroughly revised version of your manuscript and a new iteration with all the referees that I will be in position to make my final decision.") After completing the PhD, I tried to publish the papers resulting from the thesis and I didn't spent too much time on other related problems (the periodic case for example). It was only recently that I discovered some interest in the subject from other authors and I decided to upload it on arxiv.org. Use with caution as there is no revision of the text in the last twelve years.

In this paper we study a convection-diffusion equation on a star-shaped graph composed by $n$ incoming edges and $m$ outgoing edges with a nonlinearity $f\in C^1(\rr)$ satisfying some additional general conditions. First, we prove the global well-posedness of the solutions of the system under consideration. Next, in the particular case that the nonlinear convection is given by $\partial_x(f(u(t, x))$ with $f(s)=-a|s|^{q-1}s$ with $q\geq 2$ and $a\in \rr$ verifying $(n-m)a\geq 0$, we analyze the long time behavior of the solutions. For $q> 2$ we find that the asymptotic behavior of the solutions is given by some self-similar profiles of the heat equation on the considered structure. In the case $q=2$, the nonnegative/nonpositive solutions converge to the self-similar profiles of Burgers' equation. Explicit representations of the limit profiles are obtained., Comment: 32 pag, 1 figure

We investigate dispersive and Strichartz estimates for the Schr\"{o}dinger time evolution propagator $\mathrm{e}^{-\mathrm{i}tH}$ on a star-shaped metric graph. The linear operator, $H$, taken into consideration is the self-adjoint extension of the Laplacian, subject to a wide class of coupling conditions. The study relies on an explicit spectral representation of the solution in terms of the resolvent kernel which is further analyzed using results from oscillatory integrals. As an application, we obtain the global well-posedness for a class of semilinear Schr\"{o}dinger equations., Comment: 27 pages, 1 figure

In this paper we treat controllability properties for the linear Kuramoto-Sivashinsky equation on a network with two types of boundary conditions. More precisely, the equation is considered on a star-shaped tree with Dirichlet and Neumann boundary conditions. By using the moment theory we can derive null-controllability properties with boundary controls acting on the external vertices of the tree. In particular, the controllability holds if the anti-diffusion parameter of the involved equation does not belong to a critical countable set of real numbers. We point out that the critical set for which the null-controllability fails differs from the first model to the second one., Comment: 36 pages

In this paper we consider a splitting method for the augmented Burgers equation and prove that it is of first order. We also analyze the large-time behavior of the approximated solution by obtaining the first term in the asymptotic expansion. We prove that, when time increases, these solutions behave as the self-similar solutions of the viscous Burgers equation, Comment: 24 pages

We investigate certain families $X^\hbar$, $0<\hbar \ll 1$, of Gaussian random smooth functions on the $m$-dimensional torus $\mathbb{T}^m_\hbar:=\mathbb{R}^m/(\hbar^{-1}\mathbb{Z} )^m$. We show tha,t for any cube $B\subset \mathbb{R}^m$ of size $<1/2$ and centered at the origin, the number of critical points of $X^\hbar$ in the region $\hbar^{-1}B/(\hbar^{-1}\mathbb{Z} )^m\subset\mathbb{T}^m_\hbar$ has mean $\sim c_1\hbar^{-m}$, variance $\sim c_2\hbar^{-m/2}$, $c_1,c_2>0$, and satisfies a central limit theorem as $\hbar\searrow 0$., Comment: 34 pages,1 figure added a new result, improved the presentation

We prove a central limit theorem concerning the number of critical points in large cubes of an isotropic Gaussian random function on a Euclidean space., Comment: 29 pages, fixed typos, improved the presentation, added references

In this paper we analyze the dispersion property of some models involving Schr\"odinger equations. First we focus on the discrete case and then we present some results on graphs., Comment: 8 pages

In this paper we analyze the large-time behavior of the augmented Burgers equation. We first study the well-posedness of the Cauchy problem and obtain $L^1$-$L^p$ decay rates. The asymptotic behavior of the solution is obtained by showing that the influence of the convolution term $K*u_{xx}$ is the same as $u_{xx}$ for large times. Then, we propose a semi-discrete numerical scheme that preserves this asymptotic behavior, by introducing two correcting factors in the discretization of the non-local term. Numerical experiments illustrating the accuracy of the results of the paper are also presented., Comment: Accepted for publication in ESAIM: Mathematical Modelling and Numerical Analysis

We prove that a Gaussian ensemble of smooth random sections of a real vector bundle over compact manifold canonically defines a metric on the bundle together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet-Chern theorem stating that if the bundle and the manifold are oriented, then the Euler form of the above connection can be identified, as a current, with the expectation of the random current defined by the zero-locus of a random section in the above Gaussian ensemble., Comment: 24 pages, fixed typos, streamlined presentation, stregthened some results. To appear in Prob. Th. and Related Fileds

We prove that the Euler form of a metric connection on real oriented vector bundle $E$ over a compact oriented manifold $M$ can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilistically the metric and the connection on $E$ from the statistics of random sections of $E$., Comment: 35 pages, references added, fixed typos, to appear Trans. Amer. Math. Soc

To any positive number $\varepsilon$ and any nonnegative even Schwartz function $w:\mathbb{R}\to\mathbb{R}$ we associate the random function $u^\varepsilon$ on the $m$-torus $T^m_\varepsilon:=\mathbb{R}^m/(\varepsilon^{-1}\mathbb{Z})^m$ defined as the real part of the random Fourier series $$ \sum_{\nu\in\mathbb{Z}^m} X_{\nu,\varepsilon} \exp\bigl(\; 2\pi \varepsilon \sqrt{-1} \;(\nu\cdot \theta)\;\bigr),$$ where $X_{\nu,\varepsilon}$ are complex independent Gaussian random variables with variance $w(\varepsilon|\nu|)$. Let $N^\varepsilon$ denote the number of critical points of $u^\varepsilon$. We describe explicitly two constants $C, C'$ such that as $\varepsilon$ goes to the zero, the expectation of the random variable $\frac{1}{{\rm vol}\,(T^m_\varepsilon)}N^\varepsilon$ converges to $C$, while its variance is extremely small and behaves like $C'\varepsilon^{m}$., Comment: 44 pages. Fixed typos, improved presentation, added references

In this paper we analyze the large time asymptotic behavior of the discrete solutions of numerical approximation schemes for scalar hyperbolic conservation laws. We consider three monotone conservative schemes that are consistent with the one-sided Lipschitz condition (OSLC): Lax-Friedrichs, Engquist-Osher and Godunov. We mainly focus on the inviscid Burgers equation, for which we know that the large time behavior is of self-similar nature, described by a two-parameter family of N-waves. We prove that, at the numerical level, the large time dynamics depends on the amount of numerical viscosity introduced by the scheme: while Engquist-Osher and Godunov yield the same N-wave asymptotic behavior, the Lax-Friedrichs scheme leads to viscous self-similar profiles, corresponding to the asymptotic behavior of the solutions of the continuous viscous Burgers equation. The same problem is analyzed in the context of self-similar variables that lead to a better numerical performance but to the same dichotomy on the asymptotic behavior: N-waves versus viscous ones. We also give some hints to extend the results to more general fluxes. Some numerical experiments illustrating the accuracy of the results of the paper are also presented., Comment: Error corrected in main theorem in v3. Obtained results do not change in essence. Minor typos corrected in v2

In this paper we consider a model that involves nonlocal diffusion and a classical convective term. Using a scaling argument and a new compactness argument we obtain the first term in the asymptotic behavior of the solutions.

In this paper we give a new compactness criterion in the Lebesgue spaces $L^p((0,T)\times \Omega)$ and use it to obtain the first term in the asymptotic behaviour of the solutions of a nonlocal convection diffusion equation. We use previous results of Bourgain, Brezis and Mironescu to give a new criterion in the spirit of the Aubin-Lions-Simon Lemma.

We study random Morse functions on a Riemann manifold $(M^m,g)$ defined as a random Gaussian weighted superpositions of eigenfunctions of the Laplacian of the metric $g$. The randomness is determined by a fixed Schwartz function $w$ and a small parameter $\varepsilon>0$. We first prove that as $\varepsilon\to 0$ the expected distribution of critical values of this random function approaches a universal measure on $\mathbb{R}$, independent of $g$, that can be explicitly described in terms the expected distribution of eigenvalues of the Gaussian Wigner ensemble of random $(m+1)\times (m+1)$ symmetric matrices. In contrast, we prove that the metric $g$ and its curvature are determined by the statistics of the Hessians of the random function for small $\varepsilon$., Comment: 47 pages (changed the title, fixed typos, substantially revised the introduction, updated references). arXiv admin note: text overlap with arXiv:1201.4972

In this paper we obtain bounds for the decay rate in the $L^r (\rr^d)$-norm for the solutions to a nonlocal and nolinear evolution equation, namely, $$u_t(x,t) = \int_{\rr^d} K(x,y) |u(y,t)- u(x,t)|^{p-2} (u(y,t)- u(x,t)) \, dy, $$ with $ x \in \rr^d$, $ t>0$. Here we consider a kernel $K(x,y)$ of the form $K(x,y)=\psi (y-a(x))+\psi(x-a(y))$, where $\psi$ is a bounded, nonnegative function supported in the unit ball and $a$ is a linear function $a(x)= Ax$. To obtain the decay rates we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form $ T(u) = - \int_{\rr^d} K(x,y) |u(y)-u(x)|^{p-2} (u(y)-u(x)) \, dy$, with $1 \leq p < \infty$. The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole $\rr^d$: $$ \lambda_{1,p} (\rr^d) = 2(\int_{\rr^d} \psi (z) \, dz)|\frac{1}{|\det{A}|^{1/p}} -1|^p. $$ Moreover, we deal with the $p=\infty$ eigenvalue problem studying the limit as $p \to \infty$ of $\lambda_{1,p}^{1/p}$.

We obtain the asymptotic expansion of the solutions of some anisotropic heat equations when the initial data belong to polynomially weighted Lp-spaces. We mainly address two model examples. In the first one, the diffusivity is of order two in some variables but higher in the other ones. In the second one we consider the heat equation on the Heisenberg group.

We investigate the probability of detecting combinatorial Morse flows on a simplicial complex via a random search. We prove that it is really small, in a quantifiable way., Comment: 13 pages, 2 figures

We relate the distribution of eigenvalues of a random symmetric matrix in the Gaussian Orthogonal Ensemble to the distribution of critical values of a random linear combination of eigenfunctions of the Laplacian on a compact Riemann manifold. We then prove a central limit theorem describing what happens when the dimension of the manifold is very large., Comment: 23 pages, LaTex, to appear Indiana U. Math. J

We describe an algorithm that associates to each positive real number $r$ and each finite collection $C_r$ of planar pixels of size $r$ a planar piecewise linear set $S_r$ with the following additional property: if $C_r$ is the collection of pixels of size $r$ that touch a given compact semialgebraic set $S$, then the normal cycle of $S_r$ converges to the normal cycle of $S$ in the sense of currents. In particular, in the limit we can recover the homotopy type of $S$ and its geometric invariants such as area, perimeter and curvature measures. At its core, this algorithm is a discretization of stratified Morse theory., Comment: 36 pages, 13 figures, to appear in Algebraic & Geometric Topology

We formulate a precise conjecture about the universal behavior near the diagonal of the spectral function of the Laplacian of a smooth compact Riemann manifold. We prove this conjecture when the manifold and the metric are real analytic, and we also present an alternate proof when the manifold is the round sphere., Comment: 13 pages, 2 figures. v2 Added some comments and additional references. v3 Added references v4. Fixed typos

Given a compact, $m$-dimensional Riemann manifold $(M,g)$ and a large positive constant $L$ we denote by $U_L$ the subspace of $C^\infty(M)$ spanned by the eigenfunctions of the Laplacian corresponding to eigenvalues $\leq L$. We equip $U_L$ with the standard Gaussian probability measure induced by the $L^2$-metric on $U_L$, and we denote by $N_L$ the expected number of critical points of a random function in $U_L$. We prove that $N_L\sim C_m\dim U_L$ as $L\rightarrow \infty$, where $C_m$ is an explicit positive constant that depends only on the dimension $m$ and satisfying the asymptotic estimate $\log C_m\sim\frac{m}{2}\log m$ as $m\to \infty$., Comment: 37 pages, 1 figure, to appear Asian J. Math

We prove a Chern-Lashof type formula computing the expected number of critical points of smooth function on a smooth manifold $M$ randomly chosen from a finite dimensional subspace $V\subset C^\infty(M)$ equipped with a Gaussian probability measure. We then use this formula this formula to find the asymptotics of the expected number of critical points of a random linear combination of a large number eigenfunctions of the Laplacian on the round sphere, tori, or a products of two round spheres. In the case $M=S^1$ we show that the number of critical points of a trigonometric polynomial of degree $\leq \nu$ is a random variable $Z_\nu$ with expectation $E(Z_\nu)\sim 2\sqrt{0.6}\,\nu$ and variance $var(Z_\nu)\sim c\nu$ as $\nu\to \infty$, $c\approx 0.35$., Comment: 77 pages

Given a knot K in an Euclidean space E and a finite dimensional space V of smooth functions on K, we express the expected number of critical points of a random function in V in terms of an integral-geometric invariant of K and V. When V consists of the restrictions to K of homogeneous polynomials of degree d on E, this invariant takes the form of the total curvature of a certain immersion of K. In particular, when K is the unit circle in the plane centered at the origin, then the expected number of critical points of the restriction to K of a random homogeneous polynomial of degree d is $2\sqrt{3d-2}$, and the expected number of critical points on K of a trigonometric polynomial of degree d is approximately 1.549d., Comment: 13 pages, 2 figures

We present a short complete proof of the existence of the normal cycle of a compact subanalytic set. The approach is inspired by some old ides of Joseph Fu, uses Morse theoretic techniques and $o$-minimal topology., Comment: 24 pages, 1 figure. Vers 4: filled in some missing details, included an approximation result, a comparison with the approach of A. Bernig and a several appendices

We investigate the Dolbeault operator on a pair of pants, i.e., an elementary cobordism between a circle and the disjoint union of two circles. This operator induces a canonical selfadjoint Dirac operator $D_t$ on each regular level set $C_t$ of a fixed Morse function defining this cobordism. We show that as we approach the critical level set $C_0$ from above and from below these operators converge in the gap topology to (different) selfadjoint operators $D_\pm$ that we describe explicitly. We also relate the Atiyah-Patodi-Singer index of the Dolbeault operator on the cobordism to the spectral flows of the operators $D_t$ on the complement of $C_0$ and the Kashiwara-Wall index of a triplet of finite dimensional lagrangian spaces canonically determined by $C_0$., Comment: 31 pages, 3 figures; completely rewrote Section 4 using the definition of Kirk and Lesch of the Kashiwara-Wall index

We investigate the Cesaro and Abel sums of divergent series of the form $\sum_{n\geq 0} a_n T^nP(x)$, where $P$ is a real polynomial and $T$ is a translation invariant operator on the space of real polynomials., Comment: 9 pages

We analyze a bead sliding game and relate it to some classical convex inequalities., Comment: 8 pages