Search results for "FOS: Mathematics"
showing 10 items of 1448 documents
A Quantitative Analysis of Metrics on Rn with Almost Constant Positive Scalar Curvature, with Applications to Fast Diffusion Flows
2017
We prove a quantitative structure theorem for metrics on $\mathbf{R}^n$ that are conformal to the flat metric, have almost constant positive scalar curvature, and cannot concentrate more than one bubble. As an application of our result, we show a quantitative rate of convergence in relative entropy for a fast diffusion equation in $\mathbf{R}^n$ related to the Yamabe flow.
Removable singularities for div v=f in weighted Lebesgue spaces
2018
International audience; Let $w\in L^1_{loc}(\R^n)$ be apositive weight. Assuming that a doubling condition and an $L^1$ Poincar\'e inequality on balls for the measure $w(x)dx$, as well as a growth condition on $w$, we prove that the compact subsets of $\R^n$ which are removable for the distributional divergence in $L^{\infty}_{1/w}$ are exactly those with vanishing weighted Hausdorff measure. We also give such a characterization for $L^p_{1/w}$, $1<p<+\infty$, in terms of capacity. This generalizes results due to Phuc and Torres, Silhavy and the first author.
Mappings of Finite Distortion : Compactness of the Branch Set
2017
We show that an entire branched cover of finite distortion cannot have a compact branch set if its distortion satisfies a certain asymptotic growth condition. We furthermore show that this bound is strict by constructing an entire, continuous, open and discrete mapping of finite distortion which is piecewise smooth, has a branch set homeomorphic to an (n - 2)-dimensional torus and distortion arbitrarily close to the asymptotic bound. Peer reviewed
Generalized Riesz systems and quasi bases in Hilbert space
2019
The purpose of this article is twofold. First of all, the notion of $(D, E)$-quasi basis is introduced for a pair $(D, E)$ of dense subspaces of Hilbert spaces. This consists of two biorthogonal sequences $\{ \varphi_n \}$ and $\{ \psi_n \}$ such that $\sum_{n=0}^\infty \ip{x}{\varphi_n}\ip{\psi_n}{y}=\ip{x}{y}$ for all $x \in D$ and $y \in E$. Secondly, it is shown that if biorthogonal sequences $\{ \varphi_n \}$ and $\{ \psi_n \}$ form a $(D ,E)$-quasi basis, then they are generalized Riesz systems. The latter play an interesting role for the construction of non-self-adjoint Hamiltonians and other physically relevant operators.
Circular law for sparse random regular digraphs
2020
Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ for every large integer $n$. Denote by $A_n$ the adjacency matrix of a uniform random directed $d$-regular graph on $n$ vertices. We show that, as long as $d\to\infty$ with $n$, the empirical spectral distribution of appropriately rescaled matrix $A_n$ converges weakly in probability to the circular law. This result, together with an earlier work of Cook, completely settles the problem of weak convergence of the empirical distribution in directed $d$-regular setting with the degree tending to infinity. As a crucial element of our proof, we develop a technique of bounding intermediate singular values of $A_n$ based on studyi…
NUMERICAL ALGORITHMS
2013
For many systems of differential equations modeling problems in science and engineering, there are natural splittings of the right hand side into two parts, one non-stiff or mildly stiff, and the other one stiff. For such systems implicit-explicit (IMEX) integration combines an explicit scheme for the non-stiff part with an implicit scheme for the stiff part. In a recent series of papers two of the authors (Sandu and Zhang) have developed IMEX GLMs, a family of implicit-explicit schemes based on general linear methods. It has been shown that, due to their high stage order, IMEX GLMs require no additional coupling order conditions, and are not marred by order reduction. This work develops a …
On the fractional probabilistic Taylor's and mean value theorems
2016
In order to develop certain fractional probabilistic analogues of Taylor's theorem and mean value theorem, we introduce the nth-order fractional equilibrium distribution in terms of the Weyl fractional integral and investigate its main properties. Specifically, we show a characterization result by which the nth-order fractional equilibrium distribution is identical to the starting distribution if and only if it is exponential. The nth-order fractional equilibrium density is then used to prove a fractional probabilistic Taylor's theorem based on derivatives of Riemann-Liouville type. A fractional analogue of the probabilistic mean value theorem is thus developed for pairs of nonnegative rand…
Recovery of time-dependent coefficients from boundary data for hyperbolic equations
2019
We study uniqueness of the recovery of a time-dependent magnetic vector-valued potential and an electric scalar-valued potential on a Riemannian manifold from the knowledge of the Dirichlet to Neumann map of a hyperbolic equation. The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem. The proof is based on Gaussian beams and inversion of the light ray transform on Lorentzian manifolds under the assumptions that the Lorentzian manifold is a product of a Riemannian manifold with a time interval and that the geodesic ray transform is invertible on the Riemannian manifold.
Exact simulation of first exit times for one-dimensional diffusion processes
2019
International audience; The simulation of exit times for diffusion processes is a challenging task since it concerns many applications in different fields like mathematical finance, neuroscience, reliability horizontal ellipsis The usual procedure is to use discretization schemes which unfortunately introduce some error in the target distribution. Our aim is to present a new algorithm which simulates exactly the exit time for one-dimensional diffusions. This acceptance-rejection algorithm requires to simulate exactly the exit time of the Brownian motion on one side and the Brownian position at a given time, constrained not to have exit before, on the other side. Crucial tools in this study …
Radio Labelings of Distance Graphs
2013
A radio $k$-labeling of a connected graph $G$ is an assignment $c$ of non negative integers to the vertices of $G$ such that $$|c(x) - c(y)| \geq k+1 - d(x,y),$$ for any two vertices $x$ and $y$, $x\ne y$, where $d(x,y)$ is the distance between $x$ and $y$ in $G$. In this paper, we study radio labelings of distance graphs, i.e., graphs with the set $\Z$ of integers as vertex set and in which two distinct vertices $i, j \in \Z$ are adjacent if and only if $|i - j| \in D$.