Related papers: A remark on Pinney's equation
A recently proposed integral representation for permanents is rederived using only elementary combinatorics. For this proof the assumption that the matrix, for which the permanent is calculated, has an inverse is not necessary.
Thanks to the test function of Bian-Guan[2], we successfully obtain a constant rank theorem for partial convex solutions of a class partial differential equations. This is the microscopic version of the macroscopic partial convexity…
Chang's lemma is a useful tool in additive combinatorics and the analysis of Boolean functions. Here we give an elementary proof using entropy. The constant we obtain is tight, and we give a slight improvement in the case where the…
Methods are described for the solution of linear inference problems subject to deterministic constraints. The approach builds on work by Backus (1970a,b,c) and Parker (1977), but a range useful advances are suggested to address both…
New iterative methods for solving linear equations are presented that are easy to use, generalize good existing methods, and appear to be faster. The new algorithms mix two kinds of linear recurrence formulas. Older methods have either high…
We consider the solvability of the Fokker-Planck equation with both time-dependent drift and diffusion coefficients by means of the similarity method. By the introduction of the similarity variable, the Fokker-Planck equation is reduced to…
The objective of this article is to prove the necessity statement in Crawley-Boevey's conjectural solution to the (tame) Deligne-Simpson problem. We use the nonabelian Hodge correspondence, variation of parabolic weights and results of…
In this note, we prove a quantization formula for singular reductions. The main result is obtained as a simple application of an extended quantization formula proved in [TZ2].
It is well recognized that in auxiliary equation methods, the exact solutions of different types of auxiliary equations may produce new types of exact travelling wave solutions to nonlinear partial differential equations in hand. In this…
In this short exposition we provide a simplified proof of Buser's result for Cheeger's isoperimetric constant.
In the article, some bilinear evolution equations in Hilbert space driven by paths of low regularity are considered and solved explicitly. The driving paths are scalar-valued and continuous, and they are assumed to have a finite $p$-th…
In this article, we provide a modified argument for proving the conditional stability of inverse source problem for a hyperbolic equation. Our method does not require any extension of solution with respect to time and therefore simplifies…
We prove an asymptotic formula for a variant of the binary additive divisor problem with linear factors in the arguments, which has a power saving error term and which is uniform in all involved parameters.
We formulate on rectangles and on the right horizontal half-strip initial-boundary value problems for a two-dimensional Benney-Lin type equation. Existence and uniqueness of a regular solution as well as the exponential decay rate for the…
We consider a second order linear evolution equation with a dissipative term multiplied by a time-dependent coefficient. Our aim is to design the coefficient in such a way that all solutions decay in time as fast as possible. We discover…
This article studies a class of semilinear scalar field equations on the real line with variable coefficients in the linear terms. These coefficients are not necessarily small perturbations of a constant. We prove that under suitable…
A proposed solution to the Riemann Hypothesis
In this note, notwithstanding the generalization, we simplify and shorten the proofs of the main results of the third author's paper \cite{SXY} significantly. In particular, the new proof for \cite[Theorem 1.1]{SXY} is quite short and,…
A change of variables is introduced to reduce certain nonlinear stochastic evolution equations with multiplicative noise to the corresponding deterministic equation. The result is then used to investigate a stochastic porous medium…
We introduce two variants of $q$-hypergeometric equation. We obtain several explicit solutions of variants of $q$-hypergeometric equation. We show that a variant of $q$-hypergeometric equation can be obtained by a restriction of $q$-Appell…