Related papers: Optimal bounds in Bend-and-Break
We prove optimal lower bounds for the growth of the energy over balls of minimizers to the vectorial Allen-Cahn energy in two spatial dimensions, as the radius tends to infinity. In the case of radially symmetric solutions, we can prove a…
The most successful method to obtain lower bounds for the minimum distance of an algebraic geometric code is the order bound, which generalizes the Feng-Rao bound. We provide a significant extension of the bound that improves the order…
We show that all hyperbolic surfaces admit an ideal triangulation with bounded shear parameters. This upper bound depends logarithmically on the topology of the surface.
A classic result of Cook et al. (1986) bounds the distances between optimal solutions of mixed-integer linear programs and optimal solutions of the corresponding linear relaxations. Their bound is given in terms of the number of variables…
We consider the least singular value of a large random matrix with real or complex i.i.d. Gaussian entries shifted by a constant $z\in\mathbb{C}$. We prove an optimal lower tail estimate on this singular value in the critical regime where…
We propose a new method, using deformation theory, to study the maximal rank conjecture. For line bundles of extremal degree, which can be viewed as the first case to test the conjecture, we prove that maximal rank conjecture holds by our…
Given a set of endomorphisms on $\mathbb{P}^N$, we establish an upper bound on the number of points of bounded height in the associated monoid orbits. Moreover, we give a more refined estimate with an associated lower bound when the monoid…
In this paper we develop a technique for discovering (non-effective) irrational rays at the boundary of the Mori cone for linear systems on a general blowup of the plane, and give examples of such irrational rays.
We prove new endpoint bounds for the lacunary spherical maximal operator and as a consequence obtain almost everywhere pointwise convergence of lacunary spherical means for functions locally in $L\log\log\log L(\log\log\log\log…
In this paper, we present and analyze the Clenshaw-Curtis-Filon methods for computing two classes of oscillatory Bessel transforms with algebraic or logarithmic singularities. More importantly, for these quadrature rules we derive new…
Given smooth, projective, geometrically integral algebraic curves $X$ and $Y$ defined over a number field $K$, assuming that there is a non-constant $K$-morphism $\varphi \colon X \to Y$, we give an upper bound on the minimum of the degrees…
The theory of modular forms and spherical harmonic analysis are applied to establish new best bounds towards the counting and equidistribution of rational points on spheres and other higher dimensional ellipsoids, in what may be viewed as a…
In this survey paper, we study the optimal regularity of solutions to uniformly degenerate elliptic equations in bounded domains and establish the H\"older continuity of solutions and their derivatives up to the boundary.
Considering the paradigmatic driven Brownian motion, we perform extensive numerical analysis on the performance of optimal linear-response processes far from equilibrium. We focus on the overdamped regime where exact optimal processes are…
Exact lower and upper bounds on the best possible misclassification probability for a finite number of classes are obtained in terms of the total variation norms of the differences between the sub-distributions over the classes. These…
We obtain new lower and upper bounds for probabilities of unions of events.These bounds are sharp. They are stronger than earlier ones. General bounds maybe applied in arbitrary measurable spaces.We have improved the method that has been…
This paper focuses on computing the directional extremal boundary of a union of equal-radius circles. We introduce an efficient algorithm that accurately determines this boundary by analyzing the intersections and dominant relationships…
We prove that triangulations with maximum degree at most 5 satisfy the List-Edge-Coloring Conjecture.
In 1981 Beck and Fiala proved an upper bound for the discrepancy of a set system of degree d that is independent of the size of the ground set. In the intervening years the bound has been decreased from 2d-2 to 2d-4. We improve the bound to…
In this paper, we obtain an optimal $L^2$ extension theorem for continuous $L^2$-optimal Hermitian metric on bounded planer domains. As applications, we affirmatively answer a question of Deng-Ning-Wang and a question of Inayama.