Related papers: On Small Solutions to Quadratic Congruences
We prove various estimates for the mean square lattice point discrepancy for dilates of a convex body.
We consider the problem of partitioning a line segment into two subsets, so that $n$ finite measures all have the same ratio of values for the subsets. Letting $\alpha\in[0,1]$ denote the desired ratio, this generalises the PPA-complete…
Orthogonal polynomials of degree $n$ with respect to the weight function $W_\mu(x) = (1-\|x\|^2)^\mu$ on the unit ball in $\RR^d$ are known to satisfy the partial differential equation $$ [ \Delta - \la x, \nabla \ra^2 - (2 \mu +d) \la x,…
The cardinality of the set of $D\leqslant x$ for which the fundamental solution of the Pell equation $t^2-Du^2=1$ is less than $D^{\frac{1}{2}+\alpha}$ with $\alpha\in[\frac{1}{2},1]$ is studied and certain lower bounds are obtained,…
We use exact enumeration to characterize the solutions of quadratic unconstrained binary optimization problems of less than 21 variables in terms of their distributions of Hamming distances to close-by solutions. We also perform experiments…
Random constraint satisfaction problems play an important role in computer science and combinatorics. For example, they provide challenging benchmark instances for algorithms and they have been harnessed in probabilistic constructions of…
We generalize Dirichlet's diophantine approximation theorem to approximating any real number $\alpha$ by a sum of two rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2}$ with denominators $1 \leq q_1, q_2 \leq N$. This turns out to be…
Given a fractional differential equation of order $\alpha \in (0,1]$ with Caputo derivatives, we investigate in a quantitative sense how the associated solutions depend on their respective initial conditions. Specifically, we look at two…
Let $\mathscr{Q}(m,q)$ and $\mathscr{S}(m,q)$ be the sets of quadratic forms and symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$, respectively. The orbits of $\mathscr{Q}(m,q)$ and $\mathscr{S}(m,q)$ under a…
Let $F$ be a quadratic form in four variables, let $m\in\mathbb{N}$ and let $\mathbf{k}\in \mathbb{Z}^4$. We count integer solutions to $F(\mathbf{x})=0$ with $\mathbf{x}\equiv \mathbf{k}\:\mathrm{mod}(m)$. One can compare this to the…
We study linear chance-constrained problems where the coefficients follow a Gaussian mixture distribution. We provide mixed-binary quadratic programs that give inner and outer approximations of the chance constraint based on piecewise…
In this paper we develop an asymptotic analysis for formal and actual solutions of q-difference equations, under a regularity assumption. In particular, evaluations of regular solutions of regular q-difference equations have an exponential…
In this paper we explore a family of congruences over $\N^\ast$ from which one builds a sequence of symmetric matrices related to the Mertens function. From the results of numerical experiments, we formulate a conjecture about the growth of…
The goal of this paper is to relax convexity assumption on some classical results in mean curvature flow. In the first half of the paper, we prove a generalized version of Hamilton's differential Harnack inequality which holds for mean…
We give an asymptotic formula for correlations \[ \sum_{n\le x}f_1(P_1(n))f_2(P_2(n))\cdot \dots \cdot f_m(P_m(n))\] where $f\dots,f_m$ are bounded "pretentious" multiplicative functions, under certain natural hypotheses. We then deduce…
Nagata has conjectured that the following statement (N_r) holds for all $r\geq 10$: (N_r) if $P_1,...P_r \in {\mathbb P}^2$ are generic points then any plane curve $C$ satisfies $\sum_1^r mult_{P_i}(C)\leq \sqrt{r} deg(C)$. Nagata proved…
We discuss the optimal matching solution for both the assignment problem and the matching problem in one dimension for a large class of convex cost functions. We consider the problem in a compact set with the topology both of the interval…
The accuracy of the conventional treatment of quark mass thresholds in the QCD running coupling constant $\alpha_s$, based on the step approximation to the $\beta$--function, is investigated. The errors of extrapolating $\ase(\mu)$ from low…
We show that for any odd $k$ and any instance of the Max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a $\frac{1}{2} + \Omega(1/\sqrt{D})$ fraction of constraints, where…
This paper is divided in two parts. In the first part we consider irregular singular analytic q-difference equations, with q\in ]0,1[, and we show how the Borel sum of a divergent solution of a differential equation can be uniformly…