Related papers: The Discrete Lambert Map

We perform a special number field sieve discrete logarithm computation in a 1024-bit prime field. To our knowledge, this is the first kilobit-sized discrete logarithm computation ever reported for prime fields. This computation took a…

A discrete analog of the holomorphic map $z^{\gamma}$ is studied. It is given by Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding circle patterns are embedded and described by special…

This article proposes a new numerical algorithm for second order elliptic equations in non-divergence form. The new method is based on a discrete weak Hessian operator locally constructed by following the weak Galerkin strategy. The…

Discrete signatures are invariants computed from time series corresponding to the discretised version of the signature of paths. We study the algebraic varieties arising from their images, the discrete signature varieties. We introduce them…

The problem of iterated partial summations is solved for some discrete distributions defined on discrete supports. The power method, usually used as a computational approach to finding matrix eigenvalues and eigenvectors, is in some cases…

We propose a discrete form for an equation due to Gambier and which belongs to the class of the fifty second order equations that possess the Painleve property. In the continuous case, the solutions of the Gambier equation is obtained…

We discuss asymptotic properties of a family of discrete probability measures which may be used to model particle configurations with a wall on a set of discrete nodes. The correlations are shown to be determinantal and are expressed in…

In this paper, we propose two exact distributed algorithms to solve mixed integer linear programming (MILP) problems with multiple agents where data privacy is important for the agents. A key challenge is that, because of the non-convex…

The purpose of this article is to generalize some results of Vatsal on studying the special values of Rankin-Selberg L-functions in an anticyclotomic $\mathbb{Z}_{p}$-extension. Let $g$ be a cuspidal Hilbert modular form of parallel weight…

The function $y = g(x) = \mathrm{log}\big(W(e^x)\big)$, where $W()$ denotes the Lambert W function, is the solution to the equation $y + e^y = x$. It appears in various problem situations, for instance the calculation of current-voltage…

Let $\ell \geq 5$ be a prime and let $N$ be a square-free integer prime to $\ell$. For each prime $p$ dividing $N$, let $a_p$ be either $1$ or $-1$. We give sufficient criteria for the existence of a newform $f$ of weight 2 for…

We study the problem of solving linear programs of the form $Ax\le b$, $x\ge0$ with differential privacy. For homogeneous LPs $Ax\ge0$, we give an efficient $(\epsilon,\delta)$-differentially private algorithm which with probability at…

Elliptic curves with a known number of points over a given prime field with n elements are often needed for use in cryptography. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication…

We compute the PSL(2,N)-module structure of the Riemann-Roch space L(D), where D is an invariant non-special divisor on the modular curve X(N), with N > 5 prime. This depends on a computation of the ramification module, which we give…

The challenge of producing accurate statistics while respecting the privacy of the individuals in a sample is an important area of research. We study minimax lower bounds for classes of differentially private estimators. In particular, we…

In this article we prove that there exists a Dixmier map for nilpotent super Lie algebras. In other words, if we denote by Prim(U(g)) the set of (graded) primitive ideals of the enveloping algebra U(g) of g and Ad_0 the adjoint group of…

We use the p-adic local Langlands correspondence for GL_2(Q_p) to explicitly compute the reduction modulo p of crystalline representations of small slope, and give applications to modular forms.

For the principal eigenvalue of discrete weighted $p$-Laplacian on the set of nonnegative integers, the convergence of an approximation procedure and the inverse iteration is proved. Meanwhile, in the proof of the convergence, the…

We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show…

Drawings of highly connected (dense) graphs can be very difficult to read. Power Graph Analysis offers an alternate way to draw a graph in which sets of nodes with common neighbours are shown grouped into modules. An edge connected to the…