Related papers: Smooth Integer Encoding via Integral Balance
We use recurrence equations (alias difference equations) to enumerate the number of formula-representations of positive integers using only addition and multiplication, and using addition, multiplication, and exponentiation, where all the…
We present a de Bruijn type approximation for quantifying the content of m smooth numbers, derived from samples obtained through a probability measure over the set of integers less than or equal to n, with point mass function at k inversely…
Implicit neural representations (INRs) have emerged as a powerful tool for solving inverse problems in computer vision and computational imaging. INRs represent images as continuous domain functions realized by a neural network taking…
We study the numerical approximation of integrals over $\mathbb{R}^s$ with respect to the standard Gaussian measure for integrands which lie in certain Hermite spaces of functions. The decay rate of the associated sequence is specified by a…
We consider the problem of computing numerical invariants of programs by abstract interpretation. Our method eschews two traditional sources of imprecision: (i) the use of widening operators for enforcing convergence within a finite number…
Given a way to evaluate an unknown polynomial with integer coefficients, we present new algorithms to recover its nonzero coefficients and corresponding exponents. As an application, we adapt this interpolation algorithm to the problem of…
A formalism for the numerical integration of one- and two-loop integrals is presented. It is based on subtraction terms which remove the soft, collinear and some of the ultraviolet divergences from the integrand. The numerical integral is…
The eigenvalue density of a matrix plays an important role in various types of scientific computing such as electronic-structure calculations. In this paper, we propose a quantum algorithm for computing the eigenvalue density in a given…
It is natural to expect the following loosely stated approximation principle to hold: a numerical approximation solution should be in some sense as smooth as its target exact solution in order to have optimal convergence. For piecewise…
We present a mathematica package that performs the symbolic calculation of integrals of the form \int^{\infty}_0 e^{-x/u} x^n j_{\nu} (x) j_{\mu} (x) dx where $j_{\nu} (x)$ and $j_{\mu} (x)$ denote spherical Bessel functions of integer…
Exact representations of real numbers such as the signed digit representation or more generally linear fractional representations or the infinite Gray code represent real numbers as infinite streams of digits. In earlier work by the first…
We consider the geometric numerical integration of Hamiltonian systems subject to both equality and "hard" inequality constraints. As in the standard geometric integration setting, we target long-term structure preservation. We…
Optimal balance is a non-asymptotic numerical method to compute a point on the slow manifold for certain two-scale dynamical systems. It works by solving a modified version of the system as a boundary value problem in time, where the…
Noisy shuffling channels capture the main characteristics of DNA storage systems where distinct segments of data are received out of order, after being corrupted by substitution errors. For realistic schemes with short-length segments,…
The aim of this paper is to study distributional properties of integers without large or small prime factors. Define an integer to be $[y',y]$-smooth if all of its prime factors belong to the interval $[y',y]$. We identify suitable weights…
A new integrable discrete system is constructed and studied, based on the algebraization of the difference operator. The model is named the discrete generalized nonlinear Schrodinger (GNLS) equation for which can be reduced to classical…
Consider an unknown smooth function $f: [0,1] \rightarrow \mathbb{R}$, and say we are given $n$ noisy$\mod 1$ samples of $f$, i.e., $y_i = (f(x_i) + \eta_i)\mod 1$ for $x_i \in [0,1]$, where $\eta_i$ denotes noise. Given the samples…
In this paper will be presented new approach to entropy coding: family of generalizations of standard numeral systems which are optimal for encoding sequence of equiprobable symbols, into asymmetric numeral systems - optimal for freely…
The E(xplicit)I(implicit)N(null) method was developed recently to remove numerical instability from PDEs, adding and subtracting an operator $\mathcal{D}$ of arbitrary structure, treating the operator implicitly in one case, and explicitly…
This paper presents a neural network (NN)-based solver for an integro-differential equation that models shrinkage-induced fragmentation. The proposed method directly maps input parameters to the corresponding probability density function…