Related papers: Approximate expressions for mathematical constants…
One strategy to solve a nonlinear eigenvalue problem $T(\lambda)x=0$ is to solve a polynomial eigenvalue problem (PEP) $P(\lambda)x=0$ that approximates the original problem through interpolation. Then, this PEP is usually solved by…
We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution. Our main example is the Poisson boundary value problem, when…
Let $\eta_{1},\eta_2,...$ be independent (not necessarily identically distributed) zero-mean random variables (r.v.'s) such that $|\eta_i|\le1$ almost surely for all $i$, and let $Z$ stand for a standard normal r.v. Let $a_1,a_2,...$ be any…
Numerical approximate computation can solve large and complex problems fast. It has the advantage of high efficiency. However it only gives approximate results, whereas we need exact results in many fields. There is a gap between…
Many famous integer sequences including the Catalan numbers and the Motzkin numbers can be expressed in the form $ConstantTermOf\left[P(x)^nQ(x)\right]$ for Laurent polynomials $Q$, and symmetric Laurent trinomials $P$. In this paper we…
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…
Let $A(x)=A\_0+x\_1A\_1+...+x\_nA\_n$ be a linear matrix, or pencil, generated by given symmetric matrices $A\_0,A\_1,...,A\_n$ of size $m$ with rational entries. The set of real vectors x such that the pencil is positive semidefinite is a…
In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let $p$ be a prime and let $a$ be any positive integer. We determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}$ mod $p^2$ for…
In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and…
This study presents a method for constructing a sequence of approximate solutions of increasing accuracy to general equilibrium models on nonlocal domains. The method is based on a technique originated from dynamical systems theory. The…
Unlike polynomials, rational functions can represent functions having poles or branch cuts with root-exponential convergence and no Runge phenomenon. Recent developments of the AAA and greedy Thiele algorithms have sparked renewed interest…
We construct simple, explicit matrices with columns having unit $\ell^2$ norm and discrepancy approaching $1 + \sqrt{2} \approx 2.414$. This number gives a lower bound, the strongest known as far as we are aware, on the constant appearing…
We give the first almost optimal polynomial-time proper learning algorithm of Boolean sparse multivariate polynomial under the uniform distribution. For $s$-sparse polynomial over $n$ variables and $\epsilon=1/s^\beta$, $\beta>1$, our…
The Basel problem, solved by Leonhard Euler in 1734, asks to resolve $\zeta(2)$, the sum of the reciprocals of the squares of the natural numbers, i.e. the sum of the infinite series: \begin{equation}…
Present work presents a code written in the very simple programming language MATLAB, for three dimensional linear elastostatics, using constant boundary elements. The code, in full or in part, is not a translation or a copy of any of the…
We consider the problem of synthesizing programs with numerical constants that optimize a quantitative objective, such as accuracy, over a set of input-output examples. We propose a general framework for optimal synthesis of such programs…
The topic of this tutorial is Least Squares Sparse Principal Components Analysis (LS SPCA) which is a simple method for computing approximated Principal Components which are combinations of only a few of the observed variables. Analogously…
In this work we show $\zeta(3) = 4\pi^{2}\ln(B)$ with the Bendersky-Adamchik constant $B$.
We consider the solution of systems of linear algebraic equations (SLAEs) with an ill-conditioned or degenerate exact matrix and an approximate right-hand side. An approach to solving such a problem is proposed and justified, which makes it…
We study the hardness of Approximate Query Processing (AQP) of various types of queries involving joins over multiple tables of possibly different sizes. In the case where the query result is a single value (e.g., COUNT, SUM, and…