Related papers: How to compute $\sum 1/n^2$ by solving triangles
Permutations can be represented as linear combinations of natural numbers with different powers. In this paper, its coefficient matrix and inverse matrix is derived, and the results show the coefficient matrix is a lower triangular matrix…
A method is developed to compute analytically fully symmetric cubature rules on the triangle by using symmetric polynomials to express the two kinds of invariance inherent in these rules. Rules of degree up to 15, some of them new and of…
Numerical data processing is a key task across different fields of computer technology use. However, even simple summation of values is not precise due to the floating point representation use. This paper presents a practical algorithm for…
The problem of finding the number of lattice points in a triangle has a classical solution if the lattice is $\mathbf{Z}^2$ and the vertices of the triangle have integer valued coordinates. We consider what happens when we replace the…
Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation…
In this paper we suggest analytical methods and associated algorithms for determining the sum of the subsets $X_m$ of the set $X_n$ (subset sum problem). Our algorithm has time complexity $T=O(C_{n}^{k})$ ($k=[m/2]$, which significantly…
We investigate the approximation for computing the sum $a_1+...+a_n$ with an input of a list of nonnegative elements $a_1,..., a_n$. If all elements are in the range $[0,1]$, there is a randomized algorithm that can compute an…
Denote by $N(n)$ the number of integer solutions $(x_1,\,x_2,\ldots ,x_n)$ of the equation $x_1+x_2+\ldots+x_n=x_1x_2\cdot\ldots\cdot x_n$ such that $x_1\ge x_2\ge\ldots\ge x_n\ge 1$, $n \in \mathbb{Z}^+$. The aim of this paper are is…
In this paper, we present a novel algorithm and its three variations for solving the Rubik's cube more efficiently. This algorithm can be used to solve the complete $n \times n \times n$ cube in $O(\frac{n^2}{\log n})$ moves. This algorithm…
In the present paper a new mean value theorem for polynomials of special form is obtained. The case of sums on vertices of a regular polygon is studied. A criterion for a certain equation to be satisfied is obtained.
In this paper, we study universal sums of triangular numbers and squares. Specifically, we prove that a sum of triangular numbers and squares is universal if and only if it represents…
In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for…
An original regular approach to constructing special type symmetries for boundary value problems, namely renormgroup symmetries, is presented. Different methods of calculating these symmetries, based on modern group analysis are described.…
Despite extensive research on symmetric polynomial quadrature rules for triangles, as well as approaches to their calculation, few studies have focused on non-polynomial functions, particularly on their integration using symmetric triangle…
We develop a numerical method for solving a system of nonlinear integral equations involving two integral terms: at the current time t, one integral is taken from 0 to t, and a different integral is taken from t to infinity. We prove the…
We present a subdivision method to solve systems of congruence equations. This method is inspired in a subdivision method, based on Bernstein forms, to solve systems of polynomial inequalities in several variables and arbitrary degrees. The…
We use a variation of the Circle Method, along with the Saddle Point Method, to obtain an asymptotic formula for the number of partitions of a number n into integers which are sums of two squares. Unlike previous work on partitions into…
An algorithm for computing /pi(N) is presented.It is shown that using a symmetry of natural numbers we can easily compute /pi(N).This method relies on the fact that counting the number of odd composites not exceeding N suffices to calculate…
In this note, we propose simple summations for primes, which involve two finite nested sums and Bernoulli numbers. The summations can also be expressed in terms of Bernoulli polynomials.
An efficient procedure for the computation of $Li_{s}(z)$ where $s<0$ is here presented. We started with Polylogarithm $Li_{s}(z)$ where $s<0$. The summation of $n^{s}z^{n}$ is evaluated using a new method. An assumption is made that the…