Related papers: Simple Methods for Finding Actual Congruent Number…
Reasoning about real number expressions in a proof assistant is challenging. Several problems in theorem proving can be solved by using exact real number computation. I have implemented a library for reasoning and computing with complete…
The article contains some important classes of multisets. Combinatorial proofs of problems on the number of m-submultisets and m-permutations of multiset elements are considered and effective algorithms for their calculation are given. In…
A simple method called symbolic representation for piecewise linear functions on the real line is introduced and used to compute the numbers of periodic points of all periods for some such functions. Since, for every positive integer m, the…
The differences between the sets in which ideal arithmetics takes place and the sets of floating point numbers are outlined. A set of classical problems in correct numerical evaluation is presented, to increase the awareness of newcomers to…
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
Convergence results are stated for the variational iteration method applied to solve an initial value problem for a system of ordinary differential equations.
We discuss various universality aspects of numerical computations using standard algorithms. These aspects include empirical observations and rigorous results. We also make various speculations about computation in a broader sense.
The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be…
We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…
We revisit the algorithmic problem of finding a triangle in a graph (\textsc{Triangle Detection}), and examine its relation to other problems such as \textsc{3Sum}, \textsc{Independent Set}, and \textsc{Graph Coloring}. We obtain several…
In the literature, we have various ways of proving irrationality of a real number. In this survey article, we shall emphasize on a particular criterion to prove irrationality. This is called nice approximation of a number by a sequence of…
We consider the problem of minimising the number of edges that are contained in triangles, among $n$-vertex graphs with a given number of edges. We prove a conjecture of F\"uredi and Maleki that gives an exact formula for this minimum, for…
Triangles are everywhere in the virtual world. The surface of nearly every graphical object is saved as a triangular mesh on a computer. Light effects and movements of virtual objects are computed on the basis of triangulations. Besides…
Yet more candidates are proposed for inclusion in the Encyclopedia of Triangle Centers. Our focus is entirely on simple calculations.
A new simple geometrical interpretation of complex numbers is presented. It differs from their usual interpretation as points in the complex plane. From the new point of view the complex numbers are rather operations on vectors than points.…
A degree-regular triangulation is one in which each vertex has identical degree. Our main result is that any such triangulation of a (possibly non-compact) surface $S$ is geometric, that is, it is combinatorially equivalent to a geodesic…
A large multitude of scientific computing tools is available today. This article gives an overview of available tools and explains the main application fields. In addition basic principles of number representations in computing and the…
A simple iteration methodology for the solution of a set of a linear algebraic equations is presented. The explanation of this method is based on a pure geometrical interpretation and pictorial representation. Convergence using this method…
A positive integer $A$ is called a congruent number if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a congruent number if and only if the congruent number curve $y^2=x^3-A^2x$ has a rational…
We explore an algorithm for approximating roots of integers, discuss its motivation and derivation, and analyze its convergence rates with varying parameters and inputs. We also perform comparisons with established methods for approximating…