Representation Of Level Paths Of An Analytic Function
We find an arc-parameterization of the contour on which an given analytic function has constant modulus. This contour is seen to satisfy a differential equation which we explicitly give.
math.GM↗
SEARCH · Searcharxiv
Results for “math.GM”
Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
Triacontagonal coordinates for the E(8) root system
This note gives an explicit formula for the elements of the E(8) root system. The formula is triacontagonally symmetric in that one may clearly see an action by the cyclic group with 30 elements. The existence of such a formula is due to the fact that the Coxeter number of E(8) is 30.
math.GM↗
Hamiltonian Graphs and the Traveling Salesman Problem
A new characterization of Hamiltonian graphs using f-cutset matrix is proposed. Based on this new characterization, a new exact polynomial time algorithm for the traveling salesman problem (TSP) is developed. We then define the so-called ordered weighted adjacency list for given weighted complete graph and proceed to the paper's main result, namely, the exact algorithm based on the utilization of the ordered weighted adjacency list and the simple properties that any path or circuit must satisfy. This algorithm performs checking of sub-lists, containing (p-1) entries (edge pairs) for paths and p entries (edge pairs) for circuits, chosen from ordered adjacency list in a well defined sequence to determine exactly the shortest Hamiltonian path and shortest Hamiltonian circuit in a weighted complete graph of p vertices. The procedure has intrinsic advantage of landing on the desired solution in quickest possible time and even in worst case in polynomial time. A new characterization of the shortest Hamiltonian tour for a weighted complete graph satisfying triangle inequality (i.e. for tours passing through every city on a realistic map of cities where cities can be taken as points on a Euclidean plane) is also proposed. Finally, we propose a classical algorithm for unstructured search, three new quantum algorithms for unstructured search, which exponentially speed up the searching ability in the unstructured database, and one quantum algorithm for solving a K-SAT problem and indicate its effect on traveling salesman problem and other NP-complete problems.
math.GM↗
Existence and Number of Solutions of Diophantine Quadratic Equations with Two Unknowns in $Z$ and $N$
In this short note we study the existence and number of solutions in the set of integers ($Z$) and in the set of natural numbers ($N$) of Diopahntine Equations of second degree with two variables of the general form $ax^2-by^2=c$.
math.GM↗
Self-similar and self-affine sets; measure of the intersection of two copies
Let K be a self-similar or self-affine set in R^d, let μbe a self-similar or self-affine measure on it, and let G be the group of affine maps, similitudes, isometries or translations of R^d. Under various assumptions (such as separation conditions or we assume that the transformations are small perturbations or that K is a so called Sierpinski sponge) we prove theorems of the following types, which are closely related to each other; Non-stability: There exists a constant c<1 such that for every g\in G we have either μ(K\cap g(K)) 0 \iff int_K (K\cap g(K)) is nonempty (where int_K is interior relative to K). Extension: The measure μhas a G-invariant extension to R^d. Moreover, in many situations we characterize those g's for which μ(K\cap g(K) > 0, and we also get results about those $g$'s for which $g(K)\su K$ or $g(K)\supset K$ holds.
math.GM↗
The General Form Of Cyclic Orthonormal Generators In R^N
In this paper we give a definition of cyclic orthonormal generators (cogs) in R^N. We give a general canonical form for their expression. Further, we give an explicit formula for computing the canonical form of any given cog.
math.GM↗
On the Theory of Colorful Graphs
The theory of colorful graphs can be developed by working in Galois field modulo (p), p > 2 and a prime number. The paper proposes a program of possible conversion of graph theory into a pleasant colorful appearance. We propose to paint the usual black (indicating presence of an edge) and white (indicating absence of an edge) edges of graphs using multitude of colors and study their properties. All colorful graphs considered here are simple, i.e. not having any multiple edges or self-loops. This paper is an invitation to the program of generalizing usual graph theory in this direction.
math.GM↗
Congruent numbers, elliptic curves, and the passage from the local to the global
The ancient unsolved problem of congruent numbers has been reduced to one of the major questions of contemporary arithmetic: the finiteness of the number of curves over $\bf Q$ which become isomorphic at every place to a given curve. We give an elementary introduction to congruent numbers and their conjectural characterisation, discuss local-to-global issues leading to the finiteness problem, and list a few results and conjectures in the arithmetic theory of elliptic curves.
math.HO↗
On Hadwiger Conjecture
We propose an algorithm to reduce a k-chromatic graph to a complete graph of largest possible order through a well defined sequence of contractions. We introduce a new matrix called transparency matrix and state its properties. We then define correct contraction procedure to be executed to get largest possible complete graph from given connected graph. We finally give a characterization for k-chromatic graphs and use it to settle Hadwigers conjecture.
math.GM↗
On Solving General Linear Equations in the Set of Natural Numbers
In this paper one shows if the number of natural solutions of a general linear equation is limited or not. Also, it is presented a method of solving the Diophantine equation $ax-by=c$ in the set of natural numbers, and an example of solving in $N$ a Diophantine equation with three variables.
math.GM↗
The Integrals of Motion for the Deformed Virasoro Algebra
We explicitly construct two classes of infinitly many commutative operators in terms of the deformed Virasoro algebra. We call one of them local integrals and the other nonlocal one, since they can be regarded as elliptic deformations of the local and nonlocal integrals of motion obtained by V.Bazhanov, S.Lukyanov and Al.Zamolodchikov.
math-ph↗
How to release Frege's system from Russell's antinomy
The conditions for proper definitions in mathematics are given, in terms of the theory of definition, on the basis of the criterions of eliminability and non-creativity. As a definition, Russell's antinomy is a violation of the criterion of eliminability (Behmann, 1931; Bochvar, 1943). Following the path of the criterion of non-creativity, this paper develops a new analysis of Comprehension schema and, as a consequence, proof that Russell's antinomy argumentation, despite the words of Frege himself, does not hold in Grundgesetze der Arithmetik. According to Basic Law (III), the class of classes not belonging to themselves is a class defined by a function which can not take as argument its own course of value. In other words, the class of classes not belonging to themselves is a class whose classes are not identical to the class itself.
math.GM↗
On Two of Erdös's Open Problems
In this short note we present some remarks and conjectures on two of Erdös's open problems in number theory.
math.GM↗
Subjective Questions and Answers for a Mathematics Instructor of Higher Education
This article of mathematical education reflects author's experience with job applications and teaching methods and procedures to employ in the American Higher Education. It is organized as a standard questionnaire.
math.GM↗
On the Riemann zeta-function, Part I: Outline
Results of a multipart work are outlined. Use is made therein of the conjunction of the Riemann hypothesis, RH, and hypotheses advanced by the author. Let z(n) be the nth nonreal zero of the Riemann zeta-function with positive imaginary part in order of magnitude thereof. A relation is obtained, of the pair z(n) and the first derivative thereat of the zeta-function, to the preceding such pairs and the values of zeta at the points one-half plus a nonnegative multiple of four. That relation is derived from two forms of the density of the Laplace representation, on a certain vertical strip, of a meromorphic function constructed from zeta. Specific functions which play a central role therein are proven to have analytic extensions to the entire complex plane. It is established that the Laplace density is positive. That positivity implies RH and that each nonreal zero of zeta is simple. A metric geometry expression of the positivity of the density is derived. An analogous context is delineated relative to Dirichlet L-functions and the Ramanujan tau Dirichlet function.
math.GM↗
Discontinuity and Involutions on Countable Sets
For any infinite subset $X$ of the rationals and a subset $F \subseteq X$ which has no isolated points in $X$ we construct a function $f: X \to X$ such that $f(f(x))=x$ for each $x\in X$ and $F $ is the set of discontinuity points of $f$.
math.GM↗
On the complexity of solving ordinary differential equations in terms of Puiseux series
We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti differential polynomials associated to ordinary linear differential operators. In this paper, we get the same bound for arbitrary differential polynomials. The algorithm is based on a differential version of the Newton-Puiseux procedure for algebraic equations.
math.GM↗
$p$-Adic Haar multiresolution analysis and pseudo-differential operators
The notion of {\em $p$-adic multiresolution analysis (MRA)} is introduced. We discuss a ``natural'' refinement equation whose solution (a refinable function) is the characteristic function of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is a sum of $p$ characteristic functions of mutually disjoint discs of radius $p^{-1}$. This refinement equation generates a MRA. The case $p=2$ is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable function generating our Haar MRA is 1-periodic, which never holds for real refinable functions. This fact implies that there exist infinity many different 2-adic orthonormal wavelet bases in ${\cL}^2(\bQ_2)$ generated by the same Haar MRA. All of these bases are described. We also constructed multidimensional 2-adic Haar orthonormal bases for ${\cL}^2(\bQ_2^n)$ by means of the tensor product of one-dimensional MRAs. A criterion for a multidimensional $p$-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We proved also that these wavelets are eigenfunctions of the Taibleson multidimensional fractional operator. These facts create the necessary prerequisites for intensive using our bases in applications.
math-ph↗