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Richard J. Mathar

Publications and source records attributed to Richard J. Mathar.

At least 37 records · Page 2Linked to original sources
arXiv2020

A Java Math.BigDecimal Implementation of Core Mathematical Functions

The mathematical functions log(x), exp(x), root[n]x, sin(x), cos(x), tan(x), arcsin(x), arctan(x), x^y, sinh(x), cosh(x), tanh(x) and Gamma(x) have been implemented for arguments x in the real domain in a native Java library on top of the multi-precision BigDecimal representation of floating point numbers. This supports scientific applications where more than the double precision accuracy of the library of the Standard Edition is desired. The full source code is made available under the LGPL v3.0.

math.NA↗
arXiv2019

RiemCirc: A Generator of Nodes and Weights for Riemann Integration on the Circle

RiemCirc is a C++ program which allocates points inside the unit circle for numerical quadrature on the circle, aiming at homogeneous equidistant distribution. The weights of the quadrature rule are computed by the area of the tiles that surround these nodes. The shapes of the areas are polygonal, defined by Voronoi tessellation.

math.NA↗
arXiv2018

Twenty Digits of Some Integrals of the Prime Zeta Function

The double sum sum_(s >= 1) sum_p 1/(p^s log p^s) = 2.00666645... over the inverse of the product of prime powers p^s and their logarithms, is computed to 24 decimal digits. The sum covers all primes p and all integer exponents s>=1. The calculational strategy is adopted from Cohen's work which basically looks at the fraction as the underivative of the Prime Zeta Function, and then evaluates the integral by numerical methods.

math.NT↗
arXiv2018

Counting Connected Graphs without Overlapping Cycles

The simple connected graphs may be classified by their cycle composition (number and lengths of cycles). This work derives the counting series of the simple connected graphs that have cycles of unrestricted number and length, but no overlapping cycles. Cycle pairs of these graphs of interest must not have common nodes or edges. The recipe of counting these graphs is based on the counting series of the associated planted graphs, multisets of planted graphs, a recursive synthesis of enriched trees, and a generalized Otter's formula that maps the underlying rooted block graphs to the underlying block graphs.

math.CO↗
arXiv2018

Orthogonal Basis Function Over the Unit Circle with the Minimax Property

We construct an orthogonal basis of functions defined over the unit circle as the product of the common sinusoidal functions of the azimuth angle by radial functions which are essentially sines of a polynomials of the radial distance to the origin. The main impetus of this approach is to generate basis functions where the minima and maxima along both coordinates, the azimuth and the distance r to the center, have the same amplitude, akin to the Chebyshev polynomial basis of the one-dimensional unit interval. The construction is based on numerical evaluation of the overlap integrals, which have the format of generalized Fresnel integrals.

math.NA↗
arXiv2017

Statistics on Small Graphs

We create the unlabeled or vertex-labeled graphs with up to 10 edges and up to 10 vertices and classify them by a set of standard properties: directed or not, vertex-labeled or not, connectivity, presence of isolated vertices, presence of multiedges and presence of loops. We present tables of how many graphs exist in these categories.

math.CO↗
arXiv2017

Construction of Bhaskara Pairs

We construct integer solutions {a,b} to the coupled system of diophantine quadratic-cubic equations a^2+b^2=x^3 and a^3+b^3=y^2 for fixed ratios a/b.

math.NT↗
arXiv2016

Topologically Distinct Sets of Non-intersecting Circles in the Plane

Nested parentheses are forms in an algebra which define orders of evaluations. A class of well-formed sets of associated opening and closing parentheses is well studied in conjunction with Dyck paths and Catalan numbers. Nested parentheses also represent cuts through circles on a line. These become topologies of non-intersecting circles in the plane if the underlying algebra is commutative. This paper generalizes the concept and answers quantitatively - as recurrences and generating functions of matching rooted forests - the questions: how many different topologies of nested circles exist in the plane if (i) pairs of circles may intersect, or (ii) even triples of circles may intersect. That analysis is driven by examining the symmetry properties of the inner regions of the fundamental type(s) of the intersecting pairs and triples.

math.CO↗
arXiv2016

Tiling n X m rectangles with 1 X 1 and s X s squares

We consider tilings of a rectangle which is n units wide and m units long by non-overlapping 1 X 1 squares and s X s squares. Bivariate generating functions are computed with the Transfer Matrix Method for moderately large but fixed widths n as a function of the parameter m and of the number of s X s squares in the rectangle.

math.CO↗
arXiv2015

Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli

The Dirichlet characters of reduced residue systems modulo m are tabulated for moduli m <= 195. The associated L-series are tabulated for m <= 14 and small positive integer argument s accurate to 10^(-50), their first derivatives for m <= 6. Restricted summation over primes only defines Dirichlet Prime L-functions which lead to Euler products (Prime Zeta Modulo functions). Both are materialized over similar ranges of moduli and arguments. Formulas and numerical techniques are well known; the aim is to provide direct access to reference values.

math.NT↗
arXiv2015

A C++ Incarnation of Zernike Circle Functions

An explicit C++ library is provided which deals with Zernike Functions over the unit circle as the main subject. The implementation includes basic means to evaluate the functions at points inside the unit circle and to convert the radial and azimuthal parameters to Noll's index and vice versa. Advanced methods allow to expand products of Zernike Functions into sums of Zernike Functions, and to convert Zernike Functions to polynomials over the two Cartesian coordinates and vice versa.

math.NA↗
arXiv2015

Apparent Places with an Ellipsoidal Geometry of Refraction in the Earth's Atmosphere

The displacement of star images by atmospheric refraction observed by an Earth-bound telescope is dominated by a familiar term proportional to the product of the tangent of the zenith angle by the refractivity at the ground. The manuscript focuses on the torsion of the ray path through the atmosphere in a model of atmospheric layers above the ellipsoidal Earth surface, induced by the two slightly different principal curvatures along N--S and E--W pointing directions. This breaking of the azimuthal symmetry effects apparent places at the sub-milliarcsecond scale at optical and infrared wavelengths.

astro-ph.IM↗
arXiv2014

Erratum to "Solutions problem 89-2: On the principal value of a quadruple integral", SIAM Rev. 32 (1990) 143

W. B. Jordan's conclusion that the quadruple principal value integral in problem 89-2 vanishes does not hold. The error sneaks in through a contribution of a subintegral which impedes some sign symmetry with respect to the master parameter (the Fermi radius) and which was overlooked in the published solution. In summary, the original problem of solving the quadruple integral remains unsolved.

math.GM↗
arXiv2014

Four-center Integral of a Dipolar Two-electron Potential Between s-type GTO's

We reduce two-electron 4-center products of Cartesian Gaussian Type Orbitals with Boys' contraction to 2-center products of the form psi_alpha(r_i-A) psi_beta(r_j-B), and compute the 6-dimensional integral over d^3r_i d^3r_j over these with the effective potential V_{ij} = (r_i-r_j) . r_j / |r_i-r_j|^3 in terms of Shavitt's confluent hypergeometric functions.

physics.chem-ph↗
arXiv2014

ApSimon's Mint Problem with Three or More Weighings

ApSimon considered the problem of deciding by a process of two weighings on which of a known number of mints emit either coins of a known genuine weight or emit coins of a different secondary but unknown weight. The combinatorial problem consists of finding two sets of coin numbers to be loaded on the tray for each of the weighings, and then to minimize the total count of coins to be drawn from all mints for these two weighings. This work yields numerical results for the generalized problem which allows three or more weighings to settle which of the mints produce either sort of coins.

math.CO↗
arXiv2014

Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices

Step by step completion of a left-to-right tiling of a rectangular floor with tiles of a single shape starts from one edge of the floor, considers the possible ways of inserting a tile at the leftmost uncovered square, passes through a sequence of rugged shapes of the front line between covered and uncovered regions of the floor, and finishes with a straight front line at the opposite edge. We count the tilings by mapping the front shapes to nodes in a digraph, then counting closed walks on that digraph with the transfer matrix method. Generating functions are detailed for tiles of shape 1 x 3, 1 x 4 and 2 x 3 and modestly wide floors. Equivalent results are shown for the 3-dimensional analog of filling bricks of shape 1x 1 x 2, 1 x 1 x 3, 1 x 1 x 4, 1 x 2 x 2 or 1 x 2 x 3 into rectangular containers of small cross sections.

math.CO↗
arXiv2013

Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings

The number of complete tilings of m X n floors for tiles of shape 1 X 2, 1 X 3, 1 X 4 and 2 X 3 is computed numerically for floors up to width m=9 and variable floor lengths n. Counts are obtained for two classes, for fixed tile stack orientation on one hand and for counts up to rotations and reflections on the other hand. Counts are refined by the number of points on the floor where 4 tiles meet, i.e., by the degree of violation of the requirement for Tatami tilings.

math.CO↗