Searcharxiv⌕ Search

arXiv subjects

Richard J. Mathar

Publications and source records attributed to Richard J. Mathar.

At least 19 recordsLinked to original sources
arXiv2025

Simulation of Time-dependent Karhunen-Loeve Phase Screens: an Ergodic Approach

Time-dependent phase screens in ground-based astronomy are typically simulated in the so-called frozen-screen approximation by establishing a static phase screen on a large pupil and dragging an aperture equivalent to the size of the actual input pupil across this oversized phase screen. The speed of this motion sweeping through the large phase screen is equivalent to a wind speed that changes the phase screen as a function of time. The ergodic ansatz replaces this concept by constructing the structure function in a three-dimensional volume -- a sphere for reasons of computational efficiency -- , sampling phase screens by two-dimensional planar cuts through that volume, and dragging them along the surface normal at some speed which generates a video of a phase screen. This manuscript addresses the linear algebra of populating the three-dimensional volume with phase screens of the von-Karman model of atmospheric turbulence.

astro-ph.IM↗
arXiv2025

Chebyshev Expansion of the Zernike Polynomials

The even and odd Zernike Polynomials R_n^m(x) can be expanded into sums of even and odd Chebyshev Polynomials T_i(x). This manuscript provides closed forms for the rational expansion coefficients c_{n,m,i} for a set of small 0 <= n-m <= 6 and a holonomic five-term recurrence for these coefficients for all larger n.

math.CA↗
arXiv2024

Chebyshev approximation of $x^m (-\log x)^l$ in the interval $0\le x \le 1$

The series expansion of $x^m (-\log x)^l$ in terms of the shifted Chebyshev Polynomials $T_n^*(x)$ requires evaluation of the integral family $\int_0^1 x^m (-\log x)^l dx / \sqrt{x-x^2}$. We demonstrate that these can be reduced by partial integration to sums over integrals with exponent $m=0$ which have known representations as finite sums over polygamma functions.

math.CA↗
arXiv2024

Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards

The manuscript studies configurations of non-overlapping non-bonding dominoes on finite rectangular boards of unit squares characterized by row and column number. The non-bonding dominoes are defined here by the requirement that any domino on the board shares at most one point (one of its four corner points) with any other domino, but no edge. With the Transfer Matrix Method, rational generating functions are derived that solve the enumeration problem entirely, here evaluated for boards with up to six rows or columns.

math.CO↗
arXiv2023

The Foucault Pendulum: Trajectories of the Full Lagrangian

The Foucault Pendulum is a Spherical Pendulum of fixed length with two angular degrees of freedom, attached to a suspension which rotates once a day around the Earth axis at a distance essentially set by Earth radius and the geodetic latitude of the pendulum. We write the Lagrange Function in the inertial frame of the fixed Earth axis and couple it strictly to the rotating frame at which the suspension appears at rest. The Euler-Lagrange equations are a coupled system of second-order differential equations for the two coordinates of the mass projected on the local horizontal plane. These are solved numerically in a C++ program which allows to study the trajectories beyond the various standard approximations of the literature.

physics.class-ph↗
arXiv2023

Integrals Associated with the Digamma Integral Representation

The definite integral with the kernel x/(x^2+b^2)/[\exp(2πx)-1] integrated from x=0 to infinity is the main term of a representation of the Digamma-Function psi(b), the derivative of the logarithm of the Gamma-Function. We present relations within the set of integrals over x^n/(x^2+b^2)^j/[\exp(μx)-1]^s for small integer exponents n, j and s with the aim to reduce them all to Polygamma-Functions.

math.GM↗
arXiv2023

Volume of Intersection of a Cone with a Sphere

The manuscript provides formulas for the volume of a body defined by the intersection of a solid cone and a solid sphere as a function of the sphere radius, of the distance between cone apex and sphere center, and of the cone aperture angle. If the sphere center lies on the (extended) cone axis the analysis may be based on cylinder coordinates fixed at the cone axis, and the volume is the sum of the well-known volumes of finite cones and sphere caps. At the general geometry the sphere center is not on the (extended) cone axis. Our approach calculates the volume by slicing space perpendicular to the cone axis and by integrating the lens areas defined by the sphere-cone intersection. These volume integrals are rephrased with the aid of the Byrd-Friedmann tables to Elliptic Integrals of the First, Second and Third Kind.

math.CA↗
arXiv2023

2-regular Digraphs of the Lovelock Lagrangian

The manuscripts tabulates arc lists of the 1, 1, 3, 8, 25, 85, 397 ... unlabeled 2-regular digraphs on n=0, 1, 2, ..., 9 nodes, including disconnected graphs, graphs with multiarcs and/or graphs with loops. Each of these graphs represents one term of the Lagrangian of Lovelock's type -- a contraction of a product of n Riemann tensors -- once the 2 covariant and 2 contravariant indices of a tensor are associated with the in-edges and out-edges of a node.

math.GM↗
arXiv2022

Geodetic Line at Constant Altitude above the Ellipsoid

The two-dimensional surface of a bi-axial ellipsoid is characterized by the lengths of its major and minor axes. Longitude and latitude span an angular coordinate system across. We consider the egg-shaped surface of constant altitude above (or below) the ellipsoid surface, and compute the geodetic lines - lines of minimum Euclidean length - within this surface which connect two points of fixed coordinates. This addresses the common "inverse" problem of geodesics generalized to non-zero elevations. The system of differential equations which couples the two angular coordinates along the trajectory is reduced to a single integral, which is handled by Taylor expansion up to fourth power in the eccentricity.

math.MG↗
arXiv2022

A Java Program Generating Barycentric Observer Velocities from JPL Ephemerides

This works presents a program which computes velocities of an Earth-bound observatory in the reference frame of the barycenter of the solar system. It feeds from ephemerides files of the Jet Propulsion Laboratory to extract the velocity of the geocenter, optionally with corrections from Earth rotation data of the International Earth Rotation Service, takes a datum (time and geodetic location) of the observer as parameters, and processes these data with the program library of the working group `Standards of Fundamental Astronomy' of the International Astronomical Union. The prospective application of the computed velocities is to subtract their projection onto a pointing direction from observed velocities in a step of data reduction of astronomic radial velocities.

astro-ph.IM↗
arXiv2021

The series limit of sum_k 1/[k log k (log log k)^2]

The slowly converging series sum_{k=3}^infinity 1/[k * log k * (log log k)^a] is evaluated to 38.4067680928 at a=2. After some initial terms, the infinite tail of the sum is replaced by the integral of the associated interpolating function, which is available in simple analytic form. Biases that originate from the difference between the smooth area under the function and the corresponding Riemann sum are corrected by standard means. The cases a=3 and a=4 are computed in the same manner.

math.NA↗
arXiv2020

A Barometric Exponential Model of the Atmosphere's Refractive Index: Zenith Angles and Second Order Aberration in the Entrance Pupil

This report models the refractive index above a telescope site by an atmosphere with exponential decay of the refractive index (susceptibility) as a function of altitude. The air is represented as a spherical hull around the surface. We compute (i) the differential zenith angle -- the difference between the actual zenith angle at arrival of rays at the telescope and a hypothetical zenith angle without the atmosphere -- and (ii) the optical path length distribution of rays at arrival in the entrance pupil as a function of the distance to the pupil center. The key technique in this work is to expand some integrals -- that depend on the refractive index profile along the curved path of each ray from some virtual plane in the direction of the star up to the entrance pupil -- in power series of small parameters.

astro-ph.IM↗
arXiv2020

Lozenge Tilings Of the Equilateral Triangle

We consider incomplete tilings of the equilateral triangle of edge length n that is subdivided into n^2 regular equilateral smaller unit triangles. Pairs of the unit triangles that share a side may be converted into lozenges, leaving some subset of the unit triangles untouched. We count numerically these coverings by lozenges and unit triangles for edge lengths n <= 6: the total and the detailed refinement as a function of the number of lozenges.

math.CO↗
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↗