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Richard J. Mathar

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

    Tue, 10/18/2011 - 15:01 — SomNambul
    Authors: Richard J. Mathar
    Subjects: Numerical Analysis
    Abstract

    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.

  2. Corrigendum to "Universal factorization of 3n-j (j>2) symbols..." [J. Phys. A: Math. Gen. 37 (2004) 3259].

    Mon, 02/28/2011 - 16:01 — SomNambul
    Authors: Richard J. Mathar
    Subjects: Mathematical Physics
    Abstract

    Ten values of 12-j symbols of the first kind published earlier are challenged
    by values calculated with an independent Python program. The program first
    implements a limited class of square roots of rational numbers, utilizing
    Python's unlimited representation of big integers. Wigner's 3jm symbols, 6-j,
    9-j and 12-j symbols are then calculated by their familiar representations as
    sums over products of these.

  3. Point Counts of D_k and Some A_k and E_k Integer Lattices Inside Hypercubes.

    Tue, 02/23/2010 - 16:01 — SomNambul
    Authors: Richard J. Mathar
    Subjects: Geometric Topology
    Abstract

    Regular integer lattices are characterized by k unit vectors that build up
    their generator matrices. These have rank k for D-lattices, and are
    rank-deficient for A-lattices, E_6 and E_7. We count lattice points inside
    hypercubes centered at the origin for all three types, as if classified by
    maximum infinity norm in the host lattice. The results assume polynomial format
    as a function of the hypercube edge length.

  4. Numerical Evaluation Of the Oscillatory Integral over exp(i*pi*x)*x^(1/x).

    Tue, 12/22/2009 - 16:01 — SomNambul
    Authors: Richard J. Mathar
    Subjects: Classical Analysis and ODEs
    Abstract

    Real and imaginary part of the limit 2N->infinity of the integral
    int_{x=1..2N} exp(i*pi*x)*x^(1/x) dx are evaluated to 20 digits with brute
    force methods after multiple partial integration, or combining a standard
    Simpson integration over the first halve wave with series acceleration
    techniques for the alternating series co-phased to each of its points. The
    integrand is of the logarithmic kind; its branch cut limits the performance of
    integration techniques that rely on smooth higher order derivatives.

  5. Tile Count in the Interior of Regular 2n-gons Dissected by Diagonals Parallel to Sides.

    Thu, 11/19/2009 - 16:01 — SomNambul
    Authors: Richard J. Mathar
    Subjects: Combinatorics
    Abstract

    The regular 2n-gon (square, hexagon, octagon, ...) is subdivided into smaller
    polygons (tiles) by the subset of diagonals which run parallel to any of the 2n
    sides. The manuscript reports on the number of tiles up to the 78-gon.

  6. A Java Math.BigDecimal Implementation of Core Mathematical Functions.

    Mon, 08/24/2009 - 13:10 — SomNambul
    Authors: Richard J. Mathar
    Subjects: Numerical Analysis
    Abstract

    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.

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