Super Catalan Numbers and Fourier Summation over Finite Fields

We study polynomial summation over unit circles over finite fields of odd characteristic, obtaining a purely algebraic integration theory without recourse to infinite procedures. There are nonetheless strong parallels to classical integration theory over a circle, and we show that the super Catalan numbers and closely related rational numbers lie at the heart of both theories. This gives a uniform analytic meaning to these up to now somewhat mysterious numbers. Our derivation utilises the three-fold symmetry of chromogeometry between Euclidean and relativistic geometries, and we find that the Fourier summation formulas we derive in these two different settings are closely connected.Comment: 35 pages, 4 table

Super Catalan Numbers and Fourier Summations over Finite Fields

We find an algebraic interpretation of the super Catalan numbers through polynomial summation formulas over unit circles over finite fields of odd characteristic. While traditional Fourier analysis involves Riemann integration over the unit circle in the real number plane, we will develop a purely algebraic integration theory without recourse to infinite procedures, and develop an algorithm for explicitly computing such Fourier sums for general monomials. We consider three unit circles that arise from the Euclidean geometry and two relativistic geometries, and demonstrate the strong relationship between the integration theory in each geometry. The algebraic integrals in the three geometries are called the Fourier summation functionals and take values in the same finite field. The key results in this thesis are the existence and uniqueness of the Fourier summation functionals, as well as the explicit formulas for them in terms of the super Catalan numbers and their rational variants which we call the circular super Catalan numbers. This investigation not only opens up new avenues in developing finite field harmonic analysis from a completely algebraic point of view, but also highlights many similarities to the traditional integration theory over the unit circle

The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators

Linear multistep methods for integrating reversible differential equations

This paper studies multistep methods for the integration of reversible dynamical systems, with particular emphasis on the planar Kepler problem. It has previously been shown by Cano & Sanz-Serna that reversible linear multisteps for first-order differential equations are generally unstable. Here, we report on a subset of these methods -- the zero-growth methods -- that evade these instabilities. We provide an algorithm for identifying these rare methods. We find and study all zero-growth, reversible multisteps with six or fewer steps. This select group includes two well-known second-order multisteps (the trapezoidal and explicit midpoint methods), as well as three new fourth-order multisteps -- one of which is explicit. Variable timesteps can be readily implemented without spoiling the reversibility. Tests on Keplerian orbits show that these new reversible multisteps work well on orbits with low or moderate eccentricity, although at least 100 steps/radian are required for stability.Comment: 31 pages, 9 figures, in press at The Astronomical Journa

Landau singularities and singularities of holonomic integrals of the Ising class

We consider families of multiple and simple integrals of the ``Ising class'' and the linear ordinary differential equations with polynomial coefficients they are solutions of. We compare the full set of singularities given by the roots of the head polynomial of these linear ODE's and the subset of singularities occurring in the integrals, with the singularities obtained from the Landau conditions. For these Ising class integrals, we show that the Landau conditions can be worked out, either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral. The singular behavior of these integrals is obtained in the self-dual variable $w= s/2/(1+s^2)$, with $s= \sinh(2K)$, where $K=J/kT$ is the usual Ising model coupling constant. Switching to the variable $s$, we show that the singularities of the analytic continuation of series expansions of these integrals actually break the Kramers-Wannier duality. We revisit the singular behavior (J. Phys. A {\bf 38} (2005) 9439-9474) of the third contribution to the magnetic susceptibility of Ising model $\chi^{(3)}$ at the points $1+3w+4w^2= 0$ and show that $\chi^{(3)}(s)$ is not singular at the corresponding points inside the unit circle $| s |=1$, while its analytical continuation in the variable $s$ is actually singular at the corresponding points $2+s+s^2=0$ oustside the unit circle ($| s | > 1$).Comment: 34 pages, 1 figur