arXiv subjects
Publications and source records attributed to M. Lawrence Glasser.
Through the application of an evaluation technique based on cyclotomic multiple zeta values recently due to Au, we solve open problems on inverse binomial series that were included in a 2010 analysis textbook by Chen.
We introduce and prove evaluations for families of multiple elliptic integrals by solving special types of ordinary and partial differential equations. As an application, we obtain new expressions of Ramanujan-type series of level 4 and associated singular values for the complete elliptic integral $\mathbf K$ with integrals involving $\mathbf K$.
In 1998 Don Zagier introduced the modified Bernoulli numbers $B_{n}^{*}$ and showed that they satisfy amusing variants of some properties of Bernoulli numbers. In particular, he studied the asymptotic behavior of $B_{2n}^{*}$, and also obtained an exact formula for them, the motivation for which came from the representation of $B_{2n}$ in terms of the Riemann zeta function $ζ(2n)$. The modified Bernoulli numbers were recently generalized to Zagier polynomials $B_{n}^{*}(x)$. For $0<x<1$, an exact formula for $B_{2n}^{*}(x)$ involving infinite series of Bessel function of the second kind and Chebyshev polynomials, that yields Zagier's formula in a limiting case, is established here. Such series arise in diffraction theory. An analogous formula for $B_{2n+1}^{*}(x)$ is also presented. The $6$-periodicity of $B_{2n+1}^{*}$ is deduced as a limiting case of it. These formulas are reminiscent of the Fourier expansions of Bernoulli polynomials. Some new results, for example, the one yielding the derivative of the Bessel function of the first kind with respect to its order as the Fourier coefficient of a function involving Chebyshev polynomials, are obtained in the course of proving these exact formulas. The asymptotic behavior of Zagier polynomials is also derived from them. Finally, a Zagier-type exact formula is obtained for $B_{2n}^{*}\left(-\frac{3}{2}\right)+B_{2n}^{*}$.
We developed and experimentally verified an analytical model to describe diffusion of oligonucleotides from stable hydrogel beads. The synthesized alginate beads are Fe3+-cross-linked as well as polyelectrolyte-doped for uniformity and stability at physiological pH. Data on diffusion of oligonucleotides from inside the beads provide physical insights into the volume nature of the immobilization of a fraction of oligonucleotides due to polyelectrolyte cross-linking, i.e., the absence of the surface-layer barrier in this case. Furthermore, our results suggest a new simple approach to measuring the diffusion coefficient of the mobile oligonucleotide molecules inside hydrogel. The considered alginate beads provide a model for a well-defined component in drug release systems and for the oligonucleotide-release transduction steps in drug-delivering and biocomputing applications. This is illustrated by destabilizing the beads with citrate that induces full oligonucleotide release with non-diffusional kinetics.
We present a new and elegant integral approach to computing the Gardner-Fisher trigonometric power sum, which is given by $$ S_{m,v}=\left(\frac π{2m}\right)^{2v}\sum_{k=1}^{m-1}\cos^{-2v}\left(\frac{kπ}{2m}\right)\, , $$ We present a new and elegant integral approach to computing the Gardner-Fisher trigonometric power sum, which is given by $$ S_{m,v}=\left(\frac π{2m}\right)^{2v}\sum_{k=1}^{m-1}\cos^{-2v}\left(\frac{kπ}{2m}\right)\, , $$ where $m$ and $v$ are positive integers. This method not only confirms the results obtained earlier by an empirical method, but it is also much more expedient from a computational point of view. By comparing the formulas from both methods, we derive several new interesting number theoretic results involving symmetric polynomials over the set of quadratic powers up to $(v-1)^2$ and the generalized cosecant numbers. The method is then extended to other related trigonometric power sums including the untwisted Dowker sum. By comparing both forms for this important sum, we derive new formulas for specific values of the Nörlund polynomials. Finally, by using the results appearing in the tables, we consider more advanced sums involving the product of powers of cotangent and tangent with powers of cosecant and secant respectively.
We present the transformation of several sums of positive integer powers of the sine and cosine into non-trigonometric combinatorial forms. The results are applied to the derivation of generating functions and to the number of the closed walks on a path and in a cycle.
We study the kinetics of diffusion-limited coalescence, A+A-->A, and annihilation, A+A-->0, in the Bethe lattice of coordination number z. Correlations build up over time so that the probability to find a particle next to another varies from ρ^2 (ρis the particle density), initially, when the particles are uncorrelated, to [(z-2)/z]ρ^2, in the long-time asymptotic limit. As a result, the particle density decays inversely proportional to time, ρ~ 1/kt, but at a rate k that slowly decreases to an asymptotic constant value.
We propose a mechanism of long-range coherent coupling between nuclear spins to be used as qubits in solid-state semiconductor-heterojunction quantum information processing devices. The coupling is via localized donor electrons which in turn interact with the two-dimensional electron gas. An effective two-spin interaction Hamiltonian is derived and the coupling strength is evaluated. We also discuss mechanisms of qubit decoherence and consider possibilities for gate control of the interaction between neighboring qubits. The resulting quantum computing scheme retains all the gate-control and measurement aspects of earlier approaches, but allows qubit spacing at distances of order 100nm, attainable with the present-day semiconductor device technologies.
This note examines the second virial coefficient for an imperfect gas subject to a 2n-n interparticle potential in any dimension d between 0 and n. A compact analytic expression is presented for this quantity which shows that, apart from a numerical factor, its temperature dependence is a universal function parameterized by d/n.
We briefly review some common diffusion-limited reactions with emphasis on results for two-species reactions with anisotropic hopping. Our review also covers single-species reactions. The scope is that of providing reference and general discussion rather than details of methods and results. Recent exact results for a two-species model with anisotropic hopping and with `sticky' interaction of like particles, obtained by a novel method which allows exact solution of certain single-species and two-species reactions, are discussed.
We report exact results for one-dimensional reaction-diffusion models A+A -> inert, A+A -> A, and A+B -> inert, where in the latter case like particles coagulate on encounters and move as clusters. Our study emphasized anisotropy of hopping rates; no changes in universal properties were found, due to anisotropy, in all three reactions. The method of solution employed mapping onto a model of coagulating positive integer charges. The dynamical rules were synchronous, cellular-automaton type. All the asymptotic large-time results for particle densities were consistent, in the framework of universality, with other model results with different dynamical rules, when available in the literature.
One-dimensional reaction-diffusion models A+A -> 0, A+A -> A, and $A+B -> 0, where in the latter case like particles coagulate on encounters and move as clusters, are solved exactly with anisotropic hopping rates and assuming synchronous dynamics. Asymptotic large-time results for particle densities are derived and discussed in the framework of universality.
We present a method derived from Laplace transform theory that enables the evaluation of fractional integrals. This method is adapted and extended in a variety of ways to demonstrate its utility in deriving alternative representations for other classes of integrals. We also use the method in conjunction with several different techniques to derive many results that have not appeared in tables of integrals.
It appears that, along with many of my friends and colleagues, I had been brainwashed by the great and tragic lives of Abel and Galois to believe that no general formulas are possible for roots of equations higher than quartic. This seemed to be confirmed by the brilliant and arduous solution of the general quintic by Hermite. Yet, below we find a formula giving a root to any algebraic equation of degree 2-5 and any reduced equation (see below) of higher degree. This algorithm, which must have been familiar to Lagrange, resulted when I was working on a paper on the asymptotics of hypergeometric functions where Gauss' multiplication formula for the gamma function is used to reduce certain infinite series, and by a happy accident my copy of Whittaker and Watson opened at p. 133.
A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized hypergeometric function with subsequent reduction to special cases. Connection is made with Weber's second exponential integral and Laplace transforms of products of three Bessel functions.