Analytical Solving a Nonlinear Model of the Glow Discharge

A nonlinear model has been introduced for the positive column of DC glow discharge in apure sealed, or low flow, gas media by including the diffusion, recombination, attachment, detachment,process and having the two-step ionization process of the metastable excited states, too. By thecombination of the system of the nonlinear continuity equations of the system, using some physicalestimations, and degrading the resulted nonlinear PDE in polar and rectangular systems of coordinatethe steady-state nonlinear ODE have been derived. Using a series-based solution, an innovativenonlinear recursion relation has been proposed for calculating the sentence of series. Using the stateof elimination of free charge on the outer boundary of the discharge vessel, the universal equation ofthe characteristic energy of the electrons versus the similarity variable, using the maximum degree ofionization as the parameter, has been derived.

The quantum-mechanical Coulomb propagator in an L2 function representation

The quantum-mechanical Coulomb propagator is represented in a square-integrable basis of Sturmian functions. Herein, the Stieltjes integral containing the Coulomb spectral function as a weight is evaluated. The Coulomb propagator generally consists of two parts. The sum of the discrete part of the spectrum is extrapolated numerically, while three integration procedures are applied to the continuum part of the oscillating integral: the Gauss–Pollaczek quadrature, the Gauss–Legendre quadrature along the real axis, and a transformation into a contour integral in the complex plane with the subsequent Gauss–Legendre quadrature. Using the contour integral, the Coulomb propagator can be calculated very accurately from an L$$^2$$ 2 basis. Using the three-term recursion relation of the Pollaczek polynomials, an effective algorithm is herein presented to reduce the number of integrations. Numerical results are presented and discussed for all procedures.

On the Evaluation of Riemann Zeta Functions for Even Integers

A recursive method for obtaining the Zeta function for even integers is obtained starting from the Fourier series expansion of the function f(x) = x. Repeating the method after term by term integration yields the final, simplified closed form that happens to be a recursion relation. Using the obtained recursion relation, one can successively evaluate the values of zeta functions for even integers.

Notes on gravity multiplet correlators in AdS3 × S3

We present a compact formula in Mellin space for the four-point tree-level holographic correlators of chiral primary operators of arbitrary conformal weights in (2, 0) supergravity on AdS3× S3, with two operators in tensor multiplet and the other two in gravity multiplet. This is achieved by solving the recursion relation arising from a hidden six-dimensional conformal symmetry. We note the compact expression is obtained after carefully analysing the analytic structures of the correlators. Various limits of the correlators are studied, including the maximally R-symmetry violating limit and flat-space limit.

Calculations of the Average Number of Radicals per Particle in Emulsion Polymerization

In emulsion polymerization, the free radicals enter the particles intermittently from the aqueous phase. The number of radicals per particleis given by the Smith-Ewart recursion relation which balances the rate of radical entry into, the rate of radical exit from and the rate of radical termination inside the particle. Models for emulsion polymerisation are based on the 0-1 kinetics or the pseudo-bulk kinetics. Small particles, low initiator concentrations and large number of particles favour the 0–1 kinetics, whereas the large particles, high initiator concentrations and small number of particles will favour pseudo-bulk kinetics. A given polymerization system may exhibit both these kinetic behaviours, initially following the 0-1 kinetics and during the later stages of polymerization following the pseudo-bulk kinetics. The aim of this work is to calculate the time dependent values of the average number of radicals per particle in emulsion polymerization for the pseudo-bulk kinetics.

Recursion relation for instanton counting for SU(2) $$ \mathcal{N} $$ = 2 SYM in NS limit of Ω background

In this paper we investigate different ways of deriving the A-cycle period as a series in instanton counting parameter q for $$ \mathcal{N} $$ N = 2 SYM with up to four antifundamental hypermultiplets in NS limit of Ω background. We propose a new recursive method for calculating the period and demonstrate its efficiency by explicit calculations. The new way of doing instanton counting is more advantageous compared to known standard techniques and allows to reach substantially higher order terms with less effort. This approach is applied for the pure case as well as for the case with several hypermultiplets.In addition we suggest a numerical method for deriving the A-cycle period for arbitrary values of q. In the case when one has no hypermultiplets for the A-cycle an analytic expression for large q asymptotics is obtained using a conjecture by Alexei Zamolodchikov. We demonstrate that this expression is in convincing agreement with the numerical approach.

Double Hurwitz Numbers and Multisingularity Loci in Genus 0

In the Hurwitz space of rational functions on ${{\mathbb{C}}}\textrm{P}^1$ with poles of given orders, we study the loci of multisingularities, that is, the loci of functions with a given ramification profile over 0. We prove a recursion relation on the Poincaré dual cohomology classes of these loci and deduce a differential equation on Hurwitz numbers.

RECURSION FOR MASUR-VEECH VOLUMES OF MODULI SPACES OF QUADRATIC DIFFERENTIALS

We derive a quadratic recursion relation for the linear Hodge integrals of the form $\langle \tau _{2}^{n}\lambda _{k}\rangle $ . These numbers are used in a formula for Masur-Veech volumes of moduli spaces of quadratic differentials discovered by Chen, Möller and Sauvaget. Therefore, our recursion provides an efficient way of computing these volumes.