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Table Of Integrals Series And Products

Fractional Differential Equations

Tables of integrals

In this correspondence, we investigate the effect of channel estimation error on the capacity of multiple-input-multiple-output (MIMO) fading channels. We study lower and upper bounds of mutual information under channel estimation error, and show that the two bounds are tight for Gaussian inputs. Assuming Gaussian inputs we also derive tight lower bounds of ergodic and outage capacities and optimal transmitter power allocation strategies that achieve the bounds under perfect feedback. For the ergodic capacity, the optimal strategy is a modified waterfilling over the spatial (antenna) and temporal (fading) domains. This strategy is close to optimum under small feedback delays, but when the delay is large, equal powers should be allocated across spatial dimensions. For the outage capacity, the optimal scheme is a spatial waterfilling and temporal truncated channel inversion. Numerical results show that some capacity gain is obtained by spatial power allocation. Temporal power adaptation, on the other hand, gives negligible gain in terms of ergodic capacity, but greatly enhances outage performance.

Capacity and power allocation for fading MIMO channels with channel estimation error

We consider the solution of the stochastic heat equation @TZ D 1 @ 2 X ZZ P W with delta function initial condition Z.T D0;X/ DiXD0 whose logarithm, with appropriate normalization, is the free energy of the con- tinuum directed polymer, or the Hopf-Cole solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We obtain explicit formulas for the one-dimensional marginal distributions, the crossover distributions, which interpolate between a standard Gaussian dis- tribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest-descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion process with antishock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes. © 2010 Wiley Periodicals, Inc.

/pdf/probability-distribution-of-the-free-energy-of-the-continuum-2h6gr3boio.pdf

Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions

Preface. Acknowledgements. 1. Introduction and Preliminaries. 2. The Zeta and Related Functions. 3. Series Involving Zeta Functions. 4. Evaluations and Series Representations. 5. Determinants of the Laplacians. 6. Miscellaneous Results. Bibliography. Author Index. Subject Index.

Series Associated with the Zeta and Related Functions

Some q-extensions of the Apostol–Bernoulli and the Apostol–Euler polynomials of order n, and the multiple Hurwitz zeta function

Recently several authors have extended the Gamma function, Beta function, the hypergeometric function, and the con-fluent hypergeometric function by using their integral representa-tions and provided many interesting properties of their extended functions. Here we aim at giving further extensions of the above-mentioned extended functions and investigating various formulas for the further extended functions in a systematic manner. More-over, our extension of the Beta function is shown to be applied to Statistics and also our extensions find some connections with other special functions and polynomials such as Laguerre polynomials, Macdonald and Whittaker functions.

Extension of extended beta, hypergeometric and confluent hypergeometric functions

https://www.tpbin.com/Uploads/Subjects/5835a810-1ee9-4ad4-9ce1-2abd7fac4ca0.pdf

Junesang Choi

Papers

Series Associated with the Zeta and Related Functions

Some q-extensions of the Apostol–Bernoulli and the Apostol–Euler polynomials of order n, and the multiple Hurwitz zeta function

Extension of extended beta, hypergeometric and confluent hypergeometric functions

Some summation formulas involving harmonic numbers and generalized harmonic numbers

Certain Classes of Series Involving the Zeta Function