In a surprising sequence of developments, the longest increasing subsequence problem, originally mentioned as merely a curious example in a 1961 paper, has proven to have deep connections to many seemingly unrelated branches of mathematics, such as random permutations, random matrices, Young tableaux, and the corner growth model The detailed and playful study of these connections makes this book suitable as a starting point for a wider exploration of elegant mathematical ideas that are of interest to every mathematician and to many computer scientists, physicists and statisticians The specific topics covered are the Vershik-Kerov–Logan-Shepp limit shape theorem, the Baik–Deift–Johansson theorem, the Tracy–Widom distribution, and the corner growth process This exciting body of work, encompassing important advances in probability and combinatorics over the last forty years, is made accessible to a general graduate-level audience for the first time in a highly polished presentation

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The Surprising Mathematics of Longest Increasing Subsequences

Analytical modeling of PCM solidification in a shell and tube finned thermal storage for air conditioning systems

Helmholtz resonator is often used to reduce noise in a narrow frequency range. To obtain a broader noise attenuation band, combing several resonators is a possible way. This paper presents a theoretical study of sound propagation in a one-dimensional duct with identical side-branch resonators mounted periodically. The analysis of each resonator was based on a distributed-parameter model that considered multi-dimensional wave propagation in its neck-cavity interface. This model provided a more accurate prediction of the resonant frequency of the resonator than traditional lumped-parameter model. Bloch wave theory and the transfer matrix method were used to investigate wave propagation in these spatially periodic resonators. The results predicted by the theory fit well with the computer simulation using a three-dimensional finite element method and the experimental results. This study indicates that the wave coupling in this periodic system results in the dispersion of the frequency band into the stop and the pass bands. The long-term significance is that periodic resonators may more effectively control noise in ducts by broadening the bandwidth they attenuate and increasing the magnitude of sound attenuation.

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Wave propagation in a duct with a periodic Helmholtz resonators array

We provide Karhunen—Loeve expansions on (0, 1) for the processes t θ B (tρ)and t θ W (tθ)where B (•) a Brownian bridge, W(•) is a Wiener process, and ρ and θ are arbitrary real numbers such that θ > −(ρ + 1)/2. The eigenfunctions of these expansions have simple expressions in terms of Bessel functions J v(•) and Jv-1(•) of the first kind with indexes ν = ρ/(2θ + ρ + 1) and ν − 1 = −(2θ + 1)/(2θ + ρ + 1). The corresponding eigenvalues have simple expressions in terms of the positive zeros z ν,1 0, a Fourier-Bessel expansion on (0, 1) of t −ν B(t 2ν ) and a Dini expansion on (0, 1) of t —ν W(t 2ν). As a consequence, we obtain explicit forms for the distribution functions of the L2(0, 1) norms oft θ W (tρ)andt θ B (tρ)whose upper quantiles are tabulated in part. These distributions are also shown to be identical up to a multiplicative constant after the change of v into v + 1. We provide sharp evaluations for small deviations as well as for large deviations for these L2(0,1) norms. This allows us to obtain simple proofs of the Rayleigh extensions of Euler’s formula involving the positive zeros of Jv(•), an example of which is given by the identity \(\sum _{{k = 1}}^{\infty }1/z_{{\nu ,k}}^{2} = 1/(4(\nu + 1))\) for ν>−1.

https://link.springer.com/content/pdf/10.1007/978-3-0348-8059-6_4.pdf

Karhunen-Loève Expansions for Weighted Wiener Processes and Brownian Bridges via Bessel Functions

The subject of special functions is often presented as a collection of disparate results, rarely organized in a coherent way. This book emphasizes general principles that unify and demarcate the subjects of study. The authors' main goals are to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more. It shows how much of the subject can be traced back to two equations - the hypergeometric equation and confluent hypergeometric equation - and it details the ways in which these equations are canonical and special. There is extended coverage of orthogonal polynomials, including connections to approximation theory, continued fractions, and the moment problem, as well as an introduction to new asymptotic methods. There are also chapters on Meijer G-functions and elliptic functions. The final chapter introduces Painleve transcendents, which have been termed the 'special functions of the twenty-first century'.

Special Functions and Orthogonal Polynomials

Foundation of the Theory of Bessel Functions Bessel Equation. Properties of Bessel Functions. Definite and Improper Integrals. Series in Bessel Functions. Applications of Bessel Functions. Problems of the Theory of Plates and Shells. Problems of the Theory of Oscillations, Hydrodynamics and Heat Transfer. Appendix A. Brief Information on Gamma Functions.

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B G Korenev

Papers

Bessel Functions and Their Applications