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THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
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THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
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Abstracts
Some inequalities for polynomials
Richard Askey
Univ. of Wisconsin-Madison
About 40 years ago some inequalities for sums of Jacobi polynomials were proven. The subject seemed to be in a semi-finished state. Recently a surprise arose from a conjecture of Schoberl for a sum of Legendre polynomials and became much more interesting from a proof of a more general result by Pillwein. Some of this will be described, and a few other inequalities will be discussed.
Approximate solution of some boundary value problems of the generalized theory of couple-stress thermodiffusion
Manana Chumburidze
Akakai Tsereteli State University
Coauthors: Valerian Kordzadze, Mathematics and Physics College
We consider a non-classical model of a pseudo oscillations systems of partial differential equations arising in two-dimensional couple-stress thermodiffusion, which was formulated by Green-Lindsay . The matrix of fundamental and singular solutions for isotropic homogeneous elastic media are received. They are presented in elementary functions, what is important for effective construction and computation of basic solutions. Approximate solutions of some boundary value problems for a finite domain, bounded by some closed surface are constructed. They are solved explicitly by using the methods: generalized Fourier series method, potential method and singular integral equations.
Twenty plus years of the Lambert W function
Robert Corless
University of Western Ontario
The Lambert W function is now part of the normal lexicon of so-called special functions. Its origin goes back to Euler, not Lambert, though Euler did mention Lambert in his title; it’s been used and ‘reinvented’ in a wide range of applications, although until it was named and standardized this wasn’t appreciated. This talk will survey some known results and properties, and talk about some remaining puzzles.
Entropy, Eigenstructure, and 1/f Fluctuation
Priscilla Greenwood
University of British Columbia
Coauthors: Ralph Chamberlin
Entropy, Eigenstructure, and 1/f Fluctuation Terrell L. Hill (1917-2014), enjoyed almost the identical long lifespan as did Lee Lorch. Hill laid the foundations of small-system thermodynamics, or nanothermodynamics. Ralph V. Chamberlin, at Arizona State University, continuing Hill's work, has found a mechanism connecting nanothermodynamics and 1/f fluctuations using stochastic simulations. It appears that non-extensive (nonlinear) dependencies introduced by moving from the usual thermodynamic limit with an infinite thermal bath, to a small system with finite bath, are reflected in fluctuations moving from Gaussian white noise with independent increments to "1/f noise" with dependent increments. A corresponding maximum entropy condition is satisfied, which can be shown, using the eigenstructure of Markov processes, to be equivalent to the presence of 1/f fluctuations. An explanation of the ubiquity of 1/f noise in physical and biological systems is suggested in terms of this understanding of maximum entropy. This is joint work with Ralph Chamberlin.
A High Order Numerical Treatment of Operators Arising in Scattering from Periodic Penetrable Media
Michael Haslam
York University
The problem of evaluating the electromagnetic response of a periodic surface to an incident plane wave is of great importance in science and engineering. Applications of the theory exist in several fields of study including solar energy research, optical instrument design, and remote sensing. We discuss the extension of our previous methods to treat the problem of scattering from metallic surfaces with a complex refractive index. The generalization of our methods is not straight-forward, and involves the careful treatment of certain hyper-singular operators which arise in the formulation of the problem in terms of surface integral equations. We demonstrate the rapid convergence of our methods for classically difficult cases in the optical sciences.
How hard is it to compute a finite sum?
Alexey Kuznetsov
York University
It should come as no surprise that evaluating a finite sum of N numbers typically requires at least N-1 additions. Some finite sums can be computed explicitly and thus require much less effort: for example, 1+2+...+N=N(N+1)/2 requires only four arithmetic operations, while 1+x+x^2+...+x^N=(x^{N+1}-1)/(x-1) needs seven of them. In this talk I will discuss a fast algorithm for evaluating a finite quadratic-exponential sum, having N terms of the form x_n=exp(ian^2+ibn). There is no known explicit formula for this finite sum, yet it can be computed in poly-logarithmic (in N) number of arithmetic operations. I will also explain how this result fits into a simple and very efficient algorithm (due to G. Hiary) for fast evaluation of the Riemann zeta function on the critical line.
The distribution of the Riemann zeta function in the critical strip
Youness Lamzouri
York University
Coauthors: Steve Lester (KTH Sweden) and Maksym Radziwill (Rutgers University)In joint work with Steve Lester and Maksym Radziwill, we investigate the extent to which the distribution of values of the Riemann zeta function $\zeta(s)$ can be approximated by that of a corresponding probabilistic random model in the strip $1/2<\text{Re}(s)\leq 1$. As an application, we obtain the first effective error term for the number of $a$-points of $\zeta(s)$ (defined as the roots of $\zeta(s)=a$)
in a strip $1/2<\sigma_1<\sigma_2<1$. We also study the joint distribution of shifts of $\zeta(s)$ and use it to improve Voronin's celebrated universality theorem for the zeta function.
Evolution of Mathematical functions in the Wolfram Language
Oleg Marichev
Wolfram Research Inc.
The version 1.0 of the Mathematica system was released in 1988. It had 114 special functions. The current version, Version 10 has about 300 mathematical functions, more than any other software system. The Version 10 of the Wolfram Language introduced the fully integrated Wolfram Knowledge base. A variety of quantitative information about cities, people, chemicals and mathematical functions is available in computational form.
In this talk we demonstrate the main steps that lead to the current breadth and depth of computational capabilities and knowledge about special functions in the Wolfram Language: the beginnings, the Wolfram Functions Site (functions.wolfram.com), the Functions posters, Wolfram|Alpha (on iphone, etc.), and the current
MathematicalFunctionData. Our realization of the Askey scheme for orthogonal
polynomials will serve as a prototype of the descriptions of inter-relations between
mathematical functions.
Spectral theory of general Sturm-Liouville equations
Angelo B. Mingarelli
Carleton UniversityWe give a status report on a very old problem, that of completely understanding the spectrum of non-definite Sturm-Liouville problems on a finite interval. We will begin with an historical overview, present the main characters, and discuss current work in the area. Difficulties, conjectures, and open questions will be discussed. The talk should be accessible to non-specialists and graduate students alike.
The Heat Kernel of the Sub-Laplacian on the Non-Isotropic Heisenberg Groups with Multi-Dimensional Centers
Shahla Molahajloo
Institute for Advanced Studies in Basic Sciences, IRAN
Coauthors: M.W. WongWe first introduce non-isotropic Heisenberg groups with multi-dimensional centers and the corresponding Schrödingier representations. The Wigner and Weyl transforms are then defined. We compute the sub-Laplacian on the non-isotropic Heisenberg group. By taking the inverse Fourier transform with respect to the center we get the parametrized twisted Laplacians. Then by means of the special Hermite functions we are able to find the eigenfunctions and the eigenvalues of the twisted Laplacians. The explicit formulas for the heat kernels and Green functions of the twisted Laplacians can then be obtained. Then we give an explicit formula for the heat kernal and Green function of the sub-Laplacian on the non-isotropic Heisenberg group with multi-dimensional center.
The Helgason-Fourier Transform Associated to the Weighted Laplace-Beltrami Operator on the Hyperbolic Unit Ball
Lizhong Peng
School of Mathetical Sciences, Peking University, China
Coauthors: Congwen LiuThe harmonic analysis associated to the weighted Laplace-Beltrami operator on the hyperbolic unit ball is established. The associated weighted Helgason-Fourier transform and the corresponding spherical transform are defined and studied. In particular, the inversion formula and partial Plancherel theorem are obtained.
On two-one, two-three and more formulas for multiple zeta (star) values and their q-analogs
Tatiana Hessami Pilehrood
Fields InstituteIn recent years there has been intensive research on the ${\mathbb Q}$-linear relations between multiple zeta (star) values. The two-one formula for multiple zeta star values was conjectured by Ohno and Zudilin in 2007 and proved recently by J.~Zhao. Its $q$-analog was established very recently by Hessami Pilehroods. In this talk, we will discuss many families of identities involving the $q$-analogs
of zeta values, from which we can always deduce the corresponding results for classical multiple zeta values by taking $q\to 1$. We will explain how to obtain further generalizations of the two-one formula to arbitrary multiple zeta star values by building up from some base cases. Based on joint works with Khodabakhsh Hessami Pilehrood, Roberto Tauraso and Jianqiang Zhao.
Best Guaranteed Result Principle and Decision Making in Operations with Stochastic Factors and Uncertainty
Iouldouz Raguimov
Department of Mathematics and Statistics, York University
We introduce the best guaranteed result principle and study decision making in operations with stochastic factors and uncertainty. We introduce the ideas of ïnformation hypothesis" and "decision-making turn". Various scenarios of game situations with different turn sequences of decision making and voluntary information exchange between the players are investigated. We analyze decision making in operations with stochastic factors under the assumption that the probabilities are defined in terms of the corresponding frequencies. In our setting, the possibility of using stochastic information about an uncontrollable factor is connected with the possibility (and intelligibility) of transition from the original operation to an extended one with infinitely many repetitions. It is shown that if our criterion in the extended operation is the maximization of the lower limit of the average value of an original criterion, then our best guaranteed result is determined as a solution to a certain optimization problem and the optimal strategy will be split into choosing, at each step of the extended operation, some optimal solution to that optimization problem. Similar results for the problem of maximization of the probability of exceeding a given level, as well as for the problem of maximization of a level with a given guarantee are obtained. We consider applications of the results obtained to a two-stage stochastic programming problem.
Asymptotic Methods for Integrals, Special Functions and Examples for Finding their Zeros
Nico M Temme
Centrum Wiskunde & Informatica, Amsterdam, The NetherlandsWe give examples in which special functions, such as the complementary error function and Airy functions can be used as approximations in uniform asymptotic expansions of integrals. As an introduction, we start with the standard methods of asymptotic analysis for integrals, such as Watson's lemma and the method of stationary phase. We describe applications from physics. To point out connections with Lee Lorch's extensive research interest in the properties of zeros of special functions, in particular we indicate how uniform expansions can be used to obtain global and detailed information on the location of the zeros of these functions.
Properties of special cases of the Wright function arising in probability theory
Vladimir Vinogradov
Ohio University
Coauthors: Richard B. Paris (University of Abertay Dundee)We derive previously unknown properties of some special cases of the Wright function whose consideration is necessitated by our work on probability theory and the theory of stochastic processes. Specifically, we establish new asymptotic properties of the particular Wright function ${}_1\Psi_1(\rho,k;\rho,0;x)=\sum_{n=1}^{\infty}(\Gamma(k+\rho n)/\Gamma(\rho n))x^n/n!$ when the parameter $\rho\in(-1,0)\cup(0,\infty)$ and the argument $x$ is real. In the probability theory applications, which are focused on studies of the Poisson-Tweedie mixtures, the parameter $k$ is a non-negative integer. Several representations involving well-known special functions are given for certain particular values of $\rho$. The asymptotics of ${}_1\Psi_1(\rho,k;\rho,0;x)$ are obtained under numerous assumptions on the behavior of the arguments $k$ and $x$ when the parameter $\rho$ is both positive and negative. We also provide some integral representations and structural properties involving the reduced Wright function ${}_0\Psi_1(---;\rho,0;x)$ with $\rho\in(-1,0)\cup(0,\infty)$, which might be useful for the derivation of new properties of members of the power-variance family of distributions. Some of these imply a reflection principle that connects the functions ${}_0\Psi_1(---;\pm \rho,0;\cdot)$ and certain Bessel functions. Several asymptotic relationships for both particular cases of this function are also given. A few of these follow under additional constraints from probability theory results which, although previously available, were unknown to analysts.
Paper reference: arXiv:1508.00863
Existence of stationary solutions for some integro-differential equations with anomalous diffusion
Vitali Vougalter
University of Toronto, Department of Mathematics, Toronto, ON, M5S 2E4, Canada
Coauthors: Vitali Vougalter, Vitaly VolpertThe article deals with the existence of solutions of an integro-differential equation arising in population dynamics in the case of anomalous diffusion involving the negative Laplace operator raised to a certain fractional power. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for non-Fredholm elliptic operators in unbounded domains along with the Sobolev inequality for a fractional Laplacian are being used.
Pseudo-differential operators on finite abelian groups
K. L. Wong
York University
Coauthors: Shahla MolahajlooWe give the basic theory of pseudo-differential operators on finite abelian groups. In the case of a group with two elements, we give a criterion for invertibility of these operators and we also give a solution of the spectral invariance problem for these operators.
Lebesgue Constants and Szegö's Conjecture
Roderick Wong
City University of Hong Kong, China
Lebesgue constants $\rho_n$ are well known in the theory of Fourier series. In 1910, Fejer conjectured that these constants form a monotonically increasing sequence. Shortly afterwards, this conjecture was proved by Gronwall (1912). In 1926, Szegö conjectured that the Lebesgue constants $L_n$ for the Legendre series also form a monotonically increasing sequence. But this conjecture was not settled until 1988 by a very complicated asymptotic analysis. The solution of this problem has a lot to do with Professor Lee Lorch. In this talk we present a historical account of the development of this problem.
For a better life: the role of analysis in modern medicine
Hongmei Zhu
York UniversityToday the concepts and theories of mathematical analysis are being applied to many aspects of medicine from designing imaging systems to processing and analyzing medical data. In this talk, we will illustrate examples of analysis successfully applied in clinical health research, from which one can experience the increasing importance of mathematics in digital-age medical diagnosis and health care.
