Fall 2017 :

Organizer : Oleksandr Tovstolis

• Localized Transfunctions
  Tuesday, Nov. 28 , 3:00 PM ~ 4:00 PM, MSB 318
  Speaker : Jason Bentley , University of Central Florida
  Abstract : Localized transfunctions are functions that map measures in small balls from one measurable space to measures in small balls in another measurable space (e.g. measurement reading errors before and after some physical process). We treat localized transfunctions with nice properties as generalized functions between the underlying spaces and we develop some theory which gives important characterizations and approximations. We finally present potential applications where transfunctions might play a future role.
• On the Importance of Simple Functions
  Tuesday, Nov. 21 , 3:00 PM ~ 4:00 PM, MSB 318
  Speaker : Piotr Mikusinski , University of Central Florida
• Orthonormal bases generated by Cuntz isometries
  Tuesday, Nov. 14 , 3:00 PM ~ 4:00 PM, MSB 318
  Speaker : Dorin Dutkay , University of Central Florida
  Abstract : We show how some isometries (known as “Cuntz isometries”) can be used to generate some well known orthonormal bases such as Fourier series, Walsh bases, wavelet bases and some new examples which include Fourier series on fractals and generalized Walsh bases.
• “Between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain.”
  P. Painleve, 1900.
  Variations on the theme of analytic continuation
  Tuesday, Nov. 7, 3:00 PM ~ 4:00 PM, MSB 318
  Speaker : Dmitry Khavinson , University of South Florida
  Abstract : Why does the Taylor series \sum cos(\sqrt(n))z^n extend to the whole complex plane except for the point 1? And why \sum 2^{-n}z^{n^2) does not extend anywhere beyond the unit circle?
  How far does the Newtonian potential of a solid bounded by an algebraic surface extend inside the solid? Why is the celebrated Schwarz reflection principle never discussed in dimensions higher than 2? How does one find singularities of an axially symmetric harmonic function in the ball from the coefficients in its expansion by spherical harmonics? If a line intersects a domain over two disjoint segments and a harmonic function in the domain vanishes on one, does it have to vanish on the other one?
  We shall discuss these questions in the unified light of analytic continuation, and, in particular , analytic continuation of solutions to linear analytic pde. The talk will be accessible to undergraduate and graduate students majoring in math, physics and engineering.
• A boundary value problem for the 1-Laplacian in conductivity imaging
  Tuesday, Oct. 31, 3:00 PM ~ 4:00 PM, MSB 318
  Speaker : Alexandru Tamasan , University of Central Florida
• Dead core equations
  Tuesday, Oct. 24, 3:00 PM ~ 4:00 PM, MSB 318
  Speaker : Eduardo Teixeira , University of Central Florida
• Construction of Non-Atomic, Regular, Strictly-Positive Measures in Locally-Compact Non-Atomic Polish Spaces
  Tuesday, Oct. 10, 3:00 PM ~ 4:00 PM, MSB 318
  Speaker : Jason Bentley , University of Central Florida
  Abstract : We present a constructive proof of the existence of a regular non-atomic measure that is positive on all non-empty open sets of a locally compact non-atomic Polish space. We construct a sequence of finitely-additive set functions defined recursively on an ascending sequence of rings of subsets and obtain a measure with the desired properties as the extension of a pre-measure defined as the limit of the sequence of set functions.
• On the theory of nonphysical free boundaries
  Tuesday, Oct. 3, 3:00 PM ~ 4:00 PM, MSB 318
  Speaker : Eduardo Teixeira , University of Central Florida
• Asymptotics of greedy energy sequences on the unit circle
  Tuesday, Sept. 19, 3:00 PM ~ 4:00 PM, MSB 318
  Speaker : Abey Lopez-Garcia , University of Central Florida
• The Magic Power of Askey-Wilson Operator in Deriving Series Identities
  Tuesday, Sept. 5, 3:00 PM ~ 4:00 PM, MSB 318
  Speaker : Xin Li , University of Central Florida
  Abstract : I will report on some interesting infinite summation identities, some new some known, that can be discovered by a new representation formula for the Askey-Wilson operator.

Spring 2017 :

Organizer : Mourad Ismail

• Localized Transfunctions
  Tuesday, Apr. 17, 2:00 PM ~ 3:00 PM, MSB 318
  Speaker : Jason Bentley, University of Central Florida
  Abstract : A transfunction is any function between sets of finite (positive) measures from their respective measurable spaces, treated as a generalized function between the underlying sets. Transfunctions can have a variety of combinations of properties, such as additivity (strong, weak), homogeneity, norm preservation and boundedness, continuity (weak, monotone, pointwise, norm), etc.. Some topological/metric transfunctions behave locally on the spaces analogous to continuous functions. Others can behave globally without any local emphasis, such as spreading the total domain measure across the entire range space. We develop the beginning of a theory on transfunctions with local properties in order to capture how close a transfunction is to behaving like a continuous or measurable function. Certain types of transfunctions are considered as potential models for future applications. Two recent results will be discussed if time permits.
• Anisotropic Functional Laplace Deconvolution
  Tuesday, Apr. 11, 2:00 PM ~ 3:00 PM, MSB 318
  Speaker : Rasika Rajapakshage , University of Central Florida
  Abstract : The high frequency Dynamical Contrast Enhanced (DCE) imaging techniques are used for various medical assessments such as brain flows, strokes or cancer angiogenesis. The DCE imaging experiments can be described by a collection of Laplace convolution equations based on noisy observations, one equation per unit volume (voxel). Previously, available data curves were preclustered and averaged, and the inverse problem was solved with these secondary data. In the present work, we do not use pre-clustering and just analyze all available data together by expanding the curves over the Laguerre functions basis in time domain and finitely supported wavelets in spatial domain. The resulting solutions have good theoretical properties and show good precision in simulations. In addition, since for each voxel, the solution curve is represented via only few Laguerre coefficients, these curves can be clustered more efficiently than the original curves. We conducted simulation study to show that the estimation works well in finite simulations settings. Finally, we use the technique for the solution of the Laplace deconvolution problem on the basis of DCE Computerized Tomography data.
• A Bernstein-type inequality for the Askey-Wilson operator
  Tuesday, Apr. 4, 2:00 PM ~ 3:00 PM, MSB 318
  Speaker : Rajitha Ranasinghe , University of Central Florida
• Optimal Control Theory, Part I: Classical Results Revisited
  Tuesday, Mar. 21, 2:00 PM ~ 3:00 PM, MSB 318
  Speaker : Jiongmin Yong , University of Central Florida
  Abstract : In this talk, I will briefly revisit the classical theory of optimal control, starting from the branchistochrone problem. I will recall the Pontryagin’s maximum principle, the Bellman’s dynamic programming method and Hamilton-Jacobi-Bellman equation. With this base, I will be able to present results in various possible directions (in following parts).
• Biased coin toss game and Nash equilibrium
  Tuesday, Mar. 7, 2:00 PM ~ 3:00 PM, MSB 318
  Speaker : Peter Dragnev , Indiana-Purdue University
• On the $C^{p’}$-regularity conjecture
  Tuesday, Feb. 21, 2:00 PM ~ 3:00 PM, MSB 318
  Speaker : Eduardo Teixeira , University of Central Florida
  Abstract : While it has been well established that minimizers of convex functionals are of class $C^{1,\alpha}$, determining the optimal H\”older exponent $\alpha$ has been a tantalizing question with major implications to the PDE theory and its applications. For $p$-convex functionals, the explicit example, $\Delta_p (|x|^{p’}) = cte$ sets the ground for what has been termed the $C^{p’}$-regularity conjecture: The optimal H\”older continuity exponent for the gradient of a function whose $p$-laplacian is bounded should be $\frac{1}{p-1}$, at least when $p>2$. In this talk, I will discuss a recent proof of the $C^{p’}$-regularity conjecture in dimension two. If time permites, I will also discuss a program to tackle the conjecture in higher dimensions as well. This is a joint work with Araujo and Urbano.
• R-Continued Fractions, Generalized Eigenvalue Problems, and Biorthogonal Rational Functions
  Tuesday, Feb. 14, 2:00 PM ~ 3:00 PM, MSB 318
  Speaker : Mourad Ismail , University of Central Florida
• q-analogues in analysis, algebra, and combinatorics
  Colloquium, Friday, Feb. 10, 3:00 PM ~ 4:00 PM, MSB 318
  Speaker : Dennis Stanton , University of Minnesota
• Emergent phenomena in nonlinear dispersive waves: semiclassical and large time limits of integrable PDE
  Tuesday, Feb. 7, 2:00 PM ~ 3:00 PM, MSB 318
  Speaker : Robert Jenkins , University of Arizona
• Inverse Bernstein Inequality of Erdelyi-Hardin-Saff for Rational Functions
  Tuesday, Jan. 31, 2:00 PM ~ 3:00 PM, MSB 318
  Speaker : Xin Li , University of Central Florida
• Asymptotic Integration Method, A Rigorous Treatment
  Tuesday, Jan. 24, 2:00 PM ~ 3:00 PM, MSB 318
  Speaker : Mourad Ismail , University of Central Florida

Fall 2016 :

Organizer : Zhe Liu

• Transfunctions
  Monday, Nov. 28, 3:30 PM ~ 4:30 PM, MSB 318
  Speaker : Piotr Mikusinski , University of Central Florida
  Abstract : We propose a new framework for generalizing functions. The motivation for our definition is the fact that in real world the input is rarely a precise element of the domain and the same is true about the output. We prefer to think of the input as a “set of points with probabilities” and the output as a “set of outputs with probabilities”. More precisely, we are mapping measures in the domain space to measures in the target space. For mathematical convenience, we are not restricting the measures in the domain and range to probability measures, but instead we consider arbitrary finite measures. We call these objects transfunctions.
• Orthogonal Matrix Retrieval in Cryo-electron Microscopy
  Monday, Nov. 21, 3:30 PM ~ 4:30 PM, MSB 318
  Speaker : Teng Zhang , University of Central Florida
  Abstract : In single particle reconstruction (SPR) from cryo-electron microscopy (EM), the 3D structure of a molecule needs to be determined from its 2D projection images taken at unknown viewing directions. Zvi Kam showed already in 1980 that the autocorrelation function of the 3D molecule over the rotation group SO(3) can be estimated from 2D projection images whose viewing directions are uniformly distributed over the sphere. The autocorrelation function determines the expansion coefficients of the 3D molecule in spherical harmonics up to an orthogonal matrix. We will show how techniques for solving the phase retrieval problem in X-ray crystallography can be modified for the cryo-EM setup for retrieving the missing orthogonal matrices. Specifically, we present two new approaches that we term Orthogonal Extension and Orthogonal Replacement, in which the main algorithmic components are the singular value decomposition and semidefinite programming. We demonstrate the utility of these approaches through numerical experiments on simulated data. This talk is based joint works with Tejal Bhamre and Amit Singer, available athttps://arxiv.org/abs/1412.0494, https://arxiv.org/abs/1506.02217, and http://arxiv.org/abs/1602.06632.
• Noncommutative methods in several complex variables
  Monday, Nov. 14, 3:30 PM ~ 4:30 PM, MSB 318
  Speaker : Michael Jury , University of Florida
  Abstract : I will describe a “noncommutative” version of the multivariable Cauchy (Fantappie) transform, in which instead of integrating a scalar-valued kernel against a measure, we form an operator-valued kernel and take its image under a state on a C*-algebra. I will describe some classes of analytic functions on the unit ball of C^n that are representable in this way, and give some applications to function theory in the unit ball (in particular a generalization of the so-called Aleksandrov-Clark measures to this setting). This talk is based on joint work with Robert T.W. Martin.
• Riemann-Hilbert problems and Finite Hilbert Transforms with Ap-
  plications to Medical Imaging
  Monday, Nov. 7, 3:30 PM ~ 4:30 PM, MSB 318
  Speaker : Elliot Blackstone, University of Central Florida
  Abstract : In this talk, we discuss some relationships between medical imaging, finite Hilbert transforms(FHT), ODEs and Riemann-Hilbert problems(RHP).
  More specifically, take the x-axis with four points, -\infty < a1 < a2 < a3 < a4 <\infty, and consider the FHT which map L2 ([a1; a2]) –> L2 ([a3; a4]) and vice-versa. The singular value decomposition of the FHT can be obtained from both an ODE approach and RHP approach. We study the changes when the points a2 approaches a3. The notable dierence is that the spectrum of the FHT shifts from discrete to continuous as a2 –> a3.
• Propagation failure of traveling waves in lattice equations
  Monday, Oct. 24, 3:30 PM ~ 4:30 PM, MSB 318
  Speaker : Brain Moore, University of Central Florida
  Abstract : Various techniques, including Fourier transforms, Jacobi operator theory, and backward error analysis, provide means to construct traveling wave solutions for several discrete reaction-diffusion equations and discrete semi-linear wave equations. The results supply, wave speed estimates, and necessary and sufficient conditions for fronts and pulses to fail to propagate due to inhomogeneities in the medium, as well as confirmation that certain discretizations reproduce the qualitative solution behavior of the corresponding partial differential equations.
• The Asymptotic Integration Method (AIM) 
  Monday, Oct. 17, 3:30 PM ~ 4:30 PM, MSB 318
  Speaker : Mourad Ismail, University of Central Florida
  Abstract : Will survey the AIM technique developed around 2003 and mention an extension to difference equations.
• A spatial-temporal model for innate immune response to Borrelia burgdorferi 
  Analysis & Mathematical Biology Joint Seminar, Monday, Oct. 3, 3:30 PM ~ 4:30 PM, MSB 318
  Speaker : Peng Feng, Florida Gulf Coast University
  Abstract : In this talk, I will introduce the basic principles behind mathematical immunology. Then we will discuss a few temporal and spatio-temporal models that describe how our immune system responds to various pathogens. We also establish a PDE chemotaxis model for the innate response to Borrelia burgdorferi, the causative agent of Lyme disease. We illustrate the key factors that lead to the characteristic skin rash that often associated with Lyme disease. We finish the talk with a few comments regarding modeling in immunology.
• Existence of Fast-decay ground state for a weakly-coupled elliptic system 
  Lecture I (Basic Theory), Monday, Sept. 19, 3:30 PM ~ 4:30 PM, MSB 318
  Lecture II (Advanced Theory), Monday, Sept. 26, 3:30 PM ~ 4:30 PM, MSB 318
  Speaker : Yuanwei Qi, University of Central Florida
  Abstract : In this talk, I shall present some of the most recent works I have done on a nonlinear and weakly coupled Elliptic system which arises from the corresponding parabolic system when consider the self-similar solutions. (This is a joint work with Xinfu Chen of U. Pittsburgh, Zhi Zheng and Shulin Zhou of Peking U.)
• Spatially distributed sampling and reconstruction of signals on a Graph 
  Monday, Sept. 12, 3:30 PM ~ 4:30 PM, MSB 318
  Speaker : Cheng Cheng, University of Central Florida
  Abstract : A spatially distributed system contains a large amount of agents with limited sensing, data processing, and communication capabilities. Recent technological advances have opened up possibilities to deploy spatially distributed systems for signal sampling and reconstruction. In this talk, I will discuss a graph structure for a distributed sampling and reconstruction system, distributed-verifiable criterion for system stability and distributed algorithms for signal reconstruction.
• Operator Spaces – A Natural Quantization of Banach Spaces
  Colloquium, Friday, Sept. 9, 3:00 PM ~ 4:00 PM, MSB 318
  Speaker : Zhong-Jin Ruan, University of Illinois at Urbana-Champaign
  Abstract : An operator space can be (concretely) defined to be a norm closed subspace of B(H), the space of all bounded linear operators on a Hilbert space H, together with a distinguished operator matrix norm inherited from B(H). Morphisms on operator spaces are completely bounded linear maps. During the last three decades, the theory has been developed into an important research area in modern analysis. In this talk, I plan to review some fundamental results of operator spaces and show some interesting applications to related areas, such that operator algebras and abstract harmonic analysis.