Mathematical Data Science Seminar
Math Data Science
This seminar welcomes faculty and students from the Pikes Peak region who are interested in mathematical aspects of data science and machine learning, and more broadly, optimization, scientific computation, modeling and simulations. Formerly known as the AaA seminar, it is intended to have a very informal format, with several seminars in the format of Math Clinic workshops rather than lecture format, introducing the audience to topics that are of current interest in the field.
Please contact Dr. Radu Cascaval (radu@uccs.edu) if you are interested to join this seminar or need more info. Limited number of parking passes will be made available to nonUCCS individuals attending this seminar.
Fridays 12:151:30pm, ENG 247 or ENG 101, UCCS campus
(refreshments available at 12:15pm, talks start at 12:30pm)
Fall 2024  
Sept 13  Radu Cascaval UCCS  An Introduction to PINNs (PhysicsInformed Neural Networks) 
Sept 20  Troy Butler CU Denver and NSF  Transforming Displacements to Distributions with a MachineLearning Framework 
Oct 4  Mihno Kim Colorado College  An Introduction to Handling Missing Values 
Oct 11 & Oct 18  UCCS Math Clinic  Python Workshops 
Nov 1  Meng Li Data Scientist, Booz Allen Hamilton  Mathematics in Data Science: Key Concepts and Applications 
Nov 15  Jenny Russell Director, UCCS Institutional Research  How DATA Can Improve the Decision Making Process 
Dec 6  Seth Minor Applied Math, CU Boulder  Discovering ClosedForm Weather Models from Data 
Fall 2024 Abstracts:
Dec 6, 2024
Speaker: Seth Minor, Applied Math Dep, Cu Boulder
Title: Discovering ClosedForm Weather Models from Data
Abstract: The multiscale and turbulent nature of Earth's atmosphere has historically rendered accurate weather modeling a hard problem. Recently, there has been an explosion of interest surrounding datadriven approaches to weather modeling (e.g., GraphCast), which in many cases boasts both improved forecast accuracy and computational efficiency when compared to traditional methods. However, many of the new datadriven approaches employ highly parameterized neural networks, which often result in uninterpretable models and, in turn, a limited gain in scientific understanding. In this talk, we address a current research direction that addresses the interpretability problem in datadriven weather modeling. In particular, we cover a datadriven approach for explicitly discovering the governing PDEs symbolically, thus identifying mathematical models with direct physical interpretations. In particular, we use a weakform sparse regression method dubbed the Weak Sparse Identification of Nonlinear Dynamics (WSINDy) algorithm to learn models from simulated and assimilated weather data.
Nov 15, 2024
Speaker: Jenny Russell, Director, UCCS Institutional Research
Title: How DATA Can Improve the Decision Making Process
Abstract: There is no shortage of data in today’s world – but data alone is not enough. The true value of data emerges when it is communicated in ways that resonate with diverse audiences, especially those who may find data overwhelming or uninteresting. This presentation explores ways to transform data into compelling stories that engage stakeholders, facilitate understanding, and ultimately influence decisions. By utilizing various visualization techniques, narrative structures, and impactful examples, we can break down complex data and provide actionable insights. We will discuss how to make data accessible, relatable, and impactful, regardless of the industry or audience.
Nov 1, 2024
Speaker: Meng Li, Data Scientist, Booz Allen Hamilton
Title: Mathematics in Data Science: Key Concepts and Applications
Abstract: Mathematics forms the backbone of data science, providing the tools and frameworks necessary for creating predictive models, optimizing decisionmaking, and deriving actionable insights from data. This talk offers a highlevel overview of key mathematical methods integral to data science, such as linear algebra for dimensionality reduction, probability and statistics for model assessment, optimization techniques for algorithm training, and calculus for understanding model dynamics. We will explore how these methods are applied across industries, including finance, healthcare, and marketing, illustrating realworld applications from portfolio optimization to customer behavior prediction. This session is designed for both new and experienced data enthusiasts looking to understand the mathematical principles that drive data science innovation.
Oct 4, 2024
Speaker: Mihno Kim, Colorado College
Title: An Introduction to Handling MIssing Values
Abstract: Almost all reallife data contains missing values, yet most modeling techniques were developed based on having complete data. While accounting for the missing values is a popular research area, they are a greatly overlooked problem in practice because it is a preliminary step before the primary analysis. This talk will address the importance of properly handling the missing values. Different types of missing values will be introduced, along with several techniques that are easy to use.
Sept 20, 2024
Speaker: Troy Butler, CU Denver and NSF
Title: Transforming Displacements to Distributions with a MachineLearning Framework
Abstract: In general, any uncertainty quantification (UQ) analysis for a computational model of an engineered or physical system governed by principles of mechanics seeks to use principles of probability theory to identify, classify, and quantify sources of uncertainty between simulated predictions and observational data. A significant UQ challenge is that both aleatoric (i.e., irreducible) and epistemic (i.e., reducible) sources of uncertainty can plague the modeling, manufacturing, and observation of a system. Aleatoric uncertainty may arise from the intrinsic variability of material properties represented as model parameters such as Young's modulus while epistemic uncertainty may arise from an inability to perfectly measure the true state of system responses such as displacements when subjected to known forces. In this talk, we combine two novel frameworks to quantify both types of uncertainty in the modeling of engineered systems. The dataconsistent (DC) framework is utilized to quantify aleatoric uncertainties in system properties appearing as model parameters for a given Quantities of Interest (QoI) map. The Learning Uncertain Quantities (LUQ) framework is a threestep machinelearning enabled process for transforming noisy spatiotemporal data into samples of a learned QoI map to enable DCbased inversion. We focus discussion primarily on the LUQ framework to highlight key aspects of the mathematical foundations, the implications for learning QoI maps from a combination of data and simulations, and also to develop quantitative sufficiency tests for the data. Illustrative examples are used throughout the presentation to provide intuition for each step while the final two examples demonstrate the full capabilities of the methods for problems involving the manufacturing of shells of revolution motivated by realworld applications.
Sept 13, 2024
Speaker: Radu Cascaval, UCCS
Title: An Introduction to PINNs (PhysicsInformed Neural Networks)
Abstract: PINNs have gained popularity in the field of Scientific Machine Learning due to the wide range of applications where PINNs have been found to be effective. In particular solving PDEs, inverse problems and optimization problems. In this seminar we will introduce a few such application, together with a description of the DeepXDE package.
Spring 2024  
Mar 22  Radu C Cascaval UCCS  Mathematics in Data Science and the UCCS Math Clinic 
Apr 5  Justin Garrish Colorado School of Mines  Quantitative Assessment of Metabolic Health: Bayesian Hierarchical Models Uniting Dynamical Systems and Gaussian Processes 
Apr 12  Caroline Kellackey US Navy  Advanced Framework for Simulation, Integration and Modeling (AFSIM) 
Apr 19  Dustin Burchett MITRE Corp.  Network Resiliency and Optimization: The Graph Conductance Problem 
Apr 26  Jonathan Thompson UCCS  Supervised and Unsupervised Learning via the Kernel Trick 
Spring 2024 Abstracts:
April 26, 2024
Speaker: Jonathan Thompson, UCCS Math
Title: Supervised and Unsupervised Learning via the Kernel Trick
The field of machine learning has garnered extraordinary interest over the last decade as a result of powerful hardware, improved algorithms, and new theoretical insights. In this talk, we offer an elementary introduction to the study of kernel methods by way of extending supervised and unsupervised learning algorithms (such as support vector machines and principal component analysis) to support nonlinear predictive models embedded in an infinitedimensional Hilbert space. In doing so, we utilize the socalled "kernel trick", which allows us to learn reducedorder decision boundaries for highdimensional nonlinear data.
April 19, 2024
Speaker: Dustin Burchett, MITRE Corp.
Title: Network Resiliency and Optimization: The Graph Conductance Problem
This talk aims to provide an entrylevel understanding of networks and topologies, with a specific focus on analyzing resiliency as a function of the nodes present in the network. A significant portion of the talk will be dedicated to the “graph conductance problem”, one possible measurement of network resiliency. The conductance of a cut in a given topology can be easily computed. However, the conductance of a graph is the minimum conductance of all possible cuts, which is an Npcomplete nonlinear combinatorial optimization problem. An overview of optimization problems are provided, as well as candidate approaches to solving the conductance problem. Solution approximations are showcased, which highlight key nodes providing resiliency in the network.
This talk will provide attendees with a basic understanding of the relationships between network topologies and optimization problems, equipping them with the knowledge to measure and enhance network resiliency.
April 12, 2024
Speaker: Caroline Kellackey, US Navy
Title: Advanced Framework for Simulation, Integration and Modeling (AFSIM) Survivability Study
In this talk, we will explore the AFSIM Survivability Study, which focuses on utilizing the Advanced Framework for Simulation, Integration and Modeling (AFSIM) software to analyze the survivability of military systems. We will delve into the concept of survivability and how it is measured using the probability of a missile reaching its target. The Monte Carlo method will be discussed as a means to obtain numerical results, and the calculation of the number of scenarios in a study will be explored. Additionally, we will walk through the process of designing a missile using AFSIM, considering factors such as altitude, speed, flight path, and endgame maneuvers. Practical instructions on conducting a study using AFSIM will be provided, including scenario selection, input file generation, running the experiment, and data analysis. Finally, we will discuss how to interpret and present the findings, examining the relationship between speed, altitude, and the probability of survival through graphical representations.
April 5, 2024
Speaker: Justin Garrish, Colorado School of Mines
Title: Quantitative Assessment of Metabolic Health: Bayesian Hierarchical Models Uniting Dynamical Systems and Gaussian Processes
Diseases such as diabetes and cystic fibrosis disrupt the body's ability to regulate plasma glucose, resulting in chronic health issues across multiple body systems, often requiring longterm management. Through data collected under controlled settings, clinicians and researchers can utilize differential equations (DE)based models to analyze the physiological response to glucose and generate indices of metabolic health. While such models have proven invaluable in clinical metabolism research, their efficacy is often limited outside of specific conditions. To address this challenge, we propose integrating Gaussian processes into established DEbased models within a Bayesian framework to construct robust hybrid models capable of providing reliable indices of metabolic health with associated uncertainty quantification. In this presentation, we first explore the necessary background in Gaussian processes and Bayesian statistics, emphasizing their connection to positive definite kernelbased approximation methods. Then, through illustrative examples, we discuss mathematical and statistical modeling, numerical implementation, and clinical interpretation of results in human metabolism studies.
March 22, 2024
Speaker: Dr. Radu Cascaval, UCCS
Title: Mathematics in Data Science and the UCCS Math Clinic
We will provide an overview of various mathematical tools (from the field of linear algebra and optimization) in machine learning and data science, including regularizations and kernel methods. We then describe a few applications that have being investigated during the UCCS Math Clinic in recent years.

Spring 2022 abstracts:
Speaker: Dr. Radu Cascaval, UCCS
Title: Mathematical Analysis of Deep Learning and Kernel Methods
Kernel methods have become an important tool in the realm of machine learning and found a wide applicability in classification tasks such as support vector machines and deep learning. This seminar will provide an overview of such methods and how mathematical analysis can aid in understanding their success.
Feb 23, 2022
Speaker: Dr. Denis Silantyev, UCCS
Title: Obtaining Stokes wave with highprecision using conformal maps and spectral methods on nonuniform grids
Twodimensional potential flow of the ideal incompressible fluid with free surface and infinite depth has a class of solutions called Stokes waves which is fully nonlinear periodic gravity waves propagating with the constant velocity. We developed a new highly efficient method for computation of Stokes waves. The convergence rate of the numerical approximation by a Fourier series is determined by the complex singularity of the travelling wave in the complex plane above the free surface. We study this singularity and use an auxiliary conformal mapping which moves it away from the free surface thus dramatically speeding up Fourier series convergence of the solution. Three options for the auxiliary conformal map are described with their advantages and disadvantages for numerics. Their efficiency is demonstrated for computing Stokes waves near the limiting Stokes wave (the wave of the greatest height) with 100digit precision. Drastically improved convergence rate significantly expands the family of numerically accessible solutions and allows us to study the oscillatory approach of these solutions to the limiting wave in great detail.
March 30, 2022
Speaker: Dr. John Villavert, Univ Texas Rio Grande Valley
Title: Methods for superlinear elliptic problems
We give an elementary overview of several nonlinear elliptic (and parabolic) PDEs that arise from wellknown problems in analysis and geometry. We discuss existence, nonexistence (including Liouville theorems) and qualitative results for the equations and introduce some powerful geometric and topological techniques used to establishing these results.
We shall attempt to highlight the underlying ideas in the techniques and illustrate how we can refine them to handle more general problems involving differential and integral equations.
April 27, 2022
Speaker: Dr. Cory B. Scott, Colorado College
Title: Machine Learning for Graphs
The recent rise in Deep Learning owes much of its success to a small handful of techniques which made machine learning models drastically more efficient on image and video input. These techniques are directly responsible for the explosion of image filters, face recognition apps, deepfakes, etc. However, they all rely on the fact that image and video data lives on a grid of pixels (2D for images, 3D for video). If we want to analyze data that doesn't have a rigid gridlike structure  like molecules, social networks, biological food webs or traffic patterns  we need some more tricks. One of these techniques is called a Graph Neural Network (GNN). In this talk, we will talk about GNNs in general, and demonstrate a couple of cool applications of these models.
Spring 2019  
Feb 20, 2019  Daniel Appelö Applied Math, CU Boulder  What’s new with the wave equation? 
Mar 6, 2019  Richard Wellman Comp Sci, Colorado College  Scalable semisupervised learning with operators in Hilbert space 
Mar 20, 2019  Radu Cascaval UCCS Math  The mathematics of (spatial) mobility 
Apr 17, 2019  Mahmoud Hussein Aerospace Eng, CU Boulder  Exact dispersion relation for strongly nonlinear elastic wave propagation 
May 8, 2019  Michael Calvisi Mechanical and Aerospace Eng, UCCS  The Curious Dynamics of Translating Bubbles: An Application of Perturbation Methods and Potential Flow Theory 
Spring 2019 abstracts:
Feb 20, 2019
Speaker: Dr. Daniel Appelo, CU Boulder
Title: What’s new with the wave equation?
The defining feature of waves is their ability to propagate over vast distances in space and time without changing shape. This unique property enables the transfer of information and constitutes the foundation of today’s communication based society. To see that accurate propagation of waves requires high order accurate numerical methods, consider the problem of propagating a wave in three dimensions for 100 wavelengths with 1% error. Using a second order method this requires 0.2 trillion spacetime samples while a high order method requires many orders of magnitude fewer samples.
In the first part of this talk we present new arbitrary order dissipative and conservative Hermite methods for the scalar wave equation. The degreesoffreedom of Hermite methods are tensorproduct Taylor polynomials of degree m in each coordinate centered at the nodes of Cartesian grids, staggered in time. The methods achieve spacetime accuracy of order O(2m). Besides their high order of accuracy in both space and time combined, they have the special feature that they are stable for CFL = 1, for all orders of accuracy. This is significantly better than standard highorder element methods. Moreover, the large time steps are purely local to each cell, minimizing communication and storage requirements.
In the second part of the talk we present a spatial discontinuous Galerkin discretization of wave equations in second order form that relies on a new energy based strategy featuring a direct, meshindependent approach to defining interelement fluxes. Both energyconserving and upwind discretizations can be devised. The method comes with optimal a priori error estimates in the energy norm for certain fluxes and we present numerical experiments showing that optimal convergence for certain fluxes.
Mar 6, 2019
Speaker: Dr. Richard Wellman, Colorado College
Title: Scalable semisupervised learning with operators in Hilbert space
There is preponderance of semisupervised learning problems in science and industry, but there is a dearth of applicable semisupervised algorithms. The LaplaceSVM SemiSupervised Support Vector Machine is a learning algorithm that demonstrates state of the art performance on benchmark semisupervised data sets. However this algorithm does not scale. In this talk we’ll discuss the mathematical foundations of the LaplaceSVM and show the kernel is a solution of a nonhomogenous selfadjoint operator equation. It can be shown certain Galerkin spectral approximations are themselves valid reproducing kernels that encode the underlying Riemannian geometry. The spectral kernels have excellent scalability metrics and interesting mathematical properties. We discuss both the mathematics and experimental results of the resultant semisupervised alogirthm.
Apr 6, 2019
Speaker: Dr. Mahmoud Hussein, CU Boulder
Title: Exact dispersion relation for strongly nonlinear elastic wave propagation
Wave motion lies at the heart of many disciplines in the physical sciences and engineering. For example, problems and applications involving light, sound, heat or fluid flow are all likely to involve wave dynamics at some level. In this seminar, I will present our recent work on a class of problems involving intriguing nonlinear wave phenomena‒largedeformation elastic waves in solids; that is, the “largeonsmall” problem.
May 8, 2019
Speaker: Dr. Michel Calvisi, Mechanical and Aerospace Eng, UCCS
Title: The Curious Dynamics of Translating Bubbles: An Application of Perturbation Methods and Potential Flow Theory
When subject to an acoustic field, bubbles will translate and oscillate in interesting ways. This motion is highly nonlinear and its understanding is essential to the application of bubbles in diagnostic ultrasound imaging, microbubble drug delivery, and acoustic cell sorting, among others. This talk will review some of the interesting physics that occur when bubbles translate in an acoustic field, including Bjerknes forces, the added mass effect, and nonspherical shape oscillation. Such nonspherical shape modes strongly affect the stability and acoustic signature of encapsulated microbubbles (EMBs) used for biomedical applications, and thus are an important factor to consider in the design and utilization of EMBs. The shape stability of an EMB subject to translation is investigated through development of an axisymmetric model for the case of small deformations using perturbation analysis. The potential flow in the bulk volume of the external flow is modeled using an asymptotic analysis. Viscous effects within the thin boundary layer at the interface are included, owing to the noslip boundary condition. The results of numerical simulations of the evolutions equations for the shape and translation of the EMB demonstrate the counterintuitive result that, compared to a free gas bubble, the encapsulation actually promotes instability when a microbubble translates due to an acoustic wave.
Fall 2018  
Sep 12, 2018  Sarbarish Chakravarty UCCS Math  Beach waves and KP solitons 
Oct 3, 2018  Robert Carlson UCCS Math  An elementary trip from the Gauss hypergeometric function to the PoschlTeller potential in quantum mechanics 
Oct 17, 2018  Geraldo de Souza Auburn University  Fourier series, Wavelets, Inequalities, Geometry and Optimization 
Nov 14, 2018  Robert Jenkins CSU Fort Collins  Semiclassical soliton ensembles 
Dec 5, 2018  Barbara Prinari UCCS Math  Discrete solitons for the focusing AblowitzLadik equation with nonzero boundary conditions via inverse scattering transform 
Fall 2018 abstracts:
Sep 12, 2018
Speaker: Dr. Sarbarish Chakravarty, UCCS
Title: Beach Waves and KP Solitons
Abstract: In this talk, I will give a brief overview of the soliton solutions of the KP equation, and discuss how these solutions can describe shallow water wave patterns on long flat beaches.
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Oct 3, 2018
Speaker: Dr. Robert Carlson, UCCS
Title: An elementary trip from the Gauss hypergeometric function to the PoschlTeller potential in quantum mechanics
Abstract: A simple transformation takes the (G) equation for the Gauss hypergeometric function to the (J) equation for Jacobi polynomials. J has an (unusual) adjoint equation (H) (of Heun type) with an extra singular point. H has eigenfunctions that can be expressed in terms of the Gauss hypergeometric function. Another change of variables lets us rediscover a ‘solvable’ (PoschlTeller) Schrodinger equation. The methods use the kinds of techniques we often teach in Math 3400.
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Oct 17, 2018
Speaker: Dr. Geraldo de Souza, Auburn University
Title: Fourier series, Wavelets, Inequalities, Geometry and Optimization
Abstract: This talk will have two parts. In the first part, I will start with motivation and comments to some important problems in Analysis. Each problem has led to important discovery, such as Wavelets, technique of convergence of Fourier, among others. The second part I will talk about Inequalities. In general, I view the second part of this presentation as simple or perhaps an elementary approach to the subject (even though it is a new idea). On the other hand, this talk will show some interesting observations that are part of the folklore of mathematics. I will go over some very common and important inequalities in analysis that we see in the course of Analysis and even in Calculus. I will give some different views of different proofs, using Geometry, Graphing and some of them “a new analytic proof” by using optimization of functions of two variables (this is very interesting).
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Nov 14, 2018
Speaker: Dr. Robert Jenkins, Colorado State University  Fort Collins
Title: Semiclassical soliton ensembles
Abstract: Equations like the Kortewegde Vries (KdV) and the nonlinear Schroedinger equation exhibit interesting and complicated dynamics when the dispersive length scales in the problem are small compared to those of the initial wave profile; this is the relevant scaling regime for many problem is optical fibers. In this talk I'll discuss one way to analyze such problems for integrable PDEs using the inverse scattering transform (IST) that approximates initial data by an increasingly large sum of solitons. I'll talk both about NLS and some more recent work of mine on the resonant three wave interaction equations. There will be lots of pictures to help clear up the technical details!
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Dec 5, 2018
Speaker: Dr. Barbara Prinari, UCCS
Title: Discrete solitons for the focusing AblowitzLadik equation with nonzero boundary conditions via inverse scattering transform
Abstract: Soliton solutions of the focusing AblowitzLadik (AL) equation with nonzero boundary conditions at infinity are derived within the framework of the inverse scattering transform (IST). After reviewing the relevant aspects of the direct and inverse problems, explicit soliton solutions will be discussed which are the discrete analog of the TajiriWatanabe and KuznetsovMa solutions to the focusing NLS equation on a finite background. Then, by performing suitable limits of the above solutions, discrete analog of the celebrated Akhmediev and Peregrine solutions will also be presented. These solutions, which had been recently derived by direct methods, are obtained for the first time within the framework of the IST, thus providing a spectral characterization of the solutions and a description of the singular limit process.
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Spring 2018  
Apr 11, 2018  Greg Fasshauer Colorado School of Mines  An Introduction to KernelBased Approximation Methods 
Mar 14, 2018  Ethan Berkove Lafayette College  Short Paths and Long Titles: Travels through the Sierpinski carpet, Menger sponge, and beyond. 
Feb 28, 2018  Radu Cascaval UCCS Math  Traffic Flow Models. A Tutorial 
Spring 2018 abstracts:
Apr 11, 2018
Speaker: Dr. Greg Fasshauer, Colorado School of Mines
Title: An Introduction to KernelBased Approximation Methods
Abstract: I will start with a few historical remarks, and then motivate the use of kernelbased approximation as a numerical approach that generalizes standard polynomialbased methods. Examples of kernels and their use in data fitting problems will be provided along with an overview of some of the concerns and issues associated with the use of kernel methods.
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Mar 14, 2018
Speaker: Dr. Ethan Berkove, Laffayette College (Joint with Rings and Wings Seminar)
Title: Short Paths and Long Titles: Travels through the Sierpinski carpet, Menger sponge, and beyond.
Abstract: Sierpinski carpet and Menger sponge are fractals which can be thought of as two and three dimensional versions of the Cantor set. Like the Cantor set, each is formed by starting with a shape (a square for the carpet, a cube for the sponge) and then recursively removing certain subsets of it. Unlike the Cantor set, what remains is connected in the following sense: given any two points s and f in the carpet or sponge, there is a path from s to f that stays in the carpet or sponge. In this talk, we’ll discuss what we know about the shortest path from s to f in the carpet, sponge, and even higher dimensional versions of these fractals. The proofs required a surprising (at least to us) breadth of techniques, from combinatorics, geometry, and even linear programming. (Joint work with Derek Smith)
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Feb 28, 2018
Speaker: Dr. Radu Cascaval, UCCS
Title: Traffic Flow Models. A Tutorial
Abstract: We present several traffic flow models, both at the micro and macroscale, including for multilane traffic. Problems of controlling the traffic will be described and numerical simulations will illustrate possible solutions.
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Fall 2017  
Dec 8, 2017  Barbara Prinari UCCS Math  Solitons and rogue waves for a square matrix nonlinear Schrodinger equation with nonzero boundary conditions 
Nov 17, 2017  Oksana Bihun UCCS Math  New properties of the zeros of Krall polynomials 
Oct 27, 2017  Radu Cascaval UCCS Math  What do Analysis and Scientific Computation have in common ... 
Sep 29, 2017  Fritz Gesztesy Baylor Univ.  The eigenvalue counting function for Kreinvon Neumann extensions of elliptic operators 
Fall 2017 seminars:
Dec 11, 2017
Speaker: Dr. Barbara Prinari, UCCS
Title: Solitons and rogue waves for a square matrix nonlinear Schrodinger equation with nonzero boundary conditions
Abstract: In this talk we discuss the Inverse Scattering Transform (IST) under nonzero boundary conditions for a square matrix nonlinear Schrodinger equation which has been proposed as a model to describe hyperfine spin F = 1 spinor BoseEinstein condensates with either repulsive interatomic interactions and antiferromagnetic spinexchange interactions, or attractive interatomic interactions and ferromagnetic spinexchange interactions. Emphasis will be given to a discussion of the soliton and rogue wave solutions one can obtain as a byproduct of the IST.
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Nov 17, 2017 Seminar
Speaker: Dr. Oksana Bihun, UCCS
Title: New properties of the zeros of Krall polynomials
Abstract: We identify a class of remarkable algebraic relations satisfied by the zeros of the Krall orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two. Given an orthogonal polynomial family p_n(x), we relate the zeros of the polynomial p_N with the zeros of p_m for each m <=N (the case m = N corresponding to the relations that involve the zeros of pN only). These identities are obtained by exacting the similarity transformation that relates the spectral and the (interpolatory) pseudospectral matrix representations of linear differential operators, while using the zeros of the polynomial p_N as the interpolation nodes. The proposed framework generalizes known properties of classical orthogonal polynomials to the case of nonclassical polynomial families of Krall type. We illustrate the general result by proving new remarkable identities satisfied by the KrallLegendre, the KrallLaguerre and the KrallJacobi orthogonal polynomials.
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Oct 27, 2017 Seminar
Speaker: Dr. Radu C. Cascaval, UCCS
Title: What do Analysis and Scientific Computation have in common ...
Abstract: Analysis, the world of the infinitesimally small, is thought to be one of the last standing outposts where humans can fight the computational invasion. In spite of this fact, computational sciences continue to benefit greatly from advances in analysis. This talk will illustrate this relationship, in particular functional analysis connections to numerical spectral methods, meshless methods, and their applications to numerical solutions to PDEs.
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Sept 29, 2017 Seminar
Speaker: Dr. Fritz Gesztesy, Baylor University
Title: The eigenvalue counting function for Kreinvon Neumann extensions of elliptic operators
Abstract: We start by providing a historical introduction into the subject of Weylasymptotics for Laplacians on bounded domains in ndimensional Euclidean space, and a brief introduction into the basic principles of selfadjoint extensions. Subsequently, we turn to bounds on eigenvalue counting functions and derive such a bound for Kreinvon Neumann extensions corresponding to a class of uniformly elliptic second order PDE operators (and their positive integer powers) on arbitrary open, bounded, ndimensional subsets \Omega in R^n. (No assumptions on the boundary of \Omega are made; the coefficients are supposed to satisfy certain regularity conditions.) Our technique relies on variational considerations exploiting the fundamental link between the Kreinvon Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of the corresponding differential operator suitably extended to all of R^n. We also consider the analogous bound for the eigenvalue counting function for the corresponding Friedrichs extension. This is based on joint work with M. Ashbaugh, A. Laptev, M. Mitrea, and S. Sukhtaiev.
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