# Mathematical Data Science Seminar

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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 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 non-UCCS individuals attending this seminar.

**Fridays 12:15-1:15pm, ENGR 247, UCCS campus**

(refreshments available at 12:00pm, talks start at 12:15pm)

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Mar 22, 2024 | Radu C CascavalUCCS | Mathematics in Data Science and the UCCS Math Clinic |

Apr 5, 2024 | Justin GarrishColorado School of Mines | Quantitative Assessment of Metabolic Health: Bayesian Hierarchical Models Uniting Dynamical Systems and Gaussian Processes |

Apr 12, 2024 | Caroline Kellackey US Navy | Advanced Framework for Simulation, Integration and Modeling (AFSIM) |

Apr 19, 2024 | Dustin BurchettMITRE Corp. | Network Resiliency and Optimization: The Graph Conductance Problem |

Apr 26, 2024 | Jonathan ThompsonUCCS | Supervised and Unsupervised Learning via the Kernel Trick |

Fall 2024 | ||

Sep 2024 | Troy Butler, CU Denver & NSF | TBA |

Sep 2024 | Mihno Kim, Colorado College | TBA |

__Spring 2024 Abstracts:__

__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 non-linear predictive models embedded in an infinite-dimensional Hilbert space. In doing so, we utilize the so-called "kernel trick", which allows us to learn reduced-order decision boundaries for high-dimensional non-linear data.

**April 19, 2024 ****Speaker: Dustin Burchett, MITRE Corp.****Title: **Network Resiliency and Optimization: The Graph Conductance Problem

This talk aims to provide an entry-level 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 Np-complete 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 end-game 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 long-term 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 DE-based 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 kernel-based 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.

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Feb 20, 2019 | Daniel AppelöApplied Math, CU Boulder | What’s new with the wave equation? | ||||||||||||||||

Mar 6, 2019 | Richard WellmanComp Sci, Colorado College | Scalable semi-supervised learning with operators in Hilbert space | ||||||||||||||||

Mar 20, 2019 | Radu CascavalUCCS Math | The mathematics of (spatial) mobility | ||||||||||||||||

Apr 17, 2019 | Mahmoud HusseinAerospace Eng, CU Boulder | Exact dispersion relation for strongly nonlinear elastic wave propagation | ||||||||||||||||

May 8, 2019 | Michael CalvisiMechanical and Aerospace Eng, UCCS | The Curious Dynamics of Translating Bubbles: An Application of Perturbation Methods and Potential Flow Theory | ||||||||||||||||

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Sep 12, 2018 | Sarbarish ChakravartyUCCS Math | Beach waves and KP solitons | ||||||||||||||||

Oct 3, 2018 | Robert CarlsonUCCS Math | An elementary trip from the Gauss hypergeometric function to the Poschl-Teller potential in quantum mechanics | ||||||||||||||||

Oct 17, 2018 | Geraldo de SouzaAuburn University | Fourier series, Wavelets, Inequalities, Geometry and Optimization | ||||||||||||||||

Nov 14, 2018 | Robert JenkinsCSU Fort Collins | Semiclassical soliton ensembles | ||||||||||||||||

Dec 5, 2018 | Barbara PrinariUCCS Math | Discrete solitons for the focusing Ablowitz-Ladik equation with non-zero boundary conditions via inverse scattering transform | ||||||||||||||||

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Apr 11, 2018 | Greg FasshauerColorado School of Mines | An Introduction to Kernel-Based 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 CascavalUCCS Math | Traffic Flow Models. A Tutorial | ||||||||||||||||

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Dec 8, 2017 | Barbara PrinariUCCS Math | Solitons and rogue waves for a square matrix nonlinear Schrodinger equation with nonzero boundary conditions | ||||||||||||||||

Nov 17, 2017 | Oksana BihunUCCS Math | New properties of the zeros of Krall polynomials | ||||||||||||||||

Oct 27, 2017 | Radu CascavalUCCS Math | What do Analysis and Scientific Computation have in common ... | ||||||||||||||||

Sep 29, 2017 | Fritz GesztesyBaylor Univ. | The eigenvalue counting function for Krein-von Neumann extensions of elliptic operators |

**TITLES & ABSTRACTS: **

__Spring 2022 Seminars:__

__Spring 2022 Seminars:__

**Feb 9, 2022 ****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 high-precision using conformal maps and spectral methods on non-uniform grids

Two-dimensional 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 100-digit 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 well-known problems in analysis and geometry. We discuss existence, non-existence (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 grid-like 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 seminars:__

__Spring 2019 seminars:__

**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 space-time 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 degrees-of-freedom of Hermite methods are tensor-product Taylor polynomials of degree m in each coordinate centered at the nodes of Cartesian grids, staggered in time. The methods achieve space-time 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 high-order 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, mesh-independent approach to defining interelement fluxes. Both energy-conserving 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 semi-supervised learning with operators in Hilbert space

There is preponderance of semi-supervised learning problems in science and industry, but there is a dearth of applicable semi-supervised algorithms. The LaplaceSVM Semi-Supervised Support Vector Machine is a learning algorithm that demonstrates state of the art performance on benchmark semi-supervised 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 non-homogenous self-adjoint 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 semi-supervised 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‒large-deformation elastic waves in solids; that is, the “large-on-small” 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 no-slip 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 seminars:__

__Fall 2018 seminars:__

**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 Poschl-Teller 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’ (Poschl-Teller) 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 Korteweg-de 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 Ablowitz-Ladik equation with non-zero boundary conditions via inverse scattering transform **

**Abstract: **Soliton solutions of the focusing Ablowitz-Ladik (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 Tajiri-Watanabe and Kuznetsov-Ma 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 seminars:__

__Spring 2018 seminars:__

**Apr 11, 2018 **

**Speaker: Dr. Greg Fasshauer, Colorado School of Mines****Title:**** An Introduction to Kernel-Based Approximation Methods **

**Abstract: ** I will start with a few historical remarks, and then motivate the use of kernel-based approximation as a numerical approach that generalizes standard polynomial-based 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.*****************************************************************

**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 macro-scale, including for multi-lane traffic. Problems of controlling the traffic will be described and numerical simulations will illustrate possible solutions.**

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__Fall 2017 seminars:__

__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 Bose-Einstein condensates with either repulsive interatomic interactions and anti-ferromagnetic spin-exchange interactions, or attractive interatomic interactions and ferromagnetic spin-exchange 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 non-classical polynomial families of Krall type. We illustrate the general result by proving new remarkable identities satisfied by the Krall-Legendre, the Krall-Laguerre and the Krall-Jacobi 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 Krein-von Neumann extensions of elliptic operators**

**Abstract: ** We start by providing a historical introduction into the subject of Weyl-asymptotics for Laplacians on bounded domains in n-dimensional Euclidean space, and a brief introduction into the basic principles of self-adjoint extensions. Subsequently, we turn to bounds on eigenvalue counting functions and derive such a bound for Krein-von Neumann extensions corresponding to a class of uniformly elliptic second order PDE operators (and their positive integer powers) on arbitrary open, bounded, n-dimensional 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 Krein-von 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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