Lattice models and super telescoping formula

In this paper, we introduce the super telescoping formula, a natural generalization of well-known telescoping formula. We explore various aspects of the formula including its origin and the telescoping cancellations emerging from symmetric patterns. We also show that the super telescoping formula leads to the construction of exactly solvable lattice models with interesting partition functions.Comment: 14 page

Inversion and Symmetries of the Star Transform

The star transform is a generalized Radon transform mapping a function of two variables to its integrals along "star-shaped" trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in mathematical models of various imaging modalities based on scattering of elementary particles. The paper presents a comprehensive study of the inversion of the star transform. We describe the necessary and sufficient conditions for invertibility of the star transform, introduce a new inversion formula and discuss its stability properties. As an unexpected bonus of our approach, we prove a conjecture from algebraic geometry about the zero sets of elementary symmetric polynomials

Kernel Smoothing Operators on Thick Open Domains

We define the notion of a thick open set $\Omega$ in a Euclidean space and show that a local Hardy-Littlewood inequality holds in $L^p(\Omega)$, $p \in (1, \infty]$. We then establish pointwise and $L^p(\Omega)$ convergence for families of convolution operators with a Markov normalization on $\Omega$. We demonstrate application of such smoothing operators to piecewise-continuous density, velocity, and stress fields from discrete element models of sea ice dynamics.Comment: 28 pages, 5 figure

Numerical implementation of generalized V-line transforms on 2D vector fields and their inversions

The paper discusses numerical implementations of various inversion schemes for generalized V-line transforms on vector fields introduced in [6]. It demonstrates the possibility of efficient recovery of an unknown vector field from five different types of data sets, with and without noise. We examine the performance of the proposed algorithms in a variety of setups, and illustrate our results with numerical simulations on different phantoms.Comment: 38 page

Loop Group Factorization, Lattice Systems and OPUC

In this dissertation, starting from Verblunsky correspondence, we study the representationof the Gaussian free field in terms of Verblunsky sequence. We then explore connections of Verblunsky theory with loop group factorization and certain 1D Ising models. Next, we provide a new one-parameter family of Verblunsky sequences with their associated measure. At the end, we show how to translate instances of Verblunsky correspondence into partition function calculations of Ising models