MATH Research Publications

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The collection of peer-reviewed research articles (co)authored by faculty of the Department of Mathematics and Statistics.

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    A novel Bayesian computational approach for bridge-randomized quantile regression in high dimensional models
    (Taylor and Francis Ltd., 2024) Zhang, Shen; Dao, Mai; Ye, Keying; Han, Zifei; Wang, Min
    A bridge-randomized penalization that employs a prior for the shrinkage parameter, as opposed to the conventional bridge penalization with a fixed penalty, often delivers more superior performance compared to many other traditional shrinkage methods. In this paper, we develop an efficient Bayesian computational algorithm via the two-block Markov Chain Monte Carlo method for the bridge-randomized penalization in quantile regression to perform inference in the high-dimensional "large-p" and "large-p-small-n" settings. To construct a fully Bayesian formulation, we utilize the asymmetric Laplace distribution as an auxiliary error distribution and the generalized Gaussian distribution prior for the regression coefficients. Simulation studies encompassing a wide range of scenarios indicate that the proposed method performs at least as well as, and often better than, other existing procedures in terms of both parameter estimation and variable selection. Finally, a real-data application is provided for illustrative purposes. © 2024 Informa UK Limited, trading as Taylor & Francis Group.
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    Vortex shedding from bluff bodies: a conformal mapping approach
    (Springer Science and Business Media B.V., 2024) Matheswaran, Vijay; DeLillo, Thomas K.; Miller, L. Scott
    A model to calculate flow around bluff bodies of various geometries is presented. A conformal map between the plane of the bluff body and the plane of a unit circular cylinder is established by using a combination of Karman-Trefftz transformations and Fornberg's method. Flow in the circle plane is calculated using the authors' Hybrid Potential Flow (HPF) model and mapped back to the shape plane. By joining this calculated near-body flow with von Karman's model for a vortex wake, forces due to vortex shedding and shedding frequencies are calculated. In this manner, a complete solution for the flow around bluff bodies of various geometries is established. Results for two shapes are presented, along with recommendations for further work. © The Author(s) 2024.
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    Mobile disks in hyperbolic space and minimization of conformal capacity
    (Kent State University, 2024) Hakula, Harri; Nasser, Mohamed M. S.; Vuorinen, Matti
    Our focus is to study constellations of disjoint disks in the hyperbolic space, i.e., the unit disk equipped with the hyperbolic metric. Each constellation corresponds to a set E which is the union of m > 2 disks with hyperbolic radii rj > 0, j = 1, . . ., m. The centers of the disks are not fixed, and hence individual disks of the constellation are allowed to move under the constraints that they do not overlap and their hyperbolic radii remain invariant. Our main objective is to find computational lower bounds for the conformal capacity of a given constellation. The capacity depends on the centers and radii in a very complicated way even in the simplest cases when m = 3 or m = 4. In the absence of analytic methods, our work is based on numerical simulations using two different numerical methods, the boundary integral equation method and the hp-FEM method, respectively. Our simulations combine capacity computation with minimization methods and produce extremal cases where the disks of the constellation are grouped next to each other. This resembles the behavior of animal colonies minimizing heat flow in arctic areas. © 2024 Kent State University. All rights reserved.
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    The Curvature Operator of the Second Kind in Dimension Three
    (Springer, 2024) Fluck, Harry; Li, Xiaolong
    This article aims to understand the behavior of the curvature operator of the second kind under the Ricci flow in dimension three. First, we express the eigenvalues of the curvature operator of the second kind explicitly in terms of that of the curvature operator (of the first kind). Second, we prove that α-positive/α-nonnegative curvature operator of the second kind is preserved by the Ricci flow in dimension three for all α∈[1,5]. © The Author(s) 2024.
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    Towards computing the harmonic-measure distribution function for the middle-thirds Cantor set
    (Elsevier B.V., 2024) Green, Christopher C.; Nasser, Mohamed M. S.
    This paper is concerned with the numerical computation of the harmonic-measure distribution function, or h-function for short, associated with a particular planar domain. This function describes the hitting probability of a Brownian walker released from some point with the boundary of the domain. We use a fast and accurate boundary integral method for the numerical calculation of the h-functions for symmetric multiply connected slit domains with high connectivity. In view of the fact that the middle-thirds Cantor set C is a special limiting case of these slit domains, the proposed method is used to approximate the h-function for the set C. We also numerically analyze some asymptotic features of the calculated h-functions. © 2024 Elsevier B.V.
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