Physical Sciences and Mathematics Commons^™

A Generalization Of Combinatorial Identities For Stable Discrete Series Constants, Richard Ehrenborg, Sophie Moreland, Margaret Readdy

Mathematics Faculty Publications

This article is concerned with the constants that appear in Harish-Chandra’s character formula for stable discrete series of real reductive groups, although it does not require any knowledge about real reductive groups or discrete series. In Harish-Chandra’s work the only information we have about these constants is that they are uniquely determined by an inductive property. Later, Goresky–Kottwitz–MacPherson (1997) and Herb (2000) gave different formulas for these constants. In this article, we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longer make sense), and give a direct proof that the two …

Awegnn: Auto-Parametrized Weighted Element-Specific Graph Neural Networks For Molecules., Timothy Szocinski, Duc Duy Nguyen, Guo-Wei Wei

Mathematics Faculty Publications

While automated feature extraction has had tremendous success in many deep learning algorithms for image analysis and natural language processing, it does not work well for data involving complex internal structures, such as molecules. Data representations via advanced mathematics, including algebraic topology, differential geometry, and graph theory, have demonstrated superiority in a variety of biomolecular applications, however, their performance is often dependent on manual parametrization. This work introduces the auto-parametrized weighted element-specific graph neural network, dubbed AweGNN, to overcome the obstacle of this tedious parametrization process while also being a suitable technique for automated feature extraction on these internally complex …

Algebraic Graph-Assisted Bidirectional Transformers For Molecular Property Prediction, Dong Chen, Kaifu Gao, Duc Duy Nguyen, Xin Chen, Yi Jiang, Guo-Wei Wei, Feng Pan

Mathematics Faculty Publications

The ability of molecular property prediction is of great significance to drug discovery, human health, and environmental protection. Despite considerable efforts, quantitative prediction of various molecular properties remains a challenge. Although some machine learning models, such as bidirectional encoder from transformer, can incorporate massive unlabeled molecular data into molecular representations via a self-supervised learning strategy, it neglects three-dimensional (3D) stereochemical information. Algebraic graph, specifically, element-specific multiscale weighted colored algebraic graph, embeds complementary 3D molecular information into graph invariants. We propose an algebraic graph-assisted bidirectional transformer (AGBT) framework by fusing representations generated by algebraic graph and bidirectional transformer, as well as …

Mathematical Modeling Of The Candida Albicans Yeast To Hyphal Transition Reveals Novel Control Strategies, David J. Wooten, Jorge Gómez Tejeda Zañudo, David Murrugarra, Austin M. Perry, Anna Dongari-Bagtzoglou, Reinhard Laubenbacher, Clarissa J. Nobile, Réka Albert

Mathematics Faculty Publications

Candida albicans, an opportunistic fungal pathogen, is a significant cause of human infections, particularly in immunocompromised individuals. Phenotypic plasticity between two morphological phenotypes, yeast and hyphae, is a key mechanism by which C. albicans can thrive in many microenvironments and cause disease in the host. Understanding the decision points and key driver genes controlling this important transition and how these genes respond to different environmental signals is critical to understanding how C. albicans causes infections in the host. Here we build and analyze a Boolean dynamical model of the C. albicans yeast to hyphal transition, integrating …

Unveiling The Molecular Mechanism Of Sars-Cov-2 Main Protease Inhibition From 137 Crystal Structures Using Algebraic Topology And Deep Learning, Duc Duy Nguyen, Kaifu Gao, Jiahui Chen, Rui Wang, Guo-Wei Wei

Mathematics Faculty Publications

Currently, there is neither effective antiviral drugs nor vaccine for coronavirus disease 2019 (COVID-19) caused by acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Due to its high conservativeness and low similarity with human genes, SARS-CoV-2 main protease (M^pro) is one of the most favorable drug targets. However, the current understanding of the molecular mechanism of M^pro inhibition is limited by the lack of reliable binding affinity ranking and prediction of existing structures of M^pro-inhibitor complexes. This work integrates mathematics (i.e., algebraic topology) and deep learning (MathDL) to provide a reliable ranking of the binding …

Simulating Phase Transitions And Control Measures For Network Epidemics Caused By Infections With Presymptomatic, Asymptomatic, And Symptomatic Stages, Benjamin Braun, Başak Taraktaş, Brian Beckage, Jane Molofsky

Mathematics Faculty Publications

We investigate phase transitions associated with three control methods for epidemics on small world networks. Motivated by the behavior of SARS-CoV-2, we construct a theoretical SIR model of a virus that exhibits presymptomatic, asymptomatic, and symptomatic stages in two possible pathways. Using agent-based simulations on small world networks, we observe phase transitions for epidemic spread related to: 1) Global social distancing with a fixed probability of adherence. 2) Individually initiated social isolation when a threshold number of contacts are infected. 3) Viral shedding rate. The primary driver of total number of infections is the viral shedding rate, with probability of …

On Robust Colorings Of Hamming-Distance Graphs, Isaiah Harney, Heide Gluesing-Luerssen

Mathematics Faculty Publications

H_q(n, d) is defined as the graph with vertex set Zⁿ_q and where two vertices are adjacent if their Hamming distance is at least d. The chromatic number of these graphs is presented for various sets of parameters (q, n, d). For the 4-colorings of the graphs H₂(n, n − 1) a notion of robustness is introduced. It is based on the tolerance of swapping colors along an edge without destroying properness of the coloring. An explicit description of the maximally robust 4-colorings of …

When Does The Bombieri–Vinogradov Theorem Hold For A Given Multiplicative Function?, Andrew Granville, Xuancheng Shao

Mathematics Faculty Publications

Let f and g be 1-bounded multiplicative functions for which f ✻ g = 1_.=1. The Bombieri–Vinogradov theorem holds for both f and g if and only if the Siegel–Walfisz criterion holds for both f and g, and the Bombieri–Vinogradov theorem holds for f restricted to the primes.

Rook Placements And Jordan Forms Of Upper-Triangular Nilpotent Matrices, Martha Yip

Mathematics Faculty Publications

The set of n by n upper-triangular nilpotent matrices with entries in a finite field 𝔽_q has Jordan canonical forms indexed by partitions λ ⊢ n. We present a combinatorial formula for computing the number F_λ(q) of matrices of Jordan type λ as a weighted sum over standard Young tableaux. We construct a bijection between paths in a modified version of Young's lattice and non-attacking rook placements, which leads to a refinement of the formula for F_λ(q).

Matching And Independence Complexes Related To Small Grids, Benjamin Braun, Wesley K. Hough

Mathematics Faculty Publications

The topology of the matching complex for the 2 x n grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes Ind(Δ^m_n) that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain Ind(Δ^m_n). Furthermore, we determine the Euler characteristic of Ind(Δ^m_n) and prove that several homology groups of Ind(Δ^m_n) are non-zero.

Equivariant Iterated Loop Space Theory And Permutative G–Categories, Bertrand J. Guillou, J. Peter May

Mathematics Faculty Publications

We set up operadic foundations for equivariant iterated loop space theory. We start by building up from a discussion of the approximation theorem and recognition principle for V–fold loop G–spaces to several avatars of a recognition principle for infinite loop G–spaces. We then explain what genuine permutative G–categories are and, more generally, what E_∞–G–categories are, giving examples showing how they arise. As an application, we prove the equivariant Barratt–Priddy–Quillen theorem as a statement about genuine G–spectra and use it to give a new, categorical proof of the tom Dieck splitting theorem …

Matroid Configurations And Symbolic Powers Of Their Ideals, A. V. Geramita, B. Harbourne, J. Migliore, Uwe Nagel

Mathematics Faculty Publications

Star configurations are certain unions of linear subspaces of projective space that have been studied extensively. We develop a framework for studying a substantial generalization, which we call matroid configurations, whose ideals generalize Stanley-Reisner ideals of matroids. Such a matroid configuration is a union of complete intersections of a fixed codimension. Relating these to the Stanley-Reisner ideals of matroids and using methods of liaison theory allows us, in particular, to describe the Hilbert function and minimal generators of the ideal of, what we call, a hypersurface configuration. We also establish that the symbolic powers of the ideal of any matroid …

Categorical Models For Equivariant Classifying Spaces, Bertrand J. Guillou, J. Peter May, Mona Merling

Mathematics Faculty Publications

Starting categorically, we give simple and precise models for classifying spaces of equivariant principal bundles. We need these models for work in progress in equi- variant infinite loop space theory and equivariant algebraic K–theory, but the models are of independent interest in equivariant bundle theory and especially equivariant covering space theory.

Challenges In Modeling Complexity Of Neglected Tropical Diseases: A Review Of Dynamics Of Visceral Leishmaniasis In Resource Limited Settings, Swati Debroy, Olivia F. Prosper, Austin Mishoe, Anuj Mubayi

Mathematics Faculty Publications

Objectives: Neglected tropical diseases (NTD), account for a large proportion of the global disease burden, and their control faces several challenges including diminishing human and financial resources for those distressed from such diseases. Visceral leishmaniasis (VL), the second-largest parasitic killer (after malaria) and an NTD affects poor populations and causes considerable cost to the affected individuals. Mathematical models can serve as a critical and cost-effective tool for understanding VL dynamics, however, complex array of socio-economic factors affecting its dynamics need to be identified and appropriately incorporated within a dynamical modeling framework. This study reviews literature on vector-borne diseases and collects …

On ΤΣ-Quasinormal Subgroups Of Finite Groups, James C. Beidleman, Alexander N. Skiba

Mathematics Faculty Publications

Let σ = {σ_i ∣ i ∈ I} be a partition of the set of all primes P and G a finite group. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σ_i-subgroup of G for some i ∈ I and H contains exactly one Hall σ_i-subgroup of G for every i such that σ_i ∩ π(G) ≠ ∅.

Let τ_H(A) = {σ_i ∈ σ …

Blow-Up Algebras, Determinantal Ideals, And Dedekind-Mertens-Like Formulas, Alberto Corso, Uwe Nagel, Sonja Petrović, Cornelia Yuen

Mathematics Faculty Publications

We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen–Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine …

Simulating Within-Vector Generation Of The Malaria Parasite Diversity, Lauren M. Childs, Olivia F. Prosper

Mathematics Faculty Publications

Plasmodium falciparum, the most virulent human malaria parasite, undergoes asexual reproduction within the human host, but reproduces sexually within its vector host, the Anopheles mosquito. Consequently, the mosquito stage of the parasite life cycle provides an opportunity to create genetically novel parasites in multiply-infected mosquitoes, potentially increasing parasite population diversity. Despite the important implications for disease transmission and malaria control, a quantitative mapping of how parasite diversity entering a mosquito relates to diversity of the parasite exiting, has not been undertaken. To examine the role that vector biology plays in modulating parasite diversity, we develop a two-part model framework …

Optical Tomography On Graphs, Francis J. Chung, Anna C. Gilbert, Jeremy G. Hoskins, John C. Schotland

Mathematics Faculty Publications

We present an algorithm for solving inverse problems on graphs analogous to those arising in diffuse optical tomography for continuous media. In particular, we formulate and analyze a discrete version of the inverse Born series, proving estimates characterizing the domain of convergence, approximation errors, and stability of our approach. We also present a modification which allows additional information on the structure of the potential to be incorporated, facilitating recovery for a broader class of problems.

Convergence Rates In Periodic Homogenization Of Systems Of Elasticity, Zhongwei Shen, Jinping Zhuge

Mathematics Faculty Publications

This paper is concerned with homogenization of systems of linear elasticity with rapidly oscillating periodic coefficients. We establish sharp convergence rates in L² for the mixed boundary value problems with bounded measurable coefficients.

Q-Stirling Identities Revisited, Yue Cai, Richard Ehrenborg, Margaret Readdy

Mathematics Faculty Publications

We give combinatorial proofs of q-Stirling identities using restricted growth words. This includes a poset theoretic proof of Carlitz's identity, a new proof of the q-Frobenius identity of Garsia and Remmel and of Ehrenborg's Hankel q-Stirling determinantal identity. We also develop a two parameter generalization to unify identities of Mercier and include a symmetric function version.

Signed Lozenge Tilings, D. Cook Ii, Uwe Nagel

Mathematics Faculty Publications

It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. Using a …

Removable Singularities In C*-Algebras Of Real Rank Zero, Lawrence A. Harris

Mathematics Faculty Publications

Let 𝕬 be a C*-algebra with identity and real rank zero. Suppose a complex- valued function is holomorphic and bounded on the intersection of the open unit ball of 𝕬 and the identity component of the set of invertible elements of 𝕬. We give a short transparent proof that the function has a holomorphic extension to the entire open unit ball of 𝕬. The author previously deduced this from a more general fact about Banach algebras.

Estimating Propensity Parameters Using Google Pagerank And Genetic Algorithms, David Murrugarra, Jacob Miller, Alex N. Mueller

Mathematics Faculty Publications

Stochastic Boolean networks, or more generally, stochastic discrete networks, are an important class of computational models for molecular interaction networks. The stochasticity stems from the updating schedule. Standard updating schedules include the synchronous update, where all the nodes are updated at the same time, and the asynchronous update where a random node is updated at each time step. The former produces a deterministic dynamics while the latter a stochastic dynamics. A more general stochastic setting considers propensity parameters for updating each node. Stochastic Discrete Dynamical Systems (SDDS) are a modeling framework that considers two propensity parameters for updating each node …

Identification Of Control Targets In Boolean Molecular Network Models Via Computational Algebra, David Murrugarra, Alan Veliz-Cuba, Boris Aguilar, Reinhard Laubenbacher

Mathematics Faculty Publications

Background: Many problems in biomedicine and other areas of the life sciences can be characterized as control problems, with the goal of finding strategies to change a disease or otherwise undesirable state of a biological system into another, more desirable, state through an intervention, such as a drug or other therapeutic treatment. The identification of such strategies is typically based on a mathematical model of the process to be altered through targeted control inputs. This paper focuses on processes at the molecular level that determine the state of an individual cell, involving signaling or gene regulation. The mathematical model type …

A Note On The Γ-Coefficients Of The Tree Eulerian Polynomial, Rafael S. González D'León

Mathematics Faculty Publications

We consider the generating polynomial of the number of rooted trees on the set {1,2,...,n} counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered n-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. A classical product formula shows that this polynomial factors completely over the integers. From this product formula it can be concluded that this polynomial has positive coefficients …

A Model For Spheroid Versus Monolayer Response Of Sk-N-Sh Neuroblastoma Cells To Treatment With 15-Deoxy-Pgj2, Dorothy I. Wallace, Ann Dunham, Paula X. Chen, Michelle Chen, Milan Huynh, Evan Rheingold, Olivia F. Prosper

Mathematics Faculty Publications

Researchers have observed that response of tumor cells to treatment varies depending on whether the cells are grown in monolayer, as in vitro spheroids or in vivo. This study uses data from the literature on monolayer treatment of SK-N-SH neuroblastoma cells with 15-deoxy-PGJ₂ and couples it with data on growth rates for untreated SK-N-SH neuroblastoma cells grown as multicellular spheroids. A linear model is constructed for untreated and treated monolayer data sets, which is tuned to growth, death, and cell cycle data for the monolayer case for both control and treatment with 15-deoxy-PGJ₂. The monolayer …

The Spruce Budworm And Forest: A Qualitative Comparison Of Ode And Boolean Models, Raina Robeva, David Murrugarra

Mathematics Faculty Publications

Boolean and polynomial models of biological systems have emerged recently as viable companions to differential equations models. It is not immediately clear however whether such models are capable of capturing the multi-stable behaviour of certain biological systems: this behaviour is often sensitive to changes in the values of the model parameters, while Boolean and polynomial models are qualitative in nature. In the past few years, Boolean models of gene regulatory systems have been shown to capture multi-stability at the molecular level, confirming that such models can be used to obtain information about the system’s qualitative dynamics when precise information regarding …

Finite Groups In Which Pronomality And 𝔉-Pronormality Coincide, Adolfo Ballester-Bolinches, James C. Beidleman, Arnold D. Feldman, Matthew F. Ragland

Mathematics Faculty Publications

For a formation 𝔉, a subgroup U of a finite group G is said to be 𝔉-pronormal in G if for each g ∈ G, there exists x ∈ ⟨U, U^g⟩ ^𝔉 such that U^x = U^g. If 𝔉 contains 𝔑, the formation of nilpotent groups, then every 𝔉-pronormal subgroup is pronormal and, in fact, 𝔑-pronormality is just classical pronormality. The main aim of this paper is to study classes of finite soluble groups in which pronormality and 𝔉-pronormality coincide.

Molecular Network Control Through Boolean Canalization, David Murrugarra, Elena S. Dimitrova

Mathematics Faculty Publications

Boolean networks are an important class of computational models for molecular interaction networks. Boolean canalization, a type of hierarchical clustering of the inputs of a Boolean function, has been extensively studied in the context of network modeling where each layer of canalization adds a degree of stability in the dynamics of the network. Recently, dynamic network control approaches have been used for the design of new therapeutic interventions and for other applications such as stem cell reprogramming. This work studies the role of canalization in the control of Boolean molecular networks. It provides a method for identifying the potential edges …

Convergence Rates And Hölder Estimates In Almost-Periodic Homogenization Of Elliptic Systems, Zhongwei Shen

Mathematics Faculty Publications

For a family of second-order elliptic systems in divergence form with rapidly oscillating, almost-periodic coefficients, we obtain estimates for approximate correctors in terms of a function that quantifies the almost periodicity of the coefficients. The results are used to investigate the problem of convergence rates. We also establish uniform Hölder estimates for the Dirichlet problem in a bounded C^1,α domain.