Mathematics Commons^™

Stochastic Models Of Evidence Accumulation In Changing Environments, Alan Veliz-Cuba, Zachary P. Kilpatrick, Krešimir Josić

Mathematics Faculty Publications

Organisms and ecological groups accumulate evidence to make decisions. Classic experiments and theoretical studies have explored this process when the correct choice is fixed during each trial. However, we live in a constantly changing world. What effect does such impermanence have on classical results about decision making? To address this question we use sequential analysis to derive a tractable model of evidence accumulation when the correct option changes in time. Our analysis shows that ideal observers discount prior evidence at a rate determined by the volatility of the environment, and the dynamics of evidence accumulation is governed by the information …

Necessary And Sufficient Conditions For Stability Of Volterra Integro-Dynamic Equation Systems On Time Scales, Youssef Raffoul

Mathematics Faculty Publications

In this research we establish necessary and sufficient conditions for the stability of the zero solution of scalar Volterra integro-dynamic equation on general time scales. Our approach is based on the construction of suitable Lyapunov functionals. We will compare our findings with known results and provides application to quantum calculus.

Mathematical And Anthropological Analysis Of Northern Luzon Funeral Textile, Ma. Louise Antonette N. De Las Peñas, Analyn V. Salvador-Amores

Mathematics Faculty Publications

The study presents a mathematical analysis and provides an anthropological perspective of the funeral textile of the indigenous communities in northern Luzon, Philippines. In particular, a symmetry analysis is performed, based on principles of group theory and transformation geometry, on the various repeating patterns found in funeral garments and blankets. Results show that particular frieze groups and plane crystallographic groups are favored due to choice of motifs which are reflective of cultural beliefs and funeral traditions, as well as weaving style and methodology. The results of the analysis point to the depth of mathematics present in the work of the …

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.

Canonoid And Poissonoid Transformations, Symmetries And Bihamiltonian Structures, Giovanni Rastelli, Manuele Santoprete

Mathematics Faculty Publications

We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Utilizing this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler’s equations of the rigid body (on so*(3) and so*(4)) and for an integrable …

On The Relationship Between Two Notions Of Compatibility For Bi-Hamiltonian Systems, Manuele Santoprete

Mathematics Faculty Publications

Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems. Because of this, a few different notions of compatibility have been introduced. In this paper we show that, under some additional assumptions, compatibility in the sense of Magri implies a notion of compatibility due to Fass`o and Ratiu, that we dub bi-affine compatibility. We present two proofs of this fact. The first one uses the uniqueness of the connection parallelizing all the Hamiltonian vector fields tangent to the leaves of a Lagrangian foliation. …

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 …

From Subcompact To Domain Representable, William Fleissner, Lynne Yengulalp

Mathematics Faculty Publications

No abstract provided.

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.

Candy Crush Combinatorics, Dana Rowland

Mathematics Faculty Publications

In the popular game Candy Crush, differently colored candies are arranged in a grid and a player swaps adjacent candies in order to crush them by lining up three or more of the same color. At the beginning of each game, the grid cannot have three consecutive candies of the same color in a row or column, but it must be possible to swap two adjacent candies in order to get at least three consecutive candies of the same color. How many starting configurations are there? We derive recurrence relations to answer this question for a single line of candy, …

Improving The Solution Time Of Integer Programs By Merging Knapsack Constraints With Cover Inequalities, Fabio Vitor

Mathematics Faculty Publications

Integer Programming is used to solve numerous optimization problems. This class of mathematical models aims to maximize or minimize a cost function restricted to some constraints and the solution must be integer. One class of widely studied Integer Program (IP) is the Multiple Knapsack Problem (MKP). Unfortunately, both IPs and MKPs are NP-hard, potentially requiring an exponential time to solve these problems.

Utilization of cutting planes is one common method to improve the solution time of IPs. A cutting plane is a valid inequality that cuts off a portion of the linear relaxation space. This thesis presents a new class …

Random Sampling Of Skewed Distributions Implies Taylor’S Power Law Of Fluctuation Scaling, Joel E. Cohen, Meng Xu

Mathematics Faculty Publications

Taylor’s law (TL), a widely verified quantitative pattern in ecology and other sciences, describes the variance in a species’ population density (or other nonnegative quantity) as a power-law function of the mean density (or other nonnegative quantity): Approximately, variance = a(mean)b, a > 0. Multiple mechanisms have been proposed to explain and interpret TL. Here, we show analytically that observations randomly sampled in blocks from any skewed frequency distribution with four finite moments give rise to TL. We do not claim this is the only way TL arises. We give approximate formulae for the TL parameters and their …

On Spectra Of Composition Operators, Valentin Matache

Mathematics Faculty Publications

In this paper we consider composition operators Cφ on the Hilbert Hardy space over the unit disc, induced by analytic selfmaps φ. We use the fact that the operator C∗φCφ is asymptotically Toeplitz to obtain information on the essential spectrum and spectrum of Cϕ, which we are able to describe in select cases (including the case of some hypercyclic composition operators or that of composition operators with the property that the asymptotic symbol of C∗φCφ is constant a.e.). One of our tools is the Nikodym derivative of the pull-back measure induced by φ. An alternative formula for the essential norm …

Difference Equation For Tracking Perturbations In Systems Of Boolean Nested Canalyzing Functions, Elena S. Dimitrova, Oleg I. Yordanov, Mihaela Teodora Matache

Mathematics Faculty Publications

This paper studies the spread of perturbations through networks composed of Boolean functions with special canalyzing properties. Canalyzing functions have the property that at least for one value of one of the inputs the output is fixed, irrespective of the values of the other inputs. In this paper the focus is on partially nested canalyzing functions, in which multiple, but not all inputs have this property in a cascading fashion. They naturally describe many relationships in real networks. For example, in a gene regulatory network, the statement “if gene A is expressed, then gene B is not expressed regardless of …

On Color Fixing Groups Associated With Colored Symmetrical Tilings, April Lynne D. Say-Awen, Ma. Louise Antonette N. De Las Peñas, Teofina A. Rapanut

Mathematics Faculty Publications

In this paper, we contribute to the study of colored symmetrical tilings by giving formulas for their associated color fixing groups. In the second part of the paper we provide an application of the results in describing symmetry groups of nanostructures.

Open And Dense Topological Transitivity Of Extensions By Non-Compact Fiber Of Hyperbolic Systems: A Review, Viorel Nitica, Andrei Török

Mathematics Faculty Publications

Currently, there is great renewed interest in proving the topological transitivity of various classes of continuous dynamical systems. Even though this is one of the most basic dynamical properties that can be investigated, the tools used by various authors are quite diverse and are strongly related to the class of dynamical systems under consideration. The goal of this review article is to present the state of the art for the class of Hölder extensions of hyperbolic systems with non-compact connected Lie group fiber. The hyperbolic systems we consider are mostly discrete time. In particular, we address the stability and genericity …

Unit Origami: Star-Building On Deltahedra, Heidi Burgiel

Mathematics Faculty Publications

This workshop provides instructions for folding the star-building unit – a modification of the Sonobe module for unit origami. Geometric questions naturally arise during this process, ranging in difficulty from middle school to graduate levels. Participants will learn to fold and assemble star-building units, then explore the structure of the eight strictly convex deltahedra.

Qualitative Theory Of Functional Differential And Integral Equations, Muhammad Islam, Cemil Tunc, Mouffak Benchohra, Bingwen Lui, Samir H. Saker

Mathematics Faculty Publications

Functional differential equations arise in many areas of science and technology: whenever a deterministic relationship involving some varying quantities and their rates of change in space and/or time (expressed as derivatives or differences) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time varies. In some cases, this differential equation (called an equation of motion) may be solved explicitly. In fact, differential equations play an important role in modelling virtually every physical, technical, biological, ecological, and epidemiological process, from celestial motion, to bridge design, …

Bounded, Asymptotically Stable, And L^1 Solutions Of Caputo Fractional Differential Equations, Muhammad Islam

Mathematics Faculty Publications

The existence of bounded solutions, asymptotically stable solutions, and L¹ solutions of a Caputo fractional differential equation has been studied in this paper. The results are obtained from an equivalent Volterra integral equation which is derived by inverting the fractional differential equation. The kernel function of this integral equation is weakly singular and hence the standard techniques that are normally applied on Volterra integral equations do not apply here. This hurdle is overcomed using a resolvent equation and then applying some known properties of the resolvent. In the analysis Schauder's fixed point theorem and Liapunov's method have been employed. …

Symmetry Groups Associated With Tilings On A Flat Torus, Ma. Louise Antonette N. De Las Peñas, Mark L. Loyola, Grace M. Estrada, Eko Budi Santoso

Mathematics Faculty Publications

This work investigates symmetry and color symmetry properties of Kepler, Heesch and Laves tilings embedded on a flat torus and their geometric realizations as tilings on a round torus in Euclidean 3-space. The symmetry group of the tiling on the round torus is determined by analyzing relevant symmetries of the planar tiling that are transformed to axial symmetries of the three-dimensional tiling. The focus on studying tilings on a round torus is motivated by applications in the geometric modeling of nanotori and the determination of their symmetry groups.

Homogenization Of Stokes Systems And Uniform Regularity Estimates, Shu Gu, Zhongwei Shen

Mathematics Faculty Publications

This paper is concerned with uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and L^∞ estimates for the pressure as well as a Liouville property for solutions in ℝ^d. We also obtain the boundary W^1,p estimates in a bounded C¹ domain for any 1 < p < ∞.

Control, Stability, And Qualitative Theory Of Dynamical Systems 2014, Nazim I. Mahmudov, Mark A. Mckibben, Sakthivel Rathinasamy, Yong Ren

Mathematics Faculty Publications

No abstract provided.

Further Results On Vanishing Coefficients In Infinite Product Expansions, James Mclaughlin

Mathematics Faculty Publications

We extend results of Andrews and Bressoud on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if (q r−tk, qmk−(r−tk) ; q mk)∞ (q r, qmk−r; qmk)∞ =: X∞ n=0 cnq n , for certain integers k, m s and t, where r = sm+t, then ckn−rs is always zero. Our theorems also partly give a simpler reformulation of results of Alladi and Gordon, but also give results for cases not covered by the theorems of Alladi and Gordon. We also give some interpretations of the analytic results in terms …

Asymptotically Exact Local Discontinuous Galerkin Error Estimates For The Linearized Korteweg-De Vries Equation In One Space Dimension, Mahboub Baccouch

Mathematics Faculty Publications

We present and analyze a posteriori error estimates for the local discontinuous Galerkin (LDG) method for the linearized Korteweg-de Vries (KdV) equation in one space dimension. These estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We extend the work of Hufford and Xing [J. Comput. Appl. Math., 255 (2014), pp. 441-455] to prove new superconvergence results towards particular projections of the exact solutions for the two auxiliary variables in the LDG method that approximate the first and second derivatives of the solution. The order of convergence …

Sobriety In Delta Not Sober, Joe Mashburn

Mathematics Faculty Publications

We will show that the space delta not sober defined by Coecke and Martin is sober in the Scott topology, but not in the weakly way below topology.

Multi-Term Linear Fractional Nabla Difference Equations With Constant Coefficients, Paul W. Eloe, Zi Ouyang

Mathematics Faculty Publications

We shall consider a linear fractional nabla (backward) fractional difference equation of Riemann–Liouville type with constant coefficients. We apply a transform method to construct solutions. Sufficient conditions in terms of the coefficients are given so that the solutions are absolutely convergent. The method is known for two-term fractional difference equations; the method is new for fractional equations with three or more terms. As a corollary, we exhibit new summation representations of a discrete exponential function, at, t = 0; 1; : : : .

The Unimodality Of Pure O-Sequences Of Type Two In Four Variables, Bernadette Boyle

Mathematics Faculty Publications

Since the 1970's, great interest has been taken in the study of pure O-sequences, which, due to Macaulay's theory of inverse systems, have a bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these according to their shape. Macaulay's theorem immediately gives us that all Artinian algebras in two variables have unimodal Hilbert functions. Furthermore, it has been shown that all Artinian level monomial algebras of type two in three variables have unimodal Hilbert functions. This paper will classify all Artinian level monomial algebras of type two in four variables into …

Pure Injective And *-Pure Injective Lca Groups, Peter Loth

Mathematics Faculty Publications

No abstract provided.

Some Properties Of The Exchange Operator With Respect To Structured Matrices Defined By Indefinite Scalar Product Spaces, Hanz Martin C. Cheng, Roden Jason David

Mathematics Faculty Publications

The properties of the exchange operator on some types of matrices are explored in this paper. In particular, the properties of exc(A,p,q), where A is a given structured matrix of size (p+q)Ã(p+q) and exc : M ÃNÃN â M is the exchange operator are studied. This paper is a generalization of one of the results in [N.J. Higham. J-orthogonal matrices: Properties and generation. SIAM Review, 45:504â519, 2003.].

Cauchy Extension Of Mα-Integral And Absolute Mα-Integrability, Ian June L. Garces, Abraham P. Racca

Mathematics Faculty Publications

Park, Ryu, and Lee recently introduced a Henstock-type integral, which lies between the Mcshane and the Henstock integrals. This paper proves the closure property of this new integral under Cauchy extension, and presents a characterization on absolute Mα-integrability