Physical Sciences and Mathematics Commons^™

Computing Singular Values Of Large Matrices With An Inverse-Free Preconditioned Krylov Subspace Method, Qiao Liang, Qiang Ye

Mathematics Faculty Publications

We present an efficient algorithm for computing a few extreme singular values of a large sparse m×n matrix C. Our algorithm is based on reformulating the singular value problem as an eigenvalue problem for C^TC. To address the clustering of the singular values, we develop an inverse-free preconditioned Krylov subspace method to accelerate convergence. We consider preconditioning that is based on robust incomplete factorizations, and we discuss various implementation issues. Extensive numerical tests are presented to demonstrate efficiency and robustness of the new algorithm.

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

Mathematics Faculty Publications

Since the 1970’s, great interest has been taken in the study of pure O-sequences, which are in bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these by their shape. It has been shown that all monomial complete intersections, Artinian algebras in two variables and Artinian level monomial algebras with type two in both three and four variables have unimodal Hilbert functions. This paper proves that Artinian level monomial algebras of type three in three variables have unimodal Hilbert functions. We will also discuss the licciness of these algebras.

Sensitivity Analysis Of Biological Boolean Networks Using Information Fusion Based On Nonadditive Set Functions, Naomi Kochi, Tomáš Helikar, Laura Allen, Jim A. Rogers, Zhenyuan Wang, Mihaela Teodora Matache

Mathematics Faculty Publications

Background: An algebraic method for information fusion based on nonadditive set functions is used to assess the joint contribution of Boolean network attributes to the sensitivity of the network to individual node mutations. The node attributes or characteristics under consideration are: in-degree, out-degree, minimum and average path lengths, bias, average sensitivity of Boolean functions, and canalizing degrees. The impact of node mutations is assessed using as target measure the average Hamming distance between a non-mutated/wild-type network and a mutated network.

Results: We find that for a biochemical signal transduction network consisting of several main signaling pathways whose nodes represent signaling …

Isometric Weighted Composition Operators, Valentin Matache

Mathematics Faculty Publications

A composition operator is an operator on a space of functions defined on the same set. Its action is by composition to the right with a fixed selfmap of that set. A composition operator followed by a multiplication operator is called a weighted composition operator. In this paper, we study when weighted composition operators on the Hilbert Hardy space of the open unit disc are isometric. We find their Wold decomposition in select cases and apply it to the computation of numerical ranges.

Thin Sequences And The Gram Matrix, Pamela Gorkin, John E. Mccarthy, Sandra Pott, Brett D. Wick

Mathematics Faculty Publications

We provide a new proof of Volberg’s Theorem characterizing thin interpolating sequences as those for which the Gram matrix associated to the normalized reproducing kernels is a compact perturbation of the identity. In the same paper, Volberg characterized sequences for which the Gram matrix is a compact perturbation of a unitary as well as those for which the Gram matrix is a Schatten-2 class perturbation of a unitary operator. We extend this characterization from 2 to p, where 2 p ≤∞.

On The Intersection Of Certain Maximal Subgroups Of A Finite Group, Adolfo Ballester-Bolinches, James C. Beidleman, Hermann Heineken, Matthew F. Ragland, Jack Schmidt

Mathematics Faculty Publications

Let Δ(G) denote the intersection of all non-normal maximal subgroups of a group G. We introduce the class of T₂-groups which are defined as the groups G for which G/Δ(G) is a T-group, that is, a group in which normality is a transitive relation. Several results concerning the class T₂ are discussed. In particular, if G is a solvable group, then Sylow permutability is a transitive relation in G if and only if every subgroup H of G is a T₂-group such that the nilpotent residual of H …

Evaluation Of Bias-Variance Trade-Off For Commonly Used Post-Summarizing Normalization Procedures In Large-Scale Gene Expression Studies, Zhixin Wu, Rui Hu, Xing Qiu

Mathematics Faculty Publications

Normalization procedures are widely used in high-throughput genomic data analyses to remove various technological noise and variations. They are known to have profound impact to the subsequent gene differential expression analysis. Although there has been some research in evaluating different normalization procedures, few attempts have been made to systematically evaluate the gene detection performances of normalization procedures from the bias-variance trade-off point of view, especially with strong gene differentiation effects and large sample size. In this paper, we conduct a thorough study to evaluate the effects of normalization procedures combined with several commonly used statistical tests and MTPs under different …

Zp-Modules With Partial Decomposition Bases In L Δ ∞Ω, Carol Jacoby, Peter Loth

Mathematics Faculty Publications

We consider the class of mixed Zp-modules with partial decomposition bases. This class includes those modules classified by Ulm and Warfield and is closed under L∞ω-equivalence. In the context of L∞ω- equivalence, Jacoby defined invariants for this class and proved a classification theorem. Here we examine this class relative to Lδ∞ω, those formulas of quantifier rank ≤ some ordinal δ, defining invariants and proving a classification theorem. This generalizes a result of Barwise and Eklof.

Fractional Generalizations Of Filtering Problems And Their Associated Fractional Zakai Equation, Sabir Umarov, Fred Daum, Kenric Nelson

Mathematics Faculty Publications

In this paper we discuss fractional generalizations of the filtering problem. The ”fractional” nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Lévy process.

Pythagorean Triples Challenge, Thomas Moore

Mathematics Faculty Publications

No abstract provided.

Smoothing Of Commutators For A Hörmander Class Of Bilinear Pseudodifferential Operators, Árpád Bényi, Tadahiro Oh

Mathematics Faculty Publications

Commutators of bilinear pseudodifferential operators with symbols in the Hörmander class BS¹_1,0 and multiplication by Lipschitz functions are shown to be bilinear Calderón-Zygmund operators. A connection with a notion of compactness in the bilinear setting for the iteration of the commutators is also made.

A Squeeze For Two Common Sequences That Converge To E, Branko Ćurgus

Mathematics Faculty Publications

In this note, we give a direct proof that {S_n} and {P_n} converge to the same limit. The main tool in our proof is the squeeze theorem, which is probably the easiest to prove among the limit theorems. However, to use it, we need to establish a relevant squeeze, which is the main result of this note.

A Proof Of The Main Theorem On Bezoutians, Branko Ćurgus, Aad Dijksma

Mathematics Faculty Publications

We give a self-contained proof that the nullity of the Bezoutian matrix associated with a pair of polynomials f and g equals the number of their common zeros counting multiplicities.

Computing Local Constants For Cm Elliptic Curves, Sunil Chetty, Lung Li

Mathematics Faculty Publications

Let E/k be an elliptic curve with CM by O. We determine a formula for (a generalization of) the arithmetic local constant of Mazur-Rubin at almost all primes of good reduction. We apply this formula to the CM curves defined over Q and are able to describe extensions F/Q over which the O-rank of E grows.

Conjugate Points For Fractional Differential Equations, Paul W. Eloe, Jeffrey T. Neugebauer

Mathematics Faculty Publications

Let b > 0. Let 1 < α ≤ 2. The theory of u 0-positive operators with respect to a cone in a Banach space is applied to study the conjugate boundary value problem for Riemann-Liouville fractional linear differential equations D 0+α u + λp(t)u = 0, 0 < t < b, satisfying the conjugate boundary conditions u(0) = u(b) = 0. The first extremal point, or conjugate point, of the conjugate boundary value problem is defined and criteria are established to characterize the conjugate point. As an application, a fixed point theorem is applied to give sufficient conditions for existence of a solution of a related boundary value problem for a nonlinear fractional differential equation.

Existence And Comparison Of Smallest Eigenvalues For A Fractional Boundary-Value Problem, Paul W. Eloe, Jeffrey T. Neugebauer

Mathematics Faculty Publications

The theory of u0-positive operators with respect to a cone in a Banach space is applied to the fractional linear differential equations

(see PDF)

0 < t < 1, with each satisfying the boundary conditions u(0) = u(1) = 0. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained.

A Generalization Of Poincaré-Cartan Integral Invariants Of A Nonlinear Nonholonomic Dynamical System, Muhammad Usman, M. Imran

Mathematics Faculty Publications

Based on the d'Alembert-Lagrange-Poincar\'{e} variational principle, we formulate general equations of motion for mechanical systems subject to nonlinear nonholonomic constraints, that do not involve Lagrangian undetermined multipliers. We write these equations in a canonical form called the Poincar\'{e}-Hamilton equations, and study a version of corresponding Poincar\'{e}-Cartan integral invariant which are derived by means of a type of asynchronous variation of the Poincar\'{e} variables of the problem that involve the variation of the time. As a consequence, it is shown that the invariance of a certain line integral under the motion of a mechanical system of the type considered characterizes the …

Tropical Convexity Over Max-Min Semiring, Viorel Nitica, Sergei Sergeev

Mathematics Faculty Publications

No abstract provided.

A Metric On Max-Min Algebra, Jonathan Eskeldson, Miriam Jaffe, Viorel Nitica

Mathematics Faculty Publications

No abstract provided.

An Explicit Construction Of Kleinian Groups With Small Limit Sets, Andrew Lazowski

Mathematics Faculty Publications

This paper provides an explicit construction of Kleinian groups that have small Hausdorff dimension of their limit sets. It is known that such groups exist and they can be constructed by results of Patterson. The purpose here is to work out the methods of calculation.

Global Convergence Of A Posteriori Error Estimates For A Discontinuous Galerkin Method For One-Dimensional Linear Hyperbolic Problems, Mahboub Baccouch

Mathematics Faculty Publications

In this paper we study the global convergence of the implicit residual-based a posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional linear hyperbolic problems. We apply a new optimal superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L² -norm under mesh refinement. The order of convergence is proved to be k + 2, when k-degree piecewise polynomials with k ≥ 1 are used. As a consequence, we …

Almost Contact Lagrangian Submanifolds Of Nearly Kaehler 6-Sphere, Ramesh Sharma, Sharief Deshmukh, Falleh Al-Solamy

Mathematics Faculty Publications

For a Lagrangian submanifold M of S 6 with nearly Kaehler structure, we provide conditions for a canonically induced almost contact metric structure on M by a unit vector field, to be Sasakian. Assuming M contact metric, we show that it is Sasakian if and only if the second fundamental form annihilates the Reeb vector field ξ, furthermore, if the Sasakian submanifold M is parallel along ξ, then it is the totally geodesic 3-sphere. We conclude with a condition that reduces the normal canonical almost contact metric structure on M to Sasakian or cosymplectic structure.

Almost Ricci Solitons And K-Contact Geometry, Ramesh Sharma

Mathematics Faculty Publications

We give a short Lie-derivative theoretic proof of the following recent result of Barros et al. “A compact non-trivial almost Ricci soliton with constant scalar curvature is gradient, and isometric to a Euclidean sphere”. Next, we obtain the result: a complete almost Ricci soliton whose metric g is K-contact and flow vector field X is contact, becomes a Ricci soliton with constant scalar curvature. In particular, for X strict, g becomes compact Sasakian Einstein.

Sasakian Metric As A Ricci Soliton And Related Results, Ramesh Sharma, Amalendu Ghosh

Mathematics Faculty Publications

We prove the following results: (i) a Sasakian metric as a non-trivial Ricci soliton is null η-Einstein, and expanding. Such a characterization permits us to identify the Sasakian metric on the Heisenberg group H2n+1 as an explicit example of (non-trivial) Ricci soliton of such type. (ii) If an η-Einstein contact metric manifold M has a vector field V leaving the structure tensor and the scalar curvature invariant, then either V is an infinitesimal automorphism, or M is D-homothetically fixed K-contact.

Hankel Vector Moment Sequences And The Non-Tangential Regularity At Infinity Of Two Variable Pick Functions, Jim Agler, John E. Mccarthy

Mathematics Faculty Publications

A Pick function of variables is a holomorphic map from to , where is the upper halfplane. Some Pick functions of one variable have an asymptotic expansion at infinity, a power series with real numbers that gives an asymptotic expansion on non-tangential approach regions to infinity. In 1921 H. Hamburger characterized which sequences can occur. We give an extension of Hamburger's results to Pick functions of two variables.

Lyapunov Functionals That Lead To Exponential Stability And Instability In Finite Delay Volterra Difference Equations, Catherine Kublik, Youssef Raffoul

Mathematics Faculty Publications

We use Lyapunov functionals to obtain sufficient conditions that guarantee exponential stability of the zero solution of the finite delay Volterra difference equation.

Also, by displaying a slightly different Lyapunov functional, we obtain conditions that guarantee the instability of the zero solution. The highlight of the paper is the relaxing of the condition |a(t)| < 1. Moreover, we provide examples in which we show that our theorems provide an improvement of some recent results.

Small-World Properties Of Facebook Group Networks, Jason Wohlgemuth, Mihaela Teodora Matache

Mathematics Faculty Publications

Small-world networks permeate modern society. In this paper we present a methodology for creating and analyzing a practically limitless number of networks exhibiting small-world network properties. More precisely, we analyze networks whose nodes are Facebook groups sharing a common word in the group name and whose links are mutual members in any two groups. By analyzing several numerical characteristics of single networks and network aggregations, we investigate how the small-world properties scale with a coarsening of the network. We show that Facebook group networks have small average path lengths and large clustering coefficients that do not vanish with increased network …

Superconvergence And A Posteriori Error Estimates Of A Local Discontinuous Galerkin Method For The Fourth-Order Initial-Boundary Value Problems Arising In Beam Theory, Mahboub Baccouch

Mathematics Faculty Publications

In this paper, we investigate the superconvergence properties and a posteriori error estimates of a local discontinuous Galerkin (LDG) method for solving the one-dimensional linear fourth-order initial-boundary value problems arising in study of transverse vibrations of beams. We present a local error analysis to show that the leading terms of the local spatial discretization errors for the k-degree LDG solution and its spatial derivatives are proportional to (k + 1)-degree Radau polynomials. Thus, the k-degree LDG solution and its derivatives are O(h^k+2) superconvergent at the roots of (k + 1)-degree Radau polynomials. Computational results indicate …

On The Maximum Leaf Number Of A Family Of Circulant Graphs, Felix P. Muga Ii

Mathematics Faculty Publications

This paper determines the maximum leaf number and the connected domination number of some undirected and connected circulant networks which are optimal among all the maximum leaf numbers and connected domination numbers of circulant networks of the same order n and the same degree 2k. We shall tackle this problem by working on the largest possible number of vertices between two consecutive jump sizes.