Mathematics Commons^™

Note On Gradient Estimates Of Heat Kernel For Schrödinger Operators, Shijun Zheng

Mathematical Sciences: Faculty Publications

Let H = -Δ+V be a Schrödinger operator on Rⁿ. We show that gradient estimates for the heat kernel of H with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for L^p and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent proofs. We give a counterexample in one dimension to show that there exists V in the Schwartz class such that the long time gradient heat …

On The Characteristic Polynomial Of Regular Linear Matrix Pencil, Yan Wu, Phillip Lorren

Mathematical Sciences: Faculty Publications

Linear matrix pencil, denoted by (A,B), plays an important role in control systems and numerical linear algebra. The problem of finding the eigenvalues of (A,B) is often solved numerically by using the well-known QZ method. Another approach for exploring the eigenvalues of (A,B) is by way of its characteristic polynomial, P(λ)=A − λB. There are other applications of working directly with the characteristic polynomial, for instance, using Routh-Hurwitz analysis to count the stable roots of P(λ) and transfer function representation of control systems governed by differential-algebraic equations. In this paper, we …

A Unified Theory Of Function Spaces And Hyperspaces: Local Properties, Szymon Dolecki, Frédéric D. Mynard

Mathematical Sciences: Faculty Publications

Many classically used function space structures (including the topology of pointwise convergence, the compact-open topology, the Isbell topology and the continuous convergence) are induced by a hyperspace structure counterpart. This scheme is used to study local properties of function space structures on C(X,R), such as character, tighntess, fan-tightness, strong fan-tightness, the Fr{\'e}chet property and some of its variants. Under mild conditions, local properties of C(X,R) at the zero function correspond to the same property of the associated hyperspace structure at X. The latter is often easy to characterize in terms of covering properties …

Some Implications Of Chu's 10Ψ10 Generalization Of Bailey's 6Ψ6 Summation Formula, James Mclaughlin, Andrew Sills, Peter Zimmer

Mathematical Sciences: Faculty Publications

Lucy Slater used Bailey's 6ψ6 summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type.

In the present paper we apply the same techniques to Chu's 10ψ10 generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then recovered as special limiting cases of these more general pairs.

In re-examining Slater's work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find new …

Rademacher-Type Formulas For Partitions And Overpartitions, Andrew Sills

Mathematical Sciences: Faculty Publications

A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of n and the Zuckerman formula for the Fourier coefficients of ϑ4_0 | τ_−1 is presented.

Generalized Complex Hamiltonian Torus Actions: Examples And Constraints, Thomas Baird, Yi Lin

Mathematical Sciences: Faculty Publications

Consider an effective Hamiltonian torus action T×M→M on a topologically twisted,generalized complex manifold M of dimension 2n. We prove that the rank(T)≤n−2 and that the topological twisting survives Hamiltonian reduction. We then construct a large new class of such actions satisfying rank(T)=n−2, using a surgery procedure on toric manifolds.

Towards An Automation Of The Circle Method, Andrew Sills

Mathematical Sciences: Faculty Publications

The derivation of the Hardy-Ramanujan-Rademacher formula for the number of partitions of n is reviewed. Next, the steps for finding analogous formulas for certain restricted classes of partitions or overpartitions is examined, bearing in mind how these calculations can be automated in a CAS. Finally, a number of new formulas of this type which were conjectured with the aid of Mathematica are presented along with results of a test for their numerical accuracy.

Homological Dimensions And Regular Rings, Alina Iacob, Srikanth B. Iyengar

Mathematical Sciences: Faculty Publications

A question of Avramov and Foxby concerning injective dimension of complexes is settled in the affirmative for the class of noetherian rings. A key step in the proof is to recast the problem on hand into one about the homotopy category of complexes of injective modules. Analogous results for flat dimension and projective dimension are also established.

Interior-Point Algorithms For A Class Of Convex Optimization Problems, Goran Lesaja, Verlynda Slaughter

Mathematical Sciences: Faculty Publications

In this paper we consider interior-point methods (IPM) for the nonlinear, convex optimization problem where the objective function is a weighted sum of reciprocals of variables subject to linear constraints (SOR). This problem appears often in various applications such as statistical stratified sampling and entropy problems, to mention just few examples. The SOR is solved using two IPMs. First, a homogeneous IPM is used to solve the Karush-Kuhn-Tucker conditions of the problem which is a standard approach. Second, a homogeneous conic quadratic IPM is used to solve the SOR as a reformulated conic quadratic problem. As far as we are …

Lifting Bailey Pairs To Wp-Bailey Pairs, James Mclaughlin, Andrew Sills, Peter Zimmer

Mathematical Sciences: Faculty Publications

A pair of sequences (α_n(a,k,q),β_n(a,k,q)) such that α₀(a,k,q)=1 and β_n(a,k,q)=∑ⁿ_j=0 (k/a;q)_n−j (k;q)_n+j / (q;q)_n−j (aq;q)_n+j α_j (a,k,q) is termed a WP-Bailey Pair. Upon setting k=0 in such a pair we obtain a Bailey pair.

In the present paper we consider the problem of “lifting” a Bailey pair to a WP-Bailey pair, and use some …

Rogers-Ramanujan Computer Searches, James Mclaughlin, Andrew Sills, Peter Zimmer

Mathematical Sciences: Faculty Publications

We describe three computer searches (in PARI/GP, Maple, and Mathematica, respectively) which led to the discovery of a number of identities of Rogers–Ramanujan type and identities of false theta functions.

Some More Identities Of Rogers-Ramanujan Type, Douglas Bowman, James Mclaughlin, Andrew Sills

Mathematical Sciences: Faculty Publications

In this we paper we prove several new identities of the Rogers-Ramanujan-Slater type. These identities were found as the result of computer searches. The proofs involve a variety of techniques, including series-series identities, Bailey pairs, a theorem of Watson on basic hypergeometric series, generating functions and miscellaneous methods.

Interpolation Theorems For Self-Adjoint Operators, Shijun Zheng

Mathematical Sciences: Faculty Publications

We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a self adjoint operator L, without assuming the gradient estimate for its spectral kernel. The result applies to the cases where L is a uniformly elliptic operator or a Schrödinger operator with electro-magnetic potential.

Introducing Interior-Point Methods For Introductory Operations Research Courses And/Or Linear Programming Courses, Goran Lesaja

Mathematical Sciences: Faculty Publications

In recent years the introduction and development of Interior-Point Methods has had a profound impact on optimization theory as well as practice, influencing the field of Operations Research and related areas. Development of these methods has quickly led to the design of new and efficient optimization codes particularly for Linear Programming. Consequently, there has been an increasing need to introduce theory and methods of this new area in optimization into the appropriate undergraduate and first year graduate courses such as introductory Operations Research and/or Linear Programming courses, Industrial Engineering courses and Math Modeling courses. The objective of this paper is …

Gorenstein Flat Dimension Of Complexes, Alina Iacob

Mathematical Sciences: Faculty Publications

We define a notion of Gorenstein flat dimension for unbounded complexes over left GF-closed rings. Over Gorenstein rings we introduce a notion of Gorenstein cohomology for complexes; we also define a generalized Tate cohomology for complexes over Gorenstein rings, and we show that there is a close connection between the absolute, the Gorenstein and the generalized Tate cohomology.

On Fourier Parabolic And Wave Equation, Mekki Terbeche, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

This paper is devoted to a diffusion (heat) and wave equations for the function of two independent variables. We establish a criterion for existence and uniqueness of the solution of Fourier parabolic equation using Taylor series. A formal solution for a wave equation is investigated.

On Some Differential Equations, Mekki Terbeche, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

This paper investigates Cauchy and Goursat problems for partial differential operators. Successive approximation techniques for partial differential equations and the estimated results are employed to obtain the existence and the uniqueness of the solutions of such problems. An extended Darboux-Goursat-Beudon problem is studied.

Combinatorics Of Ramanujan-Slater Type Identities, James Mclaughlin, Andrew Sills

Mathematical Sciences: Faculty Publications

We provide the missing member of a family of four q-series identities related to the modulus 36, the other members having been found by Ramanujan and Slater. We examine combinatorial implications of the identities in this family, and of some of the identities we considered in Identities of the Ramanujan-Slater type related to the moduli 18 and 24.

Establishment Of Weak Conditions For Darboux-Goursat-Beudon Theorem, Mekki Terbeche, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

This paper is devoted to the establishment of weak conditions for Darboux-Goursat-Beudon (DGB) theorem in order to improve analogous results in [1, 7]. By adapting a technique proposed by [7] in another setting [8] via majorant method, we obtain the generalization of DGB theorem.

Ramanujan–Slater Type Identities Related To The Moduli 18 And 24, James Mclaughlin, Andrew Sills

Mathematical Sciences: Faculty Publications

We present several new families of Rogers–Ramanujan type identities related to the moduli 18 and 24. A few of the identities were found by either Ramanujan, Slater, or Dyson, but most are believed to be new. For one of these families, we discuss possible connections with Lie algebras. We also present two families of related false theta function identities.

Rogers-Ramanujan-Slater Type Identities, James Mclaughlin, Andrew Sills, Peter Zimmer

Mathematical Sciences: Faculty Publications

In this survey article, we present an expanded version of Lucy Slater's famous list of identities of the Rogers-Ramanujan type, including identities of similar type, which were discovered after the publication of Slater's papers, and older identities (such as those in Ramanujan's lost notebook) which were not included in Slater's papers. We attempt to supply the earliest known reference for each identity. Also included are identities of false theta functions, along with their relationship to Rogers-Ramanujan type identities. We also describe several ways in which pairs/larger sets of identities may be related, as well as dependence relationships between identities.

Disturbing The Q-Dyson Conjecture, Andrew Sills

Mathematical Sciences: Faculty Publications

I discuss the computational methods behind the formulation of some conjectures related to variants on Andrews’ q-Dyson conjecture.

Inequalities And Exponential Approximations For Residual Life Reliability Functions, Broderick O. Oluyede, Marvis Pararai

Mathematical Sciences: Faculty Publications

Given that a unit is of age t, the remaining life after time t is random. The expected value of this random residual life is called the mean residual life at time t. Specifically, if T is the life of a component with distribution function F, then δF (t) = E(T −t|T > t) is called the mean residual life function (MRLF). It is well known that the class of distributions with decreasing mean residual life (DMR) contains the class of distributions with increasing hazard rate (IHR). In this note, …

A Note On Extreme Bernoulli And Dependent Families Of Bivariate Distributions, Broderick O. Oluyede, Marvis Pararai

Mathematical Sciences: Faculty Publications

The objective and purpose of this note is to generate bivariate distributions via extreme Bernoulli distributions and obtain results on positively and negatively dependent families of bivariate binomial distributions generated by extreme Bernoulli distributions. Some distributional properties and results are presented. The factorial moment generating functions, correlation functions, conditional distributions and the regression functions are given.

A Partition Bijection Related To The Rogers-Selberg Identities And Gordon's Theorem, Andrew Sills

Mathematical Sciences: Faculty Publications

We provide a bijective map from the partitions enumerated by the series side of the Rogers–Selberg mod 7 identities onto partitions associated with a special case of Basil Gordon's combinatorial generalization of the Rogers–Ramanujan identities. The implications of applying the same map to a special case of David Bressoud's even modulus analog of Gordon's theorem are also explored.

On The Ordinary And Signed Göllnitz-Gordon Partitions, Andrew Sills

Mathematical Sciences: Faculty Publications

A partition of an integer n is a representation of n as an unordered sum of positive integers. In a recent paper [1], Andrews introduced the notion of a "signed partition," that is, a representation of a positive integer as an unordered sum of integers, some possibly negative.

Harmonic Analysis Related To Schrödinger Operators, Gestur Olafsson, Shijun Zheng

Mathematical Sciences: Faculty Publications

In this article, we give an overview of some recent developments in Littlewood-Paley theory for Schrödinger operators. We extend the Littlewood-Paley theory for special potentials considered in our previous work [J. Fourier Anal. Appl. 12 (2006), no. 6, 653–674; MR2275390]. We elaborate our approach by considering a potential in C^∞₀ or the Schwartz class in one dimension. In particular, the low energy estimates are treated by establishing some new and refined asymptotics for the eigenfunctions and their Fourier transforms. We give a maximal function characterization of the Besov spaces and Triebel-Lizorkin spaces associated with H. Then we …

On The Simplification Of Certain Q-Multisums, Andrew Sills

Mathematical Sciences: Faculty Publications

Some examples of naturally arising multisum q-series which turn out to have representations as fermionic single sums are presented. The resulting identities are proved using transformation formulas from the theory of basic hypergeometric series.

Bounds And Comparisons For Weighted Renewal-Type Integral Equations, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

In this note, inequalities and bounds for weighted renewal-type integral equations are presented. Some upper and lower bounds for the weighted renewal-type integral equations with monotone weight functions are derived. Some upper and lower bounds for the weighted renewal-type equations with monotone weight functions are derived. Bounds for the difference between two weighted renewal functions as well between the parent and weighted renewal functions are obtained in terms of the parent renewal reliability functions and their first and second moments. Relations for renewal-type integrals of the ruin probability are presented. Some inequalities, bounds and convergence results are also established.

Closure Under Transfinite Extensions, Edgar Enochs, Alina Iacob, Overtoun Jenda

Mathematical Sciences: Faculty Publications

The closure under extensions of a class of objects in an abelian category is often an important property of that class. Recently the closure of such classes under transfinite extensions (both direct and inverse) has begun to play an important role in several areas of mathematics, for example, in Quillen's theory of model categories and in the theory of cotorsion pairs. In this paper we prove that several important classes are closed under transfinite extensions.