Mathematics Commons^™

Krein Systems, Daniel Alpay, I. Gohberg, M. A. Kaashoek, L. Lerer, A. Sakhnovich

Mathematics, Physics, and Computer Science Faculty Articles and Research

In the present paper we extend results of M.G. Krein associated to the spectral problem for Krein systems to systems with matrix valued accelerants with a possible jump discontinuity at the origin. Explicit formulas for the accelerant are given in terms of the matrizant of the system in question. Recent developments in the theory of continuous analogs of the resultant operator play an essential role.

Superbimatrices And Their Generalizations, Florentin Smarandache, W.B Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The systematic study of supermatrices and super linear algebra has been carried out in 2008. These new algebraic structures find their applications in fuzzy models, Leontief economic models and data-storage in computers. In this book the authors introduce the new notion of superbimatrices and generalize it to super trimatrices and super n-matrices. Study of these structures is not only interesting and innovative but is also best suited for the computerized world. The main difference between simple bimatrices and super bimatrices is that in case of simple bimatrices we have only one type of product defined on them, whereas in case …

Rees Products Of Posets And Inequalities, Tricia Muldoon Brown

University of Kentucky Doctoral Dissertations

In this dissertation we will look at properties of two different posets from different perspectives. The first poset is the Rees product of the face lattice of the n-cube with the chain. Specifically we study the Möbius function of this poset. Our proof techniques include straightforward enumeration and a bijection between a set of labeled augmented skew diagrams and barred signed permutations which label the maximal chains of this poset. Because the Rees product of this poset is Cohen-Macaulay, we find a basis for the top homology group and a representation of the top homology group over the symmetric …

On A-Ary Subdivision For Curve Design Ii. 3-Point And 5-Point Interpolatory Schemes, Jian-Ao Lian

Applications and Applied Mathematics: An International Journal (AAM)

The a-ary 3-point and 5-point interpolatery subdivision schemes for curve design are introduced for arbitrary odd integer a greater than or equal to 3. These new schemes further extend the family of the classical 4- and 6-point interpolatory schemes.

On A-Ary Subdivision For Curve Design: I. 4-Point And 6-Point Interpolatory Schemes, Jian-Ao Lian

Applications and Applied Mathematics: An International Journal (AAM)

The classical binary 4-point and 6-point interpolatery subdivision schemes are generalized to a-ary setting for any integer a greater than or equal to 3. These new a-ary subdivision schemes for curve design are derived easily from their corresponding two-scale scaling functions, a notion from the context of wavelets.

Signed Decomposition Of Fully Fuzzy Linear Systems, Tofigh Allahviranloo, Nasser Mikaeilvand, Narsis A. Kiani, Rasol M. Shabestari

Applications and Applied Mathematics: An International Journal (AAM)

System of linear equations is applied for solving many problems in various areas of applied sciences. Fuzzy methods constitute an important mathematical and computational tool for modeling real-world systems with uncertainties of parameters. In this paper, we discuss about fully fuzzy linear systems in the form AX = b (FFLS). A novel method for finding the non-zero fuzzy solutions of these systems is proposed. We suppose that all elements of coefficient matrix A are positive and we employ parametric form linear system. Finally, Numerical examples are presented to illustrate this approach and its results are compared with other methods.

Representations Of The Temperley-Lieb Algebra, Anne Moore

Mathematics, Statistics, and Computer Science Honors Projects

This paper gives an introduction to Temperley-Lieb algebra that is easily accessible to undergraduates, presenting TL diagrams, the method for multiplying the diagrams, and the properties of the multiplication that it is necessary to preserve in a representation. The paper also gives a method for finding representations of the TL monoids (sets of diagrams classified by number of vertices) using Young tableaux, and shows that these representations are all of the irreducible representations. While ideas of Hecke algebra imply the fact that this method produces representations, this paper provides a direct proof, strictly within the field of representation theory. It …

A Classification Of Certain Maximal Subgroups Of Alternating Groups, Bret Benesh

Mathematics Faculty Publications

This paper addresses and extension of Problem 12.82 of the Kourovka notebook, which asks for all ordered pairs (n,m) such that the symmetric groups S_n embeds in S_m as a maximal subgroup. Problem 12.82 was answered in a previous paper by the author and Benjamin Newton. In this paper, we will consider the extension problem where we allow either or both of the groups from the ordered pair to be an alternating group.

Review: Lie Structure In Semiprime Superalgebras With Superinvolution, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Review: Some Remarks On Quantized Lie Superalgebras Of Classical Type, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Review: Degenerate Series Representations Of The Q-Deformed Algebra Soq′(R,S), Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Students' Perceptions Of Sense Of Community In Abstract Algebra: Contributing Factors And Benefits, Hortensia Soto-Johnson, Nissa Yestness, Casey Dalton

Mathematical Sciences Faculty Publications

In this phenomenological study, we explore how multiple assessments contribute to creating a sense of community (SOC) in an undergraduate abstract algebra course. Strike (2004) describes community as a process rather than a feeling and outlines four characteristics of community: coherence, cohesion, care, and contact. In this report, we describe contributing factors to and perceived benefits of SOC that students provided in an open-ended interview. Our findings indicate students viewed the teacher and the classroom environment as the primary sources of creating a SOC. Our findings also suggest students believed the SOC of the classroom increased classroom interaction and opened …

Chinese Remainder Theorem And Its Applications, Jacquelyn Ha Lac

Theses Digitization Project

No abstract provided.

Poincaré Duality, Christopher Michael Duran

Theses Digitization Project

This project is an expository study of the Poincaré duality theorem. Homology, cohomology groups of manifolds and other aglebraic and topological preliminaires are discussed.

Functorial Coalgebraic Logic: The Case Of Many-Sorted Varieties, Alexander Kurz, Daniela Petrişan

Engineering Faculty Articles and Research

Following earlier work, a modal logic for T-coalgebras is a functor L on a suitable variety. Syntax and proof system of the logic are given by presentations of the functor. This paper makes two contributions. First, a previous result characterizing those functors that have presentations is generalized from endofunctors on one-sorted varieties to functors between many-sorted varieties. This yields an equational logic for the presheaf semantics of higher-order abstract syntax. As another application, we show how the move to functors between many-sorted varieties allows to modularly combine syntax and proof systems of different logics. Second, we show how to associate …

Indestructible Blaschke Products, William T. Ross

Department of Math & Statistics Faculty Publications

No abstract provided.

Polynomial Extension Operators. Part I, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl

Mathematics and Statistics Faculty Publications and Presentations

In this series of papers, we construct operators that extend certain given functions on the boundary of a tetrahedron into the interior of the tetrahedron, with continuity properties in appropriate Sobolev norms. These extensions are novel in that they have certain polynomial preservation properties important in the analysis of high order finite elements. This part of the series is devoted to introducing our new technique for constructing the extensions, and its application to the case of polynomial extensions from H^½(∂K) into H¹(K), for any tetrahedron K.

A Mixed Method For Axisymmetric Div-Curl Systems, Dylan M. Copeland, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

We present a mixed method for a three-dimensional axisymmetric div-curl system reduced to a two-dimensional computational domain via cylindrical coordinates. We show that when the meridian axisymmetric Maxwell problem is approximated by a mixed method using the lowest order Nédélec elements (for the vector variable) and linear elements (for the Lagrange multiplier), one obtains optimal error estimates in certain weighted Sobolev norms. The main ingredient of the analysis is a sequence of projectors in the weighted norms satisfying some commutativity properties.

N- Linear Algebra Of Type I And Its Applications, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

With the advent of computers one needs algebraic structures that can simultaneously work with bulk data. One such algebraic structure namely n-linear algebras of type I are introduced in this book and its applications to n-Markov chains and n-Leontief models are given. These structures can be thought of as the generalization of bilinear algebras and bivector spaces. Several interesting n-linear algebra properties are proved. This book has four chapters. The first chapter just introduces n-group which is essential for the definition of nvector spaces and n-linear algebras of type I. Chapter two gives the notion of n-vector spaces and several …

A Characterization Of Schur Multipliers Between Character-Automorphic Hardy Spaces, Daniel Alpay, M. Mboup

Mathematics, Physics, and Computer Science Faculty Articles and Research

We give a new characterization of character-automorphic Hardy spaces of order 2 and of their contractive multipliers in terms of de Branges Rovnyak spaces. Keys tools in our arguments are analytic extension and a factorization result for matrix-valued analytic functions due to Leech.

Truncated Toeplitz Operators On Finite Dimensional Spaces, William T. Ross, Joseph A. Cima, Warren R. Wogen

Department of Math & Statistics Faculty Publications

In this paper, we study the matrix representations of compressions of Toeplitz operators to the finite dimensional model spaces H²ƟBH², where B is a finite Blaschke product. In particular, we determine necessary and sufficient conditions - in terms of the matrix representation - of when a linear transformation on H²ƟBH² is the compression of a Toeplitz operator. This result complements a related result of Sarason [6].

Writing Representations Over Proper Division Subrings, Stephen P. Glasby

All Faculty Scholarship for the College of the Sciences

Let �� be a division ring, and G a finite group of automorphisms of E whose elements are distinct modulo inner automorphisms of ��. Let �� = ��^G be the division subring of elements of �� fixed by G. Given a representation p : �� →��^d×d of an �� -algebra ��, we give necessary and sufficient conditions for p to be writable over ��. (Here ��^d×d denotes the algebra of d×d matrices over ��, and a matrix A writes p over �� if A⁻¹p(��)A ⊆ F^d×d.) We give an algorithm for constructing an …

Super Fuzzy Matrices And Super Fuzzy Models For Social Scientists, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Amal

Branch Mathematics and Statistics Faculty and Staff Publications

The concept of supermatrix for social scientists was first introduced by Paul Horst. The main purpose of his book was to introduce this concept to social scientists, students, teachers and research workers who lacked mathematical training. He wanted them to be equipped in a branch of mathematics that was increasingly valuable for the analysis of scientific data. This book introduces the concept of fuzzy super matrices and operations on them. The author has provided only those operations on fuzzy supermatrices that are essential for developing super fuzzy multi expert models. We do not indulge in labourious use of suffixes or …

Musical Actions Of Dihedral Groups, Alissa S. Crans, Thomas M. Fiore, Ramon Satyendra

Mathematics, Statistics and Data Science Faculty Works

The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.

Value Monoids Of Zero-Dimensional Valuations Of Rank 1, Edward Mosteig

Mathematics, Statistics and Data Science Faculty Works

Classically, Grobner bases are computed by first prescribing a set monomial order. Moss Sweedler suggested an alternative and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on k(x, y) that are suitable for this framework. For these valuations, we compute ν(k[x, y] ∗ ) and use this to perform computations concerning ideals in the polynomial ring k[x, y]. Interestingly, for these valuations, some ideals have a finite Grobner basis with respect to the valuation that is not …

Cohomology Of Categorical Self-Distributivity, J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi, Masahico Saito

Mathematics, Statistics and Data Science Faculty Works

We define self-distributive structures in the categories of coalgebras and cocommutative coalgebras. We obtain examples from vector spaces whose bases are the elements of finite quandles, the direct sum of a Lie algebra with its ground field, and Hopf algebras. The self-distributive operations of these structures provide solutions of the Yang–Baxter equation, and, conversely, solutions of the Yang–Baxter equation can be used to construct self-distributive operations in certain categories. Moreover, we present a cohomology theory that encompasses both Lie algebra and quandle cohomologies, is analogous to Hochschild cohomology, and can be used to study deformations of these self-distributive structures. All …

Iterated Aluthge Transforms: A Brief Survey, Jorge Antezana, Enrique R. Pujals, Demetrio Stojanoff

Publications and Research

Given an r × r complex matrix T, if T = U|T| is the polar decomposition of T, then the Aluthge transform is defined by

∆(T) = |T|^1/2U|T|^1/2.

Let ∆ⁿ(T) denote the n-times iterated Aluthge transform of T, i.e. ∆⁰(T) = T and ∆ⁿ(T) = ∆(∆ⁿ⁻¹(T)), n ∈ N. In this paper we make a brief survey on the known properties and applications of …

On The Reproducing Kernel Hilbert Spaces Associated With The Fractional And Bi-Fractional Brownian Motions, Daniel Alpay, David Levanony

Mathematics, Physics, and Computer Science Faculty Articles and Research

We present decompositions of various positive kernels as integrals or sums of positive kernels. Within this framework we study the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions. As a tool, we define a new function of two complex variables, which is a natural generalization of the classical Gamma function for the setting we consider.

Rational Functions Associated To The White Noise Space And Related Topics, Daniel Alpay, David Levanony

Mathematics, Physics, and Computer Science Faculty Articles and Research

Motivated by the hyper-holomorphic case we introduce and study rational functions in the setting of Hida’s white noise space. The Fueter polynomials are replaced by a basis computed in terms of the Hermite functions, and the Cauchy-Kovalevskaya product is replaced by the Wick product.

Super Linear Algebra, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, the authors introduce the notion of Super linear algebra and super vector spaces using the definition of super matrices defined by Horst (1963). This book expects the readers to be well-versed in linear algebra. Many theorems on super linear algebra and its properties are proved. Some theorems are left as exercises for the reader. These new class of super linear algebras which can be thought of as a set of linear algebras, following a stipulated condition, will find applications in several fields using computers. The authors feel that such a paradigm shift is essential in this computerized …