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Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part Ii, Carol Jacoby, Peter Loth 2012 Sacred Heart University

Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part Ii, Carol Jacoby, Peter Loth

Mathematics Faculty Publications

We consider abelian groups with partial decomposition bases in Lδ∞ω for ordinals δ. Jacoby, Leistner, Loth and Str¨ungmann developed a numerical invariant deduced from the classical global Warfield invariant and proved that if two groups have identical modified Warfield invariants and Ulm-Kaplansky invariants up to ωδ for some ordinal δ, then they are equivalent in Lδ∞ω. Here we prove that the modified Warfield invariant is expressible in Lδ∞ω and hence the converse is true for appropriate δ.


White Noise Based Stochastic Calculus Associated With A Class Of Gaussian Processes, Daniel Alpay, Haim Attia, David Levanony 2012 Chapman University

White Noise Based Stochastic Calculus Associated With A Class Of Gaussian Processes, Daniel Alpay, Haim Attia, David Levanony

Mathematics, Physics, and Computer Science Faculty Articles and Research

Using the white noise space setting, we define and study stochastic integrals with respect to a class of stationary increment Gaussian processes. We focus mainly on continuous functions with values in the Kondratiev space of stochastic distributions, where use is made of the topology of nuclear spaces. We also prove an associated Ito formula.


Space-Time Codes, Non-Associative Division Algebras, And Elliptic Curves, Steve Limburg 2012 University of Colorado at Boulder

Space-Time Codes, Non-Associative Division Algebras, And Elliptic Curves, Steve Limburg

Mathematics Graduate Theses & Dissertations

Space-time codes are used to reliably send data from multiple transmit antennas and are directly related to non-associative division algebras. While interested in classifying and building space-time codes, using this relationship this thesis considers the corresponding problem of classifying and building non-associative division algebras. The first four chapters develop the problem and give a classification of 4-dimensional non-associative division algebras. Using the classification in chapter 4, I identify a class with no previously known examples. The rest of the thesis develops the background material necessary to understand two methods to build new non-associative division algebras in the aforementioned class. The ...


Bicomplex Numbers And Their Elementary Functions, M. E. Luna-Elizarrarás, M. Shapiro, Daniele C. Struppa, Adrian Vajiac 2012 E.S.F.M

Bicomplex Numbers And Their Elementary Functions, M. E. Luna-Elizarrarás, M. Shapiro, Daniele C. Struppa, Adrian Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we introduce the algebra of bicomplex numbers as a generalization of the field of complex numbers. We describe how to define elementary functions in such an algebra (polynomials, exponential functions, and trigonometric functions) as well as their inverse functions (roots, logarithms, inverse trigonometric functions). Our goal is to show that a function theory on bicomplex numbers is, in some sense, a better generalization of the theory of holomorphic functions of one variable, than the classical theory of holomorphic functions in two complex variables.


On The Class Rsi Of J-Contractive Functions Intertwining Solutions Of Linear Differential Equations, Daniel Alpay, Andrey Melnikov, Victor Vinnikov 2012 Chapman University

On The Class Rsi Of J-Contractive Functions Intertwining Solutions Of Linear Differential Equations, Daniel Alpay, Andrey Melnikov, Victor Vinnikov

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we extend and solve in the class of functions RSI mentioned in the title, a number of problems originally set for the class RS of rational functions contractive in the open right-half plane, and unitary on the imaginary line with respect to some preassigned self-adjoint matrix. The problems we consider include the Schur algorithm, the partial realization problem and the Nevanlinna-Pick interpolation problem. The arguments rely on the one-to-one correspondence between elements in a given subclass of RSI and elements in RS. Another important tool in the arguments is a new result pertaining to the classical tangential ...


Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part I, Carol Jacoby, Katrin Leistner, Peter Loth, Lutz Strungmann 2012 Sacred Heart University

Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part I, Carol Jacoby, Katrin Leistner, Peter Loth, Lutz Strungmann

Mathematics Faculty Publications

We consider the class of abelian groups possessing partial decomposition bases in Lδ∞ω for δ an ordinal. This class contains the class of Warfield groups which are direct summands of simply presented groups or, alternatively, are abelian groups possessing a nice decomposition basis with simply presented cokernel. We prove a classification theorem using numerical invariants that are deduced from the classical Ulm-Kaplansky and Warfield invariants. This extends earlier work by Barwise-Eklof, Göbel and the authors.


An Interpolation Problem For Functions With Values In A Commutative Ring, Daniel Alpay, Haim Attia 2012 Chapman University

An Interpolation Problem For Functions With Values In A Commutative Ring, Daniel Alpay, Haim Attia

Mathematics, Physics, and Computer Science Faculty Articles and Research

It was recently shown that the theory of linear stochastic systems can be viewed as a particular case of the theory of linear systems on a certain commutative ring of power series in a countable number of variables. In the present work we study an interpolation problem in this setting. A key tool is the principle of permanence of algebraic identities.


Stochastic Processes Induced By Singular Operators, Daniel Alpay, Palle Jorgensen 2012 Chapman University

Stochastic Processes Induced By Singular Operators, Daniel Alpay, Palle Jorgensen

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we study a general family of multivariable Gaussian stochastic processes. Each process is prescribed by a fixed Borel measure σ on Rn. The case when σ is assumed absolutely continuous with respect to Lebesgue measure was stud- ied earlier in the literature, when n = 1. Our focus here is on showing how different equivalence classes (defined from relative absolute continuity for pairs of measures) translate into concrete spectral decompositions of the corresponding stochastic processes under study. The measures σ we consider are typically purely singular. Our proofs rely on the theory of (singular) unbounded operators in Hilbert ...


New Topological C-Algebras With Applications In Linear Systems Theory, Daniel Alpay, Guy Salomon 2012 Chapman University

New Topological C-Algebras With Applications In Linear Systems Theory, Daniel Alpay, Guy Salomon

Mathematics, Physics, and Computer Science Faculty Articles and Research

Motivated by the Schwartz space of tempered distributions S′ and the Kondratiev space of stochastic distributions S−1 we define a wide family of nuclear spaces which are increasing unions of (duals of) Hilbert spaces H′p,p∈N, with decreasing norms |⋅|p. The elements of these spaces are functions on a free commutative monoid. We characterize those rings in this family which satisfy an inequality of the form |f∗g|p≤A(p−q)|f|q|g|p for all p≥q+d, where * denotes the convolution in the monoid, A(p−q) is a strictly positive number and ...


Spaces Of Sections Of Banach Algebra Bundles, Emmanuel Dror Farjoun, Claude Schochet 2012 Hebrew University of Jerusalem

Spaces Of Sections Of Banach Algebra Bundles, Emmanuel Dror Farjoun, Claude Schochet

Mathematics Faculty Research Publications

Suppose that B is a G-Banach algebra over 𝔽 = ℝ or ℂ, X is a finite dimensional compact metric space, ζ : PX is a standard principal G-bundle, and Aζ = Γ(X,P xG B) is the associated algebra of sections. We produce a spectral sequence which converges to π(GLoAζ) with

E_2p,qp(X ; πq(GLoB)).

A related spectral sequence converging to K∗+1(Aζ) (the real or complex topological K-theory) allows us to conclude that if B is Bott-stable, (i.e ...


The Jordan Canonical Form For A Class Of Zero–One Matrices, David A. Cardon, Bradford Tuckfield 2011 University of Pennsylvania

The Jordan Canonical Form For A Class Of Zero–One Matrices, David A. Cardon, Bradford Tuckfield

Operations, Information and Decisions Papers

Let f:N→N be a function. Let An=(aij) be the n×n matrix defined by aij=1 if i=f(j) for some i and j and aij=0 otherwise. We describe the Jordan canonical form of the matrix An in terms of the directed graph for which An is the adjacency matrix. We discuss several examples including a connection with the Collatz 3n+1 conjecture.


Finitely Presented Modules Over The Steenrod Algebra In Sage, Michael J. Catanzaro 2011 Wayne State University

Finitely Presented Modules Over The Steenrod Algebra In Sage, Michael J. Catanzaro

Wayne State University Theses

No abstract provided.


On Consensus In A Correlated Model Of Network Formation Based On A Polya Urn Process, Arastoo Fazeli, Ali Jadbabaie 2011 University of Pennsylvania

On Consensus In A Correlated Model Of Network Formation Based On A Polya Urn Process, Arastoo Fazeli, Ali Jadbabaie

Departmental Papers (ESE)

In this paper, we consider a consensus seeking process based on local averaging of opinions in a dynamic model of social network formation. At each time step, individual agents randomly choose another agent to interact with. The interaction is one-sided and results in the agent averaging her opinion with that of her randomly chosen neighbor. Once an agent chooses a neighbor, the probabilities of interactions are updated in such a way that prior interactions are reinforced and future interactions become more likely, resulting in a random consensus process in which networks are highly correlated with each other. Using results of ...


Analyzing Common Algebra-Related Misconceptions And Errors Of Middle School Students., Sarah B. Bush 2011 University of Louisville

Analyzing Common Algebra-Related Misconceptions And Errors Of Middle School Students., Sarah B. Bush

Electronic Theses and Dissertations

The purpose of this study was to examine common algebra-related misconceptions and errors of middle school students. In recent years, success in Algebra I is often considered the mathematics gateway to graduation from high school and success beyond. Therefore, preparation for algebra in the middle grades is essential to student success in Algebra I and high school. This study examines the following research question: What common algebra-related misconceptions and errors exist among students in grades six and eight as identified on student responses on an annual statewide standardized assessment? In this study, qualitative document analysis of existing data was used ...


Completeness Of Interacting Particles, Pavel Abramski 2011 Dublin Institute of Technology

Completeness Of Interacting Particles, Pavel Abramski

Doctoral

This thesis concerns the completeness of scattering states of n _-interacting particles in one dimension. Only the repulsive case is treated, where thereare no bound states and the spectrum is entirely absolutely continuous, so the scattering Hilbert space is the whole of L2(Rn). The thesis consists of 4 chapters: The first chapter describes the model, the scattering states as given by the Bethe Ansatz, and the main completeness problem. The second chapter contains the proof of the completeness relation in the case of two particles: n = 2. This case had in fact already been treated by B. Smit (1997 ...


The Eigenvalues Of A Tridiagonal Matrix In Biogeography, Boris Igelnik, Daniel J. Simon 2011 BMI Research

The Eigenvalues Of A Tridiagonal Matrix In Biogeography, Boris Igelnik, Daniel J. Simon

Electrical Engineering & Computer Science Faculty Publications

We derive the eigenvalues of a tridiagonal matrix with a special structure. A conjecture about the eigenvalues was presented in a previous paper, and here we prove the conjecture. The matrix structure that we consider has applications in biogeography theory.


The Square Discrete Exponentiation Map, A Wood 2011 DePaul University

The Square Discrete Exponentiation Map, A Wood

Mathematical Sciences Technical Reports (MSTR)

We will examine the square discrete exponentiation map and its properties. The square discrete exponentiation map is a variation on a commonly seen problem in cryptographic algorithms. This paper focuses on understanding the underlying structure of the functional graphs generated by this map. Specifically, this paper focuses on explaining the in-degree of graphs of safe primes, which are primes of the form p = 2q + 1, where q is also prime.


Base-Free Formulas In The Lattice-Theoretic Study Of Compacta, Paul Bankston 2011 Marquette University

Base-Free Formulas In The Lattice-Theoretic Study Of Compacta, Paul Bankston

Mathematics, Statistics and Computer Science Faculty Research and Publications

The languages of finitary and infinitary logic over the alphabet of bounded lattices have proven to be of considerable use in the study of compacta. Significant among the sentences of these languages are the ones that are base free, those whose truth is unchanged when we move among the lattice bases of a compactum. In this paper we define syntactically the expansive sentences, and show each of them to be base free. We also show that many well-known properties of compacta may be expressed using expansive sentences; and that any property so expressible is closed under inverse limits and co-existential ...


Factorization Of Primes Primes Primes: Elements Ideals And In Extensions, Peter J. Bonventre 2011 Union College - Schenectady, NY

Factorization Of Primes Primes Primes: Elements Ideals And In Extensions, Peter J. Bonventre

Honors Theses

It is often taken it for granted that all positive whole numbers except 0 and 1 can be factored uniquely into primes. However, if K is a finite extension of the rational numbers, and OK its ring of integers, it is not always the case that non-zero, non-unit elements of OK factor uniquely. We do find, though, that the proper ideals of OK do always factor uniquely into prime ideals. This result allows us to extend many properties of the integers to these rings. If we a finite extension L of K and OL of OK , we find that prime ...


The Derived Category And The Singularity Category, Mehmet U. Isik 2011 University of Pennsylvania

The Derived Category And The Singularity Category, Mehmet U. Isik

Publicly Accessible Penn Dissertations

We prove an equivalence between the derived category of a variety and the equivari- ant/graded singularity category of a corresponding singular variety. The equivalence also holds at the dg level.


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