Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part Ii, 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 L^{δ}_{∞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 L^{δ}_{∞ω}. Here we prove that the modified Warfield invariant is expressible in L^{δ}_{∞ω} and hence the converse is true for appropriate *δ*.

White Noise Based Stochastic Calculus Associated With A Class Of Gaussian Processes, 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, 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, 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, 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, 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 *L ^{δ}*

_{∞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, 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, 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, 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, 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 : *P* → *X* is a standard principal *G*-bundle, and *A*_{ζ} = Γ(*X*,*P* x_{G} *B*) is the associated algebra of sections. We produce a spectral sequence which converges to π_{∗}(GL_{o}*A*_{ζ}) with

*E*_{_}^{2}_{p,q} ≅ *Ȟ ^{p}*(

*X*; π

_{q}(GL

_{o}

*B*)).

A related spectral sequence converging to *K*_{∗+1}(*A*_{ζ}) (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, 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 A_{n}=(a_{ij}) be the n×n matrix defined by a_{ij}=1 if i=f(j) for some *i* and *j* and a_{ij}=0 otherwise. We describe the Jordan canonical form of the matrix A_{n} in terms of the directed graph for which A_{n} 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, 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, 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., 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, 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, 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, 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, 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, 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, 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.