Mathematics Commons^™

2004 (Winter), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2004 Winter Colloquium

Conversations Among Women In Mathematics (Program), University Of Dayton. Department Of Mathematics

Biennial Alumni Seminar

No abstract provided.

On A Certain Class Of Bessel Integrals, Ali A. Al-Jarrah, A. Al-Momani

Turkish Journal of Mathematics

There are many old results of integrals involving Bessel functions, currently available in handbooks, but we found no recourse in the well-known references to how they were established. In this paper, we attempt to have a clear way of proving some of these results. In fact, we consider a certain class of Bessel integrals where we prove that such integrals vanish under certain conditions. To this end some theorems regarding this class of integrals with their proofs are put forward. A computer algorithm is provided to implement some of our results. The result in this paper extend the work in …

A Note On Groups With All Subgroups Subnormal, Ahmet Arikan, Tahi̇re Özen

Turkish Journal of Mathematics

We prove that if G is a periodic group with all subgroups subnormal, and if for every x, y \in G, ^{G} is an FC-group, then G is nilpotent.

On The Cotorsion Images Of The Baer-Specker Group, Brendan Goldsmith, T. Kelly, S, Wallutis

Articles

No abstract available

Quasi-Minimal Abelian Groups, Brendan Goldsmith, S. O. Hogain, S. Wallutis

Articles

An abelian group $G$ is said to be quasi-minimal (purely quasi-minimal, directly quasi-minimal) if it is isomorphic to all its subgroups (pure subgroups, direct summands, respectively) of the same cardinality as $G$. Obviously quasi-minimality implies pure quasi-minimality which in turn implies direct quasi-minimality, but we show that neither converse implication holds. We obtain a complete characterisation of quasi-minimal groups. In the purely quasi-minimal case, assuming GCH, a complete characterisation is also established. An independence result is proved for directly quasi-minimal groups.

On The Empirical Balanced Truncation For Nonlinear Systems, Marissa Condon, Rossen Ivanov

Articles

Novel constructions of empirical controllability and observability gramians for nonlinear systems for subsequent use in a balanced truncation style of model reduction are proposed. The new gramians are based on a generalisation of the fundamental solution for a Linear Time-Varying system. Relationships between the given gramians for nonlinear systems and the standard gramians for both Linear Time-Invariant and Linear Time-Varying systems are established as well as relationships to prior constructions proposed for empirical gramians. Application of the new gramians is illustrated through a sample test-system.

The Spectral Function For Sturm-Liouville Problems Where The Potential Is Of Wigner-Von Neumann Type Or Slowly Decaying, Daphne Gilbert, B.J. Harris, S.M. Riehl

Articles

We consider the linear, second-order, differential equation (∗) with the boundary condition (∗∗)

We suppose that q(x) is real-valued, continuously differentiable and that q(x)→0 as x→∞ with q∉L1[0,∞). Our main object of study is the spectral function ρα(λ) associated with () and (). We derive a series expansion for this function, valid for λ⩾Λ0 where Λ0 is computable and establish a Λ1, also computable, such that () and () with α=0, have no points of spectral concentration for λ⩾Λ1. We illustrate our results with examples. In particular we consider the case of the Wigner–von Neumann potential.

Asymptotic And Numerical Solutions For Diffusion Models For Compounded Risk Reserves With Dividend Payments, Sally S. L. Shao, C. L. Chang

Mathematics and Statistics Faculty Publications

We study a family of diffusion models for compounded risk reserves which account for the investment income earned and for the inflation experienced on claim amounts. We are interested in the models in which the dividend payments are paid from the risk reserves. After defining the process of conditional probability in finite time, martingale theory turns the nonlinear stochastic differential equation to a special class of boundary value problems defined by a parabolic equation with a nonsmooth coefficient of the convection term. Based on the behavior of the total income flow, asymptotic and numerical methods are used to solve the …

Constructing Random Probability Distributions, Theodore P. Hill, David E.R. Sitton

Research Scholars in Residence

This article surveys several classes of iterative methods for constructing random probability distributions (or random convex functions, or random homeomorphisms), and includes illustrative applications in statistics, optimal-control theory, and game theory. Computer simulations of these methods are fast and easy to implement

Algebraic Semantics For Coalgebraic Logics, Clemens Kupke, Alexander Kurz, Dirk Pattinson

Engineering Faculty Articles and Research

With coalgebras usually being defined in terms of an endofunctor T on sets, this paper shows that modal logics for T-coalgebras can be naturally described as functors L on boolean algebras. Building on this idea, we study soundness, completeness and expressiveness of coalgebraic logics from the perspective of duality theory. That is, given a logic L for coalgebras of an endofunctor T, we construct an endofunctor L such that L-algebras provide a sound and complete (algebraic) semantics of the logic. We show that if L is dual to T, then soundness and completeness of the algebraic semantics immediately yield the …

Coalgebras And Modal Expansions Of Logics, Alexander Kurz, Alessandra Palmigiano

Engineering Faculty Articles and Research

In this paper we construct a setting in which the question of when a logic supports a classical modal expansion can be made precise. Given a fully selfextensional logic S, we find sufficient conditions under which the Vietoris endofunctor V on S-referential algebras can be defined and we propose to define the modal expansions of S as the logic that arises from the V-coalgebras. As an example, we also show how the Vietoris endofunctor on referential algebras extends the Vietoris endofunctor on Stone spaces. From another point of view, we examine when a category of ‘spaces’ (X,A), ie sets X …

Uniqueness Theorems In Bioluminescence Tomography, Ge Wang, Yi Li, Ming Jiang

Mathematics and Statistics Faculty Publications

Motivated by bioluminescent imaging needs for studies on gene therapy and other applications in the mouse models, a bioluminescence tomography (BLT) system is being developed in the University of Iowa. While the forward imaging model is described by the well-known diffusion equation, the inverse problem is to recover an internal bioluminescent source distribution subject to Cauchy data. Our primary goal in this paper is to establish the solution uniqueness for BLT under practical constraints despite the ill-posedness of the inverse problem in the general case. After a review on the inverse source literature, we demonstrate that in the general case …

A Combintorial Proof Of The Sum Of Q-Cubes, Kristina Garrett, Kristin Hummel

Mathematics and Statistics Faculty Work

We give a combinatorial proof of a q-analogue of the classical formula for the sum of cubes.

The Dual Spectral Set Conjecture, Steen Pedersen

Mathematics and Statistics Faculty Publications

Suppose that Λ = (aZ + b) ∪ (cZ + d) where a, b, c, d are real numbers such that a ≠ 0 and c ≠ 0. The union is not assumed to be disjoint. It is shown that the translates Ω + λ, λ is an element of Λ, tile the real line for some bounded measurable set Ω if and only if the exponentials e_λ(x) = e^i2πλx, λ is an element of Λ, form an orthogonal basis for some bounded measurable set Ω'.

Common Cyclic Vectors For Normal Operators, William T. Ross, Warren R. Wogen

Department of Math & Statistics Faculty Publications

If μis a finite compactly supported measure on C, then the set S_μ of multiplication operators Mᵩ : L² (μ) --> L² (μ), Mᵩ f = ᵩ f, where ᵩ ϵ L ∞ (μ) is injective on a set of full μ measure, is the complete set of cyclic multiplication operators on L² (μ) In this paper, we explore the question as to whether or not S_μ has a common cyclic vector

A Comparison Of Modified Reconstructability Analysis And Ashenhurst‐Curtis Decomposition Of Boolean Functions, Anas Al-Rabadi, Marek Perkowski, Martin Zwick

Complex Systems Faculty Publications and Presentations

Modified reconstructability analysis (MRA), a novel decomposition technique within the framework of set‐theoretic (crisp possibilistic) reconstructability analysis, is applied to three‐variable NPN‐classified Boolean functions. MRA is superior to conventional reconstructability analysis, i.e. it decomposes more NPN functions. MRA is compared to Ashenhurst‐Curtis (AC) decomposition using two different complexity measures: log‐functionality, a measure suitable for machine learning, and the count of the total number of two‐input gates, a measure suitable for circuit design. MRA is superior to AC using the first of these measures, and is comparable to, but different from AC, using the second.

Reversible Modified Reconstructability Analysis Of Boolean Circuits And Its Quantum Computation, Anas Al-Rabadi, Martin Zwick

Complex Systems Faculty Publications and Presentations

Modified Reconstructability Analysis (MRA) can be realized reversibly by utilizing Boolean reversible (3,3) logic gates that are universal in two arguments. The quantum computation of the reversible MRA circuits is also introduced. The reversible MRA transformations are given a quantum form by using the normal matrix representation of such gates. The MRA-based quantum decomposition may play an important role in the synthesis of logic structures using future technologies that consume less power and occupy less space.

Modified Reconstructability Analysis For Many-Valued Functions And Relations, Anas Al-Rabadi, Martin Zwick

Complex Systems Faculty Publications and Presentations

A novel many-valued decomposition within the framework of lossless Reconstructability Analysis is presented. In previous work, Modified Recontructability Analysis (MRA) was applied to Boolean functions, where it was shown that most Boolean functions not decomposable using conventional Reconstructability Analysis (CRA) are decomposable using MRA. Also, it was previously shown that whenever decomposition exists in both MRA and CRA, MRA yields simpler or equal complexity decompositions. In this paper, MRA is extended to many-valued logic functions, and logic structures that correspond to such decomposition are developed. It is shown that many-valued MRA can decompose many-valued functions when CRA fails to do …

State-Based Reconstructability Analysis, Martin Zwick, Michael S. Johnson

Complex Systems Faculty Publications and Presentations

Reconstructability analysis (RA) is a method for detecting and analyzing the structure of multivariate categorical data. While Jones and his colleagues extended the original variable‐based formulation of RA to encompass models defined in terms of system states, their focus was the analysis and approximation of real‐valued functions. In this paper, we separate two ideas that Jones had merged together: the “g to k” transformation and state‐based modeling. We relate the idea of state‐based modeling to established variable‐based RA concepts and methods, including structure lattices, search strategies, metrics of model quality, and the statistical evaluation of model fit for analyses based …

Algebras With Inner Mb-Representation, Krzysztof Ciesielski

Faculty & Staff Scholarship

We investigate algebras of sets, and pairs (A,I) consisting of an algebra A and an ideal I, which is a subset of A, that possess an inner MB-representation. We compare inner MB-representability of (A,I) with several properties of (A,I) considered by Baldwin. We show that A is inner MB-representable if and only if A =S(A \ H (A)), where S(^.) is a Marczewski operation defined below and H consists of sets that are hereditarily in A. We study uniqueness issue of the ideal in that representation.

Continuous Images Of Big Sets And Additivity Of S0 Under Cpaprism, Krzysztof Ciesielski

Faculty & Staff Scholarship

We prove that the Covering Property Axiom CPA_prism, which holds in the iterated perfect set model, implies the following facts.

• There exists a family G of uniformly continuous functions from R to [0,1] such that G has cardinality \omega₁ < \continuum and for every subset S of R of cardinality \continuum there exists a g in G with g[S]=[0,1].
• The additivity of the Marczewski's ideal s₀ is equal to \omega₁ < \continuum.

Homology Of Holomorphs Of Free Groups, Craig A. Jensen

Mathematics Faculty Publications

Let Fn be the free group on n letters and let Aut(Fn), Out(Fn) denote the automorphism group and the outer automorphism group of Fn. In this paper the purpose is to obtain some new results on stability and to calculate the mod-p homology of the holomorph of Fn for odd primes in dimensions 1 and 2, and the rational homology in dimensions from 1 through 5.

Automorphisms Of Free Groups With Boundaries, Craig A. Jensen, Nathalie Wahl

Mathematics Faculty Publications

The automorphisms of free groups with boundaries form a family of groups An,k closely related to mapping class groups, with the standard automorphisms of free groups as An,0 and (essentially) the symmetric automorphisms of free groups as A0,k. We construct a contractible space Ln,k on which An,k acts with finite stabilizers and finite quotient space and deduce a range for the virtual cohomological dimension of An,k. We also give a presentation of the groups and calculate their first homology group.

The Growth Of Valuations On Rational Function Fields In Two Variables, Edward Mosteig, Moss Sweedler

Mathematics, Statistics and Data Science Faculty Works

Given a valuation on the function field k( x; y), we examine the set of images of nonzero elements of the underlying polynomial ring k[ x; y] under this valuation. For an arbitrary field k, a Noetherian power series is a map z : Q --> k that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on k( x; y). Although the value groups corresponding to such valuations are well-understood, the restrictions of the valuations to underlying polynomial rings have yet to be characterized. Let Lambda(n) denote the images under the valuation v of …

Tree Diagrams For String Links Ii: Determining Chord Diagrams, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term relations).

Higher Dimensional Algebra Vi: Lie 2-Algebra, John C. Baez, Alissa S. Crans

Mathematics, Statistics and Data Science Faculty Works

The theory of Lie algebras can be categorified starting from a new notion of `2-vector space', which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector spaces as objects, `linear functors' as morphisms and `linear natural transformations' as 2-morphisms. We define a `semistrict Lie 2-algebra' to be a 2-vector space L equipped with a skew-symmetric bilinear functor [ . , . ] : L x L -> L satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the `Jacobiator', which in turn must satisfy a certain law of its …

A New Proof Of A Theorem Of Phan, Curtis Bennett, Sergey Shpectorov

Mathematics, Statistics and Data Science Faculty Works

We apply diagram geometry and amalgam techniques to give a new proof of a theorem of K.-W. Phan, characterizing the special unitary group as a group generated by certain systems of subgroups SU(2, q(2)).

A Survey Of Results Involving Transforms And Convolutions In Function Space, David Skough, David Storvick

Department of Mathematics: Faculty Publications

In this paper we survey various results involving Fourier-Wiener transforms, Fourier-Feynman transforms, integral transforms and convolution products of functionals over function space that have been established since Cameron and Martin first introduced Fourier-Wiener transforms in 1945.

Computational Modeling Of Tumor Angiogenesis, Santanu Chatterjee

Masters Theses

No abstract provided.