Mathematics Commons^™

Approximation Of Nested Fixpoints, Alexander Kurz

Engineering Faculty Articles and Research

The question addressed in this paper is how to correctly approximate infinite data given by systems of simultaneous corecursive definitions. We devise a categorical framework for reasoning about regular datatypes, that is, datatypes closed under products, coproducts and fixpoints. We argue that the right methodology is on one hand coalgebraic (to deal with possible nontermination and infinite data) and on the other hand 2-categorical (to deal with parameters in a disciplined manner). We prove a coalgebraic version of Bekic lemma that allows us to reduce simultaneous fixpoints to a single fix point. Thus a possibly infinite object of interest is …

Presenting Distributive Laws, Marcello M. Bonsangue, Helle H. Hansen, Alexander Kurz, Jurriaan Rot

Engineering Faculty Articles and Research

Distributive laws of a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specification of well-behaved structural operational semantics and, more recently, also for enhancements of the bisimulation proof method. If T is a free monad, then such distributive laws correspond to simple natural transformations. However, when T is not free it can be rather difficult to prove the defining axioms of a distributive law. In this paper we describe how to obtain a distributive law for a monad with an equational presentation …

Positive Fragments Of Coalgebraic Logics, Adriana Balan, Alexander Kurz, Jirí Velebil

Engineering Faculty Articles and Research

Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and are built only from atomic propositions, conjunction, disjunction, box and diamond. In this paper, we provide a coalgebraic analysis of this theorem, which not only gives a conceptual proof based on duality theory, but also generalizes Dunn's result from Kripke frames to coalgebras for weak-pullback preserving functors. To facilitate this analysis we prove a number of category theoretic results on …

Young Tableaux, Multisegments, And Pbw Bases, John Claxton, Peter Tingley

Mathematics and Statistics: Faculty Publications and Other Works

The crystals for finite dimensional representations of sln+1 can be realized using Young tableaux. The infinity crystal on the other hand is naturally realized using multisegments, and there is a simple description of each embedding B(λ) ,→ B(∞) in terms of these realizations. The infinity crystal is also parameterized by Lusztig’s PBW basis with respect to any reduced expression for w0. We give an explicit description of the unique crystal isomorphism from PBW bases to multisegments in the case where w0 = s1s2s3 · · · sns1 · · · s1s2s1, thus obtaining simple formulas for the actions of all …

The Characteristic Polynomial Of The Adams Operators On Graded Connected Hopf Algebras, Marcelo Aguiar, Aaron Lauve

Mathematics and Statistics: Faculty Publications and Other Works

The Adams operators ‰n on a Hopf algebra H are the convolution powers of the identity of H. They are also called Hopf powers or Sweedler powers. We study the Adams operators when H is graded connected. The main result is a complete description of the characteristic polynomial — both eigenvalues and their multiplicities — for the action of the operator ‰n on each homogeneous component of H. The eigenvalues are powers of n. The multiplicities are independent of n, and in fact only depend on the dimension sequence of H. These results apply in particular to the antipode of …

The Central Hankel Transform, Matthew J. Levine

Honors Theses

This honors thesis presents the Hankel transform on an integer sequence, a function with colorful mathematical history and rich theoretical background. We then introduce a matricial Toeplitz transform that parallels some of most famous qualities of the Hankel transform, especially when in consideration of popular sequences like the Fibonacci numbers. The result is a characterization of the injectivity of this new function, a description of some of its interesting behaviors, and a discussion of a few new Fibonacci identities.

Quantization Of Analysis, Kelvin K. Lui

Honors Theses

In quantum mechanics the replacement of complex vectors with operators is essential to “quantizing” space. Nonetheless, in many physics textbooks there is no justification for this action. Therefore in this thesis I will attempt to understand the mathematical formalism that allows for such a “replacement” to be rigorous. I will approach this topic by first defining a vector spaces and its dual space, a Hilbert space and a conjugate Hilbert space, and an operator space. Next, I will look at the algebraic tensor product of two vector spaces, two Hilbert spaces, and finally two operator spaces. Ultimately we will look …

Essays On Unit Root Testing In Time Series, Xiao Zhong

Doctoral Dissertations

"Unit root tests are frequently employed by applied time series analysts to determine if the underlying model that generates an empirical process has a component that can be well-described by a random walk. More specifically, when the time series can be modeled using an autoregressive moving average (ARMA) process, such tests aim to determine if the autoregressive (AR) polynomial has one or more unit roots. The effect of economic shocks do not diminish with time when there is one or more unit roots in the AR polynomial, whereas the contribution of shocks decay geometrically when all the roots are outside …

Discrete And Dynamic Population Models With Logistic Growth Rate, Sabrina Heike Streipert

Doctoral Dissertations

"The Beverton-Holt difference equation defines a discrete relation describing a population model. Considering periodic carrying capacity and periodic inherent growth rate, a population with seasonal changing life cycle and environment is reflected. The so-called periodically forced Beverton-Holt equation is investigated and its unique periodic solution is derived. This provides the first Cushing-Henson conjecture, while a counterexample proves that the classical second Cushing-Henson conjecture is not satisfied. Modifications of the conjecture are formulated. To extend the studies, the Beverton-Holt equation is investigated in the quantum calculus time setting. The existence of the globally attracting periodic solution of the Beverton-Holt q-difference …

Aspects Of Fourier Analysis On Euclidean Space, Joseph William Roberts

Graduate Theses/Dissertations

The field of Fourier analysis encompasses a vast spectrum of mathematics and has far reaching applications in all STEM fields. Here we introduce and study the Fourier transform and Fourier series on Euclidean space. After defining the Fourier transform, establishing its basic properties, and presenting some classical results we looked into what impact the smoothness of a function has on the growth and integrability of its Fourier transform. This endeavor also involved a brief study of Bessel functions and interpolation of operators. Having established several results indicating that the behavior of a function's Fourier transform is largely dictated by the …

Discrete Analogues Of Some Classical Special Functions, Thomas Joseph Cuchta

Doctoral Dissertations

"Analogues of special functions on time scales are studied with special focus on the time scale 𝕋 = hℤ. Functions investigated in particular include complex monomials, new trigonometric functions, Gaussian bell, Hermite and Laguerre polynomials, Bessel functions, and hypergeometric series"--Abstract, page iii.

Small Sample Umpu Equivalence Testing Based On Saddlepoint Approximations, Renren Zhao

Doctoral Dissertations

"In the first section, we consider small sample equivalence tests for exponentiality. Statistical inference in this setting is particularly challenging since equivalence testing procedures typically require a much larger sample size, in comparison to classical "difference tests", to perform well. We make use of Butler's marginal likelihood for the shape parameter of a gamma distribution in our development of equivalence tests for exponentiality. We consider two procedures using the principle of confidence interval inclusion, four Bayesian methods, and the uniformly most powerful unbiased (UMPU) test where a saddlepoint approximation to the intractable distribution of a canonical sufficient statistic is used. …

Smooth Positive Solutions To An Elliptic Model With C2 Functions, Joon Hyuk Kang

Faculty Publications

Two species of animals are competing or cooperating in the same environment. Under what conditions do they coexist peacefully? Or under what conditions does either one of the two species become extinct, that is, is either one of the two species excluded by the other? We investigate this phenomena in mathematical point of view.

In this paper we concentrate on coexistence solutions of the competition or cooperation model

This system is the general model for the steady state of a competitive or cooperative interacting system depending on growth conditions for and . The techniques used in this paper are elliptic …

A Curve Satisfying T/K=S With Constant K>0, Yun Myung Oh, Ye Lim Seo

Faculty Publications

In the present paper, we investigate a space curve in which the curvature is constant and the torsion is a linear function. The aim of this paper is to find an explicit formula for this space curve when the ratio of the torsion to the curvature is a linear function when the curvature is constant.

A Glance At Tropical Operations And Tropical Linear Algebra, Semere Tsehaye Tesfay

Masters Theses

The tropical semiring is ℝ ∪ {∞} with the operations x ⊕ y = min{x, y}, x ⊕ ∞ = ∞ ⊕ x = x, x ⊙ y = x + y, x ⊙ ∞ = ∞ ⊙ y = ∞. This paper explores how ideas from classical algebra and linear algebra over the real numbers such as polynomials, roots of polynomials, lines, matrices and matrix operations, determinants, eigen values and eigen vectors would appear in tropical mathematics. It uses numerous computed examples to illustrate these concepts and explores the relationship between certain tropical matrices and graph …

Hyperbolic Geometry With And Without Models, Chad Kelterborn

Masters Theses

We explore the development of hyperbolic geometry in the 18th and early 19th following the works of Legendre, Lambert, Saccheri, Bolyai, Lobachevsky, and Gauss. In their attempts to prove Euclid's parallel postulate, they developed hyperbolic geometry without a model. It was not until later in the 19th century, when Felix Klein provided a method (which was influenced by projective geometry) for viewing the hyperbolic plane as a disk in the Euclidean plane, appropriately named the "Klein disk model". Later other models for viewing the hyperbolic plane as a subset of the Euclidean plane were created, namely the Poincaré disk model, …

Fpu Lattices In Multidimensions, Jeffrey A. Schwarz

Theses, Dissertations and Culminating Projects

In a 1955 paper, Enrico Fermi, John Pasta, and Stanislaw Ulam studied a one dimensional, nonlinear dynamical system on an early electronic computer. They were surprised to see the system’s original energy configuration recur. The experiment, which has since become known as the “FPU Problem”, initiated the field of experimental mathematics, is at the origin of the soliton concept and chaos theory, has sparked revolutions in modem science, and called into question the equipartition hypothesis.

The FPU experiment still has many open questions, not the least of which is an explanation for the recurrence.

In this work, we compare the …

The Impact Of Math Anxiety In The Primary Grades, Amanda R. Anderson-Mix

Mathematics Graduate Theses

Math anxiety is an increasingly common occurrence among individuals when working with mathematical concepts. While researching math anxiety the author came across common gender stereotypes implying: women do not perform as well as men in the field of mathematics, women experience a much higher level of math anxiety than men, and most women do not appreciate performing mathematical tasks. Although the concept of math anxiety is an interesting topic to the author, it's important to look deeper than simply the question, “What is math anxiety?” Seeing these stereotypical assumptions that women have higher math anxiety levels, it made this author …

Diagonals Of Tensor Products Of Banach Lattices With Bases., Byunghoon Lee

Electronic Theses and Dissertations

In this dissertation, we investigate diagonals of tensor products of Banach lattices with bases. We first consider questions on the positive tensor products of l_p spaces. We characterize the main diagonals of the positive projective tensor product and the positive injective tensor product of l_p space. Then by using these two main diagonals, we characterize the reflexivity, the property of being Kantorovich - Banach spaces, and the property of being order continuous of n-fold positive projective and positive injective tensor products of l_p spaces. Next, we consider the diagonals of injective tensor product of Banach lattices with bases. Let E …

Symmetries And Patterns In Non-Euclidean Settings, Sarah Elizabeth Stoops

Undergraduate Honors Thesis Collection

From the Megalithic Temples of Malta constructed over 5,500 years ago, to the Pyramid of Djoser in Egypt built some 4,700 years ago, to more recent works of architectural wonder such an the Taj Mahal, the testimonials to the innate human genius for creating beauty through symmetry, color, and patterns abound. Evidently, the mathematical underpinnings of many architectural marvels are mostly rooted in the Euclidean Geometry. Now, as marvelous as these monuments are, one may wonder what would be the concepts of beauty and symmetry in a non-Euclidean universe. It turns out that this is not a far-fetched thought. It …

Isabel Stafford Presents Research To National Security Agency, Lucero Castellanos-Aguirre

Lake Union Herald

No abstract provided.

Knowledge Sharing In Organisations: A Bayesian Analysis Of The Role Of Reciprocity And Formal Structure, Alberto Caimo, Alessandro Lomi

Articles

We examine the conditions under which knowledge embedded in advice relations is likely to reach across intraorganizational boundaries and be shared between distant organizational members. We emphasize boundary-crossing relations because activities of knowledge transfer and sharing across subunit boundaries are systematically related to desirable organizational outcomes. Our main objective is to understand how organizational and social processes interact to sustain the transfer of knowledge carried by advice relations. Using original fieldwork and data that we have collected on members of the top management team in a multiunit industrial group, we show that knowledge embedded in task advice relations is unlikely …

Modeling, Optimization, And Sensitivity Analysis Of A Continuous Multi-Segment Crystallizer For Production Of Active Pharmaceutical Ingredients, Bradley James Ridder

Open Access Dissertations

We have investigated the simulation-based, steady-state optimization of a new type of crystallizer for the production of pharmaceuticals. The multi-segment, multi-addition plug-flow crystallizer (MSMA-PFC) offers better control over supersaturation in one dimension compared to a batch or stirred-tank crystallizer. Through use of a population balance framework, we have written the governing model equations of population balance and mass balance on the crystallizer segments. The solution of these equations was accomplished through either the method of moments or the finite volume method. The goal was to optimize the performance of the crystallizer with respect to certain quantities, such as maximizing the …

A 2-Categorical Extension Of The Reshetikhin--Turaev Theory, Yu Tsumara

Open Access Dissertations

We concretely construct a 2-categorically extended topological quantum field theory that extends the Reshetikhin-Turaev TQFT to cobordisms with corners. The source category will be a well chosen 2-category of decorated cobordisms with corners and the target bicategory will be the Kapranov-Voevodsky 2-vector spaces.

Zero Triple Product Determined Generalized Matrix Algebras, Dong Han

Turkish Journal of Mathematics

In this paper, we prove that the generalized matrix algebra G = \left[ A M N B \right] is a zero triple product (resp. zero Jordan triple product) determined if and only if A and B are zero triple products (resp. zero Jordan triple products) determined under certain conditions. Then the main results are applied to triangular algebras and full matrix algebras.

Homogenization Of Stokes Systems And Uniform Regularity Estimates, Shu Gu, Zhongwei Shen

Mathematics Faculty Publications

This paper is concerned with uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and L^∞ estimates for the pressure as well as a Liouville property for solutions in ℝ^d. We also obtain the boundary W^1,p estimates in a bounded C¹ domain for any 1 < p < ∞.

Stability Of Perturbed Dynamic System On Time Scales With Initial Time Difference, Coşkun Yakar, Bülent Oğur

Turkish Journal of Mathematics

The behavior of solutions of a perturbed dynamic system with respect to an original unperturbed dynamic system, which have initial time difference, are investigated on arbitrary time scales. Notions of stability, asymptotic stability, and instability with initial time difference are introduced. Sufficient conditions of stability properties are given with the help of Lyapunov-like functions.

Spherically Symmetric Finsler Metrics With Scalar Flag Curvature, Weidong Song, Fen Zhou

Turkish Journal of Mathematics

In this paper, we study spherically symmetric Finsler metrics F= y \phi( x ,\frac{}{ y }), where x \in B^n(r) \subset R^n, y \in T_xB^n(r)\{0} and \phi:[0,r)\times R \rightarrow R. By investigating a PDE equivalent to these metrics being locally projectively flat, we manufacture projectively flat spherically symmetric Finsler metrics in terms of error functions and, using Shen's result, we give its flag curvature.

Rings With Finite Ding Homological Dimensions, Chunxia Zhang, Zhongkui Liu

Turkish Journal of Mathematics

In this paper, we study Ding homological dimensions of complexes. Special attention is paid to the dimensions of homologically bounded complexes that have nice functorial descriptions. These results are applied to give new characterizations of rings R such that l.Ggldim(R) < \infty and quasi-Frobenius rings.

A Note On Infinite Groups Whose Subgroups Are Close To Be Normal-By-Finite, Francesco De Giovanni, Federica Saccomanno

Turkish Journal of Mathematics

A group G is said to have the CF-property if the index X:X_G is finite for every subgroup X of G. Extending previous results by Buckley, Lennox, Neumann, Smith, and Wiegold, it is proven here that if G is a locally graded group whose proper subgroups have the CF-property, then G is abelian-by-finite, provided that all its periodic sections are locally finite. Groups in which all proper subgroups of infinite rank have the CF-property are also studied.