Mathematics Commons^™

Mathematics Education In A Multilingual And Multicultural Environment, Anjum Halai, Richard Barwell

Book Chapters / Conference Papers

No abstract provided.

Cd4 Deficit And Tuberculosis Risk Persist With Delayed Antiretroviral Therapy: 5-Year Data From Cipra Ht-001, S. E. Collins, M.A. Jean Juste, S. P. Koenig, R. Secours, O. Ocheretina, D. Bernard, C. Riviere, M. Calnan, A. Dunning, Sandra M. Hurtado Rua, W. D. Johnson Jr., J. W. Pape, D. W. Fitzgerald, P. Severe

Mathematics and Statistics Faculty Publications

SETTING: Port-au-Prince, Haiti. OBJECTIVE: To determine long-term effects of early vs. delayed initiation of antiretroviral therapy (ART) on immune recovery and tuberculosis (TB) risk in human immunodeficiency virus (HIV) infected individuals. DESIGN: Open-label randomized controlled trial of immediate ART in HIV-infected adults with CD4 counts between 200 and 350 cells/mm(3) vs. deferring ART until the CD4 count was <200 cells/mm(3). The primary comparisons were CD4 counts over time and risk for incident TB, with 5 years of follow-up. RESULTS: A total of 816 participants were enrolled, with 408 in each treatment arm. The early treatment group started ART within 2 weeks, while the deferred treatment group started ART a median of 1.3 years after enrollment. After 5 years, the mean CD4 count in the early treatment group was significantly higher than in the deferred treatment group (496 cells/mm(3), 95% confidence interval [CI] 477-515 vs. 373 cells/mm(3), 95%CI 357-389; P < 0.0001). TB risk was higher in the deferred treatment group (unadjusted HR 2.41, 95%CI 1.56-3.74; P < 0.0001) and strongly correlated with lower CD4 counts in time-dependent multivariate analysis. CONCLUSION: Delays in ART initiation for HIV-infected adults with CD4 counts of 200-350 cells/mm(3) can result in long-term immune dysfunction and persistent increased risk for TB. TRIAL REGISTRATION: ClinicalTrials.gov NCT00120510.

An Isomorphism Problem In Z2, Matt Noble

Theory & Applications of Graphs

We consider Euclidean distance graphs with vertex set ℚ² or ℤ² and address the possibility or impossibility of finding isomorphisms between such graphs. It is observed that for any distances d₁, d₂ the non-trivial distance graphs G(ℚ², d₁) and G(ℚ², d₂) are isomorphic. Ultimately it is shown that for distinct primes _p1, _p2 the non-trivial distance graphs G(ℤ², √p1) and G(ℤ², √p2) are not isomorphic. We conclude with a few additional questions related to this work.

Automorphisms Of Graph Curves On K3 Surfaces, Joshua C. Ferrerra

College of Graduate Studies: Theses & Dissertations

We examine the automorphism group of configurations of rational curves on $K3$ surfaces. We use the properties of finite automorphisms of $\PP^1$ to examine what restrictions a given elliptic fibration imposes on the possible finite order non-symplectic automorphisms of the $K3$ surface. We also examine the fixed loci of these automorphisms, and construct an explicit fibration to demonstrate the process.

Non-Planar On-Shell Diagrams, Sebastían Franco, Daniele Galloni, Brenda Penante, Congkao Wen

Publications and Research

We initiate a systematic study of non-planar on-shell diagrams in N = 4 SYM and develop powerful technology for doing so. We introduce canonical variables generalizing face variables, which make the d log form of the on-shell form explicit. We make significant progress towards a general classification of arbitrary on-shell diagrams by means of two classes of combinatorial objects: generalized matching and matroid polytopes. We propose a boundary measurement that connects general on-shell diagrams to the Grassmannian. Our proposal exhibits two important and non-trivial properties: positivity in the planar case and it matches the combinatorial description of the diagrams in …

Differentiability Of Correlations In Realistic Quantum Mechanics, Alejandro Cabrera, Edson De Faria, Enrique Pujals, Charles Tresser

Publications and Research

We prove a version of Bell’s theorem in which the locality assumption is weakened. We start by assuming theoretical quantum mechanics and weak forms of relativistic causality and of realism (essentially the fact that observable values are well defined independently of whether or not they are measured). Under these hypotheses, we show that only one of the correlation functions that can be formulated in the framework of the usual Bell theorem is unknown. We prove that this unknown function must be differentiable at certain angular configuration points that include the origin. We also prove that, if this correlation is assumed …

Bifurcation Analysis Of Mutually Coupled Laser Systems, Vincent Brennan

Theses

Coupled laser systems have application in everyday modern life, including for example, telecommunications and cryptography. Therefore it is important to gain insight into the dynamic processes at work and theoretical modeling can elucidate such processes. The work presented in this thesis is primarily concerned with aspects such as excitability and phase locking in mutually coupled oscillators. Phase-only delay-coupled oscillator models are introduced to demonstrate excitability and it is subsequently shown that similar processes can occur in a low-dimensional variation of these models. Further, delay differential rate equation models also exhibit excitability and these are explored and contrasted with another, similar, …

A Domain Decomposition Method For The Steady-State Navier-Stokes-Darcy Model With Beavers-Joseph Interface Condition, Xiaoming He, Jian Li, Yanping Lin, Ju Ming

Mathematics and Statistics Faculty Research & Creative Works

This paper proposes and analyzes a Robin-type multiphysics domain decomposition method (DDM) for the steady-state Navier-Stokes-Darcy model with three interface conditions. In addition to the two regular interface conditions for the mass conservation and the force balance, the Beavers-Joseph condition is used as the interface condition in the tangential direction. The major mathematical difficulty in adopting the Beavers-Joseph condition is that it creates an indefinite leading order contribution to the total energy budget of the system [Y. Cao et al., Comm. Math. Sci., 8 (2010), pp. 1-25; Y. Cao et al., SIAM J. Numer. Anal., 47 (2010), pp. 4239-4256]. In …

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 …

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 …

A New Resolvent Equation For The S-Functional Calculus, Daniel Alpay, Fabrizio Colombo, Jonathan Gantner, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

The S-functional calculus is a functional calculus for (n + 1)-tuples of non necessarily commuting operators that can be considered a higher dimensional version of the classical Riesz-Dunford functional calculus for a single operator. In this last calculus, the resolvent equation plays an important role in the proof of several results. Associated with the S-functional calculus there are two resolvent operators: the left S−1 L (s, T ) and the right one S−1 R (s, T ), where s = (s0, s1, . . . , sn) ∈ Rn+1 and T = (T0, T1, . . . , Tn) is …

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 …

Professional Formation Of Engineers: Enhancing The First Year Student Experience, Eileen Goold

Conference Papers

This paper reports on a study investigating engineering students’ perceptions of engineering practice and whether engineering students’ cognitive engagement benefits from bridging the gap between the technical issues in their education and the practical realities of modern engineering practice.