Applied Mathematics Commons^™

Enumeration Of Tilings Of Quartered Aztec Rectangles, Tri Lai

Department of Mathematics: Faculty Publications

We generalize a theorem of W. Jockusch and J. Propp on quartered Aztec diamonds by enumerating the tilings of quartered Aztec rectangles. We use subgraph replacement method to transform the dual graph of a quartered Aztec rectangle to the dual graph of a quartered lozenge hexagon, and then use Lindstr¨om-Gessel- Viennot methodology to find the number of tilings of a quartered lozenge hexagon.

Von Neumann Algebras And Extensions Of Inverse Semigroups, Allan P. Donsig, Adam H. Fuller, David R. Pitts

Department of Mathematics: Faculty Publications

In the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan MASAs using measured equivalence re- lations and 2-cocycles on such equivalence relations. In this paper, we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman-Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.

Centered-Difference Applications For Schrödinger's Equation, Matthew Thomas Murachver

Physics

This project enumerates methods utilizing discretized centered-difference approximations on the second order differential equation for quantum particles known as Schrodinger’s Equation. An eigenvalue-eigenfunction scheme is developed to sieve for valid solutions to The Time Independent Schrodinger Equation. Additionally the Crank-Nicolson method is applied to the Time Dependent Schrodinger Equation to describe wavefunction (eigenfunction) time evolution. The validity of these methods is discussed with applications to several fundamental pedagogical introductory quantum mechanic systems.

Nonsmooth Algorithms And Nesterov's Smoothing Technique For Generalized Fermat-Torricelli Problems, Nguyen Mau Nam, Nguyen Thai An, R. Blake Rector, Jie Sun

Mathematics and Statistics Faculty Publications and Presentations

We present algorithms for solving a number of new models of facility location which generalize the classical Fermat--Torricelli problem. Our first approach involves using Nesterov's smoothing technique and the minimization majorization principle to build smooth approximations that are convenient for applying smooth optimization schemes. Another approach uses subgradient-type algorithms to cope directly with the nondifferentiability of the cost functions. Convergence results of the algorithms are proved and numerical tests are presented to show the effectiveness of the proposed algorithms.

Improving Airplane Touchdown Control By Utilizing The Adverse Elevator Effect, Nihad E. Daidzic Ph.D., Sc.D.

International Journal of Aviation, Aeronautics, and Aerospace

The main objective of this original research article is to understand the short-term dynamic behavior of the transport-category airplane during landing flare elevator control application. Increasing the pitch angle to arrest the sink rate, the elevator will have to produce negative lift to rotate the airplane’s nose upward. This has an immediate adverse effect of initially accelerating airplane downward. A mathematical model of landing flare based on the flat-Earth longitudinal dynamics of rigid airplane was developed which is realistic only on very short time-scales as pitch stiffness and damping were neglected. Pilot control scenarios using impulse and step elevator pull-up …

Granularity Explains Empirical Factor-Of-Three Relation Between Probabilities Of Pulmonary Embolism In Different Patient Categories, Beverly Rivera, Francisco Zapata, Vladik Kreinovich

Departmental Technical Reports (CS)

Pulmonary embolism is a very dangerous difficult-to-detect medical condition. To diagnose pulmonary embolism, medical practitioners combine indirect signs of this condition into a single score, and then classify patients into low-probability, intermediate-probability, and high-probability categories. Empirical analysis shows that, when we move from each category to the next one, the probability of pulmonary embolism increases by a factor of three. In this paper, we provide a theoretical explanation for this empirical relation between probabilities.

Examining The Consistence Of Futures Margin Levels Using Bivariate Extreme Value Copulas, X. Gong, Hung T. Nguyen, Vladik Kreinovich, Songsak Sriboonchitta

Departmental Technical Reports (CS)

This study examines the consistence of the futures margin levels of different commodities and combinations in the CME group by Extreme Value Copula (EVC). We find that if we ignore the co-movements of the commodities, the margins become consistent with each other, and the margin violation rates hover around 0.5%. However, if we consider the co-movement of the related commodities using EVC, the margin levels are found to be not consistent anymore, especially in the combinations of strongly related commodities which are in the same category. Therefore, we suggest that the CME group should try to harmonize the margins policy …

Epistemic Uncertainty Quantification In Scientific Models, Xiaoxiao Chen

Open Access Dissertations

In the field of uncertainty quantification (UQ), epistemic uncertainty often refers to the kind of uncertainty whose complete probabilistic description is not available, largely due to our lack of knowledge about the uncertainty. Quantification of the impacts of epistemic uncertainty is naturally difficult, because most of the existing stochastic tools rely on the specification of the probability distributions and thus do not readily apply to epistemic uncertainty. And there have been few studies and methods to deal with epistemic uncertainty. A recent work can be found in [J. Jakeman, M. Eldred, D. Xiu, Numerical approach for quantification of epistemic uncertainty, …

How To Gauge Unknown Unknowns: A Possible Theoretical Explanation Of The Usual Safety Factor Of 2, Joe Lorkowski, Vladik Kreinovich

Departmental Technical Reports (CS)

To gauge the accuracy of a measuring instrument, engineers analyze possible factors contributing to the instrument's inaccuracy. In addition to known factors, however, there are usually unknown factors which also contribute to the instrument's inaccuracy. To properly gauge the instrument's accuracy -- and thus, to make sure that we do not compromise our safety by underestimating the inaccuracy -- we need to also take these "unknown unknowns" into account. In practice, this is usually done by multiplying the original estimate for inaccuracy by a "safety" factor of 2. In this paper, we provide a possible theoretical explanation for this empirical …

Transients In The Synchronization Of Oscillator Arrays, Carlos E. Cantos, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

The purpose of this note is threefold. First we state a few conjectures that allow us to rigorously derive a theory which is asymptotic in N (the number of agents) that describes transients in large arrays of (identical) linear damped harmonic oscillators in R with completely decentralized nearest neighbor interaction. We then use the theory to establish that in a certain range of the parameters transients grow linearly in the number of agents (and faster outside that range). Finally, in the regime where this linear growth occurs we give the constant of proportionality as a function of the signal velocities …

Quantitative Modeling Of Spatiotemporal Systems: Simulation Of Biological Systems And Analysis Of Error Metric Effects On Model Fitting, James Hengenius

Open Access Dissertations

Understanding the biophysical processes underlying biological and biotechnological processes is a prerequisite for therapeutic treatments and technological innovation. With the exponential growth of computational processing speed, experimental findings in these fields have been complemented by dynamic simulations of developmental signaling and genetic interactions. Models provide means to evaluate "emergent" properties of systems sometimes inaccessible by reductionist approaches, making them test beds for biological inference and technological refinement.^ The complexity and interconnectedness of biological processes pose special challenges to modelers; biological models typically possess a large number of unknown parameters relative to their counterparts in other physical sciences. Estimating these parameter …

Structured Deterministic Models Applied To Malaria And Other Endemic Diseases, Katia Patricia Vogt Geisse

Open Access Dissertations

This thesis includes modeling studies on three structured deterministic models. These models are used to study the disease dynamics of malaria or the joint disease dynamics of HIV and HSV-2. Each of the models includes multiple components containing individuals in various epidemiological classes for the purpose of addressing questions that are of interests to biologists and epidemiologists. Some of the compartments have a continuous age-structure, which is necessary for studying the specific biological questions under investigation.^ In Chapter 2 a chronological-age structured deterministic model for malaria is presented. The model includes the human and mosquito populations with the human population …

Component Trees For The Exploration Of Macromolecular Structures In Biology, Lucas Oliveira

Dissertations, Theses, and Capstone Projects

Understanding the three-dimensional structure of a macromolecular complex is essential for understanding its function. A component tree is a topological and geometric image descriptor that captures information regarding the structure of an image based on the connected components determined by different grayness thresholds. This dissertation presents a novel interactive framework for visual exploration of component trees of the density maps of macromolecular complexes, with the purpose of improved understanding of their structure. The interactive exploration of component trees together with a robust simplification methodology provide new insights in the study of macromolecular structures. An underlying mathematical theory is introduced and …

A Mentoring Program For Inquiry-Based Teaching In A College Geometry Class, Nathaniel Miller, Nathan Wakefield

Department of Mathematics: Faculty Publications

This paper describes a mentoring program designed to prepare novice instructors to teach a college geometry class using inquiry-based methods. The mentoring program was used in a medium-sized public university with approximately 12,000 undergraduate students and 1,500 graduate students. The authors worked together to implement a mentoring program for the first time. One author was an associate professor and experienced using inquiry-based learning. The other author was a graduate student in mathematics education. During the course of the year the graduate student first observed and then taught a college level inquiry-based geometry course for pre-service teachers. This article describes the …

Shifted One-Parameter Supersymmetric Family Of Quartic Asymmetric Double-Well Potentials, Haret C. Rosu, S.C. Mancas, Pisin Chen

Publications

Extending our previous work (Rosu, Mancas, Chen, Ann.Phys. 343 (2014) 87-102), we define supersymmetric partner potentials through a particular Riccati solution of the form F (x) = (x - c)^2 - 1, where c is a real shift parameter, and work out the quartic double-well family of one-parameter isospectral potentials obtained by using the corresponding general Riccati solution. For these parametric double well potentials, we study how the localization properties of the two wells depend on the parameter of the potentials for various values of the shifting parameter.

Partitioning Bipartite Graphs: A Modified Louvain, Emily Diana

Yale Day of Data

Abstract

How do we find communities in a graph? How does this change if the graph is bipartite? The Louvain method maximizes links within communities and minimizes those between in order to determine an optimal grouping. Yet, because it may fail when bipartite restrictions are introduced, we have adjusted the null model so as to improve performance in these conditions.

Conclusion

Our Bipartite Louvain is more robust with respect to permutations of vertices than the standard Louvain. For our synthetic examples, Bipartite Louvain typically yields a higher modularity and uncovers the ground truth communities with a higher probability. In the …

A Two-Light Version Of The Classical Hundred Prisoners And A Light Bulb Problem: Optimizing Experimental Design Through Simulations, Alexander S. Barrett, Cyril Rakovski

e-Research: A Journal of Undergraduate Work

We propose five original strategies of successively increasing complexity and efficiency that address a novel version of a classical mathematical problem that, in essence, focuses on the determination of an optimal protocol for exchanging limited amounts of information among a group of subjects with various prerogatives. The inherent intricacy of the problem�solving protocols eliminates the possibility to attain an analytical solution. Therefore, we implemented a large-scale simulation study to exhaustively search through an extensive list of competing algorithms associated with the above-mentioned 5 generally defined protocols. Our results show that the consecutive improvements in the average amount of time necessary …

Formalizing The Informal, Precisiating The Imprecise: How Fuzzy Logic Can Help Mathematicians And Physicists By Formalizing Their Intuitive Ideas, Olga Kosheleva, Renata Reiser, Vladik Kreinovich

Departmental Technical Reports (CS)

Fuzzy methodology transforms expert ideas -- formulated in terms of words from natural language -- into precise rules and formulas. In this paper, we show that by applying this methodology to intuitive physical and mathematical ideas, we can get known fundamental physical equations and known mathematical techniques for solving these equations. This fact makes us confident that in the future, fuzzy techniques will help physicists and mathematicians to transform their imprecise ideas into new physical equations and new techniques for solving these equations.

Generalized Least-Squares Regressions Iv: Theory And Classification Using Generalized Means, Nataniel Greene

Publications and Research

The theory of generalized least-squares is reformulated here using the notion of generalized means. The generalized least-squares problem seeks a line which minimizes the average generalized mean of the square deviations in x and y. The notion of a generalized mean is equivalent to the generating function concept of the previous papers but allows for a more robust understanding and has an already existing literature. Generalized means are applied to the task of constructing more examples, simplifying the theory, and further classifying generalized least-squares regressions.

Dressing Method And Quadratic Bundles Related To Symmetric Spaces: Vanishing Boundary Conditions, Tihomir Valchev

Articles

We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m+n)/S(U(m) x U(n)). The simplest representative of the corresponding integrable hierarchy is given by a multi-component Kaup-Newell derivative nonlinear Schroedinger equation which serves as a motivational example for our general considerations. We extensively discuss how one can apply Zakharov-Shabat's dressing procedure to derive reflectionless potentials obeying zero boundary conditions. Those could be used for one to construct fast decaying solutions to any nonlinear equation belonging to the same hierarchy. One can distinguish between generic soliton type solutions and rational solutions.