Applied Mathematics Commons^™

Thermoelastic Waves In Microstructured Solids, Arkadi Berezovski, Mihhail Berezovski

Publications

Thermoelastic wave propagation suggests a coupling between elastic deformation and heat conduction in a body. Microstructure of the body influences the both processes. Since energy is conserved in elastic deformation and heat conduction is always dissipative, the generalization of classical elasticity theory and classical heat conduction is performed differently. It is shown in the paper that a hyperbolic evolution equation for microtemperature can be obtained in the framework of the dual internal variables approach keeping the parabolic equation for the macrotemperature. The microtemperature is considered as a macrotemperature fluctuation. Numerical simulations demonstrate the formation and propagation of thermoelastic waves in …

Supervised Learning-Based Explicit Nonlinear Model Predictive Control And Unknown Input Estimation In Biomedical Systems, Ankush Chakrabarty

Open Access Dissertations

Application of nonlinear control theory to biomedical systems involves tackling some unique and challenging problems. The mathematical models that describe biomedical systems are typically large and nonlinear. In addition, biological systems exhibit dynamics which are not reflected in the model (so-called 'un-modeled dynamics') and hard constraints on the states and control actions, which exacerbate the difficulties in designing model-based controllers or observers.

This thesis investigates the design of scalable fast explicit nonlinear model predictive controllers (ENMPCs). The design involves (i) the estimation of a feasible region using Lyapunov stability methods and support vector machines; and (ii) within the estimated feasible …

Minimizing Differences Of Convex Functions With Applications To Facility Location And Clustering, Mau Nam Nguyen, R. Blake Rector, Daniel J. Giles

Mathematics and Statistics Faculty Publications and Presentations

In this paper we develop algorithms to solve generalized Fermat-Torricelli problems with both positive and negative weights and multifacility location problems involving distances generated by Minkowski gauges. We also introduce a new model of clustering based on squared distances to convex sets. Using the Nesterov smoothing technique and an algorithm for minimizing differences of convex functions called the DCA introduced by Tao and An, we develop effective algorithms for solving these problems. We demonstrate the algorithms with a variety of numerical examples.

Single Valued Neutrosophic Graphs, Florentin Smarandache, Said Broumi, Assia Bakali, Mohamed Talea

Branch Mathematics and Statistics Faculty and Staff Publications

The notion of single valued neutrosophic sets is a generalization of fuzzy sets, intuitionistic fuzzy sets. We apply the concept of single valued neutrosophic sets, an instance of neutrosophic sets, to graphs. We introduce certain types of single valued neutrosophic graphs (SVNG) and investigate some of their properties with proofs and examples.

Hpcnmf: A High-Performance Toolbox For Non-Negative Matrix Factorization, Karthik Devarajan, Guoli Wang

COBRA Preprint Series

Non-negative matrix factorization (NMF) is a widely used machine learning algorithm for dimension reduction of large-scale data. It has found successful applications in a variety of fields such as computational biology, neuroscience, natural language processing, information retrieval, image processing and speech recognition. In bioinformatics, for example, it has been used to extract patterns and profiles from genomic and text-mining data as well as in protein sequence and structure analysis. While the scientific performance of NMF is very promising in dealing with high dimensional data sets and complex data structures, its computational cost is high and sometimes could be critical for …

Newton's Law Of Cooling, Caleb J. Emmons

Journal of Humanistic Mathematics

A poem reflecting three different viewpoints on Newton's Law of Cooling.

Signal Flow Graph Approach To Efficient Dst I-Iv Algorithms, Sirani M. Perera

Publications

In this paper, fast and efficient discrete sine transformation (DST) algorithms are presented based on the factorization of sparse, scaled orthogonal, rotation, rotation-reflection, and butterfly matrices. These algorithms are completely recursive and solely based on DST I-IV. The presented algorithms have low arithmetic cost compared to the known fast DST algorithms. Furthermore, the language of signal flow graph representation of digital structures is used to describe these efficient and recursive DST algorithms having (n�1) points signal flow graph for DST-I and n points signal flow graphs for DST II-IV.

Early Afternoon Concurrent Panel Sessions: Commercial Space Industry Snapshot: Presentation: Small Catapult-Assisted Horizontal-Launch Reusable Rbcc Ssto Spaceplane For Economical Short-Duration Leo Access, Nihad E. Daidzic,

Aviation / Aeronautics / Aerospace International Research Conference

This article discusses the conceptual design, flight trajectory calculations, and utilization of the possible future horizontally-launched reusable Single-Stage-to-Orbit (SSTO) spaceplane for small payload short-duration manned/unmanned access to Low-Earth-Orbit (LEO). The 10,000 lb spaceplane would use 5,000 ft catapult-assist horizontal-launch facility and conduct powered approach and landing on conventional horizontal paved runways following the gliding atmospheric re-entry. To increase the economy of operation, the launch facility located at high elevations (4,000+ ft) equatorial region is required, such as, the plateaus in Kenya and Tanzania in Africa and/or Ecuador in South America. A 500-lb payload, including pilot-commander, is envisioned. The propulsion cycle …

Thermally Induced Aggregation Of Rigid Spheres On A Liquid Surface, Eric Forgoston, Leo Hentschker, Siobhan Soltau, Patrick Truitt, Ashuwin Vaidya

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

Fluids provide the optimal setting to explore natural patterns far from thermodynamic equilibrium. Experiments suggest that randomly dispersed particles on a liquid surface tend to aggregate on the surface of liquid over time, and the process is enhanced by an increase in the temperature of the liquid. We show that the agglomeration radii increases monotonically with temperature up until the point where all particles in the system form a single, large aggregate. The aggregation dynamics is related to changes in the material properties of the liquid including its viscosity and surface tension as well as the convection driven flow generated …

Parts Of The Whole: Teaching Quantitative Reasoning In The Predator-Prey Model, Dorothy Wallace

Numeracy

The classical predator-prey equations are in nearly every differential equations text and mathematical biology text. Usually they are presented fait accompli, leaving the student to analyze them or play with a computer program. Here we show that the process of fully understanding where these equations come from and how they are derived provides numerous opportunities to teach or reinforce quantitative reasoning skills necessary to future scientists. This example is used to invoke logic, systems thinking, causal reasoning, understanding functions of one or more variables, quantities versus rates of change, proportional reasoning, unit analysis, and comparison to data.

Modelling Subject-Specific Childhood Growth Using Linear Mixed-Effect Models With Cubic Regression Splines, Laura M. Grajeda, Andrada Ivanescu, Mayuko Saito, Ciprian Crainiceanu, Devan Jaganath, Robert H. Gilman, Jean E. Crabtree, Dermott Kelleher, Lilia Cabrera, Vitaliano Cama, William Checkley

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

Background: Childhood growth is a cornerstone of pediatric research. Statistical models need to consider individual trajectories to adequately describe growth outcomes. Specifically, well-defined longitudinal models are essential to characterize both population and subject-specific growth. Linear mixed-effect models with cubic regression splines can account for the nonlinearity of growth curves and provide reasonable estimators of population and subject-specific growth, velocity and acceleration. Methods: We provide a stepwise approach that builds from simple to complex models, and account for the intrinsic complexity of the data. We start with standard cubic splines regression models and build up to a model that includes subject-specific …

An Elementary Proof Of Dodgson's Condensation Method For Calculating Determinants, R. Corban Harwood, Mitch Main, Micah Donor

Faculty Publications - Department of Mathematics

In 1866, Charles Ludwidge Dodgson published a paper concerning a method for evaluating determinants called the condensation method. His paper documented a new method to calculate determinants that was based on Jacobi's Theorem. The condensation method is presented and proven here, and is demonstrated by a series of examples. The condensation method can be applied to a number of situations, including calculating eigenvalues, solving a system of linear equations, and even determining the different energy levels of a molecular system. The method is much more efficient than cofactor expansions, particularly for large matrices; for a 5 x 5 matrix, the …

Effect Of Spalled Particles Thermal Degradation On A Hypersonic Flow Field Environment, Raghava S. C. Davuluri, Huaibao Zhang, Alexandre Martin

Mechanical Engineering Faculty Publications

Two-way coupling is performed between a spallation code and a hypersonic aerothermodynamics CFD solver to evaluate the effect of spalled particles on the flow field. Time accurate solutions are computed in argon and air flow fields. A single particle simulations and multiple particles simulations are performed and studied. The results show that the carbon vapor released by spalled particles tend to change the composition of the flow field, particularly the upstream region of the shock.

Models Of Internal Waves In The Presence Of Currents, Alan Compelli, Rossen Ivanov

Conference papers

A fluid system consisting of two domains is examined. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. An internal wave propagating in one direction, driven by gravity, acts as a free common interface between the fluids. Various current profiles are considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behaviour is examined and compared to that of other known models. The linearised equations as well as long-wave approximations are formulated. The presented models provide potential applications to modelling of internal geophysical …

A Competitive Random Sequential Adsorption Model For Immunoassay Activity, Dana Mackey, Eilis Kelly, Robert Nooney

Conference papers

Immunoassays rely on highly specific reactions between antibodies and antigens and are used in biomedical diagnostics applications to detect biomarkers for a variety of diseases. Antibody immobilization to solid interfaces through random adsorption is a widely used technique but has the disadvantage of severely reducing the antigen binding activity and, consequently, the assay performance. This paper proposes a simple mathematical framework, based on the theory known as competitive random sequential adsorption (CRSA), for describing how the activity of immobilized antibodies depends on their orientation and packing density and generalizes a previous model by introducing the antibody aspect ratio as an …

Sphere Representations, Stacked Polytopes, And The Colin De Verdière Number Of A Graph, Lon Mitchell, Lynne Yengulalp

Mathematics Faculty Publications

We prove that a k-tree can be viewed as a subgraph of a special type of (k + 1)- tree that corresponds to a stacked polytope and that these “stacked” (k + 1)-trees admit representations by orthogonal spheres in R k+1. As a result, we derive lower bounds for Colin de Verdi`ere’s µ of complements of partial k-trees and prove that µ(G) + µ(G) > |G| − 2 for all chordal G.

Factorized Runge-Kutta-Chebyshev Methods, Stephen O'Sullivan

Conference papers

The second-order extended stability Factorized Runge-Kutta-Chebyshev (FRKC2) class of explicit schemes for the integration of large systems of PDEs with diffusive terms is presented. FRKC2 schemes are straightforward to implement through ordered sequences of forward Euler steps with complex stepsizes, and easily parallelised for large scale problems on distributed architectures.

Preserving 7 digits for accuracy at 16 digit precision, the schemes are theoretically capable of maintaining internal stability at acceleration factors in excess of 6000 with respect to standard explicit Runge-Kutta methods. The stability domains have approximately the same extents as those of RKC schemes, and are a third longer …

General Solution Of The Wind Triangle Problem And The Critical Tailwind Angle, Nihad E. Daidzic

Aviation Department Publications

A general analytical solution of the navigational wind-triangle problem and the calculation of the critical tailwind angle are presented in this study among other findings. Any crosswind component will effectively create a headwind component on fixed course tracks. The meaning of a route track is lost with excessive crosswinds representing the bifurcation point between the possible and the impossible navigational solutions. Any wind of constant direction and speed will effectively reduce groundspeed and increase time-of-flight on closed-loop multi-segment flights. Effective wind track component consists, in general, of true and induced components. The average groundspeed of multiple-leg flights is a harmonic …

Pattern Recognition In High-Dimensional Data, Matthew Dannenberg

HMC Senior Theses

Vast amounts of data are produced all the time. Yet this data does not easily equate to useful information: extracting information from large amounts of high dimensional data is nontrivial. People are simply drowning in data. A recent and growing source of high-dimensional data is hyperspectral imaging. Hyperspectral images allow for massive amounts of spectral information to be contained in a single image. In this thesis, a robust supervised machine learning algorithm is developed to efficiently perform binary object classification on hyperspectral image data by making use of the geometry of Grassmann manifolds. This algorithm can consistently distinguish between a …

The Global Stability Of The Solution To The Morse Potential In A Catastrophic Regime, Weerapat Pittayakanchit

HMC Senior Theses

Swarms of animals exhibit aggregations whose behavior is a challenge for mathematicians to understand. We analyze this behavior numerically and analytically by using the pairwise interaction model known as the Morse potential. Our goal is to prove the global stability of the candidate local minimizer in 1D found in A Primer of Swarm Equilibria. Using the calculus of variations and eigenvalues analysis, we conclude that the candidate local minimizer is a global minimum with respect to all solution smaller than its support. In addition, we manage to extend the global stability condition to any solutions whose support has a single …