Applied Mathematics Commons^™

How To Describe Measurement Errors: A Natural Generalization Of The Central Limit Theorem Beyond Normal (And Other Infinitely Divisible) Distributions, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When precise measurement instruments are designed, designers try their best to decrease the effect of the main factors leading to measurement errors. As a result of this decrease, the remaining measurement error is the joint result of a large number of relatively small independent error components. According to the Central Limit Theorem, under reasonable conditions, when the number of components increases, the resulting distribution tends to Gaussian (normal). Thus, in practice, when the number of components is large, the distribution is close to normal -- and normal distributions are indeed ubiquitous in measurements. However, in some practical situations, the distribution …

Global Analysis Of The Shadow Gierer-Meinhardt System With General Linear Boundary Conditions In A Random Environment, Kwadwo Antwi-Fordjour, Seonguk Kim, Marius Nkashama

Mathematics Faculty Publications

The global analysis of the shadow Gierer-Meinhardt system with multiplicative white noise and general linear boundary conditions is investigated in this paper. For this reaction-diffusion system, we employ a fixed point argument to prove local existence and uniqueness. Our results on global existence are based on a priori estimates of solutions.

How To Separate Absolute And Relative Error Components: Interval Case, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Usually, measurement errors contain both absolute and relative components. To correctly gauge the amount of measurement error for all possible values of the measured quantity, it is important to separate these two error components. For probabilistic uncertainty, this separation can be obtained by using traditional probabilistic techniques. The problem is that in many practical situations, we do not know the probability distribution, we only know the upper bound on the measurement error. In such situations of interval uncertainty, separation of absolute and relative error components is not easy. In this paper, we propose a technique for such a separation based …

What If We Use Almost-Linear Functions Instead Of Linear Ones As A First Approximation In Interval Computations, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, the only information that we have about measurement errors is the upper bound on their absolute values. In such situations, the only information that we have after the measurement about the actual (unknown) value of the corresponding quantity is that this value belongs to the corresponding interval: e.g., if the measurement result is 1.0, and the upper bound is 0.1, then this interval is [1.0−0.1,1.0+0.1] = [0.9,1.1]. An important practical question is what is the resulting interval uncertainty of indirect measurements, i.e., in other words, how interval uncertainty propagates through data processing. There exist feasible algorithms …

Analysis Of Dynamical Systems For Synthesis Of Phenobarbital, Mishal Ali

Annual Student Research Poster Session

The use of mathematical methods for the analysis of chemical reaction systems is one of the useful tools. Phenobarbital (a barbiturate type medication also called phenobarb) is a prescription drug used to control seizures, relieve anxiety, treat epilepsy (in some countries), and prevent withdrawal symptoms in people dependent on other barbiture drugs. We approaches it with matrix analysis and ODE system. It helps us understand the chemical stoichiometry of these synthesis reactions.

Supervisor: Prof. Seonguk Kim, PhD

Quasilinearization Applied To Boundary Value Problems At Resonance For Riemann-Liouville Fractional Differential Equations, Paul W. Eloe, Jaganmohan Jonnalagadda

Mathematics Faculty Publications

The quasilinearization method is applied to a boundary value problem at resonance for a Riemann-Liouville fractional differential equation. Under suitable hypotheses, the method of upper and lower solutions is employed to establish uniqueness of solutions. A shift method, coupled with the method of upper and lower solutions, is applied to establish existence of solutions. The quasilinearization algorithm is then applied to obtain sequences of lower and upper solutions that converge monotonically and quadratically to the unique solution of the boundary value problem at resonance.

A Data Analytic Framework For Physical Fatigue Management Using Wearable Sensors, Zahra Sedighi Maman, Ying-Ju Chen, Amir Baghdadi, Seamus Lombardo, Lora A. Cavuoto, Fadel M. Megahed

Mathematics Faculty Publications

The use of expert systems in optimizing and transforming human performance has been limited in practice due to the lack of understanding of how an individual's performance deteriorates with fatigue accumulation, which can vary based on both the worker and the workplace conditions. As a first step toward realizing the human-centered approach to artificial intelligence and expert systems, this paper lays the foundation for a data analytic approach to managing fatigue in physically-demanding workplaces. The proposed framework capitalizes on continuously collected human performance data from wearable sensor technologies, and is centered around four distinct phases of fatigue: (a) detection, where …

A Two-Stage Machine Learning Framework To Predict Heart Transplantation Survival Probabilities Over Time With A Monotonic Probability Constraint, Hamidreza Ahady Dolatsaraa, Ying-Ju (Tessa) Chen, Christy Evans, Ashish Gupta, Fadel M. Megahed

Mathematics Faculty Publications

The overarching goal of this paper is to develop a modeling framework that can be used to obtain personalized, data-driven and monotonically constrained probability curves. This research is motivated by the important problem of improving the predictions for organ transplantation outcomes, which can inform updates made to organ allocation protocols, post-transplantation care pathways, and clinical resource utilization. In pursuit of our overarching goal and motivating problem, we propose a novel two-stage machine learning-based framework for obtaining monotonic probabilities over time. The first stage uses the standard approach of using independent machine learning models to predict transplantation outcomes for each time-period …

Mathematical Modeling Of Nonlinear Problem Biological Population In Not Divergent Form With Absorption, And Variable Density, Maftuha Sayfullayeva

Acta of Turin Polytechnic University in Tashkent

В работе установлены критические и двойные критические случаи, обусловленные представлением двойного нелинейного параболического уравнения с переменной плотностью с поглощением в "радиально-симметричной" форме.Такое представление исходного уравнения дало возможность легко построить решения типа Зельдовоч-Баренбатт-Паттл для критических случаев в виде функций сравнения.

Cover Song Identification - A Novel Stem-Based Approach To Improve Song-To-Song Similarity Measurements, Lavonnia Newman, Dhyan Shah, Chandler Vaughn, Faizan Javed

SMU Data Science Review

Music is incorporated into our daily lives whether intentional or unintentional. It evokes responses and behavior so much so there is an entire study dedicated to the psychology of music. Music creates the mood for dancing, exercising, creative thought or even relaxation. It is a powerful tool that can be used in various venues and through advertisements to influence and guide human reactions. Music is also often "borrowed" in the industry today. The practices of sampling and remixing music in the digital age have made cover song identification an active area of research. While most of this research is focused …

Submesoscale Kinematic Properties In Summer And Winter Surface Flows In The Northern Gulf Of Mexico, M. Berta, A. Griffa, A. C. Haza, J. Horstmann, Helga Huntley, R. Ibrahim, B. Lund, T. M. Ozgokmen, A. C. Poje

College of Science & Mathematics Departmental Research

Statistical properties of near-surface horizontal velocity gradients are obtained from four drifter experiments conducted in the Gulf of Mexico during Summer 2012 and Winter 2016. The data density provided by the near-simultaneous deployments of 90-326 surface drifters in each allows direct, drifter-based estimates of the scale dependence of velocity gradients at separation scales ranging from 200 m to 5 km. The robustness of these estimates, derived from uniquley sampled, nearly equilateral triplets, is confirmed by comparisons with estimates produced from larger drifter clusters, and with estimates based on concurrent Eulerian X-band radar observations. The winter launches were deployed above a …

Numerical Computations Of Vortex Formation Length In Flow Past An Elliptical Cylinder, Matthew Karlson, Bogdan Nita, Ashwin Vaidya

Department of Mathematics Faculty Scholarship and Creative Works

We examine two dimensional properties of vortex shedding past elliptical cylinders through numerical simulations. Specifically, we investigate the vortex formation length in the Reynolds number regime 10 to 100 for elliptical bodies of aspect ratio in the range 0.4 to 1.4. Our computations reveal that in the steady flow regime, the change in the vortex length follows a linear profile with respect to the Reynolds number, while in the unsteady regime, the time averaged vortex length decreases in an exponential manner with increasing Reynolds number. The transition in profile is used to identify the critical Reynolds number which marks the …

Does Transition To Democracy Lead To Chaos: A Theorem, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When a country transitions to democracy, at first, many political parties appear. A natural question is whether the number of such parties feasible and reasonable -- or whether this is a complete chaos. In this paper, we formulate a simplified version of this question in precise terms and show that the number of parties will be feasible -- i.e., that transition to democracy does not lead to chaos.

A Reformulated Krein Matrix For Star-Even Polynomial Operators With Applications, Todd Kapitula, Ross Parker, Bjorn Sandstede

University Faculty Publications and Creative Works

In its original formulation the Krein matrix was used to locate the spectrum of first-order star-even polynomial operators where both operator coefficients are nonsingular. Such operators naturally arise when considering first-order-in-time Hamiltonian PDEs. Herein the matrix is reformulated to allow for operator coefficients with nontrivial kernel. Moreover, it is extended to allow for the study of the spectral problem associated with quadratic star-even operators, which arise when considering the spectral problem associated with second-order-in-time Hamiltonian PDEs. In conjunction with the Hamiltonian-Krein index (HKI) the Krein matrix is used to study two problems: conditions leading to Hamiltonian-Hopf bifurcations for small spatially …

An Accurate Solution Of The Self-Similar Orbit-Averaged Fokker-Planck Equation For Core-Collapsing Isotropic Globular Clusters: Properties And Application, Yuta Ito

Dissertations, Theses, and Capstone Projects

Hundreds of dense star clusters exist in almost all galaxies. Each cluster is composed of approximately ten thousand through ten million stars. The stars orbit in the clusters due to the clusters' self-gravity. Standard stellar dynamics expects that the clusters behave like collisionless self-gravitating systems on short time scales (~ million years) and the stars travel in smooth continuous orbits. Such clusters temporally settle to dynamically stable states or quasi-stationary states (QSS). Two fundamental QSS models are the isothermal- and polytropic- spheres since they have similar structures to the actual core (central part) and halo (outskirt) of the clusters. The …

Matrix Low Rank Approximation At Sublinear Cost, Qi Luan

Dissertations, Theses, and Capstone Projects

A matrix algorithm runs at sublinear cost if the number of arithmetic operations involved is far fewer than the number of entries of the input matrix. Such algorithms are especially crucial for applications in the field of Big Data, where input matrices are so immense that one can only store a fraction of the entire matrix in memory of modern machines. Typically, such matrices admit Low Rank Approximation (LRA) that can be stored and processed at sublinear cost. Can we compute LRA at sublinear cost? Our counter example presented in Appendix C shows that no sublinear cost algorithm can compute …

Evaluating Performance Of Openmp Tasks In A Seismic Stencil Application, Eric Raut, Jie Meng, Mauricio Araya-Polo, Barbara Chapman

Department of Applied Mathematics & Statistics Faculty Publications

Simulations based on stencil computations (widely used in geosciences) have been dominated by the MPI+OpenMP programming model paradigm. Little effort has been devoted to experimenting with task-based parallelism in this context. We address this by introducing OpenMP task parallelism into the kernel of an industrial seismic modeling code, Minimod. We observe that even for these highly regular stencil computations, taskified kernels are competitive with traditional OpenMP-augmented loops, and in some experiments tasks even outperform loop parallelism.

This promising result sets the stage for more complex computational patterns. Simulations involve more than just the stencil calculation: a collection of kernels is …

Mathematical Models And Tools To Understand Coupled Circadian Oscillations And Limit Cycling Systems, Guangyuan Liao

Dissertations

The circadian rhythm refers to an internal body process that regulates many body processes including the sleep-wake cycle, digestion and hormone release. The ability of a circadian system to entrain to the 24-hour light-dark cycle is one of the most important properties. There are several scenarios in which circadian oscillators do not directly receive light-dark forcing. Instead they are part of hierarchical systems in which, as \peripheral" oscillators, they are periodically forced by other \central" circadian oscillators that do directly receive light input. Such dynamics are modeled as hierarchical coupled limit cycle systems. Those models usually have a large population, …

Efficient Approximations For Stationary Single-Channel Calcium Nanodomains, Yinbo Chen

Dissertations

Mathematical and computational modeling plays an important role in the study of local Ca2+ signals underlying many fundamental physiological processes such as synaptic neurotransmitter release and myocyte contraction. Closed-form approximations describing steady-state distribution of Ca2+ in the vicinity of an open Ca2+ channel have proved particularly useful for the qualitative modeling of local Ca2+ signals. This dissertation presents several simple and efficient approximants for the equilibrium Ca2+ concentration near a point source in the presence of a mobile Ca2+ buffer, which achieve great accuracy over a wide range of model parameters. Such approximations provide an efficient method for estimating Ca2+ …

Global Optimization Algorithms For Image Registration And Clustering, Cuicui Zheng

Dissertations

Global optimization is a classical problem of finding the minimum or maximum value of an objective function. It has applications in many areas, such as biological image analysis, chemistry, mechanical engineering, financial analysis, deep learning and image processing. For practical applications, it is important to understand the efficiency of global optimization algorithms. This dissertation develops and analyzes some new global optimization algorithms and applies them to practical problems, mainly for image registration and data clustering.

First, the dissertation presents a new global optimization algorithm which approximates the optimum using only function values. The basic idea is to use the points …