Applied Mathematics Commons^™

Sloane’S Gap: Do Mathematical And Social Factors Explain The Distribution Of Numbers In The Oeis?, Nicolas J.-P. Gauvrit, Jean-Paul Delahaye, Hector Zenil

Journal of Humanistic Mathematics

The Online Encyclopedia of Integer Sequences (OEIS) is a catalog of integer sequences. We are particularly interested in the number of occurrences of N(n) of an integer n in the database. This number N(n) marks the importance of n and it varies noticeably from one number to another, and from one number to the next in a series. “Importance” can be mathematically objective (2^10 is an example of an “important” number in this sense) or as the result of a shared mathematical culture (10^9 is more important than 9^10 because we use a decimal notation). The concept of algorithmic complexity …

Mathematics Professor’S Publications Add Up, Kim Hill

News and Events (Discontinued Series)

No abstract provided.

On Multicomponent Derivative Nonlinear Schrodinger Equation Related To Symmetric Spaces, Tihomir Valchev

Conference papers

We study derivative nonlinear Schrodinger equations related to symmetric spaces of the type A.III. We discuss the spectral properties of the corresponding Lax operator and develop the direct scattering problem connected to it. By applying an appropriately chosen dressing factor we derive soliton solutions to the nonlinear equation. We find the integrals of motion by using the method of diagonalization of Lax pair.

Remarks On Quadratic Bundles Related To Hermitian Symmetric Spaces, Tihomir Valchev

Conference papers

We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m+n)/S(U(m)\times U(n)). We discuss the spectral properties of scattering operator, develop the direct scattering problem associated with it and stress on the effect of reduction on these. By applying a modification of Zakharov-Shabat's dressing procedure we demonstrate how one can obtain reflectionless potentials. That way one is able to generate soliton solutions to the nonlinear evolution equations belonging to the integrable hierarchy associated with quadratic bundles under study.

The Effects Of Variable Viscosity On The Peristaltic Flow Of Non-Newtonian Fluid Through A Porous Medium In An Inclined Channel With Slip Boundary Conditions, Ambreen Afsar Khan, R. Ellahi, Muhammad Usman

Mathematics Faculty Publications

The present paper investigates the peristaltic motion of an incompressible non-Newtonian fluid with variable viscosity through a porous medium in an inclined symmetric channel under the effect of the slip condition. A long wavelength approximation is used in mathematical modeling. The system of the governing nonlinear partial differential equation has been solved by using the regular perturbation method and the analytical solutions for velocity and pressure rise have been obtained in the form of stream function. In the obtained solution expressions, the long wavelength and low Reynolds number assumptions are utilized. The salient features of pumping and trapping phenomena are …

A Superposed Log-Linear Failure Intensity Model For Repairable Artillery Systems, Byeong Min Mun, Suk Joo Bae, Paul Kvam

Department of Math & Statistics Faculty Publications

This article investigates complex repairable artillery systems that include several failure modes. We derive a superposed process based on a mixture of nonhomogeneous Poisson processes in a minimal repair model. This allows for a bathtub-shaped failure intensity that models artillery data better than currently used methods. The method of maximum likelihood is used to estimate model parameters and construct confidence intervals for the cumulative intensity of the superposed process. Finally, we propose an optimal maintenance policy for repairable systems with bathtub-shaped intensity and apply it to the artillery-failure data.

Mathematical Modeling And Analysis Of Asthma Stability And Severity, Arezoo Hanifi

Electronic Theses and Dissertations

Asthma is one of the most common chronic conditions in the United States. Asthma affects about one in fifteen people. It affects children more than adults and blacks more than whites. People with asthma experience attacks of wheezing, breathlessness, chest tightness, and coughing. Asthma can be fatal and the costs for the disease (direct and indirect) are approximated to be tens of billions of dollars each year.

There is no cure for asthma. However; for most people if asthma is controlled well they can lead normal, active lives. Therefore asthma controllability is a main factor in clinical practice. In order …

Generalized Least-Squares Regressions Ii: Theory And Classification, Nataniel Greene

Publications and Research

In the first paper of this series, a variety of known and new symmetric and weighted least-squares regression methods were presented with efficient derivations. This paper continues and generalizes the previous work with a theory for deriving, analyzing, and classifying all symmetric and weighted least-squares regression methods.

Social Aggregation In Pea Aphids: Experiment And Random Walk Modeling, Christa Nilsen, John Paige, Olivia Warner, Benjamin Mayhew, Ryan Sutley, Matthew Lam '15, Andrew J. Bernoff, Chad M. Topaz

All HMC Faculty Publications and Research

From bird flocks to fish schools and ungulate herds to insect swarms, social biological aggregations are found across the natural world. An ongoing challenge in the mathematical modeling of aggregations is to strengthen the connection between models and biological data by quantifying the rules that individuals follow. We model aggregation of the pea aphid, Acyrthosiphon pisum. Specifically, we conduct experiments to track the motion of aphids walking in a featureless circular arena in order to deduce individual-level rules. We observe that each aphid transitions stochastically between a moving and a stationary state. Moving aphids follow a correlated random walk. …

Existence And Qualitative Properties Of Solutions For Nonlinear Dirichlet Problems, Alfonso Castro, Jorge Cossio, Carlos Vélez

All HMC Faculty Publications and Research

Sign-changing solutions to semilinear elliptic problems in connection with their Morse indices. To this end, we first establish a priori bounds for one-sign solutions. Secondly, using abstract saddle point principles we find large augmented Morse index solutions. In this part, extensive use is made of critical groups, Morse index arguments, Lyapunov-Schmidt reduction, and Leray-Schauder degree. Finally, we provide conditions under which these solutions necessarily change sign and we comment about further qualitative properties.

Generalized Least-Squares Regressions I: Efficient Derivations, Nataniel Greene

Publications and Research

Ordinary least-squares regression suffers from a fundamental lack of symmetry: the regression line of y given x and the regression line of x given y are not inverses of each other. Alternative symmetric regression methods have been developed to address this concern, notably: orthogonal regression and geometric mean regression. This paper presents in detail a variety of least squares regression methods which may not have been known or fully explicated. The derivation of each method is made efficient through the use of Ehrenberg's formula for the ordinary least-squares error and through the extraction of a weight function g(b) which characterizes …

Near-Optimal Compressed Sensing Guarantees For Anisotropic And Isotropic Total Variation Minimization, Deanna Needell, Rachel Ward

CMC Faculty Publications and Research

Consider the problem of reconstructing a multidimensional signal from partial information, as in the setting of compressed sensing. Without any additional assumptions, this problem is ill-posed. However, for signals such as natural images or movies, the minimal total variation estimate consistent with the measurements often produces a good approximation to the underlying signal, even if the number of measurements is far smaller than the ambient dimensionality. Recently, guarantees for two-dimensional images were established. This paper extends these theoretical results to signals of arbitrary dimension and to both the anisotropic and isotropic total variation problems. To be precise, we show that …

Simulations Of Surfactant Driven Thin Film Flow, Shreyas Kumar

HMC Senior Theses

This thesis is intended to fulfill the requirements of the Math and Physics departments at Harvey Mudd College. We begin with a brief introduction to the study of surfactant dynamics followed by some background on the experimental framework our work is related to. We then go through a derivation of the model we use, and explore in depth the nature of the Equation of State (EoS), the relationship between the surface tension on a fluid and the surfactant concentration. We consider the effect of using an empirical equation of state on the results of the simulations and compare the new …

Global Attracting Equilibria For Coupled Systems With Ceiling Density Dependence, Eric A. Eager, Mary Hebert, Elise Hellwig, Francisco Hernandez, Richard Rebarber, Brigitte Tenhumberg, Bryan Wigianto

School of Biological Sciences: Faculty Publications

In this paper, we present a system of two difference equations modeling the dynamics of a coupled population with two patches. Each patch can house only a limited number of individuals (called a carrying capacity) because resources like food and breeding sites are limited in each patch. We assume that the population in each patch is governed by a linear model until reaching a carrying capacity in each patch, resulting in map which is nonlinear and not sublinear. We analyze the global attractors of this model.

Duality Of The Weak Parallelogram Laws On Banach Spaces, Raymond Cheng, Charles B. Harris

Mathematics & Statistics Faculty Publications

This paper explores a family of weak parallelogram laws for Banach spaces. Some basic properties of such spaces are obtained. The main result is that a Banach space satisfies a lower weak parallelogram law if and only if its dual satisfies an upper weak parallelogram law, and vice versa. Connections are established between the weak parallelogram laws and the following: subspaces, quotient spaces, Cartesian products, and the Rademacher type and co-type properties.

Nonlocal Aggregation Models: A Primer Of Swarm Equilibria, Andrew J. Bernoff, Chad M. Topaz

All HMC Faculty Publications and Research

Biological aggregations such as fish schools, bird flocks, bacterial colonies, and insect swarms have characteristic morphologies governed by the group members' intrinsic social interactions with each other and by their interactions with the external environment. Starting from a simple discrete model treating individual organisms as point particles, we derive a nonlocal partial differential equation describing the evolving population density of a continuum aggregation. To study equilibria and their stability, we use tools from the calculus of variations. In one spatial dimension, and for several choices of social forces, external forces, and domains, we find exact analytical expressions for the equilibria. …

New Microarray Image Segmentation Using Segmentation Based Contours Method, Yuan Cheng

Doctoral Dissertations

The goal of the research developed in this dissertation is to develop a more accurate segmentation method for Affymetrix microarray images. The Affymetrix microarray biotechnologies have become increasingly important in the biomedical research field. Affymetrix microarray images are widely used in disease diagnostics and disease control. They are capable of monitoring the expression levels of thousands of genes simultaneously. Hence, scientists can get a deep understanding on genomic regulation, interaction and expression by using such tools.

We also introduce a novel Affymetrix microarray image simulation model and how the Affymetrix microarray image is simulated by using this model. This simulation …

Functional Data Analysis To Guide A Conditional Likelihood Regression In A Case-Crossover Study Investigating Whether Social Characteristics Modify The Health Effects Of Air Pollution, Juana Maribel Herrera Hernandez

Open Access Theses & Dissertations

In this study we are focused on exploring whether social characteristics modify the relationship between air pollution and hospitalizations due to asthma or chronic pulmonary obstructive disease (COPD) in El Paso, Tx. The case-crossover design with conditional regression analysis was used, here the controls and the case are the same subject at different

times and has the advantage of removing confounding by permanently confounding factors. Social characteristics are included in the models as interactions with the pollutants, variables included are age, sex, ethnicity and insurance status as indicator for the socio-economic status. The pollutant's lags were chosen using the historical …

On Different Techniques For The Calculation Of Bouguer Gravity Anomalies For Joint Inversion And Model Fusion Of Geophysical Data In The Rio Grande Rift, Azucena Zamora

Open Access Theses & Dissertations

Density variations in the Earth result from different material properties, which reflect the tectonic processes attributed to a region. Density variations can be identified through measurable material properties, such as seismic velocities, gravity field, magnetic field, etc. Gravity anomaly inversions are particularly sensitive to density variations but suffer from significant non-uniqueness. However, using inverse models with gravity Bouguer anomalies and other geophysical data, we can determine three dimensional structural and geological properties of the given area. We explore different techniques for the calculation of Bouguer gravity anomalies for their use in joint inversion of multiple geophysical data sets and a …

Hydrodynamic Modeling Of Ns-Laser Ablation, David Autrique, Vasilios Alexiades, Harihar Khanal

Publications

Laser ablation is a versatile and widespread technique, applied in an increasing number of medical, industrial and analytical applications. A hydrodynamic multiphase model describing nanosecond-laser ablation (ns- LA) is outlined. The model accounts for target heating and mass removal mechanisms as well as plume expansion and plasma formation. A copper target is placed in an ambient environment consisting of helium and irradiated by a nanosecond-laser pulse. The effect of variable laser settings on the ablation process is explored in 1-D numerical simulations.

An Experimental Testbed For Multi-Robot Tracking Of Manifolds And Coherent Structures In Flows, Matthew Michini, Kenneth Mallory, Dennis Larkin, M. Ani Hsieh, Eric Forgoston, Philip A. Yecko

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

In this paper, we describe the development of an experimental testbed capable of producing controllable ocean-like flows in a laboratory setting. The objective is to develop a testbed to evaluate multi-robot strategies for tracking manifolds and Lagrangian coherent structures (LCS) in the ocean. Recent theoretical results have shown that LCS coincide with minimum energy and minimum time optimal paths for autonomous vehicles in the ocean. Furthermore, knowledge of these structures enables the prediction and estimation of the underlying fluid dynamics. The testbed is a scaled flow tank capable of generating complex and controlled quasi-2D flow fields that exhibit wind-driven doublegyre …

Measure-Dependent Stochastic Nonlinear Beam Equations Driven By Fractional Brownian Motion, Mark A. Mckibben

Mathematics Faculty Publications

We study a class of nonlinear stochastic partial differential equations arising in themathematicalmodeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separableHilbert space and is studied using the tools …

Two-Dimensional Hydrodynamic Modeling Of Two-Phase Flow For Understanding Geyser Phenomena In Urban Stormwater System, Zhiyu S. Shao

Theses and Dissertations--Civil Engineering

During intense rain events a stormwater system can fill rapidly and undergo a transition from open channel flow to pressurized flow. This transition can create large discrete pockets of trapped air in the system. These pockets are pressurized in the horizontal reaches of the system and then are released through vertical vents. In extreme cases, the transition and release of air pockets can create a geyser feature.

The current models are inadequate for simulating mixed flows with complicated air-water interactions, such as geysers. Additionally, the simulation of air escaping in the vertical dropshaft is greatly simplified, or completely ignored, in …

When Cp(X) Is Domain Representable, William Fleissner, Lynne Yengulalp

Mathematics Faculty Publications

Let M be a metrizable group. Let G be a dense subgroup of M^X . If G is domain representable, then G = MX . The following corollaries answer open questions. If X is completely regular and C_p(X) is domain representable, then X is discrete. If X is zero-dimensional, T₂ , and C_p(X;D) is subcompact, then X is discrete.

An Implicit Interface Boundary Integral Method For Poisson’S Equation On Arbitrary Domains, Catherine Kublik, Nicolay M. Tanushev, Richard Tsai

Mathematics Faculty Publications

We propose a simple formulation for constructing boundary integral methods to solve Poisson’s equation on domains with smooth boundaries defined through their signed distance function. Our formulation is based on averaging a family of parameterizations of an integral equation defined on the boundary of the domain, where the integrations are carried out in the level set framework using an appropriate Jacobian. By the coarea formula, the algorithm operates in the Euclidean space and does not require any explicit parameterization of the boundaries. We present numerical results in two and three dimensions.

Sampling From The Hardcore Process, William C. Dodds

CMC Senior Theses

Partially Recursive Acceptance Rejection (PRAR) and bounding chains used in conjunction with coupling from the past (CFTP) are two perfect simulation protocols which can be used to sample from a variety of unnormalized target distributions. This paper first examines and then implements these two protocols to sample from the hardcore gas process. We empirically determine the subset of the hardcore process's parameters for which these two algorithms run in polynomial time. Comparing the efficiency of these two algorithms, we find that PRAR runs much faster for small values of the hardcore process's parameter whereas the bounding chain approach is vastly …

Invisibility: A Mathematical Perspective, Austin G. Gomez

CMC Senior Theses

The concept of rendering an object invisible, once considered unfathomable, can now be deemed achievable using artificial metamaterials. The ability for these advanced structures to refract waves in the negative direction has sparked creativity for future applications. Manipulating electromagnetic waves of all frequencies around an object requires precise and unique parameters, which are calculated from various mathemat- ical laws and equations. We explore the possible interpretations of these parameters and how they are implemented towards the construction of a suitable metamaterial. If carried out correctly, the wave will exit the metamaterial exhibiting the same behavior as when it had entered. …

On The Persistence Properties Of The Cross-Coupled Camassa-Holm System, David Henry, Darryl Holm, Rossen Ivanov

Articles

In this paper we examine the evolution of solutions, that initially have compact support, of a recently-derived system of cross-coupled Camassa-Holm equations. The analytical methods which we employ provide a full picture for the persistence of compact support for the momenta. For solutions of the system itself, the answer is more convoluted, and we determine when the compactness of the support is lost, replaced instead by an exponential decay rate.

Particle Trajectories In Extreme Stokes Waves Over Inifinte Depth, Tony Lyons

Articles

We investigate the velocity field of fluid particles in an extreme water wave over infinite depth. It is shown that the trajectories of the particles within the fluid and along the free surface do not form closed paths over the course of one period, but rather undergo a positive drift in the direction of wave propagation. In addition it is shown that the wave crest cannot form a stagnation point despite the velocity of the fluid being zero there.

The Truth About Lie Symmetries: Solving Differential Equations With Symmetry Methods, Ruth A. Steinhour

Senior Independent Study Theses

Differential equations are vitally important in numerous scientific fields. Oftentimes, they are quite challenging to solve. This Independent Study examines one method for solving differential equations. Norwegian mathematician Sophus Lie developed this method, which uses groups of symmetries, called Lie groups. These symmetries map one solution curve to another. They can be used to determine a canonical coordinate system for a given differential equation. Writing the differential equation in terms of a different coordinate system can make the equation simpler to solve. This I.S. explores techniques for finding a canonical coordinate system and using it to solve a given differential …