 | Michael Bell (2012)
Abstract: Full text: 
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 | Maria Barouti (2011)
Abstract: The Hilbert function for any graded module over a field k is defined by the dimension of all of the summands M_b, where b indicates the graded component being considered. One standard approach to computing the Hilbert function is to come up with a free-resolution for the graded module M and another is via a Hilbert power series which serves as a generating function. Using combinatorics and homological algebra we develop three alternative ways to generate the values of a Hilbert function when the graded module is a quotient ring over a field. Two of these approaches (which we've called the lcm-Lattice method and the Syzygy method) are conceptually combinatorial and work for any polynomial quotient ring over a field. The third approach, which we call the Hilbert function table method, also uses syzygies but the approach is better described in terms of homological algebra. Full text:
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 | Christina Battista (2011)
Abstract: Many mathematical models relevant to osteoporosis have been developed and studied. Although osteoclasts and osteoblasts are the crucial variables in bone resorption and bone formation, PTH can cause changes in the ratio of these cells and therefore should be studied more closely. Some of the current models for osteoporosis will be analyzed in this thesis as well as amended to account for the phenomenon that occurs with various methods of PTH administration. By administering PTH in either pulsatile or continuous doses, we obtain very different results. When administered in a continuous fashion, the body experiences a net bone loss over time, but given in daily, pulsatile doses, we increase bone mineral density. By developing a model that incorporates PTH administration, we hope to provide the building block for a broader model that is able to determine the efficacy of various osteoporotic treatments. Full text:
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 | Zois Boukouvalas (2011)
Abstract: Medical image registration has received considerable attention in medical imaging and computer vision, because of the large variety of ways in which it can impact patient care. Over the years, many algorithms have been proposed for medical image registration. Medical image registration uses techniques to create images of parts of the human body for clinical purposes. This thesis focuses on one small subset of registration algorithms: using machine learning techniques to train the similarity measure for use in medical image registration. This thesis is organized in the following manner. In Chapter 1 we introduce the idea of image registration, describe some some applications in medical imaging, and mathematically formulate the three main components of any registration problem: geometric transformation, similarity measure and optimization procedure. Finally we describe how the ideas in this thesis t into the eld of medical image registration, and we describe some related work. In Chapter 2 we introduce the concept of machine learning and we provide examples to illustrate machine learning algorithms. We then describe the knn-nearest neighbors algorithm and the relationship between Euclidean and Mahalanobis distance. Next we introduce distance metric learning and present two approaches for learning the Mahalanobis distance. Finally we provide a description and visual comparison of two algorithms for distance metric learning. In Chapter 3 we describe how distance metric learning can be applied to the problem of medical image registration. Our goal is to learn the optimal similarity measure given a training dataset of correctly registered images. To assess the performance of the two distance metric learning algorithms we test them using images from a series of patients. Moreover we illustrate the sensitivity of one of the learning algorithms by examining the variability of the resulting target registration errors. Finally we present our experimental results of registering CT and MR images. Finally in Chapter 4 we suggest some ideas for future work in order to improve our registration results and to speed up the algorithms. Full text:
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 | Stephen Wehry (2011)
Abstract: Monte Carlo Simulation is used to compare the performance of the Back-Propagation, Conjugate-Gradient, and Finite-Difference algorithms when training simple Multilayer Perception networks to solve pattern recognition and bit counting problems. Twelve individual simulations will be run for each training algorithm-test problem combination, resulting in an overall total of 72 simulations. The random elements in each Monte Carlo simulation are to be the individual synaptic weights between layers, which will be uniformly distributed. Two other factors, the size of the hidden layer and the exponent of the error function, will also be tested within the simulation plan outlined above. Full text:
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 | Marcela Munoz reales (2011)
Abstract: Spectral images provide a large amount of spectral information about a scene, but sometimes when studying images, we are interested in specific components. It is a difficult problem to separate the relevant information or what we call interesting from the background of a spectral image, even more so if our target objects are unknown. Anomaly detection is a process by which algorithms are designed to separate the anomalous (different) points from the background of an image. The data is complex and lives in a high dimension, manifold learning algorithms are used to analyze data that lives in a high dimensional space, but that can be represented as a lower dimensional manifold embedded in the high dimensional space. Laplacian Eigenmaps is a manifold learning algorithm that applies spectral graph theory to perform a non-linear dimensionality reduction that preserves local neighborhood information. We present an approach to reduce the dimension of the data and separate anomalous pixels in spectral images using Laplacian Eigenmaps. Full text:
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 | Karin Meuwissen (2011)
Abstract: Mathematical modeling is an effective tool when maximizing revenue in hotels. With over 120,000 different combinations of variables and elements to consider when maximizing revenue, the process of finding the most lucrative combination can be time consuming and costly to hotels. An effective mathematical model assists in reducing the guesswork involved. This study will demonstrate how the implementation of discounts reduces revenue loss by forcing pre-payment and increasing occupancy levels. In an effort to reduce revenue loss, hotels have implemented many strategies such as using credit card guarantees, demanding pre-payment, and offering discounts, all of which are used to reduce the no-show rate. A careful balance must be found, as offering a large discount to reduce the no-show rate can result in as much revenue loss as a high no-show rate. A mathematical model, used in combination with good judgement and managerial expertise, can assist in finding the discount necessary to maximize a hotel's daily revenue. Full text:
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 | Alex Bryce (2011)
Abstract: Turing instability in discrete replicator systems
When analyzing a discrete reaction-diffusion dynamical system, one primary area of interest is locating where in the parameter space Turing instabilities occur. It will be shown that Turing instabilities cannot occur in the react then diffuse equations if all diffusion coefficients are equal. The replicator dynamic is a system of equations that is used in evolutionary game theory applications to study behavior types in animal populations. Conditions for a Turing instability in first order discrete replicator systems will be discussed and illustrated with computer simulations of the results. Full text:
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 | Troy Winkstern (2011)
Abstract: In the past 20 years, there have been significant advances in the use of medical imaging in patient care. Today, image registration is being used by doctors all over the world to compare computed tomography (CT), Magnetic Resonance Imaging (MRI), X-rays, Ultrasound (US), Positron Emission Tomography (PET), and/or Single Photon Emission Computed Tomography (SPECT) images of their patients from different times to formulate an accurate diagnosis. Mathematically, image registration can be posed as an initial boundary value problem (IBVP). When applied to medical imaging, the current boundary conditions being used are not physically meaningful. We are proposing two new boundary conditions that are more physically meaningful because they were designed as combinations of the current boundary conditions. Therefore the image registration algorithms we employ are slight variations of the current techniques. We show how to apply our boundary conditions to the current image registration techniques and compare the results against the current boundary conditions. By comparing the results, we were able to illustrate the proposed boundary conditions created more visually pleasing and physically realistic results in both two and three dimensions. Full text:
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 | James Oleksyn (2011)
Abstract: Numerous mathematical models in applied mathematics can be expressed as a partial differential equation involving certain coefficients. These coefficients are known and they describe some physical properties of the model. The direct problem in this context is to solve the partial differential equation. By contrast, an inverse problem asks for the identification of the variable coefficients when a certain measurement of a solution of the partial differential equation is available. One of the most commonly used approaches for solving this inverse problem is by posing a constrained minimization problem which can be written as a variational inequality. The main contribution of this thesis is to employ various variants of extragradient methods to solve the inverse problem of parameter identification by posing it as a variational inequality. We present a thorough comparison of projected gradient method, scaled projected gradient method and several extragradient methods including the Marcotte variants, He-Goldstein type method, the projection- contraction methods proposed by Solodov and Tseng, and the hyperplane method developed by Iusem. We also test the performance of the extragradient methods for the image debluring problem. Full text:
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 | Michael Snook (2011)
Abstract: The concept of fully homomorphic encryption has been considered the "holy grail" of cryptography since the discovery of secure public key cryptography in the 1970s. Fully homomorphic encryption allows arbitrary computation on encrypted data to be performed securely. Craig Gentry's new method of bootstrapping introduced in 2009 provides a technique for constructing fully homomorphic cryptosystems. In this paper we explore one such bootstrappable system based on simple integer arithmetic in a manner that someone without a high level of experience in homomorphic encryption can readily understand. Further, we present an implementation of the system as well as a lattice- based attack. We present performance results of our implementation under various parameter choices and the resistance of the system to the lattice-based attack under those parameters. Unfortunately, while the system is very interesting from a theoretical point of view, the results show that it is still not feasible for use. Full text:
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 | Amanda Ziemann (2010)
Abstract: The ability to detect an object or activity -- such as a military vehicle, construction area, campsite, or vehicle tracks -- is highly important to both military and civilian applications. Sensors that process multi and hyperspectral images provide a medium for performing such tasks. Hyperspectral imaging is a technique for collecting and processing imagery at a large number of visible and non-visible wavelengths. Different materials exhibit different trends in their spectra, which can be used to analyze the image. For an image collected at n different wavelengths, the spectrum of each pixel can be mathematically represented as an n-element vector. The algorithm established in this work, the Simplex Volume Estimation algorithm (SVE), focuses specifically on change detection and large area search. In hyperspectral image analysis, a set of pixels constitutes a data cloud, with each pixel corresponding to a vector endpoint in Euclidean space. The SVE algorithm takes a geometrical approach to image analysis based on the linear mixture model, which describes each pixel in an image collected at n spectral bands as a linear combination of n+1 pure-material component spectra (known as endmembers). Iterative endmember identification is used to construct a 'volume function,' where the Gram matrix is used to calculate the hypervolume of the data at each iteration as the endmembers are considered in Euclidean spaces of increasing dimensionality. Linear algebraic theory substantiates that the volume function accurately characterizes the inherent dimensionality of a set of data, and supports that the volume function provides a tool for identifying the subspace in which the magnitude of the spread of the data is the greatest. A metric is extracted from the volume function, and is used to quantify the relative complexity within a single image or the change in complexity across multiple images. The SVE algorithm was applied to hyperspectral images for the tasks of change detection and large area search, and the results from these applications will demonstrate the feasibility of this method as a cueing tool for analysts. Full text:
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 | Carrie Lahnovych (2010)
Abstract: Quadratic matrix polynomials of the form Y^2 +τ ◦Y = B +τ ◦C , where Y , τ , B, and C are real, symmetric 3x3 matrices and the dot ◦ denotes the Schur product, arise in the Barboy-Tenne equations of statistical mechanics [1]. In this paper we discuss the number of solutions for Y , and devise and implement algorithms solving equations of this form. We will focus our attention on solving the equations in two specific cases and discuss the existence of a solution in the general case. Full text:
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 | Cyran Marek (2010)
Abstract: The structure of liquids is central to their thermodynamic properties and is described in a probabilistic manner. The structure is a consequence of the forces between the molecules and may be investigated with the use of many techniques. One of these techniques is the use of computer simulation, and in particular the techniques are called Monte Carlo Statistical Thermodynamic simulation, and Molecular Dynamics. In this thesis we construct a program that is capable of carrying out Event-Driven Molecular Dynamics simulation of mixtures of particles that have stepwise constant pair potential energies. We have implemented our simulation for the case of square-well particles that have a hard impenetrable core surrounded by a attractive potential well. Such mixtures are important for understanding the behavior of biological macromolecules at the high concentrations that occur in living cells. To test our implementation we have compared the resulting pair correlation functions with those that result from Monte Carlo simulations. While these pair correlation functions are in rather close agreement there remain discrepancies that remain to be resolved. Full text:
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 | Ryan Lewis (2010)
Abstract: Hyperspectral remote sensing is a valuable new technology that has numerous com- mercial and scientific applications. For example, it has been used to study crop health, mineral and soil composition, and pollution levels. Hyperspectral imaging also has im- portant military and intelligence applications such as the identification of man-made materials, and detection of chemical and biological plumes. The key mathematical challenges of hyperspectral imaging include image classification, anomaly detection, and target detection. Image classification is the process of grouping pixels into spec- trally similar clusters. This thesis describes a new topological and network-theoretic approach for classifying pixels in hyperspectral image data. Pixels in hyperspectral image data sets are thought of as constituting a point cloud in a high dimensional topological space, and a network structure is imposed on the data by considering the spectral distance between pairs of pixels. We use the tools of persistent homology to argue that the resulting network effectively models the com- plex nonlinear structures in the data. We then perform data clustering by applying a network based community detection algorithm called the method of maximum modu- larity. The method of maximum modularity is an unsupervised, deterministic method for detecting communities in networks where neither the number of communities nor their sizes needs to be specified in advance. Examples of real hyperspectral images that have been classified using the method of maximum modularity are provided in order to demonstrate the feasibility of the approach. Full text:
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 | Nicholas Battista (2010)
Abstract: Spectrally accurate initial data in numerical relativity
Einstein's theory of general relativity has radically altered the way in which we perceive the universe. His breakthrough was to realize that the fabric of space is deformable in the presence of mass, and that space and time are linked into a continuum. Much evidence has been gathered in support of general relativity over the decades. Some of the indirect evidence for GR includes the phenomenon of gravitational lensing, the anomalous perihelion of mercury, and the gravitational redshift. One of the most striking predictions of GR, that has not yet been confirmed, is the existence of gravitational waves. The primary source of gravitational waves in the universe is thought to be produced during the merger of binary black hole systems, or by binary neutron stars. The starting point for computer simulations of black hole mergers requires highly accurate initial data for the space-time metric and for the curvature. The equations describing the initial space-time around the black hole(s) are non-linear, elliptic partial differential equations (PDE). We will discuss how to use a pseudo-spectral (collocation) method to calculate the initial puncture data corresponding to single black hole and binary black hole systems. Full text:
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 | Grant Dietert (2010)
Abstract: We consider the problem of tiling large rectangles using smaller rectangles with the prescribed dimensions 4 x 6 and 5 x 7. Problem B-3 on the 1991 William Lowell Putnam Examination asked "Does there exist a natural number L such that if m and n are integers greater than L, then an m x n rectangle may be expressed as a union of 4 x 6 and 5 x 7 rectangles, any two intersect at most along their boundaries?" Narayan and Schwenk showed in 2002 that all rectangles with length and width at least 34 can be partitioned into 4 x 6 and 5 x 7 rectangles. We investigate necessary and sufficient conditions for an m x n rectangle to be tiled with 4 x 6 and 5 x 7 rectangles. Ashley et al. answered this question for all but 37 cases. We use an integer linear programming approach to eliminate all but five of these cases. Full text:
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 | Jacqieline McClive (2010)
Abstract: A coloring of a graph, G, is an assignment of positive integers to the vertices of the graph with one number assigned to each vertex, so that adjacent vertices are assigned different numbers. A k-ranking of a graph is a coloring that uses {1, 2, ..., k} with the requirement that every path between any two vertices labeled with the same number contains a vertex with a higher label. The rank number of a graph, denoted [chi]r(G), is the smallest k such that G has a k-ranking. In this paper we seek rank numbers for four specific families of graphs. Each of these families of graphs contains at least one cycle as a subgraph. Full text:
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 | Aaron Kaufer (2009)
Abstract: Not Available Full text: Not Available
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 | Gabriella Jaramillo (2009)
Abstract: not available. Full text:
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 | Karthik Bathena (2009)
Abstract: A Mathematical model of cutaneous leishmaniasis
A SIS model for Cutaneous Leishmaniasis is developed and analyzed. The model contains a human population of incidental hosts, along with animals that are the reservoir hosts and the sandfly vector. Reproductive rates for the persistence of the infection are derived from the model. Conditions for the existence of endemic and disease-free equilibrium are obtained. The stability analysis of the disease-free equilibrium is investigated and numerical simulations for the model are provided. Full text:
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 | Michael Margitus (2009)
Abstract: Researchers are interested in three types of error: experimental error, truncation error, and interpolation error. This thesis will study the last. Given a differential equation g"(x) = f(x) and a fixed number of interpolation grid points, an optimation problem is formulated to minimize the difference between f(x) and its interpolating function, thus reducing the error between the actual solution and the approximated solution of the ODE. Using the Nelder-Mead Simplex Method, the optimal distribution of grid points that will minimize the error between the solution g and its approximated solution will be found. This technique will then be applied to the one dimensional light scattering equation gxx = E²x/R. Using the Nelder-Mead Method, the optimal interpolation grid for a given number of grid points will be found. These numerical computations will ultimately be used to give guidance to experimenters on where to take measurements for the Rayleigh Ratio R. Full text:
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 | Aaron Kaufer (2009)
Abstract: Not available. Full text:
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 | Timothy Doster (2009)
Abstract: The topological anomaly detection (TAD) algorithm differs from other anomaly detection algorithms in that it does not rely on the data's being normally distributed. We have built on this advantage of TAD by extending the algorithm so that it gives a measure of the number of anomalous objects, rather than the number of anomalous pixels, in a hyperspectral image. We have done this by identifying and integrating clusters of anomalous pixels, which we accomplished with a graph-theoretical method that combines spatial and spectral information. By applying our method, the Anomaly Clustering algorithm, to hyperspectral images, we have found that our method integrates small clusters of anomalous pixels, such as those corresponding to rooftops, into single anomalies; this improves visualization and interpretation of objects. We have also performed a local linear embedding (LLE) analysis of the TAD results to illustrate its application as a means of grouping anomalies together. By performing the LLE algorithm on just the anomalies identified by the TAD algorithm, we drastically reduce the amount of computation needed for the computationally-heavy LLE algorithm. We also propose an application of a shifted QR algorithm to improve the speed of the LLE algorithm. Full text:
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 | Edgar Amaca (2008)
Abstract: The existence of an equivalence subset of rational functions with Fibonacci numbers as coefficents and the Golden Ratio as a fixed point is proven. The proof is based on two theorems establishing basic relationships underlying the Fibonacci Sequence, Pascal's Triangle and the Golden Ratio. Equaions from the two theorems are related to each other and seen to generate the equivalence subset of rational functions. Proof by induction on these equations constitutes the proof of the existence of this subset of rational functions. It is found that this subset of rational functions possesses interesting mathematical properties, particularly that of convergence to the Golden Ratio at the limit. Further investigation shows that this subset of rational functions possesses algebraic structures that would take us into the realms of abstract algebra and complex analysis. Full text:
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 | HyeYon Yi (2007)
Abstract: With the increasing threat of biological warfare, and the fear of an epidemic outbreak of deadly influenza, the field of epidemic modeling is becoming increasing important in science. This work focuses on a model for the spread of disease, and the qualitative changes in that spread that occur from changing parameters. The model is linearly stable when diffusion does not exist within the population. As diffusion is incorporated, Turing instabilities occur. Full text:
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 | Halyna Romanyuk (2007)
Abstract: It is our goal to investigate the global character of the positive solutions of a particular rational difference equation with initial conditions. We examine the global stability, periodic nature, boundedness and monotonicity of the positive solutions. Full text:
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 | Anne Marino (2007)
Abstract: The so-called "Inspection Paradox" refers to the fact that in a renewal process, the length of hte interarrival period which contains a fixed time t is stochastically larger than the length of an ordinary interarrival period. In this work, we use conditioning arguments to derive explicity formulas for the distributions of the lengths of interarrival periods in the continuous time case where the renewal process is a time-homogeneous Poisson Process, and in a discrete time where the renewal process is a Bernoulli Process. Full text:
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 | Chris Kerbert (2007)
Abstract: We investigate the periodic character and the boundedness nature of positive solutions of max-type difference equations whose coefficients are periodic sequences of positive real numbers. Full text:
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 | Ben Zindle (2007)
Abstract: We use the principle of inclusion-Exclusion and rook polynomials to study certain configurations of rooks on two and three dimensional boards. Some even-odd problems and certain Lottery and mystery cases are solved using rook polynomials. Full text:
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 | Jeremy Nieman (2007)
Abstract: Consider a cube ¡oating in space: how many unique ways can we color that cube with two colors uniquely? If the cube were øxed, the problem would simplify to two color choices for each face of the cube, with six faces. In other words, 2 x 2 x 2 x 2 x 2 x 2 = 26 = 64. A floating cube becomes a more di±cult problem due to the cube's geometric symmetries. We can develop a formula based on these symmetries to count the distinct colorings of a floating figure. With only a little more work, we can also obtain a generating function for the pattern inventory of the figure in question. Full text:
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 | Nathan Reff (2007)
Abstract: The Birkhoff-von Neumann Theorem tells us that a square matrix over the real numbers is doubly stochastic if and only if it is a convex linear combination of permutation matrices. In this work, we extended this theorem to a more general case involving hypermatrices of dimension 3. Further we investigate systems of distinct representatives (SDRs), lower bounds for hyperpermanents, applications to Latin squares and finally Birkhoff Polytopes. Full text:
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 | Rachel Marilley (2007)
Abstract: In the thesis we examine the dynamics of semiconductor lasers subject to optical injection. We study typical bifurcation scenarios, namely the saddle-node and Hopf bifurcations associated with this system. First, the system is investigated analytically, then numerical simulations verify our analytical conclusions. Full text:
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 | Deanna Olles (2006)
Abstract: In this work; three specific dynamical systems models, the Basic, Maki- Thompson, and Daley-Kendall, are used to model rumor transmission on social networks. Rumor flow is a measure of the time it takes for the rumor to completely pass through a specified network. Comparisons between random social networks and a small world social networks yield the faster transmission of a rumor over a small world network. Using unique adjacency matrices that define our random networks, observations of some characteristics of the random networks will be made that are specific to this type of graph. Differences in the constructs of the two networks will be illustrated by comparing these properties to those of the small world networks (created by a certain rewiring scheme of a k-regular network). Interesting comparisons are to be made about the networksÕ defining characteristics include average clustering coefficients, centrality measures, and average path lengths. The flow of a rumor through each type of network reveals the characteristics of the network. A rumor will clearly flow through a small world network faster than in a random network, mainly due to higher density, increased clustering, and better defined centrality. Full text:
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 | Mark Bellavia (2006)
Abstract: It is our goal to investigate the long term behavior of the solutions to difference equations whose coefficients are periodic sequences of positive real numbers. We will examine how the different periods of the sequence and the relationship of the terms of the sequence affect the long term behavior of the solutions. Full text:
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 | Amir Barghi (2006)
Abstract: By briefly introducing chromatic polynomials of finite graphs and their properties, we compute chromatic polynomials of some sequences of graphs. In addition to that, we introduce the ring of ordinomials which are utilized to define chromatic ordinomials of infinite graphs. Full text:
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 | Matt Ford (2006)
Abstract: Linear Programming has its origins in the 1940's, when complex planning problems needed to be solved to contribute to the wartime operations. The Diet Problem was one of those problems. My project deals with the formulation of my own diet problem tailored to my personal tastes where I minimize the cost of eating for one week while maintaining a healthy diet. I gathered data for different meals as well as nutritional information such as calorie count and number of servings. The formulation was eventually modified to introduce variety into the diet. Full text:
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 | Carol Panepinto (2005)
Abstract: A photoconductorÕs mobility is a measure of the speed at which electrons migrate through the material under the influence of an electric field. The mobility determines how long a packet of charge takes to go through the photoconductor. It also determines how much and in what manner the E-field changes during the packet transit. The problem in which we are interested is inferring the mobility of a photoconductor from time-of-flight measurements, that is, from measurements of the current produced per unit time by a known charge packet. Mathematically, the problem is an initial-boundary value problem for a nonlinear, non-local, hyperbolic conservation law that characterizes the E-field in the photoconductor. In this paper, we discuss the mathematical formulation of this problem, its solution using the method of characteristics, and the application of the solution to the problem of inferring mobilities. Full text:
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 | David Fraser (2005)
Abstract: The Necklace Illustrator Java Applet, the necklace problem is a well known problem from combinatorics and a variety of problems can be formulated as necklace counting problems. A Java Applet is presented which finds the number of unique necklaces. It does so by computing the cycle index and pattern inventory for the given input. It also draws each of these necklaces by comparing their string representations considering rotations and reflections. Full text:
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 | Gillian Gallle (2005)
Abstract: A ranking is when the vertices of a graph, G, are labeled with numbers such that every path between any two vertices labeled with the same number contains a vertex labeled with a higher number. A ranking is called minimal if decreasing any label greater than 1 violates the above ranking property. The arank number of a graph, denoted Pk(G), is defined to be the largest k such that G has a minimal k-ranking. While much work has been done concerning the arank of stars and paths, this paper explores arank trends over all the nonisomorphic graphs on 6 vertices or less paying special attention to furthering the work of Ghoshal, Laskar and Pillone. These results are followed by an investigation of the potential correlation of reduction classes to the arank number. Finally, the arank number of a specific family of trees is examined. Full text:
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 | Olin Stratton (2005)
Abstract: Not Available Full text: Not Available
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 | Chris Cappon (2005)
Abstract: Not Available Full text: Not Available
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 | Kevin Gonzales (2005)
Abstract: Not Available Full text: Not Available
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 | Joseph Rhoads (2005)
Abstract: Not Available Full text: Not Available
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