Applied Mathematics Commons^™

Limiting Subgradients Of Minimal Time Functions In Banach Spaces, Boris S. Mordukhovich, Nguyen Mau Nam

Mathematics Research Reports

The paper mostly concerns the study of generalized differential properties of the so-called minimal time functions associated, in particular, with constant dynamics and arbitrary closed target sets in control theory. Functions of this type play a significant role in many aspects of optimization, control theory: and Hamilton-Jacobi partial differential equations. We pay the main attention to computing and estimating limiting subgradients of the minimal value functions and to deriving the corresponding relations for Frechet type epsilon-subgradients in arbitrary Banach spaces.

Linearizable Feedforward Systems: A Special Class, Issa Amadou Tall

Miscellaneous (presentations, translations, interviews, etc)

We address the problem of linearizability of systems in feedforward form. In a recent paper [22] we completely solved the linearizability for strict feedforward systems. We extend here those results to a special class of feedforward systems. We provide an algorithm, along with explicit transformations, that linearizes the system by change of coordinates when some easily checkable conditions are met. We also re-analyze type II class of linearizable strict feedforward systems provided by Krstic in [9] and we show that this class is the unique linearizable among the class of quasi-linear strict feedforward systems (see Definition III.1). Our results allow …

The Roc Curves Of Fused Independent Classification Systems, Michael B. Walsh

Theses and Dissertations

The need for optimal target detection arises in many different fields. Due to the complexity of many targets, it is thought that the combination of multiple classification systems, which can be tuned to several individual target attributes or features, might lead to more optimal target detection performance. The ROC curves of fused independent two-label classification systems may be generated by the mathematical combination of their ROC curves to achieve optimal classifier performance without the need to test every Boolean combination. The monotonic combination of two-label independent classification systems which assign labels to the same target types results in a lattice …

Fast Fourier Transform Algorithms With Applications, Todd Mateer

All Dissertations

This manuscript describes a number of algorithms that can be used to quickly evaluate a polynomial over a collection of points and interpolate these evaluations back into a polynomial. Engineers define the 'Fast Fourier Transform' as a method of solving the interpolation problem where the coefficient ring used to construct the polynomials has a special multiplicative structure. Mathematicians define the 'Fast Fourier Transform' as a method of solving the evaluation problem. One purpose of the document is to provide a mathematical treatment of the topic of the 'Fast Fourier Transform' that can also be understood by someone who has an …

Numerical Analysis Of A Fractional Step Theta-Method For Fluid Flow Problems, John Chrispell

All Dissertations

The accurate numerical approximation of viscoelastic fluid flow poses two difficulties: the large number of unknowns in the approximating algebraic system (corresponding to velocity, pressure, and stress), and the different mathematical types of the modeling equations. Specifically, the viscoelastic modeling equations have a hyperbolic constitutive equation coupled to a parabolic conservation of momentum equation. An appealing approximation approach is to use a fractional step $\theta$-method. The $\theta$-method is an operator splitting technique that may be used to decouple mathematical equations of different types as well as separate the updates of distinct modeling equation variables when modeling mixed systems of partial …

Homomorphisms Of Graphs, Samuel Lyle

All Dissertations

Understanding the structure of graphs is fundamental to advances in many areas of graph theory, as well as in many applications. In many cases, an analysis of the structure of graphs follows one of two approaches; either many structural properties are considered over a restricted class of graphs, or a particular structural property is considered over many classes of graphs. Both approaches will be considered in this dissertation.
Graphs which do not contain a clique of size r, i.e., K_r-free graphs, are of fundamental importance in the area of extremal graph theory. Many results have been obtained …

Portfolio Selection Under Various Risk Measures, Hariharan Kandasamy

All Dissertations

Portfolio selection has been a major area of study after Markowitz's ground-breaking paper. Risk quantification for portfolio selection is studied in the literature extensively and many risk measures have been proposed.
In this dissertation we study portfolio selection under various risk measures. After exploring important risk measures currently available we propose a new risk measure, Unequal Prioritized Downside Risk (UPDR). We illustrate the formulation of UPDR for portfolio selection as a mixed-integer program. We establish conditions under which UPDR can be formulated as a linear program.
We study single-period portfolio selection using two risk measures simultaneously. We propose four alternate …

On Graph Isomorphism And The Pagerank Algorithm, Christopher J. Augeri

Theses and Dissertations

A graph is a key construct for expressing relationships among objects, such as the radio connectivity between nodes contained in an unmanned vehicle swarm. The study of such networks may include ranking nodes based on importance, for example, by applying the PageRank algorithm used in some search engines to order their query responses. The PageRank values correspond to a unique eigenvector typically computed by applying the power method, an iterative technique based on matrix multiplication. The first new result described herein is a lower bound on the execution time of the PageRank algorithm that is derived by applying standard assumptions …

Understanding Similarity: Bridging Geometric And Numeric Contexts For Proportional Reasoning, Dana Christine Cox

Dissertations

The concept of similarity is uniquely situated at the crossroads of geometric and numerical proportional reasoning. Although studies have documented the existence and nature of student difficulties with this topic, there exists a gap between documented visual insights of younger children and the quantitative inadequacies of older ones. Using a revised version of the Similarity Perception Test followed by 21 clinical interviews, this study investigated the visual and analytical strategies that are used by middle-school students to differentiate and construct similar figures.

New strategies for construction and differentiation were identified, and three overarching conclusions were drawn from the work. First, …

Optimization In The Undergraduate Curriculum, Allen Holder

Mathematical Sciences Technical Reports (MSTR)

A discussion of how an optimization course fits into the undergraduate mathematics curriculum

A Universal Theory Of Decoding And Pseudocodewords, Nathan Axvig, Deanna Dreher, Katherine Morrison, Eric T. Psota, Lance C. Pérez, Judy L. Walker

Department of Mathematics: Faculty Publications

The discovery of turbo codes [5] and the subsequent rediscovery of low-density parity-check (LDPC) codes [9, 18] represent a major milestone in the field of coding theory. These two classes of codes can achieve realistic bit error rates, between 10−5 and 10−12, with signalto- noise ratios that are only slightly above the minimum possible for a given channel and code rate established by Shannon’s original capacity theorems. In this sense, these codes are said to be near-capacity-achieving codes and are sometimes considered to have solved (in the engineering sense, at least) the coding problem for the additive white Gaussian noise …

Necessary Conditions For Nonsmooth Optimization Problems With Operator Constraints In Metric Spaces, Boris S. Mordukhovich, Libin Mou

Mathematics Research Reports

This paper concerns nonsmooth optimization problems involving operator constraints given by mappings on complete metric spaces with values in nonconvcx subsets of Banach spaces. We derive general first-order necessary optimality conditions for such problems expressed via certain constructions of generalized derivatives for mappings on metric spaces and axiomatically defined subdifferentials for the distance function to nonconvex sets in Banach spaces. Our proofs arc based on variational principles and perturbation/approximation techniques of modern variational analysis. The general necessary conditions obtained are specified in the case of optimization problems with operator constraints dDScribcd by mappings taking values in approximately convex subsets of …

Existence Of Multiple-Stable Equilibria For A Multi-Drug-Resistant Model Of Mycobacterium Tuberculosis, Abba B. Gumel, Baojun Song

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

The resurgence of multi-drug-resistant tuberculosis in some parts of Europe and North America calls for a mathematical study to assess the impact of the emergence and spread of such strain on the global effort to effectively control the burden of tuberculosis. This paper presents a deterministic compartmental model for the transmission dynamics of two strains of tuberculosis, a drug-sensitive (wild) one and a multi-drug-resistant strain. The model allows for the assessment of the treatment of people infected with the wild strain. The qualitative analysis of the model reveals the following. The model has a disease-free equilibrium, which is locally asymptotically …

Positive Solutions Of A Third-Order Three-Point Boundary-Value Problem, Bo Yang

Faculty Articles

We obtain upper and lower estimates for positive solutions of a third-order three-point boundary-value problem. Sufficient conditions for the existence and nonexistence of positive solutions for the problem are also obtained. Then to illustrate our results, we include an example.

Nonbinary Quantum Error-Correcting Codes From Algebraic Curves, Jon-Lark Kim, Judy L. Walker

Department of Mathematics: Faculty Publications

We give a generalized CSS construction for nonbinary quantum error-correcting codes. Using this we construct nonbinary quantum stabilizer codes from algebraic curves. We also give asymptotically good nonbinary quantum codes from a Garcia- Stichtenoth tower of function fields which are constructible in polynomial time.

Binary quantum error-correcting codes have been constructed in several ways. One interesting construction uses algebraic-geometry codes [2], [6], [7], [12], with the main idea being to apply the binary CSS construction [4], [5], [16] to the asymptotically good algebraic-geometry codes arising from the Garcia-Stichtenoth [11] tower of function fields over F_q2 (where q is a …

The Limnogeology Of The Lakes Sagara And Nyamagoma, Hudson Nkotagu

Tanzania Journal of Engineering and Technology (TJET)

The present study focuses on the limnogeological aspects of the lakes Sagara and Nyamagoma within the Malagarasi Wetland Ecosystem. Abiotic parameters from water and sediment samples were determined during the dry and wet seasons using appropriate gears and methods. The data show variation of the abiotic parameters in both lakes from the water surface to the bottom. Lake Nyamagoma is less turbid (0.7-24 NTU) than Lake Sagara (25 to 65 NTU). The water transparency was also high at Lake Nyamagoma (0.75-1.5 m) thus supporting the observed turbidity trend. The physical chemical changes may be attributed to processes such as cooling, …

Modeling Of A Spray Assisted Natural Draft Cooling Tower, Kaunde K

Tanzania Journal of Engineering and Technology (TJET)

Cooling towers are one of the largest heat and mass transfer devices that are in common use. A novel type of cooling tower has been proposed in which air flow rate into the tower is drawn by ejector action of sprays instead of fans as is done in conventional mechanical forced or induced draft cooling towers. This novel design offers the potential of saving the energy cost for driving the fan. The paper presents mathematical models for momentum transfer which is the driving force causing the entrainment of air. Also the heat transfer model for co-current flow of liquid and …

Assessment Of Integrated Water Resources Management Practice In Yatta Canal, Machakos District, Kenya, Lawrence Simitu

Tanzania Journal of Engineering and Technology (TJET)

Yatta Canal is situated in Yatta division, Machakos district about 100 km from Nairobi along Thika- Garissa road. It is about 60 Km long and supplies water for domestic, livestock and irrigation use. Over the years, the water supply has been unreliable especially during dry season leading to conflicts between upstream and downstream consumers. Sometimes the Government has been forced to intervene to restore order. The study aimed at assessing the use of Yatta canal, based on the principles of Integrated Water Resources Management (IWRM) which stipulates efficient use of water and involvement of all stakeholders in decision making for …

Sediment Yield Prediction In Tanzania: Case Study Of Dodoma District Catchments, Mathew Mulemgera

Tanzania Journal of Engineering and Technology (TJET)

Soil erosion and sedimentation is serious and is resulting in rapid sedimentation of streams and reservoirs in Tanzania. This makes sediment yield modeling important before undertaking water resources development projects, although it has so far not been done in the country. At present sediment yield prediction can be done using physically based models, empirical soil erosion and sediment yield models, and sediment rating curves. The physically based models contain numerous parameters that are difficult to measure or estimate, making them inappropriate in developing countries like Tanzania. The rating curves give rough estimates while the empirical sediment yield models need runoff …

Development Of A Genetic Algorithm Based Search Strategy Suited For Design Optimisation Of Internal Combustion Engines, Nalitolela G

Tanzania Journal of Engineering and Technology (TJET)

Engine design optimisation is a multi-objective, multi-domain problem in a discontinuous design space. The state of the art of optimisation techniques shows that only methods of direct and adaptive search are appropriate for this type of problem. These include, adaptive random search, simulated annealing, evolution strategies and genetic algorithms. Of these methods, the genetic algorithms have been shown to be the most suited for the optimisation of multi-modal response functions in a discontinuous design space. This paper considers the important characteristics of genetic algorithms and their adaptation for use in parametric design optimisation of internal combustion engines. In order to …

Accuracy Analysis For Kolmogorov Entropy Used In Studying The Chaotic Dynamics Of Cfb Reactors Based On Solids Concentration Fluctuations, Samwel V. Manyele

Tanzania Journal of Engineering and Technology (TJET)

A study on the chaotic dynamics of a high flux circulating fluidized bed (CFB) riser (10-m high and 76-mm i.d.) using the maximum likelihood estimate of the Kolmogorov entropy (KML) is reported. The signals used were measured by using a solids concentration fiber optic probe, accuracy of which is reported based on statistical analysis. The sensitivity analysis of the parameters used for computing KML was conducted to identify optimum settings, based on the standard error, s(KML) and the nature of PDF of the b-values from the reconstructed phase space trajectories. The length of time series, N, sampling frequency, fs, and …

Sedimentation Status Of Lake Victoria: Loading From River Mouth And In-Lake Sedimentation, F Mwanuzi1

Tanzania Journal of Engineering and Technology (TJET)

Lake sediments consist of the input of suspended solids by river discharges, the autochthonous produced particulate matter in the water column and the settled material on the lake bottom. The loading, transport, formation and decay of the sediments entering the Lake Victoria from Tanzanian side was monitored for the period 2000 to 2005. The sedimentation study focused on; quantification of settling fluxes of particulate nutrients and organic carbon at the off shore(Pelagic) and near-shore(Littoral) monitoring stations. Monitored data on sedimentation rate and settling velocities has been sampled and analysed in the period November 2000 to March 2005. The mean sedimentation …

Trends Of Rainfall And Maize Productivity In Malawi, T Kimaro

Tanzania Journal of Engineering and Technology (TJET)

Daily rainfall data between 1966 and 2005 across all Agricultural Development Areas (ADDs) in Malawi was investigated to determine changes in selected rainfall characteristics and their impact on Maize productivity. Rainfall indices analyzed included rainfall amounts (annual and seasonal), onset, cessation, dry spells and length of rainy season. Man Kendal’s trend test was applied to the time series of these indices to detect any trends in the time series. Water Requirement Satisfaction Index (WRSI) for maize crop was used to study the spatial-temporal variation of maize productivity and thus its linkage to rainfall. Definition of onset and cessation adopted from …

Material Perturbations To Enhance Performance Of The Theile Half-Width Leaky Mode Antenna, Jason A. Girard

Theses and Dissertations

Microstrip traveling-wave antennas, often referred to as leaky-wave antennas, have been shown to radiate when the dominant or fundamental mode is suppressed and the first higher-order mode is excited. One such microstrip variation is the Thiele Half-Width (THW) antenna, which operates from 5.9 - 8.2 GHz for this research. Increasing the bandwidth over which the THW antenna radiates is desired, as is a fundamental understanding of the propagation characteristics over this region. This dissertation seeks to vary or perturb the material and physical properties of the THW antenna, including strip-width variations and modifications of the substrate layer, to achieve these …

Radiotherapy Optimal Design: An Academic Radiotherapy Treatment Design System, R Acosta, W Brick, A Hanna, Allen Holder, D Lara, G Mcquillen, D Nevin, P Uhlig, B Salter

Mathematical Sciences Technical Reports (MSTR)

Optimally designing radiotherapy and radiosurgery treatments to increase the likelihood of a successful recovery from cancer is an important application of operations research. Researchers have been hindered by the lack of academic software that supports head-to-head comparisons of different techniques, and this article addresses the inherent difficulties of designing and implementing an academic treatment planning system. In particular, this article details the algorithms and the software design of Radiotherapy optimAl Design (RAD).

Model-Based Clustering Of Methylation Array Data: A Recursive-Partitioning Algorithm For High-Dimensional Data Arising As A Mixture Of Beta Distributions, E. Andres Houseman, Brock C. Christensen, Ru-Fang Yeh, Carmen J. Marsit, Margaret R. Karagas, Margaret Wrensch, Heather H. Nelson, Joseph Wiemels, Shichun Zheng, John K. Wiencke, Karl T. Kelsey

Harvard University Biostatistics Working Paper Series

No abstract provided.

Uniform Uncertainty Principle And Signal Recovery Via Regularized Orthogonal Matching Pursuit, Deanna Needell, Roman Vershynin

CMC Faculty Publications and Research

This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements—L1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of L1-minimization. Our algorithm, ROMP, reconstructs a sparse signal in a number of iterations linear in the sparsity, and the reconstruction is exact provided the linear measurements satisfy the uniform uncertainty principle.

Time-Frequency Analysis Of Terahertz Radar Signals For Rapit Heart And Breath Rate Detection, Melody L. Massar

Theses and Dissertations

We develop new time-frequency analytic techniques which facilitate the rapid detection of a person's heart and breath rates from the Doppler shift the movement of their body induces in a terahertz radar signal. In particular, the Doppler shift in the continuous radar return is proportional to the velocity of the person's body. Thus, a time-frequency analysis of the radar return will yield a velocity signal. This signal, in turn, may undergo a second time-frequency analysis to yield any periodic components of the velocity signal, which are often related to the heart and breath rates of the individual. One straightforward means …

On Linearizability Of Strict Feedforward Systems, Issa Amadou Tall, Witold Respondek

Miscellaneous (presentations, translations, interviews, etc)

In this paper we address the problem of linearizability of systems in strict feedforward form. We provide an algorithm, along with explicit transformations, that linearizes a system by change of coordinates when some easily checkable conditions are met. Those conditions turn out to be necessary and sufficient, that is, if one fails the system is not linearizable. We revisit type I and type II classes of linearizable strict feedforward systems provided by Krstic in [6] and illustrate our algorithm by various examples mostly taken from [5], [6].

Isoparametric And Dupin Hypersurfaces, Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

A hypersurface Mⁿ⁻¹ in a real space-form Rⁿ, Sⁿ or Hⁿ is isoparametric if it has constant principal curvatures. For Rⁿ and Hⁿ, the classification of isoparametric hypersurfaces is complete and relatively simple, but as ´Elie Cartan showed in a series of four papers in 1938–1940, the subject is much deeper and more complex for hypersurfaces in the sphere Sⁿ. A hypersurface Mⁿ⁻¹ in a real space-form is proper Dupin if the number g of distinct principal curvatures is constant on Mⁿ⁻¹, and each principal curvature function …