Applied Mathematics Commons^™

A New Partitioning Around Medoids Algorithm, Mark J. Van Der Laan, Katherine S. Pollard, Jennifer Bryan

U.C. Berkeley Division of Biostatistics Working Paper Series

Kaufman & Rousseeuw (1990) proposed a clustering algorithm Partitioning Around Medoids (PAM) which maps a distance matrix into a specified number of clusters. A particularly nice property is that PAM allows clustering with respect to any specified distance metric. In addition, the medoids are robust representations of the cluster centers, which is particularly important in the common context that many elements do not belong well to any cluster. Based on our experience in clustering gene expression data, we have noticed that PAM does have problems recognizing relatively small clusters in situations where good partitions around medoids clearly exist. In this …

An Analysis Of The Development And Application Of Orthogonal Polynomials With An Emphasis On The Legendre Polynomials, Francis Radnoti

Capstone Research Projects

No abstract provided.

Numerical Solution Of Markov Chains, Amr Lotfy Elsayad

Theses Digitization Project

This project deals with techniques to solve Markov Chains numerically.

On The Regularity Of Solutions To Fully Nonlinear Elliptic Equations Via The Liouville Property, Qingbo Huang

Mathematics and Statistics Faculty Publications

We show that any C^1,1 solution to the uniformly elliptic equation F(D²u) = 0 must belong to C^2,α, if the equation has the Liouville property.

Absolutely Continuous Jacobi Operators, Steen Pedersen

Mathematics and Statistics Faculty Publications

No abstract provided.

Domain Functionals And Exit Times For Brownian Motion, Chaocheng Huang, David Miller

Mathematics and Statistics Faculty Publications

Two domain functionals describing the averaged expectation of exit times and averaged variance of exit times of Brownian motion from a domain, respectively, are studied.

Explicit Multiplicative Relations Between Gauss Sums, Brian J. Murray

LSU Doctoral Dissertations

H.Hasse conjectured that all multiplicative relations between Gauss sums essentially follow from the Davenport-Hasse product formula and the norm relation for Gauss sums. While this is known to be false, very few counterexamples, now known as sign ambiguities, have been given. Here, we provide an explicit product formula giving an infinite class of new sign ambiguities and resolve the ambiguous sign in terms of the order of the ideal class of quadratic primes.

Group Automorphisms And The Decomposition Of Plancherel Measures, David Slay

LSU Doctoral Dissertations

In this paper, a natural action of the automorphisms of a group on the space of irreducible unitary representations is used to decompose the Plancherel measure on the dual space as an integral of measures on homogeneous spaces. Explicit decompositions are obtained for the cases of free 2 and 3-step nilpotent Lie groups. These results are obtained using direct integral decompositions, induced representations, the Mackey Machine, and measure theory on homogeneous spaces.

Dimensionality-Reducing Expansion, Boundary Type Quadrature Formulas, And The Boundary Element Method, Tian-Xiao He

Scholarship

This paper discusses the connection between boundary quadrature formulas constructed by using solutions of partial differential equations and boundary element schemes.

Flow Patterns In A Two-Roll Mill, Christopher Hills

Articles

The two-dimensional flow of a Newtonian fluid in a rectangular box that contains two disjoint, independently-rotating, circular boundaries is studied. The flow field for this two-roll mill is determined numerically using a finite-difference scheme over a Cartesian grid with variable horizontal and vertical spacing to accommodate satisfactorily the circular boundaries. To make the streamfunction numerically determinate we insist that the pressure field is everywhere single-valued. The physical character, streamline topology and transitions of the flow are discussed for a range of geometries, rotation rates and Reynolds numbers in the underlying seven-parameter space. An account of a preliminary experimental study of …

Semicontinuity Of Dimension And Measure For Locally Scaling Fractals, L. B. Jonker, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

The basic question of this paper is: If you consider two iterated function systems close to one another in an appropriate topology, are the dimensions of their respective invariant sets close to one another? It is well-known that the Hausdorff dimension (and Lebesgue measure) of the invariant set do not depend continuously on the iterated function system. Our main result is that (with a restriction on the ‘non-conformality’ of the transformations) the Hausdorff dimension is a lower semi-continuous function in the C¹- topology of the transformations of the iterated function system. The same question is raised of the …

On A Conjecture Of Furstenberg, Grzegorz Świçatek, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

The Hausdorff dimension of the set of numbers which can be written using digits 0, 1,t in base 3 is estimated. For everyt irrational a lower bound 0.767 … is found.

Single-Particle Model For A Granular Ratchet, Albert J. Bae, Welles Antonio Martinez Morgado, J. J. P. Veerman, Giovani L. Vasconcelos

Mathematics and Statistics Faculty Publications and Presentations

A simple model for a granular ratchet corresponding to a single grain bouncing off a vertically vibrating sawtooth-shaped base is studied. Depending on the model parameters, horizontal transport is observed in both the preferred and unfavoured directions. A phase diagram is presented indicating the regions in parameter space where the different regimes (no current, normal current, and current reversal) occur.

Parameters Affecting Partitioning Of 6 Pcb Congeners In Natural Sediments, Eid A. Alkhatib, Carl Weigand

Chemistry & Physics Faculty Publications

Polychlorinated biphenyls (PCBs) released by bottom sediments were determined by experiments in which the sediments were artificially resuspended using sediment contaminated with PCBs in a particle entrainment simulator (PES). Sediment cores, spiked with PCBs, were collected from the Housatonic River in Connecticut and run in the PES at simulated shear stresses from 0 to 0.6 N m(-2). Experimental results from these simulations have shown that mean concentration of PCBs in the solid phase for sites with high volatile organic carbon (VOC) were significantly greater than samples with low VOC; the reverse was true for the water phase. In addition, on …

Exotic Integral Witt Equivalence Of Algebraic Number Fields, Changheon Kang

LSU Doctoral Dissertations

Two algebraic number fields K and L are said to be exotically integrally Witt equivalent if there is a ring isomorphism W(O_K) ~ W(O_L) between the Witt rings of the number rings O_K and O_L of K and L, respectively. This dissertation studies exotic integral Witt equivalence for totally complex number fields and gives necessary and sufficient conditions for exotic integral equivalence in two special classes of totally complex number fields.

Dynamics Of A Granular Particle On A Rough Surface With A Staircase Profile, J. J. P. Veerman, F. V. Cunha Jr., G. L. Vasconcelos

Mathematics and Statistics Faculty Publications and Presentations

A simple model is presented for the motion of a grain down a rough inclined surface with a staircase profile. The model is an extension of an earlier model of ours where we now allow for bouncing, i.e., we consider a non-vanishing normal coefficient of restitution. It is shown that in parameter space there are three regions of interest: (i) a region of smaller inclinations where the orbits are always bounded (and we argue that the particle always stops); (ii) a transition region where halting, periodic and unbounded orbits co-exist; and (iii) a region of large inclinations where no halting …

A Recommendation For Determining The Efficacy Of Weight Removal Estimates For The Pacific Cod Longline Cdq Fishery, Anna L. Furniss

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

In January 2000, the Alaska Department of Community and Economic Development contacted the National Marine Fisheries Service (NMFS) regarding concerns over the methods used to determine catch estimates for the Pacific Cod Community Development Quota (CDQ) fishery. Currently, NMFS determines catch estimates for the Pacific Cod CDQ fishery based on the data collected by observers from the North Pacific Groundfish Observer Program (NPGOP).

Observer estimates for catch are based on the random sampling methods for a longline fishing vessel as described in the North Pacific Groundfish Observer Manual. These sampling methods provide an official total catch (OTC) estimate for each …

Proceedings Of The First International Conference On Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability And Statistics, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In 1960s Abraham Robinson has developed the non-standard analysis, a formalization of analysis and a branch of mathematical logic, that rigorously defines the infinitesimals. Informally, an infinitesimal is an infinitely small number. Formally, x is said to be infinitesimal if and only if for all positive integers n one has xxx < 1/n. Let &>0 be a such infinitesimal number. The hyper-real number set is an extension of the real number set, which includes classes of infinite numbers and classes of infinitesimal numbers. Let’s consider the non-standard finite numbers 1+ = 1+&, where “1” is its standard part and “&” its non-standard part, …

Healing Times For Circular Wounds On Plane And Spherical Bone Surfaces, J. A. Adam

Mathematics & Statistics Faculty Publications

A mathematical model is developed for the rate of healing of a circular wound in a spherical "skull". The motivation for this model is based on experimental studies of the "'critical size defect" (CSD) in animal models; this has been defined as the smallest intraosseous wound that does not heal by bone formation during the lifetime of the animal [1]. For practical purposes, this timescale can usually be taken as one year. In [2], the definition was further extended to a defect which has less than ton percent bony regeneration during the lifetime of the animal. CSDS can "heal" by …

The Effect Of Surface Curvature On Wound Healing In Bone, J. A. Adam

Mathematics & Statistics Faculty Publications

The time-independent nonhomogeneous diffusion equation is solved for the equilibrium distribution of wound-induced growth factor over a hemispherical surface. The growth factor is produced at the inner edge of a circular wound and stimulates healing in regions where the concentration exceeds a certain threshold value. An implicit analytic criterion is derived for complete healing of the wound. (C) 2001 Elsevier Science Ltd. All rights reserved.

Advances In Space Radiation Shielding Codes, John W. Wilson, Ram K. Tripathi, Garry D. Qualls, Francis A. Cucinotta, Richard E. Prael, John W. Norbury, John H. Heinbockel, John Tweed, Giovanni De Angelis

Mathematics & Statistics Faculty Publications

Early space radiation shield code development relied on Monte Carlo methods and made important contributions to the space program. Monte Carlo methods have resorted to restricted one-dimensional problems leading to imperfect representation of appropriate boundary conditions. Even so, intensive computational requirements resulted and shield evaluation was made near the end of the design process. Resolving shielding issues usually had a negative impact on the design. Improved spacecraft shield design requires early entry of radiation constraints into the design process to maximize performance and minimize costs. As a result, we have been investigating high-speed computational procedures to allow shield analysis from …

On Properties Of Linear Control Systems On Lie Groups, Fabiana Cardetti

LSU Doctoral Dissertations

In this work we study controllability properties of linear control systems on Lie groups as introduced by Ayala and Tirao in [AT99]. A linear control system _x0006_Σ Lie group G is defined by x' = X(x) + Σ^k_j=1 u_jY_j(x), where the drift vector field X is an infinitesimal automorphism, u_j are piecewise constant functions, and the control vectors Y_j are left-invariant vector fields. Properties for the flow of the infinitesimal automorphism X and for the reachable set defined by _x0006_Σ are presented in Chapter 3. Under a condition similar to the Kalman …

Racks, Quandles And Virtual Knots, Victor Samuel Nelson

LSU Doctoral Dissertations

We begin with a brief survey of the theory of virtual knots, which was announced in 1996 by Kauffman. This leads naturally to the subject of quandles and quandle homology, which we also briefly introduce. Chapter 2 contains a proof in terms of Gauss diagrams that the forbidden moves unknot virtual knots. This chapter includes material which has appeared in the Journal of Knot Theory and its Ramifications and is reprinted here by permission of World Scientific Publishing. In chapter 3 (cowritten with my advisor R.A.Litherland) we confirm a conjecture of J.S.Carter et.al. that the long exact sequence in rack …

A Monotone Follower Control Problem With A Nonconvex Functional And Some Related Problems, Jamiiru Luttamaguzi

LSU Doctoral Dissertations

A generalized one-dimensional monotone follower control problem with a nonconvex functional is considered. The controls are assumed to be nonnegative progressively measurable processes. The verification theorem for this problem is presented. A specific monotone follower control problem with a nonconvex functional is then considered in which the diffusion term is constant. The optimal control for this problem, which is explicitly given, can be viewed as tracking a standard Wiener process by a non anticipating process starting at 0. For some parameters values, the value function for this monotone follower control problem is shown to be C² and for other …

Observations And Modelling Of The Near Equatorial Tropical Atmospheric Boundary Layer, Aiman Abdalla Soleiman

Student Works (2000-2009)

Existing models developed to simulate the evolution of the atmospheric boundary layer (ABL) have focused on the conditions prevalent in mid-latitude regions. The problem for these models in terms of their application to tropical conditions is that the mid-latitude ABL is, in many respects, physically quite different from that of the tropical ABL. Because of this, and due also to the relative lack of relevant atmospheric observations in tropical areas, the tropical ABL is incorrectly represented in such models, and also in various applications that use the mid-latitude ABL representation (for example, air pollution dispersion models). A recent review of …

On Minimal Regular Graphs With Given Constraintsm, Che Sheng Gan

Student Works (2000-2009)

A classical question in graph theory has been the search of cages. The problem of determining the smallest (r,g)-cage is unsolved for most pairs of (r,g) and is extremely hard in the general case. Finding the crossing number of a graph is another challenging problem in graph theory. The crossing numbers of very few classes of graphs arc known. In this thesis, we would like to (a) investigate those smallest regular graphs with given girths and having small crossing numbers. In particular, we investigate those smallest regular graphs with given girths and crossing number c, for c ∈ {0, 1, …

On The Stabilization And Regularization Of Rational Approximation Schemes For Semigroups, Simone Flory

LSU Doctoral Dissertations

In this work we discuss consistency, stability and convergence of rational approximation methods for strongly continuous semigroups on Banach spaces. The Lax-Chernoff theorem shows that in this setting consistency and stability assumptions are necessary to obtain strong uniform convergence of approximation methods. We investigate rational approximation methods for strongly continuous semigroups and their consistency properties, with special emphasis on A-stable methods and Padé-type approximations. In particular, we discuss the stability and convergence properties of these schemes, including the stability of the well-known and widely used Backward-Euler and Crank-Nicolson Schemes. Furthermore, we modify stabilization techniques developed by Hansbo, Larsson, Luskin, Rannacher, …

Discrete Mathematics Topics In The Secondary School Curriculum, Aimee Beth Boyd

LSU Master's Theses

This thesis discusses two topics of discrete mathematics in a manner suitable for presentation at a secondary-school level. The introduction outlines the benefits of including discrete mathematics in the secondary-school curriculum. The two chapters which follow include detailed treatment of the two selected topics: the Traveling Salesman Problem and RSA encryption. The discussion of the Traveling Salesman Problem consists of introduction to the problem through several real-life scenarios, followed by a discussion of various methods for solving the problem. We discuss exact and approximate algorithms together with their computational complexity and practical limitations. The discussion of RSA encryption begins with …