VTechWorks Repository :: Browsing by Author "Rossi, John F."

Browsing by Author "Rossi, John F."

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Accuracy of Computer Generated Approximations to Julia Sets
Hoggard, John W. (Virginia Tech, 2000-07-31)
A Julia set for a complex function 𝑓 is the set of all points in the complex plane where the iterates of 𝑓 do not form a normal family. A picture of the Julia set for a function can be generated with a computer by coloring pixels (which we consider to be small squares) based on the behavior of the point at the center of each pixel. We consider the accuracy of computer generated pictures of Julia sets. Such a picture is said to be accurate if each colored pixel actually contains some point in the Julia set. We extend previous work to show that the pictures generated by an algorithm for the family λe² are accurate, for appropriate choices of parameters in the algorithm. We observe that the Julia set for meromorphic functions with polynomial Schwarzian derivative is the closure of those points which go to infinity under iteration, and use this as a basis for an algorithm to generate pictures for such functions. A pixel in our algorithm will be colored if the center point becomes larger than some specified bound upon iteration. We show that using our algorithm, the pictures of Julia sets generated for the family λtan(z) for positive real λ are also accurate. We conclude with a cautionary example of a Julia set whose picture will be inaccurate for some apparently reasonable choices of parameters, demonstrating that some care must be exercised in using such algorithms. In general, more information about the nature of the function may be needed.
Algebras of Toeplitz Operators
Ordonez-Delgado, Bartleby (Virginia Tech, 2006-05-03)
In this work we examine C*-algebras of Toeplitz operators over the unit ball in ℂⁿ and the unit polydisc in ℂ². Toeplitz operators are interesting examples of non-normal operators that generate non-commutative C*-algebras. Moreover, in the nice cases (depending on the geometry of the domain) of algebras of Toeplitz operators we can recover some analogues of the spectral theorem up to compact operators. In this setting, we can capture the index of a Fredholm operator which is a fundamental numerical invariant in Operator Theory.
Analysis of the BiCG Method
Renardy, Marissa (Virginia Tech, 2013-05-31)
The Biconjugate Gradient (BiCG) method is an iterative Krylov subspace method that utilizes a 3-term recurrence. BiCG is the basis of several very popular methods, such as BiCGStab. The short recurrence makes BiCG preferable to other Krylov methods because of decreased memory usage and CPU time. However, BiCG does not satisfy any optimality conditions and it has been shown that for up to n/2-1 iterations, a special choice of the left starting vector can cause BiCG to follow {em any} 3-term recurrence. Despite this apparent sensitivity, BiCG often converges well in practice. This paper seeks to explain why BiCG converges so well, and what conditions can cause BiCG to behave poorly. We use tools such as the singular value decomposition and eigenvalue decomposition to establish bounds on the residuals of BiCG and make links between BiCG and optimal Krylov methods.
Analytic Combinatorics Applied to RNA Structures
Burris, Christina Suzann (Virginia Tech, 2018-07-09)
In recent years it has been shown that the folding pattern of an RNA molecule plays an important role in its function, likened to a lock and key system. γ-structures are a subset of RNA pseudoknot structures filtered by topological genus that lend themselves nicely to combinatorial analysis. Namely, the coefficients of their generating function can be approximated for large n. This paper is an investigation into the length-spectrum of the longest block in random γ-structures. We prove that the expected length of the longest block is on the order n - O(n^1/2). We further compare these results with a similar analysis of the length-spectrum of rainbows in RNA secondary structures, found in Li and Reidys (2018). It turns out that the expected length of the longest block for γ-structures is on the order the same as the expected length of rainbows in secondary structures.
A case study of curriculum controversy: the Virginia Standards of Learning for history and the social sciences
Fore, Linda Compton (Virginia Tech, 1995-04-15)
Curriculum-making is a political exercise in which various groups in a society struggle over whose knowledge and values will be perpetuated through the school curriculum. As such, curriculum-making sometimes creates controversy. Controversy often accompanies the development of social studies curriculum because the purpose of social studies education is the preparation of the young for citizenship. Individuals disagree over what characteristics define the good citizen, as well as what knowledge and skills are necessary for effective citizenship. This study examines the political dimensions of social studies curriculum making in the controversy surrounding the development of the Virginia Standards of Learning for History and the Social Sciences. Using historical and qualitative methodology, the researcher collected and analyzed data from public documents, meetings of the Virginia Board of Education and its Advisory and Editing Committees, news articles, and transcripts from semi-structured interviews with eight key participants in the development of the social studies Standards of Learning. Analyses of these data sources showed that two primary groups struggled over control of the process of developing the standards, Governor Allen's education team and the professional social studies community under the leadership of the Virginia Consortium of Social Studies Specialists and College Educators. A third important force in the debate was the Virginia Board of Education, from which a small group of its members authored the final standards document. Further, this study showed two contextual influences on the Virginia social studies standards. The first was the Reagan rhetoric on academic crisis and educational reform through the establishment of tougher academic standards based on the traditional curriculum. The second was the recent controversy in Virginia over outcomes-based education. These two contextual influences combined to create a distrust of professional expertise. Three reciprocally related themes emerged from the data. Participants used power, rhetoric, and ideology to define the boundaries of the debate, control the process, name who could participate, and determine the outcome of the development process. Disagreements between the two major sides in the debate involved ideological differences over the nature of knowledge and learning and the nature of social studies education. There were also ideological differences among major participants over social issues like civil rights, gender issues, religion, and religious conflicts.
Characterizing Zero Divisors of Group Rings
Welch, Amanda Renee (Virginia Tech, 2015-06-15)
The Atiyah Conjecture originates from a paper written 40 years ago by Sir Michael Atiyah, a famous mathematician and Fields medalist. Since publication of the paper, mathematicians have been working to solve many questions related to the conjecture, but it is still open. The conjecture is about certain topological invariants attached to a group 𝐺. There are examples showing that the conjecture does not hold in general. These examples involve something like the lamplighter group (the wreath product ℤ/2ℤ ≀ ℤ). We are interested in looking at examples where this is not the case. We are interested in the specific case where 𝐺 is a finitely generated group in which the Prüfer group can be embedded as the center. The Prüfer group is a 𝑝-group for some prime 𝑝 and its finite subgroups have unbounded order, in particular the finite subgroups of G will have unbounded order. To understand whether any form of the Atiyah conjecture is true for 𝐺, it will first help to determine whether the group ring 𝑘𝐺 of the group 𝐺 has a classical ring of quotients for some field 𝑘. To determine this we will need to know the zero divisors for the group ring 𝑘𝐺. Our investigations will be divided into two cases, namely when the characteristic of the field 𝑘 is the same as the prime p for the Prüfer group and when it is different.
The Complete Pick Property and Reproducing Kernel Hilbert Spaces
Marx, Gregory (Virginia Tech, 2014-01-03)
We present two approaches towards a characterization of the complete Pick property. We first discuss the lurking isometry method used in a paper by J.A. Ball, T.T. Trent, and V. Vinnikov. They show that a nondegenerate, positive kernel has the complete Pick property if $1/k$ has one positive square. We also look at the one-point extension approach developed by P. Quiggin which leads to a sufficient and necessary condition for a positive kernel to have the complete Pick property. We conclude by connecting the two characterizations of the complete Pick property.
Core Entropy of Finite Subdivision Rules
Kim, Daniel Min (Virginia Tech, 2021-06-30)
The topological entropy of the subdivision map of a finite subdivision rule restricted to the 1-skeleton of its model subdivision complex, which we call textbf{core entropy}, is examined. We consider core entropy for finite subdivision rules realizing quadratic Misiurewicz polynomials and matings of such polynomials. It is shown that for a non-restrictive class of finite subdivision rules realizing quadratic Misiurewicz polynomials, core entropy equals Thurston's core entropy. We also show that the core entropy of formal and degenerate matings of Misiurewicz polynomials is determined by Thurston's core entropy of the mated polynomials.
The Discrete Hodge Star Operator and Poincaré Duality
Arnold, Rachel Florence (Virginia Tech, 2012-05-01)
This dissertation is a uniïfication of an analysis-based approach and the traditional topological-based approach to Poincaré duality. We examine the role of the discrete Hodge star operator in proving and in realizing the Poincaré duality isomorphism (between cohomology and homology in complementary degrees) in a cellular setting without reference to a dual cell complex. More specifically, we provide a proof of this version of Poincaré duality over R via the simplicial discrete Hodge star defined by Scott Wilson in [19] without referencing a dual cell complex. We also express the Poincaré duality isomorphism over both R and Z in terms of this discrete operator. Much of this work is dedicated to extending these results to a cubical setting, via the introduction of a cubical version of Whitney forms. A cubical setting provides a place for Robin Forman's complex of nontraditional differential forms, defined in [7], in the uniïfication of analytic and topological perspectives discussed in this dissertation. In particular, we establish a ring isomorphism (on the cohomology level) between Forman's complex of differential forms with his exterior derivative and product and a complex of cubical cochains with the discrete coboundary operator and the standard cubical cup product.
An Embedded Toeplitz Problem
Ordonez-Delgado, Bartleby (Virginia Tech, 2010-09-08)
In this work we investigate multi-variable Toeplitz operators and their relationship with KK-theory in order to apply this relationship to define and analyze embedded Toeplitz problems. In particular, we study the embedded Toeplitz problem of the unit disk into the unit ball in C^2. The embedding of Toeplitz problems suggests a way to define Toeplitz operators over singular spaces.
Enhanced intersection cutting plane and reformulation-linearization enumeration based approaches for linear complementarity problems
Krishnamurthy, Ravi S. (Virginia Tech, 1995-05-05)
In this research effort, we consider the linear complementarity problem (LCP) that arises in diverse areas including optimal control, economics, engineering, mechanics, and quadratic programming. This class of problems has posed a challenge to researchers for over three decades now. Most of the current algorithms designed to solve LCP are guaranteed to work only under some restrictive assumptions on the matrix M associated with LCP. In this research, we introduce two new algorithms based on an equivalent 0-1 mixed integer bilinear programming formulation of LCP. In the first approach, we develop an enhanced intersection cutting plane algorithm for solving LCP. The matrix M is not assumed to possess any special structure, except that the corresponding feasible region is assumed to be bounded. A procedure is described to generate cuts that are deeper versions of Tuy's intersection cuts, based on a relaxation of the usual polar set. The proposed algorithm then attempts to find an LCP solution in the process of generating either a single or a pair of such strengthened intersection cuts. The process of generating these cuts involves a vertex ranking scheme that either finds an LCP solution, or else, these cuts eliminate the entire feasible region leading to the conclusion that no LCP solution exists. Based on the bilinear formulation, a heuristic is also proposed to front-end the algorithm, in order to possibly solve LCP. In the second part of the dissertation, we present a global optimization algorithm based on a novel Reformulation-Linearization Technique (RLT). We do not place any restrictions on the matrix M associated with LCP in this case. This RLT scheme provides an equivalent linear, mixed integer programming formulation of LCP, that possesses a tight linear programming relaxation. The solution strategy developed is a composite impliCit enumeration-Lagrangian relaxation scheme. In addition to the bounds provided by the RLT-based relaxation, we further tighten these bounds at each node of the branch-and-bound tree through the use of strongest surrogate cuts, and strengthened intersection cuts. The heuristic developed in the previous algorithm is also invoked within this scheme. Both algorithms have been implemented and tested on randomly generated test problems. These problems include both indefinite and negative definite defining matrices. The results of these tests indicate that both methods are effective in solving the LCP with the heuristic being quite effective in recovering LCP solutions when the matrix Mis negative definite. The Lagrangian dual approach for the implicit enumeration scheme proved to be computationally more efficient than a similar simplex based lower bounding scheme.
The Exit Time Distribution for Small Random Perturbations of Dynamical Systems with a Repulsive Type Stationary Point
Buterakos, Lewis Allen (Virginia Tech, 2003-08-04)
We consider a stochastic differential equation on a domain D in n-dimensional real space, where the associated dynamical system is linear, and D contains a repulsive type stationary point at the origin O. We obtain an exit law for the first exit time of the solution process from a ball of arbitrary radius centered at the origin, which involves additive scaling as in Day (1995). The form of the scaling constant is worked out and shown to depend on the structure of the Jordan form of the linear drift. We then obtain an extension of this exit law to the first exit time of the solution process from the general domain D by considering the exit in two stages: first from the origin O to the boundary of the ball, for which the aforementioned exit law applies, and then from the boundary of the ball to the boundary of D. In this way we are able to determine for which Jordan forms we can obtain a limiting distribution for the first exit time to the boundary of D as the noise approaches 0. In particular, we observe there are cases for which the exit time distribution diverges as the noise approaches 0.
Exponential Stability for a Diffusion Equation in Polymer Kinetic Theory
Mulzet, Alfred Kenric (Virginia Tech, 1997-04-22)
In this paper we present an exponential stability result for a diffusion equation arising from dumbbell models for polymer flow. Using the methods of semigroup theory, we show that the semigroup U(t) associated with the diffusion equation is well defined and that all solutions converge exponentially to an equilibrium solution. Both finitely and infinitely extensible dumbbell models are considered. The main tool in establishing stability is the proof of compactness of the semigroup.
Finite Subdivision Rules from Matings of Quadratic Functions: Existence and Constructions
Wilkerson, Mary (Virginia Tech, 2012-04-24)
Combinatorial methods are utilized to examine preimage iterations of topologically glued polynomials. In particular, this paper addresses using finite subdivision rules and Hubbard trees as tools to model the dynamic behavior of mated quadratic functions. Several methods of construction of invariant structures on modified degenerate matings are detailed, and examples of parameter-based families of matings for which these methods succeed (and fail) are given.
First Cohomology of Some Infinitely Generated Groups
Eastridge, Samuel Vance (Virginia Tech, 2017-04-25)
The goal of this paper is to explore the first cohomology group of groups G that are not necessarily finitely generated. Our focus is on l^p-cohomology, 1 leq p leq infty, and what results regarding finitely generated groups change when G is infinitely generated. In particular, for abelian groups and locally finite groups, the l^p-cohomology is non-zero when G is countable, but vanishes when G has sufficient cardinality. We then show that the l^infty-cohomology remains unchanged for many classes of groups, before looking at several results regarding the injectivity of induced maps from embeddings of G-modules. We present several new results for countable groups, and discuss which results fail to hold in the general uncountable case. Lastly, we present results regarding reduced cohomology, including a useful lemma extending vanishing results for finitely generated groups to the infinitely generated case.
Generic Adaptive Handoff Algorithms Using Fuzzy Logic and Neural Networks
Tripathi, Nishith D. (Virginia Tech, 1997-08-21)
Efficient handoff algorithms cost-effectively enhance the capacity and Quality of Service (QoS) of cellular systems. This research presents novel approaches for the design of high performance handoff algorithms that exploit attractive features of several existing algorithms, provide adaptation to dynamic cellular environment, and allow systematic tradeoffs among different system characteristics. A comprehensive foundation of handoff and related issues of cellular communications is given. The tools of artificial intelligence utilized in this research, neural networks and fuzzy logic, are introduced. The scope of existing simulation models for macrocellular and microcellular handoff algorithms is enhanced by incorporating several important features. New simulation models suitable for performance evaluation of soft handoff algorithms and overlay handoff algorithms are developed. Four basic approaches for the development of high performance algorithms are proposed and are based on fuzzy logic, neural networks, unified handoff candidate selection, and pattern classification. The fuzzy logic based approach allows an organized tuning of the handoff parameters to provide a balanced tradeoff among different system characteristics. The neural network based approach suggests neural encoding of the fuzzy logic systems to simultaneously achieve the goals of high performance and reduced complexity. The unified candidacy based approach recommends the use of a unified handoff candidate selection criterion to select the best handoff candidate under given constraints. The pattern classification based approach exploits the capability of fuzzy logic and neural networks to obtain an efficient architecture of an adaptive handoff algorithm. New algorithms suitable for microcellular systems, overlay systems, and systems employing soft handoff are described. A basic adaptive algorithm suitable for a microcellular environment is proposed. Adaptation to traffic, interference, and mobility has been superimposed on the basic generic algorithm to develop another microcellular algorithm. An adaptive overlay handoff algorithm that allows a systematic balance among the design parameters of an overlay system is proposed. Important considerations for soft handoff are discussed, and adaptation mechanisms for new soft handoff algorithms are developed.
Geometry of Fractal Squares
Roinestad, Kristine A. (Virginia Tech, 2010-04-12)
This paper will examine analogues of Cantor sets, called fractal squares, and some of the geometric ways in which fractal squares raise issues not raised by Cantor sets. Also discussed will be a technique using directed graphs to prove bilipschitz equivalence of two fractal squares.
Groebner Finite Path Algebras
Leamer, Micah J. (Virginia Tech, 2004-06-04)
Let K be a field and Q a finite directed multi-graph. In this paper I classify all path algebras KQ and admissible orders with the property that all of their finitely generated ideals have finite Groebner bases.
Growth of functions in cercles de remplissage
Fenton, P. C.; Rossi, John F. (Cambridge University Press, 2002-02)
Suppose that f is meromorphic in the plane, and that there is a sequence z(n) --> infinity and a sequence of positive numbers epsilon(n) --> 0, such that epsilon(n)\z(n)f(#)(z(n))/log\z(n)\ --> infinity. It is shown that if f is analytic and non-zero in the closed discs Delta(n) = {z : \z - z(n)\ less than or equal to epsilon(n)\z(n)\}, n = 1, 2, 3,..., then, given any positive integer K, there are arbitrarily large values of n and there is a point z in Delta(n) such that \f(z)\ > \z\(K). Examples are given to show that the hypotheses cannot be relaxed.
High-Intensity Discharge Industrial Lighting Design Strategies for the Minimization of Energy Usage and Life-Cycle Cost
Flory, Isaac L. IV (Virginia Tech, 2008-08-26)
Worldwide, the electrical energy consumed by artificial lighting is second only to the amount consumed by electric machinery. Of the energy usage attributed to lighting in North America, approximately fifteen percent is consumed by those lighting products that are classified as High-Intensity Discharge (HID). These lighting products, which are dominated by Metal-Halide and High-Pressure Sodium technologies, range in power levels from 35 to 2000 watts and are used in both indoor and outdoor lighting applications, one category of which is the illumination of industrial facilities. This dissertation reviews HID industrial lighting design techniques and presents two luminaire layout algorithms which were developed to provide acceptable lighting performance based upon the minimum number of required luminaires as determined by the lumen method, regardless of the aspect ratio of the target area. Through the development of lighting design software tools based upon the Zonal Cavity Method and these layout algorithms, models for the quantification of energy requirements, lighting project life-cycle costs, and environmental impacts associated with conventional industrial lighting installations are presented. The software tools, which were created to perform indoor HID lighting designs for the often encountered application of illuminating general rectangular areas with non-sloped ceilings utilizing either High-Bay or Low-Bay luminaires, provide projections of minimal lighting system costs, energy consumption, and environmental impact based upon lamp selection, ballast selection, luminaire selection and lighting system maintenance practices. Based upon several industrial lighting application scenarios, lighting designs are presented using both the new software tools and a commercially available lighting design software package. For the purpose of validating this research, analyses of both designs for each scenario are presented complete with results of illuminance simulations performed using the commercially available software.

Browsing by Author "Rossi, John F."

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