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AN INTRODUCTION TO HELLMANN-FEYNMAN THEORY
Title
AN INTRODUCTION TO HELLMANN-FEYNMAN THEORY.
Creator
Wallace, David, Ismail, Mourad, University of Central Florida
Abstract / Description
The Hellmann-Feynman theorem is presented together with certain allied theorems. The origin of the Hellmann-Feynman theorem in quantum physical chemistry is described. The theorem is stated with proof and with discussion of applicability and reliability. Some adaptations of the theorem to the study of the variation of zeros of special functions and orthogonal polynomials are surveyed. Possible extensions are discussed.
Date Issued
2005
Identifier
CFE0000403, ucf:46349
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0000403
THE SHEFFER B-TYPE 1 ORTHOGONAL POLYNOMIAL SEQUENCES
Title
THE SHEFFER B-TYPE 1 ORTHOGONAL POLYNOMIAL SEQUENCES.
Creator
Galiffa, Daniel, Ismail, Mourad, University of Central Florida
Abstract / Description
In 1939, I.M. Sheffer proved that every polynomial sequence belongs to one and only one $type$. Sheffer extensively developed properties of the $B$-\emph{Type 0} polynomial sequences and determined which sets are also orthogonal. He subsequently generalized his classification method to the case of arbitrary $B$-\emph{Type k} by constructing the generalized generating function $A(t)\mathrm=\sum_^\infty$, with $H_i(t)=h_t^i+h_t^+\cdots,\phantomh_\neq 0$. Although extensive research has been...
Show moreIn 1939, I.M. Sheffer proved that every polynomial sequence belongs to one and only one $type$. Sheffer extensively developed properties of the $B$-\emph{Type 0} polynomial sequences and determined which sets are also orthogonal. He subsequently generalized his classification method to the case of arbitrary $B$-\emph{Type k} by constructing the generalized generating function $A(t)\mathrm=\sum_^\infty$, with $H_i(t)=h_t^i+h_t^+\cdots,\phantomh_\neq 0$. Although extensive research has been done on characterizing polynomial sequences, no analysis has yet been completed on sets of type one or higher ($k\geq1$). We present a preliminary analysis of a special case of the $B$-\emph{Type 1} ($k=1$) class, which is an extension of the $B$-\emph{Type 0} class, in order to determine which sets, if any, are also orthogonal sets. Lastly, we consider an extension of this research and comment on future considerations. In this work the utilization of computer algebra packages is indispensable, as computational difficulties arise in the $B$-\emph{Type 1} class that are unlike those in the $B$-\emph{Type 0} class.
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Date Issued
2009
Identifier
CFE0002551, ucf:47655
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0002551
Quasi-Gorenstein Modules
Title
Quasi-Gorenstein Modules.
Creator
York, Alexander, Brennan, Joseph, Martin, Heath, Ismail, Mourad, Kuebler, Stephen, University of Central Florida
Abstract / Description
This thesis will study the various roles that quasi-Gorenstein modules and their properties play in the study of homological dimensions and linkage of modules. To that effect we begin by studying these modules in their own right. An $R$-module $M$ of grade $g$ will be quasi-Gorenstein if $\Ext_R^i(M,R)=0$ for $i\neq g$ and there is an isomorphism $M\cong\Ext_R^g(M,R)$. Such modules have many nice properties which we will explore throughout this thesis. We will show they help extend a...
Show moreThis thesis will study the various roles that quasi-Gorenstein modules and their properties play in the study of homological dimensions and linkage of modules. To that effect we begin by studying these modules in their own right. An $R$-module $M$ of grade $g$ will be quasi-Gorenstein if $\Ext_R^i(M,R)=0$ for $i\neq g$ and there is an isomorphism $M\cong\Ext_R^g(M,R)$. Such modules have many nice properties which we will explore throughout this thesis. We will show they help extend a characterization of diagonalizable matrices over principal ideal domains to more general rings. We will use their properties to help lay a foundation for a study of homological dimensions, helping to generalize the concept of Gorenstein dimension to modules of larger grade and present a connection to these new dimensions with certain generalized Serre conditions.We then give a categorical construction to the concept of linkage. The main motivation of such a construction is to generalize ideal and module linkage into one unified theory. By using the defintion of linkage presented by Nagel \cite{NagelLiaison}, we can use categorical language to define linkage between categories. One of the focuses of this thesis is to show that the history of linkage has been wrought with a misunderstanding of which classes of objects to study. We give very compelling evidence to suggest that linkage is a tool to gain information about the even linkage classes of objects. Further, scattered among the literature is a wide array of results pertaining to module linkage, homological dimensions, duality, and adjoint functor pairs and for which we show that these fall under the umbrella of this unified theory. This leads to an intimate relationship between associated homological dimensions and the linkage of objects in a category. We will give many applications of the theory to modules allowing one to cover vast grounds from Gorenstein dimensions to Auslander and Bass classes to local cohomology and local homology. Each of these gives useful insight into certain classes of modules by applying this categorical approach to linkage.
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Date Issued
2018
Identifier
CFE0007268, ucf:52202
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007268
In Quest of Bernstein Inequalities Rational Functions, Askey-Wilson Operator, and Summation Identities for Entire Functions
Title
In Quest of Bernstein Inequalities Rational Functions, Askey-Wilson Operator, and Summation Identities for Entire Functions.
Creator
Puwakgolle Gedara, Rajitha, Li, Xin, Mohapatra, Ram, Ismail, Mourad, Xu, Mengyu, University of Central Florida
Abstract / Description
The title of the dissertation gives an indication of the material involved with the connecting thread throughout being the classical Bernstein inequality (and its variants), which provides an estimate to the size of the derivative of a given polynomial on a prescribed set in the complex plane, relative to the size of the polynomial itself on the same set. Chapters 1 and 2 lay the foundation for the dissertation. In Chapter 1, we introduce the notations and terminology that will be used...
Show moreThe title of the dissertation gives an indication of the material involved with the connecting thread throughout being the classical Bernstein inequality (and its variants), which provides an estimate to the size of the derivative of a given polynomial on a prescribed set in the complex plane, relative to the size of the polynomial itself on the same set. Chapters 1 and 2 lay the foundation for the dissertation. In Chapter 1, we introduce the notations and terminology that will be used throughout. Also a brief historical recount is given on the origin of the Bernstein inequality, which dated back to the days of the discovery of the Periodic table by the Russian Chemist Dmitri Mendeleev. In Chapter 2, we narrow down the contents stated in Chapter 1 to the problems we were interested in working during the course of this dissertation. Henceforth, we present a problem formulation mainly for those results for which solutions or partial solutions are provided in the subsequent chapters.Over the years Bernstein inequality has been generalized and extended in several directions. In Chapter \ref{Bern-ineq}, we establish rational analogues to some Bernstein-type inequalities for restricted zeros and prescribed poles. Our inequalities extend the results for polynomials, especially which are themselves improved versions of the classical Erd\"{o}s-Lax and Tur\'{a}n inequalities. In working towards proving our results, we establish some auxiliary results, which may be of interest on their own. Chapters \ref{AW-on-polynomials} and \ref{AW-on-entire} focus on the research carried out with the Askey-Wilson operator applied on polynomials and entire functions (of exponential type) respectively.In Chapter 4, we first establish a Riesz-type interpolation formula on the interval $[-1,1]$ for the Askey-Wilson operator. In consequence, a sharp Bernstein inequality and a Markov inequality are obtained when differentiation is replaced by the Askey-Wilson operator. Moreover, an inverse approximation theorem is proved using a Bernstein-type inequality in $L^2-$space. We conclude this chapter with an overconvergence result which is applied to characterize all $q$-differentiable functions of Brown and Ismail. Chapter \ref{AW-on-entire} is devoted to an intriguing application of the Askey-Wilson operator. By applying it on the Sampling Theorem on entire functions of exponential type, we obtain a series representation formula, which is what we called an extended Boas' formula. Its power in discovering interesting summation formulas, some known and some new will be demonstrated. As another application, we are able to obtain a couple of Bernstein-type inequalities.In the concluding chapter, we state some avenues where this research can progress.
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Date Issued
2018
Identifier
CFE0007237, ucf:52220
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007237