Variants of the Rogers–Ramanujan Identities

AbstractWe evaluate several integrals involving generating functions of continuous q-Hermite polynomials in two different ways. The resulting identities give new proofs and generalizations of the Rogers–Ramanujan identities. Two quintic transformations are given, one of which immediately proves the Rogers–Ramanujan identities without the Jacobi triple product identity. Similar techniques lead to new transformations for unilateral and bilateral series. The quintic transformations lead to curious identities involving primitive fifth roots of unity which are then extended to primitive pth roots of unity for odd p

The Chaucer Review: An Indexed Bibliography Vols. 1-30

The Chaucer Review is essential reading for Chaucerians at all levels of study. More than any other resource, it provides a record of most of the significant trends in medieval and Chaucer scholarship for the past three decades. It has, however, grown so rich and full that only with difficulty can we make the best use of it. Even those of us fortunate enough to have been charter subscribers and to have our own full runs of the Journal have no ready way to know what is in the more than three linear feet of the quarterly numbers of the Chaucer journal that now occupy our shelves. The purpose of this special issue of the journal is to provide scholars with a compiled list of all of the nearly 800 articles that have appeared, and, more important, a subject index to all of those articles.Subtitled A Journal of Medieval Studies and Literary Criticism, The Chaucer Review has spoken boldly and fully and long as the prominent sounding board for scholars of medieval literature, especially Chaucerians. Fundamental as this journal is, as issue after issue has come from The Pennsylvania State University Press, finding material in The Chaucer Review has been an increasingly daunting and frustrating challenge. The staff of the journal has provided at the end of each volume a list of the articles to appear in that volume and a comprehensive list at the end of each decade, but there has been no index. The titles of the articles are some guide to the contents, but every scholar knows that the titles often contain only the most subtle hint about what the article is really about. Who would know, for example, that Jackson J. Campbell's "Polonius among the Pilgrims" (7 [1972]: 40) is about the fictional teller of the Manciple's Tale? No one has before now attempted to provide a subject-matter index of the journal

Graph theory in America 1876-1950

This narrative is a history of the contributions made to graph theory in the United States of America by American mathematicians and others who supported the growth of scholarship in that country, between the years 1876 and 1950. The beginning of this period coincided with the opening of the first research university in the United States of America, The Johns Hopkins University (although undergraduates were also taught), providing the facilities and impetus for the development of new ideas. The hiring, from England, of one of the foremost mathematicians of the time provided the necessary motivation for research and development for a new generation of American scholars. In addition, it was at this time that home-grown research mathematicians were first coming to prominence. At the beginning of the twentieth century European interest in graph theory, and to some extent the four-colour problem, began to wane. Over three decades, American mathematicians took up this field of study - notably, Oswald Veblen, George Birkhoff, Philip Franklin, and Hassler Whitney. It is necessary to stress that these four mathematicians and all the other scholars mentioned in this history were not just graph theorists but worked in many other disciplines. Indeed, they not only made significant contributions to diverse fields but, in some cases, they created those fields themselves and set the standards for others to follow. Moreover, whilst they made considerable contributions to graph theory in general, two of them developed important ideas in connection with the four-colour problem. Grounded in a paper by Alfred Bray Kempe that was notorious for its fallacious 'proof' of the four-colour theorem, these ideas were the concepts of an unavoidable set and a reducible configuration. To place the story of these scholars within the history of mathematics, America, and graph theory, brief accounts are presented of the early years of graph theory, the early years of mathematics and graph theory in the USA, and the effects of the founding of the first institute for postgraduate study in America. Additionally, information has been included on other influences by such global events as the two world wars, the depression, the influx of European scholars into the United States of America, mainly during the 1930s, and the parallel development of graph theory in Europe. Until the end of the nineteenth century, graph theory had been almost entirely the prerogative of European mathematicians. Perhaps the first work in graph theory carried out in America was by Charles Sanders Peirce, arguably America's greatest logician and philosopher at the time. In the 1860s, he studied the four-colour conjecture and claimed to have written at least two papers on the subject during that decade, but unfortunately neither of these has survived. William Edward Story entered the field in 1879, with unfortunate consequences, but it was not until 1897 that an American mathematician presented a lecture on the subject, albeit only to have the paper disappear. Paul Wernicke presented a lecture on the four-colour problem to the American Mathematician Society, but again the paper has not survived. However, his 1904 paper has survived and added to the story of graph theory, and particularly the four-colour conjecture. The year 1912 saw the real beginning of American graph theory with Veblen and Birkhoff publishing major contributions to the subject. It was around this time that European mathematicians appeared to lose interest in graph theory. In the period 1912 to 1950 much of the progress made in the subject was from America and by 1950 not only had the United States of America become the foremost country for mathematics, it was the leading centre for graph theory

Partition identity bijections related to sign-balance and rank

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 81-83).In this thesis, we present bijections proving partitions identities. In the first part, we generalize Dyson's definition of rank to partitions with successive Durfee squares. We then present two symmetries for this new rank which we prove using bijections generalizing conjugation and Dyson's map. Using these two symmetries we derive a version of Schur's identity for partitions with successive Durfee squares and Andrews' generalization of the Rogers-Ramanujan identities. This gives a new combinatorial proof of the first Rogers-Ramanujan identity. We also relate this work to Garvan's generalization of rank. In the second part, we prove a family of four-parameter partition identities which generalize Andrews' product formula for the generating function for partitions with respect number of odd parts and number of odd parts of the conjugate. The parameters which we use are related to Stanley's work on the sign-balance of a partition.by Cilanne Emily Boulet.Ph.D

Identity-based cryptography from paillier cryptosystem.

Au Man Ho Allen.Thesis (M.Phil.)--Chinese University of Hong Kong, 2005.Includes bibliographical references (leaves 60-68).Abstracts in English and Chinese.Abstract --- p.iAcknowledgement --- p.iiiChapter 1 --- Introduction --- p.1Chapter 2 --- Preliminaries --- p.5Chapter 2.1 --- Complexity Theory --- p.5Chapter 2.2 --- Algebra and Number Theory --- p.7Chapter 2.2.1 --- Groups --- p.7Chapter 2.2.2 --- Additive Group Zn and Multiplicative Group Z*n --- p.8Chapter 2.2.3 --- The Integer Factorization Problem --- p.9Chapter 2.2.4 --- Quadratic Residuosity Problem --- p.11Chapter 2.2.5 --- Computing e-th Roots (The RSA Problem) --- p.13Chapter 2.2.6 --- Discrete Logarithm and Related Problems --- p.13Chapter 2.3 --- Public key Cryptography --- p.16Chapter 2.3.1 --- Encryption --- p.17Chapter 2.3.2 --- Digital Signature --- p.20Chapter 2.3.3 --- Identification Protocol --- p.22Chapter 2.3.4 --- Hash Function --- p.24Chapter 3 --- Paillier Cryptosystems --- p.26Chapter 3.1 --- Introduction --- p.26Chapter 3.2 --- The Paillier Cryptosystem --- p.27Chapter 4 --- Identity-based Cryptography --- p.30Chapter 4.1 --- Introduction --- p.31Chapter 4.2 --- Identity-based Encryption --- p.32Chapter 4.2.1 --- Notions of Security --- p.32Chapter 4.2.2 --- Related Results --- p.35Chapter 4.3 --- Identity-based Identification --- p.36Chapter 4.3.1 --- Security notions --- p.37Chapter 4.4 --- Identity-based Signature --- p.38Chapter 4.4.1 --- Security notions --- p.39Chapter 5 --- Identity-Based Cryptography from Paillier System --- p.41Chapter 5.1 --- Identity-based Identification schemes in Paillier setting --- p.42Chapter 5.1.1 --- Paillier-IBI --- p.42Chapter 5.1.2 --- CGGN-IBI --- p.43Chapter 5.1.3 --- GMMV-IBI --- p.44Chapter 5.1.4 --- KT-IBI --- p.45Chapter 5.1.5 --- Choice of g for Paillier-IBI --- p.46Chapter 5.2 --- Identity-based signatures from Paillier system . . --- p.47Chapter 5.3 --- Cocks ID-based Encryption in Paillier Setting . . --- p.48Chapter 6 --- Concluding Remarks --- p.51A Proof of Theorems --- p.53Chapter A.1 --- "Proof of Theorems 5.1, 5.2" --- p.53Chapter A.2 --- Proof Sketch of Remaining Theorems --- p.58Bibliography --- p.6

Relations between logic and mathematics in the work of Benjamin and Charles S. Peirce.

Charles Peirce (1839-1914) was one of the most important logicians of the nineteenth century. This thesis traces the development of his algebraic logic from his early papers, with especial attention paid to the mathematical aspects. There are three main sources to consider. 1) Benjamin Peirce (1809-1880), Charles's father and also a leading American mathematician of his day, was an inspiration. His memoir Linear Associative Algebra (1870) is summarised and for the first time the algebraic structures behind its 169 algebras are analysed in depth. 2) Peirce's early papers on algebraic logic from the late 1860s were largely an attempt to expand and adapt George Boole's calculus, using a part/whole theory of classes and algebraic analogies concerning symbols, operations and equations to produce a method of deducing consequences from premises. 3) One of Peirce's main achievements was his work on the theory of relations, following in the pioneering footsteps of Augustus De Morgan. By linking the theory of relations to his post-Boolean algebraic logic, he solved many of the limitations that beset Boole's calculus. Peirce's seminal paper `Description of a Notation for the Logic of Relatives' (1870) is analysed in detail, with a new interpretation suggested for his mysterious process of logical differentiation. Charles Peirce's later work up to the mid 1880s is then surveyed, both for its extended algebraic character and for its novel theory of quantification. The contributions of two of his students at the Johns Hopkins University, Oscar Mitchell and Christine Ladd-Franklin are traced, specifically with an analysis of their problem solving methods. The work of Peirce's successor Ernst Schröder is also reviewed, contrasting the differences and similarities between their logics. During the 1890s and later, Charles Peirce turned to a diagrammatic representation and extension of his algebraic logic. The basic concepts of this topological twist are introduced. Although Peirce's work in logic has been studied by previous scholars, this thesis stresses to a new extent the mathematical aspects of his logic - in particular the algebraic background and methods, not only of Peirce but also of several of his contemporaries

Computing aspects of problems in non-linear prediction and filtering

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Scientific virtues :

Chapter I describes diachronic realism and shows why it is a version of what is called 'metaphysical realism'. Consequently, I argue that recent claims that 'metaphysical realism' is incoherent are unfounded. Chapter II argues that certain anti-realist positions (those based on 'incommensurability arguments') involve an insufficient treatment of 'meaning' and 'reference' for theoretical terms. I review much of the current work on theories of reference and show that these incommensurability positions are bankrupt given either of the two most promising theories of reference. Chapter III argues that certain methodological factors ('scientific virtues') are the main considerations in historical cases of theory choice, and can warrant rational belief in a theory if it has achieved a sufficient level of virtues. Chapter IV defends the intuition that a realist interpretation of scientific theories explains their success by expanding the concepts of 'truth' and 'approximate truth'. I introduce the notions of truthlikeness and being on the right track to distinguish theories that were at least partially theoretically correct (though they failed to refer), from theories that were only successful in that they correctly organized experimental data. I use these notions with my diachronic approach to analyze two important historical examples.While there are many versions of scientific realism, most share the intuition that the remarkable success of some scientific theories is best explained by interpreting their theoretical claims as 'true' or 'approximately true'. Due to a variety of recent anti-realist objections, this intuition must be amended so that realist positions can remain conceptually and historically adequate. This dissertation defends a version of scientific realism, which I call diachronic realism, and includes these amendments

X-ray diffraction studies of reconstituted membranes

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Investigation of a hybrid switching control system

Bibliography: pages 84-86.A servo motor is to be used to position the cutting arm in a hypothetical pattern generation application. The motor is controlled in closed-loop in order to track, with zero asymptotic error, a reference signal represented by either a sinusoidal, triangular, or square wave. In addition, the schedule of reference signal type changes is not known a priori and the controlled system must achieve asymptotic tracking without operator intervention. As no simple single controller can satisfy these requirements for all setpoint types, a Hybrid Switching Control System is proposed which combines intuitive logic with standard control techniques. Under the guidance of a simple supervisor, the controller corresponding to each type of setpoint is switched in and out of the active feedback loop as required. A simple Multi-layer Perceptron neural network was selected to identify the type of signal being tracked and hence initiate controller switching. This network performed very well even in the presence of measurement noise, and the hybrid system automatically tracked each of the three types of reference signal over a wide range of signal amplitude and frequency. However, the reconfiguration interval was quite long (although still acceptable in terms of the proposed application), and the size of the neural net structure had to be limited for the system to work in real-time