DOCSLIB.ORG

Formal Calculus, Umbral Calculus, and Basic Axiomatics of Vertex Algebras

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Formal Calculus, Umbral Calculus, and Basic Axiomatics of Vertex Algebras FORMAL CALCULUS, UMBRAL CALCULUS, AND BASIC AXIOMATICS OF VERTEX ALGEBRAS BY THOMAS J. ROBINSON A dissertation submitted to the Graduate School—New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Mathematics Written under the direction of James Lepowsky and approved by New Brunswick, New Jersey October, 2009 ABSTRACT OF THE DISSERTATION Formal calculus, umbral calculus, and basic axiomatics of vertex algebras by Thomas J. Robinson Dissertation Director: James Lepowsky The central subject of this thesis is formal calculus together with certain applications to vertex operator algebras and combinatorics. By formal calculus we mean mainly the formal calculus that has been used to describe vertex operator algebras and their modules as well as logarithmic tensor product theory, but we also mean the formal calculus known as umbral calculus. We shall exhibit and develop certain connections between these formal calculi. Among other things we lay out a technique for efficiently proving certain general formal Taylor theorems and we show how to recast much of the classical umbral calculus as stemming from a formal calculus argument that calculates the exponential generating function of the higher derivatives of a composite function. This formal calculus argument is analogous to an important calculation proving the associativity property of lattice vertex operators. We use some of our results to derive combinatorial identities. Finally, we apply other results to study some basic axiomatics of vertex (operator) algebras. In particular, we enhance well known formal calculus approaches to the axioms by introducing a new axiom, “weak skew-associativity,” in order to exploit the S3-symmetric nature of the Jacobi identity axiom. In particular, we use this approach to give a simplified proof that the weak associativity and the Jacobi identity axioms for a module for a vertex algebra are equivalent, an important result ii in the representation theory of vertex algebras. iii Acknowledgements I wish to thank James Lepowsky most particularly for his long standing support, pa- tience and guidance. I also wish to thank both Yi-Zhi Huang and Haisheng Li, who, among other things, each taught me large portions of the material relevant to this work. In addition, my thanks go to Stephen Greenfield, Amy Cohen, Ovidiu Costin, Richard Lyons and Richard Wheeden for their help when I first arrived at Rutgers. I would also like to thank Antun Milas, Robert Wilson, Doron Zeilberger and Louis Shapiro. Of course, I have also benefited greatly from conversations with other members of the faculty at Rutgers, but they are too numerous to name. I further wish to thank Mark Krusemeyer and Gail Nelson, my undergraduate professors, and also Mr. Suggs and Dr. Watson, my high school math teachers. And most certainly all the staff in the Rutgers Math Department who through the years have saved me from myself. Finally, I am grateful for partial support from NSF grant PHY0901237. iv Dedication To my Mom and Dad. v Table of Contents Abstract ........................................ ii Acknowledgements ................................. iv Dedication ....................................... v 1. Introduction ................................... 1 1.1. IntroductiontoChapter2 . 6 1.2. IntroductiontoChapter3 . 9 1.3. IntroductiontoChapter4 . 16 2. Exponentiated derivations, the formal Taylor theorem, and Fa`adi Bruno’s formula ................................... 22 2.1. The formal Taylor theorem: a traditional approach . ........ 22 2.2. The formal Taylor theorem from a different point of view . ....... 26 2.3. The formal Taylor theorem for iterated logarithms and exponentials . 29 2.4. Formal analytic expansions: warmup . 30 2.4.1. Case N = 0, the case “x” ...................... 30 Method1 ............................... 30 Method2 ............................... 31 Method3 ............................... 31 Method4 ............................... 31 2.4.2. Case N = −1, the case “exp x”................... 32 Method1 ............................... 32 Method2 ............................... 32 Method3 ............................... 33 Method4 ............................... 33 vi 2.4.3. Case N = 1, the case “log x” .................... 34 Method1 ............................... 34 Method2 ............................... 35 Method3 ............................... 35 Method4 ............................... 36 2.4.4. Case N = −2, the case “exp exp x”................. 37 Method1 ............................... 37 Method2 ............................... 38 Method3 ............................... 38 Method4 ............................... 39 2.5. Arecursion .................................. 39 2.6. Formal analytic expansions: warmup continued and completed . 44 2.7. Formal analytic expansions: Cases N ≥ 0, “iterated logarithms” . 45 2.7.1. Method1 ............................... 45 2.7.2. Method2 ............................... 46 2.7.3. Method3 ............................... 47 2.7.4. Method4 ............................... 50 2.8. Somecombinatorics ............................. 52 2.9. Substitution maps for iterated logs and exponentials . .......... 53 2.10. A glimpse of Fa`adi Bruno and umbral calculus . ...... 55 3. Formal calculus and umbral calculus .................... 58 3.1. Preliminaries ................................. 58 3.2. A restatement of the problem and further developments . ........ 62 3.3. Umbralconnection .............................. 68 3.4. Some historical, contextual results . ...... 73 3.5. Umbralshiftsrevisited . 80 3.6. Sheffersequences ............................... 81 3.7. TheexponentialRiordangroup . 86 vii 3.8. Matrix multiplication via generating functions . .......... 89 3.9. Someunderlyingcombinatorics . 93 3.10.TheVirasoroalgebra. 100 3.11. Umbral shifts revisited and generalized . ........ 103 3.12.Fa`adiBruno’sformula. 110 3.13. A standard quadratic representation of the Virasoro algebra of central charge1.................................... 112 4. Replacement axioms for the Jacobi identity for vertex algebras and their modules ..................................... 119 4.1. Formalcalculussummarized. 119 4.2. Formal calculus further developed . 122 4.3. Vacuum-free vertex algebras . 127 4.4. Modules.................................... 133 4.5. Vertex algebras with vacuum . 136 4.6. Modules for a vertex algebras with vacuum . 148 References ....................................... 151 Vita ........................................... 155 viii 1 Chapter 1 Introduction The central subject of this thesis is formal calculus together with certain applications to vertex operator algebras and combinatorics. By formal calculus we mean mainly the formal calculus that has been used to describe vertex operator algebras and their modules as well as logarithmic tensor product theory, but we also mean the formal calculus known as umbral calculus. We shall exhibit and develop certain connections between these formal calculi. Among other things we lay out a technique for efficiently proving certain general formal Taylor theorems and we show how to recast much of the classical umbral calculus as stemming from a formal calculus argument that calculates the exponential generating function of the higher derivatives of a composite function. This formal calculus argument is analogous to an important calculation proving the associativity property of lattice vertex operators. We use some of our results to derive combinatorial identities. Finally, we apply other results to study some basic axiomatics of vertex (operator) algebras. In particular, we enhance well known formal calculus approaches to the axioms by introducing a new axiom, “weak skew-associativity,” in order to exploit the S3-symmetric nature of the Jacobi identity axiom. In particular, we use this approach to give a simplified proof that the weak associativity and the Jacobi identity axioms for a module for a vertex algebra are equivalent, an important result in the representation theory of vertex algebras. In the next two paragraphs we very briefly review some highlights in the development of vertex operator algebras and umbral calculus; then we describe the evolution of the present work. Following that, we give a basic chapter-by-chapter outline, itself followed by a more extensive section-by-section outline of the main body of this work. 2 In mathematics, vertex operators first arose in the work of J. Lepowsky and R. Wil- son, who were seeking concrete representations of affine Lie algebras [LW1], and using this work they were able to give a new, natural interpretation of the Rogers-Ramanujan partition identities [LW2]. Vertex operators were used by I. Frenkel, J. Lepowsky and A. Meurman [FLM1] to construct a natural “moonshine module,” whose “character” is the modular function J(q), for the largest sporadic finite simple group, the Fischer- Griess Monster group [G]. This proved a conjecture of J. McKay and J. Thompson and part of the Conway-Norton conjectures [CN]. R. Borcherds [Bor1] introduced the first mathematical definition of vertex algebra which, in particular, extended the relations for the vertex operators used to construct the moonshine module. Using this and other ideas, Borcherds [Bor2] proved the remaining Conway-Norton conjectures [CN] for the moonshine module. A variant of the notion of vertex algebra, that of vertex operator algebra, was introduced in [FLM2] using a “Jacobi identity,” which was implicit
Load more

Recommended publications
  • K-Quasiderivations
    K-QUASIDERIVATIONS CALEB EMMONS, MIKE KREBS, AND ANTHONY SHAHEEN Abstract. A K-quasiderivation is a map which satisfies both the Product Rule and the Chain Rule. In this paper, we discuss sev- eral interesting families of K-quasiderivations. We first classify all K-quasiderivations on the ring of polynomials in one variable over an arbitrary commutative ring R with unity, thereby extend- ing a previous result. In particular, we show that any such K- quasiderivation must be linear over R. We then discuss two previ- ously undiscovered collections of (mostly) nonlinear K-quasiderivations on the set of functions defined on some subset of a field. Over the reals, our constructions yield a one-parameter family of K- quasiderivations which includes the ordinary derivative as a special case. 1. Introduction In the middle half of the twientieth century|perhaps as a reflection of the mathematical zeitgeist|Lausch, Menger, M¨uller,N¨obauerand others formulated a general axiomatic framework for the concept of the derivative. Their starting point was (usually) a composition ring, by which is meant a commutative ring R with an additional operation ◦ subject to the restrictions (f + g) ◦ h = (f ◦ h) + (g ◦ h), (f · g) ◦ h = (f ◦ h) · (g ◦ h), and (f ◦ g) ◦ h = f ◦ (g ◦ h) for all f; g; h 2 R. (See [1].) In M¨uller'sparlance [9], a K-derivation is a map D from a composition ring to itself such that D satisfies Additivity: D(f + g) = D(f) + D(g) (1) Product Rule: D(f · g) = f · D(g) + g · D(f) (2) Chain Rule D(f ◦ g) = [(D(f)) ◦ g] · D(g) (3) 2000 Mathematics Subject Classification.
    [Show full text]
  • Algorithmic Factorization of Polynomials Over Number Fields
    Rose-Hulman Institute of Technology Rose-Hulman Scholar Mathematical Sciences Technical Reports (MSTR) Mathematics 5-18-2017 Algorithmic Factorization of Polynomials over Number Fields Christian Schulz Rose-Hulman Institute of Technology Follow this and additional works at: https://scholar.rose-hulman.edu/math_mstr Part of the Number Theory Commons, and the Theory and Algorithms Commons Recommended Citation Schulz, Christian, "Algorithmic Factorization of Polynomials over Number Fields" (2017). Mathematical Sciences Technical Reports (MSTR). 163. https://scholar.rose-hulman.edu/math_mstr/163 This Dissertation is brought to you for free and open access by the Mathematics at Rose-Hulman Scholar. It has been accepted for inclusion in Mathematical Sciences Technical Reports (MSTR) by an authorized administrator of Rose-Hulman Scholar. For more information, please contact [email protected]. Algorithmic Factorization of Polynomials over Number Fields Christian Schulz May 18, 2017 Abstract The problem of exact polynomial factorization, in other words expressing a poly- nomial as a product of irreducible polynomials over some field, has applications in algebraic number theory. Although some algorithms for factorization over algebraic number fields are known, few are taught such general algorithms, as their use is mainly as part of the code of various computer algebra systems. This thesis provides a summary of one such algorithm, which the author has also fully implemented at https://github.com/Whirligig231/number-field-factorization, along with an analysis of the runtime of this algorithm. Let k be the product of the degrees of the adjoined elements used to form the algebraic number field in question, let s be the sum of the squares of these degrees, and let d be the degree of the polynomial to be factored; then the runtime of this algorithm is found to be O(d4sk2 + 2dd3).
    [Show full text]
  • Relationship of Bell's Polynomial Matrix and K-Fibonacci Matrix
    American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Relationship of Bell’s Polynomial Matrix and k-Fibonacci Matrix Mawaddaturrohmaha, Sri Gemawatib* a,bDepartment of Mathematics, University of Riau, Pekanbaru 28293, Indonesia aEmail: [email protected] bEmail: [email protected] Abstract The Bell‟s polynomial matrix is expressed as , where each of its entry represents the Bell‟s polynomial number.This Bell‟s polynomial number functions as an information code of the number of ways in which partitions of a set with certain elements are arranged into several non-empty section blocks. Furthermore, thek- Fibonacci matrix is expressed as , where each of its entry represents the k-Fibonacci number, whose first term is 0, the second term is 1 and the next term depends on a positive integer k. This article aims to find a matrix based on the multiplication of the Bell‟s polynomial matrix and the k-Fibonacci matrix. Then from the relationship between the two matrices the matrix is obtained. The matrix is not commutative from the product of the two matrices, so we get matrix Thus, the matrix , so that the Bell‟s polynomial matrix relationship can be expressed as Keywords: Bell‟s Polynomial Number; Bell‟s Polynomial Matrix; k-Fibonacci Number; k-Fibonacci Matrix. 1. Introduction Bell‟s polynomial numbers are studied by mathematicians since the 19th century and are named after their inventor Eric Temple Bell in 1938[5] Bell‟s polynomial numbers are represented by for each n and kelementsof the natural number, starting from 1 and 1 [2, p 135] Bell‟s polynomial numbers can be formed into Bell‟s polynomial matrix so that each entry of Bell‟s polynomial matrix is Bell‟s polynomial number and is represented by [11].
    [Show full text]
  • Poly-Bernoulli Polynomials Arising from Umbral Calculus
    Poly-Bernoulli polynomials arising from umbral calculus by Dae San Kim, Taekyun Kim and Sang-Hun Lee Abstract In this paper, we give some recurrence formula and new and interesting identities for the poly-Bernoulli numbers and polynomials which are derived from umbral calculus. 1 Introduction The classical polylogarithmic function Lis(x) are ∞ xk Li (x)= , s ∈ Z, (see [3, 5]). (1) s ks k X=1 In [5], poly-Bernoulli polynomials are defined by the generating function to be −t ∞ n Li (1 − e ) (k) t k ext = eB (x)t = B(k)(x) , (see [3, 5]), (2) 1 − e−t n n! n=0 X (k) n (k) with the usual convention about replacing B (x) by Bn (x). As is well known, the Bernoulli polynomials of order r are defined by the arXiv:1306.6697v1 [math.NT] 28 Jun 2013 generating function to be t r ∞ tn ext = B(r)(x) , (see [7, 9]). (3) et − 1 n n! n=0 X (r) In the special case, r = 1, Bn (x)= Bn(x) is called the n-th ordinary Bernoulli (r) polynomial. Here we denote higher-order Bernoulli polynomials as Bn to avoid 1 conflict of notations. (k) (k) If x = 0, then Bn (0) = Bn is called the n-th poly-Bernoulli number. From (2), we note that n n n n B(k)(x)= B(k) xl = B(k)xn−l. (4) n l n−l l l l l X=0 X=0 Let F be the set of all formal power series in the variable t over C as follows: ∞ tk F = f(t)= ak ak ∈ C , (5) ( k! ) k=0 X P P∗ and let = C[x] and denote the vector space of all linear functionals on P.
    [Show full text]
  • On the Discriminant of a Certain Quartinomial and Its Totally Complexness
    ON THE DISCRIMINANT OF A CERTAIN QUARTINOMIAL AND ITS TOTALLY COMPLEXNESS SHUICHI OTAKE AND TONY SHASKA Abstract. In this paper, we compute the discriminant of a quartinomial n 2 of the form f(b;a;1)(t; x) = x + t(x + ax + b) by using the Bezoutian. Then, by using this result and another theorem of our previous paper, we construct a family of totally complex polynomials of the form f(b;a;1)(ξ; x) (ξ 2 R; (a; b) 6= (0; 0)). 1. Introduction The discriminant of a polynomial has been a major objective of research in algebra because of its importance. For example, the discriminant of a polynomial allows us to know whether the polynomial has multiple roots or not and it also plays an important role when we compute the discriminant of a number field. Moreover the discriminant of a polynomial also tells us whether the Galois group of the polynomial is contained in the alternating group or not. This is why there are so many papers focusing on studying the discriminant of a polynomial ([B-B-G], [D-S], [Ked], [G-D], [Swa]). In the last two papers, the authors concern the computation of the discriminant of a trinomial and it has been carried out in different ways. n In this paper, we compute the discriminant of a quartinomial f(b;a;1)(t; x) = x + t(x2 + ax + b) by using the Bezoutian (Theorem2). Let F be a field of characteristic zero and f1(x), f2(x) be polynomials over F . Then, for any integer n such that n ≥ maxfdegf1; degf2g, we put n f1(x)f2(y) − f1(y)f2(x) X B (f ; f ) : = = α xn−iyn−j 2 F [x; y]; n 1 2 x − y ij i;j=1 Mn(f1; f2) : = (αij)1≤i;j≤n: 0 The n × n matrix Mn(f1; f2) is called the Bezoutian of f1 and f2.
    [Show full text]
  • Bell Polynomials and Binomial Type Sequences Miloud Mihoubi
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Discrete Mathematics 308 (2008) 2450–2459 www.elsevier.com/locate/disc Bell polynomials and binomial type sequences Miloud Mihoubi U.S.T.H.B., Faculty of Mathematics, Operational Research, B.P. 32, El-Alia, 16111, Algiers, Algeria Received 3 March 2007; received in revised form 30 April 2007; accepted 10 May 2007 Available online 25 May 2007 Abstract This paper concerns the study of the Bell polynomials and the binomial type sequences. We mainly establish some relations tied to these important concepts. Furthermore, these obtained results are exploited to deduce some interesting relations concerning the Bell polynomials which enable us to obtain some new identities for the Bell polynomials. Our results are illustrated by some comprehensive examples. © 2007 Elsevier B.V. All rights reserved. Keywords: Bell polynomials; Binomial type sequences; Generating function 1. Introduction The Bell polynomials extensively studied by Bell [3] appear as a standard mathematical tool and arise in combina- torial analysis [12]. Moreover, they have been considered as important combinatorial tools [11] and applied in many different frameworks from, we can particularly quote : the evaluation of some integrals and alterning sums [6,10]; the internal relations for the orthogonal invariants of a positive compact operator [5]; the Blissard problem [12, p. 46]; the Newton sum rules for the zeros of polynomials [9]; the recurrence relations for a class of Freud-type polynomials [4] and many others subjects. This large application of the Bell polynomials gives a motivation to develop this mathematical tool.
    [Show full text]
  • On Multivariable Cumulant Polynomial Sequences with Applications E
    JOURNAL OF ALGEBRAIC STATISTICS Vol. 7, No. 1, 2016, 72-89 ISSN 1309-3452 { www.jalgstat.com On multivariable cumulant polynomial sequences with applications E. Di Nardo1,∗ 1 Department of Mathematics \G. Peano", Via Carlo Alberto 10, University of Turin, 10123, I-Turin, ITALY Abstract. A new family of polynomials, called cumulant polynomial sequence, and its extension to the multivariate case is introduced relying on a purely symbolic combinatorial method. The coefficients are cumulants, but depending on what is plugged in the indeterminates, moment se- quences can be recovered as well. The main tool is a formal generalization of random sums, when a not necessarily integer-valued multivariate random index is considered. Applications are given within parameter estimations, L´evyprocesses and random matrices and, more generally, problems involving multivariate functions. The connection between exponential models and multivariable Sheffer polynomial sequences offers a different viewpoint in employing the method. Some open problems end the paper. 2000 Mathematics Subject Classifications: 62H05; 60C05; 11B83, 05E40 Key Words and Phrases: multi-index partition, cumulant, generating function, formal power series, L´evyprocess, exponential model 1. Introduction The so-called symbolic moment method has its roots in the classical umbral calculus developed by Rota and his collaborators since 1964; [15]. Rewritten in 1994 (see [17]), what we have called moment symbolic method consists in a calculus on unital number sequences. Its basic device is to represent a sequence f1; a1; a2;:::g with the sequence 2 f1; α; α ;:::g of powers of a symbol α; named umbra. The sequence f1; a1; a2;:::g is said to be umbrally represented by the umbra α: The main tools of the symbolic moment method are [7]: a) a polynomial ring C[A] with A = fα; β; γ; : : :g a set of symbols called umbrae; b) a unital operator E : C[A] ! C; called evaluation, such that i i) E[α ] = ai for non-negative integers i; ∗Corresponding author.
    [Show full text]
  • Degenerate Poly-Bernoulli Polynomials with Umbral Calculus Viewpoint Dae San Kim1,Taekyunkim2*, Hyuck in Kwon2 and Toufik Mansour3
    Kim et al. Journal of Inequalities and Applications (2015)2015:228 DOI 10.1186/s13660-015-0748-7 R E S E A R C H Open Access Degenerate poly-Bernoulli polynomials with umbral calculus viewpoint Dae San Kim1,TaekyunKim2*, Hyuck In Kwon2 and Toufik Mansour3 *Correspondence: [email protected] 2Department of Mathematics, Abstract Kwangwoon University, Seoul, 139-701, S. Korea In this paper, we consider the degenerate poly-Bernoulli polynomials. We present Full list of author information is several explicit formulas and recurrence relations for these polynomials. Also, we available at the end of the article establish a connection between our polynomials and several known families of polynomials. MSC: 05A19; 05A40; 11B83 Keywords: degenerate poly-Bernoulli polynomials; umbral calculus 1 Introduction The degenerate Bernoulli polynomials βn(λ, x)(λ =)wereintroducedbyCarlitz[ ]and rediscovered by Ustinov []underthenameKorobov polynomials of the second kind.They are given by the generating function t tn ( + λt)x/λ = β (λ, x) . ( + λt)/λ – n n! n≥ When x =,βn(λ)=βn(λ, ) are called the degenerate Bernoulli numbers (see []). We observe that limλ→ βn(λ, x)=Bn(x), where Bn(x)isthenth ordinary Bernoulli polynomial (see the references). (k) The poly-Bernoulli polynomials PBn (x)aredefinedby Li ( – e–t) tn k ext = PB(k)(x) , et – n n! n≥ n where Li (x)(k ∈ Z)istheclassicalpolylogarithm function given by Li (x)= x (see k k n≥ nk [–]). (k) For = λ ∈ C and k ∈ Z,thedegenerate poly-Bernoulli polynomials Pβn (λ, x)arede- fined by Kim and Kim to be Li ( – e–t) tn k ( + λt)x/λ = Pβ(k)(λ, x) (see []).
    [Show full text]
  • A FAMILY of SEQUENCES of BINOMIAL TYPE 3 and If Jp +1= N Then This Difference Is 0
    A FAMILY OF SEQUENCES OF BINOMIAL TYPE WOJCIECH MLOTKOWSKI, ANNA ROMANOWICZ Abstract. For delta operator aD bDp+1 we find the corresponding polynomial se- − 1 2 quence of binomial type and relations with Fuss numbers. In the case D 2 D we show that the corresponding Bessel-Carlitz polynomials are moments of the− convolu- tion semigroup of inverse Gaussian distributions. We also find probability distributions νt, t> 0, for which yn(t) , the Bessel polynomials at t, is the moment sequence. { } 1. Introduction ∞ A sequence wn(t) n=0 of polynomials is said to be of binomial type (see [10]) if deg w (t)= n and{ for} every n 0 and s, t R we have n ≥ ∈ n n (1.1) w (s + t)= w (s)w − (t). n k k n k Xk=0 A linear operator Q of the form c c c (1.2) Q = 1 D + 2 D2 + 3 D3 + ..., 1! 2! 3! acting on the linear space R[x] of polynomials, is called a delta operator if c1 = 0. Here D denotes the derivative operator: D1 := 0 and Dtn := n tn−1 for n 1. We6 will write Q = g(D), where · ≥ c c c (1.3) g(x)= 1 x + 2 x2 + 3 x3 + .... 1! 2! 3! There is one-to-one correspondence between sequences of binomial type and delta ∞ operators, namely if Q is a delta operator then there is unique sequence w (t) of { n }n=0 binomial type satisfying Qw0(t) = 0 and Qwn(t)= n wn−1(t) for n 1.
    [Show full text]
  • Fitting and Graphing Discrete Distributions
    Chapter 3 Fitting and graphing discrete distributions {ch:discrete} Discrete data often follow various theoretical probability models. Graphic displays are used to visualize goodness of fit, to diagnose an appropriate model, and determine the impact of individual observations on estimated parameters. Not everything that counts can be counted, and not everything that can be counted counts. Albert Einstein Discrete frequency distributions often involve counts of occurrences of events, such as ac- cident fatalities, incidents of terrorism or suicide, words in passages of text, or blood cells with some characteristic. Often interest is focused on how closely such data follow a particular prob- ability distribution, such as the binomial, Poisson, or geometric distribution, which provide the basis for generating mechanisms that might give rise to the data. Understanding and visualizing such distributions in the simplest case of an unstructured sample provides a building block for generalized linear models (Chapter 9) where they serve as one component. The also provide the basis for a variety of recent extensions of regression models for count data (Chapter ?), allow- ing excess counts of zeros (zero-inflated models), left- or right- truncation often encountered in statistical practice. This chapter describes the well-known discrete frequency distributions: the binomial, Pois- son, negative binomial, geometric, and logarithmic series distributions in the simplest case of an unstructured sample. The chapter begins with simple graphical displays (line graphs and bar charts) to view the distributions of empirical data and theoretical frequencies from a specified discrete distribution. It then describes methods for fitting data to a distribution of a given form and simple, effective graphical methods than can be used to visualize goodness of fit, to diagnose an appropriate model (e.g., does a given data set follow the Poisson or negative binomial?) and determine the impact of individual observations on estimated parameters.
    [Show full text]
  • Uwe Krey · Anthony Owen Basic Theoretical Physics Uwe Krey · Anthony Owen
    Uwe Krey · Anthony Owen Basic Theoretical Physics Uwe Krey · Anthony Owen Basic Theoretical Physics AConciseOverview With 31 Figures 123 Prof. Dr. Uwe Krey University of Regensburg (retired) FB Physik Universitätsstraße 31 93053 Regensburg, Germany E-mail: [email protected] Dr. rer nat habil Anthony Owen University of Regensburg (retired) FB Physik Universitätsstraße 31 93053 Regensburg, Germany E-mail: [email protected] Library of Congress Control Number: 2007930646 ISBN 978-3-540-36804-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting and production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: eStudio Calamar S.L., F. Steinen-Broo, Pau/Girona, Spain Printed on acid-free paper SPIN 11492665 57/3180/YL - 5 4 3 2 1 0 Preface This textbook on theoretical physics (I-IV) is based on lectures held by one of the authors at the University of Regensburg in Germany.
    [Show full text]
  • L-Series and Hurwitz Zeta Functions Associated with the Universal Formal Group
    Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX (2010), 133-144 L-series and Hurwitz zeta functions associated with the universal formal group PIERGIULIO TEMPESTA Abstract. The properties of the universal Bernoulli polynomials are illustrated and a new class of related L-functions is constructed. A generalization of the Riemann-Hurwitz zeta function is also proposed. Mathematics Subject Classification (2010): 11M41 (primary); 55N22 (sec- ondary). 1. Introduction The aim of this article is to establish a connection between the theory of formal groups on one side and a class of generalized Bernoulli polynomials and Dirichlet series on the other side. Some of the results of this paper were announced in the communication [26]. We will prove that the correspondence between the Bernoulli polynomials and the Riemann zeta function can be extended to a larger class of polynomials, by introducing the universal Bernoulli polynomials and the associated Dirichlet series. Also, in the same spirit, generalized Hurwitz zeta functions are defined. Let R be a commutative ring with identity, and R {x1, x2,...} be the ring of formal power series in x1, x2,...with coefficients in R.Werecall that a commuta- tive one-dimensional formal group law over R is a formal power series (x, y) ∈ R {x, y} such that 1) (x, 0) = (0, x) = x 2) ( (x, y) , z) = (x,(y, z)) . When (x, y) = (y, x), the formal group law is said to be commutative. The existence of an inverse formal series ϕ (x) ∈ R {x} such that (x,ϕ(x)) = 0 follows from the previous definition.
    [Show full text]