The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences. It is freely available on the Web (http://www.research.att.com/~njas/sequences/) and is widely used.

There are several ways in which it benefits research: 1 It serves as a dictionary, to tell the user what is known about a particular sequence. There are hundreds of papers which thank the OEIS for assistance in this way.
 1 The associated Sequence Fans mailing list is a worldwide network which has evolved into a powerful machine for tackling new problems.
 1 As a direct source of new theorems, when a sequence arises in two different contexts.
 1 As a source of new research, when one sees a sequence in the OEIS that cries out to be analyzed.
 

The 40-year history of the OEIS recapitulates the story of modern computing, from punched cards to the internet.

The talk will be illustrated with numerous examples, emphasizing new sequences that have arrived in the past few months. Many open problems will be mentioned.

Because of the profusion of books and journals, volunteers play an important role in maintaining the database. If you come across an interesting number sequence in a book, journal or web site, please send it and the reference to the OEIS. (You do not need to be the author of the sequence to do this.) There is a web site for sending in "Comments" or "New sequences".

Several new features have been added to the OEIS in the past year. Thanks to the work of Russ Cox, searches are now performed at high speed, and thanks to the work of Debby Swayne, there is a button which displays plots of each sequence. Finally, a "listen" button enables one to hear the sequence played on a musical instrument (try Recamaan's sequence A005132!).

/pdf/the-on-line-encyclopedia-of-integer-sequences-okduuz5t6g.pdf

The On-Line Encyclopedia of Integer Sequences

Let SN be the group of permutations of 1,2,..., N. If 7r E SN, we say that 7(i1),... , 7F(ik) is an increasing subsequence in 7r if il < i2 < ... < ik and 7r(ii) < 7r(i2) < ...< 7r(ik). Let 1N(r) be the length of the longest increasing subsequence. For example, if N = 5 and 7r is the permutation 5 1 3 2 4 (in one-line notation: thus 7r(1) = 5, 7r(2) = 1, ... ), then the longest increasing subsequences are 1 2 4 and 1 3 4, and N() = 3. Equip SN with uniform distribution,

/pdf/on-the-distribution-of-the-length-of-the-longest-increasing-1b7d6efu2h.pdf

On the distribution of the length of the longest increasing subsequence of random permutations

We study a certain random growth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy–Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble (GUE).

/pdf/shape-fluctuations-and-random-matrices-t29l6as78h.pdf

Shape fluctuations and random matrices

The maximum clique problem is a classical problem in combinatorial optimization which finds important applications in different domains. In this paper we try to give a survey of results concerning algorithms, complexity, and applications of this problem, and also provide an updated bibliography. Of course, we build upon precursory works with similar goals [39, 232, 266].

The maximum clique problem

Group Representations.- Representations of the Symmetric Group.- Combinatorial Algorithms.- Symmetric Functions.- Applications and Generalizations.

The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions

The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with * include VAT for books; the €(D) includes 7% for Germany, the €(A) includes 10% for Austria. Prices indicated with ** include VAT for electronic products; 19% for Germany, 20% for Austria. All prices exclusive of carriage charges. Prices and other details are subject to change without notice. All errors and omissions excepted. B. Sagan The Symmetric Group

https://www.springer.com/productFlyer_978-0-387-95067-9.pdf?SGWID=0-0-1297-2108001-0

The Symmetric Group

Rotafest prgramme MacMahon's partition analysis - I the lecture hall partition theorem, George E.Andrews the cd-index of zonotypes and arrangements, Louis J. Billera, Richard Ehrenbor and margar Readdy letter-place methods and homotopy, David A. Buchsbaum classification of trivectors in 6-D space, Wendy Chan parameter augmentation for basic hypergeometric series I, William Y.C. Chen and Zhi-Guo Liu unities and negation, Henry crapo and Claude le conte de Poly-Barbut the would-be meth of targeted rings, ottavio M. D'Antona lattice walks and primary decomposition, Persi diaconis, Dav Eisenbud and Bernd Sturmfels natural exponential families and umbral calculus, A. di Bucchianico a D.E. Loeb umbral calculus in hilbert space, A.di Bucchianico, D.E. Loeb and Gian-Carlo Rota a strategy for determining polynomial orthogonality, J.M. Freeman plethystic formulas and positivity q,t-kosta coefficinets, A.M. Garsia and J. Remmel an alternative evaluation of the Andrews-Burge determinant, C. Krattenthaler the number of points in a combinatorial geometry with no 8-point-line minors, Joseph E. Bonin and Joseph P.S. Kung umbral shifts and symmetric functions of schur type, Miguel A. Mendez an axiomization for cubic algebras, Colin Bailey and Joseph Oliveira an elementar proof of Roichman's rule for irreducible characters of Iwahori-Hecke algebras of type A, Arun Ram universal constructions in umbral calculus, Nigel Ray hyperplane arrangements, parking functions an tree inversions, Richard P. Stanley more orthogonal polynomials as moments, Mourad E.H. Ismail and Dennis Stanton more orthogonal ploynomials as moments, Mourad E.H. Ismail and Dennis Stanton difference equations via the classical umbral calculus, Brian D. Taylor an analogy in geometric homology - rigidity and cofactors on geometric graphs, Walter Whiteley the umbral calculus and identities for hypergeometric functions with special arguments, Jet Wimp apologies to T.S. Eliot - rota nerds, J.S. Yang.

Mathematical Essays in honor of Gian-Carlo Rota

It is shown that by exchanging any two independent edges in any shortest cycle of the n-cube (n>or=3), its diameter decreases by one unit. This leads to the definition of a new class of n-regular graphs, denoted TQ/sub n/, with 2/sup n/ vertices and diameter n-1, which has the (n-1)-cube as subgraph. Other properties of TQ/sub n/ such as connectivity and the lengths of the disjoints paths are also investigated. Moreover, it is shown that the complete binary tree on 2/sup n/-1 vertices, which is not a subgraph of the n-cube, is a subgraph of TQ/sub n/. How these results can be used to enhance hypercube multiprocessors is discussed. >

The twisted N-cube with application to multiprocessing

rn The Wiener index is a graphical invariant that has found extensive application in chemistry. We define a generating function, which we call the Wiener polynomial, whose derivative is a q-analog of the Wiener index. We study some of the elementary properties of this polynomial and compute it for some common graphs. We then find a formula for the Wiener polynomial of a dendrimer, a certain highly regular tree of interest to chemists, and show that it is unimodal. Finally, we point out a connection with the Poincar6 polynomial of a finite Coxeter group. 0 1996 John Wiley & Sons, Inc.

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Bruce E. Sagan

Papers

The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions

The Symmetric Group

Mathematical Essays in honor of Gian-Carlo Rota

The twisted N-cube with application to multiprocessing

The Wiener polynomial of a graph