Related papers: 50 Years of Bailey's lemma

The main theme of this dissertation is retooling methods to work for different situations. I have taken the method derived by O'Hara and simplified by Zeilberger to prove unimodality of $q$-binomials and tweaked it. This allows us to create…

Presented are polynomial identities which imply generalizations of Euler and Rogers--Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by…

The primary aim of this chapter is, commemorating the 150th anniversary of Riemann's death, to explain how the idea of {\it Riemann sum} is linked to other branches of mathematics. The materials I treat are more or less classical and…

This paper, commissioned as a survey of the Riemann Hypothesis, provides a comprehensive overview of 165 years of mathematical approaches to this fundamental problem, while introducing a new perspective that emerged during its preparation.…

This paper is inspired by 1892 paper of Johnson, where he has given an axiomatization for the variety of Boolean algebras (equivalently, for classical propositional calculus). The fact that the axioms of Johnson include the associative law,…

We study cylindric partitions with two-element profiles using MacMahon's partition analysis. We find explicit formulas for the generating functions of the number of cylindric partitions by first finding the recurrences using partition…

In this paper, we prove the following identity $$ \lcm({n\brack 0}_q,{n\brack 1}_q,...,{n\brack n}_q) =\frac{\lcm([1]_q,[2]_q,...,[n+1]_q)}{[n+1]_q}, $$ where ${n\brack k}_q$ denotes the $q$-binomial coefficient and…

By generalizing Gessel-Xin's Laurent series method for proving the Zeilberger-Bressoud $q$-Dyson Theorem, we establish a family of $q$-Dyson style constant term identities. These identities give explicit formulas for certain coefficients of…

This book has seven chapters. In chapter one we give the basics needed to make this book a self contained one. Chapter two introduces the notion of interval semigroups and interval semifields and are algebraically analysed. Chapter three…

An elliptic Bailey lemma is formulated on the basis of the univariate rarefied elliptic beta integral. It leads to a generalized operator star-triangle relation and a new solution of the Yang-Baxter equation written as an integral operator…

We refine and generalise a Rogers-Ramanujan type partition identity arising from crystal base theory. Our proof uses the variant of the method of weighted words recently introduced by the first author.

The main point of this paper is to present a class of equations over integers that one can check if they have a solution by checking a set of inequalities. The prototype of such equations is the equations appearing in the well-known…

A series of lecture notes on the elementary theory of algebraic numbers, using only knowledge of a first-semester graduate course in algebra (primarily groups and rings). No prerequisite knowledge of fields is required. Based primarily on…

This introduction begins with a section on fundamental notions of mathematical logic, including propositional logic, predicate or first-order logic, completeness, compactness, the L\"owenheim-Skolem theorem, Craig interpolation, Beth's…

This is part one of a series of four methodological papers on (bi)quaternions and their use in theoretical and mathematical physics: 1- Alphabetical bibliography, 2- Analytical bibliography, 3- Notations and terminology, and 4- Formulas and…

This book is a continuation of the book n-linear algebra of type I and its applications. Most of the properties that could not be derived or defined for n-linear algebra of type I is made possible in this new structure: n-linear algebra of…

These are lecture notes for a course I gave in mid-1990s for MSc students at the University of Bath. It presents an algorithm with singly exponential complexity for the existential theory of the reals, in the spirit of J. Renegar. The aim…

Towards the end of his life Ramanujan wrote a manuscript on properties of the partition and tau functions, some parts of which remained unpublished until very recently. Nevertheless, this manuscript gave rise to a lot of subsequent work. In…

We review the accomplishments of nuclear theory during the past five years and identify the future challenges and research opportunities. [Final, substantially revised version of the nuclear theory white paper]

What makes a paper independently reproducible? Debates on reproducibility center around intuition or assumptions but lack empirical results. Our field focuses on releasing code, which is important, but is not sufficient for determining…