The aim of this text is to introduce the beginner to the theory of Riordan arrays. Starting in a simple and constructive manner, the basic structure of these arrays is explained with clear examples, before a more theoretical grounding is provided. Ordinary Riordan arrays and exponential Riordan arrays are examined, with many explicit examples, and their applications to combinatorics and other areas are explored. The production matrix of a Riordan array is shown to play a key role, along with various sequence characterizations. Formal prerequisites are kept to a minimum, in order to provide a gentle introduction to this exciting area, that involves linear algebra, group theory and combinatorics. The reader will be well positioned to further explore Riordan arrays and their applications, and to undertake their own projects. They will join a community of interested mathematicians that now spans all continents, in a growing area of research and application.

The ever-growing applications and richness of approaches to the Riordan group is captured in this comprehensive monograph, authored by those who are among the founders and foremost world experts in this field. The concept of a Riordan array has played a unifying role in enumerative combinatorics over the last three decades. The Riordan arrays and Riordan group is a new growth point in mathematics that is both being influenced by, and continuing its contributions to, other fields such as Lie groups, elliptic curves, orthogonal polynomials, spline functions, networks, sequences and series, Beal conjecture, Riemann hypothesis, to name several. In recent years the Riordan group has made links to quantum field theory and has become a useful tool for computer science and computational chemistry. We can look forward to discovering further applications to unexpected areas of research. Providing a baseline and springboard to further developments and study, this book may also serve as a text for anyone interested in discrete mathematics, including combinatorics, number theory, matrix theory, graph theory, and algebra.

This proceedings volume gathers selected, revised papers presented at the 51st Southeastern International Conference on Combinatorics, Graph Theory and Computing (SEICCGTC 2020), held at Florida Atlantic University in Boca Raton, USA, on March 9-13, 2020. The SEICCGTC is broadly considered to be a trendsetter for other conferences around the world – many of the ideas and themes first discussed at it have subsequently been explored at other conferences and symposia. The conference has been held annually since 1970, in Baton Rouge, Louisiana and Boca Raton, Florida. Over the years, it has grown to become the major annual conference in its fields, and plays a major role in disseminating results and in fostering collaborative work. This volume is intended for the community of pure and applied mathematicians, in academia, industry and government, working in combinatorics and graph theory, as well as related areas of computer science and the interactions among these fields.

This book presents methods for the summation of infinite and finite series and the related identities and inversion relations. The summation includes the column sums and row sums of lower triangular matrices. The convergence of the summation of infinite series is considered. The author’s focus is on symbolic methods and the Riordan array approach. In addition, this book contains hundreds summation formulas and identities, which can be used as a handbook for people working in computer science, applied mathematics, and computational mathematics, particularly, combinatorics, computational discrete mathematics, and computational number theory. The exercises at the end of each chapter help deepen understanding. Much of the materials in this book has never appeared before in textbook form. This book can be used as a suitable textbook for advanced courses for high lever undergraduate and lower lever graduate students. It is also an introductory self-study book for re- searchers interested in this field, while some materials of the book can be used as a portal for further research.

Combinatorics and Number Theory of Counting Sequences is an introduction to the theory of finite set partitions and to the enumeration of cycle decompositions of permutations. The presentation prioritizes elementary enumerative proofs. Therefore, parts of the book are designed so that even those high school students and teachers who are interested in combinatorics can have the benefit of them. Still, the book collects vast, up-to-date information for many counting sequences (especially, related to set partitions and permutations), so it is a must-have piece for those mathematicians who do research on enumerative combinatorics. In addition, the book contains number theoretical results on counting sequences of set partitions and permutations, so number theorists who would like to see nice applications of their area of interest in combinatorics will enjoy the book, too. Features The Outlook sections at the end of each chapter guide the reader towards topics not covered in the book, and many of the Outlook items point towards new research problems. An extensive bibliography and tables at the end make the book usable as a standard reference. Citations to results which were scattered in the literature now become easy, because huge parts of the book (especially in parts II and III) appear in book form for the first time.

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This text provides a comprehensive overview of the clinical and basic science aspects of MS. It is designed to be of practical use to clinical neurologists, and addresses all of the major issues that may occur in the management of persons with MS.

Polynomials are useful mathematical tools. They are simply defined and can be calculated quickly on computer systems. They can be differentiated and integrated easily and can be pieced together to form spline curves. After Weierstrass approximation Theorem, polynomial sequences have acquired considerable importance not only in the various branches of Mathematics, but also in Physics, Chemistry and Engineering disciplines. There is a wide literature on specific polynomial sequences. But there is no literature that attempts a systematic exposition of the main basic methods for the study of a generic polynomial sequence and, at the same time, gives an overview of the main polynomial classes and related applications, at least in numerical analysis. In this book, through an elementary matrix calculus-based approach, an attempt is made to fill this gap by exposing dated and very recent results, both theoretical and applied.

The Columbia River Treaty: A Primer is a vital work that clearly explains the nature of this complex water agreement between Canada and the United States and how its impending update will impact communities, landscapes, industry and water supplies between the two countries for many years to come. The Columbia River Treaty ratification in 1964 created the largest hydropower project in North America, with additional emphasis on flood protection for the USA. As the Treaty approaches its 60th anniversary, and the first opportunity for modification, its signatories are preparing proposals for new ways forward, and stakeholders on both sides of the border are speaking up. This primer explores the initial intent of the Treaty and its success to date, its costs to Columbia Basin residents and ecosystems, and new influences the signatories must now consider. Shifts in social norms related to the environment, equity and social justice, new views on the relevance of Indigenous traditional and local knowledge, and the economic and physical effects of a changing climate—are all considered as factors in future Treaty governance. The primer concludes with a summary of the perspectives that currently exist between and within each country with respect to Treaty benefits and outlines the next steps that will take place in the negotiation process. The authors conclude with a call to action, in the hope that a renewed Columbia River Treaty might prove a model for outstanding transboundary water agreements around the world as they strive to meet not only the challenges of the present day but also the needs of future generations.

Primer on Multiple Sclerosis was developed to provide a comprehensive overview of the clinical and basic science aspects of MS. It is designed to be of practical use to clinical neurologists, and addresses all of the major issues that may occur in the management of persons with MS. The reader is provided with the latest information on the science of MS, including immunology, genetics, epidemiology and pathology, as well as a summary of the newest directions in basic science research. Guidelines for diagnosis and appropriate use of diagnostic modalities are presented. All clinical aspects of MS are discussed, including extensive information on aspects that may be more challenging for the neurologist to manage, such as sleep disorders and pain. Disease modifying therapies, including those that are FDA approved, as well as off label and experimental therapies are discussed. Finally, there are chapters on employment and legal issues, as well as an overview of clinical trials and clinical trial outcome measures, which are helpful in reviewing the scientific literature in these areas.