Download or read book Generalised Ramsey numbers and Bruhat order on involutions written by Mikael Hansson and published by Linköping University Electronic Press. This book was released on 2015-12-03 with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis consists of two papers within two different areas of combinatorics. Ramsey theory is a classic topic in graph theory, and Paper A deals with two of its most fundamental problems: to compute Ramsey numbers and to characterise critical graphs. More precisely, we study generalised Ramsey numbers for two sets ?1 and ?2 of cycles. We determine, in particular, all generalised Ramsey numbers R(?1, ?2) such that ?1 or ?2 contains a cycle of length at most 6, or the shortest cycle in each set is even. This generalises previous results of Erdös, Faudree, Rosta, Rousseau, and Schelp. Furthermore, we give a conjecture for the general case. We also characterise many (?1, ?2)-critical graphs. As special cases, we obtain complete characterisations of all (Cn,C3)-critical graphs for n ? 5, and all (Cn,C5)-critical graphs for n ? 6. In Paper B, we study the combinatorics of certain partially ordered sets. These posets are unions of conjugacy classes of involutions in the symmetric group Sn, with the order induced by the Bruhat order on Sn. We obtain a complete characterisation of the posets that are graded. In particular, we prove that the set of involutions with exactly one fixed point is graded, which settles a conjecture of Hultman in the affirmative. When the posets are graded, we give their rank functions. We also give a short, new proof of the EL-shellability of the set of fixed-point-free involutions, recently proved by Can, Cherniavsky, and Twelbeck.

Download or read book Combinatorics and topology related to involutions in Coxeter groups written by Mikael Hansson and published by Linköping University Electronic Press. This book was released on 2018-05-21 with total page 62 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation consists of three papers in combinatorial Coxeter group theory. A Coxeter group is a group W generated by a set S, where all relations can be derived from the relations s2 = e for all s ?? S, and (ss?)m(s,s?) = e for some pairs of generators s ? s? in S, where e ?? W is the identity element and m(s, s?) is an integer satisfying that m(s, s?) = m(s?, s) ? 2. Two prominent examples of Coxeter groups are provided by the symmetric group Sn (i.e., the set of permutations of {1, 2, . . . , n}) and finite reflection groups (i.e., finite groups generated by reflections in some real euclidean space). There are also important infinite Coxeter groups, e.g., affine reflection groups. Every Coxeter group can be equipped with various natural partial orders, the most important of which is the Bruhat order. Any subset of a Coxeter group can then be viewed as an induced subposet. In Paper A, we study certain posets of this kind, namely, unions of conjugacy classes of involutions in the symmetric group. We obtain a complete classification of the posets that are pure (i.e., all maximal chains have the same length). In particular, we prove that the set of involutions with exactly one fixed point is pure, which settles a conjecture of Hultman in the affirmative. When the posets are pure, we give their rank functions. We also give a short, new proof of the EL-shellability of the set of fixed-point-free involutions, established by Can, Cherniavsky, and Twelbeck. Paper B also deals with involutions in Coxeter groups. Given an involutive automorphism ? of a Coxeter system (W, S), let ?(?) = {w ?? W | ?(w) = w?1} be the set of twisted involutions. In particular, ?(id) is the set of ordinary involutions in W. It is known that twisted involutions can be represented by words in the alphabet = { | s ?? S}, called -expressions. If ss? has finite order m(s, s?), let a braid move be the replacement of ? ? by ? ? ?, both consisting of m(s, s?) letters. We prove a word property for ?(?), for any Coxeter system (W, S) with any ?. More precisely, we provide a minimal set of moves, easily determined from the Coxeter graph of (W, S), that can be added to the braid moves in order to connect all reduced -expressions for any given w ?? ?(?). This improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types due to Hu, Zhang, Wu, and Marberg. In Paper C, we investigate the topology of (the order complexes of) certain posets, called pircons. A special partial matching (SPM) on a poset is a matching of the Hasse diagram satisfying certain extra conditions. An SPM without fixed points is precisely a special matching as defined by Brenti. Let a pircon be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Marietti’s zircons. Our main result is that every open interval in a pircon is a PL ball or a PL sphere. An important subset of ?(?) is the set ??(?) = {?(w?1)w | w ?? W} of twisted identities. We prove that if ? does not flip any edges with odd labels in the Coxeter graph, then ??(?), with the order induced by the Bruhat order on W, is a pircon. Hence, its open intervals are PL balls or spheres, which confirms a conjecture of Hultman. It is also demonstrated that Bruhat orders on Rains and Vazirani’s quasiparabolic W-sets (under a boundedness assumption) form pircons. In particular, this applies to all parabolic quotients of Coxeter groups.

Download or read book Expansion in Finite Simple Groups of Lie Type written by Terence Tao and published by American Mathematical Soc.. This book was released on 2015-04-16 with total page 319 pages. Available in PDF, EPUB and Kindle. Book excerpt: Expander graphs are an important tool in theoretical computer science, geometric group theory, probability, and number theory. Furthermore, the techniques used to rigorously establish the expansion property of a graph draw from such diverse areas of mathematics as representation theory, algebraic geometry, and arithmetic combinatorics. This text focuses on the latter topic in the important case of Cayley graphs on finite groups of Lie type, developing tools such as Kazhdan's property (T), quasirandomness, product estimates, escape from subvarieties, and the Balog-Szemerédi-Gowers lemma. Applications to the affine sieve of Bourgain, Gamburd, and Sarnak are also given. The material is largely self-contained, with additional sections on the general theory of expanders, spectral theory, Lie theory, and the Lang-Weil bound, as well as numerous exercises and other optional material.

Download or read book Affine Flag Varieties and Quantum Symmetric Pairs written by Zhaobing Fan and published by American Mathematical Soc.. This book was released on 2020-09-28 with total page 123 pages. Available in PDF, EPUB and Kindle. Book excerpt: The quantum groups of finite and affine type $A$ admit geometric realizations in terms of partial flag varieties of finite and affine type $A$. Recently, the quantum group associated to partial flag varieties of finite type $B/C$ is shown to be a coideal subalgebra of the quantum group of finite type $A$.

Download or read book Aperiodic Order Volume 2 Crystallography and Almost Periodicity written by Michael Baake and published by Cambridge University Press. This book was released on 2017-11-02 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quasicrystals are non-periodic solids that were discovered in 1982 by Dan Shechtman, Nobel Prize Laureate in Chemistry 2011. The mathematics that underlies this discovery or that proceeded from it, known as the theory of Aperiodic Order, is the subject of this comprehensive multi-volume series. This second volume begins to develop the theory in more depth. A collection of leading experts, among them Robert V. Moody, cover various aspects of crystallography, generalising appropriately from the classical case to the setting of aperiodically ordered structures. A strong focus is placed upon almost periodicity, a central concept of crystallography that captures the coherent repetition of local motifs or patterns, and its close links to Fourier analysis. The book opens with a foreword by Jeffrey C. Lagarias on the wider mathematical perspective and closes with an epilogue on the emergence of quasicrystals, written by Peter Kramer, one of the founders of the field.

Download or read book Sheaves on Graphs Their Homological Invariants and a Proof of the Hanna Neumann Conjecture written by Joel Friedman and published by American Mathematical Soc.. This book was released on 2014-12-20 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper the author establishes some foundations regarding sheaves of vector spaces on graphs and their invariants, such as homology groups and their limits. He then uses these ideas to prove the Hanna Neumann Conjecture of the 1950s; in fact, he proves a strengthened form of the conjecture.

Download or read book Encyclopedia of Distances written by Michel Marie Deza and published by Springer. This book was released on 2014-10-08 with total page 731 pages. Available in PDF, EPUB and Kindle. Book excerpt: This updated and revised third edition of the leading reference volume on distance metrics includes new items from very active research areas in the use of distances and metrics such as geometry, graph theory, probability theory and analysis. Among the new topics included are, for example, polyhedral metric space, nearness matrix problems, distances between belief assignments, distance-related animal settings, diamond-cutting distances, natural units of length, Heidegger’s de-severance distance, and brain distances. The publication of this volume coincides with intensifying research efforts into metric spaces and especially distance design for applications. Accurate metrics have become a crucial goal in computational biology, image analysis, speech recognition and information retrieval. Leaving aside the practical questions that arise during the selection of a ‘good’ distance function, this work focuses on providing the research community with an invaluable comprehensive listing of the main available distances. As well as providing standalone introductions and definitions, the encyclopedia facilitates swift cross-referencing with easily navigable bold-faced textual links to core entries. In addition to distances themselves, the authors have collated numerous fascinating curiosities in their Who’s Who of metrics, including distance-related notions and paradigms that enable applied mathematicians in other sectors to deploy research tools that non-specialists justly view as arcane. In expanding access to these techniques, and in many cases enriching the context of distances themselves, this peerless volume is certain to stimulate fresh research.

Download or read book The Calculus of Variations written by Bruce van Brunt and published by Springer Science & Business Media. This book was released on 2006-04-18 with total page 295 pages. Available in PDF, EPUB and Kindle. Book excerpt: Suitable for advanced undergraduate and graduate students of mathematics, physics, or engineering, this introduction to the calculus of variations focuses on variational problems involving one independent variable. It also discusses more advanced topics such as the inverse problem, eigenvalue problems, and Noether’s theorem. The text includes numerous examples along with problems to help students consolidate the material.

Download or read book Ergodic Theory and Fractal Geometry written by Hillel Furstenberg and published by American Mathematical Society. This book was released on 2014-08-08 with total page 82 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fractal geometry represents a radical departure from classical geometry, which focuses on smooth objects that "straighten out" under magnification. Fractals, which take their name from the shape of fractured objects, can be characterized as retaining their lack of smoothness under magnification. The properties of fractals come to light under repeated magnification, which we refer to informally as "zooming in". This zooming-in process has its parallels in dynamics, and the varying "scenery" corresponds to the evolution of dynamical variables. The present monograph focuses on applications of one branch of dynamics--ergodic theory--to the geometry of fractals. Much attention is given to the all-important notion of fractal dimension, which is shown to be intimately related to the study of ergodic averages. It has been long known that dynamical systems serve as a rich source of fractal examples. The primary goal in this monograph is to demonstrate how the minute structure of fractals is unfolded when seen in the light of related dynamics. A co-publication of the AMS and CBMS.

Download or read book Proceedings Of The International Congress Of Mathematicians 2018 Icm 2018 In 4 Volumes written by Boyan Sirakov and published by World Scientific. This book was released on 2019-02-27 with total page 5393 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.

Download or read book Dictionary of Distances written by Michel-Marie Deza and published by Elsevier. This book was released on 2006-11-16 with total page 413 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book comes out of need and urgency (expressed especially in areas of Information Retrieval with respect to Image, Audio, Internet and Biology) to have a working tool to compare data. The book will provide powerful resource for all researchers using Mathematics as well as for mathematicians themselves. In the time when over-specialization and terminology fences isolate researchers, this Dictionary try to be "centripedal" and "oikoumeni", providing some access and altitude of vision but without taking the route of scientific vulgarisation. This attempted balance is the main philosophy of this Dictionary which defined its structure and style. Key features: - Unicity: it is the first book treating the basic notion of Distance in whole generality. - Interdisciplinarity: this Dictionary is larger in scope than majority of thematic dictionaries. - Encyclopedicity: while an Encyclopedia of Distances seems now too difficult to produce, this book (by its scope, short introductions and organization) provides the main material for it and for future tutorials on some parts of this material. - Applicability: the distances, as well as distance-related notions and paradigms, are provided in ready-to-use fashion. - Worthiness: the need and urgency for such dictionary was great in several huge areas, esp. Information Retrieval, Image Analysis, Speech Recognition and Biology. - Accessibility: the definitions are easy to locate by subject or, in Index, by alphabetic order; the introductions and definitions are reader-friendly and maximally independent one from another; still the text is structured, in the 3D HTML style, by hyperlink-like boldfaced references to similar definitions. * Covers a large range of subjects in pure and applied mathematics * Designed to be easily applied--the distances and distance-related notions and paradigms are ready to use * Helps users quickly locate definitions by subject or in alphabetical order; stand-alone entries include references to other entries and sources for further investigation

Download or read book Nonstandard Finite Difference Schemes Methodology And Applications written by Ronald E Mickens and published by World Scientific. This book was released on 2020-11-11 with total page 332 pages. Available in PDF, EPUB and Kindle. Book excerpt: This second edition of Nonstandard Finite Difference Models of Differential Equations provides an update on the progress made in both the theory and application of the NSFD methodology during the past two and a half decades. In addition to discussing details related to the determination of the denominator functions and the nonlocal discrete representations of functions of dependent variables, we include many examples illustrating just how this should be done.Of real value to the reader is the inclusion of a chapter listing many exact difference schemes, and a chapter giving NSFD schemes from the research literature. The book emphasizes the critical roles played by the 'principle of dynamic consistency' and the use of sub-equations for the construction of valid NSFD discretizations of differential equations.

Download or read book The Abel Prize 2008 2012 written by Helge Holden and published by Springer. This book was released on 2017-04-30 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covering the years 2008-2012, this book profiles the life and work of recent winners of the Abel Prize: · John G. Thompson and Jacques Tits, 2008 · Mikhail Gromov, 2009 · John T. Tate Jr., 2010 · John W. Milnor, 2011 · Endre Szemerédi, 2012. The profiles feature autobiographical information as well as a description of each mathematician's work. In addition, each profile contains a complete bibliography, a curriculum vitae, as well as photos — old and new. As an added feature, interviews with the Laureates are presented on an accompanying web site (http://extras.springer.com/). The book also presents a history of the Abel Prize written by the historian Kim Helsvig, and includes a facsimile of a letter from Niels Henrik Abel, which is transcribed, translated into English, and placed into historical perspective by Christian Skau. This book follows on The Abel Prize: 2003-2007, The First Five Years (Springer, 2010), which profiles the work of the first Abel Prize winners.

Download or read book Finite Fields and Applications written by Gary L. Mullen and published by American Mathematical Soc.. This book was released on 2007 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: Finite fields Combinatorics Algebraic coding theory Cryptography Background in number theory and abstract algebra Hints for selected exercises References Index.

Download or read book Papers in Honour of Bernhard Banaschewski written by Guillaume Brümmer and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 496 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proceedings of the BB Fest 96, a conference held at the University of Cape Town, 15-20 July 1996, on Category Theory and its Applications to Topology, Order and Algebra

Download or read book Infinite Group Theory From The Past To The Future written by Paul Baginski and published by World Scientific. This book was released on 2017-12-26 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: The development of algebraic geometry over groups, geometric group theory and group-based cryptography, has led to there being a tremendous recent interest in infinite group theory. This volume presents a good collection of papers detailing areas of current interest.

Download or read book Non Obstetric Surgery During Pregnancy written by Ceana H. Nezhat and published by Springer. This book was released on 2018-10-29 with total page 446 pages. Available in PDF, EPUB and Kindle. Book excerpt: As minimally invasive surgical techniques evolve, the outlook for patients continues to brighten. This is also true for pregnant women experiencing conditions requiring surgical intervention during gestation. A physician’s ability to identify potential risk factors in pregnant patients that present pre-op directly corresponds with their success in monitoring patients in post-op for adverse obstetric outcomes from non-obstetric surgeries. The utilization of laparoscopic techniques during surgery for pregnant patients has decreased the risk of fetal complications and increased the positive outlook for the duration of the pregnancy. This text is designed to present a minimally invasive approach to surgery that is non-obstetrics related and to educate readers on the potential risk factors and negative outcomes on pregnant patients following non-obstetric surgery. The various sections of this book will address issues faced by surgeons who undertake the task of operating on the pregnant patient. When performing non-obstetric surgery on a pregnant patient the ability to identify possible risk factors for adverse obstetric outcomes is important. Risk factors can be maternal-, surgical-, obstetrical-, or disease-associated. Common adverse obstetric outcomes following non-obstetric surgery include preterm delivery, preterm labor without preterm delivery, and miscarriage. Fetal well-being is an additional risk-benefit factor to be considered when weighing options before conducting non-obstetric surgeries on pregnant patients. This book will provide physicians with the knowledge and tools to identify common risk factors and successfully apply evidence-based risk reduction. To date, no comprehensive resource is available for non-obstetric surgery during pregnancy. Types of pregnancies, potential complications, proper use of anesthesia, types of risk factors (maternal-, surgical-, obstetrical-, and disease-associated), etc. will be thoroughly explained and depicted within these pages. All chapters will be written by subject matter experts in their fields. All information communicated will be comprised of the most currently available knowledge