Download or read book Hopf Monoids and Generalized Permutahedra written by Marcelo Aguiar and published by American Mathematical Society. This book was released on 2023-09-27 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Download or read book Hopf Algebras and Tensor Categories written by Nicolás Andruskiewitsch and published by American Mathematical Soc.. This book was released on 2013-02-21 with total page 347 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the Conference on Hopf Algebras and Tensor Categories, held July 4-8, 2011, at the University of Almeria, Almeria, Spain. The articles in this volume cover a wide variety of topics related to the theory of Hopf algebras and its connections to other areas of mathematics. In particular, this volume contains a survey covering aspects of the classification of fusion categories using Morita equivalence methods, a long comprehensive introduction to Hopf algebras in the category of species, and a summary of the status to date of the classification of Hopf algebras of dimensions up to 100. Among other topics discussed in this volume are a study of normalized class sum and generalized character table for semisimple Hopf algebras, a contribution to the classification program of finite dimensional pointed Hopf algebras, relations to the conjecture of De Concini, Kac, and Procesi on representations of quantum groups at roots of unity, a categorical approach to the Drinfeld double of a braided Hopf algebra via Hopf monads, an overview of Hom-Hopf algebras, and several discussions on the crossed product construction in different settings.

Download or read book Bimonoids for Hyperplane Arrangements written by Marcelo Aguiar and published by Cambridge University Press. This book was released on 2020-03-19 with total page 854 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of this monograph is to develop Hopf theory in a new setting which features centrally a real hyperplane arrangement. The new theory is parallel to the classical theory of connected Hopf algebras, and relates to it when specialized to the braid arrangement. Joyal's theory of combinatorial species, ideas from Tits' theory of buildings, and Rota's work on incidence algebras inspire and find a common expression in this theory. The authors introduce notions of monoid, comonoid, bimonoid, and Lie monoid relative to a fixed hyperplane arrangement. They also construct universal bimonoids by using generalizations of the classical notions of shuffle and quasishuffle, and establish the Borel–Hopf, Poincaré–Birkhoff–Witt, and Cartier–Milnor–Moore theorems in this setting. This monograph opens a vast new area of research. It will be of interest to students and researchers working in the areas of hyperplane arrangements, semigroup theory, Hopf algebras, algebraic Lie theory, operads, and category theory.

Download or read book Discrete Mathematics Days 2022 written by Luis Felipe Tabera Alonso and published by Ed. Universidad de Cantabria. This book was released on 2022-07-03 with total page 335 pages. Available in PDF, EPUB and Kindle. Book excerpt: El congreso Discrete Mathematics Days (DMD20/22) tendrá lugar del 4 al 6 de julio de 2022, en la Facultad de Ciencias de la Universidad de Cantabria (Santander, España). Este congreso internacional se centra en avances dentro del campo de la Matemática discreta, incluyendo, de manera no exhaustiva: · Algoritmos y Complejidad · Combinatoria · Teoría de Códigos · Criptografía · Geometría Discreta y Computacional · Optimización Discreta · Teoría de Grafos · Problemas de localización discreta y temas relacionados Las ediciones anteriores de este evento se celebraros en Sevilla (2018) y Barcelona (2016), estos congresos heredan la tradición de las Jornadas de Matemática Discreta y Algorítmica (JMDA), el encuentro bienal en España en Matemática Discreta (desde 1998). Durante la celebración del congreso tendrán lugar cuatro conferencias plenarias, cuarenta y dos presentaciones orales y una sesión de once pósteres. Abstract The Discrete Mathematics Days (DMD20/22) will be held on July 4-6, 2022, at Facultad de Ciencias of the Universidad de Cantabria (Santander, Spain). The main focus of this international conference is on current topics in Discrete Mathematics, including (but not limited to): Algorithms and Complexity Combinatorics Coding Theory Cryptography Discrete and Computational Geometry Discrete Optimization Graph Theory Location and Related Problems The previous editions were held in Sevilla in 2018 and in Barcelona in 2016, inheriting the tradition of the Jornadas de Matemática Discreta y Algorítmica (JMDA), the Spanish biennial meeting (since 1998) on Discrete Mathematics. The program consists on four plenary talks, 42 contributed talks and a poster session with 11 contributions.

Download or read book Species and Hyperplane Arrangements written by Jose Dario Bastidas Olaya and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation has two leading characters : Hopf monoids in the category of species and the Tits algebra of a real hyperplane arrangement. The relation between these two comes from the work of Aguiar and Mahajan (2013), who showed that a (co)commutative Hopf monoid gives rise to a family of (left)right-modules over the Tits algebra of the braid arrangement in all dimensions. One goal of this thesis is to explore the representation theory of the Tits algebra of arbitrary affine arrangements to extend what is known in the case of linear arrangements and to give an insight into some unanswered questions in the field of Hopf monoids. In the first part, we extend the study of characteristic elements of a hyperplane arrangement from the linear to the affine case. We present the basic properties of these elements and apply them to derive numerous results about the characteristic polynomial of an arrangement, from Zaslavsky's formulas to more recent results of Kung and of Klivans and Swartz. We construct several examples of characteristic elements, including one in terms of intrinsic volumes of faces of the arrangement. In the second part, we study deformations $\arr$ of a linear arrangement $\arr_0$ and endow the Tits algebra of $\arr$ with a bimodule structure over the algebra of $\arr_0$. The left module structure sheds some light on the study of exponential sequences of arrangements, in the sense of Stanley. In particular, we construct the Hopf monoid of faces associated with such a sequence and use characteristic elements to deduce formulas for certain bivariate polynomial invariants of these arrangements. In the third part, we endow the polytope subalgebra of deformations of a zonotope with the structure of a module over the Tits algebra of the corresponding hyperplane arrangement. We study algebraic invariants of this module and find relations between statistics on (signed) permutations and the module structure in the case of (type B) generalized permutahedra. In type B, the module structure surprisingly reveals that any family of generators (via signed Minkowski sums) for generalized permutahedra of type B will contain at least

Download or read book Roads to Higher Dimensional Polytopic Projects written by Octavian Iordache and published by Springer Nature. This book was released on 2022-08-18 with total page 221 pages. Available in PDF, EPUB and Kindle. Book excerpt: High dimensional reference architectures presented here allows confronting and prevailing over the growing complexity of polytopic projects implementations. Such projects should be envisaged giving that conventional systems operations, equipments, methodologies or organizations will reach their limits for self-evolvability in high complexity conditions. Self-evolvable high complexity systems are based on high dimensional polytopic reference architectures. Polytope is the general term of the sequence: point, line, polygon, polyhedron and so on.The polytopic projects are targeting the artificiality, not only for materials where it is well known and applied, but also for biological, cognitive, intelligent and mathematical systems. The book highlights the polytopic projects basic similarity despite the noticeable difference as domains of application. The roads to follow and the algebra of changing roads are emphasized. The book is divided in 9 chapters. Chapter 1 introduces the Polytopic Roadmap to 4D and beyond. The role for the dialogue of processes in duality of the non-Aristotelian Logic of Contradiction and of Included Middle is emphasized for different domains. Chapter 2 refers to chemical systems. Supramolecular chemistry, metal organic frameworks, MOF, and reaction networks, are the examples considered in the frame of polytopic chemistry. Chapter 3 refers to biological systems. Biological dynamical hierarchies and quasi-species are the considered case studies. Technological and scientific projects targeting artificiality for cells and viruses are considered. Chapter 4 refers to cognitive systems. Developmental stages, formal and relational concepts analysis, and neural coding are considered here. The roles of the 4D systems of systems of systems and of conceptual 4D-cube are emphasized. Artificiality for cognitive systems is the object of study. Chapter 5 refers to mathematical systems. Modeling levels and the 4D digital twins are discussed. Hopf monoids as tools for the study of combinations and separations, dual graded graphs and V-models are informally presented. Chapter 6 refers to application of formal concept analysis, FCA, for high dimensional separations, nesting and drug delivery. Chapter 7 refers to polytopic engineering systems as multiscale transfer, distributors-collectors, cyclic operations, middle vessel columns, mixing, assembly and designs. Equipments have been characterized using Polytopic Roadmaps and classified by Periodic Tables. Chapter 8 introduces polytopic industry, economy, society and sustainability. Chapter 9 outlines new domains of interest as arts and architecture, transdisciplinarity, complex systems and unity of sciences and engineering. Polytopic Roadmaps are proposed as Method for experts from various fields to synthesize their thinking and capabilities into new projects implementation to face and surpass high complexity. A repetitive finding of this book is that self-evolvability observed in physical systems is based on the same directed sequence of reference architectures as the self-evolvability of concepts in our mind. Continuing to develop the field of self-evolvable systems and presenting the polytopic roadmaps for 4D and beyond advances in ever growing complexity domains, the book will be useful to engineers, researchers, entrepreneurs and students in different branches of production, complex systems sciences and engineering, ecology and applied mathematics.

Download or read book Handbook of the Tutte Polynomial and Related Topics written by Joanna A. Ellis-Monaghan and published by CRC Press. This book was released on 2022-07-06 with total page 743 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Tutte Polynomial touches on nearly every area of combinatorics as well as many other fields, including statistical mechanics, coding theory, and DNA sequencing. It is one of the most studied graph polynomials. Handbook of the Tutte Polynomial and Related Topics is the first handbook published on the Tutte Polynomial. It consists of thirty-four chapters written by experts in the field, which collectively offer a concise overview of the polynomial’s many properties and applications. Each chapter covers a different aspect of the Tutte polynomial and contains the central results and references for its topic. The chapters are organized into six parts. Part I describes the fundamental properties of the Tutte polynomial, providing an overview of the Tutte polynomial and the necessary background for the rest of the handbook. Part II is concerned with questions of computation, complexity, and approximation for the Tutte polynomial; Part III covers a selection of related graph polynomials; Part IV discusses a range of applications of the Tutte polynomial to mathematics, physics, and biology; Part V includes various extensions and generalizations of the Tutte polynomial; and Part VI provides a history of the development of the Tutte polynomial. Features Written in an accessible style for non-experts, yet extensive enough for experts Serves as a comprehensive and accessible introduction to the theory of graph polynomials for researchers in mathematics, physics, and computer science Provides an extensive reference volume for the evaluations, theorems, and properties of the Tutte polynomial and related graph, matroid, and knot invariants Offers broad coverage, touching on the wide range of applications of the Tutte polynomial and its various specializations

Download or read book The Space of Spaces Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces written by Karl-Theodor Sturm and published by American Mathematical Society. This book was released on 2023-11-27 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Download or read book Sur un Probl me de Compatibilit Local Global Localement Analytique written by Christophe Breuil and published by American Mathematical Society. This book was released on 2023-11-27 with total page 166 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Download or read book Potential Estimates and Quasilinear Parabolic Equations with Measure Data written by Quoc-Hung Nguyen and published by American Mathematical Society. This book was released on 2024-01-19 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Download or read book On the Boundary Behavior of Mass Minimizing Integral Currents written by Camillo De Lellis and published by American Mathematical Society. This book was released on 2024-01-17 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Download or read book Total Positivity is a Quantum Phenomenon The Grassmannian Case written by S. Launois and published by American Mathematical Society. This book was released on 2024-01-19 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Download or read book Global Existence of Small Amplitude Solutions for a Model Quadratic Quasilinear Coupled Wave Klein Gordon System in Two Space Dimension with Mildly Decaying Cauchy Data written by A. Stingo and published by American Mathematical Society. This book was released on 2023-11-27 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Download or read book Tate Duality in Positive Dimension over Function Fields written by Zev Rosengarten and published by American Mathematical Society. This book was released on 2023-11-27 with total page 230 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Download or read book Local Smoothing Estimates for Schr dinger Equations on Hyperbolic Space written by Andrew Lawrie and published by American Mathematical Society. This book was released on 2024-01-19 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Download or read book Lattice Paths and Branched Continued Fractions An Infinite Sequence of Generalizations of the Stieltjes Rogers and Thron Rogers Polynomials with Coefficientwise Hankel Total Positivity written by Mathias Pétréolle and published by American Mathematical Society. This book was released on 2024-01-19 with total page 166 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Download or read book Monoidal Functors Species and Hopf Algebras written by Marcelo Aguiar and published by American Mathematical Soc.. This book was released on 2010 with total page 784 pages. Available in PDF, EPUB and Kindle. Book excerpt: This research monograph integrates ideas from category theory, algebra and combinatorics. It is organized in three parts. Part I belongs to the realm of category theory. It reviews some of the foundational work of Benabou, Eilenberg, Kelly and Mac Lane on monoidal categories and of Joyal and Street on braided monoidal categories, and proceeds to study higher monoidal categories and higher monoidal functors. Special attention is devoted to the notion of a bilax monoidal functor which plays a central role in this work. Combinatorics and geometry are the theme of Part II. Joyal's species constitute a good framework for the study of algebraic structures associated to combinatorial objects. This part discusses the category of species focusing particularly on the Hopf monoids therein. The notion of a Hopf monoid in species parallels that of a Hopf algebra and reflects the manner in which combinatorial structures compose and decompose. Numerous examples of Hopf monoids are given in the text. These are constructed from combinatorial and geometric data and inspired by ideas of Rota and Tits' theory of Coxeter complexes. Part III is of an algebraic nature and shows how ideas in Parts I and II lead to a unified approach to Hopf algebras. The main step is the construction of Fock functors from species to graded vector spaces. These functors are bilax monoidal and thus translate Hopf monoids in species to graded Hopf algebras. This functorial construction of Hopf algebras encompasses both quantum groups and the Hopf algebras of recent prominence in the combinatorics literature. The monograph opens a vast new area of research. It is written with clarity and sufficient detail to make it accessible to advanced graduate students.