Lefschetz, Solomon
2012-01-01
An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.
Mahé, Louis; Roy, Marie-Françoise
1992-01-01
Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contribu...
Algebra-Geometry of Piecewise Algebraic Varieties
Institute of Scientific and Technical Information of China (English)
Chun Gang ZHU; Ren Hong WANG
2012-01-01
Algebraic variety is the most important subject in classical algebraic geometry.As the zero set of multivariate splines,the piecewise algebraic variety is a kind generalization of the classical algebraic variety.This paper studies the correspondence between spline ideals and piecewise algebraic varieties based on the knowledge of algebraic geometry and multivariate splines.
Intermediate algebra & analytic geometry
Gondin, William R
1967-01-01
Intermediate Algebra & Analytic Geometry Made Simple focuses on the principles, processes, calculations, and methodologies involved in intermediate algebra and analytic geometry. The publication first offers information on linear equations in two unknowns and variables, functions, and graphs. Discussions focus on graphic interpretations, explicit and implicit functions, first quadrant graphs, variables and functions, determinate and indeterminate systems, independent and dependent equations, and defective and redundant systems. The text then examines quadratic equations in one variable, system
Kendig, Keith
2015-01-01
Designed to make learning introductory algebraic geometry as easy as possible, this text is intended for advanced undergraduates and graduate students who have taken a one-year course in algebra and are familiar with complex analysis. This newly updated second edition enhances the original treatment's extensive use of concrete examples and exercises with numerous figures that have been specially redrawn in Adobe Illustrator. An introductory chapter that focuses on examples of curves is followed by a more rigorous and careful look at plane curves. Subsequent chapters explore commutative ring th
Algebra, Arithmetic, and Geometry
Tschinkel, Yuri
2009-01-01
The two volumes of "Algebra, Arithmetic, and Geometry: In Honor of Y.I. Manin" are composed of invited expository articles and extensions detailing Manin's contributions to the subjects, and are in celebration of his 70th birthday. The well-respected and distinguished contributors include: Behrend, Berkovich, Bost, Bressler, Calaque, Carlson, Chambert-Loir, Colombo, Connes, Consani, Dabrowski, Deninger, Dolgachev, Donaldson, Ekedahl, Elsenhans, Enriques, Etingof, Fock, Friedlander, Geemen, Getzler, Goncharov, Harris, Iskovskikh, Jahnel, Kaledin, Kapranov, Katz, Kaufmann, Kollar, Kont
Bochnak, Jacek; Roy, Marie-Françoise
1998-01-01
This book is a systematic treatment of real algebraic geometry, a subject that has strong interrelation with other areas of mathematics: singularity theory, differential topology, quadratic forms, commutative algebra, model theory, complexity theory etc. The careful and clearly written account covers both basic concepts and up-to-date research topics. It may be used as text for a graduate course. The present edition is a substantially revised and expanded English version of the book "Géometrie algébrique réelle" originally published in French, in 1987, as Volume 12 of ERGEBNISSE. Since the publication of the French version the theory has made advances in several directions. Many of these are included in this English version. Thus the English book may be regarded as a completely new treatment of the subject.
Linear algebra and projective geometry
Baer, Reinhold
2005-01-01
Geared toward upper-level undergraduates and graduate students, this text establishes that projective geometry and linear algebra are essentially identical. The supporting evidence consists of theorems offering an algebraic demonstration of certain geometric concepts. These focus on the representation of projective geometries by linear manifolds, of projectivities by semilinear transformations, of collineations by linear transformations, and of dualities by semilinear forms. These theorems lead to a reconstruction of the geometry that constituted the discussion's starting point, within algebra
Principles of algebraic geometry
Griffiths, Phillip A
1994-01-01
A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special top
Linear algebra, geometry and transformation
Solomon, Bruce
2014-01-01
Vectors, Mappings and Linearity Numeric Vectors Functions Mappings and Transformations Linearity The Matrix of a Linear Transformation Solving Linear Systems The Linear SystemThe Augmented Matrix and RRE Form Homogeneous Systems in RRE Form Inhomogeneous Systems in RRE Form The Gauss-Jordan Algorithm Two Mapping Answers Linear Geometry Geometric Vectors Geometric/Numeric Duality Dot-Product Geometry Lines, Planes, and Hyperplanes System Geometry and Row/Column Duality The Algebra of Matrices Matrix Operations Special Matrices Matrix Inversion A Logical Digression The Logic of the Inversion Alg
Moduli spaces in algebraic geometry
International Nuclear Information System (INIS)
This volume of the new series of lecture notes of the Abdus Salam International Centre for Theoretical Physics contains the lecture notes of the School on Algebraic Geometry which took place at the Abdus Salam International Centre for Theoretical Physics from 26 July to 13 August 1999. The school consisted of 2 weeks of lecture courses and one week of conference. The topic of the school was moduli spaces. More specifically the lectures were divided into three subtopics: principal bundles on Riemann surfaces, moduli spaces of vector bundles and sheaves on projective varieties, and moduli spaces of curves
Foliation theory in algebraic geometry
McKernan, James; Pereira, Jorge
2016-01-01
Featuring a blend of original research papers and comprehensive surveys from an international team of leading researchers in the thriving fields of foliation theory, holomorphic foliations, and birational geometry, this book presents the proceedings of the conference "Foliation Theory in Algebraic Geometry," hosted by the Simons Foundation in New York City in September 2013. Topics covered include: Fano and del Pezzo foliations; the cone theorem and rank one foliations; the structure of symmetric differentials on a smooth complex surface and a local structure theorem for closed symmetric differentials of rank two; an overview of lifting symmetric differentials from varieties with canonical singularities and the applications to the classification of AT bundles on singular varieties; an overview of the powerful theory of the variety of minimal rational tangents introduced by Hwang and Mok; recent examples of varieties which are hyperbolic and yet the Green-Griffiths locus is the whole of X; and a classificati...
Commutative algebra with a view toward algebraic geometry
Eisenbud, David
1995-01-01
Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algeb...
Quantum fields, periods and algebraic geometry
Kreimer, Dirk
2014-01-01
We discuss how basic notions of graph theory and associated graph polynomials define questions for algebraic geometry, with an emphasis given to an analysis of the structure of Feynman rules as determined by those graph polynomials as well as algebraic structures of graphs. In particular, we discuss the appearance of renormalization scheme independent periods in quantum field theory.
Primer for the algebraic geometry of sandpiles
Perkinson, David; Perlman, Jacob; Wilmes, John
2011-01-01
The Abelian Sandpile Model (ASM) is a game played on a graph realizing the dynamics implicit in the discrete Laplacian matrix of the graph. The purpose of this primer is to apply the theory of lattice ideals from algebraic geometry to the Laplacian matrix, drawing out connections with the ASM. An extended summary of the ASM and of the required algebraic geometry is provided. New results include a characterization of graphs whose Laplacian lattice ideals are complete intersection ideals; a new...
Algebra and geometry of Hamilton's quaternions
Krishnaswami, Govind S
2016-01-01
Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples, in what he considered his most significant work, generalizing the real and complex number systems. We give a motivated introduction to quaternions and discuss how they are related to Pauli matrices, rotations in three dimensions, the three sphere, the group SU(2) and the celebrated Hopf fibrations.
Cluster algebras and Poisson geometry
Gekhtman, M.; Shapiro, M.; Vainshtein, A.
2002-01-01
We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of connected components of refined open Bruhat cells in Grassmanians G(k,n) over real numbers.
Weaving Geometry and Algebra Together
Cetner, Michelle
2015-01-01
When thinking about student reasoning and sense making, teachers must consider the nature of tasks given to students along with how to plan to use the tasks in the classroom. Students should be presented with tasks in a way that encourages them to draw connections between algebraic and geometric concepts. This article focuses on the idea that it…
Methods of algebraic geometry in control theory
Falb, Peter
1999-01-01
"Control theory represents an attempt to codify, in mathematical terms, the principles and techniques used in the analysis and design of control systems. Algebraic geometry may, in an elementary way, be viewed as the study of the structure and properties of the solutions of systems of algebraic equations. The aim of this book is to provide access to the methods of algebraic geometry for engineers and applied scientists through the motivated context of control theory" .* The development which culminated with this volume began over twenty-five years ago with a series of lectures at the control group of the Lund Institute of Technology in Sweden. I have sought throughout to strive for clarity, often using constructive methods and giving several proofs of a particular result as well as many examples. The first volume dealt with the simplest control systems (i.e., single input, single output linear time-invariant systems) and with the simplest algebraic geometry (i.e., affine algebraic geometry). While this is qui...
Verburgt, Lukas M
2016-01-01
This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s.
Verburgt, Lukas M
2016-01-01
This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s. PMID:26806075
Connecting Functions in Geometry and Algebra
Steketee, Scott; Scher, Daniel
2016-01-01
One goal of a mathematics education is that students make significant connections among different branches of mathematics. Connections--such as those between arithmetic and algebra, between two-dimensional and three-dimensional geometry, between compass-and-straight-edge constructions and transformations, and between calculus and analytic…
Classification of complex simple Lie algebras via projective geometry geometry
Landsberg, J. M.; Manivel, Laurent
1999-01-01
We present a new proof of the classification of complex simple Lie algebras via the projective geometry of homogeneous varieties. Our proof proceeds by constructing homogeneous varieties using the ideals of the secant and tangential varieties of homogeneous varieties already constructed. Our algorithms make no reference to root systems. Our proofs use properties of root systems, but not their classification.
A Relationship between Geometry and Algebra
Bejarano, Jose Ricardo Arteaga
2011-01-01
The three key documents for study geometry are: 1) "The Elements" of Euclid, 2) the lecture by B. Riemann at G\\"ottingen in 1854 entitled "\\"Uber die Hypothesen welche der Geometrie zu Grunde liegen" (On the hypotheses which underlie geometry) and 3) the "Erlangen Program", a document written by F. Klein (1872) on his income as professor at the Faculty of Philosophy and the Senate of the Erlangen University. The latter document F. Klein introduces the concept of group as a tool to study geometry. The concept of a group of transformations of space was known at the time. The purpose of this informative paper is to show a relationship between geometry and algebra through an example, the projective plane. Erlangen program until today continues being a guideline of how to study geometry.
Computational algebraic geometry of epidemic models
Rodríguez Vega, Martín.
2014-06-01
Computational Algebraic Geometry is applied to the analysis of various epidemic models for Schistosomiasis and Dengue, both, for the case without control measures and for the case where control measures are applied. The models were analyzed using the mathematical software Maple. Explicitly the analysis is performed using Groebner basis, Hilbert dimension and Hilbert polynomials. These computational tools are included automatically in Maple. Each of these models is represented by a system of ordinary differential equations, and for each model the basic reproductive number (R0) is calculated. The effects of the control measures are observed by the changes in the algebraic structure of R0, the changes in Groebner basis, the changes in Hilbert dimension, and the changes in Hilbert polynomials. It is hoped that the results obtained in this paper become of importance for designing control measures against the epidemic diseases described. For future researches it is proposed the use of algebraic epidemiology to analyze models for airborne and waterborne diseases.
Lattice Landau Gauge and Algebraic Geometry
Mehta, Dhagash; von Smekal, Lorenz; Williams, Anthony G
2009-01-01
Finding the global minimum of a multivariate function efficiently is a fundamental yet difficult problem in many branches of theoretical physics and chemistry. However, we observe that there are many physical systems for which the extremizing equations have polynomial-like non-linearity. This allows the use of Algebraic Geometry techniques to solve these equations completely. The global minimum can then straightforwardly be found by the second derivative test. As a warm-up example, here we study lattice Landau gauge for compact U(1) and propose two methods to solve the corresponding gauge-fixing equations. In a first step, we obtain all Gribov copies on one and two dimensional lattices. For simple 3x3 systems their number can already be of the order of thousands. We anticipate that the computational and numerical algebraic geometry methods employed have far-reaching implications beyond the simple but illustrating examples discussed here.
PREFACE: Algebra, Geometry, and Mathematical Physics 2010
Stolin, A.; Abramov, V.; Fuchs, J.; Paal, E.; Shestopalov, Y.; Silvestrov, S.
2012-02-01
This proceedings volume presents results obtained by the participants of the 6th Baltic-Nordic workshop 'Algebra, Geometry, and Mathematical Physics (AGMP-6)' held at the Sven Lovén Centre for Marine Sciences in Tjärnö, Sweden on October 25-30, 2010. The Baltic-Nordic Network AGMP 'Algebra, Geometry, and Mathematical Physics' http://www.agmp.eu was created in 2005 on the initiative of two Estonian universities and two Swedish universities: Tallinn University of Technology represented by Eugen Paal (coordinator of the network), Tartu University represented by Viktor Abramov, Lund University represented by Sergei Silvestrov, and Chalmers University of Technology and the University of Gothenburg represented by Alexander Stolin. The goal was to promote international and interdisciplinary cooperation between scientists and research groups in the countries of the Baltic-Nordic region in mathematics and mathematical physics, with special emphasis on the important role played by algebra and geometry in modern physics, engineering and technologies. The main activities of the AGMP network consist of a series of regular annual international workshops, conferences and research schools. The AGMP network also constitutes an important educational forum for scientific exchange and dissimilation of research results for PhD students and Postdocs. The network has expanded since its creation, and nowadays its activities extend beyond countries in the Baltic-Nordic region to universities in other European countries and participants from elsewhere in the world. As one of the important research-dissimilation outcomes of its activities, the network has a tradition of producing high-quality research proceedings volumes after network events, publishing them with various international publishers. The PDF also contains the following: List of AGMP workshops and other AGMP activities Main topics discussed at AGMP-6 Review of AGMP-6 proceedings Acknowledgments List of Conference Participants
Scattering Amplitudes via Algebraic Geometry Methods
Søgaard, Mads; Damgaard, Poul Henrik
This thesis describes recent progress in the understanding of the mathematical structure of scattering amplitudes in quantum field theory. The primary purpose is to develop an enhanced analytic framework for computing multiloop scattering amplitudes in generic gauge theories including QCD without Feynman diagrams. The study of multiloop scattering amplitudes is crucial for the new era of precision phenomenology at the Large Hadron Collider (LHC) at CERN. Loop-level scattering amplitudes can be reduced to a basis of linearly independent integrals whose coefficients are extracted from generalized unitarity cuts. We take advantage of principles from algebraic geometry in order to extend the notion of maximal cuts to a large class of two- and three-loop integrals. This allows us to derive unique and surprisingly compact formulae for the coefficients of the basis integrals. Our results are expressed in terms of certain linear combinations of multivariate residues and elliptic integrals computed from products of ...
Construction and decoding of a class of algebraic geometry codes
DEFF Research Database (Denmark)
Justesen, Jørn; Larsen, Knud J.; Jensen, Helge Elbrønd;
1989-01-01
A class of codes derived from algebraic plane curves is constructed. The concepts and results from algebraic geometry that were used are explained in detail; no further knowledge of algebraic geometry is needed. Parameters, generator and parity-check matrices are given. The main result is a decod...... is a decoding algorithm which turns out to be a generalization of the Peterson algorithm for decoding BCH decoder codes......A class of codes derived from algebraic plane curves is constructed. The concepts and results from algebraic geometry that were used are explained in detail; no further knowledge of algebraic geometry is needed. Parameters, generator and parity-check matrices are given. The main result...
From geometry to algebra: the Euclidean way with technology
Ferrarello, Daniela; Flavia Mammana, Maria; Pennisi, Mario
2016-05-01
In this paper, we present the results of an experimental classroom activity, history-based with a phylogenetic approach, to achieve algebra properties through geometry. In particular, we used Euclidean propositions, processed them by a dynamic geometry system and translate them into algebraic special products.
Multiplier ideal sheaves in complex and algebraic geometry
Institute of Scientific and Technical Information of China (English)
Yum-Tong; Siu
2005-01-01
The application of the method of multiplier ideal sheaves to effective problems in algebraic geometry is briefly discussed. Then its application to the deformational invariance of plurigenera for general compact algebraic manifolds is presented and discussed.Finally its application to the conjecture of the finite generation of the canonical ring is explored, and the use of complex algebraic geometry in complex Neumann estimates is discussed.
Vanishing theorems and effective results in algebraic geometry
International Nuclear Information System (INIS)
The School on Vanishing Theorems and Effective Results in Algebraic Geometry took place in ICTP, Trieste from 25 April 2000 to 12 May 2000. It was organized by J. P. Demailly (Universite de Grenoble I) and R. Lazarsfeld (University of Michigan). The main topics considered were vanishing theorems, multiplyer ideal sheaves and effective results in algebraic geometry, tight closure, geometry of higher dimensional projective and Kahler manifolds, hyperbolic algebraic varieties. The school consisted of two weeks of lectures and one week of conference. This volume contains the lecture notes of most of the lectures in the first two weeks
Integrable systems in the realm of algebraic geometry
Vanhaecke, Pol
2001-01-01
This book treats the general theory of Poisson structures and integrable systems on affine varieties in a systematic way. Special attention is drawn to algebraic completely integrable systems. Several integrable systems are constructed and studied in detail and a few applications of integrable systems to algebraic geometry are worked out. In the second edition some of the concepts in Poisson geometry are clarified by introducting Poisson cohomology; the Mumford systems are constructed from the algebra of pseudo-differential operators, which clarifies their origin; a new explanation of the multi Hamiltonian structure of the Mumford systems is given by using the loop algebra of sl(2); and finally Goedesic flow on SO(4) is added to illustrate the linearizatin algorith and to give another application of integrable systems to algebraic geometry.
Quantum groups and algebraic geometry in conformal field theory
International Nuclear Information System (INIS)
The classification of two-dimensional conformal field theories is described with algebraic geometry and group theory. This classification is necessary in a consistent formulation of a string theory. (author). 130 refs.; 4 figs.; schemes
Performance Analysis of a Decoding Algorithm for Algebraic Geometry Codes
DEFF Research Database (Denmark)
Jensen, Helge Elbrønd; Nielsen, Rasmus Refslund; Høholdt, Tom
1998-01-01
We analyse the known decoding algorithms for algebraic geometry codes in the case where the number of errors is greater than or equal to [(dFR-1)/2]+1, where dFR is the Feng-Rao distance......We analyse the known decoding algorithms for algebraic geometry codes in the case where the number of errors is greater than or equal to [(dFR-1)/2]+1, where dFR is the Feng-Rao distance...
Recent results in the decoding of Algebraic geometry codes
DEFF Research Database (Denmark)
Høholdt, Tom; Jensen, Helge Elbrønd; Nielsen, Rasmus Refslund
1998-01-01
We analyse the known decoding algorithms for algebraic geometry codes in the case where the number of errors is [(dFR-1)/2]+1, where dFR is the Feng-Rao distance......We analyse the known decoding algorithms for algebraic geometry codes in the case where the number of errors is [(dFR-1)/2]+1, where dFR is the Feng-Rao distance...
From combinatorial optimization to real algebraic geometry and back
Directory of Open Access Journals (Sweden)
Janez Povh
2014-12-01
Full Text Available In this paper, we explain the relations between combinatorial optimization and real algebraic geometry with a special focus to the quadratic assignment problem. We demonstrate how to write a quadratic optimization problem over discrete feasible set as a linear optimization problem over the cone of completely positive matrices. The latter formulation enables a hierarchy of approximations which rely on results from polynomial optimization, a sub-eld of real algebraic geometry.
The geometry of supersymmetric coset models and superconformal algebras
Papadopoulos, G
1993-01-01
An on-shell formulation of (p,q), 2\\leq p \\leq 4, 0\\leq q\\leq 4, supersymmetric coset models with target space the group G and gauge group a subgroup H of G is given. It is shown that there is a correspondence between the number of supersymmetries of a coset model and the geometry of the coset space G/H. The algebras of currents of supersymmetric coset models are superconformal algebras. In particular, the algebras of currents of (2,2) and (4,0) supersymmetric coset models are related to the N=2 Kazama-Suzuki and N=4 Van Proeyen superconformal algebras correspondingly.
Experimental and Theoretical Methods in Algebra, Geometry and Topology
Veys, Willem; Bridging Algebra, Geometry, and Topology
2014-01-01
Algebra, geometry and topology cover a variety of different, but intimately related research fields in modern mathematics. This book focuses on specific aspects of this interaction. The present volume contains refereed papers which were presented at the International Conference “Experimental and Theoretical Methods in Algebra, Geometry and Topology”, held in Eforie Nord (near Constanta), Romania, during 20-25 June 2013. The conference was devoted to the 60th anniversary of the distinguished Romanian mathematicians Alexandru Dimca and Ştefan Papadima. The selected papers consist of original research work and a survey paper. They are intended for a large audience, including researchers and graduate students interested in algebraic geometry, combinatorics, topology, hyperplane arrangements and commutative algebra. The papers are written by well-known experts from different fields of mathematics, affiliated to universities from all over the word, they cover a broad range of topics and explore the research f...
Cox, David A; O'Shea, Donal
2015-01-01
This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry—the elimination theorem, the extension theorem, the closure theorem, and the Nullstellensatz—this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new chapter (ten), which presents some of the essentials of progress made over the last decades in computing Gröbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D). The book may serve as a first or second course in undergraduate abstract algebra and, with some supplementation perhaps, for beginning graduate level courses in algebraic geom...
Geometry, algebra and applications from mechanics to cryptography
Encinas, Luis; Gadea, Pedro; María, Mª
2016-01-01
This volume collects contributions written by different experts in honor of Prof. Jaime Muñoz Masqué. It covers a wide variety of research topics, from differential geometry to algebra, but particularly focuses on the geometric formulation of variational calculus; geometric mechanics and field theories; symmetries and conservation laws of differential equations, and pseudo-Riemannian geometry of homogeneous spaces. It also discusses algebraic applications to cryptography and number theory. It offers state-of-the-art contributions in the context of current research trends. The final result is a challenging panoramic view of connecting problems that initially appear distant.
Many rational points coding theory and algebraic geometry
Hurt, Norman E
2003-01-01
This monograph presents a comprehensive treatment of recent results on algebraic geometry as they apply to coding theory and cryptography, with the goal the study of algebraic curves and varieties with many rational points. They book surveys recent developments on abelian varieties, in particular the classification of abelian surfaces, hyperelliptic curves, modular towers, Kloosterman curves and codes, Shimura curves and modular jacobian surfaces. Applications of abelian varieties to cryptography are presented including a discussion of hyperelliptic curve cryptosystems. The inter-relationship of codes and curves is developed building on Goppa's results on algebraic-geometry cods. The volume provides a source book of examples with relationships to advanced topics regarding Sato-Tate conjectures, Eichler-Selberg trace formula, Katz-Sarnak conjectures and Hecke operators.
Eliashberg, Yakov; Maeda, Yoshiaki; Symplectic, Poisson, and Noncommutative geometry
2014-01-01
Symplectic geometry originated in physics, but it has flourished as an independent subject in mathematics, together with its offspring, symplectic topology. Symplectic methods have even been applied back to mathematical physics. Noncommutative geometry has developed an alternative mathematical quantization scheme based on a geometric approach to operator algebras. Deformation quantization, a blend of symplectic methods and noncommutative geometry, approaches quantum mechanics from a more algebraic viewpoint, as it addresses quantization as a deformation of Poisson structures. This volume contains seven chapters based on lectures given by invited speakers at two May 2010 workshops held at the Mathematical Sciences Research Institute: Symplectic and Poisson Geometry in Interaction with Analysis, Algebra and Topology (honoring Alan Weinstein, one of the key figures in the field) and Symplectic Geometry, Noncommutative Geometry and Physics. The chapters include presentations of previously unpublished results and ...
Quantum potential physics, geometry and algebra
Licata, Ignazio
2014-01-01
Recently the interest in Bohm realist interpretation of quantum mechanics has grown. The important advantage of this approach lies in the possibility to introduce non-locality ab initio, and not as an “unexpected host”. In this book the authors give a detailed analysis of quantum potential, the non-locality term and its role in quantum cosmology and information. The different approaches to the quantum potential are analysed, starting from the original attempt to introduce a realism of particles trajectories (influenced by de Broglie’s pilot wave) to the recent dynamic interpretation provided by Goldstein, Durr, Tumulka and Zanghì, and the geometrodynamic picture, with suggestion about quantum gravity. Finally we focus on the algebraic reading of Hiley and Birkbeck school, that analyse the meaning of the non-local structure of the world, bringing important consequences for the space, time and information concepts.
Misconceptions in Rational Numbers, Probability, Algebra, and Geometry
Rakes, Christopher R.
2010-01-01
In this study, the author examined the relationship of probability misconceptions to algebra, geometry, and rational number misconceptions and investigated the potential of probability instruction as an intervention to address misconceptions in all 4 content areas. Through a review of literature, 5 fundamental concepts were identified that, if…
Topological expansion of the Bethe ansatz, and quantum algebraic geometry
Chekhov, L; Marchal, O
2009-01-01
In this article, we solve the loop equations of the \\beta-random matrix model, in a way similar to what was found for the case of hermitian matrices \\beta=1. For \\beta=1, the solution was expressed in terms of algebraic geometry properties of an algebraic spectral curve of equation y^2=U(x). For arbitrary \\beta, the spectral curve is no longer algebraic, it is a Schroedinger equation ((\\hbar\\partial)^2-U(x)).\\psi(x)=0 where \\hbar\\propto (\\sqrt\\beta-1/\\sqrt\\beta). In this article, we find a solution of loop equations, which takes the same form as the topological recursion found for \\beta=1. This allows to define natural generalizations of all algebraic geometry properties, like the notions of genus, cycles, forms of 1st, 2nd and 3rd kind, Riemann bilinear identities, and spectral invariants F_g, for a quantum spectral curve, i.e. a D-module of the form y^2-U(x), where [y,x]=\\hbar. Also, our method allows to enumerate non-oriented discrete surfaces.
Self-Similarity in Geometry, Algebra and Arithmetic
Rastegar, Arash
2012-01-01
We define the concept of self-similarity of an object by considering endomorphisms of the object as `similarity' maps. A variety of interesting examples of self-similar objects in geometry, algebra and arithmetic are introduced. Self-similar objects provide a framework in which, one can unite some results and conjectures in different mathematical frameworks. In some general situations, one can define a well-behaved notion of dimension for self-similar objects. Morphisms between self-similar o...
Using concatenated algebraic geometry codes in channel polarization
Eid, Abdulla; Duursma, Iwan
2013-01-01
Polar codes were introduced by Arikan in 2008 and are the first family of error-correcting codes achieving the symmetric capacity of an arbitrary binary-input discrete memoryless channel under low complexity encoding and using an efficient successive cancellation decoding strategy. Recently, non-binary polar codes have been studied, in which one can use different algebraic geometry codes to achieve better error decoding probability. In this paper, we study the performance of binary polar code...
Chiral topological insulator on Nambu 3-algebraic geometry
Energy Technology Data Exchange (ETDEWEB)
Hasebe, Kazuki, E-mail: khasebe@stanford.edu
2014-09-15
Chiral topological insulator (AIII-class) with Landau levels is constructed based on the Nambu 3-algebraic geometry. We clarify the geometric origin of the chiral symmetry of the AIII-class topological insulator in the context of non-commutative geometry of 4D quantum Hall effect. The many-body groundstate wavefunction is explicitly derived as a (l,l,l−1) Laughlin–Halperin type wavefunction with unique K-matrix structure. Fundamental excitation is identified with anyonic string-like object with fractional charge 1/(2(l−1){sup 2}+1). The Hall effect of the chiral topological insulators turns out be a color version of Hall effect, which exhibits a dual property of the Hall and spin-Hall effects.
Clifford algebras geometric modelling and chain geometries with application in kinematics
Klawitter, Daniel
2015-01-01
After revising known representations of the group of Euclidean displacements Daniel Klawitter gives a comprehensive introduction into Clifford algebras. The Clifford algebra calculus is used to construct new models that allow descriptions of the group of projective transformations and inversions with respect to hyperquadrics. Afterwards, chain geometries over Clifford algebras and their subchain geometries are examined. The author applies this theory and the developed methods to the homogeneous Clifford algebra model corresponding to Euclidean geometry. Moreover, kinematic mappings for special Cayley-Klein geometries are developed. These mappings allow a description of existing kinematic mappings in a unifying framework. Contents Models and representations of classical groups Clifford algebras, chain geometries over Clifford algebras Kinematic mappings for Pin and Spin groups Cayley-Klein geometries Target Groups Researchers and students in the field of mathematics, physics, and mechanical engineering About...
[Geometry and algebra of branches of the middle cerebral artery].
Blinkov, S M
1986-01-01
A classification of the cortical branches of the middle cerebral artery (MCA) is suggested by means of which each branch in any hemisphere can be qualified and identified in any variant of MCA branching. The principle of the classification consists in grouping the branches into arteries and trunks of the second, third, etc. order. Branches supplying blood to a certain sector of the lateral surface of the hemisphere are designated arteries. Their number and zone of branching are constant. Branches giving rise to 2 and more arteries are named trunks. Branching of the trunks, the number of trunks of the second, third, etc. order, and the site and type of origin of the arteries are extremely variable. Each trunk can be designated by a formula stating its order and the name of the artery supplied by this trunk. The arrangement of the MCA branches on the surface of the gyri and deep in the sulci, represented on the map of the lateral surface of the hemisphere, is designated conditionally as geometry of MCA branches. The order of branching of the trunks and the type of origin of the arteries, represented on abstract maps of the lateral surface of the hemisphere, are designated conditionally as algebra of the MCA branches. The variability of the geometry and algebra of the MCA branches must be taken into consideration in operations for extra-intracranial microanastomosis and in endovasal intervention on the MCA. PMID:3811741
Energy Technology Data Exchange (ETDEWEB)
Gargiulo, Maria Vittoria; Vitiello, Giuseppe [I.N.F.N., Salerno (Italy); Universita di Salerno, Dipartimento di Fisica, Salerno (Italy); Sakellariadou, Mairi [King' s College London, University of London, Department of Physics, London (United Kingdom)
2014-01-15
We study the physical implications of the doubling of the algebra, an essential element in the construction of the noncommutative spectral geometry model, proposed by Connes and his collaborators as offering a geometric explanation for the standard model of the strong and electroweak interactions. Linking the algebra doubling to the deformed Hopf algebra, we build Bogoliubov transformations and show the emergence of neutrino mixing. (orig.)
Institute of Scientific and Technical Information of China (English)
无
2004-01-01
Through most of Greek history, mathematicians concentrated on geometry, although Euclid considered the theory of numbers. The Greek mathematician Diophantus (3rd century),however, presented problems that had to be solved by what we would today call algebra. His book is thus the first algebra text.
Tabak, John
2004-01-01
Looking closely at algebra, its historical development, and its many useful applications, Algebra examines in detail the question of why this type of math is so important that it arose in different cultures at different times. The book also discusses the relationship between algebra and geometry, shows the progress of thought throughout the centuries, and offers biographical data on the key figures. Concise and comprehensive text accompanied by many illustrations presents the ideas and historical development of algebra, showcasing the relevance and evolution of this branch of mathematics.
Limit Algebras of Differential Forms in Non-Commutative Geometry
Indian Academy of Sciences (India)
S J Bhatt; A Inoue
2008-08-01
Given a C∗-normed algebra A which is either a Banach ∗-algebra or a Frechet ∗-algebra, we study the algebras ∞A and A obtained by taking respectively the projective limit and the inductive limit of Banach ∗-algebras obtained by completing the universal graded differential algebra ∗A of abstract non-commutative differential forms over A. Various quantized integrals on ∞A induced by a K-cycle on A are considered. The GNS-representation of ∞A defined by a d-dimensional non-commutative volume integral on a d+-summable K-cycle on A is realized as the representation induced by the left action of A on ∗A. This supplements the representation A on the space of forms discussed by Connes (Ch. VI.1, Prop. 5, p. 550 of [C]).
Leikin, Mark; Waisman, Ilana; Shaul, Shelley; Leikin, Roza
2014-03-01
This paper presents a small part of a larger interdisciplinary study that investigates brain activity (using event related potential methodology) of male adolescents when solving mathematical problems of different types. The study design links mathematics education research with neurocognitive studies. In this paper we performed a comparative analysis of brain activity associated with the translation from visual to symbolic representations of mathematical objects in algebra and geometry. Algebraic tasks require translation from graphical to symbolic representation of a function, whereas tasks in geometry require translation from a drawing of a geometric figure to a symbolic representation of its property. The findings demonstrate that electrical activity associated with the performance of geometrical tasks is stronger than that associated with solving algebraic tasks. Additionally, we found different scalp topography of the brain activity associated with algebraic and geometric tasks. Based on these results, we argue that problem solving in algebra and geometry is associated with different patterns of brain activity.
Algebra and Geometry of Hamilton's Quaternions: 'Well, Papa, Can You Multiply Triplets?'
Indian Academy of Sciences (India)
2016-06-01
Inspired by the relation between the algebra ofcomplex numbers and plane geometry, WilliamRowan Hamilton sought an algebra of triples forapplication to three-dimensional geometry. Unableto multiply and divide triples, he inventeda non-commutative division algebra of quadruples,in what he considered his most significantwork, generalizing the real and complex numbersystems. We give a motivated introduction toquaternions and discuss how they are related toPauli matrices, rotations in three dimensions, thethree sphere, the group SU(2) and the celebratedHopf fibrations.
Pure L-functions from algebraic geometry over finite fields
Wan, D
2000-01-01
This is an expository paper which gives a simple arithmetic introduction to the conjectures of Weil and Dwork concerning zeta functions of algebraic varieties over finite fields. A number of further open questions are raised.
A Clifford Algebra approach to the Discretizable Molecular Distance Geometry Problem
Andrioni, Alessandro
2013-01-01
The Discretizable Molecular Distance Geometry Problem (DMDGP) consists in a subclass of the Molecular Distance Geometry Problem for which an embedding in ${\\mathbb{R}^3}$ can be found using a Branch & Prune (BP) algorithm in a discrete search space. We propose a Clifford Algebra model of the DMDGP with an accompanying version of the BP algorithm.
Nonlocality and the central geometry of dimer algebras
Beil, Charlie
2014-01-01
Let $A$ be a dimer algebra and $Z$ its center. It is well known that if $A$ is cancellative, then $A$ and $Z$ are noetherian and $A$ is a finitely generated $Z$-module. Here we show the converse: if $A$ is non-cancellative (as almost all dimer algebras are), then $A$ and $Z$ are nonnoetherian and $A$ is an infinitely generated $Z$-module. Although $Z$ is nonnoetherian, we show that it nonetheless has Krull dimension 3 and is generically noetherian. Furthermore, we show that the reduced center is the coordinate ring for a Gorenstein algebraic variety with the strange property that it contains precisely one 'smeared-out' point of positive geometric dimension. In our proofs we introduce formalized notions of Higgsing and the mesonic chiral ring from quiver gauge theory.
Non commutative geometry methods for group C*-algebras
International Nuclear Information System (INIS)
This book is intended to provide a quick introduction to the subject. The exposition is scheduled in the sequence, as possible for more understanding for beginners. The author exposed a K-theoretic approach to study group C*-algebras: started in the elementary part, with one example of description of the structure of C*-algebra of the group of affine transformations of the real straight line, continued then for some special classes of solvable and nilpotent Lie groups. In the second advanced part, he introduced the main tools of the theory. In particular, the conception of multidimensional geometric quantization and the index of group C*-algebras were created and developed. (author). Refs
The role of difficulty and gender in numbers, algebra, geometry and mathematics achievement
Rabab'h, Belal Sadiq Hamed; Veloo, Arsaythamby; Perumal, Selvan
2015-05-01
This study aims to identify the role of difficulty and gender in numbers, algebra, geometry and mathematics achievement among secondary schools students in Jordan. The respondent of the study were 337 students from eight public secondary school in Alkoura district by using stratified random sampling. The study comprised of 179 (53%) males and 158 (47%) females students. The mathematics test comprises of 30 items which has eight items for numbers, 14 items for algebra and eight items for geometry. Based on difficulties among male and female students, the findings showed that item 4 (fractions - 0.34) was most difficult for male students and item 6 (square roots - 0.39) for females in numbers. For the algebra, item 11 (inequality - 0.23) was most difficult for male students and item 6 (algebraic expressions - 0.35) for female students. In geometry, item 3 (reflection - 0.34) was most difficult for male students and item 8 (volume - 0.33) for female students. Based on gender differences, female students showed higher achievement in numbers and algebra compare to male students. On the other hand, there was no differences between male and female students achievement in geometry test. This study suggest that teachers need to give more attention on numbers and algebra when teaching mathematics.
Abstract algebra, projective geometry and time encoding of quantum information
Planat, M R P; Planat, Michel R. P.; Saniga, Metod
2005-01-01
Algebraic geometrical concepts are playing an increasing role in quantum applications such as coding, cryptography, tomography and computing. We point out here the prominent role played by Galois fields viewed as cyclotomic extensions of the integers modulo a prime characteristic $p$. They can be used to generate efficient cyclic encoding, for transmitting secrete quantum keys, for quantum state recovery and for error correction in quantum computing. Finite projective planes and their generalization are the geometric counterpart to cyclotomic concepts, their coordinatization involves Galois fields, and they have been used repetitively for enciphering and coding. Finally the characters over Galois fields are fundamental for generating complete sets of mutually unbiased bases, a generic concept of quantum information processing and quantum entanglement. Gauss sums over Galois fields ensure minimum uncertainty under such protocols. Some Galois rings which are cyclotomic extensions of the integers modulo 4 are al...
Projective BGG equations, algebraic sets, and compactifications of Einstein geometries
Cap, A; Hammerl, M
2010-01-01
For curved projective manifolds we introduce a notion of a normal tractor frame field, based around any point. This leads to canonical systems of (redundant) coordinates that generalise the usual homogeneous coordinates on projective space. These give preferred local maps to the model projective space that encode geometric contact with the model to a level that is optimal, in a suitable sense. In terms of the trivialisations arising from the special frames, normal solutions of classes of natural linear PDE (so-called first BGG equations) are shown to be necessarily polynomial in the generalised homogeneous coordinates; the polynomial system is the pull back of a polynomial system that solves the corresponding problem on the model. Thus questions concerning the zero locus of solutions, as well as related finer geometric and smooth data, are reduced to a study of the corresponding polynomial systems and algebraic sets. We show that a normal solution determines a canonical manifold stratification that reflects a...
Quantum error-correcting codes from algebraic geometry codes of Castle type
Munuera, Carlos; Tenório, Wanderson; Torres, Fernando
2016-07-01
We study algebraic geometry codes producing quantum error-correcting codes by the CSS construction. We pay particular attention to the family of Castle codes. We show that many of the examples known in the literature in fact belong to this family of codes. We systematize these constructions by showing the common theory that underlies all of them.
Performance analysis of a decoding algorithm for algebraic-geometry codes
DEFF Research Database (Denmark)
Høholdt, Tom; Jensen, Helge Elbrønd; Nielsen, Rasmus Refslund
1999-01-01
The fast decoding algorithm for one point algebraic-geometry codes of Sakata, Elbrond Jensen, and Hoholdt corrects all error patterns of weight less than half the Feng-Rao minimum distance. In this correspondence we analyze the performance of the algorithm for heavier error patterns. It turns out...
Arnold, Vladimir I; Khesin, Boris; Marsden, Jerrold E; Varchenko, AN; Vassiliev, Victor A; Viro, Oleg Yanovich; Zakalyukin, Vladimir
2013-01-01
Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This second volume of his ""Collected Works"" focuses on hydrodynamics, bifurcation theory, and algebraic geometry.
Quantum error-correcting codes from algebraic geometry codes of Castle type
Munuera, Carlos; Tenório, Wanderson; Torres, Fernando
2016-10-01
We study algebraic geometry codes producing quantum error-correcting codes by the CSS construction. We pay particular attention to the family of Castle codes. We show that many of the examples known in the literature in fact belong to this family of codes. We systematize these constructions by showing the common theory that underlies all of them.
Muntingh, Georg
2014-01-01
This book summarizes research carried out in workshops of the SAGA project, an Initial Training Network exploring the interplay of Shapes, Algebra, Geometry and Algorithms. Written by a combination of young and experienced researchers, the book introduces new ideas in an established context. Among the central topics are approximate and sparse implicitization and surface parametrization; algebraic tools for geometric computing; algebraic geometry for computer aided design applications and problems with industrial applications. Readers will encounter new methods for the (approximate) transition between the implicit and parametric representation; new algebraic tools for geometric computing; new applications of isogeometric analysis, and will gain insight into the emerging research field situated between algebraic geometry and computer aided geometric design.
Liu, Chien-Hao
2015-01-01
We lay down an elementary yet fundamental lemma concerning a finite algebraicness property of a smooth map from an Azumaya/matrix manifold with a fundamental module to a smooth manifold. This gives us a starting point to build a synthetic (synonymously, $C^{\\infty}$-algebraic) symplectic geometry and calibrated geometry that are both tailored to and guided by D-brane phenomena in string theory and along the line of our previous works D(11.1) (arXiv:1406.0929 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]).
Bicomplex holomorphic functions the algebra, geometry and analysis of bicomplex numbers
Luna-Elizarrarás, M Elena; Struppa, Daniele C; Vajiac, Adrian
2015-01-01
The purpose of this book is to develop the foundations of the theory of holomorphicity on the ring of bicomplex numbers. Accordingly, the main focus is on expressing the similarities with, and differences from, the classical theory of one complex variable. The result is an elementary yet comprehensive introduction to the algebra, geometry and analysis of bicomplex numbers. Around the middle of the nineteenth century, several mathematicians (the best known being Sir William Hamilton and Arthur Cayley) became interested in studying number systems that extended the field of complex numbers. Hamilton famously introduced the quaternions, a skew field in real-dimension four, while almost simultaneously James Cockle introduced a commutative four-dimensional real algebra, which was rediscovered in 1892 by Corrado Segre, who referred to his elements as bicomplex numbers. The advantages of commutativity were accompanied by the introduction of zero divisors, something that for a while dampened interest in this subject. ...
Numbers as functions: the development of an idea in the Moscow school of algebraic geometry
Parshin, A. N.
2009-01-01
This is expanded text of a lecture delivered by the author at the conference "Mat\\'eriaux pour l'Histoire des Math\\'ematiques au XX\\`eme si\\`ecle", which took place in Nice in January 1996. The task was to describe one area in the development of arithmetical algebraic geometry in Moscow during the 1950s and 1960s. We shall begin by explaining the meaning of the analogy between numbers and functions, starting with the simplest concepts. In the second part we study a nontrivial example: the exp...
$W$-geometry of the Toda systems associated with non-exceptional simple Lie algebras
Gervais, Jean-Loup; Gervais, Jean-Loup; Saveliev, Mikhail V.
1996-01-01
The present paper describes the $W$--geometry of the Abelian finite non-periodic (conformal) Toda systems associated with the $B,C$ and $D$ series of the simple Lie algebras endowed with the canonical gradation. The principal tool here is a generalization of the classical Pl\\"ucker embedding of the $A$-case to the flag manifolds associated with the fundamental representations of $B_n$, $C_n$ and $D_n$, and a direct proof that the corresponding K\\"ahler potentials satisfy the system of two--dimensional finite non-periodic (conformal) Toda equations. It is shown that the $W$--geometry of the type mentioned above coincide with the differential geometry of special holomorphic (W) surfaces in target spaces which are submanifolds (quadrics) of $CP^N$ with appropriate choices of $N$. In addition, these W-surfaces are defined to satisfy quadratic holomorphic differential conditions that ensure consistency of the generalized Pl\\"ucker embedding. These conditions are automatically fulfiled when Toda equations hold.
Algebraic geometry methods associated to the one-dimensional Hubbard model
Martins, M. J.
2016-06-01
In this paper we study the covering vertex model of the one-dimensional Hubbard Hamiltonian constructed by Shastry in the realm of algebraic geometry. We show that the Lax operator sits in a genus one curve which is not isomorphic but only isogenous to the curve suitable for the AdS/CFT context. We provide an uniformization of the Lax operator in terms of ratios of theta functions allowing us to establish relativistic like properties such as crossing and unitarity. We show that the respective R-matrix weights lie on an Abelian surface being birational to the product of two elliptic curves with distinct J-invariants. One of the curves is isomorphic to that of the Lax operator but the other is solely fourfold isogenous. These results clarify the reason the R-matrix can not be written using only difference of spectral parameters of the Lax operator.
Axion Experiments to Algebraic Geometry: Testing Quantum Gravity via the Weak Gravity Conjecture
Heidenreich, Ben; Rudelius, Tom
2016-01-01
Common features of known quantum gravity theories may hint at the general nature of quantum gravity. The absence of continuous global symmetries is one such feature. This inspired the Weak Gravity Conjecture, which bounds masses of charged particles. We propose the Lattice Weak Gravity Conjecture, which further requires the existence of an infinite tower of particles of all possible charges under both abelian and nonabelian gauge groups and directly implies a cutoff for quantum field theory. It holds in a wide variety of string theory examples and has testable consequences for the real world and for pure mathematics. We sketch some implications of these ideas for models of inflation, for the QCD axion (and LIGO), for conformal field theory, and for algebraic geometry.
Flanders, Harley
1975-01-01
Algebra presents the essentials of algebra with some applications. The emphasis is on practical skills, problem solving, and computational techniques. Topics covered range from equations and inequalities to functions and graphs, polynomial and rational functions, and exponentials and logarithms. Trigonometric functions and complex numbers are also considered, together with exponentials and logarithms.Comprised of eight chapters, this book begins with a discussion on the fundamentals of algebra, each topic explained, illustrated, and accompanied by an ample set of exercises. The proper use of a
Boolean Algebra. Geometry Module for Use in a Mathematics Laboratory Setting.
Brotherton, Sheila; And Others
This module is recommended as an honors unit to follow a unit on logic. There are four basic parts: (1) What is a Boolean Algebra; (2) Using Boolean Algebra to Prove Theorems; (3) Using Boolean Algebra to Simplify Logical Statements; and (4) Circuit Problems with Logic and Boolean Algebra. Of these, sections 1, 2, and 3 are primarily written…
Shafarevich, I
1994-01-01
This book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geometry and algebraic topology.
Institute of Scientific and Technical Information of China (English)
WANG Renhong; ZHU Chungang
2004-01-01
The piecewise algebraic variety is a generalization of the classical algebraic variety. This paper discusses some properties of piecewise algebraic varieties and their coordinate rings based on the knowledge of algebraic geometry.
Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination
Cohen, Cyril; Mahboubi, Assia
2012-01-01
International audience This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic properties. The theory of real algebraic numbers and more generally of semi-algebraic varieties is at the core of a number of effective methods in real analysis, including decision procedures for non linear arithmetic or optimization ...
Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination
Mahboubi, Assia; Cohen, Cyril
2012-01-01
This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic properties. The theory of real algebraic numbers and more generally of semi-algebraic varieties is at the core of a number of effective methods in real analysis, including decision procedures for non linear arithmetic or optimization methods for real valued fu...
DEFF Research Database (Denmark)
Müller, Stefan; Feliu, Elisenda; Regensburger, Georg;
2016-01-01
We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomials maps with arbitrary real exponents defined on the positive orthant. Our work relates and extends existing injectivity conditions expressed in terms of Jacobian matrices and determin...... and determinants. In the context of chemical reaction networks with power-law kinetics, our results can be used to preclude as well as to guarantee multiple positive steady states. In the context of real algebraic geometry, our results reveal the first ...
Torrente-Lujan, E
2004-01-01
The algebraic approach to the construction of the reflexive polyhedra that yield Calabi-Yau spaces in three or more complex dimensions with K3 fibres reveals graphs that include and generalize the Dynkin diagrams associated with gauge symmetries. In this work we continue to study the structure of graphs obtained from $CY_3$ reflexive polyhedra. We show how some particularly defined integral matrices can be assigned to these diagrams. This family of matrices and its associated graphs may be obtained by relaxing the restrictions on the individual entries of the generalized Cartan matrices associated with the Dynkin diagrams that characterize Cartan-Lie and affine Kac-Moody algebras. These graphs keep however the affine structure, as it was in Kac-Moody Dynkin diagrams. We presented a possible root structure for some simple cases. We conjecture that these generalized graphs and associated link matrices may characterize generalizations of these algebras.
Algebraic structures, physics and geometry from a Unified Field Theoretical framework
Cirilo-Lombardo, Diego Julio
2014-01-01
Starting from a Unified Field Theory (UFT) proposed previously by the authors, the possible fermionic representations arising from the same spacetime are considered from the algebraic and geometrical viewpoint. We specifically demonstrate in this UFT general context that the underlying basis of the single geometrical structure P (G,M) (the principal fiber bundle over the real spacetime manifold M with structural group G) reflecting the symmetries of the different fields carry naturally a biquaternionic structure instead of a complex one. This fact allows us to analyze algebraically and to interpret physically in a straighforward way the Majorana and Dirac representations and the relation of such structures with the spacetime signature and non-hermitian (CP) dynamic operators. Also, from the underlying structure of the tangent space, the existence of hidden (super) symmetries and the possibility of supersymmetric extensions of these UFT models are given showing that Rothstein's theorem is incomplete for that d...
Discrete differential geometry of triangle tiles and algebra of closed trajectories
Morikawa, Naoto
2006-01-01
This paper proposes a new mathematical framework that can be applied to biological problems such as analysis of the structures of proteins and protein complexes. In particular, it gives a new method for encoding the three-dimensional structure of a protein into a binary sequence, where proteins are approximated by a folded tetrahedron sequence. It also gives a new algebraic framework for describing molecular complexes and their interactions. For simplicity, we shall explain the framework in t...
Central simple Poisson algebras
Institute of Scientific and Technical Information of China (English)
SU; Yucai; XU; Xiaoping
2004-01-01
Poisson algebras are fundamental algebraic structures in physics and symplectic geometry. However, the structure theory of Poisson algebras has not been well developed. In this paper, we determine the structure of the central simple Poisson algebras related to locally finite derivations, over an algebraically closed field of characteristic zero.The Lie algebra structures of these Poisson algebras are in general not finitely-graded.
Guido, Daniele; Landi, Giovanni; Vassout, Stéphane
2016-07-01
This topical issue grew out of the International Conference "Noncommutative Geometry and Applications" held 16-21 June 2014 at Villa Mondragone, Frascati (Roma). The main purpose of the conference was to have a unified view of different incarnations of noncommutative geometry and its applications. The seven papers collected in the present topical issue represent a good sample of the topics covered at the workshop. The conference itself was one of the climaxes of the Franco-Italian project GREFI-GENCO, which was initiated in 2007 by CNRS and INDAM to promote and enhance collaboration and exchanges between French and Italian researchers in the area of noncommutative geometry.
Quantum Algorithms for Problems in Number Theory, Algebraic Geometry, and Group Theory
van Dam, Wim; Sasaki, Yoshitaka
2013-09-01
Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.
1998-01-01
This introduction to the recent exciting developments in the applications of model theory to algebraic geometry, illustrated by E. Hrushovski's model-theoretic proof of the geometric Mordell-Lang Conjecture starts from very basic background and works up to the detailed exposition of Hrushovski's proof, explaining the necessary tools and results from stability theory on the way. The first chapter is an informal introduction to model theory itself, making the book accessible (with a little effort) to readers with no previous knowledge of model theory. The authors have collaborated closely to achieve a coherent and self- contained presentation, whereby the completeness of exposition of the chapters varies according to the existence of other good references, but comments and examples are always provided to give the reader some intuitive understanding of the subject.
Ruthven, Kenneth
2008-01-01
This article examines three important facets of the incorporation of new technologies into educational practice, focusing on emergent usages of the mathematical tools of computer algebra and dynamic geometry. First, it illustrates the interpretative flexibility of these tools, highlighting important differences in ways of conceptualizing and…
The geometry of blueprints. Part I: Algebraic background and scheme theory
Lorscheid, Oliver
2011-01-01
A blueprint generalizes both commutative (semi-)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp. congruences for rings and monoids and leads to a common scheme theory. In particular, it bridges the gap between usual schemes and $\\mathbb{F}_1$-schemes (after Kato, Deitmar and Connes-Consani). Beside this unification, the category of blueprints contains new interesting objects as "improved" cyclotomic field extensions $\\mathbb{F}_{1^n}$ of $\\mathbb{F}_1$ and "archimedean valuation rings". It also yields a notion of semi-ring schemes. This first paper lays the foundation for subsequent projects, which are devoted to the following problems: Tits' idea of Chevalley groups over $\\mathbb{F}_1$, congruence schemes, sheaf cohomology and $K$-theory and a unified view on analytic geometry over $\\mathbb{F}_1$, adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry.
Clifford algebra, geometric algebra, and applications
Lundholm, Douglas
2009-01-01
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of the algebra. The various applications presented include vector space and projective geometry, orthogonal maps and spinors, normed division algebras, as well as simplicial complexes and graph theory.
Workshop on Commutative Algebra
Simis, Aron
1990-01-01
The central theme of this volume is commutative algebra, with emphasis on special graded algebras, which are increasingly of interest in problems of algebraic geometry, combinatorics and computer algebra. Most of the papers have partly survey character, but are research-oriented, aiming at classification and structural results.
Indian Academy of Sciences (India)
Cătălin Ciupală
2005-02-01
In this paper we introduce non-commutative fields and forms on a new kind of non-commutative algebras: -algebras. We also define the Frölicher–Nijenhuis bracket in the non-commutative geometry on -algebras.
Linear algebra meets Lie algebra: the Kostant-Wallach theory
Shomron, Noam; Parlett, Beresford N.
2008-01-01
In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.
Pedoe, Dan
1988-01-01
""A lucid and masterly survey."" - Mathematics Gazette Professor Pedoe is widely known as a fine teacher and a fine geometer. His abilities in both areas are clearly evident in this self-contained, well-written, and lucid introduction to the scope and methods of elementary geometry. It covers the geometry usually included in undergraduate courses in mathematics, except for the theory of convex sets. Based on a course given by the author for several years at the University of Minnesota, the main purpose of the book is to increase geometrical, and therefore mathematical, understanding and to he
Chisolm, Eric
2012-01-01
This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines a product that's strongly motivated by geometry and can be taken between any two objects. For example, the product of two vectors taken in a certain way represents their common plane. This system was invented by William Clifford and is more commonly known as Clifford algebra. It's actually older than the vector algebra that we use today (due to Gibbs) and includes it as a subset. Over the years, various parts of Clifford algebra have been reinvented independently by many people who found they needed it, often not realizing that all those parts belonged in one system. This suggests that Clifford had the right idea, and that geometric algebra, not the reduced version we use today, deserves to be the standard "vector algebra." My goal in these notes is to describe geometric al...
Lectures on algebraic statistics
Drton, Mathias; Sullivant, Seth
2009-01-01
How does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? Why would a statistician working on factor analysis raise open problems about determinantal varieties? Connections of this type are at the heart of the new field of "algebraic statistics". In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques. The goal of these lectures is to introduce newcomers from the different camps to algebraic statistics. The introduction will be centered around the following three observations: many important statistical models correspond to algebraic or semi-algebraic sets of parameters; the geometry of these parameter spaces determines the behaviour of widely used statistical inference procedures; computational algebraic geometry can be used to study parameter spaces and other features of statistical models.
DEFF Research Database (Denmark)
Byg din egen boomerang, kast den, se den flyve, forstå hvorfor og hvordan den vender tilbage, og grib den. Det handler om opdriften på vingerne når du flyver, men det handler også og allermest om den mærkværdige gyroskop-effekt, du bruger til at holde balancen, når du kører på cykel. Vi vil bruge...... matematik, geometri, og fysik til at forstå, hvad det er, der foregår....
Samuel, Pierre
2008-01-01
Algebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics - algebraic geometry, in particular.This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Gal
Guggenheimer, Heinrich W
1977-01-01
This is a text of local differential geometry considered as an application of advanced calculus and linear algebra. The discussion is designed for advanced undergraduate or beginning graduate study, and presumes of readers only a fair knowledge of matrix algebra and of advanced calculus of functions of several real variables. The author, who is a Professor of Mathematics at the Polytechnic Institute of New York, begins with a discussion of plane geometry and then treats the local theory of Lie groups and transformation groups, solid differential geometry, and Riemannian geometry, leading to a
A Novel Approach of High-dimensional Image Restoration Based on Geometry Algebra%基于几何代数的散焦模糊高维图像恢复
Institute of Scientific and Technical Information of China (English)
户利利
2012-01-01
The geometry algebra can compute and analyze the high-dimensional space geometry in an easy way. Taking advantage of this property, the paper denotes the color image as one point in the geometry space by using the geometry algebra. The image transform can be treated on the language of geometry algebra as action of some transform. The image transform can be treated as the movements of the point in the high-dimensional space from the view of geometry. Beginning with the original blurred image, two further blurred images are got, then the restoral image can be obtained through the regressive curve derived from the three points in the geometry space which are mapped from the images by making use of geometry algebra. Experiments are presented to prove the availability of this method.%几何代数易于对高维空间几何进行计算和分析,应用几何代数的这一特性,将彩色图像表示为高维几何空间中的点元素,利用几何代数描述图像的变换关系,将图像的散焦变换看作是高维空间中点元素的平移运动.通过分析模糊图像以及其衍生出的相关模糊图像对应在高维几何空间中点之间的分布关系的研究,计算出空间中复原图像的点分布位置.实验结果验证了该方法的有效性.
Geometric Algebras and Extensors
Fernandez, V. V.; Moya, A. M.; Rodrigues Jr., W. A.
2007-01-01
This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors Cl(V,G_{E}) and the theory of its deformations leading to met...
具有优良渐近参数的代数几何码%ALGEBRAIC GEOMETRY CODES WITH GOOD ASYMPTOTIC PARAMETERS
Institute of Scientific and Technical Information of China (English)
胡万宝
2007-01-01
In this paper,we discuss a class of algebraic geometry codes (A-G codes) with good asymptotic parameters.Based on some analyses on a relation amony divisor class number, number of rational divisors of high degrees,and parameters of A-G codes,we obtain an asymptotic bound of a class, which is better than both the Gilbert-Varshamov and the Xing bounds.Our result shows that these two bounds can be improved significantly around the two points where they intersect.%本文讨论了一类具有好的渐近参数的代数几何码.通过对除子类数、高次有理除子数以及代数几何码的参数分析,得到一类码其渐近界优于Gilbert-Varshamov界和Xing界.在这两个界的交点处,渐近界有所改进.
Categorical Algebra and its Applications
1988-01-01
Categorical algebra and its applications contain several fundamental papers on general category theory, by the top specialists in the field, and many interesting papers on the applications of category theory in functional analysis, algebraic topology, algebraic geometry, general topology, ring theory, cohomology, differential geometry, group theory, mathematical logic and computer sciences. The volume contains 28 carefully selected and refereed papers, out of 96 talks delivered, and illustrates the usefulness of category theory today as a powerful tool of investigation in many other areas.
Left Artinian Algebraic Algebras
Institute of Scientific and Technical Information of China (English)
S. Akbari; M. Arian-Nejad
2001-01-01
Let R be a left artinian central F-algebra, T(R) = J(R) + [R, R],and U(R) the group of units of R. As one of our results, we show that, if R is algebraic and char F = 0, then the number of simple components of -R = R/J(R)is greater than or equal to dimF R/T(R). We show that, when char F = 0 or F is uncountable, R is algebraic over F if and only if [R, R] is algebraic over F. As another approach, we prove that R is algebraic over F if and only if the derived subgroup of U(R) is algebraic over F. Also, we present an elementary proof for a special case of an old question due to Jacobson.
Cylindric-like algebras and algebraic logic
Ferenczi, Miklós; Németi, István
2013-01-01
Algebraic logic is a subject in the interface between logic, algebra and geometry, it has strong connections with category theory and combinatorics. Tarski’s quest for finding structure in logic leads to cylindric-like algebras as studied in this book, they are among the main players in Tarskian algebraic logic. Cylindric algebra theory can be viewed in many ways: as an algebraic form of definability theory, as a study of higher-dimensional relations, as an enrichment of Boolean Algebra theory, or, as logic in geometric form (“cylindric” in the name refers to geometric aspects). Cylindric-like algebras have a wide range of applications, in, e.g., natural language theory, data-base theory, stochastics, and even in relativity theory. The present volume, consisting of 18 survey papers, intends to give an overview of the main achievements and new research directions in the past 30 years, since the publication of the Henkin-Monk-Tarski monographs. It is dedicated to the memory of Leon Henkin.
Open problems on open algebraic varieties
Gurjar, R V; Kumar, N M; Miyanishi, M; Russell, P; Sakai, F; Wright, D; Zaidenberg, M G; Kaliman, Shulim; Kumar, N Mohan; Miyanishi, Masayoshi; Russell, Peter; Sakai, Fumio; Wright, David; Zaidenberg, Mikhail
1995-01-01
This report records a large number of open problems in Affine Algebraic Geometry that were proposed by participants in a Conference on Open Algebraic Varieties at the Centre de Recherches en Mathematiques in Montreal at December 1994.
Discrimination in a General Algebraic Setting
Directory of Open Access Journals (Sweden)
Benjamin Fine
2015-01-01
Full Text Available Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras.
Discrimination in a General Algebraic Setting.
Fine, Benjamin; Gaglione, Anthony; Lipschutz, Seymour; Spellman, Dennis
2015-01-01
Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras.
Discrimination in a General Algebraic Setting
Fine, Benjamin; Gaglione, Anthony; Lipschutz, Seymour; Spellman, Dennis
2015-01-01
Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras. PMID:26171421
Z$_3$-graded differential geometry of quantum plane
Celik, Salih
2002-01-01
In this work, the Z$_3$-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algebra and the dual algebra is given.
Significant advancement in algebraic geometry
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
@@ Supported by a grant for Distinguished Young Scholars of the National Natural Science Foundation of China,Prof.SUN Xiaotao with the CAS Academy of Mathematics and Systems Science has recently achieved a research breakthrough in revealing the deep relationship between stability of vector bundles and Frobenius morphism.It is considered as significant work with important theoretical value.
Geometric Algebra for Physicists
Doran, Chris; Lasenby, Anthony
2007-11-01
Preface; Notation; 1. Introduction; 2. Geometric algebra in two and three dimensions; 3. Classical mechanics; 4. Foundations of geometric algebra; 5. Relativity and spacetime; 6. Geometric calculus; 7. Classical electrodynamics; 8. Quantum theory and spinors; 9. Multiparticle states and quantum entanglement; 10. Geometry; 11. Further topics in calculus and group theory; 12. Lagrangian and Hamiltonian techniques; 13. Symmetry and gauge theory; 14. Gravitation; Bibliography; Index.
Grätzer, George
1979-01-01
Universal Algebra, heralded as ". . . the standard reference in a field notorious for the lack of standardization . . .," has become the most authoritative, consistently relied on text in a field with applications in other branches of algebra and other fields such as combinatorics, geometry, and computer science. Each chapter is followed by an extensive list of exercises and problems. The "state of the art" account also includes new appendices (with contributions from B. Jónsson, R. Quackenbush, W. Taylor, and G. Wenzel) and a well-selected additional bibliography of over 1250 papers and books which makes this a fine work for students, instructors, and researchers in the field. "This book will certainly be, in the years to come, the basic reference to the subject." --- The American Mathematical Monthly (First Edition) "In this reviewer's opinion [the author] has more than succeeded in his aim. The problems at the end of each chapter are well-chosen; there are more than 650 of them. The book is especially sui...
On crossed product of algebras
Borowiec, A.; Marcinek, W.
2000-01-01
The concept of a crossed tensor product of algebras is studied from a few points of views. Some related constructions are considered. Crossed enveloping algebras and their representations are discussed. Applications to the noncommutative geometry and particle systems with generalized statistics are indicated.
Villarreal, Rafael
2015-01-01
The book stresses the interplay between several areas of pure and applied mathematics, emphasizing the central role of monomial algebras. It unifies the classical results of commutative algebra with central results and notions from graph theory, combinatorics, linear algebra, integer programming, and combinatorial optimization. The book introduces various methods to study monomial algebras and their presentation ideals, including Stanley-Reisner rings, subrings and blowup algebra-emphasizing square free quadratics, hypergraph clutters, and effective computational methods.
Noncommutative geometry and Cayley-Smooth orders
Le Bruyn, Lieven
2007-01-01
Preface Introduction Noncommutative algebra Noncommutative geometryNoncommutative desingularizationsCayley-Hamilton Algebras Conjugacy classes of matrices Simultaneous conjugacy classesMatrix invariants and necklaces The trace algebraThe symmetric group Necklace relations Trace relations Cayley-Hamilton algebrasReconstructing Algebras Representation schemes Some algebraic geometry The Hilbert criterium Semisimple modules Some invariant theory Geometric reconstruction The Gerstenhaber-Hesselink theoremThe real moment mapÉtale Technology Étale topologyCentral simple algebrasSpectral sequencesTse
Robinson, Gilbert de B
2011-01-01
This brief undergraduate-level text by a prominent Cambridge-educated mathematician explores the relationship between algebra and geometry. An elementary course in plane geometry is the sole requirement for Gilbert de B. Robinson's text, which is the result of several years of teaching and learning the most effective methods from discussions with students. Topics include lines and planes, determinants and linear equations, matrices, groups and linear transformations, and vectors and vector spaces. Additional subjects range from conics and quadrics to homogeneous coordinates and projective geom
Directory of Open Access Journals (Sweden)
Sinan AYDIN
2009-04-01
Full Text Available Linear algebra is a basic course followed in mathematics, science, and engineering university departments.Generally, this course is taken in either the first or second year but there have been difficulties in teachingand learning. This type of active algebra has resulted in an increase in research by mathematics educationresearchers. But there is insufficient information on this subject in Turkish and therefore it has not beengiven any educational status. This paper aims to give a general overview of this subject in teaching andlearning. These education studies can be considered quadruple: a the history of linear algebra, b formalismobstacles of linear algebra and cognitive flexibility to improve teaching and learning, c the relation betweenlinear algebra and geometry, d using technology in the teaching and learning linear algebra.Mathematicseducation researchers cannot provide an absolute solution to overcome the teaching and learning difficultiesof linear algebra. Epistemological analyses and experimental teaching have shown the learning difficulties.Given these results, further advice and assistance can be offered locally.
Solomon, Frederick
This module applies linear algebraic methods to solve the following problem: If an object in a three-dimensional coordinate system is first rotated about a given axis through the origin by a given angle, and then rotated about another axis through the origin by another angle, there is a straightforward way to calculate the combined result of the…
Zapatrin, R R
1998-01-01
An algebraic scheme is suggested in which discretized spacetime turns out to be a quantum observable. As an example, a toy model producing spacetimes of four points with different topologies is presented. The possibility of incorporating this scheme into the framework of non-commutative differential geometry is discussed.
Introduction to algebraic independence theory
Philippon, Patrice
2001-01-01
In the last five years there has been very significant progress in the development of transcendence theory. A new approach to the arithmetic properties of values of modular forms and theta-functions was found. The solution of the Mahler-Manin problem on values of modular function j(tau) and algebraic independence of numbers pi and e^(pi) are most impressive results of this breakthrough. The book presents these and other results on algebraic independence of numbers and further, a detailed exposition of methods created in last the 25 years, during which commutative algebra and algebraic geometry exerted strong catalytic influence on the development of the subject.
Cartan calculus on quantum Lie algebras
International Nuclear Information System (INIS)
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions au into one big algebra, the ''Cartan Calculus.''
Pavelle, Richard; And Others
1981-01-01
Describes the nature and use of computer algebra and its applications to various physical sciences. Includes diagrams illustrating, among others, a computer algebra system and flow chart of operation of the Euclidean algorithm. (SK)
Izhakian, Zur; Rowen, Louis
2008-01-01
We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geomet...
Automorphisms in Birational and Affine Geometry
Ciliberto, Ciro; Flenner, Hubert; McKernan, James; Prokhorov, Yuri; Zaidenberg, Mikhail
2014-01-01
The main focus of this volume is on the problem of describing the automorphism groups of affine and projective varieties, a classical subject in algebraic geometry where, in both cases, the automorphism group is often infinite dimensional. The collection covers a wide range of topics and is intended for researchers in the fields of classical algebraic geometry and birational geometry (Cremona groups) as well as affine geometry with an emphasis on algebraic group actions and automorphism groups. It presents original research and surveys and provides a valuable overview of the current state of the art in these topics. Bringing together specialists from projective, birational algebraic geometry and affine and complex algebraic geometry, including Mori theory and algebraic group actions, this book is the result of ensuing talks and discussions from the conference “Groups of Automorphisms in Birational and Affine Geometry” held in October 2012, at the CIRM, Levico Terme, Italy. The talks at the conference high...
Representation Theory of Algebraic Groups and Quantum Groups
Gyoja, A; Shinoda, K-I; Shoji, T; Tanisaki, Toshiyuki
2010-01-01
Invited articles by top notch expertsFocus is on topics in representation theory of algebraic groups and quantum groupsOf interest to graduate students and researchers in representation theory, group theory, algebraic geometry, quantum theory and math physics
Blyth, T S
2002-01-01
Basic Linear Algebra is a text for first year students leading from concrete examples to abstract theorems, via tutorial-type exercises. More exercises (of the kind a student may expect in examination papers) are grouped at the end of each section. The book covers the most important basics of any first course on linear algebra, explaining the algebra of matrices with applications to analytic geometry, systems of linear equations, difference equations and complex numbers. Linear equations are treated via Hermite normal forms which provides a successful and concrete explanation of the notion of linear independence. Another important highlight is the connection between linear mappings and matrices leading to the change of basis theorem which opens the door to the notion of similarity. This new and revised edition features additional exercises and coverage of Cramer's rule (omitted from the first edition). However, it is the new, extra chapter on computer assistance that will be of particular interest to readers:...
Energy Technology Data Exchange (ETDEWEB)
Odesskii, A V [L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow (Russian Federation)
2002-12-31
This survey is devoted to associative Z{sub {>=}}{sub 0}-graded algebras presented by n generators and n(n-1)/2 quadratic relations and satisfying the so-called Poincare-Birkhoff-Witt condition (PBW-algebras). Examples are considered of such algebras, depending on two continuous parameters (namely, on an elliptic curve and a point on it), that are flat deformations of the polynomial ring in n variables. Diverse properties of these algebras are described, together with their relations to integrable systems, deformation quantization, moduli spaces, and other directions of modern investigations.
Algebra and Number Theory An Integrated Approach
Dixon, Martyn; Subbotin, Igor
2011-01-01
Explore the main algebraic structures and number systems that play a central role across the field of mathematics Algebra and number theory are two powerful branches of modern mathematics at the forefront of current mathematical research, and each plays an increasingly significant role in different branches of mathematics, from geometry and topology to computing and communications. Based on the authors' extensive experience within the field, Algebra and Number Theory has an innovative approach that integrates three disciplines-linear algebra, abstract algebra, and number theory-into one compr
Distribution theory of algebraic numbers
Yang, Chung-Chun
2008-01-01
The book timely surveys new research results and related developments in Diophantine approximation, a division of number theory which deals with the approximation of real numbers by rational numbers. The book is appended with a list of challenging open problems and a comprehensive list of references. From the contents: Field extensions Algebraic numbers Algebraic geometry Height functions The abc-conjecture Roth''s theorem Subspace theorems Vojta''s conjectures L-functions.
Linear Algebra and Smarandache Linear Algebra
Vasantha, Kandasamy
2003-01-01
The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and ve...
Algebraic K-theory of generalized schemes
DEFF Research Database (Denmark)
Anevski, Stella Victoria Desiree
Nikolai Durov has developed a generalization of conventional scheme theory in which commutative algebraic monads replace commutative unital rings as the basic algebraic objects. The resulting geometry is expressive enough to encompass conventional scheme theory, tropical algebraic geometry...... and geometry over the field with one element. It also permits the construction of important Arakelov theoretical objects, such as the completion \\Spec Z of Spec Z. In this thesis, we prove a projective bundle theorem for the eld with one element and compute the Chow rings of the generalized schemes Sp\\ec ZN......, appearing in the construction of \\Spec Z....
The algebraic structure of the Onsager algebra
DATE, ETSURO; Roan, Shi-shyr
2000-01-01
We study the Lie algebra structure of the Onsager algebra from the ideal theoretic point of view. A structure theorem of ideals in the Onsager algebra is obtained with the connection to the finite-dimensional representations. We also discuss the solvable algebra aspect of the Onsager algebra through the formal Lie algebra theory.
On Curvature in Noncommutative Geometry
Dubois-Violette, M.; Madore, J.; Masson, T.(Centre de Physique Théorique, Aix Marseille Université & Université de Toulon & CNRS UMR 7332, 13288, Marseille, France); Mourad, J.
1995-01-01
A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.
Assessing non-uniqueness: An algebraic approach
Energy Technology Data Exchange (ETDEWEB)
Vasco, Don W.
2002-09-16
Geophysical inverse problems are endowed with a rich mathematical structure. When discretized, most differential and integral equations of interest are algebraic (polynomial) in form. Techniques from algebraic geometry and computational algebra provide a means to address questions of existence and uniqueness for both linear and non-linear inverse problem. In a sense, the methods extend ideas which have proven fruitful in treating linear inverse problems.
Issa, A. Nourou
2010-01-01
Hom-Akivis algebras are introduced. The commutator-Hom-associator algebra of a non-Hom-associative algebra (i.e. a Hom-nonassociative algebra) is a Hom-Akivis algebra. It is shown that non-Hom-associative algebras can be obtained from nonassociative algebras by twisting along algebra automorphisms while Hom-Akivis algebras can be obtained from Akivis algebras by twisting along algebra endomorphisms. It is pointed out that a Hom-Akivis algebra associated to a Hom-alternative algebra is a Hom-M...
Garrett, Paul B
2007-01-01
Designed for an advanced undergraduate- or graduate-level course, Abstract Algebra provides an example-oriented, less heavily symbolic approach to abstract algebra. The text emphasizes specifics such as basic number theory, polynomials, finite fields, as well as linear and multilinear algebra. This classroom-tested, how-to manual takes a more narrative approach than the stiff formalism of many other textbooks, presenting coherent storylines to convey crucial ideas in a student-friendly, accessible manner. An unusual feature of the text is the systematic characterization of objects by universal
Kolman, Bernard
1985-01-01
College Algebra, Second Edition is a comprehensive presentation of the fundamental concepts and techniques of algebra. The book incorporates some improvements from the previous edition to provide a better learning experience. It provides sufficient materials for use in the study of college algebra. It contains chapters that are devoted to various mathematical concepts, such as the real number system, the theory of polynomial equations, exponential and logarithmic functions, and the geometric definition of each conic section. Progress checks, warnings, and features are inserted. Every chapter c
Holtz, Olga; Ron, Amos
2007-01-01
A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\\cal H}(X)$. This well-known line of study is particularly interesting in case $n\\eqbd\\rank X \\ll N$. We enhance this study to an algebraic level, and associate $X$ with three algebraic structures, referred herein as {\\it external, central, and internal.} Each algebraic structure is ...
Higher geometry an introduction to advanced methods in analytic geometry
Woods, Frederick S
2005-01-01
For students of mathematics with a sound background in analytic geometry and some knowledge of determinants, this volume has long been among the best available expositions of advanced work on projective and algebraic geometry. Developed from Professor Woods' lectures at the Massachusetts Institute of Technology, it bridges the gap between intermediate studies in the field and highly specialized works.With exceptional thoroughness, it presents the most important general concepts and methods of advanced algebraic geometry (as distinguished from differential geometry). It offers a thorough study
McKeague, Charles P
1986-01-01
Elementary Algebra, Third Edition focuses on the basic principles, operations, and approaches involved in elementary algebra. The book first ponders on the basics, linear equations and inequalities, and graphing and linear systems. Discussions focus on the elimination method, solving linear systems by graphing, word problems, addition property of equality, solving linear equations, linear inequalities, addition and subtraction of real numbers, and properties of real numbers. The text then takes a look at exponents and polynomials, factoring, and rational expressions. Topics include reducing ra
McKeague, Charles P
1981-01-01
Elementary Algebra 2e, Second Edition focuses on the basic principles, operations, and approaches involved in elementary algebra. The book first tackles the basics, linear equations and inequalities, and graphing and linear systems. Discussions focus on the substitution method, solving linear systems by graphing, solutions to linear equations in two variables, multiplication property of equality, word problems, addition property of equality, and subtraction, addition, multiplication, and division of real numbers. The manuscript then examines exponents and polynomials, factoring, and rational e
Cartan Calculus on Quantum Lie Algebras
Schupp, Peter; Watts, Paul; Zumino, Bruno
1993-01-01
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions all into one big algebra, the ``Cartan Calculus''. (This is an extended version of a talk presented by P. Schupp at the XXII$^{th}$ International Conference on Differential Geo...
Which multiplier algebras are $W^*$-algebras?
Akemann, Charles A.; Amini, Massoud; Asadi, Mohammad B.
2013-01-01
We consider the question of when the multiplier algebra $M(\\mathcal{A})$ of a $C^*$-algebra $\\mathcal{A}$ is a $ W^*$-algebra, and show that it holds for a stable $C^*$-algebra exactly when it is a $C^*$-algebra of compact operators. This implies that if for every Hilbert $C^*$-module $E$ over a $C^*$-algebra $\\mathcal{A}$, the algebra $B(E)$ of adjointable operators on $E$ is a $ W^*$-algebra, then $\\mathcal{A}$ is a $C^*$-algebra of compact operators. Also we show that a unital $C^*$-algebr...
Classical theory of algebraic numbers
Ribenboim, Paulo
2001-01-01
Gauss created the theory of binary quadratic forms in "Disquisitiones Arithmeticae" and Kummer invented ideals and the theory of cyclotomic fields in his attempt to prove Fermat's Last Theorem These were the starting points for the theory of algebraic numbers, developed in the classical papers of Dedekind, Dirichlet, Eisenstein, Hermite and many others This theory, enriched with more recent contributions, is of basic importance in the study of diophantine equations and arithmetic algebraic geometry, including methods in cryptography This book has a clear and thorough exposition of the classical theory of algebraic numbers, and contains a large number of exercises as well as worked out numerical examples The Introduction is a recapitulation of results about principal ideal domains, unique factorization domains and commutative fields Part One is devoted to residue classes and quadratic residues In Part Two one finds the study of algebraic integers, ideals, units, class numbers, the theory of decomposition, iner...
Computer algebra in spacetime embedding
Roque, Waldir L
2014-01-01
In this paper we describe an algorithm to determine the vectors normal to a space-time V4 embedded in a pseudo-Euclidean manifold M4+n. An application of this algorithm is given considering the Schwarzchild space-time geometry embedded in a 6 dimensional pseudo-Euclidean manifold, using the algebraic computing system REDUCE.
Towards relativistic quantum geometry
Directory of Open Access Journals (Sweden)
Luis Santiago Ridao
2015-12-01
Full Text Available We obtain a gauge-invariant relativistic quantum geometry by using a Weylian-like manifold with a geometric scalar field which provides a gauge-invariant relativistic quantum theory in which the algebra of the Weylian-like field depends on observers. An example for a Reissner–Nordström black-hole is studied.
Operator algebras and topology
International Nuclear Information System (INIS)
These notes, based on three lectures on operator algebras and topology at the 'School on High Dimensional Manifold Theory' at the ICTP in Trieste, introduce a new set of tools to high dimensional manifold theory, namely techniques coming from the theory of operator algebras, in particular C*-algebras. These are extensively studied in their own right. We will focus on the basic definitions and properties, and on their relevance to the geometry and topology of manifolds. A central pillar of work in the theory of C*-algebras is the Baum-Connes conjecture. This is an isomorphism conjecture, as discussed in the talks of Luck, but with a certain special flavor. Nevertheless, it has important direct applications to the topology of manifolds, it implies e.g. the Novikov conjecture. In the first chapter, the Baum-Connes conjecture will be explained and put into our context. Another application of the Baum-Connes conjecture is to the positive scalar curvature question. This will be discussed by Stephan Stolz. It implies the so-called 'stable Gromov-Lawson-Rosenberg conjecture'. The unstable version of this conjecture said that, given a closed spin manifold M, a certain obstruction, living in a certain (topological) K-theory group, vanishes if and only M admits a Riemannian metric with positive scalar curvature. It turns out that this is wrong, and counterexamples will be presented in the second chapter. The third chapter introduces another set of invariants, also using operator algebra techniques, namely L2-cohomology, L2-Betti numbers and other L2-invariants. These invariants, their basic properties, and the central questions about them, are introduced in the third chapter. (author)
Mirkovic-Vilonen polytopes and Khovanov-Lauda-Rouquier algebras
Tingley, Peter; Webster, Ben
2012-01-01
We describe how Mirkovic-Vilonen polytopes arise naturally from the categorification of Lie algebras using Khovanov-Lauda-Rouquier algebras. This gives an explicit description of the unique crystal isomorphism between simple representations of the KLR algebra and MV polytopes. MV polytopes, as defined from the geometry of the affine Grassmannian, only make sense for finite dimensional semi-simple Lie algebras, but our construction actually gives a map from the infinity crystal to polytopes in...
N(o)ther-type theorem of piecewise algebraic curves on quasi-cross-cut partition
Institute of Scientific and Technical Information of China (English)
ZHU ChunGang; WANG RenHong
2009-01-01
Nother's theorem of algebraic curves plays an important role in classical algebraic geome-try. As the zero set of a bivariate spline, the piecewise algebraic curve is a generalization of the classical algebraic curve. Nother-type theorem of piecewise algebraic curves is very important to construct the Lagrange interpolation sets for bivariate spline spaces. In this paper, using the characteristics of quasi-cross-cut partition, properties of bivariate splines and results in algebraic geometry, the Nother-type theorem of piecewise algebraic curves on the quasi-cross-cut is presented.
Nther-type theorem of piecewise algebraic curves on quasi-cross-cut partition
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
Nther’s theorem of algebraic curves plays an important role in classical algebraic geometry. As the zero set of a bivariate spline, the piecewise algebraic curve is a generalization of the classical algebraic curve. Nther-type theorem of piecewise algebraic curves is very important to construct the Lagrange interpolation sets for bivariate spline spaces. In this paper, using the characteristics of quasi-cross-cut partition, properties of bivariate splines and results in algebraic geometry, the Nther-type theorem of piecewise algebraic curves on the quasi-cross-cut is presented.
Hazewinkel, Michiel
2004-01-01
Two important generalizations of the Hopf algebra of symmetric functions are the Hopf algebra of noncommutative symmetric functions and its graded dual the Hopf algebra of quasisymmetric functions. A common generalization of the latter is the selfdual Hopf algebra of permutations (MPR Hopf algebra). This latter Hopf algebra can be seen as a Hopf algebra of endomorphisms of a Hopf algebra. That turns out to be a fruitful way of looking at things and gives rise to wide ranging further generaliz...
Liesen, Jörg
2015-01-01
This self-contained textbook takes a matrix-oriented approach to linear algebra and presents a complete theory, including all details and proofs, culminating in the Jordan canonical form and its proof. Throughout the development, the applicability of the results is highlighted. Additionally, the book presents special topics from applied linear algebra including matrix functions, the singular value decomposition, the Kronecker product and linear matrix equations. The matrix-oriented approach to linear algebra leads to a better intuition and a deeper understanding of the abstract concepts, and therefore simplifies their use in real world applications. Some of these applications are presented in detailed examples. In several ‘MATLAB-Minutes’ students can comprehend the concepts and results using computational experiments. Necessary basics for the use of MATLAB are presented in a short introduction. Students can also actively work with the material and practice their mathematical skills in more than 300 exerc...
Edwards, Harold M
1995-01-01
In his new undergraduate textbook, Harold M Edwards proposes a radically new and thoroughly algorithmic approach to linear algebra Originally inspired by the constructive philosophy of mathematics championed in the 19th century by Leopold Kronecker, the approach is well suited to students in the computer-dominated late 20th century Each proof is an algorithm described in English that can be translated into the computer language the class is using and put to work solving problems and generating new examples, making the study of linear algebra a truly interactive experience Designed for a one-semester course, this text adopts an algorithmic approach to linear algebra giving the student many examples to work through and copious exercises to test their skills and extend their knowledge of the subject Students at all levels will find much interactive instruction in this text while teachers will find stimulating examples and methods of approach to the subject
GOLDMAN ALGEBRA, OPERS AND THE SWAPPING ALGEBRA
Labourie, François
2012-01-01
We define a Poisson Algebra called the {\\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\\em algebra of multifractions} -- as an algebra of functions on the space of cross ratios and thus as an algebra of functions on the Hitchin component as well as on the space of $\\mathsf{SL}_n(\\mathbb R)$-opers with trivial holonomy. We relate this Poisson algebra to the Atiyah--Bott--Goldman symple...
Directory of Open Access Journals (Sweden)
G.C. Rao
2012-11-01
Full Text Available A C- algebra is the algebraic form of the 3-valued conditional logic, which was introduced by F. Guzman and C. C. Squier in 1990. In this paper, some equivalent conditions for a C- algebra to become a boolean algebra in terms of congruences are given. It is proved that the set of all central elements B(A is isomorphic to the Boolean algebra of all C-algebras Sa, where a B(A. It is also proved that B(A is isomorphic to the Boolean algebra of all C-algebras Aa, where a B(A.
Stoll, R R
1968-01-01
Linear Algebra is intended to be used as a text for a one-semester course in linear algebra at the undergraduate level. The treatment of the subject will be both useful to students of mathematics and those interested primarily in applications of the theory. The major prerequisite for mastering the material is the readiness of the student to reason abstractly. Specifically, this calls for an understanding of the fact that axioms are assumptions and that theorems are logical consequences of one or more axioms. Familiarity with calculus and linear differential equations is required for understand
Allenby, Reg
1995-01-01
As the basis of equations (and therefore problem-solving), linear algebra is the most widely taught sub-division of pure mathematics. Dr Allenby has used his experience of teaching linear algebra to write a lively book on the subject that includes historical information about the founders of the subject as well as giving a basic introduction to the mathematics undergraduate. The whole text has been written in a connected way with ideas introduced as they occur naturally. As with the other books in the series, there are many worked examples.Solutions to the exercises are available onlin
Deskins, W E
1996-01-01
This excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. These systems, which consist of sets of elements, operations, and relations among the elements, and prescriptive axioms, are abstractions and generalizations of various models which evolved from efforts to explain or discuss physical phenomena.In Chapter 1, the author discusses the essential ingredients of a mathematical system, and in the next four chapters covers the basic number systems, decompositions of integers, diop
Jacobson, Nathan
2009-01-01
A classic text and standard reference for a generation, this volume and its companion are the work of an expert algebraist who taught at Yale for two decades. Nathan Jacobson's books possess a conceptual and theoretical orientation, and in addition to their value as classroom texts, they serve as valuable references.Volume I explores all of the topics typically covered in undergraduate courses, including the rudiments of set theory, group theory, rings, modules, Galois theory, polynomials, linear algebra, and associative algebra. Its comprehensive treatment extends to such rigorous topics as L
Basic algebraic topology and its applications
Adhikari, Mahima Ranjan
2016-01-01
This book provides an accessible introduction to algebraic topology, a ﬁeld at the intersection of topology, geometry and algebra, together with its applications. Moreover, it covers several related topics that are in fact important in the overall scheme of algebraic topology. Comprising eighteen chapters and two appendices, the book integrates various concepts of algebraic topology, supported by examples, exercises, applications and historical notes. Primarily intended as a textbook, the book oﬀers a valuable resource for undergraduate, postgraduate and advanced mathematics students alike. Focusing more on the geometric than on algebraic aspects of the subject, as well as its natural development, the book conveys the basic language of modern algebraic topology by exploring homotopy, homology and cohomology theories, and examines a variety of spaces: spheres, projective spaces, classical groups and their quotient spaces, function spaces, polyhedra, topological groups, Lie groups and cell complexes, etc. T...
An introduction to Clifford algebras and spinors
Vaz, Jayme
2016-01-01
This text explores how Clifford algebras and spinors have been sparking a collaboration and bridging a gap between Physics and Mathematics. This collaboration has been the consequence of a growing awareness of the importance of algebraic and geometric properties in many physical phenomena, and of the discovery of common ground through various touch points: relating Clifford algebras and the arising geometry to so-called spinors, and to their three definitions (both from the mathematical and physical viewpoint). The main point of contact are the representations of Clifford algebras and the periodicity theorems. Clifford algebras also constitute a highly intuitive formalism, having an intimate relationship to quantum field theory. The text strives to seamlessly combine these various viewpoints and is devoted to a wider audience of both physicists and mathematicians. Among the existing approaches to Clifford algebras and spinors this book is unique in that it provides a didactical presentation of the topic and i...
Indian Academy of Sciences (India)
Tomás L Gómez
2001-02-01
This is an expository article on the theory of algebraic stacks. After introducing the general theory, we concentrate in the example of the moduli stack of vector bundles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory.
Oliver, Bob; Pawałowski, Krzystof
1991-01-01
As part of the scientific activity in connection with the 70th birthday of the Adam Mickiewicz University in Poznan, an international conference on algebraic topology was held. In the resulting proceedings volume, the emphasis is on substantial survey papers, some presented at the conference, some written subsequently.
El-Chaar, Caroline
2012-01-01
In this thesis, four realizations of the Onsager algebra are explored. We begin with its original definition as introduced by Lars Onsager. We then examine how the Onsager algebra can be presented as a Lie algebra with two generators and two relations. The third realization of the Onsager algebra consists of viewing it as an equivariant map algebra which then gives us the tools to classify its closed ideals. Finally, we examine the Onsager algebra as a subalgebra of the tetrahedron algebra. U...
Artin, E
2011-01-01
This classic text, written by one of the foremost mathematicians of the 20th century, is now available in a low-priced paperback edition. Exposition is centered on the foundations of affine geometry, the geometry of quadratic forms, and the structure of the general linear group. Context is broadened by the inclusion of projective and symplectic geometry and the structure of symplectic and orthogonal groups.
Scattering Amplitudes via Algebraic Geometry Methods
DEFF Research Database (Denmark)
Søgaard, Mads
Feynman diagrams. The study of multiloop scattering amplitudes is crucial for the new era of precision phenomenology at the Large Hadron Collider (LHC) at CERN. Loop-level scattering amplitudes can be reduced to a basis of linearly independent integrals whose coefficients are extracted from generalized...
Chirvasitu, Alex; Smith, S. Paul
2015-01-01
This paper examines a general method for producing twists of a comodule algebra by tensoring it with a torsor then taking co-invariants. We examine the properties that pass from the original algebra to the twisted algebra and vice versa. We then examine the special case where the algebra is a 4-dimensional Sklyanin algebra viewed as a comodule algebra over the Hopf algebra of functions on the non-cyclic group of order 4 with the torsor being the 2x2 matrix algebra. The twisted algebra is an "...
Nonmonotonic logics and algebras
Institute of Scientific and Technical Information of China (English)
CHAKRABORTY Mihir Kr; GHOSH Sujata
2008-01-01
Several nonmonotonie logic systems together with their algebraic semantics are discussed. NM-algebra is defined.An elegant construction of an NM-algebra starting from a Boolean algebra is described which gives rise to a few interesting algebraic issues.
Kleyn, Aleks
2007-01-01
The concept of F-algebra and its representation can be extended to an arbitrary bundle. We define operations of fibered F-algebra in fiber. The paper presents the representation theory of of fibered F-algebra as well as a comparison of representation of F-algebra and of representation of fibered F-algebra.
Dynamical systems of algebraic origin
Schmidt, Klaus
1995-01-01
Although much of classical ergodic theory is concerned with single transformations and one-parameter flows, the subject inherits from statistical mechanics not only its name, but also an obligation to analyze spatially extended systems with multidimensional symmetry groups. However, the wealth of concrete and natural examples which has contributed so much to the appeal and development of classical dynamics, is noticeably absent in this more general theory. The purpose of this book is to help remedy this scarcity of explicit examples by introducing a class of continuous Zd-actions diverse enough to exhibit many of the new phenomena encountered in the transition from Z to Zd, but which nevertheless lends itself to systematic study: the Zd-actions by automorphisms of compact, abelian groups. One aspect of these actions, not surprising in itself but quite striking in its extent and depth nonetheless, is the connection with commutative algebra and arithmetical algebraic geometry. The algebraic framework resulting...
Seminar on K-Theory, Arithmetic and Geometry
1987-01-01
This volume of research papers is an outgrowth of the Manin Seminar at Moscow University, devoted to K-theory, homological algebra and algebraic geometry. The main topics discussed include additive K-theory, cyclic cohomology, mixed Hodge structures, theory of Virasoro and Neveu-Schwarz algebras.
Semi-algebraic function rings and reflectors of partially ordered rings
Schwartz, Niels
1999-01-01
The book lays algebraic foundations for real geometry through a systematic investigation of partially ordered rings of semi-algebraic functions. Real spectra serve as primary geometric objects, the maps between them are determined by rings of functions associated with the spectra. The many different possible choices for these rings of functions are studied via reflections of partially ordered rings. Readers should feel comfortable using basic algebraic and categorical concepts. As motivational background some familiarity with real geometry will be helpful. The book aims at researchers and graduate students with an interest in real algebra and geometry, ordered algebraic structures, topology and rings of continuous functions.
Forty questions on singularities of algebraic varieties
Hauser, Herwig; Schicho, Josef
2011-01-01
The reader will find in this article a collection of problems, questions and exercises related to the singularities of algebraic and analytic varieties. Many of them are inspired by the work and mathematical conception of Hironaka: they are concrete, involve basic ideas and techniques from geometry and algebra, and they can immediately be attacked from scratch. Some problems rely on or use results proven by Hironaka. Simple and double asterisques indicate the more difficult pro...
Algebraic and geometric structures of Special Relativity
Giulini, Domenico
2006-01-01
I review, some of the algebraic and geometric structures that underlie the theory of Special Relativity. This includes a discussion of relativity as a symmetry principle, derivations of the Lorentz group, its composition law, its Lie algebra, comparison with the Galilei group, Einstein synchronization, the lattice of causally and chronologically complete regions in Minkowski space, rigid motion (the Noether-Herglotz theorem), and the geometry of rotating reference frames. Representation-theor...
Solvable quadratic Lie algebras
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.
Grabowski, Jan
2015-01-01
In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradings for finite type cluster algebras without coefficients, giving a full classification. Translating ...
MSc Thesis: Presentation of Certain New Trends in Noncommutative Geometry
Buachalla, Réamonn Ó
2011-01-01
MSc thesis of the author offering an introduction to the operator algebraic approach to noncommutative geometry, with a treatment of some more advanced elements such as the noncommutative geometry of quantum groups, fuzzy physics, and compact quantum metric spaces.
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
It is a small step toward the Koszul-type algebras. The piecewise-Koszul algebras are,in general, a new class of quadratic algebras but not the classical Koszul ones, simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases. We give a criteria theorem for a graded algebra A to be piecewise-Koszul in terms of its Yoneda-Ext algebra E(A), and show an A∞-structure on E(A). Relations between Koszul algebras and piecewise-Koszul algebras are discussed. In particular, our results are related to the third question of Green-Marcos.
Li, Haisheng; Tan, Shaobin; Wang, Qing
2012-01-01
In this paper, we study a notion of what we call vertex Leibniz algebra. This notion naturally extends that of vertex algebra without vacuum, which was previously introduced by Huang and Lepowsky. We show that every vertex algebra without vacuum can be naturally extended to a vertex algebra. On the other hand, we show that a vertex Leibniz algebra can be embedded into a vertex algebra if and only if it admits a faithful module. To each vertex Leibniz algebra we associate a vertex algebra with...
The Covariant Picard Groupoid in Differential Geometry
Waldmann, Stefan
2005-01-01
In this article we discuss some general results on the covariant Picard groupoid in the context of differential geometry and interpret the problem of lifting Lie algebra actions to line bundles in the Picard groupoid approach.
Gervais, Jean-Loup; Gervais, Jean-Loup; Matsuo, Yutaka
1992-01-01
It is shown that, classically, the W-algebras are directly related to the extrinsic geometry of the embedding of two-dimensional manifolds with chiral parametrisation (W-surfaces) into higher dimensional K\\"ahler manifolds. We study the local and the global geometries of such embeddings, and connect them to Toda equations. The additional variables of the related KP hierarchy are shown to yield a specific coordinate system of the target-manifold, and this allows us to prove that W-transformations are simply particular diffeomorphisms of this target space. The W-surfaces are shown to be instantons of the corresponding non-linear $\\sigma$-models.
Elementary differential geometry
O'Neill, Barrett
2006-01-01
Written primarily for students who have completed the standard first courses in calculus and linear algebra, ELEMENTARY DIFFERENTIAL GEOMETRY, REVISED SECOND EDITION, provides an introduction to the geometry of curves and surfaces. The Second Edition maintained the accessibility of the first, while providing an introduction to the use of computers and expanding discussion on certain topics. Further emphasis was placed on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard. This revision of the Second Edition p
Yoneda algebras of almost Koszul algebras
Indian Academy of Sciences (India)
Zheng Lijing
2015-11-01
Let be an algebraically closed field, a finite dimensional connected (, )-Koszul self-injective algebra with , ≥ 2. In this paper, we prove that the Yoneda algebra of is isomorphic to a twisted polynomial algebra $A^!$ [ ; ] in one indeterminate of degree +1 in which $A^!$ is the quadratic dual of , is an automorphism of $A^!$, and = () for each $t \\in A^!$. As a corollary, we recover Theorem 5.3 of [2].
Elementary differential geometry
Pressley, Andrew
2001-01-01
Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood Differential geometry is concerned with the precise mathematical formulation of some of these questions, and with trying to answer them using calculus techniques It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces while keeping the prerequisites to an absolute minimum Nothing more than first courses in linear algebra and multivariate calculus are required, and the most direct and straightforward approach is used at all times Numerous diagrams illustrate both the ideas in the text and the examples of curves and surfaces discussed there The book will provide an invaluable resource to all those taking a first course in differential geometry, for their lecture...
WEAKLY ALGEBRAIC REFLEXIVITY AND STRONGLY ALGEBRAIC REFLEXIVITY
Institute of Scientific and Technical Information of China (English)
TaoChangli; LuShijie; ChenPeixin
2002-01-01
Algebraic reflexivity introduced by Hadwin is related to linear interpolation. In this paper, the concepts of weakly algebraic reflexivity and strongly algebraic reflexivity which are also related to linear interpolation are introduced. Some properties of them are obtained and some relations between them revealed.
Rigidification of algebras over essentially algebraic theories
Rosicky, J
2012-01-01
Badzioch and Bergner proved a rigidification theorem saying that each homotopy simplicial algebra is weakly equivalent to a simplicial algebra. The question is whether this result can be extended from algebraic theories to finite limit theories and from simplicial sets to more general monoidal model categories. We will present some answers to this question.
The Yoneda algebra of a K2 algebra need not be another K2 algebra
Cassidy, T.; Phan, C.; Shelton, B.
2010-01-01
The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. K2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a K2 algebra would be another K2 algebra. We show that this is not necessarily the case by constructing a monomial K2 algebra for which the corresponding Yoneda algebra is not K2.
Enveloping algebras of some quantum Lie algebras
Pourkia, Arash
2014-01-01
We define a family of Hopf algebra objects, $H$, in the braided category of $\\mathbb{Z}_n$-modules (known as anyonic vector spaces), for which the property $\\psi^2_{H\\otimes H}=id_{H\\otimes H}$ holds. We will show that these anyonic Hopf algebras are, in fact, the enveloping (Hopf) algebras of particular quantum Lie algebras, also with the property $\\psi^2=id$. Then we compute the braided periodic Hopf cyclic cohomology of these Hopf algebras. For that, we will show the following fact: analog...
The three-dimensional origin of the classifying algebra
Fuchs, Jurgen; Schweigert, Christoph; Stigner, Carl
2009-01-01
It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying algebra, a semisimple commutative associative complex algebra. We show how this algebra arises naturally from the three-dimensional geometry of factorization of correlators of bulk fields on the disk. This allows us to derive explicit expressions for the str...
Dzhumadil'daev, A. S.
2002-01-01
Algebras with identity $(a\\star b)\\star (c\\star d) -(a\\star d)\\star(c\\star b)$ $=(a,b,c)\\star d-(a,d,c)\\star b$ are studied. Novikov algebras under Jordan multiplication and Leibniz dual algebras satisfy this identity. If algebra with such identity has unit, then it is associative and commutative.
National Council of Teachers of Mathematics, Inc., Reston, VA.
This is a reprint of the historical capsules dealing with algebra from the 31st Yearbook of NCTM,"Historical Topics for the Mathematics Classroom." Included are such themes as the change from a geometric to an algebraic solution of problems, the development of algebraic symbolism, the algebraic contributions of different countries, the origin and…
Probabilistic Concurrent Kleene Algebra
Directory of Open Access Journals (Sweden)
Annabelle McIver
2013-06-01
Full Text Available We provide an extension of concurrent Kleene algebras to account for probabilistic properties. The algebra yields a unified framework containing nondeterminism, concurrency and probability and is sound with respect to the set of probabilistic automata modulo probabilistic simulation. We use the resulting algebra to generalise the algebraic formulation of a variant of Jones' rely/guarantee calculus.
Generalized Quantum Current Algebras
Institute of Scientific and Technical Information of China (English)
ZHAO Liu
2001-01-01
Two general families of new quantum-deformed current algebras are proposed and identified both as infinite Hopf family of algebras, a structure which enables one to define "tensor products" of these algebras. The standard quantum affine algebras turn out to be a very special case of the two algebra families, in which case the infinite Hopf family structure degenerates into a standard Hopf algebra. The relationship between the two algebraic families as well as thefr various special examples are discussed, and the free boson representation is also considered.
El-Chaar, Caroline
2012-01-01
In this thesis, four realizations of the Onsager algebra are explored. We begin with its original definition as introduced by Lars Onsager. We then examine how the Onsager algebra can be presented as a Lie algebra with two generators and two relations. The third realization of the Onsager algebra consists of viewing it as an equivariant map algebra which then gives us the tools to classify its closed ideals. Finally, we examine the Onsager algebra as a subalgebra of the tetrahedron algebra. Using this fourth realization, we explicitly describe all its ideals.
Perturbations of planar algebras
Das, Paramita; Gupta, Ved Prakash
2010-01-01
We introduce the concept of {\\em weight} of a planar algebra $P$ and construct a new planar algebra referred as the {\\em perturbation of $P$} by the weight. We establish a one-to-one correspondence between pivotal structures on 2-categories and perturbations of planar algebras by weights. To each bifinite bimodule over $II_1$-factors, we associate a {\\em bimodule planar algebra} bimodule corresponds naturally with sphericality of the bimodule planar algebra. As a consequence of this, we reproduce an extension of Jones' theorem (of associating 'subfactor planar algebras' to extremal subfactors). Conversely, given a bimodule planar algebra, we construct a bifinite bimodule whose associated bimodule planar algebra is the one which we start with using perturbations and Jones-Walker-Shlyakhtenko-Kodiyalam-Sunder method of reconstructing an extremal subfactor from a subfactor planar algebra. We show that the perturbation class of a bimodule planar algebra contains a unique spherical unimodular bimodule planar algeb...
Linear algebra a first course with applications to differential equations
Apostol, Tom M
2014-01-01
Developed from the author's successful two-volume Calculus text this book presents Linear Algebra without emphasis on abstraction or formalization. To accommodate a variety of backgrounds, the text begins with a review of prerequisites divided into precalculus and calculus prerequisites. It continues to cover vector algebra, analytic geometry, linear spaces, determinants, linear differential equations and more.
Multiparameter Twisted Weyl Algebras
Futorny, Vyacheslav; Hartwig, Jonas T.
2011-01-01
We introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordan's simple localization of the multiparameter quantized Weyl algebra and Hayashi's q-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus, we also describe all Whittaker pairs...
Duality principle and braided geometry
Majid, S
1994-01-01
We give an overview of a new kind symmetry in physics which exists between observables and states and which is made possible by the language of Hopf algebras and quantum geometry. It has been proposed by the author as a feature of Planck scale physics. More recent work includes corresponding results at the semiclassical level of Poisson-Lie groups and at the level of braided groups and braided geometry.
Hartshorne, Robin
2000-01-01
In recent years, I have been teaching a junior-senior-level course on the classi cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa rately. The remainder of the book is an exploration of questions that arise natu rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And ...
Energy Technology Data Exchange (ETDEWEB)
Byrd, M.
1997-10-01
The group SU(3) is parameterized in terms of generalized {open_quotes}Euler angles{close_quotes}. The differential operators of SU(3) corresponding to the Lie Algebra elements are obtained, the invariant forms are found, the group invariant volume element is found, and some relevant comments about the geometry of the group manifold are made.
Institute of Scientific and Technical Information of China (English)
Jia-feng; Lü
2007-01-01
[1]Priddy S.Koszul resolutions.Trans Amer Math Soc,152:39-60 (1970)[2]Beilinson A,Ginszburg V,Soergel W.Koszul duality patterns in representation theory.J Amer Math Soc,9:473-525 (1996)[3]Aquino R M,Green E L.On modules with linear presentations over Koszul algebras.Comm Algebra,33:19-36 (2005)[4]Green E L,Martinez-Villa R.Koszul and Yoneda algebras.Representation theory of algebras (Cocoyoc,1994).In:CMS Conference Proceedings,Vol 18.Providence,RI:American Mathematical Society,1996,247-297[5]Berger R.Koszulity for nonquadratic algebras.J Algebra,239:705-734 (2001)[6]Green E L,Marcos E N,Martinez-Villa R,et al.D-Koszul algebras.J Pure Appl Algebra,193:141-162(2004)[7]He J W,Lu D M.Higher Koszul Algebras and A-infinity Algebras.J Algebra,293:335-362 (2005)[8]Green E L,Marcos E N.δ-Koszul algebras.Comm Algebra,33(6):1753-1764 (2005)[9]Keller B.Introduction to A-infinity algebras and modules.Homology Homotopy Appl,3:1-35 (2001)[10]Green E L,Martinez-Villa R,Reiten I,et al.On modules with linear presentations.J Algebra,205(2):578-604 (1998)[11]Keller B.A-infinity algebras in representation theory.Contribution to the Proceedings of ICRA Ⅸ.Beijing:Peking University Press,2000[12]Lu D M,Palmieri J H,Wu Q S,et al.A∞-algebras for ring theorists.Algebra Colloq,11:91-128 (2004)[13]Weibel C A.An Introduction to homological algebra.Cambridge Studies in Avanced Mathematics,Vol 38.Cambridge:Cambridge University Press,1995
Maps from the enveloping algebra of the positive Witt algebra to regular algebras
Sierra, Susan J.; Walton, Chelsea
2015-01-01
We construct homomorphisms from the universal enveloping algebra of the positive (part of the) Witt algebra to several different Artin-Schelter regular algebras, and determine their kernels and images. As a result, we produce elementary proofs that the universal enveloping algebras of the Virasoro algebra, the Witt algebra, and the positive Witt algebra are neither left nor right noetherian.
Higher algebraic K-theory an overview
Lluis-Puebla, Emilio; Gillet, Henri; Soulé, Christophe; Snaith, Victor
1992-01-01
This book is a general introduction to Higher Algebraic K-groups of rings and algebraic varieties, which were first defined by Quillen at the beginning of the 70's. These K-groups happen to be useful in many different fields, including topology, algebraic geometry, algebra and number theory. The goal of this volume is to provide graduate students, teachers and researchers with basic definitions, concepts and results, and to give a sampling of current directions of research. Written by five specialists of different parts of the subject, each set of lectures reflects the particular perspective ofits author. As such, this volume can serve as a primer (if not as a technical basic textbook) for mathematicians from many different fields of interest.
Cecil, Thomas E
2015-01-01
This exposition provides the state-of-the art on the differential geometry of hypersurfaces in real, complex, and quaternionic space forms. Special emphasis is placed on isoparametric and Dupin hypersurfaces in real space forms as well as Hopf hypersurfaces in complex space forms. The book is accessible to a reader who has completed a one-year graduate course in differential geometry. The text, including open problems and an extensive list of references, is an excellent resource for researchers in this area. Geometry of Hypersurfaces begins with the basic theory of submanifolds in real space forms. Topics include shape operators, principal curvatures and foliations, tubes and parallel hypersurfaces, curvature spheres and focal submanifolds. The focus then turns to the theory of isoparametric hypersurfaces in spheres. Important examples and classification results are given, including the construction of isoparametric hypersurfaces based on representations of Clifford algebras. An in-depth treatment of Dupin hy...
Boicescu, V; Georgescu, G; Rudeanu, S
1991-01-01
The Lukasiewicz-Moisil algebras were created by Moisil as an algebraic counterpart for the many-valued logics of Lukasiewicz. The theory of LM-algebras has developed to a considerable extent both as an algebraic theory of intrinsic interest and in view of its applications to logic and switching theory.This book gives an overview of the theory, comprising both classical results and recent contributions, including those of the authors. N-valued and &THgr;-valued algebras are presented, as well as &THgr;-algebras with negation.Mathematicians interested in lattice theory or symbolic logic, and computer scientists, will find in this monograph stimulating material for further research.
C*-algebras and operator theory
Murphy, Gerald J
1990-01-01
This book constitutes a first- or second-year graduate course in operator theory. It is a field that has great importance for other areas of mathematics and physics, such as algebraic topology, differential geometry, and quantum mechanics. It assumes a basic knowledge in functional analysis but no prior acquaintance with operator theory is required.
Representations of twisted current algebras
Lau, Michael
2013-01-01
We use evaluation representations to give a complete classification of the finite-dimensional simple modules of twisted current algebras. This generalizes and unifies recent work on multiloop algebras, current algebras, equivariant map algebras, and twisted forms.
Hom-alternative algebras and Hom-Jordan algebras
Makhlouf, Abdenacer
2009-01-01
The purpose of this paper is to introduce Hom-alternative algebras and Hom-Jordan algebras. We discuss some of their properties and provide construction procedures using ordinary alternative algebras or Jordan algebras. Also, we show that a polarization of Hom-associative algebra leads to Hom-Jordan algebra.
Walsh, Edward T
2014-01-01
This introductory text is designed to help undergraduate students develop a solid foundation in geometry. Early chapters progress slowly, cultivating the necessary understanding and self-confidence for the more rapid development that follows. The extensive treatment can be easily adapted to accommodate shorter courses. Starting with the language of mathematics as expressed in the algebra of logic and sets, the text covers geometric sets of points, separation and angles, triangles, parallel lines, similarity, polygons and area, circles, space geometry, and coordinate geometry. Each chapter incl
Lie Algebra of Noncommutative Inhomogeneous Hopf Algebra
Lagraa, M.; Touhami, N.
1997-01-01
We construct the vector space dual to the space of right-invariant differential forms construct from a first order differential calculus on inhomogeneous quantum group. We show that this vector space is equipped with a structure of a Hopf algebra which closes on a noncommutative Lie algebra satisfying a Jacobi identity.
Categories and Commutative Algebra
Salmon, P
2011-01-01
L. Badescu: Sur certaines singularites des varietes algebriques.- D.A. Buchsbaum: Homological and commutative algebra.- S. Greco: Anelli Henseliani.- C. Lair: Morphismes et structures algebriques.- B.A. Mitchell: Introduction to category theory and homological algebra.- R. Rivet: Anneaux de series formelles et anneaux henseliens.- P. Salmon: Applicazioni della K-teoria all'algebra commutativa.- M. Tierney: Axiomatic sheaf theory: some constructions and applications.- C.B. Winters: An elementary lecture on algebraic spaces.
Algebraic statistics computational commutative algebra in statistics
Pistone, Giovanni; Wynn, Henry P
2000-01-01
Written by pioneers in this exciting new field, Algebraic Statistics introduces the application of polynomial algebra to experimental design, discrete probability, and statistics. It begins with an introduction to Gröbner bases and a thorough description of their applications to experimental design. A special chapter covers the binary case with new application to coherent systems in reliability and two level factorial designs. The work paves the way, in the last two chapters, for the application of computer algebra to discrete probability and statistical modelling through the important concept of an algebraic statistical model.As the first book on the subject, Algebraic Statistics presents many opportunities for spin-off research and applications and should become a landmark work welcomed by both the statistical community and its relatives in mathematics and computer science.
REAL PIECEWISE ALGEBRAIC VARIETY
Institute of Scientific and Technical Information of China (English)
Ren-hong Wang; Yi-sheng Lai
2003-01-01
We give definitions of real piecewise algebraic variety and its dimension. By using the techniques of real radical ideal, P-radical ideal, affine Hilbert polynomial, Bernstein-net form of polynomials on simplex, and decomposition of semi-algebraic set, etc., we deal with the dimension of the real piecewise algebraic variety and real Nullstellensatz in Cμ spline ring.
Deficiently Extremal Gorenstein Algebras
Indian Academy of Sciences (India)
Pavinder Singh
2011-08-01
The aim of this article is to study the homological properties of deficiently extremal Gorenstein algebras. We prove that if / is an odd deficiently extremal Gorenstein algebra with pure minimal free resolution, then the codimension of / must be odd. As an application, the structure of pure minimal free resolution of a nearly extremal Gorenstein algebra is obtained.
Bases of Schur algebras associated to cellularly stratified diagram algebras
Bowman, C
2011-01-01
We examine homomorphisms between induced modules for a certain class of cellularly stratified diagram algebras, including the BMW algebra, Temperley-Lieb algebra, Brauer algebra, and (quantum) walled Brauer algebra. We define the `permutation' modules for these algebras, these are one-sided ideals which allow us to study the diagrammatic Schur algebras of Hartmann, Henke, Koenig and Paget. We construct bases of these Schur algebras in terms of modified tableaux. On the way we prove that the (quantum) walled Brauer algebra and the Temperley-Lieb algebra are both cellularly stratified and therefore have well-defined Specht filtrations.
Exceptional geometry and Borcherds superalgebras
Palmkvist, Jakob
2015-01-01
We study generalized diffeomorphisms in exceptional geometry with U-duality group E_{n(n)} from an algebraic point of view. By extending the Lie algebra e_n to an infinite-dimensional Borcherds superalgebra, involving also the extension to e_{n+1}, the generalized Lie derivatives can be expressed in a simple way, and the expressions take the same form for any n less than 8. The closure of the transformations then follows from the Jacobi identity and the grading of e_{n+1} with respect to e_n.
Institute of Scientific and Technical Information of China (English)
张彦军
2012-01-01
通过把正交几何椭圆拟合与代数拟合和椭圆定义的迭代拟合进行比较分析,考虑到原有拟合存在的缺陷,充分应用正交的概念及最小二乘法(LS)原理和方法的优点,对椭圆进行正交的几何拟合.实验表明,正交几何拟合很好地弥补了其它拟合方法的缺陷,取得良好的效果.%Passing comparing orthogonal geometry ellipse fitting method with algebra method and ellipse definition method, in consideration of the defect of original method, full applied advantage of orthogonal concept and least square (LS) principle, this paper carries orthogonal geometry fitting method on ellipse.The experiment expressed that orthogonal geometry fitting method make up other method nicely and obtained good result.
Jespers, Eric; Riley, David; Siciliano, Salvatore
2007-01-01
An algebra is called a GI-algebra if its group of units satisfies a group identity. We provide positive support for the following two open problems. 1. Does every algebraic GI-algebra satisfy a polynomial identity? 2. Is every algebraically generated GI-algebra locally finite?
A vector space approach to geometry
Hausner, Melvin
2010-01-01
The effects of geometry and linear algebra on each other receive close attention in this examination of geometry's correlation with other branches of math and science. In-depth discussions include a review of systematic geometric motivations in vector space theory and matrix theory; the use of the center of mass in geometry, with an introduction to barycentric coordinates; axiomatic development of determinants in a chapter dealing with area and volume; and a careful consideration of the particle problem. 1965 edition.
Indian Academy of Sciences (India)
Antonio J Calderón Martín; Manuel Forero Piulestán; José M Sánchez Delgado
2012-05-01
We study the structure of split Malcev algebras of arbitrary dimension over an algebraically closed field of characteristic zero. We show that any such algebras is of the form $M=\\mathcal{U}+\\sum_jI_j$ with $\\mathcal{U}$ a subspace of the abelian Malcev subalgebra and any $I_j$ a well described ideal of satisfying $[I_j, I_k]=0$ if ≠ . Under certain conditions, the simplicity of is characterized and it is shown that is the direct sum of a semisimple split Lie algebra and a direct sum of simple non-Lie Malcev algebras.
Computer algebra and operators
Fateman, Richard; Grossman, Robert
1989-01-01
The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.
Taghavi, Ali
2013-01-01
We study some properies of $Z^{*}$ algebras, thos C^* algebra which all positive elements are zero divisors. We show by means of an example that an extension of a Z* algebra by a Z* algebra is not necessarily Z* algebra. However we prove that an extension of a non Z* algebra by a non Z* algebra is again a Z^* algebra. As an application of our methods, we prove that evey compact subset of the positive cones of a C* algebra has an upper bound in the algebra.
Sati, Hisham
2015-01-01
We uncover higher algebraic structures on Noether currents and BPS charges. It is known that equivalence classes of conserved currents form a Lie algebra. We show that at least for target space symmetries of higher parameterized WZW-type sigma-models this naturally lifts to a Lie (p+1)-algebra structure on the Noether currents themselves. Applied to the Green-Schwarz-type action functionals for super p-brane sigma-models this yields super Lie (p+1)-algebra refinements of the traditional BPS brane charge extensions of supersymmetry algebras. We discuss this in the generality of higher differential geometry, where it applies also to branes with (higher) gauge fields on their worldvolume. Applied to the M5-brane sigma-model we recover and properly globalize the M-theory super Lie algebra extension of 11-dimensional superisometries by 2-brane and 5-brane charges. Passing beyond the infinitesimal Lie theory we find cohomological corrections to these charges in higher analogy to the familiar corrections for D-brane...
Quantum entanglement and geometry of determinantal varieties
Chen, H
2001-01-01
From the consideration of checking mixed states by separable pure states, we introduce algebraic sets, which are determinantal varieties in complex projective spaces or products of complex projective spaces, for mixed states in bipartite or multipartite quantum systems as their invariants under local unitary transformations. The algebraic sets of the mixed states are independent of eigenvalues and only measure the geometric positions of eigenvectors. Thus complex differential geometry and algebraic geometry of these determinantal varieties turn to be powerful tools for the study of quantum entanglement of both bipartite and multipartite mixed states. The algebraic sets have to be the sum of linear subspaces if the mixed states are separable. Examples of the entangled mixed states which are invariant under partial transposition (thus bound entanglement) are constructed systematically from this new separability criterion. The relation of these invariants and Schmidt numbers of the mixed states is anlysised and ...
Weak Lie symmetry and extended Lie algebra
Energy Technology Data Exchange (ETDEWEB)
Goenner, Hubert [Institute for Theoretical Physics, Friedrich-Hund-Platz 1, University of Goettingen, D-37077 Gottingen (Germany)
2013-04-15
The concept of weak Lie motion (weak Lie symmetry) is introduced. Applications given exhibit a reduction of the usual symmetry, e.g., in the case of the rotation group. In this context, a particular generalization of Lie algebras is found ('extended Lie algebras') which turns out to be an involutive distribution or a simple example for a tangent Lie algebroid. Riemannian and Lorentz metrics can be introduced on such an algebroid through an extended Cartan-Killing form. Transformation groups from non-relativistic mechanics and quantum mechanics lead to such tangent Lie algebroids and to Lorentz geometries constructed on them (1-dimensional gravitational fields).
Index Theorems on Torsional Geometries
Kimura, Tetsuji
2007-01-01
We study various topological invariants on a differential geometry in the presence of a totally anti-symmetric torsion H under the closed condition dH=0. By using the identification between the Clifford algebra on a geometry and the canonical quantization condition of fermion in the quantum mechanics, we construct the N=1 quantum mechanical sigma model in the Hamiltonian formalism and extend this model to N=2 system, equipped with the totally anti-symmetric tensor associated with the torsion on the target space geometry. Next we construct transition elements in the Lagrangian path integral formalism and apply them to the analyses of the Witten indices in supersymmetric systems. We improve the formulation of the Dirac index on the torsional geometry which has already been studied. We also formulate the Euler characteristic and the Hirzebruch signature on the torsional geometry.
The Boolean algebra and central Galois algebras
Directory of Open Access Journals (Sweden)
George Szeto
2001-01-01
Full Text Available Let B be a Galois algebra with Galois group G, Jg={b∈B∣bx=g(xb for all x∈B} for g∈G, and BJg=Beg for a central idempotent eg. Then a relation is given between the set of elements in the Boolean algebra (Ba,≤ generated by {0,eg∣g∈G} and a set of subgroups of G, and a central Galois algebra Be with a Galois subgroup of G is characterized for an e∈Ba.
On Derivations Of Genetic Algebras
International Nuclear Information System (INIS)
A genetic algebra is a (possibly non-associative) algebra used to model inheritance in genetics. In application of genetics this algebra often has a basis corresponding to genetically different gametes, and the structure constant of the algebra encode the probabilities of producing offspring of various types. In this paper, we find the connection between the genetic algebras and evolution algebras. Moreover, we prove the existence of nontrivial derivations of genetic algebras in dimension two
Stable endomorphism algebras of modules over special biserial algebras
Schröer, Jan; Zimmermann, Alexander
2002-01-01
We prove that the stable endomorphism algebra of a module without self-extensions over a special biserial algebra is a gentle algebra. In particular, it is again special biserial. As a consequence, any algebra which is derived equivalent to a gentle algebra is gentle.
$L_{\\infty}$ algebra structures of Lie algebra deformations
Gao, Jining
2004-01-01
In this paper,we will show how to kill the obstructions to Lie algebra deformations via a method which essentially embeds a Lie algebra into Strong homotopy Lie algebra or $L_{\\infty}$ algebra. All such obstructions have been transfered to the revelvant $L_{\\infty}$ algebras which contain only three terms
Omni-Lie Color Algebras and Lie Color 2-Algebras
Zhang, Tao
2013-01-01
Omni-Lie color algebras over an abelian group with a bicharacter are studied. The notions of 2-term color $L_{\\infty}$-algebras and Lie color 2-algebras are introduced. It is proved that there is a one-to-one correspondence between Lie color 2-algebras and 2-term color $L_{\\infty}$-algebras.
Evolution algebras and their applications
Tian, Jianjun Paul
2008-01-01
Behind genetics and Markov chains, there is an intrinsic algebraic structure. It is defined as a type of new algebra: as evolution algebra. This concept lies between algebras and dynamical systems. Algebraically, evolution algebras are non-associative Banach algebras; dynamically, they represent discrete dynamical systems. Evolution algebras have many connections with other mathematical fields including graph theory, group theory, stochastic processes, dynamical systems, knot theory, 3-manifolds, and the study of the Ihara-Selberg zeta function. In this volume the foundation of evolution algebra theory and applications in non-Mendelian genetics and Markov chains is developed, with pointers to some further research topics.
Finite-dimensional (*)-serial algebras
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
Let A be a finite-dimensional associative algebra with identity over a field k. In this paper we introduce the concept of (*)-serial algebras which is a generalization of serial algebras. We investigate the properties of (*)-serial algebras, and we obtain suficient and necessary conditions for an associative algebra to be (*)-serial.
Directory of Open Access Journals (Sweden)
R. A. Borzooei
2006-01-01
Full Text Available We study hyper BCC-algebras which are a common generalization of BCC-algebras and hyper BCK-algebras. In particular, we investigate different types of hyper BCC-ideals and describe the relationship among them. Next, we calculate all nonisomorphic 22 hyper BCC-algebras of order 3 of which only three are not hyper BCK-algebras.
On the Toroidal Leibniz Algebras
Institute of Scientific and Technical Information of China (English)
Dong LIU; Lei LIN
2008-01-01
Toroidal Leibniz algebras are the universal central extensions of the iterated loop algebras gOC[t±11 ,...,t±v1] in the category of Leibniz algebras. In this paper, some properties and representations of toroidal Leibniz algebras are studied. Some general theories of central extensions of Leibniz algebras are also obtained.
Algebraic classification of Robinson-Trautman spacetimes
Podolsky, Jiri
2016-01-01
We consider a general class of four-dimensional geometries admitting a null vector field that has no twist and no shear but has an arbitrary expansion. We explicitly present the Petrov classification of such Robinson-Trautman (and Kundt) gravitational fields, based on the algebraic properties of the Weyl tensor. In particular, we determine all algebraically special subcases when the optically privileged null vector field is a multiple principal null direction (PND), as well as all the cases when it remains a single PND. No field equations are a priori applied, so that our classification scheme can be used in any metric theory of gravity in four dimensions. In the classic Einstein theory this reproduces previous results for vacuum spacetimes, possibly with a cosmological constant, pure radiation and electromagnetic field, but can be applied to an arbitrary matter content. As non-trivial explicit examples we investigate specific algebraic properties of the Robinson-Trautman spacetimes with a free scalar field, ...
Developable algebraic surfaces
Institute of Scientific and Technical Information of China (English)
CHEN Dongren; WANG Guojin
2004-01-01
An algebraic surface can be defined by an implicit polynomial equation F(x,y,z)=0. In this paper, general characterizations of developable algebraic surfaces of arbitrary degree are presented. Using the shift operators of the subscripts of Bézier ordinates, the uniform apparent discriminants of developable algebraic surfaces to their Bézier ordinates are given directly. To degree 2 algebraic surfaces, which are widely used in computer aided geometric design and graphics, all possible developable surface types are obtained. For more conveniently applying algebraic surfaces of high degree to computer aided geometric design, the notion of ε-quasi-developable surfaces is introduced, and an example of using a quasi-developable algebraic surface of degree 3 to interpolate three curves of degree 2 is given.
Iachello, Francesco
2015-01-01
This course-based primer provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, it concisely presents the basic concepts of Lie algebras, their representations and their invariants. The second part includes a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book highlights a number of examples that help to illustrate the abstract algebraic definitions and includes a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators...
Symmetric Extended Ockham Algebras
Institute of Scientific and Technical Information of China (English)
T.S. Blyth; Jie Fang
2003-01-01
The variety eO of extended Ockham algebras consists of those algealgebra with an additional endomorphism k such that the unary operations f and k commute. Here, we consider the cO-algebras which have a property of symmetry. We show that there are thirty two non-isomorphic subdirectly irreducible symmetric extended MS-algebras and give a complete description of them.2000 Mathematics Subject Classification: 06D15, 06D30
Brouder, Christian
2002-01-01
The Laplace Hopf algebra created by Rota and coll. is generalized to provide an algebraic tool for combinatorial problems of quantum field theory. This framework encompasses commutation relations, normal products, time-ordered products and renormalisation. It considers the operator product and the time-ordered product as deformations of the normal product. In particular, it gives an algebraic meaning to Wick's theorem and it extends the concept of Laplace pairing to prove that the renormalise...
Algebraic nonlinear collective motion
Troupe, J.; Rosensteel, G.
1999-01-01
Finite-dimensional Lie algebras of vector fields determine geometrical collective models in quantum and classical physics. Every set of vector fields on Euclidean space that generates the Lie algebra sl(3, R) and contains the angular momentum algebra so(3) is determined. The subset of divergence-free sl(3, R) vector fields is proven to be indexed by a real number $\\Lambda$. The $\\Lambda=0$ solution is the linear representation that corresponds to the Riemann ellipsoidal model. The nonlinear g...
Algebraic extensions of fields
McCarthy, Paul J
1991-01-01
""...clear, unsophisticated and direct..."" - MathThis textbook is intended to prepare graduate students for the further study of fields, especially algebraic number theory and class field theory. It presumes some familiarity with topology and a solid background in abstract algebra. Chapter 1 contains the basic results concerning algebraic extensions. In addition to separable and inseparable extensions and normal extensions, there are sections on finite fields, algebraically closed fields, primitive elements, and norms and traces. Chapter 2 is devoted to Galois theory. Besides the fundamenta
Kurosh, A G; Stark, M; Ulam, S
1965-01-01
Lectures in General Algebra is a translation from the Russian and is based on lectures on specialized courses in general algebra at Moscow University. The book starts with the basics of algebra. The text briefly describes the theory of sets, binary relations, equivalence relations, partial ordering, minimum condition, and theorems equivalent to the axiom of choice. The text gives the definition of binary algebraic operation and the concepts of groups, groupoids, and semigroups. The book examines the parallelism between the theory of groups and the theory of rings; such examinations show the
Solomon, Alan D
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Boolean Algebra includes set theory, sentential calculus, fundamental ideas of Boolean algebras, lattices, rings and Boolean algebras, the structure of a Boolean algebra, and Boolean
Underwood, Robert G
2015-01-01
This text aims to provide graduate students with a self-contained introduction to topics that are at the forefront of modern algebra, namely, coalgebras, bialgebras, and Hopf algebras. The last chapter (Chapter 4) discusses several applications of Hopf algebras, some of which are further developed in the author’s 2011 publication, An Introduction to Hopf Algebras. The book may be used as the main text or as a supplementary text for a graduate algebra course. Prerequisites for this text include standard material on groups, rings, modules, algebraic extension fields, finite fields, and linearly recursive sequences. The book consists of four chapters. Chapter 1 introduces algebras and coalgebras over a field K; Chapter 2 treats bialgebras; Chapter 3 discusses Hopf algebras and Chapter 4 consists of three applications of Hopf algebras. Each chapter begins with a short overview and ends with a collection of exercises which are designed to review and reinforce the material. Exercises range from straightforw...
Relations Between BZMVdM-Algebra and Other Algebras
Institute of Scientific and Technical Information of China (English)
高淑萍; 邓方安; 刘三阳
2003-01-01
Some properties of BZMVdM-algebra are proved, and a new operator is introduced. It is shown that the substructure of BZMVdM-algebra can produce a quasi-lattice implication algebra. The relations between BZMVdM-algebra and other algebras are discussed in detail. A pseudo-distance function is defined in linear BZMVdM-algebra, and its properties are derived.
Four-manifolds, geometries and knots
Hillman, Jonathan
2002-01-01
The goal of this book is to characterize algebraically the closed 4-manifolds that fibre nontrivially or admit geometries in the sense of Thurston, or which are obtained by surgery on 2-knots, and to provide a reference for the topology of such manifolds and knots. The first chapter is purely algebraic. The rest of the book may be divided into three parts: general results on homotopy and surgery (Chapters 2-6), geometries and geometric decompositions (Chapters 7-13), and 2-knots (Chapters 14-...
Loop Quantum Gravity Vacuum with Nondegenerate Geometry
Directory of Open Access Journals (Sweden)
Hanno Sahlmann
2012-05-01
Full Text Available In loop quantum gravity, states of the gravitational field turn out to be excitations over a vacuum state that is sharply peaked on a degenerate spatial geometry. While this vacuum is singled out as fundamental due to its invariance properties, it is also important to consider states that describe non-degenerate geometries. Such states have features of Bose condensate ground states. We discuss their construction for the Lie algebra as well as the Weyl algebra setting, and point out possible applications in effective field theory, Loop Quantum Cosmology, as well as further generalizations.
Essential linear algebra with applications a problem-solving approach
Andreescu, Titu
2014-01-01
This textbook provides a rigorous introduction to linear algebra in addition to material suitable for a more advanced course while emphasizing the subject’s interactions with other topics in mathematics such as calculus and geometry. A problem-based approach is used to develop the theoretical foundations of vector spaces, linear equations, matrix algebra, eigenvectors, and orthogonality. Key features include: • a thorough presentation of the main results in linear algebra along with numerous examples to illustrate the theory; • over 500 problems (half with complete solutions) carefully selected for their elegance and theoretical significance; • an interleaved discussion of geometry and linear algebra, giving readers a solid understanding of both topics and the relationship between them. Numerous exercises and well-chosen examples make this text suitable for advanced courses at the junior or senior levels. It can also serve as a source of supplementary problems for a sophomore-level course. ...
Differential Hopf algebra structures on the universal enveloping algebra ofa Lie algebra
Hijligenberg, N.W. van den; Martini, R.
1995-01-01
We discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra structure of $U(g
Tubular algebras and affine Kac-Moody algebras
Institute of Scientific and Technical Information of China (English)
Zheng-xin CHEN; Ya-nan LIN
2007-01-01
The purpose of this paper is to construct quotient algebras L(A)C1/I(A) of complex degenerate composition Lie algebras L(A)C1 by some ideals, where L(A)C1 is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)C1/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)C1 generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)C1 generated by simple A-modules.
Tubular algebras and affine Kac-Moody algebras
Institute of Scientific and Technical Information of China (English)
2007-01-01
The purpose of this paper is to construct quotient algebras L(A)1C/I(A) of complex degenerate composition Lie algebras L(A)1C by some ideals, where L(A)1C is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)1C/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules.
Maximal and reduced Roe algebras of coarsely embeddable spaces
Spakula, Jan
2011-01-01
Gong, Wang and Yu introduced a maximal, or universal, version of the Roe C*-algebra associated to a metric space. We study the relationship between this maximal Roe algebra and the usual version, in both the uniform and non-uniform cases. The main result is that if a (uniformly discrete, bounded geometry) metric space X coarsely embeds in a Hilbert space, then the canonical map between the maximal and usual (uniform) Roe algebras induces an isomorphism on K-theory. We also give a simple proof that if X has property A, then the maximal and usual (uniform) Roe algebras are the same. These two results are natural coarse-geometric analogues of certain well-known implications of a-T-menability and amenability for group C*-algebras. The techniques used are E-theoretic, building on work of Higson-Kasparov-Trout and Yu.
Universal Algebras of Hurwitz Numbers
A. Mironov; Morozov, A; Natanzon, S.
2009-01-01
Infinite-dimensional universal Cardy-Frobenius algebra is constructed, which unifies all particular algebras of closed and open Hurwitz numbers and is closely related to the algebra of differential operators, familiar from the theory of Generalized Kontsevich Model.
Noncommutative spectral geometry, dissipation and the origin of quantization
Sakellariadou, Mairi; Stabile, Antonio; Vitiello, Giuseppe
2012-01-01
We present a physical interpretation of the doubling of the algebra, which is the basic ingredient of the noncommutative spectral geometry, developed by Connes and collaborators as an approach to unification. We discuss its connection to dissipation and to the gauge structure of the theory. We then argue, following 't Hooft's conjecture, that noncommutative spectral geometry classical construction carries implicit in its feature of the doubling of the algebra the seeds of quantization.
(Quasi-)Poisson enveloping algebras
Yang, Yan-Hong; Yuan YAO; Ye, Yu
2010-01-01
We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.
On algebraic volume density property
Kaliman, Shulim; Kutzschebauch, Frank
2012-01-01
A smooth affine algebraic variety $X$ equipped with an algebraic volume form $\\omega$ has the algebraic volume density property (AVDP) if the Lie algebra generated by completely integrable algebraic vector fields of $\\omega$-divergence zero coincides with the space of all algebraic vector fields of $\\omega$-divergence zero. We develop an effective criterion of verifying whether a given $X$ has AVDP. As an application of this method we establish AVDP for any homogeneous space $X=G/R$ that admi...
Automorphism groups of some algebras
Institute of Scientific and Technical Information of China (English)
PARK; Hong; Goo; LEE; Jeongsig; CHOI; Seul; Hee; NAM; Ki-Bong
2009-01-01
The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra Am,m+n is the universal enveloping algebra of the generalized Witt algebra W(m,m + n).
Automorphism groups of some algebras
Institute of Scientific and Technical Information of China (English)
PARK Hong Goo; LEE Jeongsig; CHOI Seul Hee; CHEN XueQing; NAM Ki-Bong
2009-01-01
The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra Am,m+n is the universal enveloping algebra of the generalized Witt algebra W(m, m+n).
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this…
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
Directory of Open Access Journals (Sweden)
Carlos C. Peña
2000-05-01
Full Text Available Topological algebras of sequences of complex numbers are introduced, endowed with a Hadamard product type. The complex homomorphisms on these algebras are characterized, and units, prime cyclic ideals, prime closed ideals, and prime minimal ideals, discussed. Existence of closed and maximal ideals are investigated, and it is shown that the Jacobson and nilradicals are both trivial.
Heinicke, C; Heinicke, Christian; Hehl, Friedrich W.
2001-01-01
We survey the application of computer algebra in the context of gravitational theories. After some general remarks, we show of how to check the second Bianchi-identity by means of the Reduce package Excalc. Subsequently we list some computer algebra systems and packages relevant to applications in gravitational physics. We conclude by presenting a couple of typical examples.
Algebraic boundaries of Hilbert's SOS cones
Blekherman, Grigoriy; Ottem, John Christian; Ranestad, Kristian; Sturmfels, Bernd
2011-01-01
We study the geometry underlying the difference between non-negative polynomials and sums of squares. The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether-Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized respectively by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.
Introduction to noncommutative algebra
Brešar, Matej
2014-01-01
Providing an elementary introduction to noncommutative rings and algebras, this textbook begins with the classical theory of finite dimensional algebras. Only after this, modules, vector spaces over division rings, and tensor products are introduced and studied. This is followed by Jacobson's structure theory of rings. The final chapters treat free algebras, polynomial identities, and rings of quotients. Many of the results are not presented in their full generality. Rather, the emphasis is on clarity of exposition and simplicity of the proofs, with several being different from those in other texts on the subject. Prerequisites are kept to a minimum, and new concepts are introduced gradually and are carefully motivated. Introduction to Noncommutative Algebra is therefore accessible to a wide mathematical audience. It is, however, primarily intended for beginning graduate and advanced undergraduate students encountering noncommutative algebra for the first time.
Elements of mathematics algebra
Bourbaki, Nicolas
2003-01-01
This is a softcover reprint of the English translation of 1990 of the revised and expanded version of Bourbaki's, Algèbre, Chapters 4 to 7 (1981). This completes Algebra, 1 to 3, by establishing the theories of commutative fields and modules over a principal ideal domain. Chapter 4 deals with polynomials, rational fractions and power series. A section on symmetric tensors and polynomial mappings between modules, and a final one on symmetric functions, have been added. Chapter 5 was entirely rewritten. After the basic theory of extensions (prime fields, algebraic, algebraically closed, radical extension), separable algebraic extensions are investigated, giving way to a section on Galois theory. Galois theory is in turn applied to finite fields and abelian extensions. The chapter then proceeds to the study of general non-algebraic extensions which cannot usually be found in textbooks: p-bases, transcendental extensions, separability criterions, regular extensions. Chapter 6 treats ordered groups and fields and...
Deformation of central charges, vertex operator algebras whose Griess algebras are Jordan algebras
Ashihara, Takahiro; Miyamoto, Masahiko
2008-01-01
If a vertex operator algebra $V=\\oplus_{n=0}^{\\infty}V_n$ satisfies $\\dim V_0=1, V_1=0$, then $V_2$ has a commutative (nonassociative) algebra structure called Griess algebra. One of the typical examples of commutative (nonassociative) algebras is a Jordan algebra. For example, the set $Sym_d(\\C)$ of symmetric matrices of degree $d$ becomes a Jordan algebra. On the other hand, in the theory of vertex operator algebras, central charges influence the properties of vertex operator algebras. In t...
New symbolic tools for differential geometry, gravitation, and field theory
Anderson, I. M.; Torre, C. G.
2012-01-01
DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, spinor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. These capabilities, combined with dramatic recent improvements in symbolic approaches to solving algebraic and differential equations, have allowed for development of powerful new tools for solving research problems in gravitation and field theory. The purpose of this paper is to describe some of these new tools and present some advanced applications involving: Killing vector fields and isometry groups, Killing tensors, algebraic classification of solutions of the Einstein equations, and symmetry reduction of field equations.
Commutative and Non-commutative Parallelogram Geometry: an Experimental Approach
Bertram, Wolfgang
2013-01-01
By "parallelogram geometry" we mean the elementary, "commutative", geometry corresponding to vector addition, and by "trapezoid geometry" a certain "non-commutative deformation" of the former. This text presents an elementary approach via exercises using dynamical software (such as geogebra), hopefully accessible to a wide mathematical audience, from undergraduate students and high school teachers to researchers, proceeding in three steps: (1) experimental geometry, (2) algebra (linear algebr...
The Planar Algebra of a Semisimple and Cosemisimple Hopf Algebra
Indian Academy of Sciences (India)
Vijay Kodiyalam; V S Sunder
2006-11-01
To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.
Graded Lie Algebra Generating of Parastatistical Algebraic Relations
Institute of Scientific and Technical Information of China (English)
JING Si-Cong; YANG Wei-Min; LI Ping
2001-01-01
A new kind of graded Lie algebra (We call it Z2,2 graded Lie algebra) is introduced as a framework for formulating parasupersymmetric theories. By choosing suitable Bose subspace of the Z2,2 graded Lie algebra and using relevant generalized Jacobi identities, we generate the whole algebraic structure of parastatistics.
Perspectives in Analysis, Geometry, and Topology
Itenberg, I V; Passare, Mikael
2012-01-01
The articles in this volume are invited papers from the Marcus Wallenberg symposium and focus on research topics that bridge the gap between analysis, geometry, and topology. The encounters between these three fields are widespread and often provide impetus for major breakthroughs in applications. Topics include new developments in low dimensional topology related to invariants of links and three and four manifolds; Perelman's spectacular proof of the Poincare conjecture; and the recent advances made in algebraic, complex, symplectic, and tropical geometry.
Semigroups and computer algebra in algebraic structures
Bijev, G.
2012-11-01
Some concepts in semigroup theory can be interpreted in several algebraic structures. A generalization fA,B,fA,B(X) = A(X')B of the complement operator (') on Boolean matrices is made, where A and B denote any rectangular Boolean matrices. While (') is an isomorphism between Boolean semilattices, the generalized complement operator is homomorphism in the general case. The map fA,B and its general inverse (fA,B)+ have quite similar properties to those in the linear algebra and are useful for solving linear equations in Boolean matrix algebras. For binary relations on a finite set, necessary and sufficient conditions for the equation αξβ = γ to have a solution ξ are proved. A generalization of Green's equivalence relations in semigroups for rectangular matrices is proposed. Relationships between them and the Moore-Penrose inverses are investigated. It is shown how any generalized Green's H-class could be constructed by given its corresponding linear subspaces and converted into a group isomorphic to a linear group. Some information about using computer algebra methods concerning this paper is given.
Algebraic points on meromorphic curves
Herblot, Mathilde
2012-01-01
The classic Schneider-Lang theorem in transcendence theory asserts that there are only finitely many points at which algebraically independent complex meromorphic functions of finite order of growth can simultaneously take values in a number field, when satisfying a polynomial differential equation with coefficients in this given number field. In this article, we are interested in generalizing this theorem in two directions. First, instead of considering meromorphic functions on C we consider holomorphic maps on an affine curve over the field C or C_p. This extends a statement of D. Bertrand, which applies to meromorphic functions on P^1(C) or P^1(C_p) minus a finite subset of points. Secondly, we deal with algebraic values taken by the functions, instead of rational values as in the classic setting, inspired by a work of D. Bertrand. We prove a geometric statement extending those two results, using the slopes method, written in the language of Arakelov geometry. In the complex case, we recover a special case...
Algebraic Systems Biology: A Case Study for the Wnt Pathway.
Gross, Elizabeth; Harrington, Heather A; Rosen, Zvi; Sturmfels, Bernd
2016-01-01
Steady-state analysis of dynamical systems for biological networks gives rise to algebraic varieties in high-dimensional spaces whose study is of interest in their own right. We demonstrate this for the shuttle model of the Wnt signaling pathway. Here, the variety is described by a polynomial system in 19 unknowns and 36 parameters. It has degree 9 over the parameter space. This case study explores multistationarity, model comparison, dynamics within regions of the state space, identifiability, and parameter estimation, from a geometric point of view. We employ current methods from computational algebraic geometry, polyhedral geometry, and combinatorics.
Algebraic Systems Biology: A Case Study for the Wnt Pathway.
Gross, Elizabeth; Harrington, Heather A; Rosen, Zvi; Sturmfels, Bernd
2016-01-01
Steady-state analysis of dynamical systems for biological networks gives rise to algebraic varieties in high-dimensional spaces whose study is of interest in their own right. We demonstrate this for the shuttle model of the Wnt signaling pathway. Here, the variety is described by a polynomial system in 19 unknowns and 36 parameters. It has degree 9 over the parameter space. This case study explores multistationarity, model comparison, dynamics within regions of the state space, identifiability, and parameter estimation, from a geometric point of view. We employ current methods from computational algebraic geometry, polyhedral geometry, and combinatorics. PMID:26645985
Differential Geometry and Lie Groups for Physicists
Fecko, Marián.
2011-03-01
Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.
Springer, T A
1998-01-01
"[The first] ten chapters...are an efficient, accessible, and self-contained introduction to affine algebraic groups over an algebraically closed field. The author includes exercises and the book is certainly usable by graduate students as a text or for self-study...the author [has a] student-friendly style… [The following] seven chapters... would also be a good introduction to rationality issues for algebraic groups. A number of results from the literature…appear for the first time in a text." –Mathematical Reviews (Review of the Second Edition) "This book is a completely new version of the first edition. The aim of the old book was to present the theory of linear algebraic groups over an algebraically closed field. Reading that book, many people entered the research field of linear algebraic groups. The present book has a wider scope. Its aim is to treat the theory of linear algebraic groups over arbitrary fields. Again, the author keeps the treatment of prerequisites self-contained. The material of t...
Kandasamy, W B Vasantha
2008-01-01
In this book, the authors introduce the notion of Super linear algebra and super vector spaces using the definition of super matrices defined by Horst (1963). This book expects the readers to be well-versed in linear algebra. Many theorems on super linear algebra and its properties are proved. Some theorems are left as exercises for the reader. These new class of super linear algebras which can be thought of as a set of linear algebras, following a stipulated condition, will find applications in several fields using computers. The authors feel that such a paradigm shift is essential in this computerized world. Some other structures ought to replace linear algebras which are over a century old. Super linear algebras that use super matrices can store data not only in a block but in multiple blocks so it is certainly more powerful than the usual matrices. This book has 3 chapters. Chapter one introduces the notion of super vector spaces and enumerates a number of properties. Chapter two defines the notion of sup...
Pottmann, Helmut
2014-11-26
Around 2005 it became apparent in the geometry processing community that freeform architecture contains many problems of a geometric nature to be solved, and many opportunities for optimization which however require geometric understanding. This area of research, which has been called architectural geometry, meanwhile contains a great wealth of individual contributions which are relevant in various fields. For mathematicians, the relation to discrete differential geometry is significant, in particular the integrable system viewpoint. Besides, new application contexts have become available for quite some old-established concepts. Regarding graphics and geometry processing, architectural geometry yields interesting new questions but also new objects, e.g. replacing meshes by other combinatorial arrangements. Numerical optimization plays a major role but in itself would be powerless without geometric understanding. Summing up, architectural geometry has become a rewarding field of study. We here survey the main directions which have been pursued, we show real projects where geometric considerations have played a role, and we outline open problems which we think are significant for the future development of both theory and practice of architectural geometry.
Generalizing the Connes Moscovici Hopf algebra to contain all rooted trees
Energy Technology Data Exchange (ETDEWEB)
Agarwala, Susama [Mathematical Institute, Radcliff Observatory Quarter, Oxford University, Woodstock Road, Oxford (United Kingdom); Delaney, Colleen [University of California Santa Barbara, South Hall, Room 6607, Santa Barbara, California 93106 (United States)
2015-04-15
This paper defines a generalization of the Connes-Moscovici Hopf algebra, H(1), that contains the entire Hopf algebra of rooted trees. A relationship between the former, a much studied object in non-commutative geometry, and the latter, a much studied object in perturbative quantum field theory, has been established by Connes and Kreimer. The results of this paper open the door to study the cohomology of the Hopf algebra of rooted trees.
Geometry, Renormalization, And Supersymmetry
Berg, G M
2001-01-01
This thesis is about understanding, applying and improving quantum field theory. We compute renormalization group flows as the evolution of a “coarse-graining” operator without the need for a Euclidean formulation. Renormalization is cast in the form of a Lie algebra of (in general infinite) matrices that generate, by exponentiation, counterterms for diagrams with subdivergences. These results may shed light on noncommutative geometry. We check our results in a scalar three-loop example. Then, we consider the renormalization of a certain supersymmetric gauge theory, the low-energy limit of a string model. We compare results to those computed directly in the string model and find agreement. Finally, we discuss the possibility of detecting quantum-mechanical phases distinguishing the two Pin groups, double covers of the full Lorentz group. Majorana fermions, if detected, would provide an important testing ground; such particles can restrict the choice of Pin group.
Equational axioms of test algebra
Hollenberg, M.
2008-01-01
We present a complete axiomatization of test algebra ([24,18,29]), the two-sorted algebraic variant of Propositional Dynamic Logic (PDL,[21,7]). The axiomatization consists of adding a finite number of equations to any axiomatization of Kleene algebra ([15,26,17,4]) and algebraic translations of the
Process algebra for Hybrid systems
Bergstra, J.A.; Middelburg, C.A.
2008-01-01
We propose a process algebra obtained by extending a combination of the process algebra with continuous relative timing from Baeten and Middelburg [Process Algebra with Timing, Springer, Chap. 4, 2002] and the process algebra with propositional signals from Baeten and Bergstra [Theoretical Computer
Process algebra for hybrid systems
Bergstra, J.A.; Middelburg, C.A.
2005-01-01
We propose a process algebra obtained by extending a combination of the process algebra with continuous relative timing from Baeten and Middelburg (Process Algebra with Timing, Springer,Berlin, 2002, Chapter 4), and the process algebra with propositional signals from Baeten and Bergstra(Theoret. Com
Maor, Eli
2014-01-01
If you've ever thought that mathematics and art don't mix, this stunning visual history of geometry will change your mind. As much a work of art as a book about mathematics, Beautiful Geometry presents more than sixty exquisite color plates illustrating a wide range of geometric patterns and theorems, accompanied by brief accounts of the fascinating history and people behind each. With artwork by Swiss artist Eugen Jost and text by acclaimed math historian Eli Maor, this unique celebration of geometry covers numerous subjects, from straightedge-and-compass constructions to intriguing configur
Symplectic algebraic dynamics algorithm
Institute of Scientific and Technical Information of China (English)
2007-01-01
Based on the algebraic dynamics solution of ordinary differential equations andintegration of ,the symplectic algebraic dynamics algorithm sn is designed,which preserves the local symplectic geometric structure of a Hamiltonian systemand possesses the same precision of the na ve algebraic dynamics algorithm n.Computer experiments for the 4th order algorithms are made for five test modelsand the numerical results are compared with the conventional symplectic geometric algorithm,indicating that sn has higher precision,the algorithm-inducedphase shift of the conventional symplectic geometric algorithm can be reduced,and the dynamical fidelity can be improved by one order of magnitude.
Schneider, Hans
1989-01-01
Linear algebra is one of the central disciplines in mathematics. A student of pure mathematics must know linear algebra if he is to continue with modern algebra or functional analysis. Much of the mathematics now taught to engineers and physicists requires it.This well-known and highly regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related t
Directory of Open Access Journals (Sweden)
J. W. Kitchen
1994-01-01
Full Text Available We study bundles of Banach algebras π:A→X, where each fiber Ax=π−1({x} is a Banach algebra and X is a compact Hausdorff space. In the case where all fibers are commutative, we investigate how the Gelfand representation of the section space algebra Γ(π relates to the Gelfand representation of the fibers. In the general case, we investigate how adjoining an identity to the bundle π:A→X relates to the standard adjunction of identities to the fibers.
Michael Roitman
2003-01-01
In this paper we prove that for any commutative (but in general non-associative) algebra $A$ with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra $V = V_0 \\oplus V_2 \\oplus V_3\\oplus ...$, such that $\\dim V_0 = 1$ and $V_2$ contains $A$. We can choose $V$ so that if $A$ has a unit $e$, then $2e$ is the Virasoro element of $V$, and if $G$ is a finite group of automorphisms of $A$, then $G$ acts on $V$ as well. In addition, the algebra $V$ can be chosen with...
Vitório Pereira, Jorge
2015-01-01
This book takes an in-depth look at abelian relations of codimension one webs in the complex analytic setting. In its classical form, web geometry consists in the study of webs up to local diffeomorphisms. A significant part of the theory revolves around the concept of abelian relation, a particular kind of functional relation among the first integrals of the foliations of a web. Two main focuses of the book include how many abelian relations can a web carry and which webs are carrying the maximal possible number of abelian relations. The book offers complete proofs of both Chern’s bound and Trépreau’s algebraization theorem, including all the necessary prerequisites that go beyond elementary complex analysis or basic algebraic geometry. Most of the examples known up to date of non-algebraizable planar webs of maximal rank are discussed in detail. A historical account of the algebraization problem for maximal rank webs of codimension one is also presented.
Geometry and the Quantum: Basics
Chamseddine, Ali H; Mukhanov, Viatcheslav
2014-01-01
Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M_2(H) and M_4(C) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these represen...
Meadow enriched ACP process algebras
J.A. Bergstra; Middelburg, C.A.
2009-01-01
We introduce the notion of an ACP process algebra. The models of the axiom system ACP are the origin of this notion. ACP process algebras have to do with processes in which no data are involved. We also introduce the notion of a meadow enriched ACP process algebra, which is a simple generalization of the notion of an ACP process algebra to processes in which data are involved. In meadow enriched ACP process algebras, the mathematical structure for data is a meadow.
Algebraic Properties of Propositional Calculus
Schuh, Bernd R.
2009-01-01
In this short note we relate some known properties of propositional calculus to purely algebraic considerations of a Boolean algebra. Classes of formulas of propositional calculus are considered as elements of a Boolean algebra. As such they can be represented by uniquely defined elements of this algebra which we call "logical primes". The algebraic notations appear useful because they make it possible to derive well known properties of propositional calculus by simple calculations or to subs...
Hom-power associative algebras
Yau, Donald
2010-01-01
A generalization of power associative algebra, called Hom-power associative algebra, is studied. The main result says that a multiplicative Hom-algebra is Hom-power associative if and only if it satisfies two identities of degrees three and four. It generalizes Albert's result that power associativity is equivalent to third and fourth power associativity. In particular, multiplicative right Hom-alternative algebras and non-commutative Hom-Jordan algebras are Hom-power associative.
On isomorphisms of integral table algebras
Institute of Scientific and Technical Information of China (English)
FAN; Yun(樊恽); SUN; Daying(孙大英)
2002-01-01
For integral table algebras with integral table basis T, we can consider integral R-algebra RT over a subring R of the ring of the algebraic integers. It is proved that an R-algebra isomorphism between two integral table algebras must be an integral table algebra isomorphism if it is compatible with the so-called normalizings of the integral table algebras.
Cameron, Peter J
2007-01-01
This Second Edition of a classic algebra text includes updated and comprehensive introductory chapters,. new material on axiom of Choice, p-groups and local rings, discussion of theory and applications, and over 300 exercises. It is an ideal introductory text for all Year 1 and 2 undergraduate students in mathematics. - ;Developed to meet the needs of modern students, this Second Edition of the classic algebra text by Peter Cameron covers all the abstract algebra an undergraduate student is likely to need. Starting with an introductory overview of numbers, sets and functions, matrices, polynomials, and modular arithmetic, the text then introduces the most important algebraic structures: groups, rings and fields, and their properties. This is followed by coverage of vector spaces and modules with. applications to abelian groups and canonical forms before returning to the construction of the number systems, including the existence of transcendental numbers. The final chapters take the reader further into the th...
Axler, Sheldon
2015-01-01
This best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions. No prerequisites are assumed other than the ...
Indian Academy of Sciences (India)
Vijay Kodiyalam; R Srinivasan; V S Sunder
2000-08-01
In this paper, we study a tower $\\{A^G_n(d):n≥ 1\\}$ of finite-dimensional algebras; here, represents an arbitrary finite group, denotes a complex parameter, and the algebra $A^G_n(d)$ has a basis indexed by `-stable equivalence relations' on a set where acts freely and has 2 orbits. We show that the algebra $A^G_n(d)$ is semi-simple for all but a finite set of values of , and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the `generic case'. Finally we determine the Bratteli diagram of the tower $\\{A^G_n(d): n≥ 1\\}$ (in the generic case).
Introduction to abstract algebra
Nicholson, W Keith
2012-01-01
Praise for the Third Edition ". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."-Zentralblatt MATH The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book's unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately be
Institute of Scientific and Technical Information of China (English)
Antonio AIZPURU; Antonio GUTI(E)RREZ-D(A)VILA
2004-01-01
In this paper we will study some families and subalgebras ( ) of ( )(N) that let us characterize the unconditional convergence of series through the weak convergence of subseries ∑i∈A xi, A ∈ ( ).As a consequence, we obtain a new version of the Orlicz-Pettis theorem, for Banach spaces. We also study some relationships between algebraic properties of Boolean algebras and topological properties of the corresponding Stone spaces.
Intermediate algebra a textworkbook
McKeague, Charles P
1985-01-01
Intermediate Algebra: A Text/Workbook, Second Edition focuses on the principles, operations, and approaches involved in intermediate algebra. The publication first takes a look at basic properties and definitions, first-degree equations and inequalities, and exponents and polynomials. Discussions focus on properties of exponents, polynomials, sums, and differences, multiplication of polynomials, inequalities involving absolute value, word problems, first-degree inequalities, real numbers, opposites, reciprocals, and absolute value, and addition and subtraction of real numbers. The text then ex
Andrilli, Stephen
2010-01-01
Elementary Linear Algebra develops and explains in careful detail the computational techniques and fundamental theoretical results central to a first course in linear algebra. This highly acclaimed text focuses on developing the abstract thinking essential for further mathematical study. The authors give early, intensive attention to the skills necessary to make students comfortable with mathematical proofs. The text builds a gradual and smooth transition from computational results to general theory of abstract vector spaces. It also provides flexbile coverage of practical applications, expl
Law, Shirley
2014-01-01
International audience A general lattice theoretic construction of Reading constructs Hopf subalgebras of the Malvenuto-Reutenauer Hopf algebra (MR) of permutations. The products and coproducts of these Hopf subalgebras are defined extrinsically in terms of the embedding in MR. The goal of this paper is to find an intrinsic combinatorial description of a particular one of these Hopf subalgebras. This Hopf algebra has a natural basis given by permutations that we call Pell permutations. The...
Holomorphically Equivalent Algebraic Embeddings
Feller, Peter; Stampfli, Immanuel
2014-01-01
We prove that two algebraic embeddings of a smooth variety $X$ in $\\mathbb{C}^m$ are the same up to a holomorphic coordinate change, provided that $2 \\dim X + 1$ is smaller than or equal to $m$. This improves an algebraic result of Nori and Srinivas. For the proof we extend a technique of Kaliman using generic linear projections of $\\mathbb{C}^m$.
Beginning algebra a textworkbook
McKeague, Charles P
1985-01-01
Beginning Algebra: A Text/Workbook, Second Edition focuses on the principles, operations, and approaches involved in algebra. The publication first elaborates on the basics, linear equations and inequalities, and graphing and linear systems. Discussions focus on solving linear systems by graphing, elimination method, graphing ordered pairs and straight lines, linear and compound inequalities, addition and subtraction of real numbers, and properties of real numbers. The text then examines exponents and polynomials, factoring, and rational expressions. Topics include multiplication and division
Spinors in Physics and Geometry
Trautman, A.; Furlan, G.
1988-11-01
The Table of Contents for the full book PDF is as follows: * Preface * Killing Spinors According to O. Hijazi and Applications * Self-Duality Conditions Satisfied by the Spin Connections on Spheres * Maslov Index and Half - Forms * Spin - 3/2 Fields on Black Hole Spacetimes * Indecomposable Conformal Spinors and Operator Product Expansions in a Massless QED Model * Nonlinear Spinor Representations * Nonlinear Wave Equations for Intrinsic Spinor Coordinates * Twistors - "Spinors" of SU(2,2), Their Generalizations and Achievements * Spinors, Reflections and Clifford Algebras: A Review * overline {SL}(n, R) Spinors for Particles, Gravity and Superstrings * Spinors on Compact Riemann Surfaces * Simple Spinors as Urfelder * Applications of Cartan Spinors to Differential Geometry in Higher Dimensions * Killing Spinors on Spheres and Projective Spaces * Spinor Structures on Homogeneous Riemannian Spaces * Classical Strings and Minimal Surfaces * Representing Spinors with Differential Forms * Inequalities for Spinors Norms in Clifford Algebras * The Importance of Spin * The Theory of World Spinors * Final List of Participants
Differential Hopf algebra structures on the universal enveloping algebra ofa Lie algebra
Hijligenberg, van den, N.W.; Martini, R.
1995-01-01
We discuss a method to construct a De Rham complex (differential algebra) of Poincar'e-Birkhoff-Witt-type on the universal enveloping algebra of a Lie algebra $g$. We determine the cases in which this gives rise to a differential Hopf algebra that naturally extends the Hopf algebra structure of $U(g)$. The construction of such differential structures is interpreted in terms of colour Lie superalgebras.
Topological ∗-algebras with *-enveloping Algebras II
Indian Academy of Sciences (India)
S J Bhatt
2001-02-01
Universal *-algebras *() exist for certain topological ∗-algebras called algebras with a *-enveloping algebra. A Frechet ∗-algebra has a *-enveloping algebra if and only if every operator representation of maps into bounded operators. This is proved by showing that every unbounded operator representation , continuous in the uniform topology, of a topological ∗-algebra , which is an inverse limit of Banach ∗-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-* algebra () of . Given a *-dynamical system (, , ), any topological ∗-algebra containing (, ) as a dense ∗-subalgebra and contained in the crossed product *-algebra *(, , ) satisfies ()=*(, , ). If $G = \\mathbb{R}$, if is an -invariant dense Frechet ∗-subalgebra of such that () = , and if the action on is -tempered, smooth and by continuous ∗-automorphisms: then the smooth Schwartz crossed product $S(\\mathbb{R}, B, )$ satisfies $E(S(\\mathbb{R}, B, )) = C^*(\\mathbb{R}, A, )$. When is a Lie group, the ∞-elements ∞(), the analytic elements () as well as the entire analytic elements () carry natural topologies making them algebras with a *-enveloping algebra. Given a non-unital *-algebra , an inductive system of ideals is constructed satisfying $A = C^*-\\mathrm{ind} \\lim I_$; and the locally convex inductive limit $\\mathrm{ind}\\lim I_$ is an -convex algebra with the *-enveloping algebra and containing the Pedersen ideal of . Given generators with weakly Banach admissible relations , we construct universal topological ∗-algebra (, ) and show that it has a *-enveloping algebra if and only if (, ) is *-admissible.
Institute of Scientific and Technical Information of China (English)
An Hui-hui; Wang Zhi-chun
2016-01-01
L-octo-algebra with 8 operations as the Lie algebraic analogue of octo-algebra such that the sum of 8 operations is a Lie algebra is discussed. Any octo-algebra is an L-octo-algebra. The relationships among L-octo-algebras, L-quadri-algebras, L-dendriform algebras, pre-Lie algebras and Lie algebras are given. The close relationships between L-octo-algebras and some interesting structures like Rota-Baxter operators, classical Yang-Baxter equations and some bilinear forms satisfying certain conditions are given also.
Axis Problem of Rough 3-Valued Algebras
Institute of Scientific and Technical Information of China (English)
Jianhua Dai; Weidong Chen; Yunhe Pan
2006-01-01
The collection of all the rough sets of an approximation space has been given several algebraic interpretations, including Stone algebras, regular double Stone algebras, semi-simple Nelson algebras, pre-rough algebras and 3-valued Lukasiewicz algebras. A 3-valued Lukasiewicz algebra is a Stone algebra, a regular double Stone algebra, a semi-simple Nelson algebra, a pre-rough algebra. Thus, we call the algebra constructed by the collection of rough sets of an approximation space a rough 3-valued Lukasiewicz algebra. In this paper,the rough 3-valued Lukasiewicz algebras, which are a special kind of 3-valued Lukasiewicz algebras, are studied. Whether the rough 3-valued Lukasiewicz algebra is a axled 3-valued Lukasiewicz algebra is examined.
On the Homogeneous Model Of Euclidean Geometry
Gunn, Charles
2011-01-01
We attach the degenerate signature (n,0,1) to the projectivized dual Grassmann algebra over R(n+1). We explore the use of the resulting Clifford algebra as a model for euclidean geometry. We avoid problems with the degenerate metric by constructing an algebra isomorphism between this Grassmann algebra and its dual, that yields non-metric meet and join operators. We review the Cayley-Klein construction of the projective (homogeneous) model for euclidean geometry leading to the choice of the signature (n,0,1). We focus on the cases of n=2 and n=3 in detail, enumerating the geometric products between simple k- and m-vectors. We establish that versor (sandwich) operators provide all euclidean isometries, both direct and indirect. We locate the spin group, a double cover of the direct euclidean group, inside the even subalgebra of the Clifford algebra, and provide a simple algorithm for calculating the logarithm of such elements. We conclude with an elementary account of euclidean rigid body motion within this fra...
Path operator algebras in conformal quantum field theories
International Nuclear Information System (INIS)
Two different kinds of path algebras and methods from noncommutative geometry are applied to conformal field theory: Fusion rings and modular invariants of extended chiral algebras are analyzed in terms of essential paths which are a path description of intertwiners. As an example, the ADE classification of modular invariants for minimal models is reproduced. The analysis of two-step extensions is included. Path algebras based on a path space interpretation of character identities can be applied to the analysis of fusion rings as well. In particular, factorization properties of character identities and therefore of the corresponding path spaces are - by means of K-theory - related to the factorization of the fusion ring of Virasoro- and W-algebras. Examples from nonsupersymmetric as well as N=2 supersymmetric minimal models are discussed. (orig.)
Connes, Alain
1994-01-01
This English version of the path-breaking French book on this subject gives the definitive treatment of the revolutionary approach to measure theory, geometry, and mathematical physics developed by Alain Connes. Profusely illustrated and invitingly written, this book is ideal for anyone who wants to know what noncommutative geometry is, what it can do, or how it can be used in various areas of mathematics, quantization, and elementary particles and fields.Key Features* First full treatment of the subject and its applications* Written by the pioneer of this field* Broad applications in mathemat
Berger, Marcel
2010-01-01
Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces,
Developments and retrospectives in Lie theory algebraic methods
Penkov, Ivan; Wolf, Joseph
2014-01-01
This volume reviews and updates a prominent series of workshops in representation/Lie theory, and reflects the widespread influence of those workshops in such areas as harmonic analysis, representation theory, differential geometry, algebraic geometry, and mathematical physics. Many of the contributors have had leading roles in both the classical and modern developments of Lie theory and its applications. This Work, entitled Developments and Retrospectives in Lie Theory, and comprising 26 articles, is organized in two volumes: Algebraic Methods and Geometric and Analytic Methods. This is the Algebraic Methods volume. The Lie Theory Workshop series, founded by Joe Wolf and Ivan Penkov and joined shortly thereafter by Geoff Mason, has been running for over two decades. Travel to the workshops has usually been supported by the NSF, and local universities have provided hospitality. The workshop talks have been seminal in describing new perspectives in the field covering broad areas of current research. Mos...
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
Over a field F of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector space A［D]=A［D] from any pair of a commutative associative algebra A with an identity element and the polynomial algebra ［D] of a commutative derivation subalgebra D of A. We prove that A[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only if A is D－simple and A［D] acts faithfully on A. Thus we obtain a lot of simple algebras.
Simple Algebras of Invariant Operators
Institute of Scientific and Technical Information of China (English)
Xiaorong Shen; J.D.H. Smith
2001-01-01
Comtrans algebras were introduced in as algebras with two trilinear operators, a commutator [x, y, z] and a translator , which satisfy certain identities. Previously known simple comtrans algebras arise from rectangular matrices, simple Lie algebras, spaces equipped with a bilinear form having trivial radical, spaces of hermitian operators over a field with a minimum polynomial x2+1. This paper is about generalizing the hermitian case to the so-called invariant case. The main result of this paper shows that the vector space of n-dimensional invariant operators furnishes some comtrans algebra structures, which are simple provided that certain Jordan and Lie algebras are simple.
Algebraically contractible topological tensor network states
Denny, S J; Jaksch, D; Clark, S R
2011-01-01
We adapt the bialgebra and Hopf relations to expose internal structure in the ground state of a Hamiltonian with $Z_2$ topological order. Its tensor network description allows for exact contraction through simple diagrammatic rewrite rules. The contraction property does not depend on specifics such as geometry, but rather originates from the non-trivial algebraic properties of the constituent tensors. We then generalise the resulting tensor network from a spin-half lattice to a class of exactly contractible states on spin-S degrees of freedom, yielding the most efficient tensor network description of finite Abelian lattice gauge theories. We gain a new perspective on these states as examples of two-dimensional quantum states with algebraically contractible tensor network representations. The introduction of local perturbations to the network is shown to reduce the von Neumann entropy of string-like regions, creating an unentangled sub-system within the bulk in a certain limit. We also show how perturbations l...
Superconformal Algebras and Supersymmetric Integrable Flows
Sachse, Christoph; Devchand, Chandrasekhar
2009-01-01
After a comprehensive review of superconformal algebras, super-diffeomorphisms and supervector fields on supercircles S^{1|n} we study various supersymmetric extensions of the KdV and Camassa-Holm equations. We describe their (super) Hamiltonian structures and their connection to bihamiltonian geometry. These are interpreted as geodesic flows on various superconformal groups. We also give an example of superintegrable systems of Ramond type. The one-parameter family of equations shown by Degasperis, Holm and Hone (DHH) to possess multi-peakon solutions is identified as a geodesic flow equation on a one-parameter deformation of the group of diffeomorphisms of the circle, with respect to a right-invariant Sobolev H^1--metric. A supersymmetrisation of the algebra of deformed vector fields on S^1 yields supersymmetric DHH equations (also known as b-field equations), which include the supersymmetric Camassa--Holm equation as a special case.
Fast decoding of codes from algebraic plane curves
DEFF Research Database (Denmark)
Justesen, Jørn; Larsen, Knud J.; Jensen, Helge Elbrønd;
1992-01-01
Improvement to an earlier decoding algorithm for codes from algebraic geometry is presented. For codes from an arbitrary regular plane curve the authors correct up to d*/2-m2 /8+m/4-9/8 errors, where d* is the designed distance of the code and m is the degree of the curve. The complexity of finding...
A concrete approach to abstract algebra from the integers to the insolvability of the quintic
Bergen, Jeffrey
2010-01-01
A Concrete Approach to Abstract Algebra begins with a concrete and thorough examination of familiar objects like integers, rational numbers, real numbers, complex numbers, complex conjugation and polynomials, in this unique approach, the author builds upon these familar objects and then uses them to introduce and motivate advanced concepts in algebra in a manner that is easier to understand for most students. The text will be of particular interest to teachers and future teachers as it links abstract algebra to many topics wich arise in courses in algebra, geometry, trigonometry, preca
Desseyn, H. O.; And Others
1985-01-01
Compares linear-nonlinear and planar-nonplanar geometry through the valence-shell electron pairs repulsion (V.S.E.P.R.), Mulliken-Walsh, and electrostatic force theories. Indicates that although the V.S.E.P.R. theory has more advantages for elementary courses, an explanation of the best features of the different theories offers students a better…
Pottmann, Helmut; Eigensatz, Michael; Vaxman, A.; Wallner, Johannes
2015-01-01
Around 2005 it became apparent in the geometry processing community that freeform architecture contains many problems of a geometric nature to be solved, and many opportunities for optimization which however require geometric understanding. This area of research, which has been called architectural
Quanta of Geometry and Unification
Chamseddine, Ali H
2016-01-01
This is a tribute to Abdus Salam's memory whose insight and creative thinking set for me a role model to follow. In this contribution I show that the simple requirement of volume quantization in space-time (with Euclidean signature) uniquely determines the geometry to be that of a noncommutative space whose finite part is based on an algebra that leads to Pati-Salam grand unified models. The Standard Model corresponds to a special case where a mathematical constraint (order one condition) is satisfied. This provides evidence that Salam was a visionary who was generations ahead of his time.
An invitation to noncommutative geometry
Marcolli, Matilde
2008-01-01
This is the first existing volume that collects lectures on this important and fast developing subject in mathematics. The lectures are given by leading experts in the field and the range of topics is kept as broad as possible by including both the algebraic and the differential aspects of noncommutative geometry as well as recent applications to theoretical physics and number theory. Sample Chapter(s). A Walk in the Noncommutative Garden (1,639 KB). Contents: A Walk in the Noncommutative Garden (A Connes & M Marcolli); Renormalization of Noncommutative Quantum Field Theory (H Grosse & R Wulke
Integral geometry and representation theory
Gel'fand, I M; Vilenkin, N Ya
1966-01-01
Generalized Functions, Volume 5: Integral Geometry and Representation Theory is devoted to the theory of representations, focusing on the group of two-dimensional complex matrices of determinant one.This book emphasizes that the theory of representations is a good example of the use of algebraic and geometric methods in functional analysis, in which transformations are performed not on the points of a space, but on the functions defined on it. The topics discussed include Radon transform on a real affine space, integral transforms in the complex domain, and representations of the group of comp
INdAM conference "CoMeTA 2013 - Combinatorial Methods in Topology and Algebra"
Delucchi, Emanuele; Moci, Luca
2015-01-01
Combinatorics plays a prominent role in contemporary mathematics, due to the vibrant development it has experienced in the last two decades and its many interactions with other subjects. This book arises from the INdAM conference "CoMeTA 2013 - Combinatorial Methods in Topology and Algebra,'' which was held in Cortona in September 2013. The event brought together emerging and leading researchers at the crossroads of Combinatorics, Topology and Algebra, with a particular focus on new trends in subjects such as: hyperplane arrangements; discrete geometry and combinatorial topology; polytope theory and triangulations of manifolds; combinatorial algebraic geometry and commutative algebra; algebraic combinatorics; and combinatorial representation theory. The book is divided into two parts. The first expands on the topics discussed at the conference by providing additional background and explanations, while the second presents original contributions on new trends in the topics addressed by the conference.
Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction
Wasserman, Nicholas H.
2016-01-01
This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by…
Rings of quotients of incidence algebras and path algebras
DEFF Research Database (Denmark)
Esparza, Eduardo Ortega
2006-01-01
We compute the maximal right/left/symmetric rings of quotients of finite dimensional incidence and graph algebras. We show that these rings of quotients are Morita equivalent to incidence algebras and path algebras respectively, with respect to simpler, well determined partially ordered sets...
The Planar Algebra Associated to a Kac Algebra
Indian Academy of Sciences (India)
Vijay Kodiyalam; Zeph Landau; V S Sunder
2003-02-01
We obtain (two equivalent) presentations – in terms of generators and relations-of the planar algebra associated with the subfactor corresponding to (an outer action on a factor by) a finite-dimensional Kac algebra. One of the relations shows that the antipode of the Kac algebra agrees with the `rotation on 2-boxes'.
Structure of Solvable Quadratic Lie Algebras
Institute of Scientific and Technical Information of China (English)
ZHU Lin-sheng
2005-01-01
@@ Killing form plays a key role in the theory of semisimple Lie algebras. It is natural to extend the study to Lie algebras with a nondegenerate symmetric invariant bilinear form. Such a Lie algebra is generally called a quadratic Lie algebra which occur naturally in physics[10,12,13]. Besides semisimple Lie algebras, interesting quadratic Lie algebras include the Kac-Moody algebras and the Extended Affine Lie algebras.
Algebraic totality, towards completeness
Tasson, Christine
2009-01-01
Finiteness spaces constitute a categorical model of Linear Logic (LL) whose objects can be seen as linearly topologised spaces, (a class of topological vector spaces introduced by Lefschetz in 1942) and morphisms as continuous linear maps. First, we recall definitions of finiteness spaces and describe their basic properties deduced from the general theory of linearly topologised spaces. Then we give an interpretation of LL based on linear algebra. Second, thanks to separation properties, we can introduce an algebraic notion of totality candidate in the framework of linearly topologised spaces: a totality candidate is a closed affine subspace which does not contain 0. We show that finiteness spaces with totality candidates constitute a model of classical LL. Finally, we give a barycentric simply typed lambda-calculus, with booleans ${\\mathcal{B}}$ and a conditional operator, which can be interpreted in this model. We prove completeness at type ${\\mathcal{B}}^n\\to{\\mathcal{B}}$ for every n by an algebraic metho...
Algebraic quantum field theory
International Nuclear Information System (INIS)
The basic assumption that the complete information relevant for a relativistic, local quantum theory is contained in the net structure of the local observables of this theory results first of all in a concise formulation of the algebraic structure of the superselection theory and an intrinsic formulation of charge composition, charge conjugation and the statistics of an algebraic quantum field theory. In a next step, the locality of massive particles together with their spectral properties are wed for the formulation of a selection criterion which opens the access to the massive, non-abelian quantum gauge theories. The role of the electric charge as a superselection rule results in the introduction of charge classes which in term lead to a set of quantum states with optimum localization properties. Finally, the asymptotic observables of quantum electrodynamics are investigated within the framework of algebraic quantum field theory. (author)
Jarvis, Frazer
2014-01-01
The technical difficulties of algebraic number theory often make this subject appear difficult to beginners. This undergraduate textbook provides a welcome solution to these problems as it provides an approachable and thorough introduction to the topic. Algebraic Number Theory takes the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to perform. The middle section considers more general theory and results for number fields, and the book concludes with some topics which are more likely to be suitable for advanced students, namely, the analytic class number formula and the number field sieve. This is the fi...
Directory of Open Access Journals (Sweden)
Michael Roitman
2008-08-01
Full Text Available In this paper we prove that for any commutative (but in general non-associative algebra A with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra V = V_0 oplus V2 oplus V3 oplus ..., such that dim V_0 = 1 and V_2 contains A. We can choose V so that if A has a unit e, then 2e is the Virasoro element of V, and if G is a finite group of automorphisms of A, then G acts on V as well. In addition, the algebra V can be chosen with a non-degenerate invariant bilinear form, in which case it is simple.
Roitman, Michael
2008-08-01
In this paper we prove that for any commutative (but in general non-associative) algebra A with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra V = V0 Å V2 Å V3 Å ¼, such that dim V0 = 1 and V2 contains A. We can choose V so that if A has a unit e, then 2e is the Virasoro element of V, and if G is a finite group of automorphisms of A, then G acts on V as well. In addition, the algebra V can be chosen with a non-degenerate invariant bilinear form, in which case it is simple.
Algebra for Gifted Third Graders.
Borenson, Henry
1987-01-01
Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)
Indian Academy of Sciences (India)
Anil K Karn
2003-02-01
Order unit property of a positive element in a *-algebra is defined. It is proved that precisely projections satisfy this order theoretic property. This way, unital hereditary *-subalgebras of a *-algebra are characterized.
Computer Program For Linear Algebra
Krogh, F. T.; Hanson, R. J.
1987-01-01
Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.
Reed, Nat
2011-01-01
For grades 3-5, our State Standards-based combined resource meets the algebraic concepts addressed by the NCTM standards and encourages the students to review the concepts in unique ways. The task sheets introduce the mathematical concepts to the students around a central problem taken from real-life experiences, while the drill sheets provide warm-up and timed practice questions for the students to strengthen their procedural proficiency skills. Included are opportunities for problem-solving, patterning, algebraic graphing, equations and determining averages. The combined task & drill sheets
Reed, Nat
2011-01-01
For grades 6-8, our State Standards-based combined resource meets the algebraic concepts addressed by the NCTM standards and encourages the students to review the concepts in unique ways. The task sheets introduce the mathematical concepts to the students around a central problem taken from real-life experiences, while the drill sheets provide warm-up and timed practice questions for the students to strengthen their procedural proficiency skills. Included are opportunities for problem-solving, patterning, algebraic graphing, equations and determining averages. The combined task & drill sheets
Recollements of extension algebras
Institute of Scientific and Technical Information of China (English)
CHEN; Qinghua(陈清华); LIN; Yanan(林亚南)
2003-01-01
Let A be a finite-dimensional algebra over arbitrary base field k. We prove: if the unbounded derived module category D-(Mod-A) admits symmetric recollement relative to unbounded derived module categories of two finite-dimensional k-algebras B and C:D-(Mod- B) ( ) D-(Mod- A) ( ) D-(Mod- C),then the unbounded derived module category D-(Mod - T(A)) admits symmetric recollement relative to the unbounded derived module categories of T(B) and T(C):D-(Mod - T(B)) ( ) D-(Mod - T(A)) ( ) D-(Mod - T(C)).
Hogben, Leslie
2013-01-01
With a substantial amount of new material, the Handbook of Linear Algebra, Second Edition provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides you from the very elementary aspects of the subject to the frontiers of current research. Along with revisions and updates throughout, the second edition of this bestseller includes 20 new chapters.New to the Second EditionSeparate chapters on Schur complements, additional types of canonical forms, tensors, matrix polynomials, matrix equations, special types of
Cooperstein, Bruce
2010-01-01
Vector SpacesFieldsThe Space FnVector Spaces over an Arbitrary Field Subspaces of Vector SpacesSpan and IndependenceBases and Finite Dimensional Vector SpacesBases and Infinite Dimensional Vector SpacesCoordinate VectorsLinear TransformationsIntroduction to Linear TransformationsThe Range and Kernel of a Linear TransformationThe Correspondence and Isomorphism TheoremsMatrix of a Linear TransformationThe Algebra of L(V, W) and Mmn(F)Invertible Transformations and MatricesPolynomialsThe Algebra of PolynomialsRoots of PolynomialsTheory of a Single Linear OperatorInvariant Subspaces of an Operator
Algebra & trigonometry super review
2012-01-01
Get all you need to know with Super Reviews! Each Super Review is packed with in-depth, student-friendly topic reviews that fully explain everything about the subject. The Algebra and Trigonometry Super Review includes sets and set operations, number systems and fundamental algebraic laws and operations, exponents and radicals, polynomials and rational expressions, equations, linear equations and systems of linear equations, inequalities, relations and functions, quadratic equations, equations of higher order, ratios, proportions, and variations. Take the Super Review quizzes to see how much y
Weiss, Edwin
1998-01-01
Careful organization and clear, detailed proofs characterize this methodical, self-contained exposition of basic results of classical algebraic number theory from a relatively modem point of view. This volume presents most of the number-theoretic prerequisites for a study of either class field theory (as formulated by Artin and Tate) or the contemporary treatment of analytical questions (as found, for example, in Tate's thesis).Although concerned exclusively with algebraic number fields, this treatment features axiomatic formulations with a considerable range of applications. Modem abstract te
Hohn, Franz E
2012-01-01
This complete and coherent exposition, complemented by numerous illustrative examples, offers readers a text that can teach by itself. Fully rigorous in its treatment, it offers a mathematically sound sequencing of topics. The work starts with the most basic laws of matrix algebra and progresses to the sweep-out process for obtaining the complete solution of any given system of linear equations - homogeneous or nonhomogeneous - and the role of matrix algebra in the presentation of useful geometric ideas, techniques, and terminology.Other subjects include the complete treatment of the structur
International Nuclear Information System (INIS)
In this paper, we begin the study of zero-dimensional field theories with fields taking values in a semistrict Lie 2-algebra. These theories contain the IKKT matrix model and various M-brane related models as special cases. They feature solutions that can be interpreted as quantized 2-plectic manifolds. In particular, we find solutions corresponding to quantizations of ℝ3, S3 and a five-dimensional Hpp-wave. Moreover, by expanding a certain class of Lie 2-algebra models around the solution corresponding to quantized ℝ3, we obtain higher BF-theory on this quantized space
Algebra & trigonometry I essentials
REA, Editors of
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Algebra & Trigonometry I includes sets and set operations, number systems and fundamental algebraic laws and operations, exponents and radicals, polynomials and rational expressions, eq
Partially ordered algebraic systems
Fuchs, Laszlo
2011-01-01
Originally published in an important series of books on pure and applied mathematics, this monograph by a distinguished mathematician explores a high-level area in algebra. It constitutes the first systematic summary of research concerning partially ordered groups, semigroups, rings, and fields. The self-contained treatment features numerous problems, complete proofs, a detailed bibliography, and indexes. It presumes some knowledge of abstract algebra, providing necessary background and references where appropriate. This inexpensive edition of a hard-to-find systematic survey will fill a gap i
Energy Technology Data Exchange (ETDEWEB)
Christian, J M; McDonald, G S [Joule Physics Laboratory, School of Computing, Science and Engineering, Materials and Physics Research Centre, University of Salford, Salford M5 4WT (United Kingdom); Chamorro-Posada, P, E-mail: j.christian@salford.ac.u [Departamento de Teoria de la Senal y Comunicaciones e Ingenieria Telematica, Universidad de Valladolid, ETSI Telecomunicacion, Campus Miguel Delibes s/n, 47011 Valladolid (Spain)
2010-02-26
We report, to the best of our knowledge, the first exact analytical algebraic solitons of a generalized cubic-quintic Helmholtz equation. This class of governing equation plays a key role in photonics modelling, allowing a full description of the propagation and interaction of broad scalar beams. New conservation laws are presented, and the recovery of paraxial results is discussed in detail. The stability properties of the new solitons are investigated by combining semi-analytical methods and computer simulations. In particular, new general stability regimes are reported for algebraic bright solitons.
Endomorphisms of graph algebras
DEFF Research Database (Denmark)
Conti, Roberto; Hong, Jeong Hee; Szymanski, Wojciech
2012-01-01
We initiate a systematic investigation of endomorphisms of graph C*-algebras C*(E), extending several known results on endomorphisms of the Cuntz algebras O_n. Most but not all of this study is focused on endomorphisms which permute the vertex projections and globally preserve the diagonal MASA D...... that the restriction to the diagonal MASA of an automorphism which globally preserves both D_E and the core AF-subalgebra eventually commutes with the corresponding one-sided shift. Secondly, we exhibit several properties of proper endomorphisms, investigate invertibility of localized endomorphisms both on C...
Automorphism groups of pointed Hopf algebras
Institute of Scientific and Technical Information of China (English)
YANG Shilin
2007-01-01
The group of Hopf algebra automorphisms for a finite-dimensional semisimple cosemisimple Hopf algebra over a field k was considered by Radford and Waterhouse. In this paper, the groups of Hopf algebra automorphisms for two classes of pointed Hopf algebras are determined. Note that the Hopf algebras we consider are not semisimple Hopf algebras.
Derivations of generalized Weyl algebras
Institute of Scientific and Technical Information of China (English)
SU; Yucai(苏育才)
2003-01-01
A class of the associative and Lie algebras A[D] = A × F[D] of Weyl type are studied, where Ais a commutative associative algebra with an identity element over a field F of characteristic zero, and F[D] isthe polynomial algebra of a finite dimensional commutative subalgebra of locally finite derivations of A suchthat A is D-simple. The derivations of these associative and Lie algebras are precisely determined.
The theory of algebraic numbers
Pollard, Harry
1998-01-01
An excellent introduction to the basics of algebraic number theory, this concise, well-written volume examines Gaussian primes; polynomials over a field; algebraic number fields; and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture. 1975 edition. References. List of Symbols. Index.
Optimal Algorithm for Algebraic Factoring
Institute of Scientific and Technical Information of China (English)
支丽红
1997-01-01
This paper presents on optimized method for factoring multivariate polynomials over algebraic extension fields defined by an irreducible ascending set. The basic idea is to convert multivariate polynomials to univariate polynomials and algebraic extension fields to algebraic number fields by suitable integer substituteions.Then factorize the univariate polynomials over the algebraic number fields.Finally,construct mulativariate factors of the original polynomial by Hensel lemma and TRUEFACTOR test.Some examples with timing are included.
Petersen, Peter
2016-01-01
Intended for a one year course, this text serves as a single source, introducing readers to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few Works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory. The book will appeal to a readership that have a basic knowledge of standard manifold theory, including tensors, forms, and Lie groups. Important revisions to the third edition include: a substantial addition of unique and enriching exercises scattered throughout the text; inclusion of an increased number of coordinate calculations of connection and curvature; addition of general formulas for curvature on Lie Groups and submersions; integration of variational calculus into the text allowing for an early treatment of the Sphere theorem using a proof by Berger; incorporation of several recent results about manifolds with posit...
The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders
Energy Technology Data Exchange (ETDEWEB)
Gurau, Razvan, E-mail: rgurau@perimeterinstitute.ca [Perimeter Institute for Theoretical Physics, 31 Caroline St. N, ON N2L 2Y5, Waterloo (Canada)
2012-12-01
Random tensor models for a generic complex tensor generalize matrix models in arbitrary dimensions and yield a theory of random geometries. They support a 1/N expansion dominated by graphs of spherical topology. Their Schwinger Dyson equations, generalizing the loop equations of matrix models, translate into constraints satisfied by the partition function. The constraints have been shown, in the large N limit, to close a Lie algebra indexed by colored rooted D-ary trees yielding a first generalization of the Virasoro algebra in arbitrary dimensions. In this paper we complete the Schwinger Dyson equations and the associated algebra at all orders in 1/N. The full algebra of constraints is indexed by D-colored graphs, and the leading order D-ary tree algebra is a Lie subalgebra of the full constraints algebra.
Meadow enriched ACP process algebras
J.A. Bergstra; C.A. Middelburg
2009-01-01
We introduce the notion of an ACP process algebra. The models of the axiom system ACP are the origin of this notion. ACP process algebras have to do with processes in which no data are involved. We also introduce the notion of a meadow enriched ACP process algebra, which is a simple generalization o
Laan, P. van der
2001-01-01
In the literature several Hopf algebras that can be described in terms of trees have been studied. This paper tries to answer the question whether one can understand some of these Hopf algebras in terms of a single mathematical construction. The starting point is the Hopf algebra of rooted trees as
Computer Algebra in Particle Physics
Weinzierl, Stefan
2002-01-01
These lectures given to graduate students in theoretical particle physics, provide an introduction to the ``inner workings'' of computer algebra systems. Computer algebra has become an indispensable tool for precision calculations in particle physics. A good knowledge of the basics of computer algebra systems allows one to exploit these systems more efficiently.
An Algebra of Reversible Computation
Wang, Yong
2014-01-01
We design an axiomatization for reversible computation called reversible ACP (RACP). It has four extendible modules, basic reversible processes algebra (BRPA), algebra of reversible communicating processes (ARCP), recursion and abstraction. Just like process algebra ACP in classical computing, RACP can be treated as an axiomatization foundation for reversible computation.
Final Report: Geometry And Elementary Particle Physics
International Nuclear Information System (INIS)
The effect on mathematics of collaborations between high-energy theoretical physics and modern mathematics has been remarkable. Mirror symmetry has revolutionized enumerative geometry, and Seiberg-Witten invariants have greatly simplified the study of four manifolds. And because of their application to string theory, physicists now need to know cohomology theory, characteristic classes, index theory, K-theory, algebraic geometry, differential geometry, and non-commutative geometry. Much more is coming. We are experiencing a deeper contact between the two sciences, which will stimulate new mathematics essential to the physicists quest for the unification of quantum mechanics and relativity. Our grant, supported by the Department of Energy for twelve years, has been instrumental in promoting an effective interaction between geometry and string theory, by supporting the Mathematical Physics seminar, postdoc research, collaborations, graduate students and several research papers.
FINAL REPORT: GEOMETRY AND ELEMENTARY PARTICLE PHYSICS
Energy Technology Data Exchange (ETDEWEB)
Singer, Isadore M.
2008-03-04
The effect on mathematics of collaborations between high-energy theoretical physics and modern mathematics has been remarkable. Mirror symmetry has revolutionized enumerative geometry, and Seiberg-Witten invariants have greatly simplified the study of four manifolds. And because of their application to string theory, physicists now need to know cohomology theory, characteristic classes, index theory, K-theory, algebraic geometry, differential geometry, and non-commutative geometry. Much more is coming. We are experiencing a deeper contact between the two sciences, which will stimulate new mathematics essential to the physicists’ quest for the unification of quantum mechanics and relativity. Our grant, supported by the Department of Energy for twelve years, has been instrumental in promoting an effective interaction between geometry and string theory, by supporting the Mathematical Physics seminar, postdoc research, collaborations, graduate students and several research papers.
Rigid geometry of curves and their Jacobians
Lütkebohmert, Werner
2016-01-01
This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field. The text starts with a survey of the foundation of rigid geometry, and then focuses on a detailed treatment of the applications. In the case of curves with split rational reduction there is a complete analogue to the fascinating theory of Riemann surfaces. In the case of proper smooth group varieties the uniformization and the construction of abelian varieties are treated in detail. Rigid geometry was established by John Tate and was enriched by a formal algebraic approach launched by Michel Raynaud. It has proved as a means to illustrate the geometric ideas behind the abstract methods of formal algebraic geometry as used by Mumford and Faltings. This book should be of great use to students wishing to enter this field, as well as those already working in it.
The Maximal Graded Left Quotient Algebra of a Graded Algebra
Institute of Scientific and Technical Information of China (English)
Gonzalo ARANDA PINO; Mercedes SILES MOLINA
2006-01-01
We construct the maximal graded left quotient algebra of every graded algebra A without homogeneous total right zero divisors as the direct limit of graded homomorphisms (of left A-modules)from graded dense left ideals of A into a graded left quotient algebra of A. In the case of a superalgebra,and with some extra hypothesis, we prove that the component in the neutral element of the group of the maximal graded left quotient algebra coincides with the maximal left quotient algebra of the component in the neutral element of the group of the superalgebra.
Geometry of strings and fields
2013-01-01
Ever since the birth of string theory, interaction with geometry has been one of the primary driving forces that has led to progress in superstring theory. On one hand, string theory has generated many new geometrical concepts; and on the other hand new ideas from geometry have often found their first applications in string theory. These topics include vertex algebras, conformal field theory, mirror symmetry, topological field theory and string theory, exact solutions of supersymmetric gauge theory and noncommutative field theory. Recent exciting developments include the matrix model approach to N=1 gauge theory, open string mirror symmetry, the derived category approach to D-branes on Calabi-Yau manifolds, geometric transitions, proof of the N=2 Seiberg-Witten solution by instanton methods, wall crossing formulas, the relation between Langlands program and supersymmetric gauge theories, indications of integrable structures in super Yang-Mills theory and AdS string theory. The program will be devoted to geome...
Geometry of Membrane Sigma Models
Vysoky, Jan
2015-01-01
String theory still remains one of the promising candidates for a unification of the theory of gravity and quantum field theory. One of its essential parts is relativistic description of moving multi-dimensional objects called membranes (or p-branes) in a curved spacetime. On the classical field theory level, they are described by an action functional extremalising the volume of a manifold swept by a propagating membrane. This and related field theories are collectively called membrane sigma models. Differential geometry is an important mathematical tool in the study of string theory. It turns out that string and membrane backgrounds can be conveniently described using objects defined on a direct sum of tangent and cotangent bundles of the spacetime manifold. Mathematical field studying such object is called generalized geometry. Its integral part is the theory of Leibniz algebroids, vector bundles with a Leibniz algebra bracket on its module of smooth sections. Special cases of Leibniz algebroids are better ...
Groups and Geometries : Siena Conference
Kantor, William; Lunardon, Guglielmo; Pasini, Antonio; Tamburini, Maria
1998-01-01
On September 1-7, 1996 a conference on Groups and Geometries took place in lovely Siena, Italy. It brought together experts and interested mathematicians from numerous countries. The scientific program centered around invited exposi tory lectures; there also were shorter research announcements, including talks by younger researchers. The conference concerned a broad range of topics in group theory and geometry, with emphasis on recent results and open problems. Special attention was drawn to the interplay between group-theoretic methods and geometric and combinatorial ones. Expanded versions of many of the talks appear in these Proceedings. This volume is intended to provide a stimulating collection of themes for a broad range of algebraists and geometers. Among those themes, represented within the conference or these Proceedings, are aspects of the following: 1. the classification of finite simple groups, 2. the structure and properties of groups of Lie type over finite and algebraically closed fields of f...
Certain associative algebras similar to $U(sl_{2})$ and Zhu's algebra $A(V_{L})$
Dong, Chongying; Li, Haisheng; Mason, Geoffrey
1996-01-01
It is proved that Zhu's algebra for vertex operator algebra associated to a positive-definite even lattice of rank one is a finite-dimensional semiprimitive quotient algebra of certain associative algebra introduced by Smith. Zhu's algebra for vertex operator algebra associated to any positive-definite even lattice is also calculated and is related to a generalization of Smith's algebra.
Observable Algebra in Field Algebra of G-spin Models
Institute of Scientific and Technical Information of China (English)
蒋立宁
2003-01-01
Field algebra of G-spin models can provide the simplest examples of lattice field theory exhibiting quantum symmetry. Let D(G) be the double algebra of a finite group G and D(H), a sub-algebra of D(G) determined by subgroup H of G. This paper gives concrete generators and the structure of the observable algebra AH, which is a D(H)-invariant sub-algebra in the field algebra of G-spin models F, and shows that AH is a C*-algebra. The correspondence between H and AH is strictly monotonic. Finally, a duality between D(H) and AH is given via an irreducible vacuum C*-representation of F.
General Geometry and Geometry of Electromagnetism
Shahverdiyev, Shervgi S.
2002-01-01
It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is geometry underlying Electromagnetism. Action for electromagnetic field and Maxwell equations are derived from curvature function of geometry underlying Electromagnetism. And it is shown that equation of motion for a particle interacting with electromagnetic...
Boiteau, Denise; Stansfield, David
This document describes mathematical programs on the basic concepts of algebra produced by Louisiana Public Broadcasting. Programs included are: (1) "Inverse Operations"; (2) "The Order of Operations"; (3) "Basic Properties" (addition and multiplication of numbers and variables); (4) "The Positive and Negative Numbers"; and (5) "Using Positive…
Questions on Algebraic Varieties
Marchionna, E
2011-01-01
P. Dolbeault: Residus et courants.- D. Mumford: Varieties defined by quadratic equations.- A. Neron: Hauteurs et theorie des intersections.- A. Seidenberg: Report on analytic product.- C.S. Seshadri: Moduli of p-vector bundles over an algebraic curve.- O. Zariski: Contributions to the problem of equi-singularity.
Algebraic topology and concurrency
DEFF Research Database (Denmark)
Fajstrup, Lisbeth; Raussen, Martin; Goubault, Eric
2006-01-01
We show in this article that some concepts from homotopy theory, in algebraic topology,are relevant for studying concurrent programs. We exhibit a natural semantics of semaphore programs, based on partially ordered topological spaces, which are studied up to “elastic deformation” or homotopy, giv...
Operation of Algebraic Fractions
Institute of Scientific and Technical Information of China (English)
2008-01-01
<正>The first step in factorizing algebraic expressions is to take out the common factors of all the terms of the expression.For example,2x~2+14x+24=2(x~2+7x+12)=2(x+3)(x+4) The three identities are also useful in factorizing some quadratic expressions:
Institute of Scientific and Technical Information of China (English)
SU; Yucai(
2001-01-01
［1］ Kawamoto, N., Generalizations of Witt algebras over a field of characteristic zero, Hiroshima Math. J., 1986, 16: 417.［2］ Osborn, J. M., New simple infinite－dimensional Lie algebras of characteristic 0, J. Alg., 1996, 185: 820.［3］ Dokovic, D. Z., Zhao, K., Derivations, isomorphisms, and second cohomology of generalized Witt algebras, Trans. of Amer. Math. Soc., 1998, 350(2): 643.［4］ Dokovic, D. Z., Zhao, K., Generalized Cartan type W Lie algebras in characteristic zero, J. Alg., 1997, 195: 170.［5］ Osborn, J. M., Zhao, K., Generalized Poisson bracket and Lie algebras of type H in characteristic 0, Math. Z., 1999, 230: 107.［6］ Osborn, J. M., Zhao, K., Generalized Cartan type K Lie algebras in characteristic 0, Comm. Alg., 1997, 25: 3325.［7］ Zhao, K., Isomorphisms between generalized Cartan type W Lie algebras in characteristic zero, Canadian J. Math., 1998, 50: 210.［8］ Passman, D. P., Simple Lie algebras of Witt type, J. Algebra, 1998, 206: 682.［9］ Jordan, D. A., On the simplicity of Lie algebras of derivations of commutative algebras, J. Alg., 2000, 206: 682.［10］ Xu, X., New generalized simple Lie algebras of Cartan type over a field with characteristic 0, J. Alg., 2000, 244: 23.［11］ Su, Y., Xu, X., Zhang, H., Derivation－simple algebras and the structures of Lie algebras of generalized Witt type, J. Alg., 2000, 233: 642.［12］ Dixmer, J., Enveloping Algebras, Amsterdam: North Holland, 1977.
On ultraproducts of operator algebras
Institute of Scientific and Technical Information of China (English)
LI Weihua
2005-01-01
Some basic questions on ultraproducts of C*-algebras and yon Neumann algebras, including the relation to K-theory of C*-algebras are considered. More specifically,we prove that under certain conditions, the K-groups of ultraproduct of C*-algebras are isomorphic to the ultraproduct of respective K-groups of C*-algebras. We also show that the ultraproducts of factors of type Ⅱ1 are prime, i.e. not isomorphic to any non-trivial tensor product.
Ockham Algebras Arising from Monoids
Institute of Scientific and Technical Information of China (English)
T.S. Blyth; H.J. Silva; J.C. Varlet
2001-01-01
An Ockham algebra (L; f) is of boolean shape if its lattice reduct L is boolean and f is not the complementation. We investigate a natural construction of Ockham algebras of boolean shape from any given monoid. Of particular interest is the question of when such algebras are subdirectly irreducible. In settling this, we obtain what is probably the first example of a subdirectly irreducible Ockham algebra that does not belong to the generalized variety Kω. We also prove that every semigroup can be embedded in the monoid of endomorphisms of an Ockham algebra of boolean shape.
Quantum algebra of $N$ superspace
Hatcher, N; Stephany, J
2006-01-01
We identify the quantum algebra of position and momentum operators for a quantum system in superspace bearing an irreducible representation of the super Poinca\\'e algebra. This algebra is noncommutative for the position operators. We use the properties of superprojectors in D=4 $N$ superspace to construct explicit position and momentum operators satisfying the algebra. They act on wave functions corresponding to different supermultiplets classified by its superspin. We show that the quantum algebra associated to the massive superparticle is a particular case described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently.
Algebraic Approach to Algorithmic Logic
Directory of Open Access Journals (Sweden)
Bancerek Grzegorz
2014-09-01
Full Text Available We introduce algorithmic logic - an algebraic approach according to [25]. It is done in three stages: propositional calculus, quantifier calculus with equality, and finally proper algorithmic logic. For each stage appropriate signature and theory are defined. Propositional calculus and quantifier calculus with equality are explored according to [24]. A language is introduced with language signature including free variables, substitution, and equality. Algorithmic logic requires a bialgebra structure which is an extension of language signature and program algebra. While-if algebra of generator set and algebraic signature is bialgebra with appropriate properties and is used as basic type of algebraic logic.
Notes on Piecewise-Koszul Algebras
Institute of Scientific and Technical Information of China (English)
Jia Feng L(U); Xiao Lan YU
2011-01-01
The relationships between piecewise-Koszul algebras and other "Koszul-type" algebras are discussed.. The Yoneda-Ext algebra and the dual algebra of a piecewise-Koszul algebra are studied, and a sufficient condition for the dual algebra A to be piecewise-Koszul is given. Finally, by studying the trivial extension algebras of the path algebras of Dynkin quivers in bipartite orientation, we give explicit constructions for piecewise-Koszul algebras with arbitrary "period" and piecewise-Koszul algebras with arbitrary "jump-degree".
The Green formula and heredity of algebras
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
[1]Green, J. A., Hall algebras, hereditary algebras and quantum groups, Invent. Math. 1995, 120: 361-377.[2]Ringel, C. M., Green's theorem on Hall algebras, in Representations of Algebras and Related Topics, CMS Conference Proceedings 19, Providence, 1996, 185-245.[3]Xiao J., Drinfeld double and Ringel-Green theory of Hall Algebras, J. Algebra, 1997, 190: 100-144.[4]Sevenhant, B., Van den Bergh, M., A relation between a conjecture of Kac and the structure of the Hall algebra,J. Pure Appl. Algebra, 2001, 160: 319-332.[5]Deng B., Xiao, J., On double Ringel-Hall algebras, J. Algebra, 2002, 251: 110-149.
Valued Graphs and the Representation Theory of Lie Algebras
Directory of Open Access Journals (Sweden)
Joel Lemay
2012-07-01
Full Text Available Quivers (directed graphs, species (a generalization of quivers and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this paper, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field. Namely, we show that the category of K -species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K -species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.
$A\\mathcal{T}$-Algebras and Extensions of $AT$-Algebras
Indian Academy of Sciences (India)
Hongliang Yao
2010-04-01
Lin and Su classified $A\\mathcal{T}$-algebras of real rank zero. This class includes all $A\\mathbb{T}$-algebras of real rank zero as well as many *-algebras which are not stably finite. An $A\\mathcal{T}$-algebra often becomes an extension of an $A\\mathbb{T}$-algebra by an -algebra. In this paper, we show that there is an essential extension of an $A\\mathbb{T}$-algebra by an -algebra which is not an $A\\mathcal{T}$-algebra. We describe a characterization of an extension of an $A\\mathbb{T}$-algebra by an -algebra if is an $A\\mathcal{T}$-algebra.
Noncommutative spectral geometry: A guided tour for theoretical physicists
Sakellariadou, Mairi
2012-01-01
We review a gravitational model based on noncommutative geometry and the spectral action principle. The space-time geometry is described by the tensor product of a four-dimensional Riemanian manifold by a discrete noncommutative space consisting of only two points. With a specific choice of the finite dimensional involutive algebra, the noncommutative spectral action leads to the standard model of electroweak and strong interactions minimally coupled to Einstein and Weyl gravity. We present the main mathematical ingredients of this model and discuss their physical implications. We argue that the doubling of the algebra is intimately related to dissipation and the gauge field structure. We then show how this noncommutative spectral geometry model, a purely classical construction, carries implicit in the doubling of the algebra the seeds of quantization. After a short review on the phenomenological consequences of this geometric model as an approach to unification, we discuss some of its cosmological consequenc...
Proposition Algebra with Projective Limits
Bergstra, J A
2008-01-01
Sequential logic deviates from propositional logic by taking into account that atomic propositions yield different Boolean values at different times during the sequential evaluation of a single proposition. Reactive valuations capture this dynamics of a proposition's environment. This logic is phrased as an equationally specified algebra rather than in the form of proof rules. It is strictly more general than Boolean algebra to the extent that the classical connectives fail to be expressively complete in the sequential case. The proposition algebra PRA is developed in a fashion similar to the process algebra ACP and the program algebra PGA via an algebraic specification which has a meaningful initial algebra for which a range of courser congruences are considered important as well. In addition infinite objects (that is propositions, processes and programs respectively) are preferably dealt with by means of an inverse limit construction which allows the transfer of knowledge concerning finite objects to facts ...
Algebraic connectivity and graph robustness.
Energy Technology Data Exchange (ETDEWEB)
Feddema, John Todd; Byrne, Raymond Harry; Abdallah, Chaouki T. (University of New Mexico)
2009-07-01
Recent papers have used Fiedler's definition of algebraic connectivity to show that network robustness, as measured by node-connectivity and edge-connectivity, can be increased by increasing the algebraic connectivity of the network. By the definition of algebraic connectivity, the second smallest eigenvalue of the graph Laplacian is a lower bound on the node-connectivity. In this paper we show that for circular random lattice graphs and mesh graphs algebraic connectivity is a conservative lower bound, and that increases in algebraic connectivity actually correspond to a decrease in node-connectivity. This means that the networks are actually less robust with respect to node-connectivity as the algebraic connectivity increases. However, an increase in algebraic connectivity seems to correlate well with a decrease in the characteristic path length of these networks - which would result in quicker communication through the network. Applications of these results are then discussed for perimeter security.
Abstract algebra structure and application
Finston, David R
2014-01-01
This text seeks to generate interest in abstract algebra by introducing each new structure and topic via a real-world application. The down-to-earth presentation is accessible to a readership with no prior knowledge of abstract algebra. Students are led to algebraic concepts and questions in a natural way through their everyday experiences. Applications include: Identification numbers and modular arithmetic (linear) error-correcting codes, including cyclic codes ruler and compass constructions cryptography symmetry of patterns in the real plane Abstract Algebra: Structure and Application is suitable as a text for a first course on abstract algebra whose main purpose is to generate interest in the subject, or as a supplementary text for more advanced courses. The material paves the way to subsequent courses that further develop the theory of abstract algebra and will appeal to students of mathematics, mathematics education, computer science, and engineering interested in applications of algebraic concepts.
On Dunkl angular momenta algebra
Feigin, Misha; Hakobyan, Tigran
2015-11-01
We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincaré-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl( N ) version of the subalge-bra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.
Handy elementary algebraic properties of the geometry of entanglement
Blair, Howard A.; Alsing, Paul M.
2013-05-01
The space of separable states of a quantum system is a hyperbolic surface in a high dimensional linear space, which we call the separation surface, within the exponentially high dimensional linear space containing the quantum states of an n component multipartite quantum system. A vector in the linear space is representable as an n-dimensional hypermatrix with respect to bases of the component linear spaces. A vector will be on the separation surface iff every determinant of every 2-dimensional, 2-by-2 submatrix of the hypermatrix vanishes. This highly rigid constraint can be tested merely in time asymptotically proportional to d, where d is the dimension of the state space of the system due to the extreme interdependence of the 2-by-2 submatrices. The constraint on 2-by-2 determinants entails an elementary closed formformula for a parametric characterization of the entire separation surface with d-1 parameters in the char- acterization. The state of a factor of a partially separable state can be calculated in time asymptotically proportional to the dimension of the state space of the component. If all components of the system have approximately the same dimension, the time complexity of calculating a component state as a function of the parameters is asymptotically pro- portional to the time required to sort the basis. Metric-based entanglement measures of pure states are characterized in terms of the separation hypersurface.
EPH-classifications in Geometry, Algebra, Analysis and Arithmetic
Rastegar, Arash
2015-01-01
Trichotomy of Elliptic-Parabolic-Hyperbolic appears in many different areas of mathematics. All of these are named after the very first example of trichotomy, which is formed by ellipses, parabolas, and hyperbolas as conic sections. We try to understand if these classifications are justified and if similar mathematical phenomena is shared among different cases EPH-classification is used.
Topics in Cohomological Studies of Algebraic Varieties Impanga Lecture Notes
Pragacz, Piotr
2005-01-01
The articles in this volume study various cohomological aspects of algebraic varieties:- characteristic classes of singular varieties;- geometry of flag varieties;- cohomological computations for homogeneous spaces;- K-theory of algebraic varieties;- quantum cohomology and Gromov-Witten theory.The main purpose is to give comprehensive introductions to the above topics through a series of "friendly" texts starting from a very elementary level and ending with the discussion of current research. In the articles, the reader will find classical results and methods as well as new ones. Num
Some topics pertaining to algebras of linear operators
Semmes, Stephen
2002-01-01
On the one hand the algebras of linear operators here act on finite-dimensional vector spaces, and on the other hand the point of view is generally an analysts'. Also, one might think of algebras as being used to add more data to basic geometry as on a graph, for instance. Of course this is a common theme which is considered in numerous settings. From an analysts' perspective, compact groups, their representations, and more general topological groups and their representations are basic object...
Finite quantum physics and noncommutative geometry
International Nuclear Information System (INIS)
Conventional discrete approximations of a manifold do not preserve its nontrivial topological features. In this article we describe an approximation scheme due to Sorkin which reproduces physically important aspects of manifold topology with striking fidelity. The approximating topological spaces in this scheme are partially ordered sets (posets). Now, in ordinary quantum physics on a manifold M, continuous probability densities generate the commutative C*-algebra C(M) of continuous functions on M. It has a fundamental physical significance, containing the information to reconstruct the topology of M, and serving to specify the domains of observables like the Hamiltonian. For a poset, the role of this algebra is assumed by a noncommutative C*-algebra A. As noncommutative geometries are based on noncommutative C*-algebra, we therefore have a remarkable connection between finite approximations to quantum physics and noncommutative geometries. Varies methods for doing quantum physics using A are explored. Particular attention is paid to developing numerically viable approximation schemes which at the same time preserve important topological features of continuum physics. (author). 21 refs, 13 figs
Free Malcev algebra of rank three
Kornev, Alexandr
2011-01-01
We find a basis of the free Malcev algebra on three free generators over a field of characteristic zero. The specialty and semiprimity of this algebra are proved. In addition, we prove the decomposability of this algebra into subdirect sum of the free Lie algebra rank three and the free algebra of rank three of variety of Malcev algebras generated by a simple seven-dimensional Malcev algebra.
Barcelona Conference on Algebraic Topology
Castellet, Manuel; Cohen, Frederick
1992-01-01
The papers in this collection, all fully refereed, original papers, reflect many aspects of recent significant advances in homotopy theory and group cohomology. From the Contents: A. Adem: On the geometry and cohomology of finite simple groups.- D.J. Benson: Resolutions and Poincar duality for finite groups.- C. Broto and S. Zarati: On sub-A*-algebras of H*V.- M.J. Hopkins, N.J. Kuhn, D.C. Ravenel: Morava K-theories of classifying spaces and generalized characters for finite groups.- K. Ishiguro: Classifying spaces of compact simple lie groups and p-tori.- A.T. Lundell: Concise tables of James numbers and some homotopyof classical Lie groups and associated homogeneous spaces.- J.R. Martino: Anexample of a stable splitting: the classifying space of the 4-dim unipotent group.- J.E. McClure, L. Smith: On the homotopy uniqueness of BU(2) at the prime 2.- G. Mislin: Cohomologically central elements and fusion in groups.
Configuration spaces geometry, topology and representation theory
Cohen, Frederick; Concini, Corrado; Feichtner, Eva; Gaiffi, Giovanni; Salvetti, Mario
2016-01-01
This book collects the scientific contributions of a group of leading experts who took part in the INdAM Meeting held in Cortona in September 2014. With combinatorial techniques as the central theme, it focuses on recent developments in configuration spaces from various perspectives. It also discusses their applications in areas ranging from representation theory, toric geometry and geometric group theory to applied algebraic topology.
Geometry of the restricted Boltzmann machine
Cueto, Maria Angelica; Morton, Jason; Sturmfels, Bernd
2009-01-01
The restricted Boltzmann machine is a graphical model for binary random variables. Based on a complete bipartite graph separating hidden and observed variables, it is the binary analog to the factor analysis model. We study this graphical model from the perspectives of algebraic statistics and tropical geometry, starting with the observation that its Zariski closure is a Hadamard power of the first secant variety of the Segre variety of projective lines. We derive a dimension formula for the ...
Differential forms and the geometry of general relativity
Dray, Tevian
2015-01-01
Differential Forms and the Geometry of General Relativity provides readers with a coherent path to understanding relativity. Requiring little more than calculus and some linear algebra, it helps readers learn just enough differential geometry to grasp the basics of general relativity.The book contains two intertwined but distinct halves. Designed for advanced undergraduate or beginning graduate students in mathematics or physics, most of the text requires little more than familiarity with calculus and linear algebra. The first half presents an introduction to general relativity that describes
Vertex Algebras, Kac-Moody Algebras, and the Monster
Borcherds, Richard E.
1986-05-01
It is known that the adjoint representation of any Kac-Moody algebra A can be identified with a subquotient of a certain Fock space representation constructed from the root lattice of A. I define a product on the whole of the Fock space that restricts to the Lie algebra product on this subquotient. This product (together with a infinite number of other products) is constructed using a generalization of vertex operators. I also construct an integral form for the universal enveloping algebra of any Kac-Moody algebra that can be used to define Kac-Moody groups over finite fields, some new irreducible integrable representations, and a sort of affinization of any Kac-Moody algebra. The ``Moonshine'' representation of the Monster constructed by Frenkel and others also has products like the ones constructed for Kac-Moody algebras, one of which extends the Griess product on the 196884-dimensional piece to the whole representation.
Testing algebraic geometric codes
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
Property testing was initially studied from various motivations in 1990’s. A code C GF (r)n is locally testable if there is a randomized algorithm which can distinguish with high possibility the codewords from a vector essentially far from the code by only accessing a very small (typically constant) number of the vector’s coordinates. The problem of testing codes was firstly studied by Blum, Luby and Rubinfeld and closely related to probabilistically checkable proofs (PCPs). How to characterize locally testable codes is a complex and challenge problem. The local tests have been studied for Reed-Solomon (RS), Reed-Muller (RM), cyclic, dual of BCH and the trace subcode of algebraicgeometric codes. In this paper we give testers for algebraic geometric codes with linear parameters (as functions of dimensions). We also give a moderate condition under which the family of algebraic geometric codes cannot be locally testable.
Combinatorics and commutative algebra
Stanley, Richard P
1996-01-01
Some remarkable connections between commutative algebra and combinatorics have been discovered in recent years. This book provides an overview of two of the main topics in this area. The first concerns the solutions of linear equations in nonnegative integers. Applications are given to the enumeration of integer stochastic matrices (or magic squares), the volume of polytopes, combinatorial reciprocity theorems, and related results. The second topic deals with the face ring of a simplicial complex, and includes a proof of the Upper Bound Conjecture for Spheres. An introductory chapter giving background information in algebra, combinatorics and topology broadens access to this material for non-specialists. New to this edition is a chapter surveying more recent work related to face rings, focusing on applications to f-vectors. Included in this chapter is an outline of the proof of McMullen's g-conjecture for simplicial polytopes based on toric varieties, as well as a discussion of the face rings of such special ...
Topological convolution algebras
Alpay, Daniel
2012-01-01
In this paper we introduce a new family of topological convolution algebras of the form $\\bigcup_{p\\in\\mathbb N} L_2(S,\\mu_p)$, where $S$ is a Borel semi-group in a locally compact group $G$, which carries an inequality of the type $\\|f*g\\|_p\\le A_{p,q}\\|f\\|_q\\|g\\|_p$ for $p > q+d$ where $d$ pre-assigned, and $A_{p,q}$ is a constant. We give a sufficient condition on the measures $\\mu_p$ for such an inequality to hold. We study the functional calculus and the spectrum of the elements of these algebras, and present two examples, one in the setting of non commutative stochastic distributions, and the other related to Dirichlet series.
Operator product expansion algebra
Energy Technology Data Exchange (ETDEWEB)
Holland, Jan [CPHT, Ecole Polytechnique, Paris-Palaiseau (France)
2014-07-01
The Operator Product Expansion (OPE) is a theoretical tool for studying the short distance behaviour of products of local quantum fields. Over the past 40 years, the OPE has not only found widespread computational application in high-energy physics, but, on a more conceptual level, it also encodes fundamental information on algebraic structures underlying quantum field theories. I review new insights into the status and properties of the OPE within Euclidean perturbation theory, addressing in particular the topics of convergence and ''factorisation'' of the expansion. Further, I present a formula for the ''deformation'' of the OPE algebra caused by a quartic interaction. This formula can be used to set up a novel iterative scheme for the perturbative computation of OPE coefficients, based solely on the zeroth order coefficients (and renormalisation conditions) as initial input.
Testing algebraic geometric codes
Institute of Scientific and Technical Information of China (English)
CHEN Hao
2009-01-01
Property testing was initially studied from various motivations in 1990's.A code C (∩)GF(r)n is locally testable if there is a randomized algorithm which can distinguish with high possibility the codewords from a vector essentially far from the code by only accessing a very small (typically constant) number of the vector's coordinates.The problem of testing codes was firstly studied by Blum,Luby and Rubinfeld and closely related to probabilistically checkable proofs (PCPs).How to characterize locally testable codes is a complex and challenge problem.The local tests have been studied for Reed-Solomon (RS),Reed-Muller (RM),cyclic,dual of BCH and the trace subcode of algebraicgeometric codes.In this paper we give testers for algebraic geometric codes with linear parameters (as functions of dimensions).We also give a moderate condition under which the family of algebraic geometric codes cannot be locally testable.
Equations defining secant varieties: geometry and computation
Sidman, Jessica
2009-01-01
In the 1980's, work of Green and Lazarsfeld helped to uncover the beautiful interplay between the geometry of the embedding of a curve and the syzygies of its defining equations. Similar results hold for the first secant variety of a curve, and there is a natural conjectural picture extending to higher secant varieties as well. We present an introduction to the algebra and geometry used in previous work of the authors to study syzygies of secant varieties of curves with an emphasis on examples of explicit computations and elementary cases that illustrate the geometric principles at work.
Differential Geometry of Microlinear Frolicher Spaces I
Nishimura, Hirokazu
2010-01-01
The central object of synthetic differential geometry is microlinear spaces. In our previous paper [Microlinearity in Frolicher spaces -beyond the regnant philosophy of manifolds-, to appear in International Journal of Pure and Applied Mathematics] we have emancipated microlinearity from within well-adapted models to Frolicher spaces. Therein we have shown that Frolicher spaces which are microlinear as well as Weil exponentiable form a cartesian closed category. To make sure that such Frolicher spaces are the central object of infinite-dimensional differential geometry, we develop the theory of vector fields on them in this paper. The central result is that all vector fields on such a Frolicher space form a Lie algebra.
Lee, Jaehoon; Wilczek, Frank
2013-11-27
Motivated by the problem of identifying Majorana mode operators at junctions, we analyze a basic algebraic structure leading to a doubled spectrum. For general (nonlinear) interactions the emergent mode creation operator is highly nonlinear in the original effective mode operators, and therefore also in the underlying electron creation and destruction operators. This phenomenon could open up new possibilities for controlled dynamical manipulation of the modes. We briefly compare and contrast related issues in the Pfaffian quantum Hall state.
Beigie, Darin
2014-01-01
Most people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,…
Fritzsch, H.; Gell-Mann, M.
2003-01-01
This talk follows by a few months a talk by the same authors on nearly the same subject at the Coral Gables Conference. The ideas presented here are basically the same, but with some amplification, some change of viewpoint, and a number of new questions for the future. For our own convenience, we have transcribed the Coral Gables paper, but with an added ninth section, entitled "Problems of light cone current algebra", dealing with our present views and emphasizing research topics that requir...
International Nuclear Information System (INIS)
Expository notes on Clifford algebras and spinors with a detailed discussion of Majorana, Weyl, and Dirac spinors. The paper is meant as a review of background material, needed, in particular, in now fashionable theoretical speculations on neutrino masses. It has a more mathematical flavour than the over twenty-six-year-old Introduction to Majorana masses [M84] and includes historical notes and biographical data on past participants in the story. (author)
Algebra & trigonometry II essentials
REA, Editors of
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Algebra & Trigonometry II includes logarithms, sequences and series, permutations, combinations and probability, vectors, matrices, determinants and systems of equations, mathematica
Redesigning linear algebra algorithms
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J.
1983-01-01
Many of the standard algorithms in linear algebra as implemented in FORTRAN do not achieve maximum performance on today's large-scale vector computers. The author examines the problem and constructs alternative formulations of algorithms that do not lose the clarity of the original algorithm or sacrifice the FORTRAN portable environment, but do gain the performance attainable on these supercomputers. The resulting implementation not only performs well on vector computers but also increases performance on conventional sequential computers. 13 references.
Redesigning linear algebra algorithms
Energy Technology Data Exchange (ETDEWEB)
Dongarra, J.J.
1983-01-01
Many of the standard algorithms in linear algebra as implemented in FORTRAN do not achieve maximum performance on today's large-scale vector computers. In this paper we examine the problem and construct alternative formulations of algorithms that do not lose the clarity of the original algorithm or sacrifice the Fortran portable environment, but do gain the performance attainable on these supercomputers. The resulting implementation not only performs well on vector computers but also increases performance on conventional sequential computers.
Fundamentals of linear algebra
Dash, Rajani Ballav
2008-01-01
FUNDAMENTALS OF LINEAR ALGEBRA is a comprehensive Text Book, which can be used by students and teachers of All Indian Universities. The Text has easy, understandable form and covers all topics of UGC Curriculum. There are lots of worked out examples which helps the students in solving the problems without anybody's help. The Problem sets have been designed keeping in view of the questions asked in different examinations.
Semisimple Metacyclic Group Algebras
Indian Academy of Sciences (India)
Gurmeet K Bakshi; Shalini Gupta; Inder Bir S Passi
2011-11-01
Given a group of order $p_1p_2$, where $p_1,p_2$ are primes, and $\\mathbb{F}_q$, a finite field of order coprime to $p_1p_2$, the object of this paper is to compute a complete set of primitive central idempotents of the semisimple group algebra $\\mathbb{F}_q[G]$. As a consequence, we obtain the structure of $\\mathbb{F}_q[G]$ and its group of automorphisms.
Lutfiyya, Lutfi A
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Modern Algebra includes set theory, operations, relations, basic properties of the integers, group theory, and ring theory.
Algebraic volume density property of affine algebraic manifolds
Kaliman, Shulim; Kutzschebauch, Frank
2009-01-01
We introduce the notion of algebraic volume density property for affine algebraic manifolds and prove some important basic facts about it, in particular that it implies the volume density property. The main results of the paper are producing two big classes of examples of Stein manifolds with volume density property. One class consists of certain affine modifications of $\\C^n$ equipped with a canonical volume form, the other is the class of all Linear Algebraic Groups equipped with the left i...
LOCAL AUTOMORPHISMS OF SEMISIMPLE ALGEBRAS AND GROUP ALGEBRAS
Institute of Scientific and Technical Information of China (English)
Wang Dengyin; Guan Qi; Zhan9 Dongju
2011-01-01
Let F be a field of characteristic not 2,and let A be a finite-dimensional semisimple F-algebra.All local automorphisms of A are characterized when all the degrees of A are larger than 1.If F is further assumed to be an algebraically closed field of characteristic zero,K a finite group,FK the group algebra of K over F,then all local automorphisms of FK are also characterized.
Exceptional Vertex Operator Algebras and the Virasoro Algebra
Tuite, Michael P.
2008-01-01
We consider exceptional vertex operator algebras for which particular Casimir vectors constructed from the primary vectors of lowest conformal weight are Virasoro descendants of the vacuum. We discuss constraints on these theories that follow from an analysis of appropriate genus zero and genus one two point correlation functions. We find explicit differential equations for the partition function in the cases where the lowest weight primary vectors form a Lie algebra or a Griess algebra. Exam...
Blyth, T S
2002-01-01
Most of the introductory courses on linear algebra develop the basic theory of finite dimensional vector spaces, and in so doing relate the notion of a linear mapping to that of a matrix. Generally speaking, such courses culminate in the diagonalisation of certain matrices and the application of this process to various situations. Such is the case, for example, in our previous SUMS volume Basic Linear Algebra. The present text is a continuation of that volume, and has the objective of introducing the reader to more advanced properties of vector spaces and linear mappings, and consequently of matrices. For readers who are not familiar with the contents of Basic Linear Algebra we provide an introductory chapter that consists of a compact summary of the prerequisites for the present volume. In order to consolidate the student's understanding we have included a large num ber of illustrative and worked examples, as well as many exercises that are strategi cally placed throughout the text. Solutions to the ex...
Energy Technology Data Exchange (ETDEWEB)
Palmkvist, Jakob, E-mail: palmkvist@ihes.fr [Institut des Hautes Etudes Scientifiques, 35 Route de Chartres, FR-91440 Bures-sur-Yvette (France)
2014-01-15
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D − 2 − p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
Cleaveland, Rance; Luettgen, Gerald; Natarajan, V.
1999-01-01
This paper surveys the semantic ramifications of extending traditional process algebras with notions of priority that allow for some transitions to be given precedence over others. These enriched formalisms allow one to model system features such as interrupts, prioritized choice, or real-time behavior. Approaches to priority in process algebras can be classified according to whether the induced notion of preemption on transitions is global or local and whether priorities are static or dynamic. Early work in the area concentrated on global pre-emption and static priorities and led to formalisms for modeling interrupts and aspects of real-time, such as maximal progress, in centralized computing environments. More recent research has investigated localized notions of pre-emption in which the distribution of systems is taken into account, as well as dynamic priority approaches, i.e., those where priority values may change as systems evolve. The latter allows one to model behavioral phenomena such as scheduling algorithms and also enables the efficient encoding of real-time semantics. Technically, this paper studies the different models of priorities by presenting extensions of Milner's Calculus of Communicating Systems (CCS) with static and dynamic priority as well as with notions of global and local pre- emption. In each case the operational semantics of CCS is modified appropriately, behavioral theories based on strong and weak bisimulation are given, and related approaches for different process-algebraic settings are discussed.
Stability of functional equations in Banach algebras
Cho, Yeol Je; Rassias, Themistocles M; Saadati, Reza
2015-01-01
Some of the most recent and significant results on homomorphisms and derivations in Banach algebras, quasi-Banach algebras, C*-algebras, C*-ternary algebras, non-Archimedean Banach algebras and multi-normed algebras are presented in this book. A brief introduction for functional equations and their stability is provided with historical remarks. Since the homomorphisms and derivations in Banach algebras are additive and R-linear or C-linear, the stability problems for additive functional equations and additive mappings are studied in detail. The latest results are discussed and examined in stability theory for new functional equations and functional inequalities in Banach algebras and C*-algebras, non-Archimedean Banach algebras, non-Archimedean C*-algebras, multi-Banach algebras and multi-C*-algebras. Graduate students with an understanding of operator theory, functional analysis, functional equations and analytic inequalities will find this book useful for furthering their understanding and discovering the l...
Green, R. M.
1997-01-01
We introduce an analogue of the $q$-Schur algebra associated to Coxeter systems of type $\\hat A_{n-1}$. We give two constructions of this algebra. The first construction realizes the algebra as a certain endomorphism algebra arising from an affine Hecke algebra of type $\\hat A_{r-1}$, where $n \\geq r$. This generalizes the original $q$-Schur algebra as defined by Dipper and James, and the new algebra contains the ordinary $q$-Schur algebra and the affine Hecke algebra as subalgebras. Using th...
Homological mirror symmetry and tropical geometry
Catanese, Fabrizio; Kontsevich, Maxim; Pantev, Tony; Soibelman, Yan; Zharkov, Ilia
2014-01-01
The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry. In combination with the subsequent work of Mikhalkin on the “tropical” approach to Gromov-Witten theory, and the work of Gross and Siebert, Tropical Geometry has now become a powerful tool. Homological Mirror Symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic (or algebraic) geometry. The central ideas first appeared in the work of Maxim Kontsevich (1993). Roughly speaking, the subject can be approached in two ways: either one uses Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow picture, further developed by Kontsevich and Soibelman) or one uses Lefschetz fibrations of symplectic manifolds (suggested by Kontsevich and further developed by Seidel). Tropical Ge...
DERIVATIONS ON DIFFERENTIAL OPERATOR ALGEBRA AND WEYL ALGEBRA
Institute of Scientific and Technical Information of China (English)
CHENCAOYU
1996-01-01
Let L be an n-dimensional nilpotent Lie algebra with a basis{x1…,xn),and every xi acts as a locally nilpotent derivation on algebra A. This paper shows that there exists a set of derivations{y1,…,yn}on U(L) such that (A#U(L))#k{y,1,…,yn] is ismorphic to the Weyl algebra An(A).The author also uses the de4rivations to obtain a necessary and sufficient condition for a finite dimesional Lie algebra to be nilpotent.
New Symbolic Tools for Differential Geometry, Gravitation, and Field Theory (extended version)
Torre, Charles G.; Anderson, Ian M.
2011-01-01
DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. These capabilities, combined with dramatic recent improvements in symbolic approaches to solving algebraic and differential equations, have allowed for development of powerful new tools for solving research problems in gravitation and field...
Assessing Algebraic Solving Ability: A Theoretical Framework
Lian, Lim Hooi; Yew, Wun Thiam
2012-01-01
Algebraic solving ability had been discussed by many educators and researchers. There exists no definite definition for algebraic solving ability as it can be viewed from different perspectives. In this paper, the nature of algebraic solving ability in terms of algebraic processes that demonstrate the ability in solving algebraic problem is…
Minimal ambient nuclear C*-algebras
Suzuki, Yuhei
2015-01-01
We provide examples of ambient nuclear C*-algebras of non-nuclear C*-algebras with no proper intermediate C*-algebras. In particular this gives the first examples of minimal ambient nuclear C*-algebras of non-nuclear C*-algebras. For this purpose, we study generic Cantor systems of infinite free product groups.
Algebraic topology of finite topological spaces and applications
Barmak, Jonathan A
2011-01-01
This volume deals with the theory of finite topological spaces and its relationship with the homotopy and simple homotopy theory of polyhedra. The interaction between their intrinsic combinatorial and topological structures makes finite spaces a useful tool for studying problems in Topology, Algebra and Geometry from a new perspective. In particular, the methods developed in this manuscript are used to study Quillen’s conjecture on the poset of p-subgroups of a finite group and the Andrews-Curtis conjecture on the 3-deformability of contractible two-dimensional complexes. This self-contained work constitutes the first detailed exposition on the algebraic topology of finite spaces. It is intended for topologists and combinatorialists, but it is also recommended for advanced undergraduate students and graduate students with a modest knowledge of Algebraic Topology.
A Workshop on Algebraic Design Theory and Hadamard Matrices
2015-01-01
This volume develops the depth and breadth of the mathematics underlying the construction and analysis of Hadamard matrices and their use in the construction of combinatorial designs. At the same time, it pursues current research in their numerous applications in security and cryptography, quantum information, and communications. Bridges among diverse mathematical threads and extensive applications make this an invaluable source for understanding both the current state of the art and future directions. The existence of Hadamard matrices remains one of the most challenging open questions in combinatorics. Substantial progress on their existence has resulted from advances in algebraic design theory using deep connections with linear algebra, abstract algebra, finite geometry, number theory, and combinatorics. Hadamard matrices arise in a very diverse set of applications. Starting with applications in experimental design theory and the theory of error-correcting codes, they have found unexpected and important ap...
On the topology of real algebraic plane curves
DEFF Research Database (Denmark)
Cheng, Jinsan; Lazard, Sylvain; Peñaranda, Luis;
2010-01-01
We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given...... coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use sub-resultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations...... also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on high-degree non-generic curves....
Maximal-acceleration phase space relativity from Clifford algebras
Castro, C
2002-01-01
We present a new physical model that links the maximum speed of light with the minimal Planck scale into a maximal-acceleration Relativity principle in the spacetime tangent bundle and in phase spaces (cotangent bundle). Crucial in order to establish this link is the use of Clifford algebras in phase spaces. The maximal proper-acceleration bound is a = c^2/ \\Lambda in full agreement with the old predictions of Caianiello, the Finslerian geometry point of view of Brandt and more recent results in the literature. We present the reasons why an Extended Scale Relativity based on Clifford spaces is physically more appealing than those based on kappa-deformed Poincare algebras and the inhomogeneous quantum groups operating in quantum Minkowski spacetimes. The main reason being that the Planck scale should not be taken as a deformation parameter to construct quantum algebras but should exist already as the minimum scale in Clifford spaces.
Quantum computation using geometric algebra
Matzke, Douglas James
This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anticommutative outer products into a real valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools.
Higher theories of algebraic structures
Matsuoka, Takuo
2016-01-01
The notion of (symmetric) coloured operad or "multicategory" can be obtained from the notion of commutative algebra through a certain general process which we call "theorization" (where our term comes from an analogy with William Lawvere's notion of algebraic theory). By exploiting the inductivity in the structure of higher associativity, we obtain the notion of "$n$-theory" for every integer $n\\ge 0$, which inductively "theorizes" $n$ times, the notion of commutative algebra. As a result, (c...
Differential Equations with Linear Algebra
Boelkins, Matthew R; Potter, Merle C
2009-01-01
Linearity plays a critical role in the study of elementary differential equations; linear differential equations, especially systems thereof, demonstrate a fundamental application of linear algebra. In Differential Equations with Linear Algebra, we explore this interplay between linear algebra and differential equations and examine introductory and important ideas in each, usually through the lens of important problems that involve differential equations. Written at a sophomore level, the text is accessible to students who have completed multivariable calculus. With a systems-first approach, t
COCLEFT EXTENSIONS OF HOPF ALGEBRAS
Institute of Scientific and Technical Information of China (English)
祝家贵
2006-01-01
Let B and H be finitely generated projective Hopf algebras over a commutative ring R,with B cocommutative and H commutative. In this paper we investigate cocleft extensions of Hopf algebras, and prove that the isomorphism classes of cocleft Hopf algebras extensions of B by H are determined uniquely by the group C(B, H) = ZC(B, H)/d(B, H) .
Algebraic Independence and Mahler's method
Zorin, Evgeniy
2011-01-01
We give some new results on algebraic independence within Mahler's method, including algebraic independence of values at transcendental points. We also give some new measures of algebraic independence for infinite series of numbers. In particular, our results furnishes, for $n\\geq 1$ arbitrarily large, new examples of sets $(\\theta_1,...,\\theta_n)\\in\\mrr^n$ normal in the sense of definition formulated by Grigory Chudnovsky (1980).
Algebraic geometric codes with applications
Institute of Scientific and Technical Information of China (English)
CHEN Hao
2007-01-01
The theory of linear error-correcting codes from algebraic geomet-ric curves (algebraic geometric (AG) codes or geometric Goppa codes) has been well-developed since the work of Goppa and Tsfasman, Vladut, and Zink in 1981-1982. In this paper we introduce to readers some recent progress in algebraic geometric codes and their applications in quantum error-correcting codes, secure multi-party computation and the construction of good binary codes.
NON-COMMUTATIVE POISSON ALGEBRA STRUCTURES ON LIE ALGEBRA sln(fCq) WITH NULLITY M
Institute of Scientific and Technical Information of China (English)
Jie TONG; Quanqin JIN
2013-01-01
Non-commutative Poisson algebras are the algebras having both an associa-tive algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on the Lie algebras sln(fCq) are determined.
Novotna, Jarmila; Hoch, Maureen
2008-01-01
Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense…
Planar Para Algebras, Reflection Positivity
Jaffe, Arthur
2016-01-01
We define the notion of a planar para algebra, which arises naturally from combining planar algebras with the idea of $\\Z_{N}$ para symmetry in physics. A subfactor planar para algebra is a Hilbert space representation of planar tangles with parafermionic defects, that are invariant under isotopy. For each $\\Z_{N}$, we construct a family of subfactor planar para algebras which play the role of Temperley-Lieb-Jones planar algebras. The first example in this family is the parafermion planar para algebra. Based on this example, we introduce parafermion Pauli matrices, quaternion relations, and braided relations for parafermion algebras which one can use in the study of quantum information. Two different reflections play an important role in the theory of planar para algebras. One is the adjoint operator; the other is the modular conjugation in Tomita-Takesaki theory. We use the latter one to define the double algebra and to introduce reflection positivity. We give a new and geometric proof of reflection positivi...
Congruence Kernels of Orthoimplication Algebras
Directory of Open Access Journals (Sweden)
I. Chajda
2007-10-01
Full Text Available Abstracting from certain properties of the implication operation in Boolean algebras leads to so-called orthoimplication algebras. These are in a natural one-to-one correspondence with families of compatible orthomodular lattices. It is proved that congruence kernels of orthoimplication algebras are in a natural one-to-one correspondence with families of compatible p-filters on the corresponding orthomodular lattices. Finally, it is proved that the lattice of all congruence kernels of an orthoimplication algebra is relatively pseudocomplemented and a simple description of the relative pseudocomplement is given.
Hochschild homology of structured algebras
DEFF Research Database (Denmark)
Wahl, Nathalie; Westerland, Craig Christopher
2016-01-01
We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any prop with A∞-multiplication—we think of such algebras as A∞-algebras “with extra structure”. As applications, we obtain an integral version of the Costello......–Kontsevich–Soibelman moduli space action on the Hochschild complex of open TCFTs, the Tradler–Zeinalian and Kaufmann actions of Sullivan diagrams on the Hochschild complex of strict Frobenius algebras, and give applications to string topology in characteristic zero. Our main tool is a generalization of the Hochschild complex....
Veronese geometry and the electroweak vacuum moduli space
Energy Technology Data Exchange (ETDEWEB)
He, Yang-Hui, E-mail: hey@maths.ox.ac.uk [Department of Mathematics, City University, London, Northampton Square, London EC1V 0HB (United Kingdom); School of Physics, NanKai University, Tianjin 300071 (China); Merton College, University of Oxford, Oxford OX1 4JD (United Kingdom); Jejjala, Vishnu, E-mail: vishnu@neo.phys.wits.ac.za [Centre for Theoretical Physics, NITheP, and School of Physics, University of the Witwatersrand, Johannesburg, WITS 2050 (South Africa); Matti, Cyril, E-mail: Cyril.Matti.1@city.ac.uk [Department of Mathematics, City University, London, Northampton Square, London EC1V 0HB (United Kingdom); Nelson, Brent D., E-mail: b.nelson@neu.edu [Department of Physics, Northeastern University, Boston, MA 02115 (United States); ICTP, Strada Costiera 11, Trieste 34014 (Italy)
2014-09-07
We explain the origin of the Veronese surface in the vacuum moduli space geometry of the MSSM electroweak sector. While this result appeared many years ago using techniques of computational algebraic geometry, it has never been demonstrated analytically. Here, we present an analytical derivation of the vacuum geometry of the electroweak theory by understanding how the F- and D-term relations lead to the Veronese surface. We moreover give a detailed description of this geometry, realising an extra branch as a zero-dimensional point when quadratic Higgs lifting deformations are incorporated into the superpotential.
2-Local derivations on matrix algebras over semi-prime Banach algebras and on AW*-algebras
Ayupov, Shavkat; Kudaybergenov, Karimbergen
2016-03-01
The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra A with the inner derivation property we prove that any 2-local derivation on the algebra M 2n (A), n ≥ 2, is a derivation. We apply this result to AW*-algebras and show that any 2-local derivation on an arbitrary AW*-algebra is a derivation.
On Some Algebraic and Combinatorial Properties of Dunkl Elements
Kirillov, Anatol N.
2013-06-01
We introduce and study a certain class of nonhomogeneous quadratic algebras together with the special set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements. This result is a further generalization of similar results obtained in [S. Fomin and A. N. Kirillov, Quadratic algebras, Dunkl elements and Schubert calculus, in Advances in Geometry (eds. J.-S. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, Boston, 1995), pp. 147-182, A. Postnikov, On a quantum version of Pieri's formula, in Advances in Geometry (eds. J.-S. Brylinski, R. Brylinski, V. Nistor, B. Tsygan and P. Xu), Progress in Math. Vol. 172 (Birkhäuser Boston, 1995), pp. 371-383 and A. N. Kirillov and T. Maenor, A Note on Quantum K-Theory of Flag Varieties, preprint]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group, cf. [E. Mukhin, V. Tarasov and A. Varchenko, Bethe Subalgebras of the Group Algebra of the Symmetric Group, preprint arXiv:1004.4248]. Also we describe a few combinatorial properties of some special elements in the associative quasi-classical Yang-Baxter algebra in a connection with the values of the β-Grothendieck polynomials for some special permutations, and on the other hand, with the Ehrhart polynomial of the Chan-Robbins polytope.
The Schroedinger-Virasoro algebra. Mathematical structure and dynamical Schroedinger symmetries
Energy Technology Data Exchange (ETDEWEB)
Unterberger, Jeremie [Henri Poincare Univ., Vandoeuvre-les-Nancy (France). Inst. Elie Cartan; Roger, Claude [Lyon I Univ., Villeurbanne (France). Dept. de Mathematiques
2012-07-01
This monograph provides the first up-to-date and self-contained presentation of a recently discovered mathematical structure the Schroedinger-Virasoro algebra. Just as Poincare invariance or conformal (Virasoro) invariance play a key role in understanding, respectively, elementary particles and two-dimensional equilibrium statistical physics, this algebra of non-relativistic conformal symmetries may be expected to apply itself naturally to the study of some models of non-equilibrium statistical physics, or more specifically in the context of recent developments related to the non-relativistic AdS/CFT correspondence. The study of the structure of this infinite-dimensional Lie algebra touches upon topics as various as statistical physics, vertex algebras, Poisson geometry, integrable systems and supergeometry as well as representation theory, the cohomology of infinite-dimensional Lie algebras, and the spectral theory of Schroedinger operators. (orig.)
Construction of complete generalized algebraic groups
Institute of Scientific and Technical Information of China (English)
WANG Dengyin
2005-01-01
With one exception, the holomorph of a finite dimensional abelian connectedalgebraic group is shown to be a complete generalized algebraic group. This result on algebraic group is an analogy to that on Lie algebra.
On triangular algebras with noncommutative diagonals
Institute of Scientific and Technical Information of China (English)
2008-01-01
We construct a triangular algebra whose diagonals form a noncommutative algebra and its lattice of invariant projections contains only two nontrivial projections. Moreover we prove that our triangular algebra is maximal.
Tilting mutation of Brauer tree algebras
Aihara, T
2010-01-01
We define tilting mutations of symmetric algebras as the endomorphism algebras of Okuyama-Rickard complexes. For Brauer tree algebras, we give an explicit description of the change of Brauer trees under mutation.
Algebraic dynamics solution and algebraic dynamics algorithm of Burgers equations
Institute of Scientific and Technical Information of China (English)
2008-01-01
Algebraic dynamics solution and algebraic dynamics algorithm of nonlinear partial differential evolution equations in the functional space are applied to Burgers equation. The results indicate that the approach is effective for analytical solutions to Burgers equation, and the algorithm for numerical solutions of Burgers equation is more stable, with higher precision than other existing finite difference algo-rithms.
Abstract Algebra to Secondary School Algebra: Building Bridges
Christy, Donna; Sparks, Rebecca
2015-01-01
The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…
Dynamical entropy of C* algebras and Von Neumann algebras
International Nuclear Information System (INIS)
The definition of the dynamical entropy is extended for automorphism groups of C* algebras. As example the dynamical entropy of the shift of a lattice algebra is studied and it is shown that in some cases it coincides with the entropy density. (Author)
On ''conformal spinor geometry'': An attempt to ''understand'' internal symmetry
International Nuclear Information System (INIS)
The natural homomorphism of pure spinors corresponding to a given Clifford algebra Csub(2n) to polarized isotropic n-planes of complex Euclidean space Esub(2n)sup(c) is taken as a starting point for the construction of a geometry called spinor geometry where pure spinors are the only elements out of which all tensors have to be constructed (analytically as bilinear polynomia of the components of a pure spinor). C4 and C6 spinor geometry are analyzed but it seems that C8 spinor geometry is necessary to construct Minkowski space Msup(3,1). C6 spinor field equations give rise in Minkowski space to a pair of Dirac equations (for conformal semispinors) presenting an SU(2) internal symmetry algebra. Mass is generated by spontaneously breaking the original O(4,2) symmetry of the spinor equation. (author)
Invariant algebraic curves and rational first integrals of holomorphic foliations in CP(2)
Institute of Scientific and Technical Information of China (English)
ZHANG; Xiang(张祥)
2003-01-01
In this paper, using the tools of algebraic geometry we provide sufficient conditions for a holomor-phic foliation in CP(2) to have a rational first integral. Moreover, we obtain an upper bound of the degreesof invariant algebraic curves of a holomorphic foliation in CP(2). Then we use these results to prove that anyholomorphic foliation of degree 2 does not have cubic limit cycles.
Homogeneous spaces, multi-moment maps and (2,3)-trivial algebras
DEFF Research Database (Denmark)
Madsen, T. B.; Swann, A.
2011-01-01
For geometries with a closed three-form we briefly overview the notion of multi-moment maps. We then give concrete examples of multi-moment maps for homogeneous hypercomplex and nearly Kahler manifolds. A special role in the theory is played by Lie algebras with second and third Betti numbers equal...... to zero. These we call (2,3)-trivial. We provide a number of examples of such algebras including a complete list in dimensions up to and including five....
Understanding geometric algebra Hamilton, Grassmann, and Clifford for computer vision and graphics
Kanatani, Kenichi
2015-01-01
Introduction PURPOSE OF THIS BOOK ORGANIZATION OF THIS BOOK OTHER FEATURES 3D Euclidean Geometry VECTORS BASIS AND COMPONENTS INNER PRODUCT AND NORM VECTOR PRODUCTS SCALAR TRIPLE PRODUCT PROJECTION, REJECTION, AND REFLECTION ROTATION PLANES LINES PLANES AND LINES Oblique Coordinate Systems RECIPROCAL BASIS RECIPROCAL COMPONENTS INNER, VECTOR, AND SCALAR TRIPLE PRODUCTS METRIC TENSOR RECIPROCITY OF EXPRESSIONS COORDINATE TRANSFORMATIONSHamilton's Quaternion Algebra QUATERNIONS ALGEBRA OF QUATERNIONS CONJUGATE, NORM, AND INVERSE REPRESENTATION OF ROTATION BY QUATERNION Grassmann's Outer Product
Ideals in the Roe Algebras of Discrete Metric Spaces with Coefficients in B(H)
Institute of Scientific and Technical Information of China (English)
Yingjie HU; Qin WANG
2009-01-01
The notion of an ideal family of weighted subspaces of a discrete metric space X with bounded geometry is introduced.It is shown that,if X has Yu's property A,the ideal structure of the Roe algebra of X with coefficients in B(H) is completely characterized by the ideal families of weighted subspaces of X,where B(H) denotes the C*-algebra of bounded linear operators on a separable Hilbert space H.
Niibori, Hidekazu; Sagaki, Daisuke
2009-01-01
Let $r \\in \\BC$ be a complex number, and $d \\in \\BZ_{\\ge 2}$ a positive integer greater than or equal to 2. Ashihara and Miyamoto introduced a vertex operator algebra $\\Vam$ of central charge $dr$, whose Griess algebra is isomorphic to the simple Jordan algebra of symmetric matrices of size $d$. In this paper, we prove that the vertex operator algebra $\\Vam$ is simple if and only if $r$ is not an integer. Further, in the case that $r$ is an integer (i.e., $\\Vam$ is not simple), we give a gene...
Semi-Hopf Algebra and Supersymmetry
Gunara, Bobby Eka
1999-01-01
We define a semi-Hopf algebra which is more general than a Hopf algebra. Then we construct the supersymmetry algebra via the adjoint action on this semi-Hopf algebra. As a result we have a supersymmetry theory with quantum gauge group, i.e., quantised enveloping algebra of a simple Lie algebra. For the example, we construct the Lagrangian N=1 and N=2 supersymmetry.
Stable flatness of nonarchimedean hyperenveloping algebras
Schmidt, Tobias
2008-01-01
Let L be a p-adic local field and g a finite dimensional Lie algebra over L. We show that its hyperenveloping algebra F(g) is a stably flat completion of its universal enveloping algebra. As a consequence the relative cohomology for the locally convex algebra F(g) coincides with the underlying Lie algebra cohomology. Final version. Some minor items corrected. Appeared in Journal of Algebra (2010).
Dimer models and Calabi-Yau algebras
Broomhead, Nathan
2008-01-01
In this thesis we study dimer models, as introduced in string theory, which give a way of writing down a class of non-commutative `superpotential' algebras. Some examples are 3-dimensional Calabi-Yau algebras, as defined by Ginzburg, and some are not. We consider two types of `consistency' condition on dimer models, and show that a `geometrically consistent' model is `algebraically consistent'. Finally we prove that the algebras obtained from algebraically consistent dimer models are 3-dimensional Calabi-Yau algebras.
Timorin, Vladlen
2002-01-01
Consider a smooth map from a neighborhood of the origin in a real vector space to a neighborhood of the origin in a Euclidean space. Suppose that this map takes all germs of lines passing through the origin to germs of Euclidean circles, or lines, or a point. We prove that under some simple additional assumptions this map takes all lines passing though the origin to the same circles as a Hopf map coming from a representation of a Clifford algebra does. We also describe a connection between ou...
Hazewinkel, M
2008-01-01
Algebra, as we know it today, consists of many different ideas, concepts and results. A reasonable estimate of the number of these different items would be somewhere between 50,000 and 200,000. Many of these have been named and many more could (and perhaps should) have a name or a convenient designation. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. If this happens, one should be able to find enough information in this Handbook to judge if it i
Fritzsch, Harald
2003-01-01
This talk follows by a few months a talk by the same authors on nearly the same subject at the Coral Gables Conference. The ideas presented here are basically the same, but with some amplification, some change of viewpoint, and a number of new questions for the future. For our own convenience, we have transcribed the Coral Gables paper, but with an added ninth section, entitled "Problems of light cone current algebra", dealing with our present views and emphasizing research topics that require study.
Cooperstein, Bruce
2015-01-01
Advanced Linear Algebra, Second Edition takes a gentle approach that starts with familiar concepts and then gradually builds to deeper results. Each section begins with an outline of previously introduced concepts and results necessary for mastering the new material. By reviewing what students need to know before moving forward, the text builds a solid foundation upon which to progress. The new edition of this successful text focuses on vector spaces and the maps between them that preserve their structure (linear transformations). Designed for advanced undergraduate and beginning graduate stud
Energy Technology Data Exchange (ETDEWEB)
Casasent, D.; Ghosh, A.
1983-01-01
Many of the linear algebra operations and algorithms possible on optical matrix-vector processors are reviewed. Emphasis is given to the use of direct solutions and their realization on systolic optical processors. As an example, implicit and explicit solutions to partial differential equations are considered. The matrix-decomposition required is found to be the major operation recommended for optical realization. The pipelining and flow of data and operations are noted to be key issues in the realization of any algorithm on an optical systolic array processor. A realization of the direct solution by householder qr decomposition is provided as a specific case study. 19 references.
Lopez, Cesar
2014-01-01
MATLAB is a high-level language and environment for numerical computation, visualization, and programming. Using MATLAB, you can analyze data, develop algorithms, and create models and applications. The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java. MATLAB Linear Algebra introduces you to the MATLAB language with practical hands-on instructions and results, allowing you to quickly achieve your goals. In addition to giving an introduction to
Corrochano, Eduardo Bayro
2010-01-01
This book presents contributions from a global selection of experts in the field. This useful text offers new insights and solutions for the development of theorems, algorithms and advanced methods for real-time applications across a range of disciplines. Written in an accessible style, the discussion of all applications is enhanced by the inclusion of numerous examples, figures and experimental analysis. Features: provides a thorough discussion of several tasks for image processing, pattern recognition, computer vision, robotics and computer graphics using the geometric algebra framework; int
Statecharts Via Process Algebra
Luttgen, Gerald; vonderBeeck, Michael; Cleaveland, Rance
1999-01-01
Statecharts is a visual language for specifying the behavior of reactive systems. The Language extends finite-state machines with concepts of hierarchy, concurrency, and priority. Despite its popularity as a design notation for embedded system, precisely defining its semantics has proved extremely challenging. In this paper, a simple process algebra, called Statecharts Process Language (SPL), is presented, which is expressive enough for encoding Statecharts in a structure-preserving and semantic preserving manner. It is establish that the behavioral relation bisimulation, when applied to SPL, preserves Statecharts semantics
Fernández, V. V.; Moya, A. M.; Rodrigues Jr., W. A.
2002-01-01
In this paper we introduce the concept of metric Clifford algebra $\\mathcal{C\\ell}(V,g)$ for a $n$-dimensional real vector space $V$ endowed with a metric extensor $g$ whose signature is $(p,q)$, with $p+q=n$. The metric Clifford product on $\\mathcal{C\\ell}(V,g)$ appears as a well-defined \\emph{deformation}(induced by $g$) of an euclidean Clifford product on $\\mathcal{C\\ell}(V)$. Associated with the metric extensor $g,$ there is a gauge metric extensor $h$ which codifies all the geometric inf...
Number theory III Diophantine geometry
1991-01-01
From the reviews of the first printing of this book, published as Volume 60 of the Encyclopaedia of Mathematical Sciences: "Between number theory and geometry there have been several stimulating influences, and this book records of these enterprises. This author, who has been at the centre of such research for many years, is one of the best guides a reader can hope for. The book is full of beautiful results, open questions, stimulating conjectures and suggestions where to look for future developments. This volume bears witness of the broad scope of knowledge of the author, and the influence of several people who have commented on the manuscript before publication ... Although in the series of number theory, this volume is on diophantine geometry, and the reader will notice that algebraic geometry is present in every chapter. ... The style of the book is clear. Ideas are well explained, and the author helps the reader to pass by several technicalities. Reading and rereading this book I noticed that the topics ...
Near-Horizon Geometry and Warped Conformal Symmetry
Afshar, Hamid; Grumiller, Daniel; Oblak, Blagoje
2015-01-01
We provide boundary conditions for three-dimensional gravity including boosted Rindler spacetimes, representing the near-horizon geometry of non-extremal black holes or flat space cosmologies. These boundary conditions force us to make some unusual choices, like integrating the canonical boundary currents over retarded time and periodically identifying the latter. The asymptotic symmetry algebra turns out to be a Witt algebra plus a twisted u(1) current algebra with vanishing level, corresponding to a twisted warped CFT that is qualitatively different from the ones studied so far in the literature. We show that this symmetry algebra is related to BMS by a twisted Sugawara construction and exhibit relevant features of our theory, including matching micro- and macroscopic calculations of the entropy of zero-mode solutions. We confirm this match in a generalization to boosted Rindler-AdS. Finally, we show how Rindler entropy emerges in a suitable limit.
Near-horizon geometry and warped conformal symmetry
Afshar, Hamid; Detournay, Stéphane; Grumiller, Daniel; Oblak, Blagoje
2016-03-01
We provide boundary conditions for three-dimensional gravity including boosted Rindler spacetimes, representing the near-horizon geometry of non-extremal black holes or flat space cosmologies. These boundary conditions force us to make some unusual choices, like integrating the canonical boundary currents over retarded time and periodically identifying the latter. The asymptotic symmetry algebra turns out to be a Witt algebra plus a twisted u(1) current algebra with vanishing level, corresponding to a twisted warped CFT that is qualitatively different from the ones studied so far in the literature. We show that this symmetry algebra is related to BMS by a twisted Sugawara construction and exhibit relevant features of our theory, including matching micro- and macroscopic calculations of the entropy of zero-mode solutions. We confirm this match in a generalization to boosted Rindler-AdS. Finally, we show how Rindler entropy emerges in a suitable limit.
Conformal geometry of surfaces in s4 and quaternions
Burstall, Francis E; Leschke, Katrin; Pedit, Franz; Pinkall, Ulrich
2002-01-01
The conformal geometry of surfaces recently developed by the authors leads to a unified understanding of algebraic curve theory and the geometry of surfaces on the basis of a quaternionic-valued function theory. The book offers an elementary introduction to the subject but takes the reader to rather advanced topics. Willmore surfaces in the foursphere, their Bäcklund and Darboux transforms are covered, and a new proof of the classification of Willmore spheres is given.
The Algebraic Formulation: Why and How to Use it
Directory of Open Access Journals (Sweden)
Ferretti Elena
2015-01-01
Full Text Available Finite Element, Boundary Element, Finite Volume, and Finite Difference Analysis are all commonly used in nearly all engineering disciplines to simplify complex problems of geometry and change, but they all tend to oversimplify. This paper shows a more recent computational approach developed initially for problems in solid mechanics and electro-magnetic field analysis. It is an algebraic approach, and it offers a more accurate representation of geometric and topological features.
Algorithmic Algebraic Combinatorics and Gröbner Bases
Klin, Mikhail; Jurisic, Aleksandar
2009-01-01
This collection of tutorial and research papers introduces readers to diverse areas of modern pure and applied algebraic combinatorics and finite geometries with a special emphasis on algorithmic aspects and the use of the theory of Grobner bases. Topics covered include coherent configurations, association schemes, permutation groups, Latin squares, the Jacobian conjecture, mathematical chemistry, extremal combinatorics, coding theory, designs, etc. Special attention is paid to the description of innovative practical algorithms and their implementation in software packages such as GAP and MAGM
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
The paper focuses on the 1-generated positively graded algebras with non-pure resolutions and mainly discusses a new kind of algebras called(s,t,d)-bi-Koszul algebras as the generalization of bi-Koszul algebras. An(s,t,d)-bi-Koszul algebra can be obtained from two periodic algebras with pure resolutions. The generation of the Koszul dual of an(s,t,d)-bi-Koszul algebra is discussed. Based on it,the notion of strongly(s,t,d)-bi-Koszul algebras is raised and their homological properties are further discussed.
Institute of Scientific and Technical Information of China (English)
SI JunRu
2009-01-01
The paper focuses on the 1-generated positively graded algebras with non-pure resolutions and mainly discusses a new kind of algebras called (s, t, d)-bi-Koszul algebras as the generalization of bi-Koszul algebras. An (s, t, d)-bi-Koszul algebra can be obtained from two periodic algebras with pure resolutions. The generation of the Koszul dual of an (s, t, d)-bi-Koszul algebra is discussed. Based on it, the notion of strongly (s, t, d)-bi-Koszul algebras is raised and their homological properties are further discussed.
New criterion for algebraic volume density property
Kaliman, Shulim
2012-01-01
A smooth affine algebraic variety $X$ equipped with an algebraic volume form $\\omega$ has the algebraic volume density property (AVDP) if the Lie algebra generated by completely integrable algebraic vector fields of $\\omega$-divergence zero coincides with the space of all algebraic vector fields of $\\omega$-divergence zero. We develop an effective criterion of verifying whether a given $X$ has AVDP. As an application of this method we establish AVDP for any homogeneous space $X=G/R$ that admits a $G$-invariant algebraic volume form where $G$ is a linear algebraic group and $R$ is a closed reductive subgroup of $G$.
Algebraic solution of master equations
R. Rangel; L. Carvalho
2003-01-01
We present a simple analytical method to solve master equations for finite temperatures and any initial conditions, which consists in the expansion of the density operator into normal modes. These modes and the expansion coefficients are obtained algebraically by using ladder superoperators. This algebraic technique is successful in cases in which the Liouville superoperator is quadratic in the creation and annihilation operators.
Linear Algebra and Image Processing
Allali, Mohamed
2010-01-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)
Toeplitz Algebras on Dirichlet Spaces
Institute of Scientific and Technical Information of China (English)
TAN Yan-hua; WANG Xiao-feng
2001-01-01
In the present paper, some properties of Toeplitz algebras on Dirichlet spaces for several complex variables are discussed; in particular, the automorphism group of the Toeplitz C* -algebra, (C1), generated by Toeplitz operators with C1-symbols is discussed. In addition, the first cohomology group of (C1) is computed.
Exploring Algebraic Patterns through Literature.
Austin, Richard A.; Thompson, Denisse R.
1997-01-01
Presents methods for using literature to develop algebraic thinking in an environment that connects algebra to various situations. Activities are based on the book "Anno's Magic Seeds" with additional resources listed. Students express a constant function, exponential function, and a recursive function in their own words as well as writing about…
Algebraic Squares: Complete and Incomplete.
Gardella, Francis J.
2000-01-01
Illustrates ways of using algebra tiles to give students a visual model of competing squares that appear in algebra as well as in higher mathematics. Such visual representations give substance to the symbolic manipulation and give students who do not learn symbolically a way of understanding the underlying concepts of completing the square. (KHR)
Algebraic Connectivity of Interdependent Networks
Martin-Hernandez, J.; Wang, H.; Van Mieghem, P.; D'Agostino, G.
2014-01-01
The algebraic connectivity UN-1, i.e. the second smallest eigenvalue of the Laplacian matrix, plays a crucial role in dynamic phenomena such as diffusion processes, synchronization stability, and network robustness. In this work we study the algebraic connectivity in the general context of interdepe
Templates for Linear Algebra Problems
Bai, Z.; Day, D.; Demmel, J.; Dongarra, J.; Gu, M.; Ruhe, A.; Vorst, H.A. van der
2001-01-01
The increasing availability of advanced-architecture computers is having a very signicant eect on all spheres of scientic computation, including algorithm research and software development in numerical linear algebra. Linear algebra {in particular, the solution of linear systems of equations and eig
Baxter Algebras and Umbral Calculus
Guo, Li
2004-01-01
We apply recent constructions of free Baxter algebras to the study of the umbral calculus. We give a characterization of the umbral calculus in terms of Baxter algebra. This characterization leads to a natural generalization of the umbral calculus that include the classical umbral calculus in a family of $\\lambda$-umbral calculi parameterized by $\\lambda$ in the base ring.
On Homomorphism of Valuation Algebras
Institute of Scientific and Technical Information of China (English)
GUAN XUE-CHONG; LI YONG-MING
2011-01-01
In this paper, firstly, a necessary condition and a sufficient condition for an isomorphism between two semiring-inducod valuation algebras to exist are presented respectively. Then a general valuation homomorphism based on different domains is defined, and the corresponding homomorphism theorem of valuation algebra is proved.
International Nuclear Information System (INIS)
PFSA (Program For Simple Algebra) is designed to be helpful to people doing algebra and calculus with polynomial expressions. It is written entirely in Fortran and hence is portable and easily modified. It is much (approximately 90 times) faster than Macsyma. PFSA uses Fortran integer arithmetic to compute coefficients, and so the occurrence of an excessively large number in a numerator or denominator during a computation bombs the computation. The program was developed to enable a computation (of a canonical transformation for a Hamiltonian system) which was too big to be run in other systems available at the time. The intent in creating PFSA was to make a program which would do the Hamiltonian computation and similar computations easily and fast. The only language available (on the Cray) was Fortran. Example C in Section III is a very simple canonical transformation. In running the problem for which PFSA was written some intermediate expressions have more than 20,000 terms and some answers have more than 1000 terms
On Clifford-algebraic dimensional extension and SUSY holography
Gates, S. J.; Hübsch, T.; Stiffler, K.
2015-03-01
We analyze the group of maximal automorphisms of the N-extended worldline supersymmetry algebra, and its action on off-shell supermultiplets. This defines a concept of "holoraumy" that extends the notions of holonomy and curvature in a novel way and provides information about the geometry of the supermultiplet field-space. In turn, the "holoraumy" transformations of 0-brane dimensionally reduced supermultiplets provide information about Lorentz transformations in the higher-dimensional space-time from which the 0-brane supermultiplets are descended. Specifically, Spin(3) generators are encoded within 0-brane "holoraumy" tensors. Worldline supermultiplets are thus able to holographically encrypt information about higher-dimensional space-time geometry.
On Clifford-Algebraic "Holoraumy", Dimensional Extension, and SUSY Holography
Gates, S J; Stiffler, K
2014-01-01
We analyze the group of maximal automorphisms of the N-extended world-line supersymmetry algebra, and its action on off-shell supermultiplets. This defines a concept of "holoraumy" that extends the notions of holonomy and curvature in a novel way and provides information about the geometry of the supermultiplet field-space. In turn, the "holoraumy" transformations of 0-brane dimensionally reduced supermultiplets provide information about Lorentz transformations in the higher-dimensional spacetime from which the 0-brane supermultiplets are descended. World-line supermultiplets are thus able to holographically encrypt information about higher dimensional spacetime geometry.
Representations of affine Hecke algebras
Xi, Nanhua
1994-01-01
Kazhdan and Lusztig classified the simple modules of an affine Hecke algebra Hq (q E C*) provided that q is not a root of 1 (Invent. Math. 1987). Ginzburg had some very interesting work on affine Hecke algebras. Combining these results simple Hq-modules can be classified provided that the order of q is not too small. These Lecture Notes of N. Xi show that the classification of simple Hq-modules is essentially different from general cases when q is a root of 1 of certain orders. In addition the based rings of affine Weyl groups are shown to be of interest in understanding irreducible representations of affine Hecke algebras. Basic knowledge of abstract algebra is enough to read one third of the book. Some knowledge of K-theory, algebraic group, and Kazhdan-Lusztig cell of Cexeter group is useful for the rest